Supplementary material

Transcription

1 Supplemenary maerial 3 curren, pa curren, pa.. 3 pdf ime, ms Fig. S. Dependence of he elecric curren j e J on he dimensionless proon moive force pdf RT (lef) and ime (righ). Full lines, Eq. (3). Doed lines, Eq. (6). proon moive force, mv bac no buff p 6 bac no buff p 7 bac no buff p 8 bac buff p 6 bac buff p 7 bac buff p 8 mio no buff p 6 mio no buff p 7 mio no buff p 8 mio buff p 6 mio buff p 7 mio buff p 8 mio buff p 7 3 curren, pa ime, ms Fig. S. Kineics of generaion of pdf (poins, lef ordinae, mv) and elecric curren j (line, righ ordinae, pa) calculaed by Eq. (6)

2 Calculaions of he rae consans for proon removal from region a The rae consans in Eq. (9) are k e, (S) U RT pka a,off,sp c c k c, (S) a,off,b a,off,b B a cb where c ps [], is he bimolecular rae consan for proon ransfer from he sur-,off,b face o he bulk and is he B concenraion. In order o esimae his consan, we will wrie down he equilibrium condiion for he proon exchange reacion beween he surface and he mobile buffer [],, (S3) pk B pka,off,b a,on,b a where is an apparen pka U RTln (S4) pk of he surface equal o.-6.9 a U 56 RT []. Assuming dn D (S5) a,on,b A B (we inroduced Avogadro number in order o express he consan in unis of M s ), d.6 nm (wo bonds), and aking he mobile buffer parameers as follows: he diffusion coefficien (S5) and (S) and N A 7 D B cm s [] and he concenraion 7 a,on,b.3 M s c.5c B B.5 mm, we obain from (S6) a k s a,off,b 4 pk B pk, (S7) which is of he same order of magniude as (S). For insance, a an inermediae value of he 6 barrier, U = RT, hey are boh 5 s. The oal rae consan for proon depopulaion of he surface, Eq. (9), can be less han (or on he order of) he rae of he iniial pmf increase, in which case appreciable local p changes should be expeced a low pmf. Mobile buffer concenraion is assumed o be much less han he oal buffer concenraion, he laer being usually abou o mm.

3 Mahemaical formulaion of he problem and is soluion The membrane is assumed o arac proons. The poenial well a he membrane surface (region a) has widh d and deph U. The bulk space ouside i is denoed region b. The free and bound proon concenraions in hese regions as funcions of ime and he space coordinae normal o he surface are represened in he form where a, a,eq, S, (S8), c z c N z,, (S9) b, b,eq, b,b, c z c N z,, (S) b,eq,b c d (S) a, a, is he surface concenraion, and sand for he volume concenraions (in M), S and N c c a, b, are he non-equilibrium proon concenraions creaed by he proon ing. The medium in general conains boh non-equilibrium free proons and non-equilibrium bound proons B, where B sands for he buffer. We will no consider hese wo non-equilibrium 3 proon pools simulaneously since i would be a significan ye unnecessary complicaion. Insead, we assume ha can be ignored when B is presen in physiological concenraions of - mm. Thus, he unknown funcions in he problem are (S8) and eiher (S9) or (S) plus dimensionless. The equilibrium concenraions obey he following relaions similar o (S) and (8): c d, (S) a,eq, a,eq, c c e, (S3) U RT a,eq, b,eq, L. (S4) c L,B c,eq,b a,eq,, b,eq, b Equaion (S) is merely he definiion of he surface concenraion. Equaion (S3) reflecs he increase of he surface concenraion due o he aracion. Equaions (S4) sae ha he free and bound proon concenraions in he bulk are proporional o each oher and o he surface concenraion of he free proons (B is assumed o be unable o penerae ino region a), being he equilibrium consans for he free and bound proons wih dimension of lengh. Combining hese equaions, we obain and L L,,B The same noaion N(z,) is used in Eqs. (S9) and (S), which does no lead o confusion since eiher free or bound proons are accouned for in he bulk, see below. 3 The equilibrium proons, neiher free nor bound, canno be ignored since hey boh are responsible for he p of he soluion. 3

