Optimal Control of Transitions between Nonequilibrium Steady States

Transcription

1 Optmal Control of Transtons between Nonequlbrum Steay States Patr R. Zulows,2 *, Dav A. Sva 3, Mael R. DeWeese,2,4 Department of Pyss, Unversty of Calforna, Bereley, Calforna, Unte States of Amera, 2 Rewoo Center for Teoretal Neurosene, Unversty of Calforna, Bereley, Calforna, Unte States of Amera, 3 Center for Systems an Syntet Bology, Unversty of Calforna San Franso, San Franso, Calforna, Unte States of Amera, 4 Helen Wlls Neurosene Insttute, Unversty of Calforna, Bereley, Calforna, Unte States of Amera Abstrat Bologal systems funamentally exst out of equlbrum n orer to preserve organze strutures an proesses. Many angng ellular ontons an be represente as transtons between nonequlbrum steay states, an organsms ave an nterest n optmzng su transtons. Usng te Hatano-Sasa Y-value, we exten a reently evelope geometral framewor for etermnng optmal protools so tat t an be apple to systems rven from nonequlbrum steay states. We alulate an numerally verfy optmal protools for a olloal partle ragge troug soluton by a translatng optal trap wt two ontrollable parameters. We offer expermental pretons, spefally tat optmal protools are sgnfantly less ostly tan nave ones. Optmal protools smlar to tese may ultmately pont to esgn prnples for bologal energy transuton systems an gue te esgn of artfal moleular manes. Ctaton: Zulows PR, Sva DA, DeWeese MR (23) Optmal Control of Transtons between Nonequlbrum Steay States. PLoS ONE 8(2): e o:.37/journal.pone Etor: Geraro Aesso, Unversty of Nottngam, Unte Kngom Reeve July 26, 23; Aepte Otober 27, 23; Publse Deember 26, 23 Copyrgt: ß 23 Zulows et al. Ts s an open-aess artle strbute uner te terms of te Creatve Commons Attrbuton Lense, w permts unrestrte use, strbuton, an reprouton n any meum, prove te orgnal autor an soure are rete. Funng: M.R.D. anowleges support from te MKngt Founaton, te Hellman Famly Faulty Fun, te MDonnell Founaton, an te Mary Elzabet Renne Enowment for Eplepsy Resear. M.R.D. an P.R.Z. were partly supporte by te Natonal Sene Founaton troug Grant No. IIS D.A.S. was supporte by NIGMS Systems Bology Center grant P5 GM8879. Te funers a no role n stuy esgn, ata olleton an analyss, eson to publs, or preparaton of te manusrpt. Competng Interests: Te autors ave elare tat no ompetng nterests exst. * E-mal: pzulows@bereley.eu Introuton Lvng systems are stnguse by ter self-organzaton. Gven te entrop rvng fore emboe n te seon law of termoynams, reatng an mantanng su organzaton requres stayng far from equlbrum [], typally by ouplng to nonequlbrum graents. For example, ATP-rven moleular motors (e.g., nesn) are fore away from equlbrum by ellular mantenane of a emal potental fferene between ATP an ADP [2], an te rotary F O {F ATP syntase operates out of equlbrum ue to ellular mantenane of an eletroemal graent aross te nner mtoonral membrane [3]. For onstant ATP an ADP onentratons, or onstant membrane potental, te ynams of an ensemble of su moleular motors wll approxmate a nonequlbrum steay state (NESS). Tus, bologal systems are often better araterze as nonequlbrum steay states rater tan equlbrum systems. Su NESS may ange n response to angng envronmental ontons. Gven tat seletve avantage may be nurre by energetally-effent operaton, evoluton may ave sulpte bologal omponents to nterat so as to reue te energy waste urng transtons between NESS. Aorngly, optmzng su transtons may offer nsgts nto te esgn prnples of bologal systems an gue te reaton of syntet