Supporting Material Background: smoothing of correlated and step-like data:

Transcription

1 Supporing Maerial Seps and bumps: precision exracion of discree saes of molecular Machines Max Andrew Lile, Bradley C Seel, Fan Bai, Yoshiyuki Sowa, homas Bilyard, David M Mueller, Richard M Berry, and Nick S Jones Background: smoohing of correlaed and sep-like daa: Sep-smoohing algorihms remove noise from sep-like signals ha is, signals for which he underlying, noise-free ime race o be recovered consiss of consan segmens wih sep changes beween segmens. he lieraure on sep-smoohing is large (-) bu approaches can be roughly grouped according o how hey process he daa. Sliding window mehods (such as he running mean or running median filer) consider a small se of coniguous samples and replace he cenral sample in ha window wih he oupu of an algorihm applied o he samples only in he window. he window is moved along he ime series by one sample a a ime so ha each sample in he series is replaced. Recursive mehods operae by successive subdivision or merging of consan regions. For example he op-down algorihm described in Kalafu e al. (7) sars by finding he locaion of he bes single sep for he whole signal. If a sopping crierion is no me, a bes sep locaion is found wihin each consan region eiher side of his sep. his successive subdivision is repeaed unil he sopping crierion is reached. Conversely, boom-up algorihms sar wih every sample as a puaive consan region, and merge regions successively unil a sopping crierion is reached. Finally, global mehods consider all he samples simulaneously, eiher fiing a model o he ime series or applying some kind of ransformaion. he range of algorihms is very diverse. For example, Haar-wavele de-noising (9) decomposes he signal ino a sum of sep-like waveforms and he noisy deails ha fall below a given ampliude are removed from he sum. Hidden Markov models (HMMs) rea he underlying consan regions as unobserved saes, obscured by noise, following each oher in an unknown sequence, and aemp o find he sae ransiion probabiliies, observaion noise probabiliies, and mos likely sequence of saes from he signal (6). All sep-smoohing mehods have advanages and disadvanages relaive o each oher, mosly consequences of he mahemaical assumpions embodied in he algorihm. Mos have algorihm parameers ha mus be se and he resuls will depend on he choice of hose parameers. Wihou any guiding heoreical principles, here is always some unavoidable level of subjeciviy in choosing parameer values. he goal should always be o minimize he number of parameers, subjec o he necessary degree of flexibiliy deermined by he naural variabiliy of he phenomenon under sudy oo few parameers can be as much as a problem as oo many. Issues due o algorihmic srucure: Sliding window mehods canno generally find consan regions larger han he window size; he window size herefore becomes a criical parameer. Recursive mehods can find long consan regions, bu require specificaion of he sopping crierion, and his may or may no be appropriae for he underlying race which is of course unobservable in principle. For example, he op-down mehod described above (7) produces differen resuls on differen-lengh signals, an uninended side-effec of he mahemaical formulaion of he sopping crierion. Global mehods do no suffer from such problems bu, for example, wavele de-noising requires he selecion of noise ampliude which is criical. HMMs are an alernaive, bu require a-priori choice of he number of saes, and he number of free parameers increases wih square of he number of saes. his can cause considerable uncerainy in he reproducibiliy of he resuls for experimenal daa no used o rain he HMM. Issues due o free parameers: he majoriy of sep-smoohing algorihms make he (explici or implici) assumpion ha he observed noise is independen or specifically, uncorrelaed. However, general molecular moion occurs in poenial wells cenred on each discree sae wih correlaed, random noise, and his ype of moion is ofen modelled by Langevin dynamics which incorporaes general ime lag in he moion caused by elasic and fricion forces () (for example, drag caused by a bead aached o he elasic flagellar hook (3)); he ransiions beween saes are hen smooh. Sep-smoohing under hese circumsances is quie differen from he sep-smoohing problem as invesigaed in he exising lieraure.

