Thermodynamic and kinetic properties of electrogenic ion pumps

Biochimica etBiophysicaActa, 779 (1984) 3 0 7 - 3 4 1

307

Elsevier

B B A 85265

THERMODYNAMIC

AND

KINETIC

PROPERTIES

OF

ELECTROGENIC

ION

PUMPS

P. L , ~ U G E R

Department of Biology, University of Konstanz, D- 7750 Konstanz (F. R. G.) ( R e c e i v e d D e c e m b e r 6th, 1983)

Contents I.

Introduction ............................................................................

3O8

II.

I o n p u m p s as c h a n n e l s w i t h m u l t i p l e c o n f o r m a t i o n a l states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A T P - d r i v e n p r o t o n p u m p w i t h single b i n d i n g - s i t e a n d t w o r a t e - l i m i t i n g b a r r i e r s . . . . . . . . . . . . . . . . . . . . . . . . . . B. P r o t o n p u m p w i t h n b i n d i n g sites a n d i d e a l b a r r i e r a s y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. A l t e r n a t i v e m e c h a n i s m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

308 309 310 313

III.

Electrical p r o p e r t i e s o f ion p u m p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. E l e c t r o m o t i v e a n d p r o t o n m o t i v e force; reversal p o t e n t i a l s o f t r a n s l o c a t i o n p r o c e s s a n d c h e m i c a l r e a c t i o n . . . . . . . . . B. C u r r e n t - v o l t a g e c h a r a c t e r i s t i c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. E q u i v a l e n t - c i r c u i t r e p r e s e n t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. E x p e r i m e n t a l studies o f c u r r e n t - v o l t a g e b e h a v i o u r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 313 314 318 319

IV.

Behaviour near equilibrium ..................................................................

320

V.

E q u i v a l e n c e of electric a n d o s m o t i c d r i v e n forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

VI.

U n i d i r e c t i o n a l fluxes a n d e x c h a n g e flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

VII.

Energetics .............................................................................. A. O p e r a t i o n m o d e s o f i o n p u m p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. F r e e - e n e r g y t r a n s d u c t i o n a n d f r e e - e n e r g y d i s s i p a t i o n ; t h e r m o d y n a m i c efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . C. E n e r g y levels o f t r a n s p o r t p r o t e i n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

322 322 323 325

VIII. L i g h t - d r i v e n i o n p u m p s

....................................................................

329

IX.

Type I ATPases ..........................................................................

333

X.

T r a n s i e n t b e h a v i o u r a n d electrical n o i s e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

334

XI.

Conclusion .............................................................................

335

Acknowledgements ............................................................................

336

Appendices A-D

336

.............................................................................

References ..................................................................................

0 3 0 4 - 4 1 5 7 / 8 4 / $ 0 3 . 0 0 © 1984 Elsevier Science P u b l i s h e r s B.V.

339

308

1. Introduction Active transport systems capable of accumulating or extruding ions against a gradient of electrochemical potential are widespread in cellular membranes. Depending on the nature of the driving force, different types of active transport may be distinguished. In co- or countertransport systems uphill movement of a particle species A is coupled to downhill movement of a second species B, such as in the Na ÷,Ca z+-exchange system of nerve and muscle cells. This process, sometimes referred to as 'secondary active transport', will not be considered here. In contrast to co- and countertransport, which are driven by electrochemical gradients, primary active transport systems (or 'ion pumps') utilize light, redox energy or ATP hydrolysis as energy sources. Two different classes of transport ATPase are known to exist [1,2]. Class I contains the proton translocating FoF1-ATPases of mitochondria, chloroplasts and bacteria which are distinguished by their complex subunit structure and which, in the physiological range, may work in either direction (ATP hydrolysis as well as synthesis). The second class of transport ATPases includes the ( N a ÷ + K ÷)-ATPase of animal cells, the Ca2÷-ATPase of sarcoplasmic reticulum, and the proton-translocating ATPase of fungi (Neurospora, Saccharomyces). Type II ATPases have a much simpler subunit structure and, under physiological conditions, function only in the ATP hydrolysis mode, acting as ion pumps [3]. Furthermore, type II ATPases are known to go through a cycle of phosphorylation and dephosphorylation steps coupled to conformational transitions, whereas a phosphorylated intermediate seems not to be involved in the functioning of type I enzymes [2]. Most ion pumps studied so far are electrogenic, i.e., they translocate net electric charge across the membrane. The ( N a + + K÷)-ATPase [4-6] normally transports two potassium ions inward and three sodium ions outward per ATP hydrolysed and thus translocates one elementary charge during each cycle. Other well-known examples of electrogenic ion pumps are the light-driven proton p u m p of halobacteria [7] and the proton translocating ATPase of Neurospora [1,8-10]. The proton pump of Neurosporais capable of maintaining

a large membrane potential of about - 2 0 0 mV (8.9) and consumes more than a third of the metabolically produced ATP of the cell [11]. From a mechanistic point of view, electrogenic ion pumps are particularly interesting transport systems. They convert chemical into electrical energy and they depend on an additional variable, the membrane potential. Furthermore, by means of the transmembrane electric field which they create, they may modulate other transport processes and metabolic activities of the cell. At present, a strictly 'microscopic' description of p u m p function is not possible, since the essential structural and mechanistic properties are incompletely known for all the active transport systems mentioned above. So far, the theoretical analysis of active ion transport has to be based on certain general concepts which are valid irrespective of the special mechanism. In this article, the action of ion pumps is discussed on the basis of a generalized channel model. An ionic channel is a transmembrane pathway of low potential energy. Inside the channel the ion interacts with polar ligands such as water molecules or peptide carbonyls. The potential energy profile of the ion along the channel consists of a sequence of energy wells separated by activation barriers. An ion channel functions as a pump when the channel molecule goes through a cycle of conformational transitions during which the barrier structure is transiently modified in an appropriate way. Absorption of a light quantum, transition to another redox state or phosphorylation of the channel protein may alter the binding constant of an ion-binding site in the channel and, at the same time, change the height of adjacent barriers. In this way an ion may be preferentially released to one side, while after transition back to the original conformation of the channel another ion is taken up from the opposite site. The following treatment explicitly deals with ATP-driven proton pumps of type II, but (as discussed later) a similar analysis can be applied also to other active transport systems. 11. Ion pumps as channels with multiple conformationai states Ion pumps, as far as is known, are large transmembrane proteins. In the presence of an energy

309

source the protein goes through a cycle of conformational transitions and thereby translocates one or several ions through the membrane. In order to function as a pump, the protein must have binding sites for ions, sites where an ion is in an energetically favourable interaction with the protein matrix. In a given conformational state an ion in a binding site may have easy access to the cytoplasm but may be prevented from escaping to the extracellular medium by a high energy barrier. In another state the ion may be virtually trapped inside the protein. For a formal description of the pumping process the pump protein may be treated as an ionic channel, and a potential energy profile (a sequence of energy wells and barriers) may be assigned to the channel in each conformational state (Fig. 1). The idea of representing a pump as a channel with multiple conformational states had been introduced by Patlak in 1957 [12] and has since been discussed in various forms in the literature [13-21,59]. An advantage of a channel model is the possibility of describing ion translocation within the pump molecule by the method of rate theory [22-27]. In particular, as will be discussed later, the voltage dependence of transport rates can be treated in a straightforward way using rate-theory analysis.

IIA. A TP-driven proton pump with single bindingsite and two rate-limiting barriers We first consider a channel with a single proton-binding site and two rate-limiting barriers on either side (Fig. 1). Towards the aqueous phases a number of smaller barriers may be present which allow fast diffusion of the proton, so that the outer energy wells are always in equilibrium with the respective aqueous phases. We assume that in the dephosphorylated state of the pump (state HC) the protonation site is mainly accessible from the left (cytoplasmic) side (Fig. 2). Phosphorylation of the channel protein results in a transition to a conformation H E in which the binding site is exposed to the right (extracellular) side. State H E may either return directly to HC (which would short-circuit the pump) or through the intermediate states E and C (HE ~ E ~ C ---, HC). During this cycle the proton is (preferentially) released to the extracellular side and another proton is taken up (preferen-

cyIoplasm

membrane

extracellular medium

(.E/I

(Jr)" C"

C'

,

i

i

i

,,

Fig. 1. Channel model for an ion pump. The pathway of the ion through the channel is described as a series of energy wells and barriers. Well i represents the (main) ion binding site which (in conformational state C) is easily accessible from the left-hand side but separated from the fight-hand side by a high energybarrier. Toward the aqueous phases a number of smaller barriers are present which allow fast diffusion of ions. ~k is the electric potential and c the concentration of the transported ion. K~ and r ~ are the rate constants for the release of the ion from the binding site, and )k'c and X'~ are the entry rate constants.

tially) from the cytoplasmic side. The reaction scheme depicted in Fig. 2 represents a highly simplified model of an ATP-driven proton pump. In reality the pumping cycle is likely to include additional conformational states, as discussed in subsection IIB. The model accounts for the possibility of passive leakage of ions over the barriers, if the transition rate constants K', K", ~', X" (Fig. 1) in states H C / C and H E / E are assumed to be finite. Under this condition, coupling be-

Fig. 2. M i n i m u m model for an ATP-driven proton p u m p with a single proton binding site. In the dephosphorylated state ( H C / C ) , the proton binding site is (mainly) accessible from the left (cytoplasmic) side, in the phosphorylated state ( H E / E ) from the fight (extracellular) side. During the cycle HC ---, HE ---, E ~ C ~ HC, a proton is translocated from the cytoplasm to the extracellular medium, k~: E and K ~ c are the rate constants for the conformational transitions in the protonated form, kcE and kEC are the rate constants in the deprotonated form.

310

tween ATP hydrolysis and ion flow is incomplete. The flux equations which are obtained in the general case of barriers of arbitrary height are given in Appendix C. Since barrier leakage reduces the efficiency of a pump, it is likely that real pumps function in such a way that the barriers in states H C / C and H E / E are strongly asymmetric. Accordingly, it is assumed in the following sections that in state H C / C the barrier to the right is extremely high (x~. = 0, Xc - O) and in state H E / E the barrier to the left

(,~ = 0, ~"E = 0). IIB. Proton pump with n binding sites and ideal barrier asymmetry

As a basis of further analysis of pump behaviour, we introduce a model which is thought to contain some of the essential features of real proton pumps, yet is sufficiently simple to permit a concise representation of the flux equations. In this model, which is represented in Fig. 3, phosphorylation-dephosphorylation reactions and con-

HnE

~. ~

E

+ HnX~

~:~

ADP..~, ATP~I kxC H,C

Y

kcyllkyc ~

~"

C

Fig. 3. ATP-driven proton pump with n = 2 proton binding sites. In state H , C / C , the binding sites are in equilibrium with the left-hand (cytoplasmic) side but are separated from the right-hand (extracellular) side by a high energy barrier; in state H , E / E the sites are in equilibrium with the extracellular side, but completely separated from the cytoplasm. Phosphorylated states of the protein are designated by the label P. K c and K E are the equilibrium constants for the protonation-deprotonation reactions. Phosphorylation-dephosphorylation reactions and conformational transitions are assumed to occur only in the fully protonated and fully deprotonated states.

formational transitions are treated as separate processes. This requires the introduction of two additional states ( H , X and Y) into the reaction scheme. Besides this, the model is based on the following assumptions: (1) In states H , C / C and H , X the binding sites are accessible only from the cytoplasm and in states H , E / E and Y only from the extracellular medium. (2) The pK values of states X and Y are such that X is always fully protonated and Y always fully deprotonated. (3) The rate constants of protonation and deprotonation are large, so that in state H n C / C the binding sites are always in equilibrium with the cytoplasm and in state H , E / E with the extracellular medium. This assumption is consistent with the fact that protonation reactions are generally very fast. (4) Conformational transitions can take place only in the fully protonated ( H , X ~ H , E ) or in the fully deprotonated states (E ~ Y). (5) For generality, we assume that the enzyme in state H,C may be phosphorylated not only by ATP but also by direct reaction with inorganic phosphate (the latter process may often be negligible, however). Correspondingly, dephosphorylation of state H , X can take place either directly, or by reaction with ADP (leading to ATP synthesis). Thus, the phosphorylation/ dephosphorylation reactions may be summarized by: H . C + ATP ~ H . X + ADP

(1)

H.C+P i ~ H.X

(2)

E~v+v~

(3)

Denoting the concentrations of ATP, ADP and inorganic phosphate (P~) by c T, Co and co, respectively, the rate constants of the phosphorylationdephosphorylation reactions (Fig. 3) may be written as: kcx = pC-r + qcp

(4)

k x c = rCO+ S

(5)

kvE = wcp

(6)

311

p, q, r, s, w and k e y (Fig. 3) are concentration-independent quantities. According to assumptions 1 and 3, the protonation reactions HnC ~ C + n H and HnE ~ E + n H are described by equilibrium constants K c and KE: Kc

U~(c')"

u~(¢")"

r E

NH.C

NH.E

(7)

N z is the number of p u m p molecules in state Z per unit area of the membrane and c' and c" are the proton concentrations in the cytoplasm and the extracellular medium, respectively. The principle of detailed balance (or microscopic reversibility) requires that in the absence of driving forces each partial reaction is in equilibrium separately. This means that certain relations between the rate constants must be fulfilled. Introducing the equilibrium constant K-~CD~'p/CT of ATP hydrolysis (CT, CO, CV being a set of equilibrium concentrations), these relations are obtained as: ps qr KE Kc