4 where is he oal concenraion of he buffer, exp, L d U RT, (S5) L c K b,eq, B L, (S6) c,b, is he equilibrium consan of he buffer, and B, c c z c z, (S7) B B B pk K B B M (S8) pb,eq c b M,eq, (S9) is he equilibrium concenraion of he free proons in he bulk. The lengh scale of (S5) or (S6) (shorly L ) combined wih he respecive diffusion coefficiens (shorly D ) provides for a ime scale of or L D (S),, b,,b L,B D b,b (S) b (shorly ) for dissipaion of he non-equilibrium free or bound proons in he bulk [-3], which should be compared wih he proon dwell ime a he surface, k dw a,off. (S) ere, k k k (S3) a,off a,off,sp a,off,b is he rae wih which he proons leave region a eiher sponaneously or due o collisions of B wih he surface. The dwell ime is he ime required for a ed proon a he P side o leave he surface for he firs ime. If he diffusion is fas, so ha d w, he proon released will rapidly escape o he bulk wih no chance o reurn back o he surface. owever, ypically, he siuaion is jus opposie: afer leaving he surface, he proon says long nex o i and moves back and forh 4 making k (S4) a,off jumps before he full equilibraion wih he bulk is reached. In oher words, due o slow equilibraion wih he bulk, he non-equilibrium proons in he bulk creae back flow o he surface, 4 We called his he fas-exchange limi, as opposed o he slow-exchange one, < []. 4

5 hereby rearding he decay of he surface populaion. The back flow largely compensaes he forh one, so ha quasi-equilibrium is esablished beween he surface and bulk non-equilibrium proon concenraions a every momen of ime similar o (S4), S L N,. (S5) Similar consideraions apply o he N side as well 5 where he proon populaion depleed due o he ing is slowly resored because of slow diffusion from he bulk owards he surface. The above argumens imply ha ransien non-equilibrium proon populaions can be creaed by proon ing and mainained for a while a boh sides of he membrane. The obvious condiions for his o occur are ha he ing of local p changes in regions a, firs, be faser han he buildup of because oherwise he s will sop working before any significan p is creaed, and second, be faser han he proon dissipaion along he wo above-menioned pahways, i.e., he removal of he non-equilibrium proons from region a wih he rae k a,off and heir ulimae escape o he bulk wih he rae prevening heir reurn o he surface. In order o pu quaniaive crieria, we inroduce characerisic raes and [o be specified laer, a,p see Eqs. (S66) and (S69)] ha are required for he s o change p and by uniy. The ing will be efficien in producing local p changes when, firs,, a,p p increases faser han, (S6) and second, faser han a leas one of he above wo dissipaion raes, and. In he fasexchange limi (S4), wo scenarios are conceivable, and a,p a,off k a,off k (S7) k a,off a,p, (S8) which correspond o mechanisms and menioned in he main ex. Mechanism is he mos efficien due o higher proon ing rae. owever, as will be seen laer, based on esimaes of he parameers, i is expeced o operae only in srongly basic media whereas mechanism is no subjec o such a resricion. Afer hese preliminary consideraions we urn o he derivaion and soluion of he basic equaions. The proon diffusion in he bulk is described by he equaion 5 We acily assume ha parameers L, D,, and hence and b ka,off, are idenical a boh sides of he membrane. 5

6 , N z, N z D b z, (S9) where eiher free or bound proons are implied depending on he soluion. The boundary condiion a infiniy is N z,. (S3) A he membrane surface, he proon ing and he proon flow beween he surface and he bulk (proons per second) resul in changes of he oal number of proons in region a, N z, SmembNADb J SmembNAS, (S3) z z where and he meaning of he erms are explained below. Since he s ransfer proon numbers, n V N c S N, (S3) A a, a a, memb A a, raher han concenraions c a, in volume V S d, i was convenien for us o wrie (S3) in a memb erms of proon numbers. Now, dividing i by D b S memb N A, J N z z S N z memb, we obain a more compac equaion A S. (S33) The firs erm is he proon flow due o gradien of he proon concenraion in he bulk. The second erm is he proon ing due o curren J defined in he main ex as a funcion of pmf [see Eq. (6)], which in urn depends on. For he sake of argumen, we consider cyo- chrome-c-oxidase as a model proon. Then, for he P side and for he N side because only one half of he proons upaken a he N side reach he P side; S memb he membrane. In he righ-hand side, we negleced he erm due o recombinaion of is he surface of O or dissociaion of waer because he assumed aracion of proons implies he reduced concenraion of hydroxyl, so ha he hydroxyl conribuion is small in neural and acidic media 6. I is useful o define he oal number of proons in he bulk region, wih No N z, dz. (S34) The conservaion of he oal number of proons is easy o derive from Eqs. (S9), (S3), and (S33), d J N o S d S N memb A. (S35) 6 owever, his is no rue in alkaline media. 6