moleular-sale manes. Mu reent attenton as fouse on pretng optmal protools to rve systems between equlbrum states wt mnmal expene wor [4 ]. In partular, Ref. [4] proposes a lnear response framewor for protools tat mnmze te sspaton urng nonequlbrum perturbatons of mrosop systems. Ts ea s evelope furter n [5] were te utlty of Remannan geometry suggeste n [4] s explote to fn explt optmal protools for a paragmat olloal partle moel. Our ultmate am s to exten te geometr framewor of [4] to optmal transtons between steay states so tat te tools utlze n [5] may be apple to more bologally relevant moels. In ts artle, we tae a frst step towars ts goal by optmzng te Hatano-Sasa Y-value, a quantty smlar to sspate wor, for te paragmat moel system teste n [] an analyze n [2] wt an eye towars expermental tests. We alulate lose-form expressons for bot te geoes optmal protool an te optmal stragt-lne protool an test tese protools numerally va a system of equatons erve from te Foer-Plan equaton. Fnally, we propose a regme of valty of our approxmaton base on ts numeral wor. By measurng te average wor requre to rve ts system along eter optmal or nave pats troug ontrol parameter spae, our results an be teste expermentally n a stragtforwar way usng exstng expermental tenques. Results Te moel system an ts nverse ffuson tensor We onser a partle wt spatal oornate x ffusng uner Langevn ynams subjet to a one-mensonal armon potental, wt equaton of moton. PLOS ONE Deember 23 Volume 8 Issue 2 e82754

2 _x~{ (t) xzg(t){v(t), for Gaussan wte nose g(t).here s te Cartesan frton oeffent, s te trap stffness, v s te trap enter veloty n te lab frame an x s te oornate of te olloal partle n te frame o-movng wt te trap. Te partle s ntally n NESS ue to onstant trap veloty v. As efne n [3], te Hatano-Sasa Y-value Y: t Lw Š: (x(t); l(t)) Ll ð2þ arses n NESS transtons wen te ontrol parameters l are ange raply ompare to te system s relaxaton tmesale. Here w(x; l):{ ln r ss (x; l) were r ss (x; l) s te steay state probablty strbuton an t s te protool uraton. In some smple ases ts orrespons to te system laggng ben te angng ontrol parameters. For transtons between equlbrum states ts measure reues to te stanar sspaton governe by te Clausus nequalty [4]. Ts measure of rreversblty (2) obeys a sgnfant NESS flutuaton teorem tat as been expermentally observe n our partular moel system []. We may erve an approxmate seme, exat n te lnear response regme [5], for optmzng ts Y-value urng fnte-tme transtons between fferent nonequlbrum states. Te ensemble average of te Y-value s SYT L : t Š: v Lw Ll (x; l(t))w L Durng te rvng proess, te system s probablty strbuton over mrostates funamentally epens on te story of te ontrol parameters l, w we enote by te ontrol parameter protool L. We assume te protool to be suffently smoot to be twe-fferentable. Applyng lnear response teory [4,5,5] an assumng tat te protool vares suffently slowly [4], we arrve at an expresson for te average Y-value. SYT L & t Š: f(l(t)) : ½ l t Š, n terms of te ontrol parameter velotes l=t an te nverse ffuson matrx f(l) wt entres f j (l): ð? t v Lw Ll (t ) Lw Ll j ()w L Te angle braets S...T L represent an average over nose followe by a statonary state average over ntal ontons usng te strbuton r ss (x; l). Note tat, f r ss (x; l) s te equlbrum strbuton, te nverse ffuson tensor of [5] s reovere. In general, te etale balane onton s volate n NESS an so te matrx vl l w(t )L l j w()w L may be asymmetr. Eq. (4) sows te use of Eq. (5) (spefally ts symmetr part) as a