~D~p r

AG = A G O + R T In CDCP

=

RTIn

¢T

CT//CDCp ¢T//CDCP

(12)

A/2 n is the electrochemical potential difference of the proton (cytoplasmic side minus extracellular side). AG is the Gibbs free energy of ATP hydrolysis (AG < 0 under physiological conditions), and AG o = - R T In K = - R T ln(3D3P/~T) is the standard value of AG. ~'n and ¢~ may be calculated in a straightforward way for a p u m p molecule operating under steady-state conditions. Using the abbrevations: p exp(A~n/RT)

=

exp( - A G / R T ) =

S~-exp(u)-= v c

CT/CD%

CT/CDFp

m

Y

(13)

(14)

The result reads (Appendix B): ¢kH=nA1[(Y + s/ rcD ) ( V' - - I ) +( Y - - 1 ) ]

(15)

q,¢= A, [(v"- 1 ) + v " B i ( Y - 1 ) ]

(16) (17)

(8)

CT

kxEkEykycq kExkcysW

_ ~ ' - ~"_

exp(nu)

v

(9)

(10)

R r / F - RT/F

(compare Appendix A). u is the voltage across the membrane, expressed in units of R T / F --- 25.7 mV (at 25 °C), R is the gas constant, T the absolute temperature and F the Faraday constant. The voltage dependence of the single rate constants is discussed in subsection IIIB. The activity of the p u m p is described by two flux quantities, the net proton flux, ~H, and the net ATP hydrolysis rate (or 'chemical flux'), ~ , which are both referred to a single p u m p molecule. ~H is the number of protons translocated from the cytoplasm to the extracellular solution per unit time and ~ is the number of A T P molecules hydrolysed per unit time (d?¢ is negative when A T P is synthesized). ~H and ~bc depend on the thermodynamic quantities A/2 H and AG: ¢P A~H = ~t'H -- g'fi = RTIn---S, + F ( ~l,"- tp") c

(11)

B1------1+ s

kExkvE + k E x k v c + k E y k v c K E / C ''n

(18)

k x E k E y k y c K E / C '''~ D 1 =-(kvc + k y E ) ( k c x k x E + k c x k E x + k x c k E x )

+ k¢~k,~E(kx~ + k ~ , , ) ] ( K U c ' " ) + [kcxkxE(kyE + kvc)+ kcxkEv(kxE + kvc )

+ k,,ck~,~(kxE + k x c ) ] ( X ~ / c " ) + (k,,c + kx~) )< ( k v E k c v + k E v k c v + k E v k v c ) ( K c K E / C ' " c ' '

)

(19)

Eqns. 15-19 describe the fluxes t~H and q~c as functions of the thermodynamic driving forces (o n - 1 ) = [ e x p ( n A g H / R T ) - 1] and ( Y - 1 ) = [ e x p ( - - A G / R T ) - - 1 ] . Both fluxes vanish for A/2 H = AG = 0, as must happen. The properties of q~u and q~c will be discussed in detail in Sections I I - V . The coupling ratio, 0, of the p u m p is defined as

312

the number of protons transported per ATP hydrolysed: 0~ en

(20)

According to Eqns. 15 and 16 the coupling ratio is a function of the driving forces and is, in general, different from the number, n, of protonbinding sites. Evidence for variable coupling ratio has been obtained for several active transport systems [8,21,28,48]. 8 becomes equal to n if the rates of direct phosphorylation and dephosphorylation are negligible (q = 0, s = 0). In this case, where ion transport and chemical reaction are completely coupled, Eqns. 15 and 16 reduce to: . [

q>u=nCk=nA,(v"y-1)=nA,[exp~

{nA~H--AG)

-~-

-1

]

(21)

--(flASH--AG) is the free-energy change of the coupled reaction n H +' + ATP ~- n H +'' + A D P + PiThe s t o i c h i o m e t r i c ratio, n, which is equal to the number of ion-binding sites directly involved in transport, should be distinguished from the coupling ratio, 0. Whereas the c o u p l i n g ratio is variable and depends on the driving forces, the stoichiometric ratio has a fixed value which is determined by the transport mechanism. Only in the limit of perfect coupling do 0 and n become identical. The m a x i m u m t u r n o v e r rate, f, of the pump is the limiting ion flux, q~n, which is reached at optimum substrate concentration for vanishing electrochemical gradient:

f~(*H)max(CT--~OO,CD,Cp---~O,ct=¢tt=C,u=O)

(22)

For a completely coupled pump (Eqn. 21), the maximum turnover rate becomes: +

+

+

+~

+

k~E = pc T + qcp

(24)

k~c = rc D + s

(25)

kcE = WCp

(26)

In this case, the following relations hold instead of Eqns. 9 and 15-19:

KE kEcq K(.

(23a)

sw

exp(nu)

(27)

dpH=nA 2[(Y+s/rcD) (v n - 1 ) + ( Y - I ) ]

[

~c=A2 ( v " - l ) + o " A

(28)

(,,n) ]sc

l + k E ~ KE ( Y - l )

(29)

1 Kc

(30)

2~ D~ C'~kCErcD

KE D2=-k~E + k~. + (/,~c + k~-E)~Kcc + ( k ~ + kEc) ,,~

KcKE

+(kcE + k E C ) ~

(31)

The simplified mechanism depitected in Fig. 4 represents a minimal model for an ion pump. It is useful as a framework for the discussion of pump properties, in particular, if experimental information on rate constants is incomplete.

(23)

If the pK in state E is sufficiently low for the binding site to be mostly deprotonated ( K E >> c"), Eqn. 23 reduces to: f =n

Eqn. 23a expresses the fact that at optimum ion and substrate concentrations the flux is limited by the rate constants of conformational transitions. A simplified version of the mechanism depicted in Fig. 2 is obtained if the phosphorylation/ dephosphorylation reactions and subsequent conformational transitions are combined into a single step (Fig. 4). In analogy to Eqns. 4 - 6 the transition rate constants are then given by:

HnE

4--KE.

EC ATP~

E

kCE

EG

i•

HnC

Kc.

C

Fig. 4. Simplified version of the mechanism depicted in Fig. 3. Phosphorylation/dephosphorylation reactions and conformational transitions are combined into a single step.

313

HC. Alternative mechanisms In the models discussed above, it has been assumed that phosphorylation by ATP occurs in state HC. This is by no means a necessary condition for pumping, since the external driving force can be coupled to the cycle at any intermediate step. For instance, phosphorylation by ATP could occur in state E, driving the protein into state C with the binding site exposed to the cytoplasm. A further possibility lies in the assumption that transitions between left-exposed and right-exposed states of the binding site do not require phosphorylation/dephosphorylation, but that the reaction with A T P is necessary in order to change the p K values of states H n C / C and H ~ E / E . These different possibilities are summarized in the following reaction scheme in which phosphorylation reactions have been marked by heavy arrows and phosphorylated states by a squiggle. Phosphorylation preceding conformational transition: HnC =, Hnl~ ~ I~ ~ C ---' HnC, or

Phosphorylation preceding protonation-deprotonation:

electrochemical potential difference A/a builds up which counteracts the chemical driving force. At a certain value of A/a, ion flow through the p u m p ceases, and if a higher A/a is imposed, q~H reverses its sign. The value of A/art (~H = 0), divided by the Faraday constant, is the so-called protonmotive force (PMF) of the p u m p *: PMF---(A/iH/F),.-

o

(32)

z&/a/F = (/a'H --/a'~)/F has the dimension of a voltage. It should be clear that P M F is the counterforce at which ion flow, q~n, through the pump molecule vanishes. In the presence of leakage pathways, the total ion flow through the membrane will differ from zero, even for q'n = 0. The membrane voltage at which ion flow (or p u m p current) reverses direction is usually referred to as the reversal potential or the electromotive force (EMF) of the pump. The E M F is the open-circuit voltage which can be generated by the p u m p if leakage pathways through the membrane are negligible. According to the definition of A/a H (Eqn. 13) the E M F is the difference between the protonmotive force and the Nernst potential of the transported ion: EMF---(+'- f f " ) , . _ o = P M F - R~Tln c-F c"

(33)

H . C ---, H,,E ~, E ~ C --* HnC , or H.(~ --* H.I~ -* E --, C ~ H . C

(Note that intermediate steps such as H C ~ H(~ --* HI~ have been omitted for simplicity.) These reaction schemes may be analysed in the same way as discussed in the previous subsection. The resulting flux equations have the same general properties as Eqns. 28 and 29, i.e., the fluxes ~'n and @c are expressions in the driving forces (v n - 1) and ( Y - 1).

III. Electrical properties of ion pumps I l i A . Electromotive and protonmotive force," reversal potentials of translocation process and chemical reaction If an ion p u m p starts to work under initially symmetrical conditions (c' = c " , ~k' =~k"), an

The protonmotive force of a p u m p with n binding sites is obtained from Eqn. 15 as RT PMF= - ~ln

Y + s/rc D 1 + s/rc D

(34)

In the limiting case of perfect coupling (s, q ---, 0) the protonmotive force is directly related to the free energy, AG, of ATP hydrolysis: PMF=

-- R--T-Tln Y = a G nF nF

(35)

Under physiological conditions, AG is of the order of - 5 0 to - 6 0 k J / m o l in cells with aerobic metabolism [29,30], which corresponds to a P M F of about - 5 0 0 to - 6 0 0 mV. For incomplete

* The term p r o t o n m o t i v e force is used in the literature also in a m o r e general sense to d e s i g n a t e arbitrary values of A g H / F .

314

coupling, IP M F I is always smaller than I A G / n F I, as seen from Eqn 34. According to Eqn. 34, if the rate of spontaneous dephosphorylation is large (s >> rc D) the protonmotive force goes to zero, meaning that the enzyme acts as an uncoupled ATPase [49]. This illustrates the fact that efficient operation of an ion p u m p requires that certain reaction steps are kinetically inhibited. It is interesting to note that the expression for the PMF (Eqn. 34) does not contain the proton dissociation constants K c and K E. This means that a p K difference between states C and E is not critical for the thermodynamic efficiency of the p u m p [16,49,62]. The essential feature of the pumping mechanism considered here is an ATPdriven conformational change which switches the binding sites from a left-exposed to a right-exposed state. On the other hand, as seen from Eqn. 23, a low p K of state E ( K E >> c n) is favourable for a high turnover rate of the pump. Since the rate, 'he, of ATP hydrolysis (or synthesis) depends on A/2 H, a reversal potential may be expected to exist for which ~¢ becomes equal to zero. From Eqn. 16, the reversal potential (expressed as A/2H) of the chemical reaction is obtained as: ( A g H ) ¢ , , - O = -- - - ~ l n [ 1 + B , ( Y - 1 ) ]

(36)

where B 1 > 1 is defined by Eqn. 18. The right-hand side of this equation assumes real values whenever Y > 1 - 1 / B r Thus, under physiological conditions where Y is larger than unity, a reversal potential of the chemical reaction always exists. At the value of A/~ H given by Eqn. 36 hydrolysis of ATP ceases, and at more negative values ATP is synthesised. ATP synthesis driven by a gradient of electrochemical potential has been demonstrated both for the ( N a + + K+)-ATPase [31] and the Ca2+-ATPase [32,33]. Eqn. 36 may be compared with the electrochemical reversal potential F (PMF) of the translocation reaction:

(A/~rl)q,._0 =-

~-

In

Y + s/rCD l+s/rc o

(37)

It is obvious from Eqns. 36 and 37 that the rever-

sal potentials of the chemical reaction and the translocation process are, in general, different (they become identical in the limit of perfect coupling, i.e., for s = 0). For Y > 1 the following relations hold: ( , ~ . ) , . _ o >~- -RTln Y = -AG n ?/

(38)

( a / 2 . ) , ~ _ o ~ - R----TInr = n

(39)

AG n

Thus, the reversal potential of the transport process is always less negative than the thermodynamic limit A G / n < 0, whereas a more negative value of A/2 H is needed in order to bring the rate of the chemical reaction to zero (Fig. 7). This is plausible, since A T P hydrolysis still proceeds for Cn = 0 (if coupling is incomplete). IIIB. Current-voltage characteristic The current-voltage behaviour of the p u m p is determined by the voltage dependence of the individual reaction steps. It is often assumed that only a single step, the translocation of the binding site, is voltage-dependent [8,11,36,37]. This assumption, if taken literally, would mean that the ion-binding site has to move over the whole membrane dielectric. A more realistic approach may be based on the assumption that the binding site is located inside a channel so that part of the voltage drops between the external phase and the binding site [16,21,35,71]. We first consider the translocation of the ion within the channel in state C (Fig. 1). According to the assumption of free diffusion within the entrance parts of the channel, the energy minima (i - 1) and (i + 1) are always in equilibrium with the respective aqueous phases. If "( and 3," are the fractions of total voltage u = F ( ~ k " - ~ " ) / R T dropping across the freely accessible parts of the channel (Fig. 1) and if PJ-1 and Pi+l are the probabilities that minima ( i - 1) and (i + 1) are occupied by an ion, the relations:

p,_,=h'c' exp(-f~u);pi+l=h"c" exp(-3,~'u)

