Myac i/o error 105

myac i/o error 105

Gritsayeve will speak to the MYAC's history and provide an update on to the Ontario Municipal Board for failure on the municipality's. MM new est rate of 5 [:i hi e19cr, Cllendale resicicmts have beeTi This practice adds sprers.eu level of potelitial error to Llw szles. of error (for more details on the refinement see [30,. Chapter ]). Table 2 shows the simulation times of the performance. SELECT myac.

Myac i/o error 105 - not

The stochastic control of the F-8C aircraft using the multiple model adaptive control (MMAC) method

See discussions, stats, and author profiles for this publication at: sprers.eu The stochastic control of the F-8C aircraft using the multiple model adaptive control /MMAC/ method ARTICLE · FEBRUARY DOI: /CDC · Source: NTRS CITATIONS READS 24 8 AUTHORS, INCLUDING: Keh-Ping Dunn C. Greene Massachusetts Institute of Technology University of St. Thomas 26 PUBLICATIONS CITATIONS 8 PUBLICATIONS CITATIONS SEE PROFILE SEE PROFILE Available from: Keh-Ping Dunn Retrieved on: 04 February General Disclaimer One or more of the Following Statements may affect this Document This document has been reproduced from the best copy furnished by the organizational source. It is being released in the interest of making available as much information as possible. This document may contain data, which exceeds the sheet parameters. It was furnished in this condition by the organizational source and is the best copy available. This document may contain tone-on-tone or color graphs, charts and/or pictures, which have been reproduced in black and white. This document is paginated as submitted by the original source. Portions of this document are not fully legible due to the historical nature of some of the material. However, it is the best reproduction available from the original submission. Produced by the NASA Center for Aerospace Information (CASI) AFAtA TR  Z 5- 1 6 N 0 to  A PPfovad f^- T ^- Z caae . ^ 41a: till. 22 I11C ;rXHASTI C WMIA ( 1. Or  THE F-% AIIC RAIT­INr E f!t I . T I 1 L! MOU L : F LhA ": f VL QONT M I L 1 MwC) ` a It r.— ^if M.^Atliara,  r- R/:,irhn. C 11,reene. W. H. Lee !I. R. Sandell, Jr  - 4- 4egil L - tit, rct, c 3yaEam ratr^ry Massachusetts sprers.eu of Technology V 22  sprers.eu, Massachusetts  i -  -0 - LEU — A sprers.eu r act -- - rowrlhevi r e+rin int o! mode na, it is ob- vious the the dynamic ic state equati of an air- ` Me putpose c! t:.is sprers.eu is t•^ suwarize craft involve nonlinear differential equation. results ubtaired f- • tt.e ac:a) tine crntrcl of t"t. (see Fticin !11) . Howe ver, the information given F-8C aircraft us: :. sc-called rVAC -etnod. by NASA Langley Fesear h Center (LRC) to the MIT/ Tice a:sprers.eu 4.e selection c! the per- MI. team consl%ted in the specification of the sprers.eu criteria : tetn the lateral and •-e uncoiled, linear time-invariant oiler.-loop let sprers.eua: . ra-. :.e deKigr cf the Ka;.-an iongrtudina: and lateral dynamics of the F-8c filter+ for ffe.•r ,:.t crn. .tions, tt.e aircraft as%cciated with eeghrilibriur fligft. "ider,t1!ieat_ ` t!.o design using a-:• Table I gives a list of tl.e flight conditions hypc • ! . esis te • t.r.; i!.•as, and tr.e Ierforrarce of that were avaiiatle for the design. ,thus, the the closed loaf adap.t:ve systc-.­ general structure of the 'custions were of the form i,(t)-Ax ( t)•Bu(t). ". he numerica: valises of the elements of the  A. and B matrices ca n. be found • report by Gera (21, based upon win' 1. Ir.t roduct tor, tunnel tents, and a report by Mooley and Evan, (31, based upon linearization of the sprers.eur The Furrose of tnis paper is to present dynamics sprers.eu  ty NXSA/LRC for their nonlinear preliminary resultF _r. a study which. irvclves sirulation of the F-8C aircraft. We rerark at the arili(atiot. cf a:sprers.eu adaptive control this point that the numerical values f-­r the techniq.:es tc the .resign of a stability aucren- and P matrices aiven in   and ( 31 are not ider- tation :.yster ir. Lcth the : sprers.eunal and tical reflecting the fact that different sources lateral dyra.­.ica cf the F-AC aircraft. NAFA nas were used to obtain them. The design reported in beer using the F-8C aircraft as a test vehicle this paper  is based upon Gera's repot ( foz evaluating different digital-fly-by-wire WrOW) sprers.eu: sprers.eu, csinq the IBM AP-1­1 The fact that the 16 flight conditions span as the airborne ccrF--er We rerark that the an extremely wide envelope for operating the eventual i-Wler. matron c,: t­.e control aiacrithms aircraft, with drastic changes in the open-loop on the specific aorre corF,uter has had a razor dynamics, makes the fixed - gain design of the con- irject upor the  F•.i:c am -.r ::y adopted for the design trol system unrealistic. Furthermore, handlinq of the control syst,•r In view of the obvious sto- qualities requirements, such as the C • criterion, rage and zeal-tire sprers.eu constraints. In indicate that pilots desire different closed-loop addition, the deli,:, was crucially dependent upon dynamics at different flight conditions. Thus, the sensors that could be utilizec in the sense some sort of  " adaptive" gain-scheduling contro: that sensors that utilized externs' aerodynamic system was required. However, straight - forwtrd measurements, e.g., airspeed, altitude, angle of gain schedu:: .ig based upon sprers.eu such as attack and •:idesliF vanes should  not be employed velocity, altitude, and dynamic pressure was not in the candidate design. Thus, the design quide- persiitted it view cf the sensor r­-atrictions men- lines required that the sensors asr,hciated with tioned above. Hence, the adaptive control system the adaptive control systen shoul •i be limited to had to designed in a novel way. accelerometers, rate gyros, and Ferh.:os attitude sensors (although the latter were deemed undesir- An additional restriction on the design was able in view of their  errors wher the aircraft teat the sensor noise and wind disturbanc s had underwent severe pilot induced sprers.eus). to be incorporated. This led to the neec for employing Kalwan filters, with constant coeffi- cients because of the computer memory limitations. ••fie theory and initial elgorichm developrrnt associatyd with this study were developed with The above problem overview sets the ground sprers.euom NASA / Ares kesearch Center under for the specific adaptive control technique which grant' l­cLO(  an:! from ArCSR under grant we selected to investigate in great detail. We  The specific application to the F-8C call the adaptive control technique the Multiple-_ vas supported by ;rA Langley Research Center Model-Adaptive-Control  ( MMAC) method, and we shall under grant. HSG-1 8. Proceedings  IEEE Conference on Decision and Control, Houston, Texas, December  discos• it  in ss re sprers.eu in Section S of this (foc(t)) which was introduced to a firrt order paler. It is only -,r.e of several techniques servo with a time constant of  seconds to based upon develcysr• ntt in modern (ontrol theory ,anerate the actual deviation of the elevator (wee tl.e survey article ty Athans and Varaiya 6 0 (t) from its trismsed value. the elevator was (41) an4 :t has its origins  in -o+mLininq then related to the four 'natural' longitudinal hypothesis- • estinq and stochastic ccntrni ideas state variables namely pitch -ate, 4 W (red/sec), (se e references   to (BI) ft was selected fcr velocity error v(t)(ft/sec), perturi•ed angle of this study because of its lvtential Iroeise in attack from its trimmed value, *It)(rad), and academic examples  Isl-(B:,  and Lecause its memory pitch attitude deviation from its trimmed value ar..l seal - time co^ F,.tatior , al requ . revwnts could be p (t)(rad),  In addition, a wind disturbance readily assessed it, view of its nor-iterative state w(t) was included (see Appendix A). Thus sprers.eu .re. the state vector x(t) for  t.:ie long itudinal dynam- ics was characterized by seven components As esplairud in sore detail  in Sectic.n 5, the MW C met?%od requires that a full blown steady () x'(t) %ig(t), v(t), a(t), 43(t), 6 & (t) state linear - Quadratic- aussiar. (L(x.) controller 6 lt), w(t)) 4c imp)sprers.eu  fcr each fli , ,ht condition. :T'. is ec sprers.eutated the dervel,.lment of suitaLle quad- and the control variable u(t) was the commanded ratic performance criteria for both the longitud- inal and the lateral dynamics: these are ale-iator rate descriLed for tt.e _,mtiruous tine case (9)  in A () u(t);4eC(t) Sections , and 1, respectively. Fcr :mFleraenta- tlon, one neeL! i a discrete time  LQG controller 'Min led to a linear-time invariant characteriza- ). This :s described :r. Section 4, together tion for each flight condition of the form witn the discussio of sprers.eu errors. The MYAC algorithr• is sprers.eu ir. E ection S. Tt.e simula- () ir(t)-A x(t)+Bu(t)•Li­- (t) tion results using the nonlinear F-9C dynamics are described in Section C. Section  7 presents where ­(t) was zero mean white noise, sprers.eug the m&)cr conclusion cf our studies so far. the wind disturbance and accountino for randorr actuator errors. The elements of A, and Ii We rerArk that in .hi_ japer we -;hall only changed with each flight condition while focus our sprers.eu  to the f-gulatiin aspects of the protler, i.e., return to equiiiL• rium flight fror some initial cordit:ons and in the pre- () (i 0 0 0 0 1 0)' sence of stochastic wind disturbances. In our  The longitudinal Cost Functional study we are considering the proper way of incor- porating huran pilot irputs for both the longi- In order to apply the standard steady state tudinal and lateral -axe. However, we shall not iQG procedure   a quadratic performance index present in this paper any of the approaches and has to be selected. The general structure of the preliru rary results for the pilot input case. index was () J-/rx_(t)2,x(t)+u (t)R.u(t)dt 2. Longitudina . l  Dynamics Note that the weighting matrices  , % had to b.,  Introduction different from flight condition to flight condi- tion reflecting in a natural way that the pilot In this section we present an overview of wants different handling qualities as the speee the LQG philosophy adopted for designing the req- (an ,l dynamic pressure) changes. ulator for the sprers.eunal dynamics. Attention is given in the development of the quadratic per- In the initial design it was decided that formance index and the subsequent model simplifi- one should relate the maximum deviations of cation using a short period approximation. The • pitch attitude, 9max main concept that we wish to stress is that the • pitch rate, gmax quadratic performance criteria employed changed • normal acceleration, aTzmax in a natural way with each flight condition. ':he • maximum commanded elevator rate, 6eCmax surprising result was that the short period poles resulting in the following structure of of the resultant longitudinal, closed-loop system the performance criterion were characterized fcr all flight conditions by : ^TaRzW 2 () J1,  ,ON ^ + 2(t) 62(t) 6ec(t) two constant damming rstios, one associated with nzmax q max + 9 2 max + aT--dt all subsonic flight conditions and one associated with all supersonic flight conditions. ecma.x ACCT.  The Longitudinal State Description The normal acceleration  a nz (t), in g's, was not ITIS used as a state variable. However, it is linearly an Because of a rate constraint saturation on related to some of the longitudinal state var.- INARt _ "1CFE the elevator rate, the control variable selected ables according to the formula lusilr ICA)IJ'1 was the time rate of change of the coasnanded elevator rate (6 ec (t)). This wcs integrated to 1 () a nz (t) 1- 9 1  v(t)+ K 2 a(t)+Kr6 e (t 1 1 I  in g's generate the actual commanded elevator position 41 ""Ol5UT; A^Al1Af;Ui Ilse. AIltIL ^sprers.eu . 2 ve being the equilibriu/ slated. The constants tive tradeoff between the maximum normal accelara- k i , lip, kr can tx calculated tro p the Open loot tton, a n sax, and maximum patch  rate, g max . This A matrices, and hems sprers.eu with  flight condi- Is consistent with the C O criterion (12). tion. Lftectively the structure of the criterion () implies that it at t-0 the masaau p values when the short-period closed-loot poles ware of sprers.eu, pit,.: rate, or pitch attitude evaluated for both do-signs using the nusrertcal sprers.eu, tiien one  sprers.eu be wirv; tc saturate Values gzven by (2.P), we found the unexpected the elevator rate to remove them. For the pre- result that the ciastinq ratio  was constant lo,4e0%) livtr. ary Design the following nwAarical valueir, for all 11 sprers.eu fli g ht conditions, and also were selected sprers.eu)s the help of T. Elliott and constant (0.)61) for  the SUI sonic flignt J. Gera) condition, The closed-loop naturally freouency increased wilt. d y naeic piesnure. () anrmax-6g's, gmax-ICq Nt, ? max -bg/V0aIr Since no pole-Placement techniques  +ty re em- 6 -u rod/sec ployed (i.e., the mathematics were not told tc, e c-mas place the closed-!oor. poles on a constant dasg->- where a i r is tk-e (3,3) -:rraent of the open loop ing ratio line), we constructed a tradeoff by lo"Itudlnal A matrix. charw)ing (Jecreasanri) the maximum pitch rate gmax . This would increase the patch rate penalty rw+ughly speaking, this criterion means Ciat in the cost functional, and one would expect a one is willing to saturate the elevator rate higher damping ratio. The followinq values of ( rod/sec for that F-BC) if a normal accel- gm & x were employed eration of Cq was felt, or a pitch rate equ;va- lent to 1Cq*%. or a pitch .rror which if () qUax -IOq/V., B9/Vt. 6g/Ve. 4g/Va translated to angle cf attack would also generate a 69 normal acceleraticr. s ince more the constant damping ratio phenoftnon was nbserved, i.e., for eaci-. value o.  `{max the Tt,e above nums Goal values were translated short period closed loop poles for all subsonic into the sprers.eu ratrix (nco diagonal pos- flight condi a ons fell cn a constant dampinq ratio itive semidefinite; Which changed frcrr flignt line, and similarly for all supersonic flio!­t condition to flight conditio:,, while V -P-1/ conditions. This was further verified by consid- () 2 for all flight conditions. Hence,  the ering an additional 13 different flight condi- resulting LQG problem could be solved using tions. availaole computer sutrou•ines ill;. The numerical results are presented in Table  f4duced Longitudinal Design II. The reason for this regularity of the solu- tion of the LQ problem is under investigation. The design was modified for two reasons. first the Bair. from the velocity state variable v(t) was extremely small. Second, it was det'.r- .l. Lateral Dynamics able to avoid usinc the pitch sensor. The pitch 0(t) is weakly ooservable from the system dynam-  Introduction ics so that even if a Kalman filter was used in the absence of pitch measurements, large estimm- In this section we present the parallel tion errors would be  ,­tt_iined wnicn would adveisly philosophy for the development of the control affect the perforrd. he control system since system for the lateral dynamics. In this case there is significant .ce—ack from the estimated the development of a performance criterion was pitch attitul­. At any rate, since is pilot would not as straight-forward as in the case of the fly the aircraft he .ould be able to control longitudinal dynamics. For an extensive discus- pitch himself. sion see the S.M. thesis by -reene (13). This led us to eliminating the velocity error  The iAteral Dynamic.; State Model v(t) and pitch Oft) from the state equati gns and obtaining the "short period" approximation (5 The control variables selected for lateral i state variables). Since pitch did not appear the control were criteria () was modified to  2 ji j anz2 (t) . q 2 (t) ; 76 ec (t) dt () u 1 (t)-6 ac (t)-commanded aileron rate () J2  a q sprers.eu C — (rad/sec) n ramax a cmax () u=it)-A rc (t)-commanded rudder rate and the resultant LQG problem was resolved. (rad/sec)  Summary of Results to that the control vector is defined to to () uI (t)-(ui(t) ur(.')I From the viewpoint of transient. responses Lu the variables of interest (normal a:celeration, pitch rate, angle of attack) the transient re- Rie sa­w• ssechanics were taken into account. The sponses to initial conditions were almost identi- comeranded aileron and rudder rates were integrat- cal for both designs. Thus, the Short period ad to generate the commanded aileron  ( 6 ac (t)) and motion of the aircraft was dominated by the rela- rudder (6 rc (t)) positions, respectively. For the r a1 (el , ^ p  lt! , 8 r (t) ^ rlt) F-ef ar-(taft the cueaunded amlet ' n rate 6ac(t) dtmves a l tst order la q sprers.eu, with a tier con- (3.e) LAT o  a yuax T Max WAR Max strut of 1/3U se. now , tc yenerate the actual aIlerun position t,(rsJ•). + a  ac(t)  I, Irrc(t) dt TT,e cosmtanded rudder rate b r c(t)(rads) drmKs ar ar • first order lay ser:c, wrtr,  a tine cunstant ,,f ac'ax rWea= 1/2 1S secune •., to .#sprers.eu the actual rw?r'er posi- Ths f o llwtnq maximum values Were •rsed tion 5 r (t)(r ado, ). Tne actual aiAeror anti rudder posttror. 0,(t) and ^ r (t), sprers.eu exerts t).r four Maxts+. am lateral acceleration, ay—xq's 'natural ' IatemaI e,.rarics state variables, name- ly roll-rate )(t)mrad sec), y w-rate r(tI(rod/ Maximus roll rate, p I­t (a -o) sec), srdecl ^ { ar. y le : i tfirad), er.d wnk an3/e max IOq 31n *W $ rod?. e.i.:ttton, a sprers.eu d,sturbance state var y i,le wit),  ,ee A"vnd:x t., drives the equa- Mart io Lm sideslip angle, P RWLX ^-vs a ] l tions In the same way as the trdeslip varrat,le. V Og Thus, the state eq-Attons for the lateral Maximus bank angle, I .X'O.R red (') dynamics are cLaracterized ty a t•-dimensional state vector x(i) w.u, components Maximum commanded aileron rate rad/sec () x'W t 11'(t) r(t) Pw :(t) .` a (t) br(t) raxis,um commanded rudder rate rad/sec d (t) ' r (t) w(t)1 ac re Set (13) for an extensive discus!. , on of how this performance criterion was dertvei; a ll and all and the overall lateral sprers.eu take the form are obtained from the open loop A, matrices ( x(t A aIt) • h ult) '. '(t) There is no netural ray of arriving at a simplified model for the lateral dvnarics, as was vnc re the zcrt. 'mar. A3rite racist vector  =(t) qen- the case with the longitudinal dynamics. Hence rrates tLe vird sprers.eu and ccr l wnsates for the bank angle cannot be eliminated. Althou g h a modelling errors. ::nce core the matrices A, Li change wit', flight ccr:dttion- (2), ) while bank angle sensor was de?6Ed undesirable, the weak observability of the bank an g le caused large () B•. ro0 0 0 0 0 1 0 0  0 0 0 0 0 0 1 0 state estimation errors, using Kalman filters, in the bank angle and the sideslip angle if a bank angle sensor was not I ncluded. For these reasons,  T?.e Lateral Cost Functional it was decidt-d to employ a bank an g le sensor and to penalize bank angle deviation, because bank The lateral performance index used (after angles larger than 20' can introduce significant several iterations) vr:sprers.eu the following nonlinearities through trigonn•etric functions variables: (1). • lateral acceleration, a (t) (in g's) Once sure, the IQ can be solved. Notice that • roll rate, p(t) y (in rad/sec) the use of the performance criterion () results • sideslip angle, ?(t) (in red) in a state weighting matrix (non-diagonal) • bank angle, C(t) (in rod) which changes rrith flight con tion. VS o commanded a-leror. rate, §ac(t)  Simmary of Results o comrsnanded rudder rate, 6rc(t) The abrve performance criterion gave reason- The lateral acceleration, a (t), is not a state able responses for a variety of initial conditions. variable. However, for sraYl perturbations from Its main characteristic is to reduce any lateral equilibrium. flight, it can be expresssd as a accelerations (by forcing the aircraft to go in linear cork ination of the lateral state variables coordinated turns) and to null out bank angle and the trim angle of attack,  ao, by the following errors in a slower manner. relation () ay(t)­ 9 )(km-ao)p(t)+(k­ + l)r(t)+krB(t) oncemore we observed a constant damping ratio (  ) for all supersonic conditions and a relatively constant damping ratio (  ) for +k.6(t)+ks6 (t))-.(t) all subsonic flight conditions. No additional r tradeoff studies were conducted by changing the where the constants km,, ks can be found from weights in the cost functional. the lateral open loop A i matrix, and change with the flight condition. The followir.g structure of the quadratic performance criterion was established: S 4. Sensors, Kalsan rilters and for the longitudinal models and lateral models Discrete sprers.eu kompen ^ ators were computed for each flight condition denoted 0 by 1. As we shall see those are i"et:nt in the  Introduction limieration of the POW variables. Tfie digital im3lementation of the control  The Design of the Discrete LQG Comtensators system requires the discrete-time solution of the W; problem  Its we shall see )n the next Through the use of the separation :heoren one sectic .n, the MMAC approach requires the construc- can design the discrete LCC compensators. This tion of a Lank ai UC Lnntrolleis. each of which implied that the fQ problem defined In continuous contains a discrete Kalman filter (whose resid- time in Sections 2 and 3 had to be correctly uals are used in lrobaiility a lcuiations and transformed into the equivalent discrete-time wLue.c state estlrtes are used to generate the problem in view of -he slew meascremert rate. adaptive control signals). Hence, in this section Effectively, we hive used the trans formwtions we present an overview of the issues involved in given in referen . , (13) to (15). the design of the IQG controllers based upon the noisy sensor measurements. Recapitulation 4.,e The Sampling Interval ror each flight condition, indexed by i, a complete discrete-time, steady state, LQG comfen- A sampling rate of 8 measurements/second sstor was designed for both the longitudinal and was established. Such a slow sampling rate was lateral dynamics. Each compensator generated selected so as to be able to carry out in real every 1/8 • econd the optimal control, rarely the time the multitude of real time operations re- optimal commanded elevator rate •_ e (t) for the quire-' by the M sprers.eu method. longitudinal dynamics. and the optimal commanded aileron rate % ac (t) and rudder rate o rc (t), Lased  Sensors and Norse Characteristics upon the noisy measurements of the appropriate sensors (See Table III) every 1/6 second. As explained in tt.e introduction, the guide- lines for design excluce• d the use of air data Because of the appropriate tra:.,, format ions sensors. Thus, measurements of altitude, speed, of the continuous time LQC proalei to the discrete angle! of attack., and sideslip angle were not one, we noted no significant degradation in per- available. After some preliminary investigations formance at this low sampling rate. it was decided that sensors that depend on trim variables (elevator angle and pitch attitude) The need for adaptive control is obvious be- should not be used so as tc avoid estimating trim cause if we assume that the aircraft is in flight parameters. -,able III lists the sensors and their condition i, but we use the LQG compensator ob- accuracy characteristics that were used in this tained for flight condition j for feedback con- study. We stress that the sensors measure the trol, this mismatching may generate either an true variables every 1/8 seconds in the presence unstable system or, often, a system with degraded of discrete-zero rean white noise with the stan- performance. dard deviations given in Table III. Finally, we remark ;.hat in this study we 5. 'fie MAC Method assumed that all ;enscrs were located at the C.G. of the Arcraft.  Introduction  The Design of Kalman Filters In this section we present the basic idea behind the M?sprers.eu method, and discuss how it was F,r each flight condition the steady-state in the F-8C context. In particular, we discrete-time Kalman filter, with constant gains demonst:ste how the information generated by the was calculated, for both the longitudinal lateral and longitudinal sensors is blended to- and lateral dynamic models. The level of the gether. Finally we make some remarks associated plant white noise associated with the wind dis- with the MMAC method and its general applicability turbance generation was selected so that we to the design of adaptive control systems. assumed that the aircraft was flying in cumulus clouds. (See Appendix A.)  71he 3asic Idea The decision to use steady state constant sunpnse one has N linear, discrete-time r gain Kalman filters was made so as to stinimize stochastic lime-invariant dynamic systems, indexed the computer memory requirements. by , 2, , N, generating discrete-time measurement ,. corrupted by white noise Finally, we remark that in view of the slow Suppose that at t-0 "nature" selects one of these sampling rate, the continuous time filtering systems and places it inside a "black box." The problem was carefully translated into the equiva- true system generates a discrete set of measure- lent discrete problem  1  to (15). ments z(t). The objective is to apply a control signal u(t) to the true model. The constant covariance matrices of the Kalman filter residuals, denoted by Sj LON , -SLAT S The version of tt,e MtAC method emp loy ed is Then Cha probabilities at time t, P i (t), 1 . 1, 2, as follows. one cor. • tructa a .fascrete-time steady M  computed recursively from the  are probabil- state L..r, controller f,,r ea-h model. thus, one itles at time t-1, P i (t-1), by  the tursrila has a tank of N  %,mpeer%aturs. As shown in P a lci"esp{- n iltl/21 rigor• 1, each LQI, netensat,r is driver. by  tt• () Pi(c) N actual sprers.eu app!ied to tr.e system., u(t), and driven by the actual nol sy rwasurenent vector, r (t i. :?lore are tr, signals of interest that each IQ, ttw4 en%atci .#vreratos at time t with the Ir(tial probabilities, P  r 0) given. It (1) the control vector  u It), which would to has teen claimed that  , , (bl. under suit- the optiral control it indeed the system able assumptions that asyaptotieally the true in the black box (via. aircraft) was model is identified with probability 1. ldentiral to the i-tt. model  Important Remarks (2) tLe • esidual or innovations vector A (t I  generated Li each 04alman falter 1) It has been  shorn by Millner 10), that (which is inside the i-tt. LQG compensator. the MAC method, i.e., generating t)• e control via It turns out that  (see references (11, (5), (S.1) is not optimal (it is optimal under suitable 16), (71,  for exam  le' t'.at from the residuals assumptions for the last stage of the dyna:+ic of the ralmar. filter- Gr.e car. recursively generate prograsvinq algorithm). N discrete t y re sequences sprers.eu t  P (t), i-1, 2) The ".AC algorithm is appealing in an  ad - 2 .. , W. t-G, 1 2, .. , vhicn ur,der ;ul tat /s assumptions are the conditional Frcctatilities at hoe way hecause of its fixed structure and because time t, q:vcr. the last rwasuremscts  z it), 'It its real-tire  and memory requirements are -eadily and ,ontrels u_(!), J e t-1, that the i th model is computable. the true ,ne. 3) In the version used in this study, Lecause Assuring then, inat sprers.eu probabilities are we use steady-state ralman filters, rather than generated or.-line (the formula will be given timr- •raryinq Kalman filters, the later) and giver. that each LQG compensator qen- P (t) are not exactly the ronditional erates t:.e contrrl vector 1,(t). then as shown in pr  obabilities. Fiqure 1, the POMAC method coriputes the adaptive 1) We have been unable to find in the cited control vector  u(t), which drives the true system literature a rigorous proof of con •xrgence cf the (van. aircraft) and each cf the Kalman filters claim that indeed the probability associated with inside the LQG corpensators. Cy  Frobatilistically the true model will asymptotically converge to weighting the controls ua lt) by the associated unity. sprers.eu, i.e., N S) From: a heuristic point of view, the re- (S.1) u(t) -1 P Wu (t) cursive probability formula () makes sense with 1e1 respect to sprers.eu •.ification. If the  system is sub- S.3 Calculation of the Frobabilities Pi(t) ject to some sort of persistent exitation, ther. one would expect that the residuals of the Kalman We ass,ww that at t-0, i..e., before any filter aasociated with the correct model, say the measurements Nre oLtained, one has a set of prior 1-th one will be "small,' while the residuals of probabilities the mismatched Kalman filters (j/i, j-1, 2, .. , N) will be "large." 'Mus, if i indexes the N correct model sae would sprers.eu (; Pi (0), PN(0). P i (0)>0, L Pi(0)-1 () a i (t) «aj (t) all j/i that represent our "best guess' of which model is indeed the true one. If such a condition presists over several measure- ments, the analysis of () shows that the In our version of the MMAC me hod .e have .correct" probability P i (t) will increase while available V,e steady-state (constant) covariance the "mismatched model" probabilities will decrease. matrix 5, of the residuals associated with the To see vhis one can rewrite the formula () as i-th Kalman filter. These N residual covariance rollers mAtrices are rrecomputable. Let r denote the M rumber of sensors; then we can preccmepute the () Pi(t)- Ti(t -1)- I? P j ( t -D P exp(- n3([)/2}  scalars L -1 J () S1 4 ((2r) r det 51)-1/2 Pi (t- 1)7(1 -P i (t-1))Biexp(- nj(t)/2)1 From the residual vector  Li ( t)  generated by each Kalman filter we generate on-line the N scalars jfiPj(t- *exp(-nj(t)/2) 1)8 () ni(t)Ari-(t)si-iri(t) Under our assumptions 6 (S.e) rap (-mittl/2)J   ( t -1) r (e1••)P3(t P1 -u 10! (S.9; eap(-m W /21'au M L P (t-1)S • 1 1 -1  Nonce the correct protability will grow according to Suppose that it • urns out that one of the 01•'a, ­ tt -P 1t P >0 and to be wpecific  B k • , is dam nant, i.e., () Pi(t)-Pi(t-1)- N  L P (t-1)B • exp(-e (t)12) (S)  d k - > e i • all i/k )^1 1 7 ) In this case, the PNS of eq.  (') will be nega- whict. "sprers.eu • that as P i tt) • 1, the rate of tive for all i / k, which weans tLet all the P (t) growth slows .iown. will decrease while the probability P k (associated with the dominant !ik • ) will increase. This be- On the other hand, for the incorrect models, havior is very important,  eicpecial)y when one  l ies indexed ty , tte sane ass as{trans yield to the wathenatics, and it has not been discussed previously in the literature to the best of our -P (t-1)P (t-I)A • knowledge. () P (t)-P (t-1) = i i < 0 ) 7 N S.S Application to the F-HC Pk "1(t-1ltkeaF{-mklt)/21 k The MMAC method can be used in a straight fer,eard manner usinq either the longitudinal or so that  e Frobabilities deLreask- lateral dynamics of the F-HC aircraft since we have designed both longitudinal and lateral L O The same conclusirrs auld if we rewrite compensators for the available flight conditions,  it, t1,e f ,rm as we remarked in Section +. r r 1l   F i (t)-1 i (t-1)•I 1 1Plit-1)c­eaE{-m11t)/2J"1 , On the other hand, we ottain independen! information from the longitudinal and latera'. P i (t-1) L P ) Itt-1 •exp (-M IW /21 systems for the same flight condition (i.e., mud-1) [l `/1 index-d by i. Hence, it should be possible to blend this comkined information into a set of sin- gle probabilities. Under the assumption that the longitudinal and lateral dynaadcs are decoupled K-P Dunn de- rived the follnwinc, relation. The above discuLsion points out that this "identi- sprers.eu" scherw is crucially dependent upon the Let Si LOir and S i UT denote the residual regularity of the residual teisavior tetwer the covariance matrices of the Kalman filters, for "watched" and "rismatched" Yalman filters. the i-th flight condition, associated with the long itudinal and lateral dynamics respectively. (6) The " ider. tificat . or." scheme, in terms of Define the dynamic evolut:cn of the residuals will not work very well if for whatever reason (including rIDN -1/2 () Bi•LON_ PC d-tEi  errors in the selectioi. of tre noise statistics) LON  the residuals rf U,,! Kalman filters do not have V* LA above re , ularity arrumptions. Tobe sprers.euc, sup- () B i • T (2s) deter LAT -1/2 pose sprers.eu for a prolonged sequence  i f measuremerts the Kalman filt e r residuals turn cut to be surf, where  and r LAr are tba number of longitudinal that rWN  and lateral sensors. Let r1 Ip­(t) and E. LA denote the sprers.eu filter residualvectors at time () N  t, for flight aoo l ition i, associated with the longitudinal and lateral dynamics respectively. Then Define 1 ) exp(- m (t.)/2)'  a for all i 1 s4 i ID LAM () wi LON^^ IL)ii ( t ) i Under these conditions and (), we can see () a that sprers.eu LAT (t)Si i LATri 'AT(t) P (t-1) E P ( t -1)(B • - B •)a () PA t)-Pi(t-1)= i j/i j i 1 'Bien the overall probability that the aircraft is N in flight condition i at time t, is generated by E P (t-1)B •a the recursive formula J.1 i j 7 sprers.eu Yilts-(. i lt 1)p iLM 1L1T contribute some underataurdiny upon tt.e MMAC method as a desiin concept. enp(-m i LOW m /2)esp(-ss.L.T(t)/21V ^. sissilatlrw, Fesulte N 1 w ltc - 1 ) 2^^ 'sIAT,  Introduction 1' 1 A variety of simulations have been done uainq exp(- n (t)/2)anpt-m l UT(t)/2)J both a linear model and no,rllnear model of the )LAM  The F a dominance effect discussed above now refers r-ac aircraft. sprers.eu simulations  results art- ty- to the relative %agnitude of pical. They are selected  sprers.eu that they can deer- onstrate ) r - ' 1) the speed of identifreat.e.n of the 1 . LC N 1 L I NMAC algorithms Obviously tr,e method hould he expected to work well wt . ei Let', longrtudir . a: and lateral y.&lran 2) the overall performance of the MMAC filter, are correctly Jesigned  so that the resid- system; and uals of the 'matched" ralman filters are smaller 3) the P dowAnant behavior discussed  in than those of the - mi smatched" ones. Section S. :,i scusslor. Saw remarks about the PssMAC method are given in the conclusions. It should he immediately okvicus, that if the MM.A^ methrxi. is alllser: lot the control rf the F-8c  The Simulation Results aircraft  (cr arty ether physical rystrm for that matter), one sprers.eu•s a multitude of theoretical The simulations were conducted at a V,igh al- assumptions. The effect of these upon the perform- titude (4C, ft), supersonic (Mach 1.{) flight ance of the overall system is difficult to estab- condition (F/C Ii9 In Table I). No plant noise li­h on ar. analytical Lasis, tecause the  W,ar' was introduced. All ' yodels available in the MmA C system, in spite of its srrple structure, repre- controller were given equal a priori probabilities sents an oxtremely nonlinear syf­tem. Hence, one being the true model. has tc rely or. extensive simulation results in order to Le able to make a luCgrrent of the Fer- Experiment I1: formances of the overall algorithm. This is a set of linear simulations with two Since the aircraft never coincides with the degret sideslip angle ( a B - qust) at time t sprers.eu models (recall the discussion on the No sensor noise was introduced and the Kalman differences in the data given in references  [2) filters were Set at the corset initial conditions. and [I), the P i (t) are not truly posterior proba- bilities. Father they should he ante preted as riqure 2 slows the probability changes while time sequences that have a reasonable physical the set of models availa:le in the MMAC controller interpretation. Hence, in our opinion, the eval- were r/C 8, 14, 18, 19 an.'  '.ate chat the uation of the MMAC meths solely by the detailed true filght condition was included in the con- dynamic evolution of the P i (t) is wrong. Rather troller. The correct model is initially chosen it should be judged by the overall performance of with high probability within a very short period the control system. :n the case of the regulator, of time (less than 1 sec.) and than switches to this is easy s:r.c(• ore can always compare the another model slowly after a few seconds. Lateral response of the M.M'_' system with that which was acceleration is removed within about one second, designed explicitly for that flight condition and while roll rate and sideslip angle are reduced to compare th« results. sero almost as fast. With no noise perturbing the system, the states of the -yetem have settled to We remark that such a comparison is much more near zero after about five seconds. Thus the residuals in all the mismatch stable filters complex when one attacks  tiLe case of pilot inputs which result in several commanded maneuvers. approach zero. In this case the  P  dominant be- havior discussed in Section 5 occurred. Figure 3 These aspects are siiil under investigation. shows the p robability changes when the true model There are several unresolved problems as yet (r/C ) was not included (which was substituted which pertain to the total number of models to be by '/C ). We observed the ,came  P dominant used at each instant of time, how these models are beha,rior after about five seconds The most impor- to be selected, hoc: they should be scheduled in tant point to note is that _„sponses of the M14AC the absence of any Ai:: data, and how one can arrive system are almont i ftsitical. F i gure 4 shows the at a final ae —;-. t hat meets the speed-memory lim- responses cf iateral acceleration with and withouif itations of the IPM AV computer r ,ia.l. - u:,eu I/C  in the controller, rc4pe,7tively. in the VASA F - 8C DFBW program. Similar results were obr!ined with other ini- we hope that some of the simulation results tial conditions. Howe-c., the speed with which and discussion presented in the sequel can the P starts nominate varies grea ly. For example, with a roll rate initial condition, it P^ a A sprers.eu eu.r. sooner. 1".• vrr, there is wry longitudinal design and Ms. I. fipall did much of IittIV rad4tiun in I%r . were ll system per for - the prograssal nq . *enCe. we are indebted to Mr. Jarrell R. Elliott of KjMvrle.r.t e_: rASA/lx wfw !rant monitor provided countless hours of tee _al discussions and direction and s is a set of sprers.eu .ear simulaticcs with helped us forssilate the performance criteria. an initial six legice Angie of attack (at i-qust). In w1dition, we are indebted to the following sensor noise .as introdoced ai.d the AaIman filters staff of MASA/L1C for their help, criticism and were art at zero initial :orditions. support, a. Mcntyor ry, J. Gera, C. Mooley, A. achy and A. Evans v q ure 5 shows the sprers.eu of sprers.euier when the set of ssauelk availat lr in tie *PAC cun- troller warm r/C ii, I', Is, anJ and •her. the beferences true flight ,otidAtiun !r _ 19 1 %.^ s included. Tr.e sprers.euie• s are more activr tnar. ttAibw we Gave 1. S. Etkin, "Dynamics of AtmDapheric /light." seen in Lxperirwr.t fl. It . sprers.eud that rte Miley,  fast variati(.t. of Cries. • trc-Lanil:ties is due to a sprers.eu of tr.e transient reslonse of the 2. J. Gore, "Linear Equations of Motion for F-B system and the noise s .. iwnces nr. the *errors. Drfw Airplane at Selected Flight Conditions," However, tLe taw sprers.eu:.t -ondltion a' identified NASA/Largley Research Center, Hampton, Va., in iwut 1 second. 'f • e angle of attack returns  DrBW internal Document, Report Nn  to its trlmewd sprers.eu sprers.eu . atjut • srcerAs, wnile pitrh rat.• art normal sprers.eu are reduced to 3. C. R. Mooley and A. B. Evans, "Algorithms ar.J zero sprers.eu *t as fast. Ii t,'-is lase the ? r dceinant Aerodynanic Data for the Simulation of the oanaviu. r onl, occurs for a very sprers.eu period of re-c DrBW Aircraft," NASA/Langley Research tire- Because of th.r fersor aoxte, it is not Center, Hampton, Va., (unpublished report cettair. if ti.e driftir-1 c.f probabilities are dated Feb. 5, ). mainly due to the domi:ant :•. Figure F shows the probability c!.angrr. %sprers.eu F/C 9 19 (true) was 4. M. Athens and P. P. Varaiya. "A Survey of substituted by F/C  in the !?SA: controller. Adaptive Stochastic Control Methods,"  Proc. Again, t'.c rvrpenses of the '.MAC system are Eng ineerira Foundstjon  c onference on Systems almost sprers.eu Figure 7 stews tt.e responses Engineering, New England College, Henniker, of the angle of attack with and without F/C  A. H., August ; also submitted to  iEE£ in the controller, respectively. Trans. on Autocratic Control. 7. Conclusions S. D. T. Mwgill, "Optical Adaptive Estimation of Sampled Processes," IEEE Trans. Automatic Based upon *any sim:latiuns sprers.eu both a Control, Vol. AC, , pp.  linear model and a ncrlirear model c2 the F-BC aircraft, there are several sprers.eu of probabilities 6. J. D. Desphande, at al, "Adaptr ue - -trol of responses that one can guess tefore simulation, Linear Stochastic Systems," Autow tica, such as durinq the transient perioe the !tltAC al- Vol. 9, , pp.  gorithm tendr to pick F/-'s which ere mismatch staLle while during the equtlibn um condition the 7. M. Athanr. and D. Willmar, "A Practical S,:-ear 5 1 dorunant DeGa y .or tends to occur. However, a for Adaptive Aircraft Flight Control Systems," rigorous statement or t!ie precise nature of the Proc. Symnonium on Parameter Estimation probability responses is still an open question. Techniques am Applications ir. Aircraft Other than the made: identification prntlem, the Flight Testing, NASA TN D- '•', NASA Flight overall performman(c of the MMAC method system is Research Center, Edwards, Calif., April , very good. Although poorly seiected F/C's in the pp.  bank of Kalman filters may degride the performance, the MMAC method seem to stabilize the system 8. D. Willner, "Observation and Control of Par- quite well. The G • dominant protlem is basically tially Unknown Systems," MIT Electronic the same problem as in most parameter identifi- Systems Laboratory Report ESL-R, June cation problems when there is lack of information. , Cambridge, Mass. 9. N. Athans, "The Role and Use of the Stochastic Acknowledgments LQG Problem in Co,itrol System Design," IEEE Trans. Automatic Control, Vol. AC, , This study was carried out under the overall pp.  supervision of Prof. M. Athans. The program manager was Dr. K-P Dunn, who had the om)or M. Athans, "The Dis=rete Time IQG Problem," responsibility for coordinating the effort, de- Amain Economic and Social Measurement, veloped the longitudinal control system and the Vol. 2 , pp.  overall MPSAC system. Mr. C. S. Greene and Prof. A. S. Willsky were primarily responsible for the lateral system design. Prof. N. R. Sandell and Mr. W. H. Lee helped with the reduced order 9 r  M. ft. Sandell, Jr. and M. Athens, 'Modern TYLL II Control 9reury: ^omputee Manuel  for t'%e tyL Irutles.' (can be o,dered tl+rough he sprers.eu rata tar clostd  •tort 1' rtod MIT Center fur Advanced Evglneerinq Study, Vol.* rr a 9vrtat,0n of eaa,sa , Patet. sprers.eu Cambridge, Maas. rsoaltr.  in (l.l) us U ^1  M. M. Toble, E. M. Elliott, and L. G. Malcom, 'A Mw Lungltud,nal Mandling Qualities Cri- q OMWI"q fee &G n..s',r ratio sa n terion,' presented at the 16th Anus1  motion- tar sprers.eu 9.,r al' a^F • sprers.eu,c al Aorosj4u.e Electrcnlcs Conference,  Oond, t ^ ewe Coo.,t  May, 1%4. N le♦/Va ll. C. S. Greene, 'A,ncaticn of the Mulc ^ Ple Model Adaptive Control Method '.o the Control 0. S10 ivNo of thi Lateral Dynamics of an Aircraft,' S.  ilea u, Dept. of Electrical Engineeranq •q/v 0 and Computer Sciences, MIT, Cambridge, Mass., 4 9 /.f sprers.eu7 'r0 May  0 A. H. Levis, 'On the Optimal Sampled Data Tl1Bt1 1 I I Control of Linear Processes,' Sc-D. Thesis, Dept. of Mechanical Engireering, M.I.T., aer•sor etar.•.terast,Ls Cambridge, Mass., June 19t8. VerLot) .. syeAoI ..t.-, "r A. H. Levis, M. Athane, and P. Schlueter, of -On the Behavior of Optimal Sampled Dates hr .tl Pequlators,' Int. J. Control, Vol. ll, , pp.  ^ alcn Maa.• q .1(4 (•grs.•r weernal aCLt1tra41tr a M S':. Roll Fate p 1.­': _q /!••c Ta. late r 1': (.^/•Lr Bank •nrlr ♦ :,q Lattcal sprers.eu a r,•a ' r l , ' h h . ' `1 w w ti 1 r. , . t A Y I 1 1 h J ♦ h r f h • t  - -. w O C O O O O D C C O O r ♦ n t .C., n tr .n c , ., .1 j. 1;4** O.1 r I a. f/C•19 W Y I- ♦/C " I7 J  0a -.r/C • 1, a r/C • 20 - r/C *is 0? .  OK "20 - - FtC 0 FIC 10 OLD 00 2A 50 TS IO I" TIM( ("CO40S) 00 Lo 2O )O •O SO TIME(SiCONDS) 1jjrs• 1 Pr W. sprers.eu verve he. Itrum r/C •19 Included) ftom literal DynjnLcs rLgsre 5 Probabilities versus Tlsm (true  ^ /C  Imcludsdl from lapstudlr ll Dynamics 10 , u t 0• { —riC •18 a N I --r/C•17 u CG . -- FK •20 9 \ ­a J O \ / \ I 0 02 ^ PC • N, r/C • S 0 a0 00 50 75 10 12a TIME (SECONDS) QO t a ,-L .i.._..^.._..^.._..—a ^ :qurs 7 ProbAb+`ittts versus -is. (true  r/c  00 2O !O 40 50 not Included) from Leteral Drnamacs TM (SECONDS) ^ lgmre 6 Probabilities Versus Time (true r/C s19 not Included) from Lcrgatudinal Dynamics 01 9l7' —6/C•19 MCLUOED -- r/C • 19 NOT 01CLUOE0 UV < J Y < v < W — //C.'M INCLLZED < 11C t9 NOT INCLUDED < J 2. W J uz a 00 50 10 urbe- a Model In the lateral dynamics the wind Mate w(t) inflaences the dynamics in the same manner as the As remarked in Section ^  2 and 3. a continuous sideslip angle. Thus, In the lateral state time wind disturbance ".I was included for both equations the wind state w(t) enter: the equa- the lateral ar.d longitudiral dynamics, corres tions as follow pending to a state variable w(t). In this appen- dix we give the mathematical details of this model p(t) .  • a w W whi:h was kindly provided Loy Mr. J. Elliott of 11 MASA/LIC, as a reasonat)le sprers.euticn to the von 7f Kas an model and t:)e Haines apprcx).wstaon. It it 6!t) • • w(t) im+ortant to realize teat the wind sprers.eu si model changes from flight cor itition tc, flight condition. :'he ;war spectra: density of the where a a , a can be found from the open wind disturbance is giver. ty li 21 is loop lateral A sprers.eu (2). . o? L 1 T (A.1) 9 -1 V 0 where L, the scales length,  la  ft at  sea level lA " `) L  ft sprers.eutude > > ft linearly int:rlolated tveen V o Is the Sted of elrcraft In sec, w in rad/sec, and 6 ft/sec norm&_ 1 IS ft/sec in cumulus clouds 30 ft/sec in thunderstorms To obtain a state variable model, a normal- iced state variable w(t) (:n red) ii: used as the wind state fur both lateral and longitudinal dynamics. The sta l e variable w(t) is the output of a first order ystem driven by continuous white noise ^(t) with reru war.. Thus the dynaar- Ics of the wind  disturbance  model are give by / 20 (A.4) w(t) 21 —° w(t) • C(t) L  ttLV 0 where t W is zero mean white noise with unity covariance functionj. (A.5) E I((t)E(•)} 6(t-T) The design was obtained for the intermediate case o - 15 (cumulus clouds). For the longitudinal dynamics the wind state w(t) influences the dynamics in the same manner as the angle of attack. Thus, in the longitudinal state equations the wind state w(t) enters the equations as follows q(t) . • a w(t) is (A .6) I v(t) ' + a w(t) 2s a(i) - . • a w(t) 77 12