ðþ ð3þ ð4þ ð5þ metr tensor s not prelue. However,vL l w(t )L l j w()w L s not a ovarane matrx an so a general proof of postveefnteness s lang [4]. Tese onseratons o not affet te moel onsere ere but future wor s neee to aress ts ssue for te general ase. rffffffffff Te steay-state strbuton s gven by r ss (x; l): 2p expf{ b 2 ðxzvþ2 g [,2]. Te nverse ffuson tensor s gven by 4 4 z4b ð v f(,v)~ {bv B Þ2 {bv 3 b 3 C 2 A Optmal protools Toug one an wrte own te geoes equatons for te metr [Eq. (6)] n te (,v) oornate system, more nsgt s gane by fnng a sutable ange of oornates. A ret alulaton of ts metr s R salar yels R~, emonstratng tat te unerlyng geometry s Eulean [6]. Wt te oornate transformaton j~ v, x~ p ffffffffff, te lne element s s 2 ~b 3 (j 2 zx 2 ) In ts oornate system, geoess are stragt lnes of onstant spee. By te ovarane of te geoes equaton, te optmal protools are gven by ð6þ ð7þ (t)~½p ffffffff ({T)z p ffffffffff TŠ {2, ð8aþ f v(t)~(t)½ v ({T)zT v f f Š, were T~ t. Sample optmal protools are pture n Fg.. t ð8bþ Optmal stragt-lne protools In te absene of any partular nformaton about te system s ynamal propertes, a nave ontrol strategy woul ange te ontrol parameters at a onstant rate, proung a stragt lne n ontrol parameter spae. Te nverse ffuson tensor approxmaton [Eq. (4)] proves a repe for oosng bot a potentally nonlnear pat troug ontrol parameter spae, as well as a tmeourse along tat pat. Te nverse ffuson tensor formalsm an alternatvely be use to optmze te tme-ourse along a stragtlne ontrol parameter pat. Su a protool proves a benmar aganst w an optmal protool [Eq. (8)] an be ompare. For te moel onsere ere, we wll fn tat an optmal stragt-lne protool an be substantally better tan te most nave (onstant-spee) stragt-lne protool. Furtermore, stragt-lne protools are relatvely stragtforwar to test expermentally. PLOS ONE 2 Deember 23 Volume 8 Issue 2 e82754

3 Fgure. Geoess esrbe protools tat outperform nave (onstant-spee) stragt-lne pats n parameter spae. Geoess between fxe pars of ponts n te (,v)-plane an aompanyng stragt-lne protools are pture n (a). Te flle rles represent ponts separate by equal tmes. Te open rles orrespon to te optmal parametrzaton along te respetve stragt pat. All mean Y- values were alulate usng te Foer-Plan system, Eq. (5). Here, ~:pns=mm an b { ~4:6pNnm to approxmate te experments of Ref. []. Te protool uraton s osen to be t~s to ensure tat te relatve error j{syt approx =SYT L L j s less tan :4% for all protools. Protool enponts were selete for expermental aessblty [2]. Te relatve performane of nave stragt-lne, optmal stragt-lne, an geoes protools are summarze n (b). o:.37/journal.pone g Wen (t) s el fxe, a stragtforwar applaton of varatonal alulus emonstrates tat a stragt-lne protool n v(t) s exatly optmal an agrees wt te pretons of te lnear response approxmaton [Eq. (4)]. In Ref. [], te average Y- value was measure for tree stnt expermental trals nvolvng protools wt onstant. As summarze n Fg. 2, te optmal protool, namely te nave stragt lne n te ase of onstant, sows sgnfantly reue Y-value ompare wt te protools use n ea expermental tral. However, n terms of testng te performane of te optmal protools [Eq. (8)], f = s te more general ase. As n te ase of fnng globally optmal protools, te problem of fnng optmal stragt lne protools smplfes ramatally n (j,x) oornates. Usng Eq. (7), we fn. for vyw L &b 3 t½zb 2 x 2 x 2 (t)š, ð9þ t b:2b 2 f v { v f f { Te Euler-Lagrange equaton mples x t ~ ð xf t x z pffffffffffffffffffffffffffffffff zb 2 z 2 ðþ p