(40)

hold, where h' and h'" are voltage-independent constants. Eqn. 40 means that the probability of

315

occupancy of the ( i - 1)th potential minimum increases with voltage u. The notion that in a channel a voltage change can be converted into a change of occupancy probability (or concentration) is the basis of the so-called proton-well concept [34,35,45,50]. The voltage dependence of the rate constant o~ for transitions from the (i - 1)th to the ith potential well may be described by the rate-theory expression [24,38]: ~ = 6~ exp(-/~_,u/2)

(41)

5~ is the value of o~ at zero voltage and 3%~ the fraction of voltage dropping across the barrier (Fig. 1). The overall rate constant p~ for transitions from the cytoplasmic phase to the binding site is than obtained from the relation: A~=p~c'=o~p,_ I

(42)

This yields (with ~ = h'#~): P'c-- P'c exp [ ('t~ + %_ 1 / 2 ) u]

(43)

For the other rate constants one finds in a similar

3"-=~" + ~i-l; d'-= ¢" + ¢i -

_

-Pt

t

-v

.

pt

(44)

-tv

" c = r c exp(Yiu/2), gc = ~c exp( - % _ , u / 2 )

(45)

y~ + 'yi_ 1+ yi + y~' = 1

(46)

Analogous expressions hold for state E if %', ~/', ~'i-~ and ~5 are replaced by the quantities %, %, ci-~, ¢i which obey the relation t

¢'* + ' i - I + ' , + ¢~' = 1

tP

pt

Pc

(48)

i

K E - - -- •Epv = / ( E e x p ( c " u ) OE

__

~t

~it

av = - eE

q, a x , = - V e o ~ n , a x , / a

i

(51)

i

The second part of this relation is based on the assumption that the electric field is constant within the membrane so that E = V / d (d is the membrane thickness or, strictly speaking, the length over which the external voltage drops in the pump molecule). The conformational transition from state C to E may be described by a reaction

(47)

In the following, we consider the simplified model of subsection II B (Fig. 4) under the assumption of a single binding site (n = 1). The proton dissociation constants in states C and E then become Kc = ~ = ~c exp(- ~'.)

-

K c = r c / p c and K E = r ~ / p E are the values of K c and K E at zero voltage. Since conformational transitions of the pump molecule in general involve displacements of charged groups and rotation of dipoles, the transition rate constants become voltage-dependent. For instance, a-helices have large dipolar moments corresponding to about half an elementary charge at either end of the helix [58]. In order to describe the contribution of such charge movements to the voltage dependence of rate constants, we consider the pump molecule as an assembly of point charges qi (Fig. 5). The qi contain free charges as well as charges from dipolar groups. During a transition from conformation C to E the charge q~ = ,/ie0 is displaced over a certain distance AIi (e 0 is the elementary charge). If Ax i is the component of AIi in the direction of the external field E, the contribution of membrane voltage V = ~ ' - ~ " to the total energy 'change during the conformational transition is given by

way:

p~ = h ' b / c " = ~'~ exp[ - ( ~,;'+ "ti/2) u]

~t

(50)

(49)

k0' ~

ko"

Fig. 5. For the description of voltage-dependence of transition rate constants the pump molecule is considered as an assembly of point charges qi. During a transition from conformation C to E charge q~ is displaced over a distance AL A x i is the component of AI~ in the direction of the external field E. d is the membrane thickness.

316

coordinate )~ (0 ~< 7~~< 1). If X~c and xm are the coordinates of point charge q~ in states C and E, the position x, during the transition is given by: xi = X,c + ~( x,E - Xic) = X,c + Xax,

(52)

According to the theory of absolute reaction rates [39], the conformational transition may be treated as a passage over an energy barrier (Fig. 6). In the presence of an electric field, the height of the barrier is modified. Assuming a sharp, symmetrical barrier (peak position at ~, = 1/2) the peak height changes by approx. A U / 2 so that the transition rate constants become: kc~ = kCE exp( - AU/2kT) = kCE exp(*lu/2)

(53)

kEC = kEc exp( - "qu/2)

(54)

*l=--~,n, A x , / d i

(55)

During the conformational transition C--, E, the proton binding site in general moves a certain distance, X" d, in the direction of the external field ( - 1 < X < 1). This leads to an additional chargedisplacement term in the expressions for the voltage dependence of the rate constants k ~ v -----pcT + qCp and k~c = rc D + s, which describe the transitions with occupied binding site (HC ~ HE). Thus, in addition to Eqns. 53 and 54, one has: p=pexp[(n+ X)U/2];q=glexp[(,q+ X)u/2 ]

(56)

r = ~ e x p [ - ( ' O + X ) U / 2 ] ; s = g e x p [ - ( ' O + X)U/2]

(57)

~ v AU2 / =0

Furthermore, using Eqns. 26, 27, 48, 49, 53-57, it is seen that the quantities 3", C' and X are connected by the following relation: y'+g'+x=l

Thus, the voltage dependence of ~u and ~,¢ may be described by three independent parameters, ¥', C' and 7- Depending on the values of y', c" and ~, the current-voltage characteristic of the p u m p may assume a variety of different shapes. This is illustrated by the following examples.

y'=c"= 1/Z 7=0 This corresponds to the case in which the ion binding site does not move during the conformational transition from H C to H E (Fig. 7) and in which all transition rate constants are voltage independent ( * / = X = 0). The only voltage-dependent quantities are the equilibrium dissociation constants K c and K e (Eqns. 48 and 49). In Fig. 7 the net p u m p current (given as the proton flux ~ u) as well as the net rate of ATP hydrolysis, ~c, are plotted as functions of voltage V under the condition c ' = c" for incomplete coupling of the p u m p (q, s > 0). As predicted by Eqns. 38 and 39, the reversal potential, V0H, of the proton flux is smaller, and the reversal potential, V~, of the chemical reaction is larger (in absolute magnitude) than the electromotive force, , ~ G / F , of an ideally coupled pump. As may be expected for non-ideal coupling of the pump. the short-circuit hydrolysis rate, ¢?c(0), is larger than the short-circuit ion flux, ~H(0). It is also seen from Fig. 7 that ~ H ( V ) as well as ~c(V) saturate both at high and at low voltage, the saturation values being given by q~n(oo)= ,k~ekEC ;qart(--o0 ) k~E + kEc ~,c(o~ )

8

0

1

A~

Fig. 6. Conformational transition from state C to E, described by a reaction coordinate h. In the presence of an external voltage V, the profile of conformational energy is modified. For a nearly symmetrical barrier the height of the barrier is reduced (or increased) by approximately half of the total electrostatic energy change AU.

(58)

kcEk~c kCE + '~C

PcrkEc :~, ( _ ~ ) = ~CDkcE pCT + kE~--~ -- ~co + kCE

(59)

(60)

Saturation always occurs if the rate of at least one reaction step is independent of V, so that at high voltage the voltage-independent processes become rate-limiting. 3"=C'=l/4,

rl=m

In this case it is assumed that the ion-binding

317

60- -

t

HE ~ l

4O 20

TIIi

0 -20

+:+/4

.c

-40

-3~o-~

-16o

o

loo

200

3oo

V/mY Fig. 7. Proton flux ~H and ATP hydrolysis rate ~c of a single p u m p molecule as functions of voltage V for identical proton concentrations on both sides ( c ' = c ''= - c). It is assumed that the proton-binding site remains fixed during the conformational transition HC ~ HE and that all transition rate constants are voltage independent, eH and ~c have been calculated from Eqns. 28, 29 and 53-58 using the following values of the kinetic parameters: n = 1, K c / c = 1, I ( E / c = 10, kcE = kEc = 100 s - ] , /SCT = 1 0 3 S- ] , @ D = 1 0 S-], ?CD = 1 0 0 S-1, ,~ = 300 S-1, Y = I O 0 , 7 ' = c " = 0.5, 7/= 0. In accordance with Eqns. 38 and 39, the reversal potential, V~, of the proton flux is smaller, and the reversal potential, V~0,of the chemical reaction is larger (in absolute magnitude) than the electromotive force A G / F of an ideally coupled pump.

site moves over a distance corresponding to half the dielectric thickness of the membrane during the conformational transition and that, at the same time, m elementary changes are translocated across the dielectric in the same direction (Fig. 8). The quantity rn (which can be positive or negative) accounts for any charge displacement in the protein other than the movement of the transported ion. If rn is equal to - 1 / 4 (corresponding to the translocation of one negative charge over onefourth of the membrane) all forward rates are increased and all backward rates are decreased at positive values of V. This results in a superlinear current-voltage characteristic, as shown in Fig. 9. For large (positive or negative) values of m, the membrane voltage exerts a strong driving force which tends to lock the p u m p in one particular state ( H C / C or H E / E ) . For instance, if m is positive, transitions HC--* H E and C - ~ E are favoured for V > 0; with increasing voltage more and more p u m p molecules accumulate in states H E and E so that the turnover rate goes to zero. This is illustrated in Fig. 9 for a p u m p with m = 2.

Fig. 8. Charge displacements in the protein during the pumpi ng cycle. In the course of the conformational transition H C ~ H E the ion-binding site is assumed to move over a distance corresponding to half the dielectric thickness of the mebrane. At the same time, m elementary charges are translocated across the dielectric in the same direction, m accounts for any charge displacement in the protein other than the movement of the transported ion.

The current-voltage curve has two regions with negative differential resistance where inactivation of the p u m p occurs. The voltage-dependence of conformational transitions represents a possible mechanism by which the activity of an ion p u m p m a y be regulated. The possibility of a voltage-dependent gating mechanism in the H + - A T P a s e of bacteria has been discussed by Kagawa [40]. -150 500 400

-100

-5O

0

50 /'

t

100 '

150 200

300 - 1 100

200

50

100 0

0

-100

-50

-200 -300

-3~

' -~

' -~

'

,

, V/mV---~, 100

200

40o 300

Fig. 9. Pump current (given as proton flux, '/'H) as a function of voltage, V, for the situation shown in Fig. 8 ( y ' = c " = 1 / 4 ) ; identical proton concentrations on both sides (c' = c"-~ c). The curve for ?/= m = - 1 / 4 was calculated using the same values of the kinetic parameters (except for 7', ¢" and *1) as in Fig. 7. The curve for m = 2 was calculated with n = l , / ~ c / c = l , KE/C=10, kcE=kEc=PCT=~Cp=PCD=~=10 3 S- 1 , y = 10. The left-hand ordinate and the lower abscissa scales refer to m = - 1 / 2 , the right-hand ordinate and the upper abscissa scales to m = 2.

318

IIIC. Equivalent-circuit representation

dl c o n s t a n t - c u r r e n t source: Gp = ~ --* 0

It is sometimes useful to describe the behaviour of an electrogenic ion pump by an equivalent circuit. A simple possibility consists in representing the pump as a voltage, Vp, in series with a conductance, Gp (Fig. 10). The left part of Fig. 10 schematically depicts an experiment in which the voltage across the cell membrane is held at a fixed value, V, by an external device and the pump current, I, is measured. (In an actual experiment, corrections for leakage pathways have to be made). Vp may be identified with the electromotive force of the pump, since V = V0 for I = 0. It is clear that an equivalent-circuit representation such as shown in Fig. 10 with a voltage-independent pump conductance, Go, can be valid only in a rather limited voltage-range if the overall current-voltage characteristic of the pump is nonlinear. In certain cases it may be convenient, however, to use the equivalent circuit in the whole voltage range considering Go as a voltage-dependent quantity. According to the equivalent circuit of Fig. 10, two limiting cases in the behaviour of an electrogenic pump may be distinguished. If the pump conductance Gp = d I / d V is high, a small change of membrane voltage V will result in a large change of pump current I, meaning that the pump acts as a constant-voltage source. On the other hand, if Gp is very small, the pump current is virtually independent of the external voltage V and the pump may be considered as a constant-current source:

In other words, one and the same value of pump current I may result from a small Fp combined with a large Gp (pump acting as a constant-voltage source) or from a large Fv combined with a small Gp (pump acting as a constant-current source). This is illustrated by Fig. 11, in which two different transport systems are compared which yield approximately the same short-circuit current Isc = e0q~~. In system A, the chemical driving force, Y, and, accordingly, the electromotive force, V0H, are large ( Y = 1 0 6, V0H = - 3 5 5 m V ) . In system B, the driving force is much smaller (Y---10, V0rl= -26mV) but the rate constants are assumed to be large. The later condition corresponds to a high value of the pump conductance, Gp. As seen from Fig. 11, system A behaves almost as a constantcurrent source in the physiological voltage-range ( V - - - 5 0 to - 2 0 0 mV) whereas system B approaches the behaviour of a constant-voltage source in the same range.

loo l ¢H ,5-I

5O

,H

0

dl c o n s t a n t - v o l t a g e source: Gp = ~-~ -* oo -50 I

V

I

V

Fig. 10. Equivalent-circuit r e p r e s e n t a t i o n of a n electrogenic p u m p . O n the left side a n e x p e r i m e n t is s c h e m a t i c a l l y depicted in w h i c h the voltage across the cell m e m b r a n e is held at a fixed value, V, b y a n external device a n d the p u m p current 1 is measured. The e q u i v a l e n t circuit on the r i g h t - h a n d side describes the p u m p as a n electromotive force Vp in series with a c o n d u c t a n c e Gp.