2 Issues due o compuaional burden: Addiionally, many sep-smoohing echniques are compuaionally onerous in pracice, eiher because he number of inensive calculaions scales poorly wih he lengh of he signal, or because here is no way of knowing wheher a soluion consiues he opimal se of seps oher han by exhausive (brue-force) compuaion. For example, mos recursive mehods require an exhausive search for he opimal sep locaion a each ieraion (, 7), and, alhough here are some incidenal compuaional efficiencies ha can be exploied, for his reason such algorihms are inrinsically slow. For he HMM, here are no known algorihms ha can evaluae he probabiliies in heir enirey wihou infeasible exhausive compuaion so he bes one can achieve is a soluion ha may be sub-opimal. Given hese consideraions, we did no find any exising sep-smoohing mehods o be enirely saisfacory, and indeed our experimens on simulaed daa bear his ou (see ables in he main ex and Supporing Maerial). Background: finding modes (bumps) in disribuions: Mode-finding (also known as bump-huning) is he aciviy of finding peak values of a disribuion of a random variable. ypically he random variable represens ime series of discree saes of a molecular complex or machine. If he saes are disinc, hen, in heory, he disribuion will exhibi a clear se of spikes locaed a each unique sae (see Figures c, 4b, 6b in main ex). hese separae saes can hen be idenified from he disribuion, and his requires an esimae of he disribuion of he ime series. erhaps he simples and mos immediaely accessible esimae is he hisogram: divide he full range of he random variable ino equal-sized bins and coun he number of ime series values ha fall in each bin (4). Assuming each bin is sufficienly small ha here are a leas wo bins per discree sae, he maxima of he hisogram can locae esimaes of he discree saes of he molecular machine. However, here are problems wih hisograms because he separaion beween saes is usually no known in advance, molecular moion is generally obscured by observaional noise, and we only have a finie number of samples. As he precision of he peak locaion esimaes is increased by decreasing he bin size, he bins become more sensiive o hese confounding facors and spurious peaks sar o emerge. Similarly, increasing he bin size o make he peaks less spurious decreases he precision of he peak locaion esimaes and may cause wo or more saes o become inseparable. Also, locaion esimaes will end o be sensiive o he choice of bin edges, and non-equal bin sizes appear mainly o inroduce complicaions wih no special advanages (4). Averaging over differen bin edges has been proposed (average shifed hisograms), bu i can be shown ha his is a special case of kernel densiy esimaion, where a smooh funcion (usually wih one mode unimodal) of fixed widh is cenred on each ime series sample, and he disribuion a every locaion is esimaed as he equally weighed sum of all hese funcions (4). Kernel densiy esimaes are an improvemen over hisograms because smooh disribuion esimaes sabilise peak-finding in he presence of noise. he choice of kernel widh is however crucial, because, if he widh is oo small, spurious peaks will emerge, and if oo large, disinc saes may become merged erroneously. Incorporaing addiional informaion abou he discree saes may lead o improvemens. If he number of saes is known in advance, mixure modelling may be used o find he bes combinaion of a weighed sum of componen disribuions wih arbirary locaions and widhs (5). Wih unimodal componen disribuions, one disribuion will be, ideally, locaed a each individual sae. he main difficuly wih mixure modelling is ha he simulaneous esimaion of he componen locaions and widhs is no a convex problem, so ha we canno guaranee ha any soluion we find is he bes one, and he compuaions quickly become onerous as he number of componens grows. Similar issues apply o k-means clusering which aemps o cluser he ime series samples ino a given number of saes. An algorihm for solving he problem exiss (5), and alhough he problem is simpler han mixure modelling because only he sae locaions need o be esimaed, he resuls can ofen depend quie sensiively on he iniial choice of assignmens required o sar he search for a soluion. In esimaing discree saes herefore, as wih sep-smoohing, exising bump-huning approaches described above are problemaic (see ables S and S). hysically-based sep-smoohing, quasi-periodic bump-huning and disribuion esimaion. We seek algorihms for sep-smoohing, bump-huning and disribuion esimaion ha incorporae elemenary physical knowledge of molecular conformaional dynamics. In he following, we have a ime-posiion race, =,