ffffffffffffffffffffffffffffffffffffffffff, ðþ zb 2 x 2 (t) Fgure 2. Optmal protools outperform onstant- protools teste n Ref. []. Experment (left, re) use a quarter-sne wave protool to vary te trap spee; Experments 2 (mle, blue) an 3 (rgt, orange) use an nverte tree-quarters sne wave. Spefally, v(t)~8:2z4:3 snðpt=2tþ,~4:25,t~:6,q~:2 for Experment, v(t)~9:93{3:63 snð3pt=2tþ,~4:5,t~:6,q~:2 for Experment 2, an v(t)~7:53{2:67 snð3pt=2tþ,~4:9,t~:8,q~:23 for Experment 3. Here, veloty s measure n mm=s, t s te protool uraton measure n s, s te trap stffness measure n pn=mm, an q:=b s measure n pn mm=s. Te Y-value for tese protools (lgt olor bar) an for te optmal protools (sol olor bar) were obtane numerally assumng b { ~4:6pNnm (re), b { ~4:45pN nm (blue), b { ~4:35pN nm (orange) respetvely. Tese effetve temperatures were osen to gve te best mat between experment an numeral alulaton, an may ffer from room temperature (b { ~4:4pN nm) beause of loal eatng by te optal trap [22]. We pret a sgnfant reuton n Y-value for optmal protool rvng uner te ontons of te tree experments esrbe n Ref. []. o:.37/journal.pone g2 w etermnes an mplt expresson for x(t): 2b t ð x f pffffffffffffffffffffffffffffffff z zb t 2 z 2 ~b(x(t) x qffffffffffffffffffffffffffffffff {x zb 2 x 2 )z sn { bx(t) Te relaton x~ p ffffffffff (t), an ene for v(t). p ffffffffffffffffffffffffffffffffffffffffff zb 2 x 2 (t) ½ Š{ sn { ½bx Š ð2þ etermnes an mplt expresson for Computng te Y-value numerally We valate te optmalty of te geoess [Eq. (8)] an ompare wt optmal stragt-lne protools by alulatng te average Y-value retly by ntegratng n tme te Foer-Plan equaton esrbng te ynamal evoluton of te partle probablty strbuton [2], Lt ~ (t) L Lx ðxr Þzv(t) Lx z L 2 r b Lx 2 ð3þ In full generalty, te mean Y-value as a funtonal of te protool l(t)~((t),v(t)) s t½{ _ 2 { b v 2 _z b Sx 2 2 _ 2 T L zb_vsxt L zb 2 v _vš ð4þ PLOS ONE 3 Deember 23 Volume 8 Issue 2 e82754

4 Here angle braets enote averages over te nonequlbrum probablty ensty r(x,t). By ntegratng Eq. (3) aganst x an x 2, we fn a system of equatons for relevant nonequlbrum averages: t SxT L~{ (t) SxT L{v(t), t Sx2 T L ~{ 2(t) Sx 2 T L {2v(t)SxT L z 2 b, supplemente by ntal ontons vxw L ()~{ v, ð5aþ ð5bþ ð6aþ vx 2 w L ()~ z v 2 ð6bþ Note tat for a more omplex system te frst an seon moments SxT an Sx 2 T are not suffent to araterze te probablty strbuton, but tme-epenent solutons are stll aessble troug stanar (but more omputatonally ntensve) numeral ntegraton of te full Foer-Plan equaton (3) [7]. We solve tese equatons numerally an ompare te performane of optmal stragt lnes aganst geoess [Eq. (8)] an nave (onstant-spee) stragt-lne protools n Fg.. We selete enponts an pysal onstants base on tose use n te experments of Ref. [] w may be foun n te apton of Fg. 2. In ts near-equlbrum regme te nverse ffuson tensor approxmaton proues small relatve error n Y-value. Toug tere s only a margnal fferene n performane between te optmal stragt-lne protool an te geoes for bot sets of enponts, tere s a substantal beneft n usng eter over te nave stragt lne protool. Te nverse ffuson tensor arses naturally from te Foer-Plan equaton If we neglet terms nvolvng seon- an ger-orer temporal ervatves (an alternatve near-equlbrum approxmaton), we obtan an approxmate soluton to te Foer-Plan system: vxw L &{ v z 2 v 2 _v{ _, vx 2 w L & 2 v2 2 z z _ 2 3 z 23 v 2 _ 4 { 23 v_v 3 ð7aþ ð7bþ Substtutng ts nto te expresson for te mean Y-value [Eq. (4)], we reover Eq. (28). Te argument above suggests tat te