/ -400

V/rnV -300

-200

-100

0

'

1~o

Fig. 11. Two electrogenic t r a n s p o r t systems which yield app r o x i m a t e l y the same short-circuit current I ~ = e0@S~. In system A, the electromotive force is large ( V ~ - - - - 3 5 5 mV), whereas in s y s t e m B the electromotive force is m u c h smaller ( VH = - 26 mV), b u t the rate c o n s t a n t s are a s s u m e d to be large. System A acts as a c o n s t a n t - c u r r e n t source in the voltage-range above - 100 mV; system B a p p r o a c h e s the b e h a v i o u r of a c o n s t a n t - v o l t a g e source in the same range. The curves have been c a l c u l a t e d for c' = c"=- c, n = 1, K c / c = 1, K E / c = 100, q=0, s=0, y'=("=l/4, 7 1 = 0 from Eqns. 28 and 5 3 - 5 8 using the following p a r a m e t e r values: (A) Y = 106, '~CE = 7¢EC = ~'c o = 1 0 0 s -1, ,~c T = 1 0 6 s - l ; (B) Y = 1 0 , k c E = ~:EC = r C D = 3500 s - l , , b C T = 350 s -1.

319

A more straightforward definition of pump conductance Gp is possible for an ideally coupled pump in the vicinity of equilibrium, i.e., for A ~ H / F = PMF =AG/nF. In this case the pump current is given by I

=

e0~

H ~

)

[ ~#H _ PMF ~ UPt F

20 .A~om~

....

_~.._

_

h

10

(61) 0

For identical proton concentrations on both sides (c' = c"), Eqn. 66 assumes the more familiar form 1 = Gp( V - E M F )

(62)

According to Eqns. 19 and 32, Gp is a function of the rate constants and therefore contains implicitly the voltage V.

IIID. Experimental studies of current-voltage behaviour Experimental studies of current-voltage behaviour of ion pumps are faced with the problem of separating the pump current from other membrane currents. The most detailed information on 1 - V properties of proton pumps so far came from studies of the H+-ATPase in the plasma membrane of the fungus Neurospora crassa [8,9,11,37]. The current-voltage characteristic of the pump was obtained by measuring the I - V curve of the membrane under normal conditions and after ATP depletion. The I - V difference curve (Fig. 12) which was assumed to represent the I - V characteristic of the pump showed saturation behaviour with a maximal current of approx. 20 btA/cm 2 and an extrapolated reversal potential around - 4 0 0 mV. Under the given experimental conditions, this value is close to the maximal reversal potential of an ideally coupled pump operating at 1 : 1 stoichiometry. Similar experiments with the Na,K-pump could be performed so far only in a rather limited voltage range between 0 and about - 1 0 0 mV. In this range, the pump current was virtually voltageindependent, meaning that the Na,K-pump acts as a constant-current source under these conditions [51-54]. Voltage-dependent transport rates of the lightdriven proton pump of Halobacterium halobium have been studied by incorporating purple-membrane fragments into planar bilayer membranes

-10

-20

-30 -400

t

II -300

[ ~

I -200

I

I -100

I 0

v/my

Fig. 12. C u r r e n t - v o l t a g e curve of the A T P - d r i v e n p r o t o n p u m p in the p l a s m a m e m b r a n e of Neurospora crassa (redrawn from Fig. 2 of Ref. 8). The I - V characteristic of the p u m p was o b t a i n e d by m e a s u r i n g the 1-V curve of the m e m b r a n e u n d e r n o r m a l c o n d i t i o n s a n d after A T P d e p l e t i o n (in the latter case 1 m M K C N was a d d e d to the medium).

[55]. The pump current (defined as the difference between the current under illumination and in the dark) was found to be a linear function of voltage between - 1 5 0 and 150 mV with an extrapolated reversal potential of about - 2 0 0 mV. Reversal potentials (A/2H/F) around - 2 0 0 mV have been estimated from experiments with whole cells of H. halobium (56] and plasma-membrane vesicles containing purple-membrane fragments [57], Since shunt pathways for H ÷ are difficult to exclude in such experiments, the value of [ A # H / F [ ----200 mV represent a lower limit of the true reversal potential of the pump. Evidence for the electrogenicity of the calcium pump from sarcoplasmic reticulum mainly comes from studies with reconstituted vesicle systems [63,64]. The contribution of the pump to the voltage across the sarcoplasmic reticulum membrane is probably small under physiological conditions since the membrane seems to have a high leak conductance for monovalent ions [65].

320 IV. Behaviour near equilibrium

po=_(dd~)

=RTLH/C

(71)

A/2H ~ AG ~ 0

If both the chemical driving force, AG, and the electrochemical potential gradient, A/2 H, are small, the pump operates near equilibrium. Under this condition the relations Y = 1 - A G / R T and v ~ = 1 + n A [ t H / R T hold. Insertion into Eqns. 28 and 29 and using Eqns. 8, 25 and 27 yields (with c ' = C~t-~- C):

(63)

g}. = L . A ~ H -- L:qcAG dpc = LclqA/~H --

LH =-

LcAG

n2KEk~EkEc D2RTc ~

L H c = LcH ~

(c" = c" = c; c' - c" = Ac). The introduction of G O and P0 accounts for the fact that any pump acts as a passive transport system at vanishing chemical driving force. From Eqns. 63 and 64 the ratio of the electrochemical reversal potentials of ion flow and chemical reaction is obtained as:

npCTKEkEc D2RTen

( A / 2 H ) e~. = 0

LH~LcH

(64)

(A#H), =0

LHLc

(65)

From the general relation [42]:

(66)

0 ~< L~LcH ~ L ~ <~1

(67)

+ k~/~")

(74)

(68)

Eqns. 63 and 64, which represent the fluxes ~H and q,c as linear combinations of the driving forces A/214 and - A G , have the form of the so-called phenomenological equations of nonequilibrium thermodynamics [41,42]. The 'cross-coefficients' LH~ and L~n obey Onsager's symmetry relation LH = L ~ m This means that in the vicinity of equilibrium the fluxes are connected by the relation: q'u q'~

According to the postulates of nonequilibrium thermodyanmics Eqn. 69 has to be regarded as a general relation which holds for any reaction mechanism, provided that AG and A/2 n are sufficiently small. The coefficient L n is related to the conductance, Go, and the permeability, Po, which characterise the transport properties of the p u m p at zero chemical driving force (AG = 0): G

[ dl ]

° ~ d-V J a # . - a c - o = e°FLH

(73)

it follows that ( A / 2 . ) , . _ o and ( A ~ . ) , = 0 have always the same sign and obey the relation:

D~=-(1 + r ~ / ~ " ) ( k ~'c + k~cK~/~") + 0 + ~/~-)(k~

(72)

(70)

According to Eqns. 38 and 39, this relation holds at any driving force for the transport mechanism discussed in subsection IIB. In the limiting case of complete coupling the proton flux, q~H, is always equal to n.q, c. This is only possible if L H = nLnc = nZL~. (This relation is easily verified from Eqns. 65-67 with s = 0 and k~E =pCT. ) Eqns. 63 and 64 then reduce to: q~14= nq~c= Ltt(A~tH -- AG/n )

(75)

When the p u m p is completely coupled, the system is in equilibrium whenever A/2 u = A G / n , but now A#H and AG may be arbitrarily large. For values of A/2 H in the vicinity of A G / n , one may then still use a relation of the form of Eqn. 75, although L H is no longer given by Eqn. 65. Since under physiological conditions the chemical driving force is usually large, the linearized flux equations have a rather limited range of application. On the other hand, as the behaviour of the p u m p becomes particularly simple in the linear range, experimental determinations of the coefficients Lr~, Lc, LHc at variable concentrations of H +, ATP etc. may be valuable for distinguishing between different reaction mechanisms.

321

V. Equivalence of electric and osmotic driving forces The electrochemical potential difference, A ~ H , is the sum of an osmotic term A#H = R T l n ( c ' / c " ) and an electric term F V (Eqn. 11):

(76)

A ~ H = Al£ H "Jr F V

Under equilibrium or near-equilibrium conditions A/~H and F V are interchangeable. This is immediately evident from the relation:

AG

(A/2H)+,_,c_0=--~-

(77)

which describes the equilibrium state of an ideally coupled pump. The electrochemical potential difference in Eqn. 77 may consist either of a purely electric ( F V ) or a purely osmotic force (A/xH) or of any combination of both. This may be expressed by the relation: (A/IH).,-.o-0, v=0 = (FV)q,n-,o=o,a~H=O which states that

A#H

and F V are thermodynami-

that an increment of A/t H produces the same change of ~H as an increment of FV:

]

[ a,. ]

= t a(Fv) J ~ .

_(a,.]

~"(, a*")..v=ac" , a,, ,~,~,,

(79a)

must be fulfilled ( u = F V / R T ) . Introducing CH from Eqn. 28, it can be shown that Eqn. 79a holds if it is assumed that all rate constants of conformational transitions as well as the equilibrium constant K c are voltage-independent and that the equilibrium constant, KE, is given by K E = KEexp(u ). This is only possible if the binding site is located very close to the cytoplasmic surface so that the full voltage drops between the binding site and the extracellular medium.

(78)

cally equivalent. Kinetic equivalence of A/xH and F V would mean

~ J v

where the pump molecule behaves as an ideal ' p r o t o n well'. In order to show this, we assume that A/~H is varied by varying the extracellular proton concentration, c", while the cytoplasmic concentration, c', is held constant. In this case kinetic equivalence of A/~H and F V means that the relation:

(79)

This relation is fulfilled in the vicinity of equilibrium, as seen from Eqns. 63 and 76. At arbitrary forces, however, the flux epn is given by Eqns. 15 or 28 which contain not only A/~H and V but also the ion concentrations c' and c" separately. A given change of A#H can be produced in an infinite number of ways by changing c' and c'. This means that '#H is not a unique function of the driving forces alone, and for this reason Eqn. 79 no longer holds [43]. The range in which Eqn. 63 is valid depends, of course, on the particular system, and in some cases an approximate equivalence of A#H and F V has been observed far from equilibrium [35,37,44-46]. Kinetic equivalence of electric and osmotic driving forces at arbitrarily large values of A#H and F V is predicted by Eqn. 28 under conditions

Vl. Unidirectional fluxes and exchange flows On a macroscopic level, active ion transport manifests itself as a steady-state uphill ion flow. Microscopically, however, an ion pump operates in a stochastic way, carrying out a sort of biased random walk among the states of the reaction cycle. This means that, although there is a net flow in one direction, cycles will always occur in which an ion moves in the 'wrong' direction through the pump. While this problem can be treated by the method of stochastic reaction kinetics, a much simpler (still macroscopic) approach consists in representing the net (outward) ion flow, ~bH, as the difference of an efflux, @ , and an influx, ~'~:

,. =~-¢fi

(80)

The unidirectional fluxes, ¢'n and ,~'~, are defined by a hypothetical isotope flow experiment in which the left (cytoplasmic) solution contains only isotopic species 1 and the right (extracellular) solution only isotopic species 2: CH~*1;~H m -- ¢2 (~; = ¢', 4' = c", ¢;'= d = 0)

(81)

322

In a similar way, the net rate @~of ATP hydrolysis can be written as the difference of a hydrolysis rate ~'~ and a synthesis rate @": ~=<-<'

(82)

For instance, @~can be obtained by measuring the rate of formation of labelled inorganic phosphate in an experiment with 32p-labelled ATP. For the minimal model (Fig. 4) the unidirectional fluxes are obtained as:

~°-----(~b'¢)eqq= (q~")eq

The subscript 'eq' denotes equilibrium conditions (A/2 H = AG = 0). U s i n g the relation fn ~ ptn kcEkEcKc/c =kEckcEKE/C , which is valid under equilibrium conditions, one obtains from Eqns. 83 and 85: q~o =

(83) (84)

#~ = pCTPHc = ~ f

( k~c + k E c K E / C " " )

@'~'=rCDPHE= red (k~E + kcEKc/c" ) D2

(85)

PCTrCD PCT(I+KE/e"")+rCo(I+Kc/c"

(91)

)

(86)

p u m p molecule is in states H , C and H , E , respectively, and D 2 is given by Eqn. 31. It may be verified (using Eqns. 8, 13, 14 and 24-27) that q e u - @'~ is identical with t~H from Eqn. 28 and @'~- @~' identical with @¢ from Eqn. 29. Eqn. 84 predicts that there is always a finite ion flow in the direction opposite to net flux. If chemical reaction and ion transport are completely coupled (q---0, s = 0) the relations @'n = nq,'-@' and q,'~= n ~ ' - @ " hold. In the limit of low ion concentrations (c', c " ~ 0), the ratio of the unidirectional fluxes is directly related to the free-energy change - ( n A ~ H -- AG) of the coupled reaction n H +' + ATP ~ n H +'' + A D P + Pi [60]: (87)

This relation (which is obtained using Eqns. 8, 13, 14, 24-27, 83 and 84) no longer holds when the binding sites become saturated. Important quantities in the description of p u m p kinetics are the 'exchange flows' which are the unidirectional flows in the limit of zero driving forces: =

(90)

+ k c(1 + Xc/C'")

q,o, @'H and @'H are related to the ohmic conductance, Go, of the p u m p (Eqn. 70) and to the coefficient, L H, of the phenomenological equations. Differentiating q~H/~'~ with respect to V and using the relation @n = q~'n - @'~ = 1/% yields [47]: (92)

PHC and PHE are the probabilities that a given

q" ,,, = exp[(na~n - AG)/RT]

nk~-k~c

k E(1 + ~Oc=

k*

(89)

(88)

\q~H / J ,,',#a= ~G = 0

In a similar way, one obtains from Eqns. 63 and 64: L

<=-¢

o [ 0(q'n/q'~) ]

a(
(93)

(94)

These relations which connect the unidirectional fluxes with the phenomenological coefficients are generally valid, irrespective of the particular reaction mechanism. A similar connection between phenomenological coefficients and one-way cyclefluxes in equilibrium has recently been discussed by Hill [101,102].