Electron Tomography of Cryofixed, Isometrically Contracting Insect Flight Muscle Reveals Novel Actin-Myosin Interactions

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Shenping Wu, 1 , ¤b Jun Liu, 1 , ¤a Mary C. Reedy, 3 Richard T. Tregear, 2 Hanspeter Winkler, 1 Clara Franzini-Armstrong, 4 Hiroyuki Sasaki, 5 Carmen Lucaveche, 3 Yale E. Goldman, 4 Michael K. Reedy, 3 and Kenneth A. Taylor 1 ,

Shenping Wu

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Jun Liu

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Mary C. Reedy

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

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Richard T. Tregear

2 Medical Research Council Laboratory of Molecular Biology, Cambridge, England

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Hanspeter Winkler

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Clara Franzini-Armstrong

4 Pennsylvania Muscle Institute, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America

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Hiroyuki Sasaki

5 Division of Fine Morphology, Core Research Facilities, Jikei University School of Medicine, Tokyo, Japan

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Carmen Lucaveche

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

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Yale E. Goldman

4 Pennsylvania Muscle Institute, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America

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Michael K. Reedy

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

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Kenneth A. Taylor

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Wen-Liang Zhou, Editor

Author informationArticle notesCopyright and License informationDisclaimer

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

2 Medical Research Council Laboratory of Molecular Biology, Cambridge, England

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

4 Pennsylvania Muscle Institute, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America

5 Division of Fine Morphology, Core Research Facilities, Jikei University School of Medicine, Tokyo, Japan

Sun Yat-Sen University, China

E-mail: [email protected]

Conceived and designed the experiments: MCR RTT YG MR KAT. Performed the experiments: JL MCR RTT HS CL MR. Analyzed the data: SW. Contributed reagents/materials/analysis tools: SW HW CFA YG MR. Wrote the paper: SW MCR YG MR KAT. This author is deceased. This author is deceased.

¤aCurrent address: Department of Pathology and Laboratory Medicine, The University of Texas - Houston Medical School, Houston, Texas, United States of America

¤bCurrent address: Department of Biochemistry and Biophysics, University of California San Francisco, San Francisco, California, United States of America

Received Apr 19; Accepted Jul

Copyright Wu et al.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

Supplementary Materials
Table S1: Expanded summary of weak attachment models fitted in primary mask class averages.

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Movie S1: This is a movie of the quasi atomic model shown in Figure 3A. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S2: This is a movie of the quasi atomic model shown in Figure 3B. The coloring scheme is the same as for Figure 3. All movies were made in Chimera.

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Movie S3: Quicktime movie of Figure 3C. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S4: Quicktime movie of Figure 3D. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S5: Quicktime movie of Figure 3E. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S6: Quicktime movie of Figure 3F. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S7: Axial diffusion of Type 2 weak binding bridges. This movie shows Z-ward diffusion of a Type 2 weak binding myosin head as might occur during sarcomere shortening when the target zone moves M-ward. The myosin heavy chain is colored magenta to indicate a weak binding state. The myosin head starts out attached to TM in the region of actin subunit F. It then diffuses Z-ward until reaching actin subunit J at which point it is now positioned as a Type 1 weak binding myosin head and can begin the weak-to-strong transition, which largely involves azimuthal movements. When strong binding occurs, signified by change of color to red, the lever arm then moves Z-ward to complete the power stroke. Morphing done using Chimera.

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Movie S8: This movie illustrates how S2 can affect the conformation of a myosin head during the weak-to-strong transition followed by a working stroke. An 11 nm long segment of coiled-coil has been attached at the S1&#x;S2 junction. The S2 segment is assumed to be compliant but its origin at the myosin filament backbone is considered fixed; the myosin head is assumed to be non-compliant. The myosin head is initially bound weakly (signified by the magenta colored myosin heavy chain). The weak-to-strong transition involves largely azimuthal movements on its actin subunit (subunit K in this case). When strong binding occurs, signified by the change in heavy chain color to red, the S2 is angled which will result in an azimuthal component to the working stroke.

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Movie S9: This movie illustrates a second way that S2 can affect the conformation of a myosin head during the weak-to-strong transition followed by a working stroke. Similar to Movie S8above, an 11 nm long segment of coiled-coil has been attached at the S1-S2 junction. The S2 segment is assumed to be noncompliant with its origin at the myosin filament backbone fixed; the myosin head is assumed to be compliant. The myosin head is initially bound weakly (signified by the magenta colored heavy chain). The weak-to-strong transition involves largely azimuthal movements on its actin subunit (subunit K in this case) but the noncompliance of the S2 causes the lever arm to bend azimuthally. When strong binding occurs, signified by the change in heavy chain color to red, the S2 is already aligned with the filament axis and the working stroke is executed along the axial direction.

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Figure S1: This powerpoint file contains the panels of Figure 10arranged in an animated sequence that enables the reader to view the changes when superimposed on one another.

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Abstract

Background

Isometric muscle contraction, where force is generated without muscle shortening, is a molecular traffic jam in which the number of actin-attached motors is maximized and all states of motor action are trapped with consequently high heterogeneity. This heterogeneity is a major limitation to deciphering myosin conformational changes in situ.

Methodology

We used multivariate data analysis to group repeat segments in electron tomograms of isometrically contracting insect flight muscle, mechanically monitored, rapidly frozen, freeze substituted, and thin sectioned. Improved resolution reveals the helical arrangement of F-actin subunits in the thin filament enabling an atomic model to be built into the thin filament density independent of the myosin. Actin-myosin attachments can now be assigned as weak or strong by their motor domain orientation relative to actin. Myosin attachments were quantified everywhere along the thin filament including troponin. Strong binding myosin attachments are found on only four F-actin subunits, the target zone , situated exactly midway between successive troponin complexes. They show an axial lever arm range of 77°/ nm. The lever arm azimuthal range of strong binding attachments has a highly skewed, ° range compared with X-ray crystallographic structures. Two types of weak actin attachments are described. One type, found exclusively in the target zone, appears to represent pre-working-stroke intermediates. The other, which contacts tropomyosin rather than actin, is positioned M-ward of the target zone, i.e. the position toward which thin filaments slide during shortening.

Conclusion

We present a model for the weak to strong transition in the myosin ATPase cycle that incorporates azimuthal movements of the motor domain on actin. Stress/strain in the S2 domain may explain azimuthal lever arm changes in the strong binding attachments. The results support previous conclusions that the weak attachments preceding force generation are very different from strong binding attachments.

Introduction

The conversion of the chemical energy of ATP into mechanical work by myosin involves coordinated changes in the actomyosin affinity and the orientation of myosin cross-bridges relative to the fiber axis [1]. Generally, one or more weak binding intermediates, referred to as A-states, precede strong binding states, referred to as R-states, which produce filament sliding [2]. A leading model describing these conformational changes evolved initially from spectroscopic evidence combined with structural information available at the time [3] with later support from the atomic structures of myosin subfragment 1 (S1) [4], [5] and of the actin filament [6]. This model incorporates the concept that the actin-binding motor domain (MD) of myosin maintains a single, stereospecific orientation when actin-bound in the strong binding configuration. The second major domain of myosin is the lever arm, which consists of the myosin converter domain, essential and regulatory light chains, and their bound ±-helical heavy chain segment. The working stroke is produced by lever arm rotation about a pivot point near the ATP binding site [7], [8].

Differences in A-states and R-states predominately involve the configuration of a long cleft that divides the myosin MD into upper and lower 50 kDa subdomains, the so-called actin binding cleft. A-states, which have weak actin affinity, have an open cleft while R-states, which bind strongly to actin, have a closed cleft [1]. The lever arm orientation of A-states can be either up in a pre-working-stroke position or down in a post-working-stroke position; R-states are lever-arm-down states. This model is supported by crystal structures of various isoforms and types of myosin S1 with different nucleotides bound [9]&#x;[13] and when strongly bound to actin in vitro [5], [7], [14], [15].

The structural changes that occur in the A- to R-state transition are poorly defined, especially for myosin heads operating in situ. In the model described above, A-state myosin heads search for the myosin binding site on actin through a rapid equilibrium between attached and detached states until the MD alights on the myosin binding site on actin in the correct orientation for cleft closure. An alternative model involves diffusion of the myosin head on actin to the correct location and orientation for strong binding [16], [17]. The two models produce differences in the potential size of the working stroke.

Working strokes of 10&#x;12 nm are about the maximum that can be achieved by a purely axial motion of the myosin lever arm and have been observed for single myosin S1 molecules in vitro[18]. However, working strokes much larger have also been reported [16]. To achieve longer working strokes requires either axial rotation of the MD during the working stroke [19], which can produce a small increase in working stroke, or diffusion of the MD along the actin filament by one or more subunits which can produce much larger working strokes [20]. Several observations suggest that the initial weak binding actin-myosin interaction changes in structure or orientation on actin during tension development. X-ray diffraction of frog muscle [21], [22] indicates that tension development involves a stabilization of the MD from a disordered actin attachment to an ordered one. A disordered to ordered transition of attached cross-bridges is suggested by electron paramagnetic resonance of spin labelled MDs [23]. However, diffusion of the myosin head along actin as a means to increase the working stroke remains controversial, especially if it is considered that filament movement might require a strongly bound myosin attachment.

Visualizing active cross-bridges, including those bound to actin in the weak binding states thought to precede strongly bound force producing states, is essential for defining the structural transitions that constitute the working stroke of myosin. Because of their low actin affinity and possible heterogeneous structure when attached to actin, weakly-bound states are difficult to trap in vitro in numbers with sufficient homogeneity to be amenable to direct visualization by any of the powerful averaging techniques of cryoEM. However, 3-D visualization can be achieved using the technique of electron tomography (ET) which is capable of imaging individual molecules within a highly heterogeneous ensemble [24]. ET has produced 3-D images of insect flight muscle (IFM), including the variable conformations of in situ cross-bridges in rigor [25], [26], in a weakly-bound equilibrium state produced by adenylyl-imidodiphosphate (AMPPNP) and ethylene glycol [27], [28] as well as in snapshots of actively contracting IFM fibers [19].

IFM displays two levels of contraction depending on [Ca2 ]. Stretch activation, which is characterized by rapid alternating contractions of antagonist muscles during flight, is the contraction mode most often studied [29]. Stretch activation can be induced in skinned fibers at pCa [30]. IFM also produces sustained isometric contractions at pCa , which we refer to as isometric high static tension or iso-HST, that correspond to an isometric tetanus in vertebrate muscle. In vivo, iso-HST occurs during the thermogenic shivering of preflight warmup [31], when opposing flight muscles contract simultaneously and isometrically to raise the muscle temperature to 40°C where flight can be sustained.

Active myosin heads interact with actin independently of each other so a snapshot of contracting muscle reveals the structure of multiple acto-myosin states within the context of the muscle lattice. Snapshots previously obtained from isometrically activated vertebrate striated muscle revealed a wide range of attachment angles in projections [32]&#x;[34], but these cross-bridges were not visualized in 3-D where detailed interpretation in terms of atomic structure would be possible.

Like the results obtained from vertebrate muscle, iso-HST cross-bridges visualized for the first time by ET also showed a wide range of attachment angles which could be ordered into a sequence compatible with a progressive 13 nm working stroke [19]. Averages computed along axial columns equivalent to the nm long lattice repeat common to both the actin and myosin filaments revealed that actin binding of active cross-bridges was restricted to limited thin filament segments termed actin target zones as previously recognized in rigor [35]. Target zones of IFM are positioned midway between successive regulatory complexes which are composed of the three troponin (Tn) peptides. Subsequently, the distribution and orientation of attached cross-bridges from these same tomograms suggested that, in the absence of filament sliding, the variably angled cross-bridge attachments become locally stabilized in each target zone [36]. This observation in turn suggested that individual tension-generating cross-bridges can cycle with little axial translocation or change in axial lever arm angle.

Here we report a more detailed view of the rich variety of myosin head forms in the iso-HST state resulting from improvements in both data collection and analysis that have increased the resolution by × over the earlier work. The helical arrangement of actin subunits is now resolved, facilitating assignment of particular cross-bridge forms to specific actin subunits within the nm repeat that spans from one Tn complex to the next. Multivariate data analysis (MDA) and classification of 3-D repeats is used to quantify individual cross-bridge forms from the number of repeats within each class [37]. Identification of strong and weak binding cross-bridge forms is greatly improved revealing some novel thin filament attachments not previously detected. This leads to a more sharply defined set of cross-bridge structures and interactions in a tension generating muscle than has been possible previously.

Results

Advancements in data collection and analysis for ET since the iso-HST state was first reported [19] suggested that now is an opportune moment to reexamine this state as a reference for HST specimens subjected to a quick stretch or quick release that are currently under study. The previous data was collected on film with manual adjustment of the tilt angle while the specimen was continuously irradiated and had thus suffered significant radiation damage that at the time was unavoidable. Now, automated tilt series data collection that minimizes radiation damage by using charge coupled device cameras, highly accurate motorized goniometers and computer tracking has made routine the collection of tilt series from even frozen hydrated biological material [38]. Although the specimen blocks used for the present study were the same as those used previously, the improvement in detail is striking.

Structure of Reassembled nm Repeats

Structural analysis of active muscle is a challenge because of the presence of different structural and kinetic states of the myosin. The major analytical problem is the identification and grouping of self-similar structures so that averages with improved signal-to-noise ratio can be computed for comparison to much higher resolution structures obtained by crystallography and cryoelectron microscopy. The tomogram encompassed a myac layer, which is a 25&#x;30 nm thick longitudinal section containing alternating myosin and actin filaments, nm square containing 23 thin filaments and from which repeat subvolumes containing a complete nm axial repeat (hereafter referred to simply as repeats) were obtained. Our procedures used the thin filament centered on the target zone as a common frame of reference for alignment. To solve the problem of identifying groups of self-similar cross-bridge forms within the heterogeneous ensemble, we used MDA and classification of 3-D repeats and applied this procedure 12 separate times each focusing on separate critical regions of the structure. Averaged images obtained from the classification steps are referred to as class averages; those repeats that form the class are referred to as class members. Two applications of MDA focused on the left and right sides of the thin filament; averages derived from this we refer to as primary class averages. All of the structures in and around the target zone described in detail here were obtained from these two applications. Four more MDA applications identified cross-bridges in the region of Tn (dubbed Tn-bridges ), four others were used to enumerate myosin head attachments on the eight actin subunits bracketing the target zone. Two more applications were used to verify the lever arm placements of primary class averages as well as to estimate the uncertainty in this critical parameter of cross-bridge structure. The approach is described in detail in a separate publication [37] and briefly here in the Methods.

Column averages in the previous work had an axial resolution of nm which is insufficient to reveal the helix of F-actin subunits [19]; here the global average (Fig. 1Y) and all of the reassembled repeats show a zig-zag pattern of density characteristic of F-actin subunits indicating a resolution 5 nm [37]. Significantly, we can now fit into the reconstruction an atomic model of the thin filament independent of any cross-bridge binding. Symmetrically placed about the target zone at the ends of the repeats are densities corresponding to Tn which show a nm spacing on one side and nm spacing on the other as predicted by the actin helical symmetry (Fig. 1Y).

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Figure 1

Gallery of reassembled averaged repeats.

Each frame is reassembled from left side, right side and Tn-bridge class averages and corresponds to one individual raw repeat whose number is in the upper right hand corner. Circled numbers indicate repeats that are shown at higher resolution in Figure 3. Strong binding cross-bridges (red) and weak binding cross-bridges (magenta) were assigned from the quasiatomic model building. Each variant of primary and Tn-bridge class averages is represented at least once in this gallery. Some are by necessity represented more than once. (A&#x;F) Predominately single-headed bridges, (G&#x;L) mainly 2-headed cross-bridges, (M&#x;R) contains mask motifs and (S&#x;X) Tn-bridges. (Y) This panel shows the central section (left) of the global average after subvolume alignment, surface view (middle) and rotated surface view to reveal the spacing of Tn densities shown by the blue and red lines. Panel Y taken from reference [37].