3 obained from experimenal measuremens of molecular dynamics. he series is he unknown series of posiions corresponding o conformaions of he molecular machine o be deermined (and is assumed o be piecewise consan), and he series m is an esimae of given he ime-posiion race. We require ha any algorihm resuls can be obained wih reasonable compuaional cos, and ha hey are guaraneed o converge on he globally opimal soluion. Algorihm L-WC: L-regularized global sep-smoohing wih independen noise. he smoohed esimae is consruced by minimizing he negaive log poserior (NL) cos funcion wih respec o a possible series of posiions m: m arg min NL m m arg min m m m m (S) he implici physical model is in he form, where is a ime series consising of consan segmens wih abrup jumps (seps), and is a ime series of independen Gaussian noise. he problem is o find he series m which consiss of he piecewise consan seps buried in he noise, bu ha is simulaneously a good approximaion o he recorded ime series θ. he firs erm in he NL represens he error (negaive loglikelihood) of he approximaion. he second erm represens he oal absolue difference beween consecuive approximaion samples (in he Bayesian inerpreaion his is he negaive log prior). When he penaly (regularizaion) erm 0, he soluion becomes m and no smoohing occurs. hus, he useful behaviour of his algorihm occurs when 0, so ha increasing weigh is placed on minimizing he second erm a he expense of he firs. here is a maximum useful value for his regularizaion parameer, (see below), and if hen he soluion becomes m, i.e. all approximaion samples ake on he mean of he recorded ime series. Assuming ha here are only a small number of seps in amouns o imposing a sparsiy condiion on m m, ha is, only a few erms in his expression are non-zero. Under his condiion i is (wih very high probabiliy) possible o recover an approximaion o μ ha finds he rue locaions of he seps (6). Increasing γ forces mos of he differences beween consecuive samples of m o zero. his NL cos funcion Eq. (S) is convex and quadraic so ha he opimal approximaion can be obained by minimizing NL wih respec o m using sandard quadraic programming echniques (7). his algorihm is similar o opimal piecewise linear smoohing (8). In he main ex, we demonsrae he use of his algorihm o approximae sep-like moion in experimenal bacerial flagellar moor ime-angle races where he sampling rae is sufficienly low ha he sepping is effecively insananeous. Algorihm L-WC-AR: L-regularized global sep-smoohing wih known correlaed noise. his algorihm is an adapaion of he L-WC algorihm ha incorporaes a general, discree-ime Langevin model wihin he filer srucure. I minimizes he following cos funcion: m arg min NL m m m (S) m arg min ai i m m i Here he implici model is i ai i which capures general Gaussian, linear, discree-ime sochasic dynamics. he real-valued coefficiens a i represen he discree-ime feedback of pas values of he recorded ime series on he curren value. his model incorporaes he special case of discree-ime Langevin dynamics wih linear drif and diffusion erms, in widespread use as models for general molecular dynamics (). he forcing erm μ consiss of piecewise consan segmens wih jumps. Again, minimizing his cos funcion is convex and quadraic and can be solved for m using a sandard quadraic programming algorihm. he underlying consan sep approximaion is hen recovered as m a (noe ha i is no he i i unmodified m because any consan signal inpu o an AR model is amplified by he feedback effec of he model). In paricular, in he main ex we demonsrae use of he special case wih = (L-WC-AR) o capure Langevin moion in experimenal F -Aase angle-ime race:

4 NL m a m m m (S3) he useful range of regularizaion parameer can be deermined by reference o general principles of convex opimizaion (8). If where: DD D (S4) hen, upon opimizing Eq. (S), m will be consan; here he noaion indicaes he elemenwise maximum, and he marix D is he firs difference marix wih ones on he main diagonal, and on he nex diagonal above he main one and zeros elsewhere. hus, Eq. (S4) ses he maximum useful value of he regularizaion parameer. Furhermore, using knowledge ha a uni ime sep of heigh h is flaened away when h (6), allows us o sugges an esimae for he minimum useful value: if he noise has sandard deviaion σ hen seing will flaen away 99% of he noise. hus, he meaningful range of he regularizaion parameer is, and seing he parameer jus above he lower bound reains hose seps ha are jus large enough o be deecable above he noise. In pracice, we need o know σ in order o deermine his range: his can be esimaed from known consan dwells in if he noise is uncorrelaed. Where he noise is correlaed, he correlaion can firs be removed and hen he uncorrelaed signal would be used o esimae σ. In he case of simulaions, we of course know σ a-priori. Algorihm ECF-Bump: Nonparameric bump-huning. he characerisic funcion is an alernaive represenaion of he disribuion p(m) of he sep-smoohed ime series of molecular saes: f ifmpm exp d m (S5) In his conex, p(m) is unknown. However, (f) can be esimaed from he ime series m, using he empirical characerisic funcion (ECF): f expif m j j Here, he ECF is evaluaed over a se of chosen frequencies f j, j =, K. he disribuion funcion p(m) can be reconsruced from he coefficiens (f j ), and covers he range [0, π]: K m if m f p exp (S7) jk j j where he overbar denoes complex conjugaion (and f j = f j o ensure ha p(m) is a real probabiliy). Each frequency f j corresponds o a poenial symmery (periodiciy) of he molecular machine. In he special case of a roary machine i corresponds o he number of seps per complee revoluion. his characerisic funcion is closely relaed o he Fourier ransform of he disribuion of he ime series. hus, he (f j ) can be inerpreed as Fourier coefficiens of he disribuion funcion, and are calculaed over a range of experimenally relevan symmeries f j. he advanage of his represenaion of he disribuion is ha i is usually he case for a molecular machine ha is has repeaing molecular srucures and so undergoes moion in a series of repeaing seps. his implies ha he disribuion of saes is boh bounded and periodic, so he reconsrucion converges on he exac disribuion exremely rapidly as more frequency componens are inroduced (K increases). his is because he Fourier ransform is a sparse represenaion (8) for smooh, bounded, periodic funcions. Sparse represenaions have he desirable propery ha only a few of he coefficiens are large, and hese are he ones ha conribue mos o he shape of he disribuion. he res of he coefficiens will flucuae due o experimenal noise and conribue lile, if anyhing. hus, considering he coefficiens as he sum of rue molecular machine symmeries and experimenal arefacs and disorions, we can apply nonlinear hresholding by ranking coefficiens by heir (S6)

5 power (f j ) and reaining a small fracion φ of he larges coefficiens (we reain around 0-0% in his paper, and his represens a good compromise beween reaining he main shape of he disribuion ye summarising i only by he mos imporan periodiciies). I has been more recenly shown ha his simple procedure for recovering he noisy disribuion is saisically opimal (in he minimax sense) if he disribuion is bounded, smooh and periodic (8), for an appropriae choice of φ relaed o he amoun of experimenal noise. his nonlinear hresholding procedure conrass wih linear hresholding where only a cerain number of he lowes frequency componens are reained (in his conex, kernel densiy smoohing is effecively a linear hresholding operaion, and i herefore gives us no opporuniy o reain high frequency componens if hey are acually imporan). Because we only have a small number of samples of m, he higher frequency coefficiens flucuae due o saisical finie sample effecs. herefore, we can also apply linear hresholding by reaining only hose