VII. Energetics VIIA. Operation modes of ion pumps Depending on the magnitude of the chemical and osmotic driving forces, an ion p u m p may hydrolyse ATP and transport ions uphill, or the p u m p may run backward, utilising energy from

323

downhill transport for ATP synthesis. If chemical reaction and transport process are only loosely coupled, the enzyme may operate in a purely dissipative mode (ATP hydrolysis associated with downhill transport). The range of conditions under which these different modes of operation may be observed is only restricted by the thermodynamic requirement that the entropy production must be positive (or zero). In the following we treat the membrane together with the adjacent solutions as a closed system which is kept at constant temperature and pressure (a closed system may exchange energy but not matter with the surroundings). For such a system the entropy production, d+S/dt, is equal to the rate of free-energy dissipation, divided by temperature [61]. Thus, Tdi S

~ = ~HA~H-@cAG ~ 0

(95)

This means that thermodynamically allowed modes of the pump have to fulfill the condition @tAG ~<(~HA#H

(PHA~H

-epAG is the rate of free-energy liberation by the chemical reaction and --dpHA/2H is the rate of free-energy storage in the electrochemical gradient. It is convenient to represent the different operation modes of a pump in a diagram with @AG and @HA/2H as cartesian coordinates (Fig. 13). The allowed modes of the pump are located to the left of the straight line ¢HA/2H = ~AG. Uphill transport of H ÷ (¢HA/~H < 0) requires that the chemical reaction is exergonic ( ¢ A G < 0); for all practical purposes this means that the reaction must be poised toward spontaneous ATP hydrolysis (spontaneous ATP generation would require extreme concentrations of ATP, ADP and Pi)- An endergonic chemical reaction (ATP synthesis) may be driven by downhill transport of H ÷ . If the pump is weakly coupled, downhill ion transport and exergonic chemical reaction may occur at the same time; in this case the pump operates in a purely dissipative fashion. VIIB. Free-energy transduction and free-energy dissipation," thermodynamic efficiency

(96)

~d7 "

I(pc

endergonicj

purely dissipative

/

(Pc exergonic ~ [ ,,,,c.~ "A (PHuphill-~.,~'~

(PcAG ---

/-

Fig. 13. Thermodynamically allowed operation modes of a proton pump. AG is the free energy of ATP hydrolysis and A/~H--=/2~ - - / ~ is the electrochemical potential difference of the proton in the cytoplasm (phase ') and in the extracellular medium (phase "). The chemical flux, @¢,is taken to be positive for ATP hydrolysis and negative for ATP synthesis. Free energy may be stored either by uphill transport of H ÷ (driven by an exergonic chemical reaction) or by an endergonic chemical reaction (driven by downhill transport of H ÷ ).

If ~ mol ATP are hydrolysed per unit time, an amount -e&AG of free energy becomes available. Part of this energy (--~HA/2H) is stored in the electrochemical gradient, the rest is dissipated. The thermodynamic efficiency, ~ H, for uphill ion transport may therefore be defined by: ~ln

+IaA#H ~A#H ocAG

o-'~-

((hHAP'H< 0)

(97)

0 is the coupling ratio which has been introduced before (Eqn. 20). When the pump operates under 'level flow' conditions (A/2 H = 0), the efficiency becomes zero. Maximum efficiency (*/H = 1) is achieved by a completely coupled pump (0 equal to the stoichiometric ratio, n) in the vicinity of the reversal potential A/2H = AG/n (Eqn. 35). In a similar way, the thermodynamic efficiency, *L, of ATP synthesis (driven by downhill ion transport) may be defined by: ~pcAG ~,~ ~HAp.H

(~¢AG > 0)

(98)

Again, for a completely coupled pump the efficiency, ~L, may approach unity.

324 The free-energy change, AG, is composed of an enthalpy term AH and an entropy term - T A S according to the thermodynamic relation AG = A H - TAS. Under physiological conditions, AG is about - 5 0 to - 6 0 k J / t o o l and A H about - 2 0 kJ/mol. The enthalpy, A H , of ATP hydrolysis determines (in part) the amount of heat that is exchanged with the surroundings during the pumping process. For a more detailed discussion we again describe the membrane with the adjacent solutions as a closed system which is kept at constant pressure P and which is in contact with a heat bath of temperature T (Fig. 14). If I mol ATP is hydrolysed at constant temperature an amount of heat of magnitude A Q is absorbed by the system from the heat bath. For a closed system at constant P and T that exchanges only heat and pressure-volume work with the surroundings, A Q is equal to the total enthalpy change A H t which is the sum of the contributions of the chemical reaction, A H , and of the transport process, AHtr. AHtr in turn contains an 'osmotic' term AHos m due to the transfer of 1 mol H + from concentration c' to concentration c" (at zero voltage) plus an electric term - F ( ~ k ' - + " ) = - F V which represents the

electric work associated with the movement of the charge F against the potential gradient. Thus, for the hydrolysis of 1 mol ATP and the transport of 0 mol H + from the cytoplasm to the extracellular medium: AQ =AH t =AH+ 0(AHosm FV) -

PD

+

PP

Fig. 14. Hydrolysisof 1 tool ATP associated with uphill transport of 0 tool H+. The free energy of hydrolysis is given by AG =/~D +/xr'-/'tT, where /t T, /~D and /~p are the chemical potentials of ATP, ADP and Pi, respectively. - AQ is the heat which is delivered to the surroundings per mol ATP hydrolysed.

(99)

AHos m should be negligible in many cases (for

ideal solutions A a o s m is zero). If a completely coupled pump (0 = n) operates under the condition c ' = c" close to the reversal potential V = EMF = A G / n F , the heat absorbed by system is: ( AQ )v_E~F = A H - nFV= A H - AG = ?'As

(100)

In this limiting case where the pump operates reversibly and performs purely electrical work, dissipation of chemical energy ( A H ) is minimal. (Actually, since A H ~ - 20 k J / m o l and AG = - 50 k J/tool, a positive amount of heat is absorbed from the surroundings.) If, on the other hand, the pump operates irreversibly under 'level flow' conditions ( c ' = c", V = 0), the full amount of the reaction enthalpy, A H, is dissipated as heat: (AQ)v=o=AH

13T

-

(101)

It is interesting to note that, even for reversible operation of the pump, the dissipated heat is different depending on whether the electrochemical gradient consists in a voltage alone ( c ' = c") or in a concentration difference alone ( V = 0). In the first case, where the pump performs purely electric work, the relation AQ = A H - - n F V holds (Eqn. 100), whereas in the second case (purely osmotic work), Eqn. 99 yields A Q = A H + 0 A H o s m = A H . This means that a voltage and a concentration difference which are thermodynamically equivalent in terms of free energy are nonequivalent in terms of enthalpy. This is also evident from the relation A/2 H = R T l n ( c ' / c " ) + F V where the first term represents the contribution of entropy and the second term the contribution of enthalpy. Furthermore, it is clear that dissipation of chemical free energy AG in a level-flow system does not mean that an amount of heat equal to AG is liberated. For a level-flow system A Q is always equal to A H, and only for A S = 0 does the relation A Q = AG hold.

325 VIlC. Energy levels of transport proteins In the pumping cycle, energy supplied by A T P or light is converted in a stepwise fashion into free energy of an electrochemical gradient. During this process the transport protein undergoes a series of transformations (conformational changes, binding and unbinding of ligands such as Pi or H ÷ , etc.). For an understanding of the energetics of the pumping process it is important to know the energy changes associated with the single reaction steps. It should be clear, as Hill [66,67] has emphasised, that energy transduction is, strictly speaking, not the result of a single step in the cycle but rather the outcome of the cycle as a whole. In other words, it is meaningless to ask: 'in what particular step (or steps) of the cycle is the chemical free energy converted into osmotic free energy?'. Despite this reservation, it is obvious that some steps may be more important in terms of energy change than others and that much insight into the transduction mechanism can be gained from an analysis of the energy levels of intermediate states [66-69]. Important questions for instance are: To what extent is free energy transiently stored in the form of conformational energy [70,74] of the p u m p protein? Is the free energy mainly used for switching the binding site from a left-exposed to a rightexposed state or rather for creating a difference in the affinity of ion binding?" In qrd~r to introduce the. notion of free-energy levels of a transport protein [66], we consider N identical, non-interacting protein molecules embedded in a rrletnbrane4~he protein molecule.may exist in a number of eonformational states corresponding to the states of the pumping cycle. Following Hill [66], we assume that each conformational state consists of a large number of substates differing, for instance, in the orientation of amino-acid side-chains, in the orientiation of water molecules on the polar surface of the protein, in the vibrational modes of the backbone, and so forth. Transitions between such substates are usually fast, occurring in the s u b n a n o s e c o n d time-range. On the other hand, turnover rates of ion p u m p s such as the (Na ÷ + K+)-ATPase or the proton p u m p of halobacteria are of the order of 100 s-1 and thus are extremely small on a molecu-

lar time-scale. We may therefore assume that the states of the cycle have a sufficiently long lifetime so that each state is in internal equilibrium with respect to its substates. (In fact, the states of the cycle are defined by the condition that they are not in equilibrium with each other during steady-state operation of the pump, but remain in an equilibrium state internally [66].) Accordingly, a p u m p molecule in a given (long-lived) conformational state may be treated as a distinct chemical species with a well-defined chemical potential. If N z protein molecules out of totally N are in conformation Z, the chemical potential of the quasi-species, Z, is given by ttz = t~°z+ RTin( N z / N )

(102)

#z is the free energy (per mole) of the protein in state Z. (It should be noted that in a condensed system at ordinary pressures, the difference between Gibbs and Helmholtz free-energy and also between enthalpy and energy is immaterial.) The standard value of the chemical potential, /~z, 0 is a true molecular quantity of the protein (~o is equal to - R T In qz where qz is the partition function of the protein in state Z [72]). ~z, on the other hand, contains the mole fraction N z / n and thus depends on the state of the whole system. Energy levels of states in the cycle may be defined on the basis of both /~z and of ~z. 0 This may be illustrated by considering as a simple example a cycle starting with a light-induced transition from a ground state A to a (long-lived) excited state B, followed by a series of monomolecular dark reactions: A ~h~_~B ~---~C -~__I.D ... ~...._~A

(103)

(The primary photoproduct may be a short-lived vibrationally excited state, B*, which quickly desactivates to B; in this case a chemical potential can not be assigned to state B*). The difference /t ° - / x ° depends only on the molecular properties of states A and B and determines how much free energy can be stored in state B with respect to the ground state A. At° - / ~ °A is usually smaller than the photon energy h~,. The difference #a -/~A, on the other hand, contains the concentrations NA and N B and thus depends on light intensity J and on all

326

transition rates in the cycle. For J = 0 the system is in thermal equilibrium ( # A = / t a = / t C = . . . ) and /tB--/tA vanishes. If the rates of the dark reactions B ~ A and B ~ C are much larger than the rate of light absorption, / t n - / t A will remain small even at large light intensity. Since spontaneous reactions are associated with a decrease of free energy, the differences/tc - / t B , / t o - / t c , -.are always negative. If the rate constants of a particular reaction, say C ~ D, are very large, this reaction will remain near equilibrium a n d / t D - - / t c will be close to zero. Thus, the differences # v - / t x reflect both the magnitude of the driving force as well as the kinetic properties of the whole cycle. In a real p u m p cycle, the transport protein binds and releases ligands such as Pi or H ÷ in some of the transitions. For instance, in the transition HC---, H E (Fig. 2) the protein becomes phosphorylated and therefore states H C and H E differ not only in the conformation but also in the chemical composition of the macromolecule. This means that the free energies /t0HC and /to E are not directly comparable (there is no c o m m o n reference state for /t° c and /tOE). However, a free-energy difference between H C and H E m a y be defined in an unambiguous way taking into account that inorganic phosphate is present in the solution and that the phosphorylated state H E m a y be generated (at least in principle) by direct reaction of H C with Pi" The free-energy change ( / t o _ / t o ) corresponding to the transition H C ~ H E is obtained by defining two pseudoisomeric states 1 and 2 in the following way [66,73]: (1) protein molecule in c o n f o r m a t i o n H C plus phosphate ion in solution at concentration Cp; (2) phosphorylated protein molecule in conformation HE. This means that the free-energy difference has to be chosen to be /tHE 0 - (/t° c - / t p ) where /tp = /tO + R T In cp is the chemical potential of P~ in the cytoplasm. The free-energy difference /to E - (/t° c + / t p ) is connected with the equilibrium constant of the pseudomonomolecular transition H C ~ H E by the relation ( NHE ] N H C ] eq

k~E=exp[-(#°E--#°c--l,

tp)/RT]