7 This relaion will be used laer. Following our previous approach [, 4], we approximae he firs erm in Eq. (S33) by he so-called Langmuir kineics, N z, D k L c z, b a,off b a, z, (S36) c b which represens he ne proon flow from he bulk o he surface as a difference beween he incoming and ougoing proon fluxes, boh being proporional o he respecive proon concenraions ( is concenraion of eiher free or bound proons in he bulk). In equilibrium, he flow vanishes according o (S4). Insering his ino (S33) and using (S8)-(S), we obain, J. (S37) ka,off L N S S SmembNA A his juncion, i is worhwhile o commen on he applicaion of he fas-exchange limi (S4). In his limi, boh incoming and ougoing fluxes in he righ-hand side of (S36) are large whereas heir difference is small, so ha relaion (S5) is approximaely fulfilled. owever, heir small difference mus be reained in Eq. (S33) because i is he slow diffusion in he bulk ha is responsible for he slow decay of he surface populaion. The diffusion equaion (S9) wih boundary condiions (S3), (S33) wihou he erm bu wih he iniial condiion solved in our BJ publicaion []; we will denoe i BJ S S, was. A more general problem, i.e., wihou invoking he fas-exchange approximaion, bu in he seady-sae case, was also solved in our JCP publicaion [4]. ere, we are ineresed o find a soluion of he ime-dependen problem in he general case because we need o consider shor imes afer he proon ing sars, a which no back flow is ye developed. We will show how our presen soluion denoed as BJ is relaed o S. Thus, following our mehod [, 5], we apply he Laplace ransform 7, o Eqs. (S9), (S34), (S35), and (S37): f f e d JCP S, (S38) d Db N zn dz No, N z z, (S39) dz, (S4) 7 The funcions f and f() are called he Laplace ransform and he original, respecively. 7

8 No, S J S,, (S4) memb, S ka,off L,BN S SmembN A N A J. (S4) The general soluion o Eq. (S39) obeying condiion (S3) is exp b N z N z D. (S43) Insering his ino Eq. (S4), we obain N o, N D b L N. (S44) Thus, we have hree equaions (S4), (S4), and (S44) for hree unknowns, from which we find S J where he righ-hand ideniy is he definiion of ino J,, JCP S SmembN A S memb NA JCP S. In he fas-exchange limi BJ S, (S45) (S4) i urns, (S46) which was obained in []. Finally, we have o perform he inverse Laplace ransform [6], f a i fe d i, (S47) where a is an arbirary posiive number. The produc of he Laplace ransforms in (S45) urns ino he convoluion of he originals [6], memb A a i JCP S S J d S N, (S48) where he dimensionless curren has been inroduced, J J. (S49) The oher funcion in he inegrand is a i JCP JCP S S e d i a i. (S5) This funcion represens he soluion o Eq. (S37) wih no bu wih he iniial condiion I is easy o show ha S JCP. (S5) 8

9 S JCP (S5) [see Eq. (S45)] is regular everywhere in he complex plane of excep for a cu along he negaive real half-axis. To be more specific, we make he replacemens so ha ka,off, k a,off, (S53) a i JCP S e d i. (S54) a i We draw he cu along he negaive real half-axis and pu i se,. (S55) The denominaor in he inegrand of (S54) vanishes only a Re, which is obained by solving a quadraic equaion. owever, according o (S55), Re s cos, (S56) so ha no oher singulariies are presen in addiion o he cu. Then, he inegraion conour can be displaced o he lef o circumven he cu, JCP i s s i s s S s e ds s e ds i i s i i s. (S57) A simple calculaion, afer replacemen s x, gives This expression is suiable when and, i.e., when mechanism is operaing. In paricular, when, he inegral comes from a viciniy of x. Replacing he inegraion variable, x JCP dx x S e x x. (S58) y, and exending inegraion over y from o, resuls in JCP dy S e e y. (S59) This resul is easy o undersand. In he original Eq. (S37), he incoming flux is absen because means rapid dissipaion (due o small ) of he non-equilibrium proons in he bulk, and no is working since iniial condiion (S5) is assumed insead. Then, he exponenial decay (S59) is immediaely obained. In he opposie case,, which gives, we rewrie Eq. (S58) by replacing x wih x and wih 9

10 JCP dx x S e x x,. (S6) In paricular, in he fas-exchange limi of Eq. (S4), his urns ino he expression BJ dx x S e x (S6) obained in [], which can be also derived by aking inverse Laplace ransform of Eq. (S46). I is suiable for reaing mechanism. Now we are in posiion o calculae he local ime-dependen variaions of p wihin region a. According o Eqs. (9), (S8), and (S) hey are given by where q a, q a, pa, (S6) ln a, S ln RT a,eq, is he change of he local dimensionless chemical poenial wih respec o is equilibrium value. Subsiuing (S48) for S, we obain (S63) for he N side and for he P-side, where JCP q ln S J d a, a,p (S64) JCP q ln S J d a, a,p (S65) n, n a,p,p a,eq, a a,eq, (S66) are he imescale for proon ing a wo sides of he membrane menioned earlier and JCP or S JCP are wih,, or,. S The rans-membrane difference of he elecric poenial obeys he equaion e JCP S ej, e. (S67) CmembSmemb Inroducing he dimensionless poenial, and he ime-scale for generaion, F Fe, e, (S68) RT RT