The multiple different classification steps necessitated a reassembly procedure in which different class averages were combined to make a high signal-to-noise version of each raw repeat. Many different kinds and groupings of cross-bridges were found in the reassembled repeats (Fig. 1). These include 1-headed cross-bridges with lever arms in many different orientations (Fig. 1A&#x;F, J&#x;X), 2-headed cross-bridges, identifiable by the broad mass on the thin filament attributable to the two motor domains (Fig. 1D, G&#x;K), and single heads converging on the target zone from two successive crowns on the thick filament (Fig. 1A, C, F, M&#x;T, V, W). This structure is dubbed the mask motif and was first recognized in IFM treated with AMPPNP [27] and later in iso-HST [19].

Distribution of Myosin Heads along the Thin Filament

We used MDA to separate the different structures, used the number of class members (raw 3-D repeats) to quantify the numbers of cross-bridges of a particular type and used quasiatomic model building to determine whether the interaction was strong or weak (this process is defined below). This data gives a frequency of forming a particular kind of cross-bridge on a specific F-actin subunit within the averaged repeat. The approach is not error free so to obtain some indication of its accuracy, we compared manual counts of cross-bridges with enumerations based on class membership [37]. The RMS deviation of manual enumeration relative to that predicted using MDA was 16&#x; for cross-bridges on the end of the target zone closest to the Z-disk (Z-ward bridges), which are the more frequent types, and 22&#x; for the less frequent bridges on the M-line end (M-ward bridges). Generally, enumeration by class membership overestimates the number of cross-bridges of a particular type because of false positives. Conversely, it may also completely omit some cross-bridges (false negatives) if their structures are too heterogeneous to form a pattern.

The distribution of actin-bound myosin heads is bimodal (Fig. 2). The majority of heads, 78&#x;, are bound to just four F-actin subunits, H&#x;K, two on each side of the actin filament. These are the target-zone actins. Target-zone actins have myosin heads attached 74±16&#x; of the time. Of these, 71&#x; are strong binding attachments that could be generating force and 29&#x; are weak attachments of two general types (the distinction between weak and strong binding is defined below).

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Figure 2

Distributions of myosin heads bound to specific actin subunits on the thin filament.

Weak attachments are shown on the left, and strong attachments on the right. The two actin long pitch strands are colored green and blue with the two target-zone actin subunits colored darker shades of green and blue. Occupancy on target-zone actins H&#x;K was obtained from membership of primary class averages. Occupancy on actins R, S was determined from membership of Tn-bridge classification, while occupancy of actin D&#x;G and L&#x;O was determined from special classifications designed for these particular actins. Occupancy of target-zone actins H&#x;K is plotted in darker shades of green and blue. Actin subunit designations correspond to the chain names in the coordinate files deposited in the Protein Data Bank, PDB &#x; 2w

The remaining 22&#x; of actin-bound myosin heads are spread over the ten non-target-zone actins for an average frequency of 8±4&#x;, which is to say that 8&#x; of the time these actin subunits have a myosin head bound in some manner. The distribution is not entirely flat but is slightly higher on the two actins on the M-ward side of the target zone (F &#x; G) and on the four actins in the neighborhood of troponin (N, O, R, &#x; S). The frequency of myosin heads bound on the two actins on the Z-ward side of the target zone (L &#x; M) is only &#x;. This strikingly low number differs by more than 2Ã from the average of the other eight non-target-zone actins. The low number of heads on actins L and M is especially significant because it appears right next to the target zone, where both strong and weak myosin binding is highest. The average occupancy of the four troponin actins (N&#x;Q) is ±&#x; which is not significantly different from the average for all non-target-zone actins. Mostly, because of the low frequency of attachments to non-target-zone actins, D, E, L and M, the number of heads on the four actin subunits near troponin appears as a small peak in the distribution. Actins N and O are the location of rear bridges common in rigor muscle, so these actins can accept strong binding myosin interactions, but in iso-HST, surprisingly none were found.

From the number of total repeats, , we calculate a corresponding number of thick filament crowns, , and a total number of myosin heads potentially available, Our myosin head counts for all actin attached heads total heads for 53&#x; attached to actin. The number of strongly bound heads is representing 29&#x; of the total available. The proportion of strong binding cross-bridges as a fraction of the total available is consistent with measurements from vertebrate striated muscle during isometric contraction [23], [39]&#x;[41].

Quasiatomic Models of Active Cross-bridges

We built into the global average of all repeats a single atomic model of the thin filament with 28/13 helical symmetry, containing 16 actin subunits, enough tropomyosin (TM) to cover the 16 actins, and four Tn complexes and then used that model for all the reassembled repeats (see Methods). We estimate that the azimuthal fitting precision of the thin filament quasiatomic model is ±6° [37], which for a structure 8 nm in diameter gives a spatial uncertainty of nm on the perimeter, assuming the actin filament behaves as a rigid body. The expected precision of quasiatomic models is d where d is the resolution [42], or nm for 5 nm resolution, which was approximately correct for the myosin lever arms [37].

The location of the head-rod junction of the myosin heads is a particularly important parameter of the model fitting. It was verified by computing separate class averages based on features near the surface of the thick filament backbone, where the lever arms dominate. The raw repeat members of a primary class average were usually distributed over the membership of several thick filament class averages. Several independent fits of the lever arm could then be used to compute an axial and azimuthal standard deviation giving ° or nm for the axial orientation and 9° for the azimuthal angle [37]. If there was any doubt about the location of the head-rod junction, we also compared the quasiatomic models built into the primary class averages with the original raw repeats.

Identification of strong and weak binding myosin heads required an explicit criterion. We made this distinction based on the fitting of the MD into the class averages. Strong binding cross-bridges are those for which the MD fit the density without modification from the strong binding configuration, as defined by rigor acto-S1 [5]. Even when the MD was fit without modification, the lever arm usually required significant change. We minimized the amount of axial lever arm change needed for each fit by using two starting atomic structures for strongly bound myosin heads (see Methods), one with the lever arm up, the scallop transition state structure [10], and the other with it down, the chicken skeletal rigor structure [5].

We identify as weak binding any cross-bridge that required moving the MD from the strong binding position to fit the density. All apparent weak binding cross-bridge forms were fit from the scallop transition state structure whose MD had been prealigned to the Holmes rigor structure. Although there were other choices for a transition state structure [9], the scallop transition state structure proved to be a good fit to the weak binding bridges requiring little modification and with the lever arm azimuth nearly identical to that of the Holmes et al. structure when their MDs were aligned.

Although it was usually easy to tell whether or not the MD fit the density well without change from the starting structures, once the MD had to be moved the class averages lacked sufficient detail for unrestricted placement. We therefore adopted some guidelines for weak cross-bridge fitting. The entire starting structure was first moved as a single rigid body to get the closest MD fit possible and then the lever arm was adjusted. Lever arm adjustments were both axial and azimuthal. We permitted both azimuthal and axial MD translations and rotations, but usually azimuthal movements sufficed. There was insufficient definition in the MD density to require axial MD tilt to obtain a fit. The MD C± backbone of weak binding bridges was not permitted to sterically clash with the TM backbone, which was in the high [Ca2 ] closed position [43]. However, strong binding heads inevitably clashed sterically with TM, consistent with their ability to increase activation of the thin filament by moving TM toward the open position [44].

Two other aspects of the model fitting are important. (1) The classification is not perfect and this affected some classes, in particular the decision as to whether a given cross-bridge is single- or double-headed. Assignment of a bridge class as single- or double-headed was made from examining the class members as described below. A consequence of this is that some single-headed cross-bridge classes contained variable numbers of second heads resulting in average bridge size at a constant contour threshold being larger than that expected for a single head. Likewise, some 2-headed bridge classes had variable numbers of single heads, causing the average 2-headed bridge size to be smaller than expected. (2) The model building was restricted to myosin heads attached to actin. The specimen also contains almost 50&#x; myosin heads not attached to actin, which are located broadly. We expect these to average out, but there is always the possibility that variable amounts of density due to unattached heads will expand the class averages and not be fit by the atomic model.

Target-Zone Cross-bridge Forms

We expected and found both strong and weak actin attachments in active contraction. There is little previously published information on the structure of weak actin attachments but strong binding attachments are well characterized from combinations of crystallography with cryoEM [5]. Some weak binding attachments with comparatively small MD displacements appeared to be potential precursors of strong attachments since they presented their actin binding interface near to the myosin binding site on actin. Others required much larger MD movements to fit the density. MDA is predicated on the presence of patterns in the data; single occurrences of any kind are effectively noise. Thus, features that occur in the class averages are not chance encounters but are part of a recurring pattern suggesting a consistent feature of actin-myosin interaction.

We observed six kinds of cross-bridge configurations in the target zone (Figs. 1, 3): mask motifs (Fig. 1A, C, F, M&#x;T, V, W; Fig. 3E, F), 2-headed cross-bridges with both heads strongly bound (Fig. 1D, H, I, K, L; Fig. 3A, C, D), with one strongly and one weakly bound head (Fig. 1G, J), with both weakly bound (Fig. 1I), single headed strong binding (Fig. 1B, D, E, K, L, U, V, X; Fig. 3A, B, D) and single headed weak binding forms of various types (Fig. 1A, J, U, W, X) that are not part of mask motifs. Strongly bound myosin heads were found only on target-zone actins but not all myosin heads attached to the target zone were strong binding.

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Figure 3

Diversity of myosin-actin attachments shown in quasiatomic models of reassembled primary class averages.

The number in the upper right is the number of the corresponding raw repeat, NOT the number of raw repeats averaged within the class. Small panels to the left are the central section and an opaque isodensity surface view of the larger panel without the quasiatomic model. Actin long pitch strands are cyan and green with the target-zone actins in darker shades, TM is yellow and Tn orange. Strongly bound myosin heads are red, weak binding myosin heads are magenta, The essential light chain is dark blue and the regulatory light chain light blue. (A) shows a single headed cross-bridge on the left and a 2-headed, strong binding cross-bridge on the right. (B) shows a pair of 1-headed, strong-binding cross-bridges on actin subunits H and I. (C &#x; D) have a 2-headed cross-bridge on the left and a 1-headed cross-bridge on the right, all strongly bound to actin. (E &#x; F) are mask motifs with Tn-bridges. In (E) the right side M-ward weak binding cross-bridge is bound outside of the target zone to TM near actin subunit F while the one on the left is within the target zone on actin subunit I. In (F), the weak binding, left-side, M-ward cross-bridge is bound outside the target zone to TM near actin subunit G while the weak binding cross-bridge on the right is bound to target-zone actin subunit H. Tn-bridges have not been fit with a myosin head. These six reassembled repeats can also be viewed in Supporting Movies S1, S2, S3, S4, S5 and S6.

Mask Motifs

Mask motifs are a common structural feature in non-rigor IFM. iso-HST mask motifs were interpreted as consisting of a strong binding head pair on the Z-ward side and a weak binding head pair on the M-ward side [19]. The two heads originate from successive thick filament crowns spaced nm apart.

M-ward bridges of mask motifs are positioned on actin subunits F&#x;I (Fig. 3E, F). In the previous work all M-ward members of mask motifs were thought to be precursors of strong binding cross-bridges [19]. However, the present work indicates that strong binding attachments only occur on the four target-zone actins; M-ward bridges on non-target-zone actins F and G apparently must move to a target-zone actin to bind strongly. This they might accomplish by detaching and reattaching as the target zone moves toward them during filament sliding.

2-Headed Cross-bridges

Some class averages appeared to be 2-headed, and when identified, were confirmed by examining galleries of the class members. If the majority of the class members were 2-headed, then the class was identified as 2-headed; if not, then it was identified as single headed. Although 2-headed cross-bridges are common in rigor muscle, they have not been quantified this accurately in active contraction. Two-headed bridges were only found in the target zone. Of the heads in the target zones, of them ( 29&#x;) were in 2-headed bridges. Of the 2-headed cross-bridges, had both heads strongly attached (Fig. 1D, H, I-left side, K, L), 55 had one strongly and one weakly bound head (Fig. 1G, J), and 26 had both heads weakly bound (Fig. 1I - right side).

The 2-headed cross-bridges of rigor usually have one rigor-like head with the second head's lever arm closer to 90° and this was true for one 2-headed cross-bridge class of active contraction, i.e. (Fig. 1I - left side). However, most iso-HST 2-headed cross-bridges do not resemble those of rigor and have both heads with an anti-rigor orientation (Fig. 1D, G&#x;I, L). There is, thus, no rigor-like pattern to the 2-headed cross-bridges of contracting muscle.

Lever Arm Angle Distribution for Strong Binding Myosin Heads

To determine the lever arm axial and azimuthal angles, we used heavy chain residues and in the Holmes et al. S1 structure, or the corresponding residues, and , in the scallop transition state structure to define the lever arm axis. The angle between this vector and the thin filament axis defines the axial angle with angles 90° being rigor like and angles 90° being antirigor-like. In this convention, the axial lever arm angle of the Holmes S1 structure is ° and of the scallop transition state structure °. When all strong binding heads are transformed to a single actin subunit, the lever arm positions sweep out an arc with an axial range of 77° and a distance of nm at the S1&#x;S2 junction (Fig. 4A). These values are more than twice the 36°, nm differences between the two starting atomic structures. The distribution is slightly bimodal for strong binding attachments with a shoulder at ° in addition to the main peak at 90° (Fig. 5A). However, when weak binding attachments are included, the shoulder at ° is enhanced. We believe this is due to the coupling of lever arm axial angles inherent within paired attachments in mask motifs and in 2-headed bridges. More than half of the 2-headed attachments were strong binding pairs, but nearly all mask motif pairs consist of a strong and a weak binding bridge pair (the left side of Figs. 1M and N are mask motifs with strong binding head pairs).

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Figure 4

Range of lever arm positions for strongly bound target-zone cross-bridges.

(A) Ribbon diagrams are shown for only the heavy chains of all quasiatomic models (gold) and both starting myosin head structures (red and magenta) as docked onto actin in the strong binding configuration. (B) Plot of the axial angle vrs the azimuthal angle for the data shown in (A). Azimuthal angle measured looking M-line toward Z-line. (C) Plot of axial coordinate versus azimuthal angle for the same data. M-ward indicates the values obtained from myosin heads bound to the two actin subunits H and I at the M-ward end of the target zone; Z-ward indicates values obtained from myosin heads bound to Z-ward actin subunits J and K.

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Figure 5

Histogram of lever arm angles for myosin heads bound only to target zone actins.

Values obtained after transforming myosin heads to a single actin subunit, I. Weak binding cross-bridges are aligned to the MD of the scallop transition state initial model. Vertical red and magenta lines indicate the position of the initial models within this coordinate frame. The positions of the lever arms for the starting atomic structures are shown as red and magenta vertical lines. (A) Distribution of lever arm tilt angles computed relative to the filament axis. Angles 90° are rigor-like and angles 90° are antirigor-like. The green curve is a Gaussian fit to the data with µ = °, Ã = °. The fit for strong binding heads alone (not shown) is µ = °, Ã = °. (B) Distribution of lever arm azimuths relative to the inter-filament axis. The inset gives the angular convention given a direction of view from M-line toward Z-disk. Red and magenta vertical lines show the azimuths of the starting atomic structures, which are very similar. Clearly, if starting-structure azimuth were the only influence, then the final azimuths in B should center around these vertical lines at °, as indeed the weak-binding target-zone bridges tend to do here. Direct Z-ward views of the unexpected azimuthal skewing observed for strong-binding bridges are shown in Figure 6B&#x;C, in contrast to the starting-structure azimuths for target-zone myosin heads shown in Figure 6A. Figure 10 depicts possible torsional effects that might contribute to the skewing.

We determined the azimuthal angle from the projection of the lever arm vector defined above, onto the equatorial plane of the filament lattice. A lever arm azimuth of 90° would be aligned parallel to the inter-thick-filament axis. The azimuthal angles thus defined are dependent on the actin subunit to which the myosin quasiatomic models are transformed, in this case subunit I, but the angular range is not. Surprisingly, the azimuthal range of all strong binding myosin heads is ° (Figs. 4B, 5B). There is no correlation between axial tilt and azimuth (Figs. 4B, C). The azimuthal angle distribution is bimodal with M-ward bridges being spread over a 52°&#x;° range (mean 86°±24°) and Z-ward bridges spread over a 16°&#x;73° range (mean 41°±17°). The total spread is unexpectedly wide given that the two starting structures are azimuthally only 4° apart, Holmes rigor, °, scallop transition state, °. Very few of the fitted strong binding myosin heads have lever arms positioned like the starting structures. The departures are not random but systematic. When viewed Z-ward, nearly all are positioned anticlockwise (with respect to the thin filament) from the starting structures.

The bimodal azimuthal angular distribution can be attributed to the 26° difference in azimuth presented by the myosin binding site on actin of the two target-zone subunits on each side of the thin filament as a consequence of its helical structure. When the starting structures are placed on the target-zone actins, their S1&#x;S2 junctions are positioned clockwise from the line that connects the centers of the thick and thin filaments (Fig. 6A). The S1&#x;S2 junctions of the myosin heads on the Z-ward actins J and K are positioned further from this line than those on the M-ward actins H and I. The S1&#x;S2 junctions for all but two of the quasiatomic models for strongly bound heads are positioned anticlockwise from this line. The two exceptions are apparently early working-stroke attachments. The range of positions of the S1&#x;S2 junctions at the thick filament surfaces for heads bound strongly to the M- and Z-ward actin subunits almost completely overlap (Fig. 6B&#x;D) despite the 26° difference in azimuth between their bound actin subunits. Thus, cross-bridges strongly bound to either target-zone actin appear as if they originate only from a restricted region of the thick filament independent of two different actin azimuths. Noteworthy is the fact that the S1&#x;S2 junctions of the starting structures placed on actin subunits F and G fall on the same anticlockwise side of the line as the observed strong-binding attachments even though no strong-binding cross-bridges were found on F and G.

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Figure 6

Models of strong binding bridges superimposed and displayed on their bound actin subunits.

To provide a spatial reference, the models are displayed with the map of the global average. The horizontal dashed line represents the inter-thick-filament axis. All views are looking from the M-line toward the Z-line. (A) Scallop transition state starting model on actin subunits F&#x;K. The S1&#x;S2 junctions are positioned clockwise from the interfilament axis for all starting models on actins H&#x;K. The S1&#x;S2 junctions of starting models on F and G are located anticlockwise from the interfilament axis. However, no strong binding attachments occur on actins F and G. (B) Bridge models strongly bound to M-ward actin subunits H and I. The lever arms of the only two models that fall above the inter-thick-filament axis are bound to actin subunit H and have the appearance of early beginning-working-stroke conformations. (C) Bridge models strongly bound to Z-ward actin subunits J and K. (D) All strong binding models on their bound actin subunits showing azimuthal distribution skewed notably anti-clockwise from hypothetical dispersions centered around starting model positions in A.

Finally, the orientation of the hook of the myosin heavy chain in the starting structures (before rebuilding), which connects the myosin head to the S2 domain, is oriented away from the direction that the lever arm has to be bent to fit the strong binding bridges. This would suggest that the forces bending the lever arm azimuthally in situ, might also cause the lever arm to be twisted, a phenomenon which has been observed spectroscopically [45].

Section compression, which in the present data reduced the inter-thick-filament spacing from 52 nm to 47 nm, may affect both the axial and azimuthal lever arm angles. We estimated the effect of section compression using a simplified model (Fig. 7). We assumed that the thick and thin filaments are incompressible and that most of the compression occurs in the space between filaments, precisely where the lever arms are located. Section compression of the magnitude observed here would broaden the range of azimuthal angles by ±9°. The effect as modeled would be non-linear as lever arms that were more angled with respect to the inter-filament axis would be affected more. For the axial angle, the effect is smaller, and would broaden the distribution by ±6°. If filaments are compressible so that compression is spread uniformly across all structures, the effect on lever arm angle would be less as the space between filaments is a relatively small proportion of the distance separating the filament centers. Thus, accounting for section compression in the worst case scenario would reduce the azimuthal spread from ° to ° and the axial spread from 77° to 65°.

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Figure 7

Schematic model of the effect of section compression on the angle of the lever arm.

Left illustrates the initial state, prior to sectioning and section compression; right side illustrates the effect of section compression. Top row is the view looking down the filament axis; bottom row is the view looking perpendicular to the filament axis. Color scheme has the thick filament red, thin filament magenta, motor domain blue and lever arm black. The region of the target zone is colored cyan. We assume a worst case scenario, in which the thick and thin filament as well as the myosin motor domain are unaffected by compression and the entire effect is concentrated on the lever arm. Section compression decreases the interfilament spacing with a corresponding increase in section thickness. Widening of the section is assumed to be minimal since the reconstructions are scaled to the axial periodicities. See text for the values obtained from this model.

Weak Binding Cross-bridges

Many weak binding cross-bridge forms were identified in primary class averages, which revealed cross-bridges attached to actin subunits F&#x;K. We divided weak binding bridges into two types, depending on the displacement of their MD center of mass from that of the strong binding attachments, and on whether they contact actin or TM (Table 1, Table S1). Type 1 weak binding cross-bridges had MD displacements of nm from the starting structure and their MD contacted the actin subunit; Type 2 attachments were displaced by &#x; nm and their MD contacted TM rather than actin. All Type 1 attachments were within the target zone; five of seven Type 2 attachments were outside of the target zone on actins F and G with the other two on actin H, the most M-ward target-zone actin subunit. MD displacements show a trend from the largest on actin subunits F and G (closest to the M-line), to smallest on actin subunit K (closest to the Z-line).

Table 1

Summary of weak attachment models fitted in primary mask class averages.

Repeat ## of MembersActin LabelTypeTotal Displacement (nm)
22F2
26F2
15F2
33G2
24G2
19H2
24H2
26H1
27H1
46H1
27I1
7324I1
22I1
32I1
18I1
26J1
16J1
21J1
46J1
34J1
21J1
23K1

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We transformed all weak binding cross-bridge quasiatomic models to align their MDs to the starting scallop S1 atomic structure placed on actin subunit I (Fig. 8A) which places them in the same frame of reference as the strong binding cross-bridges described in Figures 4 and 5. One weak binding class, visible in panels W and Q in Figure 1, had a lever arm orientation toward rigor. This class is possibly a post-rigor structure and contained 46 members compared with a total of weak binding bridges. For all other weak binding bridges, the azimuthal angle range is 89° to ° (Table 2). The ranges for Types 1 and 2 are only partially overlapping, with Type 2 quasiatomic models being distributed to one side of the starting scallop structure (Fig. 8A) and the Type 1 models being distributed on both sides of it.

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Figure 8

Composite view of weak binding cross-bridge models.

(A) Axial and azimuthal views of all weak binding cross-bridges aligned on the motor domain of the scallop transition state structure. This view illustrates the variations in lever arm compared with the starting scallop S1 structure. All weak binding bridges were built starting from the scallop transition state atomic structure, which is shown as a magenta colored ribbon diagram. Type 1 bridges are shown in gray and Type 2 bridges in gold, both rendered as chain traces. The single post-rigor conformation is colored light brown. (B) All weak binding cross-bridges superimposed on actin subunit I. This view illustrates the variations in MD position when referred to a single actin subunit. Coloring scheme is the same as for panel A. Note the relatively small axial dispersion of the Type 1 MDs compared to the broad dispersion of the Type 2 MDs.

Table 2

Summary of Weak Binding Lever Arm Parameters§.

Crossbridge TypeNumber of classesAxial AngleAxial Displacement (nm)Azimuthal Angle
Scallop Transition State-° °
Holmes Rigor-°&#x;°
Type 11392°&#x;° &#x; 89°&#x;°
Type 1 153°&#x;°
Type 2988°&#x;93° &#x; °&#x;°
All Weak Binding2353°&#x;°&#x;&#x; 89°&#x;°

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The average of the axial angles excluding the post-rigor model, °±10°, is close to that of the starting scallop transition state model, °, and roughly evenly distributed around it. This axial range is less than twice the estimated uncertainty of the quasiatomic models and far less than the range for strong binding heads. Within the confidence limits of the model building, this range represents the inherent flexibility limits between the lever arm and the weakly bound MD.

Type 1 attachments are the most likely candidates for pre-working-stroke forms. When all Type 1 weak cross-bridges are transformed (aligned) onto actin subunit I, as opposed to a strong binding MD on actin subunit I, they suggest a progression in a clockwise direction (looking Z-ward) toward the strong binding structure (Fig. 8B). The MD displacements suggested by this alignment are azimuthal in character and would suggest a clockwise azimuthal rotation of the MD (looking Z-ward) to reach the strong binding orientation on actin. A characteristic of the Type 1 cross-bridges is a narrow distribution of MD displacements in the axial direction. A narrow range of lever arm axial angles is present within these models, which is apparently uncorrelated with the azimuthal angles.

When all Type 2 weak binding bridges are aligned to actin subunit I, their MDs are all placed well beyond the azimuthal strong binding position on actin (Fig. 8B) and would require an anticlockwise movement (looking Z-ward) along the thin filament to reach a position where strong binding is possible. In contrast to Type 1 weak binding bridges, there is also a significant axial smear of the Type 2 MDs, indicating a significant difference in the ordering of their interaction with the thin filament compared with Type 1 bridges. Thus, the Type 2 weak binding bridges are a new and fundamentally different type of cross-bridge contact (or interaction).

Discussion

Comparison with Previous Work

Here we combine modifications in both data collection and analysis to achieve richer and finer detail of the variety of myosin head forms in iso-HST than was possible previously [19]. The improvement is most pronounced in the averages which are important for identifying structural variation in the cross-bridges. Column averages as used previously were only effective at filtering patterns of cross-bridges that repeated axially every nm and is limited in resolution to nm, the highest resolution layer line visible in the transform of those tomograms. MDA as used here is effective at identifying different cross-bridge structures regardless of their distribution in the filament lattice. The use of MDA to identify self-similar structures within an ensemble is not limited in resolution by long range order, but is more limited by the number and homogeneity of the structures being averaged.

Previous work utilizing X-ray diffraction and ET concluded that 28&#x;32&#x; of the myosin heads are attached to the target zone for a frequency of heads/target zone [19]. A subsequent analysis [36] of the same data obtained a binding stoichiometry of myosin heads/target zone and heads/actin (assuming 5 actins/target) or heads/actin (assuming 4 actins/target). The new results indicate myosin heads are bound per target zone and averaging heads per target-zone actin. The two results are not necessarily incompatible. The earlier results were derived based on two spatial averaging techniques, X-ray diffraction and column averages of the tomogram, both of which observe only repeating patterns in the structure. The present analysis does not require spatial ordering. Irregularly distributed structures, such as the M-ward bridges of mask motifs identified here by MDA would contribute little to X-ray reflections that are enhanced by target-zone bridges.

Actin Target Zones

The term target zone is defined as the segment of the thin filament where actin subunits are best oriented to form strong attachments to myosin heads projecting from adjacent, parallel thick filaments. Though defined initially for IFM [35], the concept is entirely general not only for organized filament arrays such as striated muscle [33], [34], [46], [47] but also for in vitro single-molecule experiments using various myosin isoforms [48]&#x;[53]. However, up to now the target-zone size has not been so precisely defined.

Previously, we inferred that the target zone was on average actins long on each long pitch helical strand [19], [36], three actins on one strand and two on the other, alternating sides with successive crossovers. The present result, in which actin subunits and Tn's are resolved, and weak and strong binding attachments are distinguished, shows that the target zone comprises just two subunits on each actin strand, positioned exactly midway between two successive Tn's on that strand. The frequency of all myosin attachments increases 10 to fold on target-zone actins relative to the nearest neighbors. Moreover, strong binding heads disappear abruptly outside of the target zone. The lack of any strong binding cross-bridges on actin subunits F and G, excludes these actins from the target zone where previously they were included [36].

We think that the highly restricted target zone cannot be due to the assumptions used to distinguish strong and weak binding attachments. Only Type 2 cross-bridges are found on M-ward actins F and G; even Type 1 attachments are not found on these actin subunits. Moreover, virtually no attachments of any kind were found on Z-ward actins L and M.

The relative orientation of actin subunits with respect to myosin head origins is the most obvious factor defining the target zone [35], [54] but the size of the target zone can vary. In rigor the target zone encompasses eight successive actin subunits along the thin filament genetic helix (actin subunits H&#x;O), i.e. four on each long pitch strand. Two-headed lead cross-bridges (leading toward the M-line) occupy the four M-ward subunits (H&#x;K) and the usually single-headed rear cross-bridges are bound to the most Z-ward pair (N, O) [55]. The extensive deformation of rigor rear bridges and their complete absence in AMPPNP [27], [56], suggests that the high actin affinity of nucleotide-free myosin extends the limit of acceptable actin azimuth.

Our sharply defined target zone implies a restrictive geometrical constraint on myosin head binding that is contradicted by the highly variable shape of strong binding cross-bridges, in particular their widely varying azimuthal lever arm orientations. If our two extremes for lever arm azimuth of strong binding myosin heads reflected intrinsic, state independent, myosin head flexibility, some heads should be able to bind strongly at all but two actins, D and E, while staying connected to the thick filament. Moreover, if actin azimuth alone were the limiting factor, we would expect a more gradual tapering of strong binding attachments from the center of the target zone. This we also do not see.

Type 1 weak binding attachments show a smaller azimuthal lever arm variation (Figs. 4A, 5B) than the strong binding heads that they evolve toward. However, when the two azimuthal lever arm extremes of Type 1 weak actin attachments are transformed to the position of strong binding motor domains on each actin subunit, some heads should be able to bind strongly at all but four actins, D, E, R, &#x; S, while staying connected to the thick filament. The initial myosin crystal structures sit near one azimuthal extreme of our strong binding attachments (Fig. 5B). When placed on actin in the strong binding configuration, simultaneous myosin-actin attachments can be made only at six actins F&#x;K (Fig. 6A), which includes the entire target zone. It is notable that the lever arm azimuth in this case would exclude actin subunits L and M from strong myosin attachments but our results show that even non-specific, weak attachments are exceptionally low on those subunits.

Based on the two initial crystal structures, we would expect that strong binding bridges would be found on actins F and G, but none are found. Even Type 1 weak binding bridges are not found on actins F and G. This suggests that an additional factor is limiting the target zone which may be the dynamic properties of TM. The Tn complex adds additional actin affinity to TM and holds it relatively securely at the ends of the actin repeat period but motion of TM would be expected to be highest midway between Tn complexes, exactly where the target zone is located. Reconstructions of actin-TM-Tn done by single particle methods revealed the weakest TM density midway between Tn complexes [57] consistent with the idea of high mobility in this region which coincides with the IFM target zone.

Distribution of Actin-Bound Myosin Heads

Our results show that myosin head attachments occur all along the thin filament, although with much lower frequency than the target-zone attachments. Target-zone attachments account for 78&#x; of all attachments but utilize only 28&#x; of the actin subunits. The other 22&#x; of attachments are distributed among the remaining 72&#x; of actin subunits. Many of these attachments are non-specific with respect to the myosin binding site on actin. Some may represent collision complexes, but others, such as the Type 2 weak binding bridges on actin subunits F and G are both numerous and have a well defined appearance in the averages suggestive of some kind of specific interaction, even if novel. Outside of the target zone, actin subunits are labelled only 8&#x; of the time on average (range of 3&#x; to 13&#x;). Nevertheless, these non-specific attachments occur with enhanced frequency on the 4 actin subunits near Tn and with strikingly low frequency on two actin subunits (L and M) adjacent to the Z-ward side of the target zone.