(104)

k ~C

Since the charge distribution of the protein molecule is, in general, different in states H C and HE,

the free energies /t0HC and /to E are functions of voltage. Comparison with Eqns. 56 and 57 shows that the voltage-dependent part of (/tHE0 _ / t o c /tp)/RT is equal to - - ( 7 / + X)U. Similar considerations apply to the binding and release of the transported ion. The transition from state C to state H C by binding of a proton from the cytoplasmic side (C + H + ' ~ HC) is associated with a change of free energy of magnitude /t0HC_(/t ° +/2~). At a given cytoplasmic concentration of H ÷, the transition C ~ H C can be treated as a pseudomonomolecular reaction with an equilibrium constant NHc ] = c' = e x p [ Nc ]¢q K c

(#OHc- t~Oc--p.'H)/RT]

(105)

(Eqn. 7, n = 1). Again, the free energies /t° c and /t0c as well as the electrochemical potential /2'H = / t ' n + F ~ ' of the proton in the cytoplasm are functions of voltage. The voltage dependence of 0 /t0C a n d / 2 ' , has to be referred, of course, to a /tHC, c o m m o n zero point of the electric potential, e.g., ~p"=0. The free-energy difference /t° c - / t o _ / 2 , H is then a unique function of voltage. Under the assumptions of subsection I|IB, the voltage-dependent part of (/t° c - ~ t o _ ~,H)/R T is simply given by - ~ , ' u (Eqn. 48). Introducing the notation: O-- 0 . 0-- 0 #1 = ~tH( - + # p , ~ 2 = # H E

~0__-~,o + V~, ~,o__-~,o + ~,p + V;,

(106)

N] ---NHC; N2=--NHE; N3~ NE; N4-= Nc

it is easy to show that under conditions of thermod y n a m i c equilibrium ( A G = A / 2 H = 0 ) the p u m p states are distributed according to exp( - i.t°/ R T )

(107)

J

If #i and /tj are the chemical potentials of two subsequent states i and j in the cycle [/ti =/t0 + R T I n ( N J N ),/tg = #jo + R T I n ( N J N ) ] , the change of free energy in the transition i --,j is given by ILJ _ ~, = #o _ #o + RTIn( N j / N , )

(108)

327

If the pump operates in forward direction, the chemical-potential differences,/~j - / t i, of all transitions must be negative except for the first transition 1 ~ 2, which is driven by ATP hydrolysis. The condition # j - / x , < 0 is met if /~j0 < #0 and Nj < N~, but even if the transition is energetically unfavourable in terms of #0 and /~o (/~0 > #0), spontaneous reaction i - - , j still takes place provided that N/ is sufficiently larger than Nj. Since under equilibrium conditions the relation #i = #j holds, Eqn. 108 may be written in the form

~/N, i~j -

i~ = RTIn

,

(109)

The free-energy levels of the protein states in the pumping cycle may be represented in a diagram as shown in Fig. 15. The example given in Fig. 15 is based on the sequence HC ---, HE ~ E C ---, HC (Fig. 2). All energy levels are referred to # ° c + #p as an arbitrary zero-point (level 1). Level l a represents the chemical driving force --AG = #ATP --/~ADP -- #P of the cycle and level 2 the free

~'~ (~)Pi*PT-PD

freeenergyl

]- @lJ~c.pp,P~-p~

P'r-PD-~P

P~c*PP Fig. 15. Free-energy levels of the pump model depicted in Fig. 2, based on standard values of chemical potential. #°n¢, #OE, /t° and #o are the free energies of the pump molecule in the states HC, HE, E and C, respectively (Eqn. 102). #T, #D and #p are the chemical potentials of ATP, ADP and Pi. All energy 0 - /~p as an arbitrary zero point (level levels are referred to/~nc 1). Level la represents the free energy supplied by ATP hydrolysis and level 1 b the electrochemical potential difference of H + between extracellular and cytoplasmic side. The pump cycle involves four consecutive steps: 1 ~ 2, phosphorylation of HC; 2 ~ 3, release of H + to the extracellular medium; 3 ~ 4, dephosphorylation of E; 4---, lb, rebinding of H ÷ from the cytoplasm. State l b is identical to state 1 except for the transfer of a proton from the cytoplasm to the extracellular medium.

energy of the phosphorylated state (HE) of the protein with protonated binding site. For an efficient functioning of the pump level l a should be above level 2, otherwise the chemical driving-force would be too small tO raise the ratio NHE/NHc significantly above its equilibrium value, and the turnover rate of the pump would be low. On the other hand, if level 2 is too low, state HE will be appreciably populated already in the absence of a chemical driving force ( A G = 0 ) and therefore phosphorylation by ATP (AG < 0) could not increase NHE much further. Taken together with the previous statement, this means that for efficient operation of the pump the free-energy level of the primary phosphorylated state should approximately match the free energy of ATP hydrolysis. In Fig. 15, it is assumed that b o t h protonation/deprotonation steps HE ~ E + H +'' (level 2 ~ level 3) and C + H ÷' ~ HC (level 4 level lb) represent transitions from lower to higher free energy states of the protein at the given extracellular and cytoplasmic proton concentrations. This means that, if the pump is driven through the cycle in forward direction, Nr~E must be larger than N E and N c larger than Nnc. After completion of one cycle the system is in level l b which differs from the ground level 1 by the electrochemical potential difference /2'~-/2'n resulting from the transfer of a proton from the cytoplasm to the extracellular medium. Two limiting cases of the pumping mechanism are conceivable. In the first case the p K values of the proton binding site in conformations H C / C and H E / E are such that p K c approximately matches the pH of the cytoplasm and p K E the p H of the extracellular medium under physiological conditions of uphill transport [68,75]. As discussed in subsection IIIA, such a change of binding strength (easy release of the ion to the side of high electrochemical potential, easy rebinding from the side of low electrochemical potential) is by no means a prerequisite of active transport, but it is favourable for a high turnover rate of the pump. Under the condition p K c = pH', p K E = p H " levels 2 and 3, as well as levels 4 and l b have approximately the same energy (Fig. 16) according to the relations #° b -/~o__ - 2 . 3 0 R T ( p K c - p H ' ) and /x° - #02 = 2.30 R T ( p K E - p H " ) . Steady-state forward operation of the pump (requiring/hb -- ~1~4< 0

328

freeenergyl P~-&G

0

0

-AG

~ ADP~, ATP-J

-zx~H

pKC ~ pH' pKE -~ pH"

(~111-AG

~

R p~

C) Fig. 16. Free-energy levels (as in Fig. 15) of a c o m p l e t e l y c o u p l e d p u m p o p e r a t i n g near its reversal p o t e n t i a l (A/~ H = AG). The n o t a t i o n of the energy levels is the same as in Fig. 15 ( c o m p a r e Eqns. 106). The p K values of the p r o t o n b i n d i n g site in states H C / C and H E / E are a s s u m e d to m a t c h approxim a t e l y the p H values of the c y t o p l a s m and of the extracellular medium, respectively ( p K c -~ p H ' , p K E ~ p H " ) .

and #3 - F2 < 0) is then possible at concentration ratios Nuc/Nc and NHE/NE of the order of unity, as seen from Eqn. 108. In the second case, the p K of the proton binding site remains unchanged in the transition from the left-exposed to the right-exposed state. This means that one or both protonation/deprotonation steps (transitions 2 ~ 3 and 4-", lb) are energetically unfavourable in terms of the standard values, /~z, 0 of chemical potential if the pump operates in the presence of an electrochemical gradient. This situation is illustrated in Fig. 15, where it has been assumed that both transitions lead to states of higher free energy. According to Eqn. 109, stationary forward operation of the p u m p is still possible provided that the ratios N H E / / N E and Nc/NHc a r e raised sufficiently above their equilibrium values by the energy-supplying reaction. Free-energy levels defined in terms of standard values, Fz, 0 of chemical potential have been termed 'basic flee-energy levels' by Hill [66], As discussed above, basic flee-energy levels are particularly useful for a mechanistic understanding of pump function, since they represent intrinsic properties of the p u m p molecule. Energy levels may also be based on the (concentration dependent) chemical potentials,/z z, defined by Eqn. 102 [66,68]. Free-energy differences #j - #~ for spontaneous transitions i --,j

I"

4

-

ADP-~ ATP" I

freet energy]

113

~H

-&PH I11

"~

Fig. 17. Energy-level diagram, based on (concentration-dependent) chemical p o t e n t i a l s t~i = btl° + RT ln(Ni/N), for the reaction model shown in Fig. 2, /Zl------~Hc, + ~p; / Z 2 ~ tI, H E ; ~ 3 - - ~ E 4_ /Y~; #4--=#C + /2'~ + p.p ( c o m p a r e Eqns. 106). The p u m p runs in forward direction ( t r a n s p o r t rate '#H > 0). Except for the ATPdriven reaction 1 ~ 2, all transitions are energetically d o w n ward. T r a n s i t i o n 2 ~ 3 is a s s u m e d to be near equilibrium. Free-energy dissipation occurs m a i n l y in transitions 3 ---, 4 and 4 - . lb.

(Eqn. 108) are always negative at finite pump rates, and for this reason energy levels derived from #z values contain less information than basic free-energy levels. On the other hand the energylevel diagram based on ~z values better reflects the kinetic properties of the cycle. This is illustrated in Fig. 17, where the values of/L, are plotted for the same reaction cycle as in Fig. 15 assuming that the p u m p runs in forward direction (q'u > 0). If the rate constants for a particular transition are very large, this reaction remains near equilibrium. In Fig. 17 it has been a s s u m e d that the protonation/deprotonation reaction 2---, 3 is a near-equilibrium reaction. This means that the free energy difference A/~23 = t~2 - #3 is close to zero. Since ffHAtt u is the amount of free energy which is dissipated per unit time in the transition i -~j, the energy diagram based on tti levels directly indicates which steps of the cycle are critical in terms of free-energy dissipation. In Fig. 17, most of the free energy is dissipated in steps 3 ~ 4 and 4 ~ lb; in other words, the total amount, --(e&AGffnA/2H), of dissipated free energy is chiefly used to accelerate reactions 3 ~ 4 and 4---, lb. If the p u m p operates close to the reversal potential (q~H = 0), levels 2, 3, 4 and l b are all at the same height and free-energy dissipation (if any) occurs only between levels l a and 2.

329

VIII. Light-driven ion pumps Bacteriorhodopsin, a retinal-containing protein in the plasma membrane of Halobacterium halobium is known to act as a light-driven, electrogenic proton pump [7]. Recently, evidence has been obtained that another retinal-containing protein of H. halobium utilises light energy for uphill chloride transport [77]. The detailed mechanism by which H + is translocated after photoexcitation of bacteriorhodopsin is still unknown. The general properties of light driven ion pumps may be discussed on the basis of the concept which has already been used for the description of ATPdriven pumps (Section II). A reaction scheme based on the notion that the external energy source serves to switch an ion binding site from a left-exposed to a right-exposed conformation [16,17] is represented in Fig. 18. In the dark the pump molecule is mainly in the ground state H , C with the binding site accessible only from the left (cytoplasmic) side. Absorption of a light quantum of energy, hp, leads to a short-lived excited state, H , X , which quickly desactivates to H , E . State H , E , which has the binding site accessible only from the external medium, may directly return to the ground state H , C (whereby the stored energy is dissipated) or it may go through the cycle H , E E ~ C ~ HnC in which n protons are released to the extracellular side and n other protons taken up from the cytoplasm. State HnE may be tentatively assigned 'to the L intermediate in the bacteriorhodopsin photocycle, state E to the de-

h~ll

HnE

-

"- E

all\ iI k~E k~c

kCE kEc

HnC

C

" Kc

Fig. 18. Reaction scheme for a light-driven proton pump. The pump states are similar to those in Fig. 2, except for the additional state H,X, representing a short-lived excited state. States H~X and HnE are assumed to be in equilibrium with each other.

protonised M product and state C to the O product which is reprotonated to yield the original R protein [7]. The model of Fig. 18 is similar to the reaction schemes depicted in Fig. 2 and 4, except for the addition of state HnX which has been introduced in order to account for the fact that only a small fraction of the quantum energy is stored in long-lived states of bacteriorhodopsin [78]. Transition of a protein molecule to an excited state by absorption of a light quantum is superficially similar to the process of phosphorylation by an energy-rich phosphate, but the analogy cannot be taken very far. The interaction of matter with a radiation field is governed by special laws and this has the consequence that light energy and chemical energy are fundamentally different from a thermodynamic point of view. A thermodynamically consistent treatment of processes such as photosynthesis or generation of photoelectric power becomes possible on the basis of Planck's [79] concept that any radiation of given wavelength and intensity may be related to the radiation of a black body kept at a well-defined temperature [80-86]. The energetics of light-driven ion transport may be discussed starting with the thought experiment depicted in Fig. 19 [80]. Light from a black body kept at temperature T~(acting as a radiation source) impinges under a solid angle, 12, on a membrane containing the light-dependent transport system. The black body at temperature T~ is part of a large spherical cavity surrounding the membrane, and the rest of the cavity (solid angle 4 ~ r - 12) is a black body kept at ambient temperature T. The membrane and the adjacent solutions are in contact with a heat bath of the same temperature T. Between the radiation source and the membrane a filter is interposed that transmits light in a narrow frequency band between p and p + Ap and reflects light at all other frequencies. The photosystem in the membrane absorbs light isotropically in the same frequency interval between J, and v + A v and is transparent at all other frequencies; the rest of the system (solutions, heat bath, etc.) is completely transparent at all frequencies. Each isotropically absorbing chromophore may be thought of as being surrounded by a small sphere of radius p. According to the law of black-body radiation [79], the flux density, Js, of quanta emitted by the

