Mathematical formulation of the F 0 motor model

Transcription

1 negy Tnsduction in TP Synthse: Supplement Mthemticl fomultion of the F 0 moto model. Mkov chin model fo the evolution of the oto stte The fou possible potontion sttes of the two oto sp61 sites t the otostto intefce e: oth sites empty () : (o o) The ight site potonted () : (o ) oth sites potonted (F) : ( ) The left site potonted () : ( o) s potons hop into nd out of the chnnel, the stte of the two exposed oto sites chnges. Figue 1 shows the tnsitions between these oto sttes due to poton hops. The Mkov tnsition mtix coesponding to the tnsition digm in Figue 1 is: é k + k k 0 k ( + ) K = k k kf kf 0 0 kf ( kf + kf ) kf ë k 0 k k + k F F We ssume tht potons cnnot hop between the sp61 sites. t ny fixed oto position, it tkes two poton hops to go fom stte (o ) to stte ( o) (i.e. one poton hops out nd nothe one hops in). So thee is no diect Mkov tnsition between nd in Figue 1. Howeve the tnsition between nd cn be done by ottion of the oto (shown s dshed line in Figue 1). Fo exmple, when the oto is in stte (o ), ottion to the ight cn push the potonted site on the ight into the membne nd pull potonted site out of the membne on the left, which chnges the oto into stte ( o). In Figue, we plot the fee enegy chnge, G(, s), in the system s function of ottion ngle,, nd oto stte, s, when one poton psses though the moto poducing ottion of p/1 din ginst lod toue of 41 pnnm. The cycle shown is the powe cycle. The potonmotive foce coss the membne is p = 0 mv (= 8.8 k T). The wok done by the poton ginst the lod toue is 41 pnnm x (p/1) = 5. k T. The enegy dissipted in the pocess is = 3.6 k T. In Figue, the fee enegy cuve fo stte t the end of the cycle is 3.6 k T lowe thn the one fo stte t the beginning of the cycle. ù û (1) 1

2 ottion k F k F k F Full ottion k F ight eft k mpty k k k Figue 1 Mkov chin descibing the fou possible oto sttes nd thei tnsition tes, k i j. = mpty, F = F ull, = eft site occupied, = ight site occupied. ( o ) = unpotonted sp61 sites, ( ) = potonted sites G(,s) k T p 1 p 1

3 Figue Fee enegy chnge, G(, s), in the system s function of ottion ngle,, nd oto stte, s, when one poton psses though the moto poducing ottion of p/1 din ginst lod toue of 41 pnnm. The eflecting membne boundies e denoted by solid ectngles. The cycle shown is the powe cycle ; othe cycles shown in Figue 1 e not powe poducing, nd involve oto slip nd/o poton lekge. The potonmotive foce coss the membne is p = 0 mv» 8.8 k T. The wok done by the poton ginst the lod toue is 41 pnnm x (p/1) = 5. k T. The enegy dissipted in the pocess is = 3.6 k T, so the fee enegy cuve fo stte t the end of the cycle is 3.6 k T lowe thn the one fo stte t the beginning of the cycle.. ngevin eution fomultion Since ineti is negligible, euting the viscous dg on the oto to the foces cting on it yields the ngevin eution 1, whee: d z d y ( ; s) dfh s = ( ; ) t + (), t s =,,,F dt d d () z = dg coefficient of the coto. Ä(t) = the ndom foce due to bownin fluctutions. The bownin foce is modeled in the usul wy by Gussin distibution with unit men nd mplitude z kt t, whee t is the time step in the numeicl simultion. t = lod toue fom F 1. s = (,, F, ). The stte of the two exposed oto sites evolves ccoding to the tnsition tes k ij shown in Figue 1. y (; s) = the potentil ffecting the oto in stte s. y (; s) contins the intections between the oto sites nd the fixed stto chges (g10, Glu19, nd His45). f H (; s) = the hydophobic potentil bie peventing the ottion of unpotonted oto sites into the membne. The enegy bie ginst otting n unpotonted sp61 site into the bilye depends on the diffeence in the dielectic constnts 1 1 between the two medi: G» 00ç 3. Tking emem» 3 nd e è emembne estto stto» 10, we hve G» 45 k T. Note tht y (; s) nd f H (; s) could be combined into one potentil in eution (). Howeve, we keep them septe becuse in the FokkePlnck fomultion, the effect of the membne potentil bie, f H (; s), is modeled by boundy conditions, while 3