Initially, we thought actin subunits L and M never had myosin attachments, but a classification designed specifically for this region found some attachments, the density of which was generally poorly defined. Although it is possible that this represents statistical uncertainty, it may also indicate something special about these actin subunits. Non-specific attachments occur everywhere outside the target zone so the especially low frequency here may indicate an adaptation for IFM.

One in every seven subunits of IFM thin filaments is a ubiquinated form of actin named arthrin [58], [59]. Although arthrin is regularly spaced every nm along Drosophila IFM thin filaments [60], its location is close to Tn, and not to the target zone [61]. Thus, arthin located at positions L and M is an unlikely explanation.

Limitations on Cross-bridge Attachment

Tregear et al. [36] reanalyzed the earlier data [19] combining column averaged images to define target-zone locations and cross-bridge origins on the thick filament with the raw tomogram to determine the angles and positions of all individual attached cross-bridges. Tregear et al. concluded that the binding probability of cross-bridges to the target zone followed a Gaussian distribution that depended on the axial offset between target-zone center and the shelf of cross-bridge origin (Fig. 9A). Their methods could not discriminate between weak- and strong-binding attachments in and around the target zone and thus deemed the target zone to be larger than found here.

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Figure 9

Probability of target-zone cross-bridge formation as a function of cross-bridge origin.

(A) Data reproduced from Tregear et al. () in which the target zone is assumed to be three actin subunits on one side and two actin subunits on the other. (B) Data from the present study, which includes only target-zone cross-bridges on two actin subunits from each side. Although the Tregear et al. data, which were measured by hand, have an overall Gaussian shape, the present measurements, which are based on quasiatomic model fitting, do not follow a strictly Gaussian distribution. The continuous line is a Gaussian fit with µ &#x; nm and à = nm.

For comparison, we have replotted our fitting data shown in Fig. 9B by referring the axial coordinate of all lever arm C-termini (P for Holmes and P for scallop) to the center of the target zone. Data from left and right hand sides of the target zone are combined, but referred to the center of the individual side, not the overall center of both. The shape of the resulting distribution is not Gaussian but when fit to a Gaussian, the average is &#x; nm (angling towards rigor). All of the measured points fall within ±10 nm of the target-zone center and 84&#x; fall within ± nm. If the present data are referred to the overall center of the paired target zones making it comparable to Tregear et al., the distribution would be spread an additional nm on either side which would mean that all data fall within ± nm. For the Tregear et al. data, 85&#x; of all attached cross-bridges originate within ± nm of the center of the target zone. The present analysis is therefore broadly consistent with Tregear et al. and differs primarily in its details that are better defined due to the higher resolution of the reconstruction.

The Asymmetry of Actin Attachments

Our results show that the number of actin-myosin attachments is relatively symmetric within the target zone, but is very asymmetric just outside the target zone, where the asymmetry in numbers is matched by an asymmetry in structure. The two actins M-ward of the target zone (F and G) are comparatively well occupied, while attachments to actins L and M on the Z-ward side of the target zone are rare. Weak attachments on actin subunits F and G (Type 2) differ the most from strong binding attachments, while those on actin subunit K (Type 1) have the smallest MD displacements from strong binding and are fewer in number.

Muscles are designed to generate tension while shortening and accomplish this by using filaments that are polar within the half sarcomere. Thus, the asymmetry we observe is not unexpected given the filament structures themselves. When a muscle shortens the target zone on the actin filament moves toward the M-line. On the M-line side of the target zone, on actin subunits F&#x;H, we find the most unusual of the weak binding cross-bridge forms, Type 2. Although considered weak attachments, their structure is well defined in the averages and they are the only attachments found on those actin subunits. While not positioned on actin in a way that readily converts to strong binding, Type 2 attachments are placed so that if they remained stationary while the actin target zone moved M-ward by 1 or 2 actin subunits, they would be positioned to bushwhack the target-zone actins and form Type 1 weak attachments (Supporting Movie S7). If the movement were larger, we can identify no weak attachments that would be in a position to quickly bind the target zone.

Conversely, if the muscle is stretched, which is to say the target zone moves Z-ward, there are literally no myosin heads positioned to bind the target zone. Adaptation to stretch would therefore require a different mechanism for rapidly placing more myosin heads on target-zone actins. It has been suggested that when a muscle is stretched, two-headed attachments increase by recruitment of second heads to single-headed bridges [62]. Our results show that two-headed cross-bridges exist in isometrically contracting muscle even without stretch which shows directly that this mechanism for adapting to stretch is mechanically feasible in active IFM.

The two types of weak binding attachments identified here may play differing roles in the recovery of tension after a quick release. Type 1 attachments are closest to the strong binding configuration and would thus appear to be best suited for contributing to Phase 2 rapid force recovery if the release distance was 5 nm [63], [64]. Lombardi et al. [65] observed that the stiffness attributable to attached bridges was constant up through Phase 2, while Bershitsky et al. [22] found that stiffness of weak binding bridges that were not stereospecifically attached to actin was unchanged during the weak-to-strong transition that brought about stereospecific attachment and force development. Therefore the Type 1 bridges seen here that are not stereospecifically attached could well be capable of contributing to the stiffness attributed by Bershitsky et al. to weak binding bridges, yet transform without stiffness change to stereospecific strong binding force generating bridges during the phase 2 rapid force recovery observed by Lombardi et al. The Type 2 attachments, which are not attached to actin at all but rather contact TM, may instead be responsible for a more delayed Phase 3 tension recovery.

What exactly is holding the Type 2 bridges in place is not visible in the reconstructions. Their position is variable with respect to TM, both azimuthally and axially which argues against a specific interaction. One possibility is the N-terminal extension of the regulatory light chain. The sequence of this extension is known in Drosophila, which has led to the suggestion that it positions myosin heads for attachment to the thin filament [66]. Its validity for Lethocerus will depend on the demonstration of a similar N-terminal extension.

2-Headed Cross-Bridges Are Not Rare in Isometric Contraction

Based on column averages, it was concluded that the great majority of cross-bridge attachments were single-headed [19], and a subsequent analysis of the same iso-HST tomograms drew the same conclusion [36]. The present analysis indicates that 2-headed attachments are in the minority, but are not rare. Approximately 14&#x; of the cross-bridges (all in the target zone) contain both heads of one myosin molecule. For the single-headed cross-bridges, we believe the companion head is either mobile or remains close to the relaxed state docking position on the thick filament shaft producing the residual density of the myosin shelves spaced nm apart. These shelves are prominent in iso-HST but not in rigor where 2-headed binding predominates [55].

Single headed myosin attachments are consistent with a variety of other structural evidence such as X-ray modelling of relaxed insect thick filaments [67] and various experiments on vertebrate muscle, both in situ [32], [68], [69] and in vitro [70], [71]. Conibear et al. have argued on energetic grounds that 2-headed binding is unlikely during active cycling of myosin [44]. However, 2-headed attachments are consistent with several sets of experiments involving single molecule motility [72]. The present result indicates that at least in isometric contraction, two-headed myosin attachments are significant but are not the major cross-bridge form.

The two stage working stroke

Previous analysis of iso-HST tomograms elucidated a working-stroke sequence by arranging all the quasiatomic models of cross-bridge fittings in a sequence based on the coordinate of the C-terminal residue at the S1&#x;S2 junction [19]. The sequencing suggested that the working stroke consisted of two stages, one in which both the MD and the lever arm move from an M-ward tilt toward a Z-ward tilt in Stage 1, with Stage 2 beginning when the MD settled into the strong binding configuration and only the lever arm moved. The present results are not easily compared with that work for several reasons. (1) Cross-bridge features seen earlier had to be spatially repetitive; averages would be computed over heterogeneous structures if this were not true. The present result is not so limited and averages have been computed over structures that are more homogeneous even though irregularly placed in the filament lattice. (2) The independent fitting of a thin filament quasiatomic model into the present maps provides an improved criterion for distinguishing weak from strong binding cross-bridges. (3) The greater detail of the weak binding target-zone bridges and the availability of two crystal structures, one a transition state with the lever arm elevated and the other a rigor structure, reduced the amount by which the starting models had to be altered to fit the density. Thus, previously the motor domain placement of weak binding cross-bridges involved both axial and azimuthal changes in the MD, but the present fittings of weak binding bridges could be fit using only azimuthal MD changes.

In the sequential ordering of structures from weak-to-strong binding [19], the azimuthal range of lever arm positions narrowed as the lever arm moved from the initial stage to the final, rigor-like stage of the working stroke. The present work differs in that a large azimuthal range of the lever arms persists at all stages. Thus, the present work does not support the conclusion that the azimuthal distribution of the lever arms is wide in the weak binding states and narrows as the myosin heads go through the working stroke. The distribution of lever arm azimuths is widest for strongly bound heads and smaller for weakly bound heads. However, the number of different weak binding forms does suggest new details of the transition from weak-to-strong binding.

Stage 1- weak attachments

We characterized two types of weak attachments between myosin heads and the thin filament, with Type 1 attachments being closest in appearance to strong binding cross-bridges and with Type 2 being decidedly different. We think that Type 1 could be pre-working-stroke attachments because their MD position is closest to the strong binding configuration and they occur only within the target zone. Type 1 weak attachments are consistent with the idea that initial binding need not be precise [73] but rather is disordered as observed by cryoEM [74], Electron Paramagnetic Resonance [23], and X-ray fibre diffraction [22]. Random thermal motions would then align the MD into the strong binding configuration possibly generating force [75].

Our class averages at the moment lack the resolution and signal-to-noise ratio to be unambiguous but they suggest the following properties of the weak binding attachment that precedes strong binding. Type 1 cross-bridges can be arranged into a weak-to-strong binding sequence (Fig. 8B), but this would embody mostly azimuthal movements of the MD across the actin surface, toward TM and the strong myosin binding site on actin. The present data lack the resolution to exclude small changes in MD tilt as part of the transition. In this sequence, MD progress toward the strong binding site on actin displays a corresponding but uncorrelated set of comparatively small axial and azimuthal lever arm movements.

With the exception of one Type 1 weak binding cross-bridge, a possible post rigor conformation, all the weak binding forms on actin subunits F&#x;K display a narrow range of axial and azimuthal lever arm orientations when aligned to the MD (Figs. 5B, 8A). The azimuthal lever arm range of the weak binding bridges is much smaller than the range found for the strong binding bridges, and is rather symmetric with respect to the lever arm position of the crystal structures consistent with inherent flexibility. Conversely, for strong binding cross-bridges, the distribution is not only large, but is strongly biased toward one direction, suggestive of a genuine characteristic of isometric force generation in situ.

Stage 2 strong binding

The fitting of strong binding cross-bridges generally required considerable axial alteration of the lever arm for both crystal structures, which differ by 36°/ nm compared with the 77°/ nm range observed here. Our observed range is consistent with previous estimates made by comparing crystal structures of myosin catalytic intermediates from different isoforms [8], [10] and is also similar to that observed earlier [19]. Section compression as modelled here could broaden the spread by about ±6° leaving a maximum working stroke of 10 nm. Despite these factors, our range is smaller than the ° observed by comparison of reconstructions obtained from actin filaments decorated in AMPPNP or ADP with a myosin V construct containing two IQ motifs with bound calmodulin [15].

The average axial lever arm angle obtained from a Gaussian fit to the data in Figure 5, gave a mean angle of 93°±20° for the strong binding cross-bridges, nearly perpendicular to the filament axis and in good agreement with values that were found to fit the meridional X-ray reflection from isometrically contracting vertebrate muscle fibers [76], [77].

A consistent and surprising feature of the strong binding cross-bridges is the large, asymmetric distribution of the lever arm azimuths relative to the starting models. This is a feature of the data that has been extensively checked against different applications of MDA as well as against the original raw tomograms so we believe it to be real. The width of the distribution is affected by section compression, but even accounting for an additional 9° at either extreme, still gives a range of °. These azimuthal lever arm changes relative to the two initial structures gives the cross-bridges a more straightened appearance.

Cross-bridges with a straightened appearance have been observed previously in situ. They were first reported in EMs of IFM following AMPPNP treatment [78], and in EMs of quick-frozen, contracting vertebrate muscle [32], [33]. IFM reconstructions of rigor [25], [26] and following AMPPNP treatment [27], [28], [56] also show this effect. Thus, we think that this feature of the cross-bridge structure is not unprecedented. In the previous IFM work, it was uncertain whether the observation was an isoform difference or was revealing a novel feature of in situ cross-bridges, and there was no control group in the reconstructions in the form of unattached or weakly attached heads for comparison. That this observation is not a species difference is supported by the recent 3-D reconstruction of actin filaments decorated with IFM myosin S1, which have some small differences, but largely are very similar to rigor acto-S1 from other isoforms of myosin II [79]. The weak binding bridges in the present reconstruction are a control group. Their smaller and more symmetrical distribution about the initial models suggests that for weak binding bridges we are observing flexibility and for strong binding bridges we are observing an aspect of the working stroke of myosin heads functioning in situ.

Correlations with X-ray Fiber Diffraction

Correlated changes in the actin layer line intensities with characteristic myosin reflections using an applied jump in temperature during isometric contraction [21], [22] show that tension increased with the rise in temperature indicating an increase in strong binding cross-bridges. However, fiber stiffness and the intensity of the 1,1 equatorial reflection did not change, indicating constancy of numbers of actin attached cross-bridges. Moreover, the intensity of the nm layer line, which measures stereospecific attachment to actin rose, while the intensity of the 1,0 reflection, which measures mass ordered on the thick filament decreased. These changes were interpreted to indicate that the structure of attached cross-bridges changed and that these changes are largely azimuthal with respect to both thick and thin filaments. An increase in stereospecific actin-myosin attachments would increase the nm actin layer line as observed, but cannot be accounted for if the myosin head attaches weakly to actin in the stereospecific orientation in which case strong binding simply closes the actin binding cleft. The observed increase requires a more dramatic change in disposition of the MD on actin during the weak-to-strong transition, as observed here.

In vertebrate striated muscle, the intensity ratio of the 1,0 and 1,1 equatorial reflections is a measure of the change in myosin heads moving from thick filaments to thin filaments and these changes normally correlate with tension generation in isometric contractions [80], [81]. Equatorial X-ray diagrams at low ionic strength (µ = 50 mM), which enhances the population of weakly attached cross-bridges even in the relaxed state to nearly 80&#x; of those available, revealed little change in the 1,1 equatorial reflection but a large decrease in the 1,0 equatorial intensity when Ca2 activated, indicative of movement of mass away from the thick filament [82]. This observation was attributed to a pronounced structural change in the already attached cross-bridges; this could not be explained if the MD of weak binding bridges were already oriented as in rigor but with the actin binding cleft open, needing only to be closed for strong binding, a structural change too limited to account for the X-ray change.

Meaning of the Azimuthal Changes in Strong Binding Cross-bridges

Accepting that our skewed azimuthal lever arm distribution is a property of active cross-bridges, the obvious question is what does it mean for myosin function in situ? Is it a reflection of intrinsic myosin head flexibility, is it an aspect of the weak-to-strong transition, or is it an aspect of the myosin working stroke? The broad lever arm azimuthal angular range seen for strong binding bridges is a conundrum. Placement of strong binding cross-bridges on actin can be limited only if the myosin head is given limited flexibility as suggested by the two crystal structures used for the fitting. Yet the changes observed in the strong (and weak) binding cross-bridges compared with the crystal structures imply substantial azimuthal flexibility that, if intrinsic to myosin heads, would permit strong binding bridges to form virtually anywhere along the actin nm repeat. Even in rigor, where actin affinity might increase the target-zone size, cross-bridges are still largely confined to the target zone of contracting muscle. Intrinsic myosin head flexibility would seem to be an insufficient explanation for the strongly biased azimuthal change.

If the azimuthal lever arm distribution for strong binding cross-bridges were an aspect of the working stroke, we might expect to see a relationship between axial angle, representing progress through the working stroke, with increasing, or decreasing azimuthal angle. However, graphs of axial tilt angle versus azimuthal angle for the strong binding cross-bridges fail to show the obvious correlation (Fig. 4B, C) expected if the two motions were coupled.

That there is no coupling between axial and azimuthal lever arm angles may be an effect of S2. Cross-bridges do not emerge from the thick filament backbone at the S1&#x;S2 junction; they originate where S2 emerges from the thick filament backbone. S2 is widely thought to provide a flexible tether that can bend radially, as well as azimuthally around the thick filament surface, to facilitate actin-myosin attachment. If S2 swings azimuthally during cross-bridge attachment so that it becomes angled with respect to the filament axis when force is initiated down the filament axis, that force will have both an azimuthal component (a torque) as well an axial component. If the S2 swing is anticlockwise (looking Z-wards), then the angled S2 would produce a torque that bends the lever arm clockwise with respect to the thick filament. In this case, the torque would straighten the myosin head compared with the starting crystal structures, in accord with our observation. If the S2 swing was in the opposite direction (clockwise), the lever arm would bend anticlockwise with respect to the thick filament thereby making the myosin head more bent than the already bent S1 crystal structures contrary to our observation.

In iso-HST, we do not observe where S2 emerges from the thick filament backbone, we only see the S1&#x;S2 junction of the lever arm and so cannot directly evaluate this effect. Where S2 has been observed in swollen IFM fibers, the range of directions suggests that S2 swings equally well both clockwise and anticlockwise, about the thick filament surface [26]. A symmetrical distribution of angles for S2 with respect to the filament axis at the beginning of force production might explain the lack of coupling between axial and azimuthal lever arm angles of strong binding bridges, but it does not explain the strong directional bias (Fig. 4).

We can think of two mechanisms that would bias the azimuthal bend of the lever arm to produce a cross-bridge that appears azimuthally straightened in comparison to the crystal structures. One of these involves the weak-to-strong transition, the other invokes an active azimuthal component to the working stroke. Either mechanism would enable the S1&#x;S2 junction to be positioned anticlockwise of the inter-filament axis as depicted in Figures 6 and

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Figure 10

Two mechanisms to account for azimuthal skewing of lever arms of strongly bound cross-bridges.

(A&#x;C) Conversion from weak-to-strong binding according to Scenario 1. (D&#x;F) Active azimuthal component to the working stroke. View direction is M-line toward Z-line. Myosin is colored either red (strong binding) or magenta (weak binding). Actin subunits are green and blue. Three successive levels of S2 origins are shown in shades of brown that darken with distance from the observer. The lever arm is the line originating on the red (or magenta) MD while S2 is shown as a short segment when oriented nearly parallel with the filament axis and becomes longer when angled with respect to the filament axis. The horizontal line is the inter-filament axis. Arrows show the direction of the torques (not their magnitude) produced during the weak-to-strong transition or as a component of force generation and filament sliding. The direction of thin filament movement during sarcomere shortening is toward the observer. (A &#x; D) Initial weak binding is shown, which in (A) begins away from the strong binding orientation and in (D) begins in the strong binding orientation but with actin binding cleft open. (B) Conversion to strong binding involves diffusion of the MD clockwise on actin which swings the S2 anticlockwise about the thick filament. (C) Force production realigns S2 with the filament axis while bending the lever arm azimuthally. (E) Transition from weak to strong binding involves no change in myosin orientation on actin, just a closing of the actin binding cleft. The lever arm in (D &#x; E) is in the same orientation suggested by the crystal structures. (F) An azimuthal component to the working stroke moves the lever arm clockwise around the thick filament. This figure can be seen as an animated sequence in Supporting File S1.

The weak-to-strong transition

The range of structures of Type 1 cross-bridges (Fig. 8B) suggests that the weak-to-strong transition involves a concerted azimuthal translation and rotation of the MD in a clockwise direction (looking Z-wards) about the thin filament (this direction is also toward TM). Two scenarios that involve different compliance between the lever arm and the S2 may serve to illustrate.

In the first scenario (Supporting Movie S8, Fig. 10A&#x;C, Fig. S1), the myosin head is assumed initially noncompliant and so does not change as it moves azimuthally across the target-zone actin during the weak-to-strong binding transition. Instead, S2 is assumed compliant and begins this transition aligned parallel with the filament axis (Fig. 10A), but the myosin head movement swings S2 azimuthally (anticlockwise about the thick filament looking Z-ward) angling it with respect to the filament axis (Fig. 10B). When the working stroke applies axial force to the lever arm, the angled S2 will produce a torque that will pull the lever arm clockwise with respect to the thick filament as S2 aligns with the filament axis (Fig. 10C). The myosin head at the end has become straightened.

In the second scenario (Supporting Movie S9; Fig. 10A,C), S2 is non-compliant and so resists being swung azimuthally during the weak-to-strong transition. Instead, the lever arm of the myosin head is compliant and bends azimuthally clockwise (looking Z-wards) with respect to the thick filament during the transition. Essentially, the transition would go from Figure 10A to Figure 10C, bypassing the state depicted in Figure 10B. When strong binding occurs and the working stroke is initiated the force is applied axially with a minimal torque but the lever arm is already bent azimuthally.

The first scenario is supported by the two strong binding structures which fell above the inter-filament axis in Fig. 6B and could be interpreted as very early working-stroke attachments. The second scenario is supported by the lever arm azimuthal distribution of Type 1 weak binding bridges which are somewhat straightened azimuthally, but the amount is small compared to strong binding cross-bridges. Thus, there are structures within the ensemble to support either scenario.

The first scenario bears some resemblance to the roll and lock mechanism described by Ferenczi et al. [83]. This detailed model incorporates the concept that myosin in kinetic states with ATP or ADP&#x;Pi bound make variable attachments to actin that differ not only in the azimuthal position on actin, as suggested by the present data, but also with variations in axial orientation as suggested by earlier work on the I-HST state of IFM [19]. The present work shows highly variable attachments all along the thin filament, but only weak attachments within the target zone (Type 1) are likely candidates to transition to strong binding. The variation in their orientation on actin of this group is much smaller than envisioned in roll and lock . In addition, the present work does not require MD tilt in the Type 1 weak attachments, but we think the data at the moment do not definitively rule out some MD tilt as part of the weak to strong transition. Whether Type 2 attachments can roll on actin is doubtful because their MDs do not appear to attach actin at all. Nevertheless, the present data support at least some aspects of the roll and lock model for the weak to strong transition.

An azimuthal component to the working stroke

Several observations made with in vitro motility assays have shown a torque component to the working stroke, dubbed twirling , inferred from rotations of the actin filament in gliding assays [84], [85]. Twirling as generally observed with myosin II involves a left-handed actin filament rotation during filament sliding. With an active azimuthal component to the working stroke, azimuthal diffusion in the weak-to-strong transition described above is not required. Initial weak binding could place the myosin head on actin oriented as in strong binding but with the actin binding cleft opened (Fig. 10D) and the lever arm azimuth positioned as in the crystal structures. The weak-to-strong transition would simply involve closure of the actin binding cleft without a change in orientation of the MD (Fig. 10E). If the working stroke involved an inherent clockwise rotation of the lever arm about the thick filament when looking Z-wards, the S1&#x;S2 junction would move clockwise about the thick filament (Fig. 10F) and the cross-bridge would be straightened. A consequence of this mechanism is that the myosin origin (i.e. where S2 emerges from the thick filament backbone) could be positioned initially above the inter-filament axis (Fig. 10D), where the crystal structures predict it should be, and the S1&#x;S2 junction would, after tension developed, appear below the inter-filament axis where we observe it in situ.

The azimuthal movement of the lever arm under this mechanism would not only straighten the cross-bridge but would also impose a clockwise torque on both the thick and thin filament, opposite the one imposed by the weak-to-strong transition. If filament sliding was possible, the applied torque would impose a left handed rotation to the thin filament, matching the observation in vitro [84]. For myosin II, Beausang et al. observed a screw pitch of nm, about half the length of thin filament observed in our tomogram. This is large enough that a rotation or change in twist of the actin filament should be visible in our reconstructions if it occurred, but it is not.

Generally, motility assays are performed with the motor bound to a substrate via the rod domain (S2 plus LMM), which would argue that the S1&#x;S2 junction is comparatively immobile and the actin filament free to move during these assays. If the actin filament were fixed as it appears to be in IFM and the S1&#x;S2 junction free to move on its S2 tether, then the same conformational change in the myosin head that causes left hand twirling of free F-actin would rotate the lever arm anticlockwise or right handed with respect to the thin filament to produce myosin head straightening as observed.

Beausang et al. [84] indicated that the visibility of actin filament rotation in motility assays depends on the number of rigor bound heads providing a drag force to slow filament movement. In isometrically contracting muscle, not only are there many strongly bound heads, but there is an external load to prevent the fibers from shortening. The effect of a torque on the actin filament as a component of the working stroke would by inference be most visible in isometric contraction.

Effects on the Thick and Thin Filaments

Forces or components of forces that alter the lever arm azimuth must also impose torsional forces on the thick and thin filaments. Neither the thin filaments, anchored at the Z-disk nor the thick filaments, which are bipolar, can rotate as rigid bodies; they can only change their helical twist in response to an applied torque. For the thin filaments this is not as absolute as for thick filaments since it depends on the compliance of actin crosslinks in the Z-disk. The rotation of the MD on actin during the weak-to-strong transition produces an anticlockwise torque (looking Z-wards) on both the thick and thin filaments (Fig. 10A&#x;C). This torque has opposite effects on the thick and thin filaments because their fixed points are at opposite ends of the half sarcomere. An anticlockwise torque on the thick filament during the weak-to-strong transition, would unwind a right handed helix, but the same torque on the actin filament would over wind a right handed helix. Conversely, an active azimuthal component to the working stroke would apply a clockwise torque to the thick filaments which would over wind a right handed helix and if applied to the thin filament unwind a right handed helix.

There is no evidence in the present tomograms or in any published data from IFM of a change in the half pitch of the thin filament which would be a necessary consequence of any torsional rotations in situ. The thin filament azimuth in our tomograms of active contraction appears virtually identical to that found in other states of IFM investigated by electron microscopy alone [86], [87] and by ET [27], [28]. If a change in thin filament pitch occurred in the present data, the repeat alignment scheme, which utilized only ° azimuthal rotations, would have obliterated the Tn density which is congruent with the half pitch, whereas in fact, the alignment scheme if anything enhanced it. Lack of evidence that the thin filament in IFM rotates as a rigid body or undergoes changes in helical structure does not mean that the thin filament must be unmodified by strong binding myosin attachments. It only means that any changes appear to be local and not global.

There is some evidence from X-ray diffraction of helical changes in the thin filament of vertebrate striated muscle during isometric contraction [88], [89]. The changes are much smaller than suggested by the present data. Another report describes helical changes in the thin filament in low tension rigor [90] and suggests that these changes are the result of strong binding itself causing local distortions that are measured on average over the entire filament. No publications have reported changes in helical twist of the thick filament during isometric contraction although changes in axial spacings in vertebrate striated muscle are widely known [88], [89].

Perhaps the lack of visible effects on the helical structure of the filaments is due to the differing sense of the two hypothesized torques; the weak to strong transition produces a clockwise torque, twirling results from an anticlockwise torque. More likely the thick and thin filaments are too stiff to be significantly altered by these forces and most of the torsional force is dissipated by S2 and the myosin lever arm.

Conclusion

We have shown multiple myosin head structures in isometrically contracting muscle consistent with both weak binding and force producing cross-bridges. We can infer from these structures that the weak to strong transition involves largely azimuthal movements of the myosin MD and consequent changes in the lever arm. This agrees with X-ray diffraction observations of the weak to strong transition in which changes in intensity occur largely on the nm actin layer line which is sensitive to increases in stereospecific binding, on the equatorial reflections which are sensitive to azimuthal and radial changes in structure and with minimal changes in the nm meridional reflection, which is sensitive to lever arm or MD axial orientation [22]. Both our weak binding and strong binding bridges have a symmetrical lever arm axial angle distribution with a mean near 90° while the MD orientation differences are largely azimuthal, qualities that would translate to little change in the meridional intensity and large changes in the nm actin layer line as tension increases. The 1,1 reflection which measures myosin attachment to actin could remain roughly constant but the large azimuthal changes in the lever would affect the 1,0 intensity. Our strongly biased azimuthal lever arm distribution can be explained by an azimuthal movement of the MD combined with some azimuthal component to the working stroke. That the transition from weak to strong binding is largely azimuthal makes it possible for cross-bridges to cycle in place with little change in axial angle when converting from weak to strong binding [36].

Materials and Methods

Rapid Freezing and Freeze Substitution

Rapid freezing with simultaneous monitoring of fiber tension and stiffness was performed on a Heuser Cryopress freezing head [33]. Specific modifications made to the freezing head for this work have been described in detail as have specifics of the specimen manipulation prior to and subsequent to freezing [19], [91].

Electron Tomography

Details of all the data analysis procedures have been described [37] but a brief summary is given here. Two tilt series covering ±72° at 90° relative orientation were recorded from half sarcomeres on a FEI CMFEG electron microscope using a Gatan Model High Tilt Analytical Holder and a TVIPS F 2k×2k charge coupled device camera. Each tilt series consisting of images was recorded from regions where longitudinal thin sections contained single myosin and actin filament (myac) layers [35]. Tilt angles within each tilt series were calculated according to the Saxton scheme [92]. The two tilt series were first independently aligned using marker-free alignment [93] and then merged by patch correlation and volume warp using IMOD [94] to produce the raw, dual axis tomogram. The pixel size was internally calibrated using the axial nm period. Although we collected and merged two dual-axis raw tomograms, only one contained an appreciable number, , of well centered repeats.

Repeat Subvolume Processing

Repeats spaced nm apart axially and containing a nm axial length of the actin filaments, their bound cross-bridges and adjacent thick filament segments were centered on the actin target zones. Alignment error between the raw repeats was minimized by choosing as the reference the single large structure, the thin filament, which was in common to all repeats. A global average of all extracted raw repeats was used as the initial reference and was followed by MDA and multireference alignment. After several cycles of multireference alignment, the final alignment used a single reference as it was desirable to fit one atomic model to the thin filament for all class averages and raw repeats that would be used for all subsequent model building.

Multivariate Data Analysis and Repeat Reassembly

The purpose of MDA is to sort the highly variable cross-bridge forms within the repeats into self-similar groups. A key part of MDA is the generation of Boolean masks, which select a set of contiguous voxels defining a region of the repeat within which patterns of density will be identified. To retain the greatest variation in structure possible, we generated several classification masks. Two of these selected myosin heads bound to either the left or right side of the actin target zones; these are the primary class averages. Four masks were used for the troponin region, four masks were specific for actins outside of the target zone and two were used for the surface of the thick filament to verify the lever arm positions. Class averages from each classification were subsequently reassembled to make composite class averages [37]. The classification clusters repeats according to the features within the specific mask, but averaging was always carried out using the entire repeat.

Because of the complex manner in which the individual repeats are reassembled, conventional methods of resolution determination, such as the Fourier Shell Correlation are meaningless. However, a qualitative measure of the resolution can be obtained by the fact that the helix of actin subunits can be observed, which requires at least nm resolution. The resolution of the previous work was limited to 9 nm so the improvement is at least a factor of 2&#x;3.