emergene of te nverse ffuson tensor from te Foer-Plan equaton may follow from a perturbaton expanson n small parameters [5]. Dsusson We ave taen te frst step towars extenng te geometr framewor for alulatng optmal protools presente n [4,5] to systems relaxng to NESS. As energy-transung bologal systems are more fatfully esrbe by NESS tan by equlbrum statsts, ts brngs reent teoretal evelopments loser to te beavor of n vvo bologal systems. Usng a lnear response approxmaton, we foun te optmal mean Y-value for a moel system of a olloal partle (ntally n NESS) ragge troug soluton an subjet to a tme-epenent armon potental. We too as our ontrol parameters te veloty an sprng onstant of te armon potental. As n [5], tools from Remannan geometry reveale a useful oornate transformaton w greatly smplfe te onstruton of optmal stragt-lne protools as well as geoes protools. Tese optmal protools were teste numerally an te small relatve error n te Y-value approxmaton for expermentally relevant oes of parameters s enouragng. Our pretons may be teste expermentally wt exstng arware an metos. In Ref. [], te autors report on experments performe wt mron-sze polystyrene beas n soluton. Te armon potental s reate by superposng te fo of two ounterpropagatng laser beams. Te loaton of ts trap was translate usng a steerable mrror. Te veloty v of te trap loaton was altere by angng te mrror s angular rate of rotaton, an te trap stffness an be manpulate by ynamally angng te ntensty of te laser beam [8] or by passng te laser beam troug a polarzaton flter an ynamally angng te polarzaton of te laser beam. Fore s nferre from te rate of ange of te momentum of lgt measure by poston-senstve potoetetors. Comparson of te average wor nurre urng fferent protools woul prove an expermental test of te optmal protools prete n ts manusrpt. Usng te nverse ffuson tensor approxmaton n general allows us aess to te full power of Remannan geometry n alulatng optmal protools. However, su expermental tests an assess te range of valty of te approxmaton. Our alternate ervaton of te nverse ffuson tensor va a ervatve-trunaton expanson [5] suggests a greater robustness of te approxmaton. In ts paper we prove onrete teoretal pretons for experments spefally, we fn tat geoess, optmal stragtlnes, an nave stragt-lne protools all are substantally more effent tan te protools teste n Trepagner, et al. Moreover, we emonstrate tat for smultaneous ajustment of an v, optmal stragt-lne protools an perform substantally better tan nave (onstant-spee) stragt-lne protools. Te neessary metoology an expermental apparatus are well-establse [] to not only verfy our pretons but to pus beyon te nearsteay-state regme. Gven te greater generalty emboe by te extenson to NESS, an te auray of ts approxmaton for a stanar moel system, optmal rvng protools erve n ts framewor promse greater applablty to moels of bomoleular manes. Neverteless, mportant urles reman: our moel system experenes fores lnear n poston an as a steay-state strbuton fferng from te equlbrum one only n ts average splaement. Moleular manes feature nonlnear fore profles, potentally nontrval steay-state strbutons, an often operate far from equlbrum. Tus our omparatvely smple teoretal framewor may nee furter elaboraton to aress te ynams an effeny of moleular manes wt reasonable felty. PLOS ONE 4 Deember 23 Volume 8 Issue 2 e82754