4 the effect of otostto chge intections e cied by the potentil y(; s), which esides in the eutions of motion. To compute the toue geneted by the moto, eutions () must be solved simultneously with the Mkov pocess govening the hopping of potons on nd off of the oto sites. lthough we dw the ottion of the oto s link (dshed line) between sttes nd, the ottion of the oto fom to +p/1 cnnot be teted simply s single Mkov step fo the esons listed below: The ottion of the oto is continuous. The ottion of the oto fom to +d cn be teted s single Mkov step only if the diffusion of the oto in [, +d] is much fste thn the net ottion of the oto fom to +d. The time scle fo diffusion in [, +d] is ~d / nd the time scle of the net ottion is ~d/<v>. Theefoe, if d is smll enough, the diffusion is lwys fste thn the net ottion. This is the essence of the numeicl discetiztion of the model. Howeve, d = p/1 is not smll enough, so the ottion fom to +p/1 cnnot be teted s single Mkov step; insted it hs to be teted s seuence of smlle Mkov steps. The effect of the stto chge, g10, on the oto is stongly ngledependent. When potonted oto site psses g10, the stto chge educes the effective pk of the oto site nd foces the oto site to elinuish its poton. When oto site is f fom the stto chge, its lge pk cn hold the poton tightly. This pevents mny futile potontion nd depotontion cycles. If we tet the ottion fom to +p/1 s single Mkov step, then we cnnot model the ngledependence of the intections between the stto nd oto chges. C. FokkePlnck eution fomultion In the FokkePlnck fomultion coesponding to eutions (), the stte of the system is descibed by the fou pobbility density functions: é (, t) = ë F (, t) (, t) (, t) (, t) ù û whee s (,t) is the pobbility density tht the moto is in stte s nd the oto is t loction t time t. These distibutions evolve ccoding to the convective diffusion eutions 1 ì dy = t z îè d ý þ + + K (4) whee = k T/z is the diffusion coefficient of the coto. The electosttic potentil mtix is (3) 4

5 = Y éy ù 0 y ( ) y ( ) F 0 ë y ( ) û whee y s () is the potentil cting on the oto due to the electosttic intections between the oto sites nd the stto chges when the two oto sites e in stte s. The boundy conditions fo eution (4) e given by oundy condition fo ((o o): eflecting t = 0 nd t d = p/1): ì1 dy îz è d oundy condition fo ((o ): eflecting t = 0): ì1 dy îz è d ì1 dy + ý = 0 ; + 0 þ 0 z è d ý = (6) = î þ= d + ý = 0 þ = 0 oundy condition fo F (( ): peiodic with peiod = d): ì1 dy F F 1 dy F F ì F + t F z è d ý = + + î þ 0 z è d ý = î þ= d oundy condition fo (( o): eflecting t = d): ì1 dy îz è d + ý = 0 þ = d The pobbility flux leving the ight end ( = d) of stte is eul to the flux enteing the left end ( = 0) of stte : (5) (7) (8) (9) ì1 dy 1 dy ì + t z è d ý = + + î þ d z è d ý = î þ= 0. Numeicl Clcultions (10).1 Moto without g10: pue ownin tchet In this model, we ssume: k in = the te of poton hopping into the chnnel fom the cidic side (low ph) of the membne. It is independent of the oto position,, nd independent of the stte of the othe site. 5

6 k in = the te of poton hopping into the chnnel fom the bsic side (high ph) of the membne. It is independent of the oto position,, nd independent of the stte of the othe site. k out = the te of poton hopping off the site to the cidic side. It is independent of the oto position,, nd independent of the stte of the othe site. k out = the te of poton hopping off the site to the bsic side. It is independent of the oto position,, nd independent of the stte of the othe site. These tnsition tes depend on the pmetes listed below: ph = poton concenttion on the cidic side (low ph) of the membne. ph = poton concenttion on the bsic side (high ph) of the membne. y = potentil dop coss the membne. f = the potentil dop t sufce (due to sufce chges) t the cidic side of the membne. f = the potentil dop t sufce (due to sufce chges) t the bsic side of the membne. p = poton diffusion coefficient. = dius of the chnnel leding to the oto sites.. Tnsition tes fo tchet model In this subsection we fist use k in nd k out s n exmple to demonstte ou stepbystep pocedue fo clculting the tnsition tes. Then we give the te fomuls fo the model without g10. The ection on the oto sites is epesented by t euilibium, sp61 + H + «sp61 H k [ sp61 ] = k sp61 H (11) in out sp H y definition, pk = ph + 61 log 10, fom which sp61 sp61 H sp61 = 10 pk ph The poton hopping tes, k in nd k out, e elted to pk by combining (11) nd (1): (1) 6