coefficiens below a hreshold. In pracice however, nonlinear hresholding ends o remove mos irrelevan high frequency componens anyway, and he linear hresholding has negligible or no effec. Analysis of he characerisic frequency domain reveals imporan informaion abou he symmeries of he molecular machine, since probabiliy calculaions wih combinaions of random variables become very simple in his domain. If we change he scale of a random variable X by muliplying i by a scaling facor and adding a consan, he new random variable Y X has he following characerisic funcion: Y f Eexpif X expiff (S8) Hence, shifing he locaion of he peak of a disribuion corresponds simply o muliplying he characerisic funcion by exp(ifμ). Similarly, increasing he spread of he peak corresponds o decreasing he widh of he characerisic funcion. In he case when he disribuion is composed of superposed bumps of widh scaled by σ and wih period N, we have ha: inf exp( if ) (S9) bump bump bump N n0 n N n0 N N exp if N N N f f expif f exp f For he purposes of analysis, he power of he characerisic funcion is usually more convenien o work wih han he characerisic funcion, and we are ineresed in whole ineger symmeries f only: cos(f ) f bump bump f kn (S0) N cos f N 0 oherwise f f where k =,, is he muliple and σ is he spread due o noise of he bump a each molecular sae. his shows ha he power of he characerisic funcion for a periodic disribuion consiss of a series of non-zero coefficiens a ineger muliples of he period N, he res are zero. he absolue square magniude of hese nonzero coefficiens is proporional o he absolue square magniude of he characerisic funcion of he bump disribuions, so ha he spikes are aenuaed in magniude as he muliple increases. he characerisic funcion of many well-known disribuions can be found exacly. For example, if he bump disribuions are Laplacian, since bump (f) = /( + σ f ), we obain, a he peaks: f k N k N (S) Similarly, for Gaussian bumps (f = kn) = exp( σ [kn] ). Qualiaively, as he noise spread increases, he non-zero coefficiens in he characerisic funcion diminish in magniude. herefore, he sharper he bumps in he disribuion, he easier i will be o idenify he period above he background of finie sample variabiliy and experimenal arefacs. In some cases, here will be an arrangemen of molecular saes ha has more han one superimposed period, in general, a se of Q differen periods N q for q =, Q which are no muliples of each oher. In his case, he characerisic funcion power will be: Q q bump q N q N r (S) r f 0, f kn k N

6 and (f) = 0 oherwise. hus he power of he characerisic funcion has a series of non-zero coefficiens a every ineger muliple of each of he Q consiuen periods. he non-zero coefficiens for period N will have Q q r absolue square magniude proporional o N N r, so ha larger periods have larger absolue square magniude. Again, he non-zero coefficiens will be aenuaed in magniude by he characerisic funcion of he bump disribuion, and his will ypically decrease faser wih increasing noise spread. herefore, superimposed symmeries in he molecular machine can be readily deeced from analysis of he larges peaks in he power specrum. Algorihm ML-eaks: maximum likelihood reconsrucion of discree sae ime race from disribuion peaks. Using algorihm ECF-Bump and applying he inverse Fourier ransform o he coefficiens (f j ), we can reconsruc he disribuion p(m) of molecular saes. his disribuion may have some small peaks ha are due o finie sampling effecs or inaccuracies in he reconsrucion of he molecular sae ime race m. However, he larges peaks are associaed wih he mos dominan, and also mos likely, posiions of he molecular saes. hus, if he known dominan symmery is M seps per revoluion, his informaion can be used o selec he M larges peaks in he disribuion as he dominan discree molecular sae dwell locaions. Having locaed hese peaks, he sep-smoohed ime race m can be used o find an esimae of he rue sep-like conformaional sae signal ˆ by classificaion of each of he m o he neares reained peak in he disribuion. his classificaion is he maximum likelihood reconsrucion of μ (see main ex) if he noise around he dwells is Laplace