The stochastic control of the F-8C aircraft using the multiple model adaptive control (MMAC) method

See discussions, stats, and author profiles for this publication at: sprers.eu The stochastic control of the F-8C aircraft using the multiple model adaptive control /MMAC/ method ARTICLE · FEBRUARY DOI: /CDC · Source: NTRS CITATIONS READS 24 8 AUTHORS, INCLUDING: Keh-Ping Dunn C. Greene Massachusetts Institute of Technology University of St. Thomas 26 PUBLICATIONS CITATIONS 8 PUBLICATIONS CITATIONS SEE PROFILE SEE PROFILE Available from: Keh-Ping Dunn Retrieved on: 04 February General Disclaimer One or more of the Following Statements may affect this Document This document has been reproduced from the best copy furnished by the organizational source. It is being released in the interest of making available as much information as possible. This document may contain data, which exceeds the sheet parameters. It was furnished in this condition by the organizational source and is the best copy available. 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Lcth the : sprers.eunal and tical reflecting the fact that different sources lateral dyra.­.ica cf the F-AC aircraft. NAFA nas were used to obtain them. The design reported in beer using the F-8C aircraft as a test vehicle this paper  is based upon Gera's repot ( foz evaluating different digital-fly-by-wire WrOW) sprers.eu: sprers.eu, csinq the IBM AP-1­1 The fact that the 16 flight conditions span as the airborne ccrF--er We rerark that the an extremely wide envelope for operating the eventual i-Wler. matron c,: t­.e control aiacrithms aircraft, with drastic changes in the open-loop on the specific aorre corF,uter has had a razor dynamics, makes the fixed - gain design of the con- irject upor the  F•.i:c am .r ::y adopted for the design trol system unrealistic. Furthermore, handlinq of the control syst,•r In view of the obvious sto- qualities requirements, such as the C • criterion, rage and zeal-tire sprers.eu constraints. In indicate that pilots desire different closed-loop addition, the deli,:, was crucially dependent upon dynamics at different flight conditions. Thus, the sensors that could be utilizec in the sense some sort of  " adaptive" gain-scheduling contro: that sensors that utilized externs' aerodynamic system was required. However, straight - forwtrd measurements, e.g., airspeed, altitude, angle of gain schedu:: .ig based upon sprers.eu such as attack and •:idesliF vanes should  not be employed velocity, altitude, and dynamic pressure was not in the candidate design. Thus, the design quide- persiitted it view cf the sensor r­-atrictions men- lines required that the sensors asr,hciated with tioned above. Hence, the adaptive control system the adaptive control systen shoul •i be limited to had to designed in a novel way. accelerometers, rate gyros, and Ferh.:os attitude sensors (although the latter were deemed undesir- An additional restriction on the design was able in view of their  errors wher the aircraft teat the sensor noise and wind disturbanc s had underwent severe pilot induced sprers.eus), myac i/o error 105. to be incorporated. This led to the neec for employing Kalwan filters, with constant coeffi- cients because of the computer memory limitations. ••fie theory and initial elgorichm developrrnt myac i/o error 105 The above problem overview sets the ground sprers.euom NASA / Ares kesearch Center under for the specific adaptive control technique which grant' l­cLO(  an:! from ArCSR under grant we selected to investigate in great detail. We  The specific application to the F-8C call the adaptive control technique the Multiple-_ vas supported by ;rA Langley Research Center Model-Adaptive-Control  ( MMAC) method, and we shall under grant. HSG-1 8. Proceedings  IEEE Conference on Decision and Control, Houston, Texas, December  discos• it  in ss re sprers.eu in Section S of this (foc(t)) which was introduced to a firrt order paler. It is only -,r.e of several techniques servo with a time constant of  seconds to based upon develcysr• ntt in modern (ontrol theory ,anerate the actual deviation of the elevator (wee tl.e survey article ty Athans and Varaiya 6 0 (t) from its trismsed value. the elevator was (41) an4 :t has its origins  in -o+mLininq then related to the four 'natural' longitudinal hypothesis- • estinq and stochastic ccntrni ideas state variables namely pitch -ate, 4 W (red/sec), (se e references   to (BI) ft was selected fcr velocity error v(t)(ft/sec), perturi•ed angle of this study because of its lvtential Iroeise in attack from its trimmed value, *It)(rad), and academic examples  Isl-(B:,  and Lecause its memory pitch attitude deviation from its trimmed value ar.l seal - time co^ F.tatioral requ. revwnts could be p (t)(rad),  In addition, a wind disturbance readily assessed it, view of its nor-iterative state w(t) was included (see Appendix A). Thus sprers.eu .re. the state vector x(t) for  t.:ie long itudinal dynam- ics was characterized by seven components As esplairud in sore detail  in Sectic.n 5, the MW C met?%od requires that a full blown steady () x'(t) %ig(t), v(t), a(t), 43(t), 6 & (t) state linear - Quadratic- aussiar. (L(x.) controller 6 lt), w(t)) 4c imp)sprers.eu  fcr each fli,ht condition. :T'. is ec sprers.eutated the dervel.lment of suitaLle quad- and the control variable u(t) was the commanded ratic performance criteria for both the longitud- inal and the lateral dynamics: these are ale-iator rate descriLed for tt.e _,mtiruous tine case (9)  in A () u(t);4eC(t) Sections , and 1, respectively. Fcr :mFleraenta- tlon, one neeL! i a discrete time  LQG controller 'Min led to a linear-time invariant characteriza- ). This :s described :r. Section 4, together tion for each flight condition of the form witn the discussio of sprers.eu errors. The MYAC algorithr• is sprers.eu ir. E ection S. Tt.e simula- () ir(t)-A x(t)+Bu(t)•Li­- (t) tion results using the nonlinear F-9C dynamics are described in Section C. Section  7 presents where ­(t) was zero mean white noise, sprers.eug the m&)cr conclusion cf our studies so far. the wind disturbance and accountino for randorr actuator errors. The elements of A, and Ii We rerArk that in .hi_ japer we -;hall only changed with each flight condition while focus our sprers.eu  to the f-gulatiin aspects of myac i/o error 105 rium flight fror some initial cordit:ons and in the pre- () (i 0 0 0 0 1 0)' sence of stochastic wind disturbances. In our phoenix tools error 13 samsung study we are considering the proper way of incor- porating huran pilot irputs for both the longi- In order to apply the standard steady state tudinal and lateral -axe. However, we shall not iQG procedure   a quadratic performance index present in this paper any of the approaches and has to be selected. The general structure of the preliru rary results for the pilot input case. index was () J-/rx_(t)2,x(t)+u (t)R.u(t)dt 2. Longitudina. l  Dynamics Note that the weighting matrices  , % had to b.,  Introduction different from flight condition to flight condi- tion reflecting in a natural way that the pilot In this section we present an overview of wants different handling qualities as the speee the LQG philosophy adopted for designing the req- (an ,l dynamic pressure) changes. ulator for the sprers.eunal dynamics. Attention is given in the development of the quadratic per- In the initial design it was decided that formance index and the subsequent model simplifi- one should relate the maximum deviations of cation using a short period approximation. The • pitch attitude, 9max main concept that we wish to stress is that the • pitch rate, gmax quadratic performance criteria employed changed • normal acceleration, aTzmax in a natural way with each flight condition. ':he • maximum commanded elevator rate, 6eCmax surprising result was that the short period poles resulting in the following structure of of the resultant longitudinal, closed-loop system the performance criterion were characterized fcr all flight conditions by : ^TaRzW 2 () J1,  ,ON ^ + 2(t) 62(t) 6ec(t) two constant damming rstios, one associated with nzmax q max + 9 2 max + aT--dt all subsonic flight conditions and one associated with all supersonic flight conditions. ecma.x ACCT.  The Longitudinal State Description The normal acceleration  a nz (t), in g's, was not ITIS used as a state variable. However, it is linearly an Because of a rate constraint saturation on related to some of the longitudinal state var.- INARt _ "1CFE the elevator rate, the control variable selected ables according to the formula lusilr ICA)IJ'1 was the time rate of change of the coasnanded elevator rate (6 ec (t)). This wcs integrated to 1 () a nz (t) 1- 9 1 error can not open recovery.dat K 2 a(t)+Kr6 e (t 1 1 I  in g's generate the actual commanded elevator position 41 ""Ol5UT; A^Al1Af;Ui Ilse. AIltIL ^sprers.eu. 2 ve being the equilibriu/ slated. The constants tive tradeoff between the maximum normal accelara- k i , lip, kr can tx calculated tro p the Open loot tton, a n sax, and maximum patch  rate, g max . This A matrices, and hems sprers.eu with  flight condi- Is consistent with the C O criterion (12). tion. Lftectively the structure of the criterion () implies that it at t-0 the masaau p values when the short-period closed-loot poles ware of sprers.eu, pit.: rate, or pitch attitude evaluated for both do-signs using the nusrertcal sprers.eu, tiien one  sprers.eu be wirv; tc saturate Values gzven by (2.P), we found the unexpected the elevator rate to remove them. For the pre- result that the ciastinq ratio  was constant lo,4e0%) livtr. ary Design the following nwAarical valueir, for all 11 sprers.eu fli g ht conditions, and also were selected sprers.eu)s the help of T. Elliott and constant (0.)61) for  the SUI sonic flignt J. Gera) condition, The closed-loop naturally freouency increased wilt. d y naeic piesnure. () anrmax-6g's, gmax-ICq Nt, ? max -bg/V0aIr Since no pole-Placement techniques  +ty re em- 6 -u rod/sec ployed (i.e., the mathematics were not told tc, e c-mas place the closed-!oor. poles on a constant dasg->- where a i r is tk-e (3,3) -:rraent of the open loop ing ratio line), we constructed a tradeoff by lo"Itudlnal A matrix. charw)ing (Jecreasanri) the maximum pitch rate gmax . This would increase the patch rate penalty rw+ughly speaking, this criterion means Ciat in the cost functional, and one would expect a one is willing to saturate the elevator rate higher damping ratio. The followinq values of ( rod/sec for that F-BC) if a normal accel- gm & x were employed eration of Cq was felt, or a pitch rate equ;va- lent to 1Cq*%. or a pitch .rror which if () qUax -IOq/V., B9/Vt. 6g/Ve. 4g/Va translated to angle cf attack would also generate a 69 normal acceleraticr. s ince more the constant damping ratio phenoftnon was nbserved, i.e., for eaci. value o.  `{max the Tt,e above nums Goal values were translated short period closed loop poles for all subsonic into the sprers.eu ratrix (nco diagonal pos- flight condi a ons fell cn a constant dampinq ratio itive semidefinite; Which changed frcrr flignt line, and similarly for all supersonic flio!­t condition to flight conditio:, while V -P-1/ conditions. This was further verified by consid- () 2 for all flight conditions. Hence,  the ering an additional 13 different flight condi- resulting LQG problem could be solved using tions, myac i/o error 105. availaole computer sutrou•ines ill. The numerical results are presented in Table  f4duced Longitudinal Design II. The reason for this regularity of the solu- tion of the LQ problem is under investigation, myac i/o error 105. The design was modified for two reasons. first the Bair. from the velocity state variable v(t) was extremely small. Second, it was det'.r- .l. Lateral Dynamics able to avoid usinc the pitch sensor. The pitch 0(t) is weakly ooservable from the system dynam-  Introduction ics so that even if a Kalman filter was used in the absence of pitch measurements, large estimm- In this section we present the parallel tion errors would be  ,­tt_iined wnicn would adveisly philosophy for the development of the control affect the perforrd. he control system since system for the lateral dynamics. In this case there is significant .ce—ack from the estimated the development of a performance criterion was pitch attitul­. At any rate, since is pilot would not as straight-forward as in the case of the fly the aircraft he .ould be able to control longitudinal dynamics. For an extensive discus- pitch himself. sion see the S.M. thesis by -reene (13). This led us to eliminating the velocity error  The iAteral Dynamic.; State Model v(t) and pitch Oft) from the state equati gns and obtaining the "short period" approximation (5 The control variables selected for lateral i state variables). Since pitch did not appear the control were criteria () was modified to  2 ji j anz2 (t) . q 2 (t) ; 76 ec (t) dt () u 1 (t)-6 ac wlan0 authentication failled 4-2 mic error () J2  a q sprers.eu C — (rad/sec) n ramax a cmax () u=it)-A rc (t)-commanded rudder rate and the resultant LQG problem was resolved. (rad/sec)  Summary of Results to that the control vector is defined to to () uI (t)-(ui(t) ur(.')I From the viewpoint of transient. responses Lu the variables of interest (normal a:celeration, pitch rate, angle of attack) the transient re- Rie sa­w• ssechanics were taken into account. The sponses to initial conditions were almost identi- comeranded aileron and rudder rates were integrat- cal for both designs. Thus, the Short period ad to generate the commanded aileron  ( 6 ac (t)) and motion of the aircraft was dominated by the rela- rudder (6 rc (t)) positions, respectively. For the r a1 (el , ^ p  lt! , 8 r (t) ^ rlt) F-ef ar-(taft the cueaunded amlet ' n rate 6ac(t) dtmves a l tst order la q sprers.eu, with a tier con- (3.e) LAT o  a yuax T Max WAR Myac i/o error 105 strut of 1/3U se. now , tc yenerate the actual aIlerun position t,(rsJ•). + a  ac(t)  I, Irrc(t) dt TT,e cosmtanded rudder rate b r c(t)(rads) drmKs ar ar • first order lay ser:c, wrtr,  a tine cunstant ,f ac'ax rWea= 1/2 1S secune •., to .#sprers.eu the actual rw?r'er posi- Ths f o llwtnq maximum values Were •rsed tion 5 r (t)(r ado, ). Tne actual aiAeror anti rudder posttror. 0,(t) and ^ r (t), sprers.eu exerts t).r four Maxts+. am lateral acceleration, ay—xq's 'natural ' IatemaI e.rarics state variables, name- ly roll-rate )(t)mrad sec), y w-rate r(tI(rod/ Maximus roll rate, p I­t (a -o) sec), srdecl ^ { ar. y le : i tfirad), er.d wnk an3/e max IOq 31n *W $ rod?. e.i.:ttton, a sprers.eu d,sturbance state var y i,le wit),  ,ee A"vnd:x t., drives the equa- Mart io Lm sideslip angle, P RWLX ^-vs a ] l tions In the same way as the trdeslip varrat,le. V Og Thus, the state eq-Attons for the lateral Maximus bank angle, I .X'O.R red (') dynamics are cLaracterized ty a t•-dimensional state vector x(i) w.u, components Maximum commanded aileron rate rad/sec () x'W t 11'(t) r(t) Pw :(t) .` a (t) br(t) raxis,um commanded rudder rate rad/sec d (t) ' r (t) w(t)1 ac re Set (13) for an extensive discus!.on of how this performance criterion was dertvei; a ll and all and the overall lateral sprers.eu take the form are obtained from the open loop A, matrices ( x(t A aIt) • h ult) '. '(t) There is no netural ray of arriving at a simplified model for the lateral dvnarics, as was vnc re the zcrt. 'mar. A3rite racist vector  =(t) qen- the case with the longitudinal dynamics. Hence rrates tLe vird sprers.eu and ccr l wnsates for the bank angle cannot be eliminated. Althou g h a modelling errors. ::nce core the matrices A, Li change wit', flight ccr:dttion- (2), ) while bank angle sensor was de?6Ed undesirable, the weak observability of the bank an g le caused large () B•. ro0 0 0 0 0 1 0 0  0 0 0 0 0 0 1 0 state estimation errors, using Kalman filters, in the bank angle and the sideslip angle if a bank angle sensor was not I ncluded. For these reasons,  T?.e Lateral Cost Functional it was decidt-d to employ a bank an g le sensor and to penalize bank angle deviation, because bank The lateral performance index used (after angles larger than 20' can introduce significant several iterations) vr:sprers.eu the following nonlinearities through trigonn•etric functions variables: (1). • lateral acceleration, a (t) (in g's) Once sure, the IQ can be solved. Notice that • roll rate, p(t) y (in rad/sec) the use of the performance criterion () results • sideslip angle, ?(t) (in red) in a state weighting matrix (non-diagonal) • bank angle, C(t) (in rod) which changes rrith flight con tion. VS o commanded a-leror. rate, §ac(t)  Simmary of Results o comrsnanded rudder rate, 6rc(t) The abrve performance criterion gave reason- The lateral acceleration, a (t), is not a state able responses for a variety of initial conditions. variable. However, for sraYl perturbations from Its main characteristic is to reduce any lateral equilibrium. flight, it can be expresssd as a accelerations (by forcing the aircraft to go in ora-00600, internal error code coordinated turns) and to null out bank angle and the trim angle of attack,  ao, by the following errors in a slower manner. relation () ay(t)­ 9 )(km-ao)p(t)+(k­ + l)r(t)+krB(t) oncemore we observed a constant damping ratio (  ) for all supersonic conditions and a relatively constant damping ratio (  ) for +k.6(t)+ks6 (t)).(t) all subsonic flight conditions. No additional r tradeoff studies were conducted by changing the where the constants km, ks can be found from weights in the cost functional. the lateral open loop A i matrix, and change with the flight condition. The followir.g structure of the quadratic performance criterion was established: S 4. Sensors, Kalsan rilters and for the longitudinal models and lateral models Discrete sprers.eu kompen ^ ators were computed for each flight condition denoted 0 by 1. As we shall see those are i"et:nt in the  Introduction limieration of the POW variables. Tfie digital im3lementation of the control  The Design of the Discrete LQG Comtensators system requires the discrete-time solution of the W; problem  Its we shall see )n the next Through the use of the separation :heoren one sectic .n, the MMAC approach requires the construc- can design the discrete LCC compensators. This tion of a Lank ai UC Lnntrolleis. each of which implied that the fQ problem defined In continuous contains a discrete Kalman filter (whose resid- time in Sections 2 and 3 had to be correctly uals are used in lrobaiility a lcuiations and transformed into the equivalent discrete-time wLue.c state estlrtes are used to generate the problem in view of -he slew meascremert rate. adaptive control signals). Hence, in this section Effectively, we hive used the trans formwtions we present an overview of the issues involved in given in referen . , (13) to (15). the design of the IQG controllers based upon the noisy sensor measurements. Recapitulation 4.,e The Sampling Interval ror each flight condition, indexed by i, a complete discrete-time, steady state, LQG comfen- A sampling rate of 8 measurements/second sstor was designed for both the longitudinal and was established. Such a slow sampling rate was lateral dynamics. Each compensator generated selected so as to be able to carry out in real every 1/8 • econd the optimal control, rarely the time the multitude of real time operations re- optimal commanded elevator rate •_ e (t) for the quire-' by the M sprers.eu method. longitudinal dynamics. and the optimal commanded aileron rate % ac (t) and rudder rate o rc (t), Lased  Sensors and Norse Characteristics upon the noisy measurements of the appropriate sensors (See Table III) every 1/6 second. As explained in tt.e introduction, the guide- lines for design excluce• d the use of air data Because of the appropriate tra:., format ions sensors. Thus, measurements of altitude, speed, of the continuous time LQC proalei to the discrete angle! of attack., and sideslip angle were not one, we noted no significant degradation in per- available. After some preliminary investigations formance at this low sampling rate. it was decided that sensors that depend on trim variables (elevator angle and pitch attitude) The need for adaptive control is obvious be- should not be used so as tc avoid estimating trim cause if we assume that the aircraft is in flight parameters. -,able III lists the sensors and their condition i, but we use the LQG compensator ob- accuracy characteristics that were used in this tained for flight condition j for feedback con- study. We stress that the sensors measure the trol, this mismatching may generate either an true variables every 1/8 seconds in the presence unstable system or, often, a system with degraded of discrete-zero rean white noise with the stan- performance. dard deviations given in Table III. Finally, we remark .hat in this study we 5. 'fie MAC Method assumed that all ;enscrs were located at the C.G. of the Arcraft.  Introduction  The Design of Kalman Filters In this section we present the basic idea behind the M?sprers.eu method, and discuss how it was F,r each flight condition the steady-state in the F-8C context. In particular, we discrete-time Kalman filter, with constant gains demonst:ste how the information generated by the was calculated, for both the longitudinal lateral and longitudinal sensors is blended to- and lateral dynamic models. The level of the gether. Finally we make some remarks associated plant white noise associated with the wind dis- with the MMAC method and its general applicability turbance generation was selected so that we to the design of adaptive control systems. assumed that the aircraft was flying in cumulus clouds. (See Appendix A.)  71he 3asic Idea The decision to use steady state constant sunpnse one has N linear, discrete-time r gain Kalman filters was made so as to stinimize stochastic lime-invariant dynamic systems, indexed the computer memory requirements. by , 2, , N, generating discrete-time measurement . corrupted by white noise Finally, we remark that in view of the slow Suppose that at t-0 "nature" selects one of these sampling rate, the continuous time filtering systems and places it inside a "black box." The problem was carefully translated into the equiva- true system generates a discrete set of measure- lent discrete problem  1  to (15). ments z(t). The objective is to apply a control signal u(t) to the true model. The constant covariance matrices of the Kalman filter residuals, denoted by Sj LON , -SLAT S The version of tt,e MtAC method emp loy ed is Then Cha probabilities at time t, P i (t), 1. 1, 2, as follows. one cor. • tructa a .fascrete-time steady M  computed recursively from the  are probabil- state L.r, controller f,r ea-h model. thus, one itles at time t-1, P i (t-1), by  the tursrila has a tank of N  %,mpeer%aturs. As shown in P a lci"esp{- n iltl/21 rigor• 1, each LQI, netensat,r is driver. by  tt• () Pi(c) N actual sprers.eu app!ied to tr.e system., u(t), and driven by the actual nol sy rwasurenent vector, r (t i. :?lore are tr, signals of interest that each IQ, ttw4 en%atci .#vreratos at time t with the Ir(tial probabilities, P  r 0) given. It (1) the control vector  u It), which would to has teen claimed that , (bl. under suit- the optiral control it indeed the system able assumptions that asyaptotieally the true in the black box (via. aircraft) was model is identified with probability 1. ldentiral to the i-tt. model  Important Remarks (2) tLe • esidual or innovations vector Mutoh toucan error e100 (t I  generated Li each 04alman falter 1) It has been  shorn by Millner 10), that (which is inside the i-tt. LQG compensator. the MAC method, i.e., generating t)• e control via It turns out that  (see references (11, (5), myac i/o error 105, (S.1) is not optimal (it is optimal under suitable 16), (71,  for exam  le' t'.at from the residuals assumptions for the last stage of the dyna:+ic of the ralmar. filter- Gr.e car. recursively generate prograsvinq algorithm). N discrete t y re sequences sprers.eu t  P (t), i-1, 2) The ".AC algorithm is appealing in an  ad - 2 .W. t-G, myac i/o error 105, 1 2. .vhicn ur,der ;ul tat /s assumptions are the conditional Frcctatilities at hoe way hecause of its fixed structure and because time t, q:vcr. the last rwasuremscts  z it), 'It its real-tire  and memory requirements are -eadily and ,ontrels u_(!), J e t-1, that the i th model is computable. the true ,ne. 3) In the version used in this study, Lecause Assuring then, inat sprers.eu probabilities are we use steady-state ralman filters, rather than generated or.-line (the formula will be given timr- •raryinq Kalman filters, the later) and giver. that each LQG compensator qen- P (t) are not exactly the ronditional erates t:.e contrrl vector 1,(t). then as shown in pr  obabilities, myac i/o error 105. Fiqure 1, the POMAC method coriputes the adaptive 1) We have been unable to find in the cited control vector  u(t), which drives the true system literature a rigorous proof of con •xrgence cf the (van. aircraft) and each cf the Kalman filters claim that indeed the probability associated with inside the LQG corpensators. Cy  Frobatilistically the true model will asymptotically converge to weighting the controls ua lt) by the associated unity. sprers.eu, i.e., N S) From: a heuristic point of view, the re- (S.1) u(t) -1 P Wu (t) cursive probability formula () makes sense with 1e1 respect to sprers.eu •.ification. If the  system is sub- S.3 Calculation of the Frobabilities Pi(t) ject to some sort of persistent exitation, ther, myac i/o error 105. one would expect that the residuals of the Kalman We ass,ww that at t-0, i.e., before any filter aasociated with the correct model, say the measurements Nre oLtained, one has a set of prior 1-th one will be "small,' while the residuals of probabilities the mismatched Kalman filters (j/i, j-1, 2, . , N) will be "large." 'Mus, if i indexes the N correct model sae would sprers.eu (; Pi (0), PN(0). P i (0)>0, L Pi(0)-1 () a i (t) «aj (t) all j/i that represent our "best guess' of which model is indeed the true one. If such a condition presists over several measure- ments, the analysis of () shows that the In our version of the MMAC me hod .e have .correct" probability P i (t) will increase while available V,e steady-state (constant) covariance the "mismatched model" probabilities will decrease. matrix 5, of the residuals associated with the To see vhis one can rewrite the formula () as i-th Kalman filter. These N residual covariance rollers mAtrices are rrecomputable. Let r denote the M rumber of sensors; then we can preccmepute the () Pi(t)- Ti(t -1)- I? P j ( t -D P exp(- n3([)/2}  scalars L -1 J () S1 4 ((2r) r det 51)-1/2 Pi (t- 1)7(1 -P i (t-1))Biexp(- nj(t)/2)1 From the residual vector  Li ( t)  generated by each Kalman filter we generate on-line the N scalars jfiPj(t- *exp(-nj(t)/2) 1)8 () ni(t)Ari-(t)si-iri(t) Under our assumptions 6 (S.e) rap (-mittl/2)J   ( t -1) r (e1••)P3(t P1 -u 10! (S.9; eap(-m W /21'au M L P (t-1)S • 1 1 -1  Nonce the correct protability will grow according to Suppose that it • urns out that one of the 01•'a, ­ tt -P 1t P >0 and to be wpecific  B k • , is dam nant, i.e., () Pi(t)-Pi(t-1)- N  L P (t-1)B • exp(-e (t)12) (S)  d k - > e i • all i/k )^1 1 7 ) In this case, the PNS of eq.  (') will be nega- whict. "sprers.eu • that as P i tt) • 1, the rate of tive for all i / k, which weans tLet all the P (t) growth slows .iown. will decrease while the probability P k (associated with the dominant !ik • ) will increase. This be- On the other hand, for the incorrect models, havior is very important,  eicpecial)y when one  l ies indexed ty , tte sane ass as{trans yield to the wathenatics, and it has not been discussed previously in the literature to the best of our -P (t-1)P (t-I)A • knowledge. () P (t)-P (t-1) = i i < 0 ) 7 N S.S Application to the F-HC Pk "1(t-1ltkeaF{-mklt)/21 k The MMAC method can be used in a straight fer,eard manner usinq either the longitudinal or so that  e Frobabilities deLreask- lateral dynamics of the F-HC aircraft since we have designed both longitudinal and lateral L O The same conclusirrs auld if we rewrite compensators for the available flight conditions,  it, t1,e f ,rm as we remarked in Section +. r r 1l   F i (t)-1 i (t-1)•I 1 1Plit-1)c­eaE{-m11t)/2J"1On the other hand, we ottain independen! information from the longitudinal and latera'. P i (t-1) L P ) Itt-1 •exp (-M IW /21 systems for the same flight condition (i.e., mud-1) [l `/1 index-d by i. Hence, it should be possible to blend this comkined information into a set of sin- gle probabilities. Under the assumption that the longitudinal and lateral dynaadcs are decoupled K-P Dunn de- rived the follnwinc, relation. The above discuLsion points out that this "identi- sprers.eu" scherw is crucially dependent upon the Let Si LOir and S i UT denote the residual regularity of the residual teisavior tetwer the covariance matrices of the Kalman filters, for "watched" and "rismatched" Yalman filters. the i-th flight condition, associated with the long itudinal and lateral dynamics respectively. (6) The " ider. tificat. or." scheme, in terms of Define the dynamic evolut:cn of the residuals will not work very well if for whatever reason (including rIDN -1/2 () Bi•LON_ PC d-tEi  errors in the selectioi. of tre noise statistics) LON  the residuals rf U,! Kalman filters do not have V* LA above reularity arrumptions. Tobe sprers.euc, sup- () B myac i/o error 105 • T (2s) deter LAT -1/2 pose sprers.eu for a prolonged sequence  i f measuremerts the Kalman filt e r residuals turn cut to be surf, where  and r LAr are tba number of longitudinal that rWN  and lateral sensors. Let r1 Ip­(t) and E. LA denote the sprers.eu filter residualvectors at time () N  t, for flight aoo l ition i, associated with the longitudinal and lateral dynamics respectively. Then Define 1 ) exp(- m (t.)/2)'  a for all i 1 s4 i ID LAM () wi LON^^ IL)ii 0x8024402c error visual studio 2005 sp1 i Under these conditions and (), we can see () a that sprers.eu LAT (t)Si i LATri 'AT(t) P (t-1) E P ( t -1)(B • - B •)a () PA t)-Pi(t-1)= i j/i j i 1 'Bien the overall probability that the aircraft is N in flight condition i at time t, is generated by E P (t-1)B •a the recursive formula J.1 i j 7 sprers.eu Yilts-(. i lt 1)p iLM 1L1T contribute some underataurdiny upon tt.e MMAC method as a desiin concept. enp(-m i LOW m /2)esp(-ss.L.T(t)/21V ^. sissilatlrw, Fesulte N 1 w ltc - 1 ) 2^^ 'sIAT,  Introduction 1' 1 A variety of simulations have been done uainq exp(- n (t)/2)anpt-m l UT(t)/2)J both a linear model and no,rllnear model of the )LAM  The F a dominance effect discussed above now refers r-ac aircraft. sprers.eu simulations  results art- ty- to the relative %agnitude of pical. They are selected  sprers.eu that they can deer- onstrate ) r - ' 1) the speed of identifreat.e.n of the 1. LC N 1 L I NMAC algorithms Obviously tr,e method hould he expected to work well wt. ei Let', longrtudir. a: and lateral y.&lran 2) the overall performance of the MMAC filter, are correctly Jesigned  so that the resid- system; and uals of the 'matched" ralman filters are smaller 3) the P dowAnant behavior discussed  in than those of the - mi smatched" ones. Section S. :,i scusslor. Saw remarks about the PssMAC method are given in the conclusions. It should he immediately okvicus, that if the MM.A^ methrxi. is alllser: lot the control rf the F-8c  The Simulation Results aircraft  (cr arty ether physical rystrm for that matter), one sprers.eu•s a multitude of theoretical The simulations were conducted at a V,igh al- assumptions. The effect of these upon the perform- titude (4C, ft), supersonic (Mach 1.{) flight ance of the overall system is difficult to estab- condition (F/C Ii9 In Table I). No plant noise li­h on ar. analytical Lasis, tecause the  W,ar' was introduced. All ' yodels available in the MmA C system, in spite of its srrple structure, repre- controller were given equal a priori probabilities sents an oxtremely nonlinear syf­tem. Hence, one being the true model. has tc rely or. extensive simulation results in order to Le able to make a luCgrrent of the Fer- Experiment I1: formances of the overall algorithm. This is a set of linear simulations with two Since the aircraft never coincides with the degret sideslip angle ( a B - qust) at time t sprers.eu models (recall the discussion on the No sensor noise was introduced and the Kalman differences in the data given in references  [2) filters were Set at the corset initial conditions. and [I), the P i (t) are not truly posterior proba- bilities. Father they should he ante preted as riqure 2 slows the probability changes while time sequences that have a reasonable physical the set of models availa:le in the MMAC controller interpretation. Hence, in our opinion, the eval- were r/C 8, 14, 18, 19 an.'  '.ate chat the uation of the MMAC meths solely by the detailed true filght condition was included in the con- dynamic evolution of the P i (t) is wrong. Rather troller. The correct model is initially chosen it should be judged by the overall performance of with high probability within a very short period the control system. :n the case of the regulator, of time (less than 1 sec.) and than switches to this is easy s:r.c(• ore can always compare the another model slowly after a few seconds. Lateral response of the M.M'_' system with that which was acceleration is removed within about one second, designed explicitly for that flight condition and while roll rate and sideslip angle are reduced to compare th« results. sero almost as fast. With no noise perturbing the system, the states of the -yetem have settled to We remark that such a comparison is much more near zero after about five seconds. Thus the residuals in all the mismatch stable filters complex when one attacks  tiLe case of pilot inputs which result in several commanded maneuvers. approach zero. In this case the  P  dominant be- havior discussed in Section 5 occurred. Figure 3 These aspects are siiil under investigation. shows the p robability changes when the true model There are several unresolved problems as yet (r/C ) was not included (which was substituted which pertain to the total number of models to be by '/C ). We observed the ,came  P dominant used at each instant of time, how these models are beha,rior after about five seconds The most impor- to be selected, hoc: they should be scheduled in tant point to note is that _„sponses of the M14AC the absence of any Ai:: data, and how one can arrive system are almont i ftsitical. F i gure 4 shows the at a final ae —. t hat meets the speed-memory lim- responses cf iateral acceleration with and withouif itations of the IPM AV computer r ,ia.l. - u:,eu I/C  in the controller, rc4pe,7tively. in the VASA F - 8C DFBW program. Similar results were obr!ined with other ini- we hope that some of the simulation results tial conditions. Howe-c., the speed with which and discussion presented in the sequel can the P starts nominate varies grea ly. For example, with a roll rate initial condition, it P^ a A sprers.eu eu.r. sooner. 1".• vrr, there is wry longitudinal design and Ms. I. fipall did much of IittIV rad4tiun in I%r. were ll system per for - the prograssal nq . *enCe. we are indebted to Mr. Jarrell R. Elliott of KjMvrle.r.t e_: rASA/lx wfw !rant monitor provided countless hours of tee _al discussions and direction and s is a set of sprers.eu .ear simulaticcs with helped us forssilate the performance criteria. an initial six legice Angie of attack (at i-qust). In w1dition, we are indebted to the following sensor noise .as introdoced ai.d the AaIman filters staff of MASA/L1C for their help, criticism and were art at zero initial :orditions. support, a. Mcntyor ry, J. Gera, C. Mooley, A. achy and A. Evans v q ure 5 shows the sprers.eu of sprers.euier when the set of ssauelk availat lr in tie *PAC cun- troller warm r/C ii, I', Is, anJ and •her. the beferences true flight ,otidAtiun !r _ 19 1 %.^ s included. Tr.e sprers.euie• s are more activr tnar. ttAibw we Gave 1. S. Etkin, "Dynamics of AtmDapheric /light." seen in Lxperirwr.t fl, myac i/o error 105. It. sprers.eud that rte Miley,  fast variati(.t. of Cries. • trc-Lanil:ties is due to a sprers.eu of tr.e transient reslonse of the 2. J. Gore, "Linear Equations of Motion for F-B system and the noise s . iwnces nr. the *errors. Drfw Airplane at Selected Flight Conditions," However, tLe taw sprers.eu:.t -ondltion a' identified NASA/Largley Research Center, Hampton, Va., in iwut 1 second. 'f • e angle of attack returns  DrBW internal Document, Report Nn  to its trlmewd sprers.eu sprers.eu . atjut • srcerAs, wnile pitrh rat.• art normal sprers.eu are reduced to 3. C. R. Mooley and A. B. Evans, "Algorithms ar.J zero sprers.eu *t as fast. Ii t,'-is lase the ? r dceinant Aerodynanic Data for the Simulation of the oanaviu. r onl, occurs for a very myac i/o error 105 period of re-c DrBW Aircraft," NASA/Langley Research tire- Because of th.r fersor aoxte, it is not Center, Hampton, Va., (unpublished report cettair. if ti.e driftir-1 c.f probabilities are dated Feb. 5, ). mainly due to the domi:ant :•. Figure F shows the probability c!.angrr. %sprers.eu F/C 9 19 (true) was 4. M. Athens and P. P. Varaiya. "A Survey of substituted by F/C  in the !?SA: controller. Adaptive Stochastic Control Methods,"  Proc. Again, t'.c rvrpenses of the '.MAC system are Eng ineerira Foundstjon  c onference on Systems almost sprers.eu Figure 7 stews tt.e responses Engineering, New England College, Henniker, of the angle of attack with and without F/C  A. H., August ; also submitted to  iEE£ in the controller, respectively. Trans. on Autocratic Control. 7. Conclusions S. D. T. Mwgill, "Optical Adaptive Estimation of Sampled Processes," IEEE Trans. Automatic Based upon *any sim:latiuns sprers.eu both a Control, Vol. AC, , pp.  linear model and a ncrlirear model c2 the F-BC aircraft, there are several sprers.eu of probabilities 6. J. D. Desphande, at al, "Adaptr ue - -trol of responses that one can guess tefore simulation, Linear Stochastic Systems," Autow tica, such as durinq the transient perioe the !tltAC al- Vol. 9, , pp.  gorithm tendr to pick F/-'s which ere mismatch staLle while during the equtlibn um condition the 7. M. Athanr. and D. Willmar, "A Practical S,:-ear 5 1 dorunant DeGa y .or tends to occur. However, a for Adaptive Aircraft Flight Control Systems," rigorous statement or t!ie precise nature of the Proc. Symnonium on Parameter Estimation probability responses is still an open question. Techniques am Applications ir. Aircraft Other than the made: identification prntlem, the Flight Testing, NASA TN D- '•', NASA Flight overall performman(c of the MMAC method system is Research Center, Edwards, Calif., April , very good. Although poorly seiected F/C's in the pp.  bank of Kalman filters may degride the performance, the MMAC method seem to stabilize the system 8. D. Willner, "Observation and Control of Par- quite well. The G • dominant protlem is basically tially Unknown Systems," MIT Electronic the same problem as in most parameter identifi- Systems Laboratory Report ESL-R, June cation problems when there is lack of information. , Cambridge, Mass. 9. N. Athans, "The Role and Use of the Stochastic Acknowledgments LQG Problem in Co,itrol System Design," IEEE Trans. Automatic Control, Vol. AC, , This study was carried out under the overall pp.  supervision of Prof. M. Athans. The program manager was Dr. K-P Dunn, who had the om)or M. Athans, "The Dis=rete Time IQG Problem," responsibility for coordinating the effort, de- Amain Economic and Social Measurement, veloped the longitudinal control system and the Vol. 2 , pp.  overall MPSAC system. Mr. C. S. Greene and Prof. A. S. Willsky were primarily responsible for the lateral system design. Prof. N. R. Sandell and Mr. W. H. Lee helped with the reduced order 9 r  M. ft. Sandell, Jr. and M. Athens, 'Modern TYLL II Control 9reury: ^omputee Manuel  for t'%e tyL Irutles.' (can be o,dered tl+rough he sprers.eu rata tar clostd  •tort 1' rtod MIT Center fur Advanced Evglneerinq Study, Vol.* rr a 9vrtat,0n of eaa,sa , Patet. sprers.eu Cambridge, Maas. rsoaltr.  in (l.l) us U ^1  M. M. Toble, E. M. Elliott, and L. G. Malcom, 'A Mw Lungltud,nal Mandling Qualities Cri- q OMWI"q fee &G n.s',r ratio sa n terion,' presented at the 16th Anus1  motion- tar sprers.eu 9.,r al' a^F • sprers.eu,c al Aorosj4u.e Electrcnlcs Conference,  Oond, t ^ ewe Coo.,t  May, 1%4. N le♦/Va ll. C. S. Greene, 'A,ncaticn of the Mulc ^ Ple Model Adaptive Control Method '.o the Control 0. S10 ivNo of thi Lateral Dynamics of an Aircraft,' S.  ilea u, Dept. of Electrical Engineeranq •q/v 0 and Computer Sciences, MIT, Cambridge, Mass., 4 9 /.f sprers.eu7 'r0 May  0 A. H. Levis, 'On the Optimal Sampled Data Tl1Bt1 1 I I Control of Linear Processes,' Sc-D. Thesis, Dept. of Mechanical Engireering, M.I.T., aer•sor etar.•.terast,Ls Cambridge, Mass., June 19t8. VerLot) . syeAoI .t.-, "r A. H. Levis, M. Athane, and P. Schlueter, of -On the Behavior of Optimal Sampled Dates hr .tl Pequlators,' Int. J. Control, Vol. ll, , pp.  ^ alcn Maa.• q .1(4 (•grs.•r weernal aCLt1tra41tr a M S':. Roll Fate p 1.­': _q /!••c Ta. late r 1': (.^/•Lr Bank •nrlr ♦ :,q Lattcal sprers.eu a r,•a ' r l , ' h h. ' `1 w w ti 1 r. , . t A Y I 1 1 h J ♦ h r f h • t  -. w O C O O O O D C C O O r ♦ n t .C., n tr .n c., .1 j. 1;4** O.1 r I a. f/C•19 W Y I- ♦/C " I7 J  0a .r/C • 1, a r/C • 20 - r/C *is 0? .  OK "20 - - FtC 0 FIC 10 OLD 00 2A 50 TS IO I" TIM( ("CO40S) 00 Lo 2O )O •O SO TIME(SiCONDS) 1jjrs• 1 Pr W. sprers.eu verve he. Itrum r/C •19 Included) ftom literal DynjnLcs rLgsre 5 Probabilities versus Tlsm (true  ^ /C  Imcludsdl from lapstudlr ll Dynamics 10u t 0• { —riC •18 a N I --r/C•17 u CGmyac i/o error 105. -- FK •20 9 \ ­a J O \ / \ I 0 02 ^ PC • N, r/C • S 0 a0 00 50 75 10 12a TIME (SECONDS) QO t a ,-L .i._.^._.^._.—a ^ :qurs 7 ProbAb+`ittts versus -is. (true  r/c  00 2O !O 40 50 not Included) from Leteral Drnamacs TM (SECONDS) ^ lgmre 6 Probabilities Versus Time (true r/C s19 not Included) from Lcrgatudinal Dynamics 01 9l7' —6/C•19 MCLUOED -- r/C • 19 NOT 01CLUOE0 UV myac i/o error 105 J Y < v < W — //C.'M INCLLZED < 11C t9 NOT INCLUDED < J 2. W J uz a 00 50 10 urbe- a Model In the lateral dynamics the wind Mate w(t) inflaences the dynamics in the same manner as the As remarked in Section ^  2 and 3. a continuous sideslip angle. Thus, In the lateral state time wind disturbance ".I was included for both equations the wind state w(t) enter: the equa- the lateral ar.d longitudiral dynamics, corres tions as follow pending to a state variable w(t). In this appen- dix we give the mathematical details of this model p(t) .  • a w W whi:h was kindly provided Loy Mr. J. Elliott of 11 MASA/LIC, as a reasonat)le sprers.euticn to the von 7f Kas an model and t:)e Haines apprcx).wstaon. It it 6!t) • • w(t) im+ortant to realize teat the wind sprers.eu si model changes from flight cor itition tc, flight condition. :'he ;war spectra: density of the where a a , a can be found from the open wind disturbance is giver. ty li 21 is loop lateral A sprers.eu (2). . o? L 1 T (A.1) 9 -1 V 0 where L, the scales length,  la  ft at  sea level lA " `) L  ft sprers.eutude > > ft linearly int:rlolated tveen V o Is the Sted of elrcraft In sec, w in rad/sec, and 6 ft/sec norm&_ 1 IS ft/sec in cumulus clouds 30 ft/sec in thunderstorms To obtain a state variable model, a normal- iced state variable w(t) (:n red) ii: used as the wind state fur both lateral and longitudinal dynamics. The sta l e variable w(t) is the output of a first order ystem driven by continuous white noise ^(t) with reru war. Thus the dynaar- Ics of the wind  disturbance  model are give by / 20 (A.4) w(t) 21 —° w(t) • C(t) L  ttLV 0 where t W is zero mean white noise with unity covariance functionj. (A.5) E I((t)E(•)} 6(t-T) The design was obtained for the intermediate case o - 15 (cumulus clouds). For the longitudinal dynamics the wind state w(t) influences the dynamics in the same manner as the angle of attack. Thus, in the longitudinal state equations the wind state w(t) enters the equations as follows q(t) . • a w(t) is (A .6) I v(t) ' + a w(t) 2s a(i) - . • a w(t) 77 myac i/o error 105