330

light source. A highly directional laser beam (~2 small) has a high radiation temperature T,, whereas isotropically scattered radiation (I2---4~r) of the same intensity and bandwidth has a much lower

• ~

~'~J

If o is the absorption cross-section (m 2) of the p u m p molecule in state H . C for radiation of frequency v, the overall rate constant, a, for transitions from state H . C to state H . X (Fig. 18) may be expressed by:

~- heat bath

a=a0+o(S,+S) Fig. 19. Thought experiment for the thermodynamic analysis of light-driven ion transport. A membrane containing light-dependent pump molecules is surrounded by a large spherical cavity. A part of the cavity subtending a solid angle, ~2, is kept at a temperature T~ and acts as a black-body radiation source; the rest of the cavity (solid angle 4~r - ~2) is a black body kept at ambient temperature T. The membrane and the adjacent solutions (not shown) are in contact with a (completely transparent) heat bath of the same temperature T. A filter is interposed between the radiation source and the membrane. The filter transmits light in a narrow frequency band between v and v + A~, and reflects light at all other frequencies.

radiation source (temperature T~) and impinging on the surface of the sphere is given by (110)

2OAr~h2 =OAv.~ "Is exp(hv/kT~)-I

Js(m - 2 . s - l ) is referred to unit cross section, q'rp2, of the sphere. Js is the flux density per unit frequency interval and unit solid angle and 2~= c / n mv is the wavelength (c is the light velocity and n m the refractive index of the medium). In the same way, the contribution, J, to the total flux density from the rest of the cavity (ambient temperature T ) is obtained as: 2(4~r

-

12)av/,k 2

.

.

.

. (111)

In an actual experiment, an arbitrary light source is used, for instance a laser beam which is characterized by its intensity, Js, frequency, u, bandwidth, Av, and solid angle, 12. The importance of Eqn. 110 resides in the possibility of assigning an equivalent black-body source with a well-defined radiation temperature, T~, to any (narrow-band)

(n2)

where a 0 is the contribution from radiationless transitions. For simplicity we assume that H , C is the only absorbing species in the system. Transitions from the excited state H , X to the ground state H , C (overall rate constant b) arise from radiationless processes (rate constant b0), by spontaneous emission of photons (rate constant bf) and by induced emission. Since induced emission, at a given light intensity, has the same probability as absorption [87], b is given by: b = b o + b f + o ( J S+ J )

(113)

(The contribution of induced emission which has been introduced here for sake of consistency can usually be neglected, since the actual rate of induced emission is very low at normal light intensities.) The principle of detailed balance requires that in thermal equilibrium (T = T~; A/2 H = 0) radiative as well as nonradiative processes are in_equilibrium separately. This means that _a0PHc= b0ffH__Xand 4rroAvJPnc = ( b f + 4 r r o A v J ) P n x . PHC and PHX are the equilibrium probabilities of finding a p u m p molecule in states H . C or H X. Assuming that H . C and H . X are single quantum states, the ratio f f . x / P H c is e q u a l to exp( -- h v / k T ) . Together with Eqn. 111 this yields: Oo

PHX

b0

PHC

exp( -- h p / k T ) =-Q

bf = 8¢roAv/h2 = 4~roAvj(1 - Q ) / Q

(114)

(115)

In analogy to Eqn. 27, the principle of detailed balance further requires that:

331

Q-= k~E B k~c

KE k~EkEc . . . . Kc k~ckcE

exp(nu)

(116)

where B - PHx//PHE is the equilibrium constant of the reaction H , E ~ H , X . The equilibrium constants K c and K E for the deprotonation reactions are defined by Eqn. 7. Under nonequilibrium conditions, net absorption of light quanta takes place in the photosystem and protons are translocated across the membrane. The net rate of proton transport, q~H, and the net absorption rate of radiation, ~ , may be obtained using the same method as in the derivation of Eqns. 28 and 29 [16]. Assuming that the protonation/deprotonation reactions as well as the reaction H , X ~ H , E are always in equilibrium, one obtains (Appendix D):

,t,,, = ,,A3 [ ~ (ao + l,*E)(o" - 1)

+ o~a,4(1- Q)(X-1)]

) (x- 1)]

a~-= a 0 + 4~roAp.]

(120) (121)

D3-~(a + k~E)(1 + B)+k~c g C

+ bB + ~-~-[ k~*c + bB + kcE(l+ B)]

¢ c KE + Kc ..... ( kCE + kEC) ¢

dpr=A3[bfQ(vn__l)+oJs{1 +(ao+k~E) " . cv---~CEI ' / K c ) ]]

(124)

(A/TtH)~,H_0= -- R~TIn ae+k~E+°flAv(']s--]) (118)

(119)

1 KE

(123)

The reversai potential of the pump is obtained f r o m Eqn. 117 using the relation v = exp(A/2 H / R T ) :

X=__J_g2 e x p ( h v / k T ) - I J exp(hv/kT~)-I

"43=="D3 c~72""kvc '""

,. = ,~3[~(ao + k,~)(v"- 1)+ oJs]

(117)

q, = A 3 [4~r0,4 vJ 1 ( v" - 1) + ol2a t,J[1 - Q + (1 - Q )

x (ao+k,~)~

quantity X - 1 = ( J s - J ) / J now appears in the flux equations, which is proportional to the difference of the radiation intensity, J,, of the light source and the intensity, J, of black-body radiation at ambient temperature T. The quantity ae which appears in Eqn. 117 is the overall rate constant for spontaneous excitation (radiative and nonradiative) in thermal equilibrium. In practical cases, the energy difference, hv, between the ground state and the primary excited state is much larger than the average thermal energy, meaning that Q << 1. Furthermore, under the usual experimental conditions the intensity, J~, of the radiation source is much higher than J. In this case Eqns. 117 and 118 simplify to

(122)

Eqns. 117 and 118 are analogous to Eqns. 28 and 29 describing ATP-driven proton transport. Instead of the chemical driving force Y - 1 , the

"

(125)

"o + k ~ + oa'a"( J s - J ) O

Under most experimental conditions, the term ol2AvarsQ = oJsQ in Eqn. 125 is negligibly small. W i t h a m o l a r e x t i n c t i o n c o e f f i c i e n t of bacteriorhodopsin of c -- 5 • 104 M -1 • c m - ! at the absorption peak of the ground state (560 nm) [88], the absorption cross-section is o = 1000(c/L) In 10 -- 2- 10 -16 cm 2 ( L is the Avogadro constant). In experiments with steady illumination, the light intensity is usually less than 1 W - c m -2, corresponding (at 560 nm) to a quantum flux density of Js < 3 • 1018 cm -2 • s -1, so that the absorption rate constant, oats, is of the order of 103 s -1 or smaller. On the other hand, the rate constant b 0 - - a o / Q for nonradiative deactivation of an excited singlet state is usually many orders of magnitude larger. This means oJs << b0, or QoJ~ << a 0 < a e. Introducing further the conditions Q << 1 and J~ >> J as discussed above, Eqn. 125 simplifies to: (A/XH),._O = - _~ln(1 +

aJ~ a . + k~E

)

(126)

332

Several interesting conclusions may be drawn from Eqns. 125 and 126. First, it is evident that the reversal potential does not depend on the p K values of the binding site (pK c and PKE). This result has also been obtained in the case of ATPdriven pumps (subsection IliA). Second, at moderate light intensities (Js << bo/°), the reversal potential is a function of quantum flux density, Js, not of quantum energy hr. At very high intensities where the term o12AvJsQ in Eqn. 125 becomes important, the reversal potential reaches the thermodynamic limit: ( A~r~)~,._ o =

1 R T l n ( 1 / Q ) = -- n L h V

(127)

For bacteriorhodopsin (;~ = 560 nm), the thermodynamic limit of the reversal potential (expressed as a voltage) is h v / e o = 2.2 V if n = 1 is assumed. This voltage is much larger than experimentally observed A/2n values, meaning that the pump normally operates far from thermodynamic equilibrium. If the rate constants of nonradiative transitions between the excited states ( H , X and H , E ) and the ground state (H,C) are negligibly small (a 0, b0, k*CE, k~c ~ 0) and if the radiation is isotropic (12 = 41r) the pump becomes completely coupled (q~n=nq~) and works as a reversible machine transforming light energy into electric and osmotic energy. (The condition 12 = 4~r is necessary for reversible operation, since otherwise part of the light quanta emitted by the photosystem would be absorbed at the lower temperature T with dissipation of free energy). Introducing the condition (a0, bo, k*cE, k~c ~ 0, 12 = 4~r) into Eqn. 125 and using Eqns. 110, 111 and 120, the reversal potential is obtained in the form: n

1

T

the net rate OrLhu of energy absorption: (~HA~.H *1~ -

q~rLh~

(129)

The thermodynamic efficiency is highest if the pump is completely coupled (q'H = nq~r) and operates close to the reversal potential. Under this condition Eqn. 128 may be used, yielding T T/max = 1 -- Tss

(130)

This is the well-known relation for the thermodynamic efficiency of a reversible Carnot machine operating between an upper heat reservoir of temperature T~ and a lower heat reservoir of temperature T. Eqn. 128 means that the energy of light quanta can never be completely converted into free energy, even when a reversibly working thermodynamic machine is used [80,81,83,84]. The fraction of hu which can be converted into free energy depends on the radiation temperature, ~ , of the source and the ambient temperature, T, and is equal to 1 - T/Ts. In a typical experiment with bacteriorhodopsin [89], the light intensity may be of the order of 10 m W - c m -2 in the wavelength band between 560 and 580 nm (in vacuo), corresponding to Js = 3 • 1016 cm -2. s -1. Using a refractive index of n = 1.33 for the aqueous medium and assuming, for simplicity, isotropic radiation (82 = 4~r), a radiation temperature of T~ = 1600 K is calculated from Eqn. 110. This means that at an ambient temperature of T ~ 300 K the maximum thermodynamic efficiency is about 0.81. The quantum efficiency, qH, for proton transport may be conveniently defined under 'level flow' conditions (v = 1):

(128) qH -=

This equation defines an upper limit for the reversal potential at arbitrary light intensity Js (i.e., at arbitrary radiation temperature Ts). In the limit of infintely high radiation temperature (Ts>> T), Eqn. 128 becomes identical with Eqn. 127, as it must be. The thermodynamic efficiency ,/ of the lightdriven pump may be defined as the ratio of work performed per unit time (--q~nA/2u), divided by

(131) v-I

Using Eqns. 117 and 118 one finds: qn =

nkcE

kcE + (ao + ,~E)c"/I¢c

(132)

For a completely coupled pump (a 0 = 0, k~E = 0), qH becomes equal to n, the number of protons translocated per cycle.

333

Any ion p u m p can, at least in principle, be driven in backward direction by artificially imposing an electrochemical gradient which is larger than the reversal potential. In the case of lightdriven ion pumps, backward operation should lead to net emission of light quanta ( q ~ < 0 ) . The quantum yield, q~, of light emission may be defined for the condition that no external radiation source is present ( X = 1): qr~ -- ( ~ 2 ~ ) X_ 1

(133)

According to Eqns. 117 and 118, q~ is obtained as (with Q << 1): qr ----

bf/n

(134)

bo + b t + k ~ E / Q

Since Q is of the order of 10 -38 for X = 570 nm at room temperature, the quantum yield, qr, of fluxdriven light emission is extremely small. In fact, such light emission from bacteriorhodopsin has never been observed. A further quantity of interest is the fluorescence quantum yield, qf, which may be defined as the ratio of the rate of spontaneous emission, divided by the absorption rate under the condition J~ >> J and v = 1:

iological conditions the enzyme usually synthesises A T P from A D P and Pi, utilising an electrochemical proton gradient. The mode of operation can be easily reversed, however, so that, depending on the magnitude of the driving forces, the enzyme may also act as an ATP-dependent proton pump. The H+-ATPases of mitochondria, chloroplasts and bacteria (often referred to as type I ATPases) probably differ in their mechanism from other transport ATPases, since recent experimental results strongly argue that a phosphorylated intermediate is not involved in proton translocation [91]. A model for the functioning of type I ATPases which does not require phosphorylation of the protein may be based on the assumption that binding of H ÷ from one side and release to the other side drives the protein through a cycle of conformational transitions which in turn are coupled to the binding and release of ATP, A D P and Pi [2]. Such a mechanism may be described by the reaction scheme of Fig. 4 if it is assumed that during the conformational transition H , C ~ HnE an A T P molecule is bound to the protein and that during the transition E ~ C A D P and Pi are released: k~E H n C + A T P ~ HnE

(137)

k~c PHX bf qf = PHco.J s

(135)