disribued, since we are minimizing he absolue difference beween he neares peak and he sae esimae. In fac, his Laplace disribuion arises as a consequence of solving Eq. (S-S) which has he absolue difference penaly erm (7). Simulaions of molecular machines. Here we describe a model of Brownian moion in a poenial well for periodic sepping moion of a molecular machine wih fricional drag and elasic energy sorage. We se up a simple linear sochasic differenial equaion (SDE) for a ypical experimen. We measure he machine conformaion hrough a small load aached o he machine whose observed posiion is θ. he spring poenial of he srucure aaching he machine o he load is: U (S3) where κ is he spring siffness consan. he load causes drag on he machine, represened using he linear fricion model: F (S4) where ξ is he fricion coefficien. Assuming ha he machine execues random moion abou he equilibrium posiion θ = 0, a Langevin equaion of moion for he experimen can be wrien as: U k M F (S5) B where k B is Bolzmann s consan, is emperaure, M is he machine mass, and ε is an independen, Gaussian random driving force wih mean zero and uni sandard deviaion. Because he raio M/ξ is very small, he inerial erm is negligible and we obain he equaions of moion: F U k B k B Including he average machine posiion we obain he following sochasic differenial equaion: k B d d dw (S7) q (S6)

7 his is an Ornsein-Uhlenbeck process wih mean μ, drif, diffusion k B and Wiener process W(). Focusing on he moion of one sep o he posiion μ, we assume ha he machine sars a ime = 0 a posiion θ = 0, hen he resuling moion is he sum of a deerminisic exponenial and correlaed random flucuaion erms. he load evenually seles ino correlaed random moion of sandard deviaion around he machine dwell conformaion μ. he effec of he load drag and siffness is o delay he ransiions by rounding off he insananeous sep ransiion in μ() wih sep ime consan s. herefore, o be effecive, a sep-smoohing algorihm mus ake ino accoun his smooh ransiion. his is a coninuous-ime sochasic process, bu he experimenal angular measuremens are available a he sampling inerval. herefore, we need o find a discree-ime version of he model. he simple Euler mehod obains: k k B B (S8) where for he ime index = 0, wih 0 0 (we noe ha alhough here are a range of generally more accurae mehods for discreising such SDEs, mos are no more accurae for his paricular model and so in his conex here is no paricular advanage o using a higher order inegraion scheme). his is also a discree-ime, firs order auoregressive (AR) model in he form: a a (S9) where is an independen, zero-mean, consan variance Gaussian process. herefore, his model is a special case of he implici model in algorihm L-WC-AR wih ( = ) described above, and we can esimae he quaniy a direcly from experimenal ime series using he auocorrelaion a one ime lag Δ of measured bead ime races θ. Sep-smoohing and bump-huning algorihm performance comparisons. Figure 3 (main ex) describes nine simulaed es cases produced by varying: he symmery of he discree sae locaions (ha is, by randomly displacing he sae locaions from equal spacing), he disribuion of dwell imes (by changing he gamma shape parameer k), he dominan symmery (e.g. he number of discree saes), he average speed of roaion (ha is, he number of revoluions per second, conrolled by scaling he dwell imes), and he siffness parameer κ. Figure (main ex) shows he ypical oupu from he discree-ime model, and Figure 3 (main ex) shows he performance of a range of sep-smoohing algorihms applied o his es daa. We es L-WC, L-WC-AR, median filering (9), he Chung-Kennedy filer (3), and he Kalafu-Visscher sep-finding mehods (7). We compare he performance of hese mehods in erms of he accuracy of heir abiliy o exrac he simulaed, bu unobserved moor posiion using he mean absolue error: MAE m (S0) and smaller is beer. Also, he relaive absolue roughness: RAR m m (S) idenifies over- and under-smoohing relaive o he known, moor posiion ime series, he closer o uniy he beer. Noe ha if he MAE = 0, hen he RAR = (alhough RAR = does no necessarily give MAE = 0, hus, i is imporan o inerpre he performance wih respec o boh quaniies).