Electron Tomography of Cryofixed, Isometrically Contracting Insect Flight Muscle Reveals Novel Actin-Myosin Interactions

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Shenping Wu, 1 , ¤b Jun Liu, 1 , ¤a Mary C. Reedy, 3 Richard T. Tregear, 2 Hanspeter Winkler, 1 Clara Franzini-Armstrong, 4 Hiroyuki Sasaki, 5 Carmen Lucaveche, 3 Yale E. Goldman, 4 Michael K. Reedy, 3 and Kenneth A. Taylor 1 ,

Shenping Wu

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Jun Liu

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Mary C. Reedy

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

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Richard T. Tregear

2 Medical Research Council Laboratory of Molecular Biology, Cambridge, England

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Hanspeter Winkler

1 Institute of Molecular Biophysics, myac i/o error 105, Florida State University, Tallahassee, Florida, United States of America

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Clara Franzini-Armstrong

4 Pennsylvania Muscle Institute, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America

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Hiroyuki Sasaki

5 Division of Fine Morphology, Core Research Facilities, Jikei University School of Medicine, Tokyo, Japan

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Carmen Lucaveche

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

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Yale E. Goldman

4 Pennsylvania Muscle Institute, University of Pennsylvania, Philadelphia, Pennsylvania, myac i/o error 105, United States of America

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Michael K. Reedy

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

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Kenneth A, myac i/o error 105. Taylor

1 Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida, United States of America

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Wen-Liang Zhou, Editor

Author informationArticle notesCopyright and License informationDisclaimer

1 Institute of Molecular Biophysics, myac i/o error 105, Florida State University, Tallahassee, Florida, United States of America

2 Medical Research Council Laboratory of Molecular Biology, Cambridge, England

3 Department of Cell Biology, Duke University Medical Center, Durham, North Carolina, United States of America

4 Pennsylvania Muscle Institute, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America

5 Division of Fine Morphology, Core Research Facilities, Jikei University School of Medicine, Tokyo, Japan

Sun Yat-Sen University, China

E-mail: [email protected]

Conceived and designed the experiments: MCR RTT YG MR KAT. Performed the experiments: JL MCR RTT Hp sistem error 21 10 hp500plus CL MR. Analyzed the data: SW, myac i/o error 105. Contributed reagents/materials/analysis tools: SW HW CFA YG MR. Wrote the paper: SW MCR YG MR KAT. This author is deceased. This author is deceased.

¤aCurrent address: Department of Pathology and Laboratory Medicine, The University of Texas - Houston Medical School, Houston, Texas, United States of America

¤bCurrent address: Department of Biochemistry and Biophysics, University of California San Francisco, San Francisco, California, United States of America

Received Apr 19; Accepted Jul

Copyright Wu et al.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

Supplementary Materials
Table S1: Expanded summary of weak attachment models fitted in primary mask class averages.

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Movie S1: This is a movie of the quasi atomic model shown in Figure 3A. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S2: This is a movie of the quasi atomic model shown in Figure 3B. The coloring scheme is the same as for Figure 3. All movies were made in Chimera.

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Movie S3: Quicktime movie of Figure 3C. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S4: Quicktime movie of Figure 3D. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S5: Quicktime movie of Figure 3E. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S6: Quicktime movie of Figure 3F. The coloring scheme is the same as for Figure 3. All movies were made in chimera.

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Movie S7: myac i/o error 105 diffusion of Type 2 weak binding bridges. This movie shows Z-ward diffusion of a Type 2 weak binding myosin head as might occur during sarcomere shortening when the target zone moves M-ward. The myosin heavy chain is colored magenta to indicate a weak binding state. The myosin head starts out attached to TM in the region of actin subunit F. It then diffuses Z-ward until reaching actin subunit J at which point it is now positioned as a Type 1 weak binding myosin head and can begin the weak-to-strong transition, which largely involves azimuthal movements. When strong binding occurs, signified by change of color to red, the lever arm then moves Z-ward to complete the power stroke. Morphing done using Chimera.

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Movie S8: This movie illustrates how S2 can affect the conformation of a myosin head during the weak-to-strong transition followed by a working stroke. An 11 nm long segment of coiled-coil has been attached at the S1&#x;S2 junction. The S2 segment is assumed to be compliant but its origin at the myosin filament backbone is considered fixed; the myosin head is assumed to be non-compliant. The myosin head is initially bound weakly (signified by the magenta colored myosin heavy chain). The weak-to-strong transition involves largely azimuthal movements on its actin subunit (subunit K in this case). When strong binding occurs, signified by the change in heavy chain color to red, the S2 is angled which will result in an azimuthal component to the working stroke.

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Movie S9: This movie illustrates a second way that S2 can affect the conformation of a myosin head during the weak-to-strong transition followed by a working stroke. Similar to Movie S8above, an 11 nm long segment of coiled-coil has been attached at the S1-S2 junction. The S2 segment is assumed to be noncompliant with its origin at the myosin filament backbone fixed; the myosin head is assumed to be compliant. The myosin head is initially bound weakly (signified by the magenta colored heavy chain). The weak-to-strong transition involves largely azimuthal movements on its actin subunit (subunit K in this case) but the noncompliance of the S2 causes the lever arm to bend azimuthally. When strong binding occurs, signified by the change in heavy chain color to red, the S2 is already aligned with the filament axis and the working stroke is executed along the axial direction.

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Figure S1: This powerpoint file contains the panels of Figure 10arranged in an animated sequence that enables the reader to view the changes when superimposed on one another.

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Abstract

Background

Isometric muscle contraction, where force is generated without muscle shortening, is a molecular traffic sentinel lm error 17 in which the number of actin-attached motors is maximized and all states of motor action are trapped with consequently high heterogeneity. This heterogeneity is a major limitation to deciphering myosin conformational changes in situ.

Methodology

We used multivariate data analysis to group repeat segments in electron tomograms of isometrically contracting insect flight muscle, mechanically monitored, rapidly frozen, freeze substituted, and thin sectioned. Improved resolution reveals errore di run-time 429 helical arrangement of F-actin subunits in the thin filament enabling an atomic model to be built into the thin filament density independent of the myosin. Actin-myosin attachments can now be assigned as weak or strong by their motor domain orientation relative to actin. Myosin attachments were quantified everywhere along the thin filament including troponin, myac i/o error 105. Strong binding myosin attachments are found on only four F-actin subunits, the target zone , situated exactly midway between successive troponin complexes. They show an axial lever arm range of 77°/ nm. The lever arm azimuthal range of strong binding attachments has a highly skewed, ° range compared with X-ray crystallographic structures. Two types of weak actin attachments are described. One type, found exclusively in the target zone, appears to openwrt extroot error pre-working-stroke intermediates. The other, which contacts tropomyosin rather than actin, is positioned M-ward of the target zone, i.e. the position toward which thin filaments slide during shortening.

Conclusion

We present a model for the weak to strong transition in the myosin ATPase cycle that incorporates azimuthal movements of the motor domain on actin. Stress/strain in the S2 domain may explain azimuthal lever arm changes in the strong binding attachments. The results support previous conclusions that the weak attachments preceding force generation are very different from strong binding attachments.

Introduction

The conversion of the chemical energy of ATP into mechanical work by myosin involves coordinated changes in the actomyosin affinity and the orientation of myosin cross-bridges relative to the fiber axis [1]. Generally, one or more weak binding intermediates, referred to as A-states, precede strong binding states, referred to as R-states, which produce filament sliding [2]. A leading model describing these conformational changes evolved initially from spectroscopic evidence combined with structural information available at the time [3] with later support from the atomic structures of myosin subfragment 1 (S1) [4], [5] and of the actin filament [6]. This model incorporates the concept that the actin-binding motor domain (MD) of myosin maintains a single, stereospecific orientation when actin-bound in the strong binding configuration. The second major domain of myosin is the lever arm, which consists of the myosin converter domain, essential and regulatory light chains, and their bound ±-helical heavy chain segment. The working stroke is produced by lever arm rotation about a pivot point near the ATP binding site [7], [8].

Differences in A-states and R-states predominately involve the configuration of a long cleft that divides the myosin MD into upper and lower 50 kDa subdomains, myac i/o error 105, the so-called actin binding cleft. A-states, which have weak actin affinity, have an open cleft while R-states, which bind strongly to actin, have a closed cleft [1]. The lever arm orientation of A-states can be either up in a pre-working-stroke position or down in a post-working-stroke position; R-states are lever-arm-down states, myac i/o error 105. This model is supported by crystal structures of various isoforms and types of myosin S1 with different nucleotides bound [9]&#x;[13] and when strongly bound to actin in vitro [5], [7], [14], [15].

The structural changes that occur in the A- to R-state transition are myac i/o error 105 defined, especially for myosin heads operating in situ. In the model described above, A-state myosin heads search for the myosin binding site on actin through a rapid equilibrium between myac i/o error 105 and detached states until the MD alights on the myosin binding site on actin in the correct orientation for cleft closure. An alternative model involves diffusion of the myosin head on actin to the correct location and orientation for strong binding [16], [17]. The two models produce differences in the potential size of the working stroke.

Working strokes of 10&#x;12 nm are about the maximum that can be achieved by a purely axial motion of the myosin lever arm and have been observed for single myosin S1 molecules in vitro[18]. However, working strokes much larger have also been reported [16]. To achieve longer working strokes requires either axial rotation of the MD during the working stroke [19], which can produce a small increase in working stroke, or diffusion of the MD along the actin filament by one or more subunits which can produce much larger working strokes [20]. Several observations suggest that the initial weak binding actin-myosin interaction changes in structure or orientation on actin during tension development. X-ray diffraction of frog myac i/o error 105 [21], [22] indicates that tension development involves a stabilization of the MD from a disordered actin attachment to an ordered one. A disordered to ordered transition of attached cross-bridges is suggested by electron paramagnetic resonance of spin labelled MDs [23]. However, myac i/o error 105, diffusion of the myosin head along actin as a means to increase the working stroke remains controversial, especially if it is considered that filament movement might require a strongly bound myosin attachment.

Visualizing active cross-bridges, including those bound to actin in the weak binding states thought to precede strongly bound force producing states, is essential for defining the structural transitions that constitute the working stroke of myosin. Because of their low actin affinity and possible heterogeneous structure when attached to actin, weakly-bound states are difficult to trap in vitro in numbers with sufficient homogeneity to be amenable to direct visualization by any of the powerful averaging techniques of cryoEM. However, 3-D visualization can be achieved using the technique of electron tomography myac i/o error 105 which is capable of imaging individual molecules within a highly heterogeneous ensemble [24], myac i/o error 105. ET has produced 3-D images of insect flight muscle (IFM), including the variable conformations of in situ cross-bridges in rigor [25], [26], in a weakly-bound equilibrium state produced by adenylyl-imidodiphosphate (AMPPNP) and ethylene glycol [27], [28] as well as in snapshots of actively contracting IFM fibers [19].

IFM displays two levels of contraction depending on [Ca2 ]. Stretch activation, which is characterized by rapid alternating contractions of antagonist muscles during flight, is the contraction mode most often studied [29]. Stretch activation can be induced in skinned fibers at pCa [30]. IFM also produces sustained isometric contractions at pCa , which we refer to as isometric high static tension or iso-HST, that correspond to an isometric tetanus in vertebrate muscle. In vivo, iso-HST occurs during the thermogenic shivering of preflight warmup [31], when opposing flight muscles contract simultaneously and isometrically to raise the muscle temperature to 40°C where flight can be sustained.

Active myosin heads interact with actin independently of each other so a snapshot of contracting muscle reveals the structure of multiple acto-myosin states within the context of the muscle lattice. Snapshots previously obtained from isometrically activated vertebrate striated muscle revealed a wide range of attachment angles in projections [32]&#x;[34], but these cross-bridges were not visualized in 3-D where detailed interpretation in terms of atomic structure would be possible.

Like the results obtained from vertebrate muscle, iso-HST cross-bridges visualized for the first time by ET also showed a wide range of attachment angles which could be ordered into a sequence compatible with a progressive 13 nm working stroke [19]. Averages computed along axial columns equivalent to the nm long lattice repeat common to both the actin and myosin filaments revealed that actin binding of active cross-bridges was restricted to limited thin filament segments termed actin target zones as previously recognized in rigor [35]. Target zones of IFM are positioned midway between successive regulatory complexes which are composed of the three troponin (Tn) peptides. Subsequently, the distribution and orientation of attached cross-bridges from these same tomograms suggested that, in the absence of filament sliding, the variably angled cross-bridge attachments become locally stabilized in each target zone [36]. This gta 4 terrorist mod in turn suggested that individual tension-generating cross-bridges can cycle with little lacie start.exe error translocation or change in axial lever arm angle.

Here we report a more detailed view of the rich variety of myosin head forms in the iso-HST state resulting from improvements in both data collection and analysis that have increased the resolution by × over the earlier work. The helical arrangement of actin subunits is now resolved, facilitating assignment of particular cross-bridge forms to specific actin subunits within the nm repeat that spans from one Tn complex to the next. Multivariate data analysis (MDA) and classification of 3-D repeats is used to quantify individual cross-bridge forms from the number of repeats within each class [37]. Identification of strong and weak binding cross-bridge forms is greatly improved revealing some novel thin filament attachments not previously detected. This leads to a more sharply defined set of cross-bridge structures and interactions in a tension generating muscle than has been possible previously.

Results

Advancements in data collection and analysis for ET since the iso-HST state was first reported [19] suggested that now is an opportune moment to reexamine this state as a reference for HST specimens subjected to a quick stretch or quick release that are currently under study. The previous data was collected on film with manual adjustment myac i/o error 105 the tilt angle while the specimen was continuously irradiated and had thus suffered significant radiation damage that at the time was unavoidable. Now, myac i/o error 105, automated tilt series data collection that minimizes radiation damage by using charge coupled device cameras, highly accurate motorized goniometers and computer tracking has made routine the collection of tilt series from even frozen hydrated biological material [38]. Although the specimen blocks used for the present study were the same as those used previously, the improvement in detail is striking.

Structure of Reassembled nm Repeats

Structural analysis of active muscle is a challenge because of the presence of different structural and kinetic states of the myosin. The major analytical problem is the identification and grouping of self-similar structures so that averages with improved signal-to-noise ratio can be computed for comparison to much higher resolution structures obtained by crystallography and cryoelectron microscopy. The tomogram encompassed a myac layer, which is a 25&#x;30 nm thick longitudinal section containing alternating myosin and actin filaments, nm square containing 23 thin filaments and from which repeat subvolumes containing a complete nm axial repeat (hereafter referred to simply as repeats) were obtained. Our procedures used the thin filament centered on the target zone as a common frame of reference for alignment. To solve the problem of identifying groups of self-similar cross-bridge forms within the heterogeneous ensemble, we used MDA and classification of 3-D repeats and applied this procedure 12 separate times each focusing on separate critical regions of the structure. Averaged images obtained from the classification steps are referred to as class averages; those repeats that form the class are referred to as class members. Two applications of MDA focused on the left and right sides of the thin filament; averages derived from this we refer to as primary class averages. All of the structures in and around the target zone described in detail here were obtained from these two applications. Four more MDA applications identified cross-bridges in the region of Tn (dubbed Tn-bridges ), four others were used to enumerate myosin head attachments on the eight actin subunits bracketing the target zone. Two more applications were used to verify the lever arm placements of primary class averages as well as to estimate the uncertainty in this critical parameter of cross-bridge structure. The approach is described in detail in a separate myac i/o error 105 [37] and briefly here in the Methods.

Column averages in the previous work had an axial resolution of nm which is insufficient to reveal the helix of F-actin subunits [19]; here the global average (Fig. 1Y) and all of the reassembled repeats show a zig-zag pattern of density characteristic of F-actin subunits indicating a resolution 5 nm [37]. Significantly, we can now fit into the reconstruction an atomic model of the thin filament independent of any cross-bridge binding. Symmetrically placed about the target zone at the ends of the repeats are densities corresponding to Tn which myac i/o error 105 a nm spacing on one side and nm spacing on the other as predicted by the actin helical symmetry (Fig. 1Y).

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Figure 1

Gallery of reassembled averaged repeats.

Each frame is reassembled from left side, right side and Tn-bridge class averages and corresponds to one individual raw repeat whose number is in the upper right hand corner. Circled numbers indicate repeats that are shown at higher resolution in Figure 3. Strong binding cross-bridges (red) and weak binding cross-bridges (magenta) were assigned from the quasiatomic model building. Each variant of primary and Tn-bridge class averages is represented at least once in this gallery. Some are by necessity represented more than once. (A&#x;F) Predominately single-headed bridges, (G&#x;L) mainly 2-headed cross-bridges, (M&#x;R) contains mask motifs and (S&#x;X) Tn-bridges. (Y) This panel shows the central section (left) of the global average after subvolume alignment, surface view (middle) and rotated surface view to reveal the spacing of Tn densities shown by the blue and red lines. Panel Y taken from reference [37].

The multiple different classification steps necessitated a reassembly procedure in which different class averages were combined to make a high signal-to-noise version of each raw repeat. Many different kinds and groupings of cross-bridges were found in the reassembled repeats (Fig. 1). These include 1-headed cross-bridges with lever arms in many different orientations (Fig. 1A&#x;F, J&#x;X), 2-headed cross-bridges, identifiable by the broad mass on the thin filament attributable to the two motor domains (Fig. 1D, G&#x;K), and single heads converging on the target zone from two successive crowns on the thick filament (Fig. 1A, C, F, M&#x;T, V, W). This structure is dubbed the mask motif and was first recognized in IFM myac i/o error 105 with AMPPNP [27] and later in iso-HST [19].

Distribution of Myosin Heads along the Thin Filament

We used MDA to separate the different structures, used the number of class members (raw 3-D repeats) to quantify the numbers of cross-bridges of a particular type and used quasiatomic model building to determine whether the interaction was strong or weak (this process is defined below). This data gives a frequency of forming a particular kind of cross-bridge on a specific F-actin subunit within the averaged repeat. The approach is not error free so to obtain some indication of its accuracy, we compared manual counts of cross-bridges with enumerations based on class membership [37]. The RMS deviation of manual enumeration relative to that predicted using MDA was 16&#x; for cross-bridges on the end of the target zone closest to the Z-disk (Z-ward bridges), which are the more frequent types, and 22&#x; for the less frequent bridges on the M-line end (M-ward bridges). Generally, enumeration by class membership overestimates the number of cross-bridges of a particular type because of false positives. Conversely, it may also completely omit some cross-bridges (false negatives) if their structures are too heterogeneous to form a pattern.

The distribution of actin-bound myosin heads is bimodal (Fig. 2). The majority of heads, 78&#x;, are bound to just four F-actin subunits, H&#x;K, two on each side of the actin filament. These are the target-zone actins. Target-zone actins have myosin heads attached 74±16&#x; of the time. Of these, 71&#x; are strong binding attachments that could be generating force and 29&#x; are weak attachments of two general types (the distinction between weak and strong binding is defined below).