7 k k in out = 10 pk ph Fist conside the cse whee the potentil dop coss the membne is zeo nd thee is no sufce chge on the membne. Fo the cidic esevoi we hve k k ( 0) in pk ph 10 out ( 0) = The te k in(0) is limited by the te potons cn jump into the chnnel. We compute the poton te constnts fo enty into the chnnels by the Smoluchowski fomul 4 : The te k in(0) is given by (13) k in = è ç sufce poton è ç bsoption te to pefectly concenttion bsobing disk of dius (14) = ph kin nm p 13 poton concenttion t the cidic side bsoption te (15) Fom eution (13), the te k out(0) is given by pk k nm 10 4 (16) out = Using the pmete vlues listed in Tble 1 (ph = 7, = 0.5 nm nd p = 9.3x10 9 nm /s), we hve k in(0) = 1.1x10 3 /s. This is the te of potons hopping into the chnnel fom the cidic side when no sufce chge is on the membne. This poton inte is too smll to chieve the TP synthesis te of 400 TP/s. The poton inte cn be incesed by incesing the poton concenttion ne the membne sufce. The sufce chges on the membne incese (o decese) the sufce concenttion of potons, nd chnge the potentil dop inside the membne. The potentil dop inside the membne is the sum of the contibution fom membne potentil nd the contibution fom the sufce chges on the membne. The potentil dop inside the membne is y f f If thee is eul mount of sufce chges on ech side (i.e., f = f ), the contibution of the sufce chges is zeo. The potentil dop inside the membne does not ffect the te of potons jumping into the chnnel. It only chnges the te p (17) 10 ph is the poton concenttion in mole/lite. 1 mole/lite = 0.6 molec / nm 3 7

8 of potons jumping off the site. So the inte nd the offte fo the model without g10 e given by ph kin = 0. 6 nm 10 4 p f pk ( ) kout = 0. 6 nm 10 4 p exp f f y ç ph kin = 0. 6 nm 10 4 p f pk kout = 0. 6 nm 10 4 p exp y f f ç (18) (19) (0) (1) Fo the model without g10, the tnsition mtix enties e given by k = k = k F in k = k = k F out k = k = k F in k = k = k F out ().3 Moto including g10: lectostticlly ssisted tchet The tnsition tes fo the model with g10 e given by ph f k ( ) = 0. 6 nm 10 4 p pk ( f f ) y y y k ( ) = nm p ç exp expç ph f k ( ) = 0. 6 nm 10 4 p y f f pk y y k ( ) = nm p ç exp expç ph f kf ( ) = 0. 6 nm 10 4 p f f y pk yf y kf ( ) = nm p ç exp expç (3) (4) (5) (6) (7) (8) 8

9 ph f kf ( ) = 0. 6nm 10 4 p y f f pk y F y kf ( ) = nm p ç exp expç (9) (30). The electosttic foces ct ginst the oto motion The vege toue geneted diectly by the electosttic intections between the oto nd stto cn be computed fom p 1 tlectosttic = å S( ) y S ( ) d ò S = F,,, 0 t the opeting point fo TP synthesis, the vege toue geneted diectly by the electosttic intections between the oto nd stto is negtive. Howeve this is moe thn compensted by the effect of the electosttic intections on the oto sites' pk 's, which tightly couples the poton flux to the oto motion, nd incese the effectiveness of ectifying the otoõs diffusion. Tble 1 lists the pmete vlues used in the numeicl simultions. The complete simultion code in Mtlbª 5 is vilble on euest. (31) 9

10 PMT Vlue p = poton diffusion coefficient nm /s = oty diffusion coefficient of the oto 10 4 /s * e c = dielectic constnt of chnnel 10 e m = dielectic constnt of the membne 3 h = bilye viscosity h = height of the oto 1 poise 6 nm 1 / l = shielding length of stto chges 1.1 nm ph = bulk ph of the cidic esevoi moto = 7 pump = 6.6 ph = bulk ph of the bsic esevoi moto = 8.4 pump = 7.6 = 'dius' of the poton chnnel = dius of oto x = distnce between sp61 esidues = p/1 ph = ph diffeence coss the membne y = membne potentil f = potentil dop by sufce chges t the cidic side f = potentil dop by sufce chges t the bsic side Tble 1. Pmete vlues. 0.5 nm 5 nm.6 nm 80 mv = 3. k T 140 mv = 5.6 k T.3 k T (without g10).0 k T (with g10).3 k T (without g10).0 k T (with g10) * The diffusion coefficient of the oto ws computed fom = k T/(6phh ), nd ws modified to eflect the fct tht pt of the oto is not subject to the membne viscosity. Potentil dop of.3 k T by sufce chges cn educe the sufce ph vlue (incese the sufce concenttion) by 1. 10

11 efeences 1. isken, H. The FokkePlnck ution (SpingeVelg, New Yok, 1989).. oeing, C. in 1990 ectues in Complex Systems (eds. Ndel,. & Stein,.) 351 (ddisonwesley, edwood City, C, 1990). 3. Iselchvili, J. Intemolecul nd Sufce Foces (cdemic Pess, New Yok, 199). 4. eg, H. ndom Wlks in iology (Pinceton Univesity Pess, Pinceton, N.J., 1983). 11