8 Sep-smoohing algorihm parameers are opimized on his es daa o achieve he bes MAE and RAR values. For he L-WC algorihm he opimal parameer values were γ = 50, and for L-WC-AR, γ =, = and a = κδ / ξ (see above for a descripion of how we choose hese values). For he median filer, he only parameer is he window size, and as expeced he opimum size was found o be he average dwell ime. For he Chung-Kennedy filer, he maximum size of all forward/backward moving average predicors plays a similar role o he window size in he median filer. In our implemenaion, we included predicors of all window sizes up o his maximum window size, and exensive experimenaion found ha seing his o half he average dwell ime opimized performance. Because of he non-insananeous sepping of he Langevin dynamics, for he nonlineariy p > 0, we found ha his algorihm inroduced numerous spurious seps and non-smoohness ha degraded he performance considerably. herefore, we found ha having no nonlineariy (i.e seing p = 0) led o he bes performance overall, because i was he smoohes possible filer and so was able o perform well for he longer dwell imes. he Kalafu-Visscher filer has no explicily unable parameers, alhough we have found ha he resuls depend heavily on he lengh of he ime series. Bump-huning algorihm comparisons were made in erms of he median and inerquarile range (5% 75% range) of he recovered number of discree saes (see Supplemenary ables and ). Algorihm parameers were opimized o achieve he bes recovery performance. For he ECF-Bump algorihm, he analysis symmeries (frequencies) ranged from zero o 0 seps per revoluion, and he nonlinear hreshold was se o reain he op 0-0% larges square magniude frequencies. he linear hreshold was se a 80 seps per revoluion. he hisogram FF algorihm used 8 hisogram bins and 8-poin FF. he kernel densiy peakpicking algorihm had a Gaussian kernel wih bandwidh parameer of 0.0 rads, and peaks smaller han 0% of he maximum peak ampliude were discarded. Disribuion fiing o dwell imes. Sandard maximum likelihood echniques minimizing he negaive loglikelihood have been used o fi each disribuion model o he dwell imes obained from bacerial flagellar moor and F -Aase ime-angle races (see below for explici deails of he double exponenial model). For disribuion model comparison, he Bayesian Informaion Crierion is calculaed as (5): BIC L p log N (S) where p is he number of free parameers in each disribuion model, N is he number of dwell imes, and L is he log-likelihood of he disribuion model. For he exponenial, p =, gamma, lognormal and double exponenial, p =, and for he generalized areo, p = 3. For he gamma wih fixed k, p =. When here are muliple subses of dwell imes ha require separae disribuion models per dwell sae, he oal BIC is obained by adding he BIC for each separae model his is consisen wih assumpion ha each dwell sae is independen of he ohers. L-WC-AR auoregressive parameer esimaion. o use he L-WC-AR algorihm, we use he sandard covariance mehod for auocorrelaion analysis o esimae he parameer a, which is he auocorrelaion a a ime lag of one sample. his requires manual idenificaion of a sufficienly long secion of he signal where he molecular machine is saionary. In he real F daa we sudied, unambiguous, long dwells are frequen so ha his approach is sraighforward. Double exponenial disribuion. For more han one reacion cascaded ogeher, a more complex process han he simple oisson process is usually a beer model for he observed discree sae dwell imes. Assuming ha one reacion has o wai for he oher o finish, he oal dwell ime will be a random variable ha is he sum of wo exponenially-disribued dwell imes = + wih rae parameers k, k. hen he disribuion of is he convoluion of he disribuion of and. his becomes he produc of he momen generaing funcions of he individual disribuions: M s k s k (S3) k s k