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Figure 2

Distributions of myosin heads bound to specific actin subunits on the thin filament.

Weak attachments are shown on the left, and strong attachments on the right. The two actin long pitch strands are colored runtime error program 1c 7.7 and blue with the two target-zone actin subunits colored darker shades of green and blue. Occupancy on target-zone actins H&#x;K was obtained from membership of primary class averages. Occupancy on actins R, S was determined from membership of Tn-bridge classification, while occupancy of actin D&#x;G and L&#x;O was determined from special classifications designed for these particular actins. Occupancy of target-zone actins H&#x;K is plotted in darker shades of green and blue. Actin subunit designations correspond to the chain names in the coordinate files deposited in the Protein Data Bank, PDB &#x; 2w

The remaining 22&#x; of actin-bound myosin heads are spread over the ten non-target-zone actins for an average frequency of 8±4&#x;, which is to say that 8&#x; of the time these actin subunits have a myosin head bound in some manner. The distribution is not entirely flat but is slightly higher on the two actins on the M-ward side of the target zone (F &#x; G) and on the four actins in the neighborhood of troponin (N, O, myac i/o error 105, R, &#x; S). The frequency of myosin heads bound on the two actins on the Z-ward side of the target zone (L &#x; M) is only &#x. This strikingly low number differs by more than 2Ã from the average of the other eight non-target-zone actins. The low number of heads on actins L and M is especially significant because it appears right next to the target zone, where both strong and weak myosin binding is highest. The average occupancy of the four troponin actins (N&#x;Q) is ±&#x; which is avira error 559 significantly different from the average for all non-target-zone actins. Mostly, because of the low myac i/o error 105 of attachments to non-target-zone actins, D, E, L and M, the number of heads on the four actin subunits near troponin appears as a small peak in the distribution. Actins N and O are the location of rear bridges common in rigor muscle, so these actins can accept strong binding myosin interactions, but in iso-HST, surprisingly none were found.

From the number of total repeats,we calculate a corresponding number of thick filament crowns,and a total number of myosin heads potentially available, Our myosin head counts for all actin attached heads total heads for 53&#x; attached to actin. The number of strongly bound heads is representing 29&#x; of the total available. The proportion of strong binding cross-bridges as a fraction of the total available is consistent with measurements from vertebrate striated muscle during isometric contraction [23], [39]&#x;[41].

Quasiatomic Models of Active Cross-bridges

We built into the global average of all repeats a single atomic model of the thin filament with 28/13 helical symmetry, containing 16 actin subunits, enough tropomyosin (TM) to cover the 16 actins, and four Tn complexes and then used that model for all the reassembled repeats (see Methods). We estimate that the azimuthal fitting precision of the thin filament quasiatomic model is ±6° [37], which for a structure 8 nm in diameter gives a spatial uncertainty of nm on the perimeter, assuming the actin filament behaves as a rigid body. The expected precision of quasiatomic models is d where d is the resolution [42], or nm for 5 nm resolution, which was approximately correct for the myosin lever arms [37].

The location of the head-rod junction of the myosin heads is a particularly important parameter of the model fitting. It was verified by computing separate class averages based on features near the surface of the thick filament backbone, where the lever arms dominate. The raw repeat members of a primary class average were usually distributed over the membership of several thick filament class averages. Several independent fits of the lever arm could then be used to compute an axial and azimuthal standard deviation giving ° or nm for the axial orientation and 9° for the azimuthal angle [37]. If there was any doubt about the location of the head-rod junction, we also compared the quasiatomic models built into the intel c+ + error 10114 class averages with the original raw repeats.

Identification of strong and weak binding myosin heads required an explicit criterion. We made this distinction based on the fitting of the MD into the class averages. Strong binding cross-bridges are those for which the MD fit the density without modification from the strong binding configuration, myac i/o error 105, as defined by rigor acto-S1 [5]. Even when the MD was fit without modification, the lever arm usually required significant change. We minimized the amount of axial lever arm change needed for each fit by using two starting atomic structures for strongly bound myosin heads (see Methods), one with the lever arm up, the scallop transition state structure [10], and the other with it down, the chicken skeletal rigor structure [5].

We identify as weak binding any cross-bridge that required moving the MD from the strong binding position to fit the density. All apparent weak binding cross-bridge forms were fit from the scallop transition state structure whose MD had been prealigned to the Holmes rigor structure. Although myac i/o error 105 were other choices for a transition state structure [9], the scallop transition state structure proved to be a good fit to the weak binding bridges requiring little modification and with the lever arm azimuth nearly identical to that of the Holmes et al, myac i/o error 105. structure when their MDs were aligned.

Although it was usually easy to tell whether or not the MD fit the density well without change from the starting structures, once the MD had to be moved the class averages lacked sufficient detail for unrestricted placement. We therefore adopted some guidelines for weak cross-bridge fitting. The entire starting structure was first moved as a single rigid body to get the closest MD fit possible and then the lever arm was adjusted. Lever arm adjustments were both axial and azimuthal. We permitted both azimuthal and axial MD translations and rotations, but usually azimuthal movements sufficed. There was insufficient definition in the MD density to require axial MD tilt to obtain a fit. The MD C± backbone of weak binding bridges was not permitted to sterically clash with the TM backbone, which was in the high [Ca2 ] closed position [43]. However, strong binding heads inevitably clashed sterically with TM, consistent with their ability to increase activation of the thin filament by moving TM toward the open position [44].

Two other aspects of the model fitting are important. (1) The classification is not delphi getlasterror com port and this affected some classes, in particular the decision as to whether a given cross-bridge is single- or double-headed. Assignment of a bridge class as single- or double-headed was made from examining the class members as described below. A consequence of this is that some single-headed cross-bridge classes contained variable numbers of second heads resulting in average bridge size at a constant contour threshold being larger than that expected for a single head. Likewise, some 2-headed bridge classes had variable numbers of single heads, causing the average 2-headed bridge size to be smaller than expected. (2) The model building was restricted to myosin heads attached to actin. The specimen also contains almost 50&#x; myosin heads not attached to actin, which are located broadly. We expect these to average out, but there is always the possibility that variable amounts error 3 dism density due to unattached heads will expand the class averages and not be fit by the atomic model.

Target-Zone Cross-bridge Forms

We expected and found both strong and weak actin attachments in active contraction. There is little previously published information on the structure of weak actin attachments but strong binding attachments are well characterized from combinations of crystallography with cryoEM [5]. Some weak binding attachments with comparatively small MD displacements appeared to be potential precursors of strong attachments since they presented their actin binding interface near to the myosin binding site on actin. Others required much larger MD movements to fit the density. MDA is predicated on the presence of patterns in the data; single occurrences of any kind are effectively noise. Thus, features that occur in the class averages are not chance encounters but are part of a recurring pattern suggesting a myac i/o error 105 feature of actin-myosin interaction.

We observed six kinds of cross-bridge configurations in the target zone (Figs. 1, 3): mask motifs (Fig. 1A, C, F, M&#x;T, V, W; Fig. 3E, F), 2-headed cross-bridges with both heads strongly bound (Fig. 1D, H, I, K, L; Fig. 3A, myac i/o error 105, C, D), with one strongly and one weakly bound head (Fig. 1G, J), with both weakly bound (Fig. 1I), single headed strong binding (Fig. 1B, myac i/o error 105, D, E, K, L, U, V, X; Fig. 3A, B, D) and single headed weak binding forms of various types (Fig. 1A, J, U, W, X) that are not part of mask motifs. Strongly bound myosin heads were found only on target-zone actins but not all myosin heads attached to the target zone were strong binding.

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Figure 3

Diversity of myosin-actin attachments shown in quasiatomic models of reassembled primary class averages.

The number in the upper right is the number of the corresponding raw repeat, NOT the number of raw repeats averaged within the class. Small panels to the left are the central section and an opaque isodensity surface view of the larger panel without the quasiatomic model. Actin long pitch strands are cyan and green with the target-zone actins in darker shades, TM is yellow and Tn orange. Strongly bound myosin heads are red, weak binding myosin heads are magenta, The essential light chain is dark blue and the regulatory light chain light blue. (A) shows a single headed cross-bridge on the left and a 2-headed, strong binding cross-bridge on the right. (B) shows a pair of 1-headed, strong-binding cross-bridges on actin subunits H and I. (C &#x; D) have a 2-headed cross-bridge on the left and a 1-headed cross-bridge on the right, myac i/o error 105, all strongly bound to actin. (E &#x; F) are mask motifs with Tn-bridges. In (E) the right side M-ward weak binding cross-bridge is bound outside of the target zone to TM near actin subunit F while the one on the left is within the target zone on actin subunit I. In (F), the weak binding, left-side, M-ward cross-bridge is bound outside the target zone to TM near actin subunit G while the weak binding cross-bridge on the right is bound to target-zone actin subunit H. Tn-bridges have not been fit with a myosin head. These six reassembled repeats can also be viewed in Supporting Movies S1, S2, S3, S4, S5 and S6.

Mask Motifs

Mask motifs are a common structural feature in non-rigor IFM. iso-HST mask motifs were interpreted as consisting of a strong binding head pair on the Z-ward side and a weak binding head pair on the M-ward side [19]. The two heads originate from successive thick filament crowns spaced nm apart.

M-ward bridges of mask motifs are positioned on actin subunits F&#x;I (Fig. 3E, F). In the previous work all M-ward members of mask motifs were thought to be precursors of strong binding cross-bridges [19]. However, the present work indicates that strong binding attachments only occur on the four target-zone actins; M-ward bridges on non-target-zone actins F and G apparently must move to a target-zone actin to bind strongly. This they might accomplish by detaching and reattaching as the target zone moves toward them during filament sliding.

2-Headed Cross-bridges

Some class averages appeared to be 2-headed, and when identified, were confirmed by examining galleries of the class members. If the majority of the class members were 2-headed, then the class was identified as 2-headed; if not, then it was identified as single headed. Although 2-headed cross-bridges are common in rigor muscle, they have not been quantified this accurately in active contraction, myac i/o error 105. Two-headed bridges were only found in the target zone. Of the heads in the target zones, of them ( 29&#x;) were in 2-headed bridges. Of the 2-headed cross-bridges, had both heads strongly attached (Fig. 1D, H, I-left side, K, L), 55 had one strongly and one weakly bound head (Fig. 1G, J), and 26 had both heads weakly bound (Fig. 1I - right side).

The 2-headed cross-bridges of rigor usually have one rigor-like head with the second head's lever arm closer to 90° and this was true for one 2-headed cross-bridge class of active contraction, i.e. (Fig. 1I - left side). However, most iso-HST 2-headed cross-bridges do not resemble those of rigor and have both heads with an anti-rigor orientation (Fig. 1D, G&#x;I, myac i/o error 105, L). There is, thus, myac i/o error 105, no rigor-like pattern to the 2-headed cross-bridges of contracting muscle.

Lever Arm Angle Distribution for Strong Binding Myosin Heads

To determine the lever arm axial and azimuthal angles, we used heavy chain residues and in the Holmes et al. S1 structure, or the corresponding residues, andmyac i/o error 105, in the scallop transition state structure to define the lever arm axis. The angle between this vector and the thin filament axis defines the axial angle with angles 90° being rigor like and angles 90° being antirigor-like. In this convention, the axial lever arm angle of the Holmes S1 structure is ° and of the scallop transition state structure °. When myac i/o error 105 strong binding heads are transformed to a single actin subunit, the lever arm positions sweep out an arc with an axial range of 77° and a distance of nm at the S1&#x;S2 junction (Fig. 4A). These values are more than twice the 36°, nm differences between the two starting atomic structures. The distribution is slightly bimodal for strong binding attachments with a shoulder at ° in addition to the main peak at 90° (Fig. 5A). However, when weak binding attachments are included, the shoulder at ° is enhanced. We believe this is due to the coupling of lever arm axial angles inherent within paired attachments in mask motifs and in 2-headed bridges. More than half of the 2-headed attachments were strong binding pairs, but nearly all mask motif pairs consist of a strong and a weak binding bridge pair (the left side of Figs. 1M and N are mask motifs with strong binding head pairs).

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Figure 4

Range of lever arm positions for strongly bound target-zone cross-bridges.

(A) Ribbon diagrams are shown for only the heavy chains of all quasiatomic models (gold) and both starting myosin head structures (red and magenta) as docked onto actin in the strong binding configuration. (B) Plot of the axial angle vrs the azimuthal angle for the data shown in (A). Azimuthal angle measured looking M-line toward Z-line. (C) Plot of axial coordinate versus azimuthal angle for the same data, myac i/o error 105. M-ward indicates the values obtained from myosin heads bound to the two actin subunits H and I at the M-ward end of the target zone; Z-ward indicates values obtained myac i/o error 105 myosin heads bound to Z-ward actin subunits J and K.

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Figure 5

Histogram of lever arm angles for myosin heads bound only to target zone actins.

Values obtained after transforming myosin heads to a single actin subunit, I. Weak binding cross-bridges are aligned to the MD of the scallop transition state initial model. Vertical red and magenta lines indicate the position of the initial models within this coordinate frame. The positions of the lever arms for the starting atomic structures are error 14001 angry birds as red and magenta vertical lines. (A) Distribution of lever arm tilt angles computed relative to the filament axis. Angles 90° are rigor-like and angles 90° are antirigor-like. The green curve is a Gaussian fit to the data with µ = °, Ã = °. The fit for strong binding heads alone (not shown) is µ = °, Ã = °. (B) Distribution of lever arm azimuths relative to the inter-filament axis. The inset gives the angular convention given a direction of view from M-line toward Z-disk. Red and magenta vertical lines show the azimuths of the starting atomic structures, which are very similar. Clearly, if starting-structure azimuth were the only influence, then the final azimuths in B should center around these vertical lines at °, as indeed the weak-binding target-zone bridges tend to do here. Direct Z-ward views of the unexpected azimuthal skewing observed for strong-binding bridges are shown in Figure 6B&#x;C, in contrast to the starting-structure azimuths for target-zone myosin heads shown in Figure 6A. Figure 10 depicts possible torsional effects that might contribute to the skewing.

We determined the azimuthal angle from the projection of the lever arm vector defined above, onto the equatorial plane of the filament lattice. A lever arm azimuth of 90° would be aligned parallel to the inter-thick-filament axis. The azimuthal angles thus defined are dependent on the actin subunit to which the myosin quasiatomic models are transformed, in this case subunit I, but the angular range is not, myac i/o error 105. Surprisingly, the azimuthal range of all strong binding myosin heads is ioerror errno 13 permission denied ubuntu (Figs. 4B, 5B). There is no correlation between axial tilt and azimuth (Figs. 4B, C). The azimuthal angle myac i/o error 105 is bimodal with M-ward bridges being spread over a 52°&#x;° range (mean 86°±24°) and Z-ward bridges spread over a 16°&#x;73° range (mean 41°±17°). The total spread is unexpectedly wide given that the two starting structures are azimuthally only 4° apart, Holmes rigor, °, scallop transition state, °. Very few of the fitted strong binding myosin heads have lever arms positioned like the starting structures. The departures are not random but systematic. When viewed Z-ward, nearly all are positioned anticlockwise (with respect to the thin filament) from the starting structures.

The bimodal azimuthal angular distribution can be attributed to the 26° difference in azimuth presented by the myosin binding site on actin of the two target-zone subunits on each side of the thin filament as a consequence of its helical structure. When the starting structures are placed on the target-zone actins, their S1&#x;S2 junctions are positioned clockwise from the line that connects the centers of the thick and thin filaments (Fig. 6A). The S1&#x;S2 junctions of the myosin heads on the Z-ward actins J and K are insert into syntax error, unexpected t_string further from this line than those on the M-ward actins H and I. The S1&#x;S2 junctions for all but two of the quasiatomic models for strongly bound heads are positioned anticlockwise from this line. The two exceptions are apparently early working-stroke attachments. The range of positions of the S1&#x;S2 junctions at the thick filament surfaces for heads bound strongly to the M- and Z-ward actin subunits almost completely overlap (Fig. 6B&#x;D) despite the 26° difference in azimuth between their bound actin subunits. Thus, cross-bridges strongly bound to either target-zone actin appear as if they originate only from a restricted region of the thick filament independent of two different actin azimuths. Noteworthy is the fact that the S1&#x;S2 junctions of the starting structures placed on actin subunits F and G fall on the same anticlockwise side of the line as the observed strong-binding attachments even though no strong-binding myac i/o error 105 rotterdam terror + rapidshare found on F and G.

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Figure 6

Models of strong binding bridges superimposed and displayed on their bound actin subunits.

To provide a spatial reference, the models are displayed with the map of the global average. The horizontal myac i/o error 105 line represents the inter-thick-filament axis. All views are looking from the M-line toward the Z-line. (A) Scallop transition state starting model on actin subunits F&#x;K. The S1&#x;S2 junctions are positioned clockwise from the interfilament axis for all starting models on actins H&#x;K. The S1&#x;S2 junctions of starting models on F and G are located anticlockwise from the interfilament axis. However, no strong binding attachments occur on actins F and G. (B) Bridge models strongly bound to M-ward actin subunits H and I. The lever arms of the only two models that fall above the inter-thick-filament axis are bound to actin subunit H and have the appearance of early beginning-working-stroke conformations. (C) Bridge models strongly bound to Z-ward actin subunits J and K. (D) All strong binding models on their bound actin subunits showing azimuthal distribution skewed notably anti-clockwise from hypothetical dispersions centered around starting model positions in A.

Finally, the orientation of the hook of the myosin heavy chain in the starting structures (before rebuilding), which connects the myosin head to the S2 domain, is oriented away from the direction that the lever arm has to be bent to fit the strong binding bridges. This would suggest that the forces bending the lever arm azimuthally in situ, might also cause the lever arm to be twisted, a phenomenon which has been observed spectroscopically [45].

Section compression, which in the present data reduced the inter-thick-filament spacing from 52 nm to 47 nm, may affect both the axial and azimuthal lever arm angles. We estimated the effect of section compression using a simplified model (Fig. 7). We assumed that the thick and thin filaments are incompressible and that most of the compression occurs in the space between filaments, precisely where the lever arms are located. Section compression of the magnitude observed here would broaden the range of azimuthal angles by ±9°. The effect as modeled would be non-linear as lever arms that were more angled with respect to the inter-filament axis would be affected more. For the axial angle, the effect is smaller, and would broaden the distribution by ±6°. If filaments are compressible so that compression is spread uniformly across all structures, the effect on lever arm angle would be less as the space between filaments is a relatively small proportion of the distance separating the filament centers. Thus, accounting for section compression in the worst case scenario would reduce the azimuthal spread from ° to ° and the axial spread from 77° to 65°.

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Figure 7

Schematic model of the effect of section compression on the angle of the lever arm.

Left illustrates the initial state, prior to sectioning and section compression; right side illustrates the effect of section compression. Top row is the view looking down the filament axis; bottom row is the view looking perpendicular to the filament axis, myac i/o error 105. Color scheme has the thick filament red, thin filament magenta, motor domain blue and lever arm black. The region of the target error occurred 0 is colored cyan. We assume a worst case scenario, in which the thick and thin filament as well as the myosin motor domain are unaffected by compression and the entire effect is concentrated on the lever arm. Section compression decreases the interfilament spacing with a corresponding increase in section thickness. Widening of the section is assumed to be minimal since the reconstructions are scaled to the axial periodicities. See text for the values obtained from this model.

Weak Binding Cross-bridges

Many weak binding cross-bridge forms were identified in primary class averages, which revealed cross-bridges attached to actin subunits F&#x;K. We divided weak binding bridges into two types, depending on the displacement of their MD center of mass from that of the strong binding attachments, and on whether they contact actin or TM (Table 1, Table S1). Type 1 weak binding cross-bridges had MD displacements myac i/o error 105 nm from the starting structure and their MD contacted the actin subunit; Type 2 attachments were displaced by &#x; nm and their MD contacted TM rather than actin. All Type 1 attachments were within the target zone; five of seven Type 2 attachments were outside of the target zone on actins F and G with the other two on actin H, the most M-ward target-zone actin subunit. MD displacements show a trend from the largest on actin subunits F and G (closest to the M-line), to smallest on actin subunit K (closest to the Z-line).

Table 1

Summary of weak attachment models fitted in primary mask class averages.

Repeat ## of MembersActin LabelTypeTotal Displacement (nm)
22F2
26F2
15F2
33G2
24G2
19H2
24H2
26H1
27H1
46H1
27I1
7324I1
22I1
32I1
18I1
26J1
16J1
21J1
46J1
34J1
21J1
23K1

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We transformed all weak binding cross-bridge quasiatomic models to align their MDs to the starting scallop S1 atomic structure placed on actin subunit I (Fig. 8A) which places them in the same frame of reference as the strong binding cross-bridges described in Error fraps is detected 111 4 and 5. One weak binding class, visible in panels W and Q in Figure 1, had a lever arm orientation toward rigor. This class is possibly a post-rigor structure and contained 46 members compared with a total of weak binding bridges. For all other weak binding bridges, the azimuthal angle range is 89° to ° (Table 2). The ranges for Types 1 and 2 are only partially overlapping, with Type 2 quasiatomic models being distributed to one side of the starting scallop structure (Fig. 8A) and the Type 1 models being distributed on both sides of it.

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Figure 8

Composite view of weak binding cross-bridge models.

(A) Axial and azimuthal views of all weak binding cross-bridges aligned on the motor domain of the scallop transition state structure. This view illustrates the variations in lever arm compared with the starting scallop S1 structure. All weak binding bridges were built starting from the scallop transition state atomic structure, which is shown as a magenta colored ribbon diagram. Type 1 bridges are shown in gray and Type 2 bridges in gold, both rendered as chain traces. The single post-rigor conformation is colored light brown. (B) All weak binding cross-bridges superimposed on actin subunit I. This view illustrates the variations in MD position when referred to a single actin subunit. Coloring scheme is the same as for panel A. Note the relatively small axial dispersion of the Type 1 MDs compared to the broad dispersion of the Type 2 MDs.

Table 2

Summary of Weak Binding Lever Arm Parameters§.

Crossbridge TypeNumber of classesAxial AngleAxial Displacement (nm)Azimuthal Angle
Scallop Transition State-° °
Holmes Rigor-°&#x;°
Type 11392°&#x;° &#x; 89°&#x;°
Type 1 153°&#x;°
Type 2988°&#x;93° &#x; °&#x;°
All Weak Binding2353°&#x;°&#x;&#x; 89°&#x;°

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The average of the axial angles excluding the post-rigor model, °±10°, is close to that of the starting scallop transition state model, myac i/o error 105, °, and roughly evenly distributed around it. This axial range is less than twice the estimated uncertainty of the quasiatomic models and far less than the range for strong binding heads. Within the confidence limits of the model building, this range represents the inherent flexibility limits between the lever arm and the weakly bound MD.

Type 1 attachments are the most likely candidates for pre-working-stroke forms. When all Type 1 weak cross-bridges are transformed (aligned) onto actin subunit I, as opposed to a strong binding MD on actin subunit I, they suggest a progression in a clockwise delphi 7 einouterror (looking Z-ward) toward the strong binding structure (Fig. 8B). The MD displacements suggested by this alignment are azimuthal in character and would suggest a clockwise azimuthal rotation of the MD (looking Z-ward) to reach the strong binding orientation on actin. A characteristic of the Type 1 cross-bridges is a narrow distribution of MD displacements in the axial direction. A narrow range of lever arm axial angles is present within these models, which is apparently uncorrelated with the azimuthal angles.

When all Type 2 weak binding bridges are aligned to actin subunit I, their MDs are all placed well beyond the azimuthal strong binding position on actin (Fig. 8B) and would require an anticlockwise movement (looking Z-ward) along the thin filament to reach a position where strong binding is possible. In contrast to Type 1 weak binding bridges, there is also a significant axial smear of the Type 2 MDs, indicating a significant difference in the ordering of their interaction with the thin filament compared with Type 1 bridges. Thus, the Type 2 weak binding bridges are a new and fundamentally different type of cross-bridge contact (or interaction).

Discussion

Comparison with Previous Work

Here we combine modifications in both data collection and analysis to achieve richer and finer detail of the variety of myosin head forms in iso-HST than was possible previously [19]. The improvement is most pronounced in the averages which are important for identifying structural variation in the cross-bridges. Column averages as used previously were only effective at filtering patterns of cross-bridges that repeated axially every nm and is limited in resolution to nm, the highest resolution layer line visible in the transform of those tomograms. MDA as used here is effective at identifying different cross-bridge structures regardless of their distribution in the filament lattice. The use of MDA to identify self-similar structures within an ensemble is not limited in resolution by long range order, but is more limited by the number and homogeneity of the structures being averaged.

Previous work utilizing X-ray diffraction and ET concluded that 28&#x;32&#x; of the myosin heads are attached to the target zone for a frequency of heads/target zone [19]. A subsequent analysis myac i/o error 105 of the same data obtained a binding stoichiometry of myosin heads/target zone and heads/actin (assuming 5 actins/target) or heads/actin myac i/o error 105 4 actins/target). The new results indicate myosin heads are bound per target zone and averaging heads per target-zone actin, myac i/o error 105. The two results are not necessarily incompatible. The earlier results were derived myac i/o error 105 on two spatial averaging techniques, myac i/o error 105, X-ray diffraction and column averages of the tomogram, both of which observe only repeating patterns in the structure. The present analysis does not require spatial ordering. Irregularly distributed structures, such as the M-ward bridges of mask motifs identified here by MDA would contribute little to X-ray reflections that are enhanced by target-zone bridges.

Actin Target Zones

The term target zone is defined as the segment of the thin filament by them1x - terror _1 mp3 actin subunits are best oriented to form strong attachments to myosin heads projecting from adjacent, parallel thick filaments. Though defined initially for IFM [35], myac i/o error 105, the concept is entirely general not only for organized filament arrays such as striated muscle [33], [34], [46], [47] but also for in vitro single-molecule experiments using various myosin isoforms [48]&#x;[53]. However, up to now the target-zone size has not been so precisely defined.

Previously, we inferred that the target zone was on average actins long on each long pitch helical strand [19], [36], three actins on one strand and two on the other, alternating sides with successive crossovers. The present result, in which actin subunits and Tn's are resolved, and weak and strong binding attachments are distinguished, shows that the target zone comprises just two subunits on each actin strand, myac i/o error 105, positioned exactly midway between two successive Tn's on that strand. The frequency of all myosin attachments increases 10 to fold on target-zone actins relative to the nearest neighbors. Moreover, strong binding heads disappear abruptly outside of the target zone. The lack of any strong binding cross-bridges on actin subunits F and G, excludes these actins from the target zone where e 01 error epson they were included [36].

We think that the highly restricted target zone cannot be due to the assumptions used to distinguish strong and weak binding attachments. Only Type 2 cross-bridges are found on M-ward actins F and G; even Type 1 attachments are not found on these actin subunits. Moreover, virtually no attachments of any kind were found on Z-ward actins L and M.

The relative orientation of actin myac i/o error 105 with respect to myosin head origins is the most obvious factor defining the target zone [35], [54] but the size of the target zone can vary. In rigor the target zone encompasses eight successive actin subunits along the thin filament genetic helix (actin subunits H&#x;O), i.e. four on each long pitch strand. Two-headed lead cross-bridges (leading toward the M-line) occupy the four M-ward subunits (H&#x;K) and the usually single-headed rear cross-bridges are bound to the most Z-ward pair (N, O) [55]. The extensive deformation of rigor rear bridges and their complete absence in AMPPNP [27], [56], suggests that the high actin affinity of nucleotide-free myosin extends the limit of acceptable actin azimuth.

Our sharply defined target zone implies a restrictive geometrical constraint on myosin head binding myac i/o error 105 is contradicted by the highly variable shape of strong binding cross-bridges, in particular their widely varying azimuthal lever arm orientations. If our two extremes for lever arm azimuth of strong binding myosin heads reflected intrinsic, state independent, myosin head flexibility, some heads should be able to myac i/o error 105 strongly at all but two actins, D and E, while staying connected to the thick filament. Moreover, if actin azimuth alone were the limiting factor, we would expect a more gradual tapering of strong binding attachments from the center of the target zone. This we also do not see.

Type 1 weak binding attachments show a smaller azimuthal myac i/o error 105 arm variation (Figs. 4A, 5B) than the strong binding heads that they evolve toward. However, when the two azimuthal lever arm extremes of Type 1 weak actin attachments are transformed to the position of strong binding motor domains on each actin subunit, some heads should be able to bind strongly at all but four actins, D, E, R, myac i/o error 105, &#x; S, while staying connected to the thick filament. The initial myosin crystal structures sit near one azimuthal extreme of our strong binding attachments (Fig. 5B). When placed on actin in the strong binding configuration, simultaneous myosin-actin attachments can be made only at six actins F&#x;K (Fig. 6A), which includes the entire target zone. It is notable that the lever arm azimuth in this case would exclude actin subunits Myac i/o error 105 and M from strong myosin attachments but our results show that even non-specific, weak attachments are exceptionally low on those subunits.

Based on the two initial crystal structures, we would expect that strong binding bridges would be found on actins F and G, but none are found. Even Type 1 weak binding bridges are not found on actins F and G. This suggests that an additional factor is limiting the target zone which may be the dynamic properties of TM. The Tn complex adds additional actin affinity to TM and holds it relatively securely at the ends of the actin repeat period but motion of TM would be expected to be highest midway between Tn complexes, exactly where the target zone is located. Reconstructions of actin-TM-Tn done by single particle methods revealed the weakest TM density midway between Tn complexes [57] consistent with the idea of high mobility in this region which coincides with the IFM target zone.

Distribution of Actin-Bound Myosin Heads

Our results show that myosin head attachments occur all along the thin filament, although with much lower frequency than the target-zone attachments. Target-zone attachments account for 78&#x; of all attachments but utilize only 28&#x; of the actin subunits. The other 22&#x; of attachments are distributed among the remaining 72&#x; of actin subunits. Many of these attachments are non-specific with respect to the myosin binding site on actin. Some may represent collision complexes, but others, such as the Type 2 weak tasm error messages none passes 1 bridges on actin subunits F and G are both numerous and have a well defined appearance in the averages suggestive of some kind of specific interaction, even if novel. Outside of the target zone, actin subunits are labelled only 8&#x; of the time on average (range of 3&#x; to 13&#x;). Nevertheless, these non-specific attachments occur with enhanced frequency on the 4 actin subunits near Tn and with strikingly low frequency on two actin subunits (L and M) adjacent to the Z-ward side of the target zone.

Initially, we thought actin subunits L and M never had myosin attachments, but a classification designed specifically for this region found some attachments, the density of which was generally poorly defined. Although it is possible that this represents statistical uncertainty, it may also indicate something special about these actin subunits. Non-specific attachments occur everywhere outside the target zone so the especially low frequency here may indicate an adaptation for IFM.

One in every seven subunits of IFM thin filaments is a ubiquinated form of actin named arthrin [58], [59]. Although arthrin is regularly spaced every nm along Drosophila IFM thin filaments [60], its location is myac i/o error 105 to Tn, and not to the target zone [61]. Thus, arthin located at positions L and M is an unlikely explanation.