5 Furtermore, te relatvely smple moel system we treat n ts manusrpt represents a new fronter for te analytal soluton of optmal protools uner te nverse ffuson tensor approxmaton. For sgnfantly more omplate moels of greater bologal nterest, a smple general approa (n leu of a sear for an analytal soluton) woul be a fully numeral meto, nvolvng te alulaton of te nverse ffuson tensor at a gr of ponts n ontrol parameter spae, analogous to te approa n [9]. Fnally, tere remans te mportant open queston of wat quantty or quanttes are to be optmze n fatful moels of bologal proesses. In ts paper, we mae te oe of optmzng te Y-value w as been expermentally stue n ts partular moel system [] an may be optmze by te same geometr framewor as n [4]. Tese qualtes were avantageous to begn a lear an matematally tratable frst step towars optmzaton of steay state transtons. However, t s possble an peraps lely tat a properly efne average sspate eat wll be te bologally relevant quantty to optmze rater tan te Y-value. We antpate tat a geometr approa to optmzaton wll be applable to tese more general systems an notons of eat prouton n a relevant regme of parameter values an protool uratons. However, a more general onstruton wll ave to tae nto aount te soalle ouseeepng eat [3,2] w s generate n mantanng te steay state at gven ontrol parameter values. Future wor s neee to aress tese ssues properly. Metos Our moel onssts of a partle wt spatal oornate x ffusng uner Langevn ynams subjet to a one-mensonal armon potental, wt equaton of moton _x~{ (t) xzg(t){v(t), ð8þ SYT L & t Š: f(l(t)) : ½ l t Š, ð22þ n terms of te ontrol parameter velotes l=t an te nverse ffuson matrx f(l) wt entres ð? f j (l): t v Lw Ll (t ) Lw Ll j ()w ð23þ L Te angle braets S...T L represent an average over nose followe by a statonary state average over ntal ontons usng te strbuton r ss (x; l). rffffffffff Te steay-state strbuton s gven by r ss (x; l): 2p expf{ b 2 ðxzvþ2 g [,2]. Te parameter spae ervatve of w s gven by Lw Lw :( Ll L, Lw Lv ) ð24aþ ~({ 2 z b 2 x2 { b v 2,b v xz ) ð24bþ 2 In orer to alulate te tme orrelaton funtons n Eq. (23), we solve Eq. (8) for onstant an v, gvng x(t)~x e { t z s e { (t{s) g(s){ v {e{ t Reallng tat g(t) s Gaussan nose, Eq. (25) mples ð25þ for Gaussan wte nose g(t) satsfyng Sg(t)T~, vg(t)g(t )w~ 2 b (t{t ) ð9þ vl w(t)l w()w l ~ bv ð Þ2 3 e { t z e{2 t 2 2, ð26aþ Here s te Cartesan frton oeffent, s te trap stffness, v s te trap enter veloty n te lab frame an x s te oornate of te olloal partle n te frame o-movng wt te trap. Te partle s ntally n NESS ue to onstant trap veloty v. As efne n [3], te Hatano-Sasa Y-value s gven by Y: t Lw Š: (x(t); l(t)) Ll ð2þ Here w(x; l):{ ln r ss (x; l) were r ss (x; l) s te steay state probablty strbuton an t s te protool uraton. Te ensemble average of te Y-value s vl w(t)l v w()w l ~{bv 2e { t, vl v w(t)l w()w l ~{bv 2e { t, vl v w(t)l v w()w l ~ b2 e{ t Integratng over tme yels te nverse ffuson tensor: ð26bþ ð26þ ð26þ SYT L : t Š: v Lw Ll (x; l(t))w L ð2þ Applyng lnear response teory [4,5,5] an assumng tat te protool vares suffently slowly [4], we arrve at an expresson for te average Y-value 4 4 z4b ð v f(,v)~ {bv B Þ2 {bv 3 b 3 C 2 A ð27þ PLOS ONE 5 Deember 23 Volume 8 Issue 2 e82754