9 Invering he momen generaing funcion gives he disribuion: k k (S4) k k exp k exp k p wih mean k k k and variance k k k k. k o fi he rae parameers given a se of dwell imes i, i =, N, we can maximize he likelihood, which is equivalen o minimizing he negaive log-likelihood: k k N kˆ, kˆ arg min N log logexp k, k k k i i exp (S5) k k his can be solved using a variey of generic nonlinear opimizaion echniques, wih he consrain k, k > 0. Confidence inervals for he rae parameers are obained by 000 boosrap resampling operaions (5). he degenerae case where k = k is he gamma disribuion wih scale parameer k =. i

10 esimaed disribuion. ECF- Bump Kernel densiy wih peak finding Defaul (6) 6 (0.0) 7 (.5) 0% dwell aperiodiciy (6) 6 (0.5) 7 (.5) Gamma dwell imes, k = (6) 6 (0.0) 9 (.3) Gamma dwell imes, k = 0 (6) 6 (0.0) 9 (.5) 30 dwell locaions (30) 30 (0.0) 3 (3.3) 40 dwell locaions (40) 40 (0.0) 46 (5.0) 50 revs/sec (6) 6 (0.0) 39 (5.3) 00 revs/sec (6) 6 (0.0) 45 (3.8) Flagellar siffness, κ = 50 (6) 6 (0.0) 47 (8.8) able S: erformance of wo differen bump-huning mehods a recovering he dominan symmery in he disribuion of saes for he nine es cases of simulaed bacerial flagellar moor roaion ime series described in he main ex. he figures are he median dominan symmery over five replicaions, and he associaed inerquarile range (difference beween 5h 75h percenile) in brackes. he brackeed number in he firs column is he rue symmery. Algorihm ECF-Bump is described above. Kernel densiy wih peak finding esimaes he disribuion of discree saes using he kernel densiy mehod, hen couns he number of peaks in he

11 Average exponenial dwell ime Number of dwell locaions.5ms.5ms.0ms 0.75ms 0.50ms ECF-Bump (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) (0.0) 30.0 (0.0) 30.0 (0.0) 30.0 (0.0) 30.0 (0.0) (0.0) 40.0 (0.0) 40.0 (0.0) 40.0 (0.0) 40.0 (0.0) (0.0) 50.0 (0.0) 50.0 (.5) 50.0 (0.0) 54.0 (8.0) (0.0) 60.0 (0.0) 60.0 (4.0) 60.0 (0.0) (no resul) Hisogram wih FF (3) (see capion) (0.0) 9.0 (.0) 0.0 (.0) 0.0 (0.0) 0.0 (0.0) (0.0) 9.0 (0.0) 9.0 (0.0) 9.0 (0.0) 30.0 (.0) (0.0) 39.0 (0.0) 39.0 (34.0) 8.0 (6.0) 4.5 (3.0) (45.0) 4.5 (9.0) 4.0 (4.0) 5.5 (5.0) 5.5 (4.0) (47.0) 3.0 (6.0) 7.5 (9.0) 4.0 (7.0) 4.0 (8.0) Kernel densiy wih peak finding (see capion) 0.0 (0.0).0 (.0).0 (.0).0 (.0).0 (.0) (.0) 30.0 (.0) 3.0 (.0) 3.5 (3.0) 34.0 (.0) (.0) 38.0 (3.0) 38.0 (.0) 37.0 (.0) 39.0 (.0) (3.0) 37.0 (.0) 37.0 (.0) 36.0 (4.0) 38.0 (.0) (3.0) 37.0 (4.0) 37.0 (4.0) 37.5 (.0) 36.0 (4.0) able S: erformance of differen bump-huning mehods a finding he dominan symmery in he disribuion of saes of simulaed bacerial flagellar moor roaion ime series wih exponenial dwell imes, over a wide range of dominan symmeries. Each enry shows he median esimaed sae periodiciy, wih he inerquarile range (5h 75h percenile) in brackes. No resul indicaes ha no dominan peak in he ECF could be found. he hisogram wih FF mehod firs esimaes he disribuion of saes using a hisogram, hen finds he fas Fourier ransform of ha hisogram; he larges peak in he specrum is he esimaed periodiciy (mehod used in Sowa e al. 005). Kernel densiy wih peak finding esimaes he disribuion of saes using he kernel densiy mehod, hen couns he number of peaks in he esimaed disribuion.

12 Figure S: Nine es cases of simulaed bacerial flagellar moor ime races; pink line is measured bead angular posiion θ, black line he (unobservable) moor posiion μ. Doed horizonal lines are he discree sae locaions. (a) Defaul case: 6 regularly spaced saes, exponenial dwell imes, flagellar hook siffness κ = 00k B / rad, 0 revs/sec. (b) As defaul, bu wih 0% dwell locaion asymmery (see Supplemenary Mehods). (c) Wih gamma-disribued dwell imes, k =. (d) Gamma dwell imes, k = 0. (e) 30 saes. (f) 40 saes. (g) 50 revs/sec. (h) 00 revs/sec. (i) Flagellar hook siffness κ = 50 k B /rad.

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