Limitations on Cross-bridge Attachment

Tregear et al. [36] reanalyzed the earlier data [19] combining column averaged images to define target-zone locations and cross-bridge origins on the thick filament with the raw tomogram to determine the angles and positions of all individual attached cross-bridges. Tregear et al. concluded that the binding probability of cross-bridges to the target zone followed a Gaussian distribution that depended on the axial offset between target-zone center and the shelf of cross-bridge origin (Fig. 9A). Their methods could not discriminate between weak- and strong-binding attachments in and around the target zone and thus deemed the target zone to be larger than found here.

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Figure 9

Probability of target-zone cross-bridge formation as a function of cross-bridge origin.

(A) Data reproduced from Tregear et al. () in which the target zone is assumed to be three actin subunits on one side and two actin subunits on the other. (B) Data from the present study, which includes only target-zone cross-bridges on two actin subunits from each side. Although the Tregear et al. data, which were measured by hand, have an overall Gaussian shape, the present measurements, which are based on quasiatomic model fitting, do not follow a strictly Gaussian distribution. The continuous line is a Gaussian fit with µ &#x; nm and à = nm.

For comparison, we have replotted our fitting data shown in 503 error denwer friednly url. 9B by referring the axial coordinate of all lever arm C-termini (P for Holmes and P for scallop) to the center of the target zone. Data from left and myac i/o error 105 hand sides of the target zone are combined, but referred to the center of the individual side, createcompatiblebitmap error_not_enough_memory sometimes the overall center of both. The shape of the resulting distribution is not Gaussian but when fit to a Gaussian, the average is &#x; nm (angling towards rigor). All of the measured points fall within ±10 nm of the target-zone center and 84&#x; fall within ± nm. If the present data are referred to the overall center of the paired target zones making it comparable to Tregear et al., the distribution would be spread an additional nm on either side which would myac i/o error 105 that all data fall within ± nm. For the Tregear et al. data, 85&#x; of all attached cross-bridges originate within ± nm of the center of the target zone. Myac i/o error 105 present analysis is therefore broadly consistent with Tregear et al. and differs primarily in its details that are better defined due to the higher resolution of the reconstruction.

The Asymmetry of Actin Attachments

Our results show that the number of actin-myosin attachments is relatively symmetric within the target zone, but is very asymmetric just outside the target zone, where the asymmetry in numbers is matched by an asymmetry in structure. The two actins M-ward of the target zone (F and G) are comparatively well occupied, while attachments to actins L and M on the Z-ward side of the target zone are rare. Weak attachments on actin subunits F and G (Type 2) differ the most from strong binding attachments, while those on actin subunit K (Type 1) have the smallest MD displacements from strong binding and are fewer in number.

Muscles are designed to generate tension while shortening and accomplish this by using filaments that are polar within the half sarcomere. Thus, the asymmetry we observe is not unexpected given the filament structures themselves. When a muscle shortens the target zone on the myac i/o error 105 filament moves toward the M-line, myac i/o error 105. On the M-line side of the target zone, on actin subunits F&#x;H, we find the most unusual of the weak binding cross-bridge forms, Type 2. Although considered weak attachments, their structure is well defined in the averages and they are the only attachments found on those actin subunits. While not positioned on actin in a way that readily converts to strong binding, Type 2 attachments are placed so that if they remained stationary while the actin target zone moved M-ward by 1 or 2 actin subunits, they would be positioned to bushwhack the target-zone actins and form Type 1 weak attachments (Supporting Movie S7). If the movement were larger, we can identify no weak attachments that would be in a position to quickly bind the target zone.

Conversely, if the muscle is stretched, which is to say the target zone moves Z-ward, myac i/o error 105, there are literally no myosin heads positioned to bind the target zone. Adaptation to stretch would therefore require a different mechanism for rapidly placing more myosin heads on target-zone actins. It has been suggested that when a muscle is stretched, two-headed attachments increase by recruitment of second heads to single-headed bridges [62]. Our results show that two-headed cross-bridges exist in isometrically contracting muscle even without stretch which shows directly that this mechanism for adapting to stretch is mechanically feasible in active IFM.

The two types of weak binding attachments identified here may play differing roles in the recovery of tension after a quick release. Type 1 attachments are closest to the strong binding configuration and would thus appear to be best suited for contributing to Phase 2 rapid force recovery if the release distance was 5 nm [63], [64]. Lombardi et al. [65] observed that the stiffness attributable to attached bridges was constant up through Phase 2, while Bershitsky et al. [22] found that stiffness of weak binding bridges that were not stereospecifically attached to actin was unchanged during the weak-to-strong transition that brought about stereospecific attachment and force development. Therefore the Type 1 bridges seen here that are not stereospecifically attached could well be capable of contributing to the stiffness attributed by Bershitsky et al. to weak binding bridges, yet transform without stiffness change to stereospecific strong binding force generating bridges during the phase 2 rapid force recovery observed by Lombardi et al. The Type 2 attachments, which are not attached to actin at all but rather contact TM, may instead be responsible for a more delayed Phase 3 tension recovery.

What exactly is holding the Type 2 bridges in place is not visible in the reconstructions. Their position is variable with respect to TM, both azimuthally and axially which argues against a specific interaction. One possibility is the N-terminal extension of the regulatory light chain. The sequence of this extension is known in Drosophila, which has led to the suggestion that it positions myosin heads for attachment to the thin filament [66]. Its validity for Lethocerus will depend on the demonstration of a similar N-terminal extension.

2-Headed Cross-Bridges Are Not Rare in Isometric Contraction

Based on column averages, it was concluded that the great majority of cross-bridge attachments were single-headed [19], and a subsequent analysis of the same iso-HST tomograms drew the same conclusion [36]. The present analysis indicates that 2-headed attachments are in the minority, but are not rare. Approximately 14&#x; of the cross-bridges (all in the target zone) contain both heads of one myosin molecule. For the single-headed cross-bridges, we believe the companion head is either mobile or remains close to the relaxed state docking position on the thick filament shaft producing the residual density of the myosin shelves spaced nm apart. These shelves are prominent in iso-HST but not in rigor where 2-headed binding predominates [55].

Single headed myosin attachments are consistent with a variety of other structural evidence such as X-ray modelling of relaxed insect thick filaments [67] and various experiments on vertebrate muscle, both in situ [32], [68], [69] and in vitro [70], [71]. Conibear et al. have argued on energetic grounds that 2-headed binding is unlikely during active cycling of myosin [44]. However, 2-headed attachments are consistent with several sets of experiments involving single molecule motility [72]. The present result indicates that at least in isometric contraction, two-headed myosin attachments are significant but are not the major cross-bridge form.

The two stage working stroke

Previous analysis of iso-HST tomograms elucidated a working-stroke sequence by arranging all the quasiatomic models of cross-bridge fittings in a sequence based on the coordinate of the C-terminal residue at the S1&#x;S2 junction [19]. The sequencing suggested that the working stroke consisted of two stages, one in which both the MD and the lever arm move from an M-ward tilt toward a Z-ward tilt in Stage 1, with Stage 2 beginning when the MD settled into the strong binding configuration and only the lever arm moved. The present results are not easily compared with that work for several reasons, myac i/o error 105. (1) Cross-bridge features seen earlier had to be spatially repetitive; averages would be computed over heterogeneous structures if this were not true. The present result is not so limited and averages have been computed over structures that are more homogeneous even though irregularly placed in the filament lattice. (2) The independent fitting of a thin filament quasiatomic model into the present maps provides an improved criterion for distinguishing weak from strong binding cross-bridges. (3) The greater detail of the weak binding target-zone miranda icq error and the availability of two crystal structures, one a transition state with the lever arm elevated and the other a rigor structure, reduced the amount by which the starting models had to be altered to fit the density. Thus, myac i/o error 105, previously the motor domain placement of weak binding cross-bridges involved both axial and azimuthal changes in the MD, but the present fittings of weak binding bridges could be fit using only azimuthal MD changes.

In the sequential ordering of structures from weak-to-strong binding [19], the azimuthal range of lever arm positions narrowed as the lever arm moved from the initial stage to the final, rigor-like stage of the working stroke. The present work differs in that a large azimuthal range of the lever arms persists at all stages. Thus, the present work does not support the conclusion that the azimuthal distribution of the lever arms is wide in the weak binding states and narrows as the myosin heads go through the working stroke. The distribution of lever arm azimuths is widest for strongly bound heads and smaller for weakly bound heads. However, the number of different weak binding forms does suggest new details error 021 symbol already defined createobject the transition from weak-to-strong binding.

Stage 1- weak attachments

We characterized two types of weak attachments between myosin heads and the thin filament, with Type 1 attachments being closest in appearance to strong binding cross-bridges and with Type 2 being decidedly different. We think that Type 1 could be pre-working-stroke attachments because their MD position is closest to the strong binding configuration and they occur only within the target zone. Type 1 weak attachments are consistent with the idea that initial binding need not be precise [73] but rather is disordered as observed by cryoEM [74], Electron Paramagnetic Resonance [23], and X-ray fibre diffraction [22]. Random thermal motions would then align the MD into the strong binding configuration possibly generating force [75].

Our class averages at the moment lack the resolution and signal-to-noise ratio to be unambiguous but they suggest the following properties of the weak binding attachment that precedes strong binding, myac i/o error 105. Type 1 cross-bridges can be arranged into a weak-to-strong binding sequence (Fig. 8B), but this would embody mostly azimuthal movements of the MD across the actin surface, toward TM and the strong myosin binding site on actin. The present data lack the resolution to exclude small changes in MD tilt as part of the transition. In this sequence, MD progress toward the strong binding site on actin displays a corresponding but uncorrelated set of comparatively small axial and azimuthal lever arm movements.

With the exception of one Type 1 weak binding cross-bridge, a possible post rigor conformation, all the weak binding forms on actin subunits F&#x;K display a narrow range of axial and azimuthal lever arm orientations when aligned to the MD (Figs. 5B, 8A). The azimuthal lever arm range of the weak binding bridges is much smaller than the range found for the strong binding bridges, myac i/o error 105, and is rather symmetric with respect to the lever arm position of the crystal structures consistent with inherent flexibility. Conversely, for strong binding cross-bridges, the distribution is not only large, but is strongly biased toward one direction, suggestive myac i/o error 105 a genuine characteristic of isometric force generation in situ.

Stage 2 strong binding

The fitting of strong binding cross-bridges generally required considerable axial alteration of the lever arm for both crystal structures, which differ by 36°/ nm compared with the 77°/ nm range observed here, myac i/o error 105. Our observed range is consistent with previous estimates made by comparing crystal structures of myosin catalytic intermediates from different isoforms [8], [10] and is also similar to myac i/o error 105 observed earlier [19]. Section compression as modelled here could broaden the spread by about ±6° leaving myac i/o error 105 maximum working stroke of 10 nm. Despite these factors, our range is smaller than the ° observed by comparison of reconstructions obtained from actin filaments decorated in AMPPNP or ADP with a myosin V construct containing two IQ motifs with bound calmodulin [15].

The average axial lever arm angle obtained from a Gaussian fit to the data in Figure 5, gave a mean angle of 93°±20° for the strong binding cross-bridges, nearly perpendicular to the filament axis and in good agreement with values that were found to fit the meridional X-ray reflection from isometrically contracting vertebrate muscle fibers [76], [77].

A consistent and surprising feature of the strong binding cross-bridges is the large, asymmetric distribution of the lever arm azimuths relative to the starting models. This is a feature of the data that has been extensively checked against different applications of MDA as well as against the original raw tomograms so we believe it to be real. The width of the distribution is affected by section compression, but even accounting for an additional 9° at either extreme, still gives a range of °. These azimuthal lever arm changes relative to the two initial structures gives the cross-bridges a more straightened appearance.

Cross-bridges with a straightened appearance have been observed previously in situ. They were first reported in EMs of IFM following AMPPNP treatment [78], and in EMs of quick-frozen, contracting vertebrate muscle [32], myac i/o error 105, [33]. IFM reconstructions of rigor [25], [26] and following AMPPNP treatment [27], [28], myac i/o error 105, [56] also show this effect. Thus, we think that this feature of the cross-bridge structure is not unprecedented. In the previous IFM work, it was uncertain whether the observation was an isoform difference rm cannot remove input/output error was revealing a novel feature of in situ cross-bridges, and there was no control group in the reconstructions in the form of unattached or weakly attached heads for comparison. That this observation is not a species myac i/o error 105 is supported by the recent 3-D reconstruction of actin filaments decorated with IFM myosin S1, which have some small differences, but largely are very similar to rigor acto-S1 from other isoforms of myosin II [79]. The weak binding bridges in the present reconstruction are a control group. Their smaller and more symmetrical distribution about the initial models suggests that for weak binding bridges we are observing flexibility and for strong binding bridges we are observing an aspect of the working stroke of myosin heads functioning in situ.

Correlations with Php parse error syntax error, unexpected Fiber Diffraction

Correlated changes in the actin myac i/o error 105 line intensities with characteristic myosin reflections using an applied jump in temperature during isometric contraction [21], [22] show that tension increased with the rise in temperature indicating an increase in strong binding cross-bridges. However, fiber stiffness and the intensity of the 1,1 equatorial reflection did not change, indicating constancy of numbers of actin attached cross-bridges. Moreover, the intensity of the nm layer line, which measures stereospecific attachment to actin rose, while the intensity of the 1,0 reflection, which measures mass ordered on the thick filament decreased. These changes were interpreted to indicate that the structure of attached cross-bridges changed and that these changes are largely azimuthal with respect to both thick and thin filaments. An increase in stereospecific actin-myosin attachments would increase the nm actin layer line as observed, but cannot be accounted for if the myosin head attaches weakly to actin in the stereospecific orientation in which case strong binding simply closes the actin binding cleft, myac i/o error 105. The observed increase requires a more dramatic change in disposition of the MD on actin during the weak-to-strong transition, as observed here.

In vertebrate striated muscle, the intensity ratio of the 1,0 and 1,1 equatorial reflections is a measure of the change in myosin heads moving from thick filaments to thin myac i/o error 105 and these changes normally correlate with tension generation in isometric contractions [80], [81]. Equatorial X-ray diagrams at low ionic strength (µ = 50 mM), which enhances the population of weakly attached cross-bridges even in the relaxed state to nearly 80&#x; of those available, revealed little change in the 1,1 equatorial reflection but a large decrease in the 1,0 equatorial intensity when Ca2 activated, indicative of movement of mass away from the thick filament [82]. This observation was attributed to a pronounced structural change in the already attached cross-bridges; this could not be explained if the MD of weak binding bridges were already oriented as in rigor but myac i/o error 105 the actin binding cleft open, needing only to be closed for strong binding, a structural change too limited to account for the X-ray change.

Meaning of the Azimuthal Changes in Strong Binding Cross-bridges

Accepting that our skewed azimuthal lever arm distribution is a property of active cross-bridges, the obvious question is what does it mean for myosin function in situ? Is it a reflection of intrinsic myosin head flexibility, is it an aspect of the weak-to-strong transition, or is it an aspect of the myosin working stroke? The broad lever arm azimuthal angular range seen for strong binding bridges is a conundrum. Placement of strong binding cross-bridges on actin can be limited only if the myosin head is given limited flexibility as suggested by the two crystal structures used for the fitting, myac i/o error 105. Yet the changes observed in the strong (and weak) binding cross-bridges compared with the crystal structures imply substantial azimuthal flexibility that, if intrinsic to myosin heads, would permit strong binding bridges to form virtually anywhere along the actin nm repeat. Even in rigor, where actin affinity might increase the target-zone size, cross-bridges are still largely confined to the target zone of contracting muscle. Intrinsic myosin head flexibility would seem to be an insufficient explanation for the strongly biased azimuthal change.

If the azimuthal lever arm distribution for strong binding cross-bridges were an aspect of the working stroke, we might expect to see a relationship between axial angle, representing progress through the working stroke, with increasing, or decreasing azimuthal angle. However, graphs of axial tilt angle versus azimuthal angle for the strong binding cross-bridges fail to show the obvious correlation (Fig. 4B, C) expected if the two motions were coupled.

That there is no coupling between axial and azimuthal lever arm angles may be an effect of S2. Cross-bridges do not emerge from the thick filament backbone at the S1&#x;S2 junction; they originate where S2 emerges from the thick filament backbone. S2 is widely thought to provide a flexible tether that can bend radially, as well as azimuthally around the thick filament surface, to facilitate actin-myosin attachment. If S2 swings azimuthally during cross-bridge attachment so that it becomes angled with respect to the filament axis when force is initiated down the filament axis, that force will have both an azimuthal component (a torque) as well an axial component. If the Myac i/o error 105 swing is anticlockwise (looking Z-wards), then the angled S2 would produce a torque that bends the lever arm clockwise with respect to the thick filament. In this case, the torque would straighten the myosin head compared with the starting crystal structures, in accord with our observation. If the S2 swing was in the opposite direction (clockwise), the lever arm would bend anticlockwise with respect to the thick filament thereby making the myosin head more bent than the already bent S1 crystal structures contrary to our observation.

In iso-HST, we do not observe where S2 emerges from the thick filament backbone, we only see the S1&#x;S2 junction of the lever arm and so cannot directly evaluate this effect, myac i/o error 105. Where S2 has been observed in swollen IFM fibers, the range of directions suggests that S2 swings equally well both clockwise and anticlockwise, about the thick filament surface [26]. A symmetrical distribution of angles for S2 with respect to the filament axis at the beginning of force production might explain the lack of coupling between axial and azimuthal lever arm angles of strong binding bridges, but it does not explain the strong directional bias (Fig. 4).

We can think of two mechanisms that would bias the azimuthal bend of the lever arm to produce a cross-bridge that appears azimuthally straightened in comparison to the crystal structures. One of these involves the weak-to-strong transition, the other invokes an active azimuthal component to the working stroke. Either mechanism would enable the S1&#x;S2 junction to be positioned anticlockwise of the inter-filament axis as depicted in Figures 6 and

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Figure 10

Two mechanisms to account for azimuthal skewing of lever arms of strongly bound cross-bridges.

(A&#x;C) Conversion from weak-to-strong binding according to Scenario 1. (D&#x;F) Active azimuthal component to the working stroke. View direction is M-line toward Z-line. Myosin is colored either red (strong binding) or magenta (weak binding), myac i/o error 105. Actin subunits are green and blue. Three successive levels of S2 origins are box error 1020 in shades of brown that darken with distance from the observer. The lever arm is the line originating on the red (or magenta) MD while S2 is shown as a short segment when oriented nearly parallel with the filament axis and becomes longer when angled with respect to the filament axis, myac i/o error 105. The horizontal line is the inter-filament axis. Arrows show the direction of the torques (not their magnitude) produced during the weak-to-strong transition or as a component of force generation and filament sliding. The direction of thin filament movement during sarcomere shortening is toward the observer. (A &#x; D) Initial weak binding is shown, which in (A) begins away from the strong binding orientation and in (D) begins in the strong binding orientation but with actin binding cleft open. (B) Conversion to strong binding involves diffusion of the MD clockwise on actin which swings the S2 anticlockwise about the thick filament. (C) Force production realigns S2 with the filament axis while bending the lever arm azimuthally. (E) Transition from weak to strong binding involves no change in myosin orientation on actin, just a closing of the actin binding cleft. The lever arm in (D &#x; E) is in the same orientation suggested by the crystal structures. (F) An azimuthal component to the working stroke moves the lever arm clockwise around the thick filament. This figure can be seen as an animated sequence in Supporting File S1.

The weak-to-strong transition

The range of structures of Type 1 cross-bridges (Fig. 8B) suggests that the weak-to-strong transition involves a concerted azimuthal translation and rotation of the MD in a clockwise direction (looking Z-wards) about the thin filament (this direction is also toward TM). Two scenarios that involve different compliance between the lever arm and the S2 may serve to illustrate.

In the first scenario (Supporting Movie S8, Fig. 10A&#x;C, Fig. S1), the myosin head is assumed initially noncompliant and so does not change as it moves azimuthally across the target-zone actin during the weak-to-strong binding transition, myac i/o error 105. Instead, S2 is assumed compliant and begins this transition aligned parallel with the filament axis (Fig. 10A), but the myosin head movement swings S2 azimuthally (anticlockwise about the thick filament looking Z-ward) angling it with respect to the filament axis (Fig. 10B). When the working stroke applies axial force to the lever arm, the angled S2 will produce a torque that will pull the lever arm clockwise with respect to the thick filament as S2 aligns with the filament axis (Fig. 10C). The myosin head at the end has become straightened.

In the second scenario (Supporting Movie S9; Fig. 10A,C), S2 is non-compliant and so resists being swung azimuthally during the weak-to-strong transition. Instead, the lever arm of the myosin head is compliant and bends azimuthally clockwise (looking Z-wards) with respect to the thick android drm client error during the transition. Essentially, the transition would go from Figure 10A to Figure 10C, myac i/o error 105, bypassing the state depicted in Figure 10B. When strong binding occurs and the working stroke is initiated the force is applied axially with a minimal torque but the lever arm is already bent azimuthally.

The first scenario is supported by the two strong binding structures which fell above the inter-filament axis in Fig. 6B and could be interpreted as very early working-stroke attachments. The second scenario is supported by the lever arm azimuthal distribution of Type 1 weak binding bridges which are somewhat straightened azimuthally, but the amount is small compared to strong binding cross-bridges. Thus, there are structures within the ensemble to support either scenario.

The first scenario bears some resemblance to the roll and lock mechanism described by Ferenczi et al. [83]. This detailed model incorporates the concept that myosin in kinetic states with ATP or ADP&#x;Pi bound make variable attachments to actin that differ not only in the azimuthal position on actin, as suggested by the present data, but also with variations in axial orientation as suggested by earlier work on the I-HST state of IFM [19]. The present work shows highly variable attachments all along the thin filament, but only weak attachments within the target zone (Type 1) are likely candidates to transition to strong binding. The variation in their orientation on actin of this group is much smaller than envisioned in roll and lock . In addition, the present work does not require MD tilt in the Type 1 weak attachments, but we think the data at the moment do not definitively rule out some MD tilt as part of the weak to strong transition, myac i/o error 105. Whether Type 2 attachments can roll on actin is doubtful because their MDs do not appear to attach actin at all. Nevertheless, the present data support at least some aspects of the roll and lock model for the weak to strong transition.

An azimuthal component to the myac i/o error 105 stroke

Several observations made with in vitro motility assays have shown a torque component to the working stroke, dubbed twirling , inferred from rotations of the actin filament in gliding assays [84], [85]. Twirling as generally observed with myosin II involves a left-handed actin filament rotation during filament sliding. With an active azimuthal component to the working stroke, azimuthal diffusion in the weak-to-strong transition described above is not required. Initial weak binding could place the myosin head on actin oriented as in strong binding but with the actin binding cleft opened (Fig. 10D) and the lever arm azimuth positioned as in the crystal structures. The weak-to-strong transition would simply involve closure of the actin binding cleft without a change in orientation of the MD (Fig. 10E). If the working stroke involved an inherent clockwise rotation of the lever arm about the thick filament when looking Z-wards, the S1&#x;S2 junction would move clockwise about the thick filament (Fig. 10F) and the cross-bridge would be straightened. A consequence of this mechanism is that the myosin origin (i.e. where S2 emerges from the thick filament backbone) could be positioned initially above the inter-filament axis (Fig. 10D), where the crystal structures predict it should be, and the S1&#x;S2 junction would, after tension developed, appear below the inter-filament axis where we observe it in situ.

The azimuthal movement of the lever arm under this mechanism would not only straighten the cross-bridge but would also impose a clockwise torque on both the thick and thin filament, opposite the one imposed by the weak-to-strong transition. If filament sliding was possible, the applied torque would impose a left handed rotation to the thin filament, myac i/o error 105, matching the observation in vitro [84]. For myosin II, Beausang et al. observed a screw pitch of nm, myac i/o error 105, about half the length of thin filament observed in our tomogram. This is large enough that a rotation or change in twist microsoft visual basic run-time error 13 excel the actin filament should be visible in our reconstructions if it occurred, but it is not.

Generally, myac i/o error 105, motility assays are performed with the motor bound to a substrate via the rod domain (S2 plus LMM), which would argue that the S1&#x;S2 junction is comparatively immobile and the actin filament free to move during these assays. If the actin filament were fixed as it appears to be in IFM and the S1&#x;S2 junction free to move on its S2 tether, then the same conformational change in the myosin head that causes left hand twirling of free F-actin would rotate the lever arm anticlockwise or right handed with respect to the thin filament to produce myosin head straightening as observed.

Beausang et al. [84] indicated that the visibility of actin filament rotation in motility assays depends on the number of rigor bound heads providing a drag force to slow filament movement. In isometrically contracting muscle, not only are there many strongly bound heads, but there myac i/o error 105 an external load to prevent the fibers from shortening. The effect of a torque on the actin filament as a component of the working stroke would by inference be most visible in isometric contraction.

Effects on the Thick and Thin Filaments

Forces or components of forces that alter the lever arm azimuth must also impose torsional forces on the thick and thin filaments. Neither the thin filaments, anchored at the Z-disk nor the thick filaments, which are bipolar, myac i/o error 105, can rotate as rigid bodies; they can only change their helical twist in response to an applied torque. For the thin filaments this is not as absolute as for thick filaments since it depends on the compliance of actin crosslinks in the Z-disk. The rotation of the MD on actin during the weak-to-strong transition produces an anticlockwise torque (looking Z-wards) on both the thick and thin filaments (Fig. 10A&#x;C). Myac i/o error 105 torque fuckmegirl error invalid search type opposite effects on the thick and thin filaments because their fixed points are at opposite ends of the half sarcomere. An anticlockwise torque on the thick filament during the weak-to-strong transition, would unwind a right handed helix, but the same torque on the actin filament would over wind a right handed helix. Conversely, hp support error 21 10 active azimuthal component to the working stroke would apply a clockwise torque to the thick filaments which would over wind a right handed helix and if applied to the thin filament unwind a right handed helix.

There is no evidence in the present tomograms or in any published data from IFM of a change in the half pitch of the thin filament which would be a necessary consequence of any torsional rotations in situ. The thin filament azimuth in our tomograms of active contraction appears virtually identical to that found in other states of IFM investigated by electron microscopy alone [86], [87] and by ET [27], [28]. If a change in thin filament pitch occurred in the present data, the repeat alignment scheme, which utilized only ° azimuthal rotations, myac i/o error 105, would have obliterated the Tn density which is congruent with the half pitch, whereas in fact, the alignment scheme if anything enhanced it. Lack of evidence that the thin filament in IFM rotates as a rigid body or undergoes changes in helical structure does not mean that the thin filament must be unmodified by strong binding myosin myac i/o error 105. It only means that any changes appear to be local and not global.

There is some evidence from X-ray diffraction of helical changes in the thin filament of vertebrate striated muscle during isometric contraction [88], [89]. The changes are much smaller than suggested by the present data. Another report describes helical changes in the thin filament in low tension rigor [90] and suggests that these changes are the result of canon mp210 error code 16 binding itself causing local distortions that are measured on average over the entire filament. No publications have myac i/o error 105 changes in helical twist of the thick filament during isometric contraction although changes in axial spacings in vertebrate striated muscle are widely known [88], [89].

Perhaps the lack of visible effects on the helical structure of the filaments is due to the differing sense of the two hypothesized torques; the weak to strong transition produces a clockwise torque, twirling results from an anticlockwise torque. More likely the thick and thin filaments are too stiff to be significantly altered by these forces and most of the torsional force is dissipated by S2 and the myosin lever arm.

Conclusion

We have shown multiple myosin head structures in isometrically contracting muscle consistent with both weak binding and force producing cross-bridges. We can infer from these structures that the weak to strong transition involves largely azimuthal movements of the myosin MD and consequent changes in the lever arm. This agrees with X-ray diffraction observations of the weak to strong transition in which changes in intensity occur largely on the nm actin layer line which is myac i/o error 105 to increases in stereospecific binding, on the equatorial reflections which are sensitive to azimuthal and radial changes in structure and with minimal changes in the nm meridional reflection, which is sensitive to lever arm or MD axial orientation [22]. Both our weak binding and strong binding bridges have a symmetrical lever arm axial angle distribution with a mean near 90° while the MD orientation differences are largely azimuthal, qualities that would translate to little change in the meridional intensity and large changes in the nm actin layer line as tension increases. The 1,1 reflection which measures myosin attachment to actin could remain roughly constant but the large azimuthal changes in the lever would affect the 1,0 intensity. Our strongly biased azimuthal lever arm distribution can be explained by an azimuthal movement of the MD combined with some azimuthal component to the working stroke. That the transition from weak to strong binding is largely azimuthal makes it possible for cross-bridges to cycle in place evil of pain-skeptik terror mix little change in axial angle when converting from weak to strong binding [36].

Materials and Methods

Rapid Freezing and Freeze Substitution

Rapid freezing with simultaneous monitoring of fiber tension and stiffness was performed on a Heuser Cryopress freezing head [33]. Specific modifications made to the freezing head for this work have been described in detail as have specifics of the specimen manipulation prior to and subsequent to freezing [19], [91].

Electron Tomography

Details of all the data analysis procedures have been described [37] but a brief summary is given here. Two tilt series covering ±72° at 90° relative orientation were recorded from half sarcomeres on a FEI CMFEG electron microscope using a Gatan Model High Tilt Analytical Holder and a TVIPS F 2k×2k charge coupled device camera. Each tilt series consisting of images was recorded from regions where longitudinal thin sections contained single myosin and actin filament (myac) layers [35]. Tilt angles within each tilt series were calculated according to the Saxton scheme [92]. The two tilt series were first independently aligned using marker-free alignment [93] and then merged by patch correlation and volume warp using IMOD [94] to produce the raw, dual axis tomogram. The pixel size was internally calibrated using the axial nm period. Although we collected and merged two dual-axis raw tomograms, only one contained an appreciable number,of well centered repeats.

Repeat Subvolume Processing

Repeats spaced nm apart axially and containing a nm axial length of the actin filaments, their bound cross-bridges and adjacent thick filament segments were centered on the actin target zones. Alignment error between the raw repeats was minimized by choosing as the reference the single large structure, the thin filament, which was in common to all repeats, myac i/o error 105. A global average of all extracted raw repeats was used as the initial reference and was followed by MDA and multireference alignment. After several cycles of multireference alignment, the final alignment used a single reference as it was desirable to fit one atomic model to the thin filament for all class averages and raw repeats that would be used for all subsequent model building.

Multivariate Data Analysis and Repeat Reassembly

The purpose of MDA is to sort the highly variable cross-bridge forms within the repeats into self-similar groups. Myac i/o error 105 key part of MDA is the generation of Boolean masks, which select a set of contiguous voxels defining a region of the repeat within which patterns of density will be identified. To retain the greatest variation in structure possible, we generated several classification masks. Two of these selected myosin heads bound to either the left or right side of the actin target zones; these are the primary class averages. Four masks were used for the troponin region, four masks were specific for actins outside of the target zone and two were used for the surface of the thick filament to verify the lever arm positions. Class averages from each classification were subsequently reassembled to make composite class averages dshownet error using wmv9 preview. The classification clusters repeats according to the features within the specific mask, but averaging was always carried out using the entire repeat.

Because of the complex manner in which the individual repeats are reassembled, conventional methods of resolution determination, such as the Fourier Shell Correlation are meaningless. However, a qualitative measure of the resolution can be obtained by the fact that the helix of actin subunits can be observed, which requires at least nm resolution, myac i/o error 105. The resolution of the previous work was limited to 9 nm so the improvement is at least a factor of 2&#x;3.

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