6 Te lne element orresponng to te metr n Eq. (27) s s 2 ~ 4 4 z4b ð v Þ2 2 {2bv 3 3 vzb 2 v2 : ð28þ To fn te explt oornate transformaton mang te Eulean geometry manfest, we wrte te lne element as s 2 ~b 3 f½ v Š 2 z( p 2 ffffff ) 2 g 3 b 2 ð29þ Ts suggests te oornate transformaton j~ v, x~ p ffffffffff, so tat s 2 ~b 3 (j 2 zx 2 ): ð3þ In ts oornate system, geoess are stragt lnes of onstant spee. To fn optmal protools n (,v) spae, one smply transforms te oornates of te enponts nto (j,x) spae, onnets tese ponts by a stragt lne, an uses te nverse transformaton to map te lne onto a urve n (,v) spae. Ts follows from te nvarane of te geoes equaton [6]. Expltly, te optmal protool jonng (,v ) an ( f,v f ) s (t)~½p ffffffff ({T)z p ffffffffff TŠ {2, ð3aþ f v(t)~(t)½ v ({T)zT v f f Š, ð3bþ were T~ t t. We valate te optmalty of te geoess [Eq. (3)] numerally va te Foer-Plan equaton [2], In full generalty, te mean Y-value as a funtonal of te protool l(t)~((t),v(t)) s t½{ _ 2 { b v 2 _z b Sx 2 2 _ 2 T L zb_vsxt L zb 2 v _vš ð33þ Here angle braets enote averages over te nonequlbrum probablty ensty r(x,t). By ntegratng Eq. (32) aganst x an x 2, we fn a system of equatons for relevant nonequlbrum averages: t SxT L~{ (t) SxT L{v(t), t Sx2 T L ~{ 2(t) Sx 2 T L {2v(t)SxT L z 2 b, supplemente by ntal ontons Anowlegments vxw L ()~{ v, ð34aþ ð34bþ ð35aþ vx 2 w L ()~ z v 2 ð35bþ We tan Crstoper Jarzyns for a useful susson of ts wor, an Vlaslav Belyy, Amet Ylz, an Jeff Mofftt for nformaton regarng optal traps. Autor Contrbutons Analyze te ata: PRZ. Wrote te paper: PRZ DAS MRD. Lt ~ (t) L Lx ðxr Þzv(t) Lx z L 2 r b Lx 2 ð32þ Referenes. Srönger E (992) Wat s Lfe? Cambrge: Cambrge Unversty Press. 2. Howar J (2) Means of Motor Protens an te Cytoseleton. Sunerlan, Massausetts: Snauer. 3. Alberts B, Jonson A, Lews J, Raff M, Roberts K, et al. (22) Moleular Bology of te Cell. New Yor: Garlan Sene. 4. Sva DA, Croos GE (22) Termoynam metrs an optmal pats. Pys Rev Lett 8: Zulows PR, Sva DA, Croos GE, DeWeese MR (22) Geometry of termoynam ontrol. Pys Rev E 86: Senfel DK, Xu H, Eastwoo MP, Dror RO, Saw DE (29) Mnmzng termoynam lengt to selet ntermeate states for free-energy alulatons an repla-exange smulatons. Pys Rev E 8: Broy DC, Hoo DW (29) Informaton geometry n vapour-lqu equlbrum. J Pys A 42: Gomez-Marn A, Smel T, Sefert U (28) Optmal protools for mnmal wor proesses n unerampe stoast termoynams. J Cem Pys 29: Smel T, Sefert U (27) Optmal fnte-tme proesses n stoast termoynams. Pys Rev Lett 98: 83.. Aurell E, Mejía-Monastero C, Muratore-Gnannes P (2) Optmal protools an optmal transport n stoast termoynams. Pys Rev Lett 6: Trepagner EH, Jarzyns C, Rtort F, Croos GE, Bustamante CJ, et al. (24) Expermental test of atano an sasa s nonequlbrum steay-state equalty. Pro Natl Aa S USA : Mazona O, Jarzyns C (999) Exatly solvable moel llustratng far-fromequlbrum pretons. ArXv webste. Avalable: ttp://arxv.org/abs/onmat/9922. Aesse 23 Nov Hatano T, Sasa S (2) Steay-state termoynams of Langevn systems. Pys Rev Lett 86: Karar M (27) Statstal pyss of partles. Cambrge: Cambrge Unversty Press. 5. Zwanzg R (2) Nonequlbrum statstal means. New Yor: Oxfor Unversty Press. 6. o Carmo MP (992) Remannan geometry. Boston: Bräuser. 7. Rsen H (996) Te Foer-Plan equaton. Berln: Sprnger-Verlag 3r eton. PLOS ONE 6 Deember 23 Volume 8 Issue 2 e82754

7 8. Joyutty J, Matur V, Venataraman V, Natarajan V (25) Dret Measurement of te Osllaton Frequeny n an Optal-Tweezers Trap by Parametr Extaton. Pys Rev Lett 95: Senfel DK, Xu H, Eastwoo MP, Dror RO, Saw DE (29) Mnmzng termoynam lengt to selet ntermeate states for free-energy alulatons an repla-exange smulatons. Pysal Revew E 8: Sefert U (22) Stoast termoynams, flutuaton teorems an moleular manes. Reports on Progress n Pyss 75: Neuman K, Blo S (24) Optal trappng. Revew of Sentf Instruments 75: Peterman EJ, Gttes F, Smt CF (23) Laser-nue eatng n optal traps. Bopysal Journal 84: PLOS ONE 7 Deember 23 Volume 8 Issue 2 e82754