11 , (S69) e we obain he following equaion describing he kineics of generaion of he rans-membrane poenial difference: or, in he inegral form, J (S7) J d. (S7) Using Eqs. (), (), (7), and (S68), he dimensionless curren (S49) as funcion of he pmf componens is rewrien as J. (S7) exp 6 q q a, a, Thus, we have hree non-linear inegral equaions (S64), (S65), and (S7) o find hree unknown funcions q, q. If he curren J was a known a,, and a, funcion of ime, he above equaions would provide hese funcions. This implies ha he problem could be solved by ieraions. Namely, assuming some iniial (zero-order) ieraion, say,,, ln q q, (S73) a, a, we calculae sep-by-sep he s, he nd, ec. ieraions by insering he curren found from Eq. (S7) a he previous sep ino Eqs. (S64), (S65), and (S7). Convergence of he ieraions is shown in Figs. S3 and S4. A significan problem wih his mehod is ha can occur a he N side a an inermediae ieraion because of a bad selecion of he iniial ieraion, making complex, which prevens developing subsequen ieraions. The naural idea o inroduce q a, buffering funcion due o hydroxyl does no work, unforunaely, because our mehod is no applicable. We overcame his difficuly by dividing S (which is negaive a he N side) by a,eq, a, is maximum value (and subracing addiional. o avoid occasional zero). owever, ofen his mehod did no work, e.g., a p > 7. or cb mm. Therefore we also employed a second mehod where he unknown funcions are propagaed in small ime seps saring a =. Equaion (S7) can be rewrien alernaively in he form ln e J d, (S74) which can be used o conrol he accuracy of he calculaions.

12 Calculaion of he global p In he absence of he surface effec, and assuming fas equilibraion of he proons ed wih he bulk soluion, we can calculae he kineics of generaion of he global p. The concenraions of free and bound proons are changing wih ime according o he equaions J c c B c, (S75) O VN A J c c B c, (S76) O VN A J, (S77), where VV m 3 J, (S78) exp6q q 3 for membrane M and V m, V for membrane B. A every momen of ime, he concenraions of bound proons and hydroxyl obey he condiions of hermodynamic equilibrium, K. M B B B, O c c c c c Insering his ino Eqs. (S75) and (S76) gives where he buffering funcion is VN Ac VN Ac. (S79) J J c, c, (S8). M ck B B c c c K We divide Eqs. (S8) by equilibrium concenraions and inroduce he following definiions: c eq, B. (S8) q c c eq, ln ln p p eq, A, (S8) Vc N. (S83) Then, q J, q J p p. (S84)

13 Thus, we have o solve hree equaions (S77) and (S84) wih he iniial condiions The resuls are shown in Figs. S7-S9. () q () q (). (S85) 3

14 Fig. S3. Convergence of he ieraions for membrane M. Shown are he 3 rd (red), 4 h (green), and 5 h (blue) ieraions in p a () and p a (). The final ieraion is shown in Fig.. Fig. S4. Convergence of he ieraions for membrane M. Shown are he 3 rd (red), 4 h (green), and 5 h (blue) ieraions in j () and (). 4

15 Fig. S5. Kineics of p a (line ), p a (line ), and pmf (line 3) in membrane B a U 8RT. Fig. S6. Kineics of p a (line ), p a (line ), and pmf (line 3) in membrane B a U RT. 5

16 Fig. S7. Kineics of generaion of global p (line ), p (line ), and pmf (line 3) in membrane M. Buffer concenraion is mm. 6

17 Fig. S8. Same as Fig. S7, buffer concenraion is mm. Fig. S9. Same as Fig. S7, buffer concenraion is 8 mm. 7

18 Fig. S. Dependence of maximum p a (), minimum p a (), relaxaion ime (3) upon free-buffer concenraion (3) for miochondrial membrane. 8

19 Table S. Calculaed ime parameers (mks) for membrane B wih various barriers U RT 8 k a,off a,p 7 9

20 . Medvedev, E. S., and Suchebrukhov, A. A. (3) FEBS Le., 587, Georgievskii, Y., Medvedev, E. S., and Suchebrukhov, A. A. () Biophys. J., 8, Medvedev, E. S., and Suchebrukhov, A. A. () J. Phys. Condens. Ma., 3, Georgievskii, Y., Medvedev, E. S., and Suchebrukhov, A. A. () J. Chem. Phys., 6, Medvedev, E. S., and Suchebrukhov, A. A. (6) J. Mah. Biol., 5, Lavren ev, M. A., and Schaba, B. V. (987) Mehods for he Complex Funcion Theory [in Russian], 5h Edn., Nauka, Moscow (Lawrenjew, M. A., and Schaba, B. V. (967) Mehoden der Komplexen Funkionenheorie, Berlin VEB).