For large J~ and Q << 1 the fluorescence quantum yield is obtained as: bf qf = bo + bt + (k~E + k c E K c / c , . ) / Q

(136)

Comparison with Eqn. 134 shows that qf is always smaller than nqr. For bacteriorhodopsin, the fluorescence quantum yield is found to be extremely low (qf ~< 10 -3) [90]. IX. Type I ATPases

k EC E ~:~ C + A D P + P ~ kcE

This means that k~c and kEC now are true monomolecular rate constants, whereas k~E and kCE (Eqns. 24 and 25) have to be replaced by: k~E = fCT; kCE = gCD¢ P

(139)

( f and g are concentration-independent quantities). With these definitions of kcE,kEc,k~E and k*EC, Eqn. 27 assumes the form (with K - ?D?P/?T): KE

.fkEc

Kc gk~c The proton-translocating ATPases of mitochondria, chloroplasts and bacteria have been extensively studied over the past 20 years (for a recent survey of the literature, see Ref. 103). Under phys-

(138)

K-exp(nu)

(140)

The proton flux, ~H' and the ATP-hydrolysis rate, q'c, are obligatorily coupled under the assumptions of the model and are given by

334 '/'n = nq,c = D~ n Kc c "-~kcEk ~c( Yv" -

1)

(141)

The quantity D 2 is defined by Eqn. 31. X. Transient

behaviour and electrical noise

From stationary measurements of ion flux (or electric current) it is usually not possible to determine the single rate constants of an ion transport system. Much more information may be obtained from relaxation experiments in which the system is disturbed by a sudden change of an external parameter, such as temperature, pressure or electric field strength, and the approach toward a new stationary state is recorded. Electrical relaxation techniques have been extensively used for kinetic studies of ionophores in artificial bilayer membranes [92]. Little work of this kind has been done so far with electrogenic ion pumps. The time-dependence of photocurrent and photovoltage after flash excitation has been studied with bacteriorhodopsin in artificial membrane systems [93-95] and in suspensions of purple membranes [96,97]. From such experiments, information can be obtained not only on the rate constants of the pumping cycle but also on the magnitude of charge displacements associated with the single reaction steps. If the pumping cycle starts at the ground state P0 and goes through p intermediate states P~, P2 . . . . . P~ before returning to P0, the time-dependence of the electric current, I = e0q~n, after a sudden perturbation at time t = 0 may be represented by the sum of p exponentials: l(t)=

~

Ciexp(-t/r,)+loo

(142)

i=l The relaxation times, ~i, are combinations of all rate constants of the cycle, whereas the 'amplitudes' C, contain, in addition, the initial conditions and the so-called displacement coefficients [92,94] describing the magnitude of charge displacement in the single transitions. As a specific example, we consider the following light-driven pumping cycle: hl' kl k2 k3 Po -~ Pl -" P2 ~ P3 ~ ao

(143)

assuming that in the dark only state P0 is popu-

lated and that the primary photoproduct P1 decays in three virtually irreversible dark reactions back to P0. If, at time zero, N p u m p molecules in the membrane are excited to state P1 by a flash of light, the transient current which is measured in the external circuit [94] is given by Eqn. 142 with u = 3 and

I klk2 klk2k3 ] Cl=eo N alkl+a2k2_k~+a3 (k2_kl)(k3_kl) (144) [ klk2 C2 ~ eoN [~2 kl _ k~ C3

eoS~3(k,-

"ri = 1 / k l ; 1~ = 0

klk2k 3 + ~ 3 ( k I - k2)( k3 - k2 )

klk2k 3

k3)(k2-

k3)

]

1

(145)

(146) (147)

The 'displacement coefficients' a i are defined in the following way: if a single transition Pi ~ Pi+l occurs in the membrane, the charge e o a ~ is displaced in the external circuit (e 0 is the elementary charge). If n protons are transferred from one aqueous phase to the other during the photocycle, then a o + a~ + a 2 + a 3 = n. Thus, the relaxation experiment yields two kinds of information: from the relaxation times, ~, the rate constants, k~, may be obtained, and from the amplitudes, Ci, the displacement coefficients, a i (provided that N is known). The quantities ai, which are closely related to the voltage-dependence of the transitions rate constants [98], contain valuable information on the microscopic nature of the conformational transitions [93,94,97]. An entirely different experimental approach which (in principle) yields the same kinetic information as relaxation experiments consists in analysing the e l e c t r i c a l n o i s e which is associated with steady-state operation of an ion pump. The pumping cycle is a sequence of elementary events which occur at random intervals and which are, in general, associated with charge movements. Accordingly, the pump current fluctuates around an average value. A convenient method for analysing such current noise consists in measuring its spectral inensity S l as a function of frequency f (or angular frequency t o = 2 ~ r f ) [99,100]. For the light-driven p u m p cycle considered above (Eqn. 143) the spectral intensity of current noise under

335

low-level steady illumination is predicted to be of the form: S I ( . ) = SI(O) [ Or2+ ~I2 + ~22 + OI2

4-

2a2a s

2a0aa

2a,a2 -

-

1 -- ~ 2 ~

1 + . 2 7 +2a°a2 (1 + J , ? ) ( l +

+ 2 ~tla3

F

27)

1 - w2r2~,s

2 2 (1+-,5)(1+.24)

1 - ~2(~1~ + ~ +2aoa~(1+~-~2~~

+ rl~)

] )]

(148)

The time constants, ,j, which appear in this equation are identical to the relaxation time constants % = l / k , of Eqn. 142. In the derivation of Eqn. 148 it has been assumed that one proton is translocated in the cycle, so that a 0 + a 1 + a 2 + ~3 = 1. The low-frequency limit, SI(0), of the spectral intensity is given by the average p u m p current I (which depends on light intensity): SI(0 ) = 2 % ]

(149)

Eqn. 148 predicts frequency-independent ('white') noise both at low and at high frequencies. Eqn. 149 is identical with the relation describing the spectral intensity of current noise in a thermionic diode [99]. XI. Conclusion

A microscopic description of active ion transport may be based on the notion that ion p u m p s are channels with multiple conformational states. Energy from an external source drives the p u m p molecule through a cycle of c0nformational transitions whereby the ion binding site is alternately exposed to the cytoplasm and to the extracellular side. The analysis accounts for incomplete coupling (sometimes called 'slippage') between the energy-supplying reaction (ATP hydrolysis, light absorption) and the translocation process. Incomplete coupling m a y result from back reactions (e.g., spontaneous dephosphorylation) or from barrier leakage.

Most of the ion pumps studied so far are electrogenic, i.e., they transfer net electric charge across the membrane. This leads to a number of interesting phenomena. The p u m p may act, depending on the conditions, as a current source or as a voltage source. Furthermore, pumping activity may be modulated by an electric field which is present in the membrane. ATP-driven proton pumps are characterised by two fluxes, the proton flux, g'Ft, and the net rate, g'c, of ATP hydrolysis, and two conjugate driving forces which depend on the electrochemical potential difference, A/2H, of the proton and the free energy, AG, of ATP hydrolysis. In the case of light-driven ion pumps, ~c is replaced by the net rate of light absorption, and the conjugate force is given by the intensity of the radiation source. The coupling ratio ~ H / ~ is variable, depending, in general, on the driving forces. ~ H / ~ should be distinguished from the stoichiometric ratio n, which is equal to the limiting value of ~ n / ~ c corresponding to perfect coupling. The protonmotive force (PMF) of the p u m p is the value of A/2H//F at which ~ n vanishes ( F is the Faraday constant). For an ATP-driven proton p u m p PMF depends on AG and on the rate constant of spontaneous dephosphorylation; only in the limit of complete coupling does P M F become equal to AG/F. The expression for the protonmotive force does not contain the equilibrium constants of proton binding. This means that a p K shift is not essential for the thermodynamic efficiency of the pump. A change of p K upon phosphorylation (or light absorption) is important, however, for a high turnover rate of the pump. At a certain value of A ~ H ATP hydrolysis ceases (ff¢ = 0); if larger values of A/TtH are imposed, A T P is synthesised from A D P and Pi. The reversal potential, Ap.H/F, of ~ is, in general, different from the reversal potential of ion flux, ~H. Both reversal potentials become equal in the limit of complete coupling. In the vicinity of equilibrium, the fluxes may be represented as linear combinations of the driving forces. The coefficients of the 'phenomenological' equations obey Onsager's reciprocal relations. Furthermore, the phenomenological coefficients are related to the unidirectional fluxes at equilibrium (the so-called exchange flows).

336

The current-voltage characteristic of the pump is determined by the voltage dependence of the single reaction steps. Depending on the location of the binding site with respect to the mebrane surface, the rate constants of binding and release of the ion will be affected by voltage. During a conformational transition the ion may move together with the binding site and, furthermore, polar residues of the protein may be reoriented. In this way, the rate constants of conformational transition become functions of voltage. A variety of different shapes of current-voltage curves is feasible. If a voltage-independent reaction step becomes rate limiting, the current-voltage curve saturates at voltages far from the reversal potential. The pump then acts as constant-current source. Under certain conditions the current-voltage curve may exhibit regions of negative resistance, meaning that the pump becomes inactivated at increasing voltage. An ion pump transforms chemical free energy (or energy from a radiation field) into osmotic and electric energy. The process of transduction of free-energy may be described on the basis of energy levels of the pump protein in the different states of the cycle. Two types of free-energy level diagram may be used. Energy levels based on standard values of chemical potentials reflect intrinsic properties of the pump molecule in the different conformational states. On the other hand, the properties of the pumping cycle as a whole (e.g., the nature of the rate-limiting steps) are more easily discussed on the basis of the concentrationdependent total values of chemical potentials.

Acknowledgements I wish to thank Dr. G. Adam and Dr. H.-J. Apell for interesting discussions. This work has been financially supported by Deutsche Forschungsgemeinschaft.

pump molecules in state Z per unit area of the membrane, then detailed balance requires that the following relations hold: Nc

Kc

NE

K E

NH~E

C"m

(A-l)

NH~ (.

C"n '

NH. E

kx E .

NH.X

pC T

qCp

NH. x

kEx '

NH. C

rc D

s

Ny

kcv

=

NE kyE WCp Nv -- kE v -- kE¥

.

kv~'

(A-2)

(A-3)

Eqn. 8 then follows directly from Eqn. A-2. Combination of Eqns. A - l - A - 3 yields: KE

Kc

kxEkEvkycq kExkcyws

[

=~-)

"

= exp(nuo )

(A-4)

Since any value of the equilibrium voltage u 0 can be obtained by suitable choice of the ion concentrations c" and c", and since the rate constants do not explicitly depend on c' and c", Eqn. A-4 must hold at any voltage. This proves Eqn. 9.

Appendix B: Derivation of Eqns. 15-19 If Pz is the probability that a given pump molecule is in state Z, the stationary fluxes q~n and q~c are given by (with H , C - - H C , etc.): ~H = k c x P n c -- k x c P H x

(B-l)

¢hc = p C T P H c -- r C D P H x

(B-2)

Since states H C / C and H E / E are always in equilibrium, the relations P c / P n c = K c / c ' " and PE//PHE = KE/C 'm hold. Introducing the probabilities P ' - Pc + PHC and P"=- PE + PHE, one obtains: P'c'"

P'Kc

Pc

K c + c'" ,

PHC

K c + C, n

(a-3)

Appendix A: Derivation of Eqns. 8 and 9 We consider the situation where chemical reaction and transport reaction are both at equilibrium. This means that CDCp/CT = ODOP/~T = K and u = l n ( c " / c ' ) = u o. If N z is the number of

PE

P"K E g E + c p'n

,

PHE

P"c""

(B-4)

K E + Cr'n

The probabilities Pz may be calculated from the condition that the time derivatives d P z / d t vanish

337 p

in the steady state:

dP'

-dt

= -- k c x P H c - k c v P C + k x c P H x + k v c P Y = 0

dPHx dt

.(kxc+kxE)PHx+kcxPc+kExPE=O

(B-5)

(B-6)

A n a l o g o u s equations hold for d P " / d t and d P v / d t . Together with P' + P " + PHX + P v = 1, one finds:

kExkxc( kvc + kvE)

1+

+ k~k~c~7;,. (kxc + kx~)

(B-7)

C

are denoted by x c' and x c" for state H C and by x E " •E' and XE " and t¢~ for state HE. Whereas r c' , r c, are monomolecular rate constnats, the binding rate constants X'c, A'~, ~kPE and A'~ are proportional to the proton concentrations c' and c" in the aqueous solutions. Assuming that the ( i - 1)th energy well is always in equilibrium with the cytoplasmic phase and the (i + 1)th well in the equilibrium with the extracellular phase (Fig. 1), we m a y simply write h' c = p'cc' and X'~ = p~c" where p~ and p~ are concentration-independent quantities. Denoting the total rate constants for protonation and deprotonation (in state H C / C ) by Xc and x o we have: ~c = lic + li~ liE

, ,,

=

(C-l)

lie' "~ liE"

(C-2)

1 [

(c-3)

~c = ~'c + x'b = p~c'+/%'c Kc

+ k c y k y E T ( k x c + kXE )

]

In analogy to Eqns. 4 - 6 , the rate constants for phosphorylation and dephosphorylation are given by:

P.× = ~ [ k ~ × k ~ ( k c x + k c ~ C c. )

+ kcxkvc( kEx + k E v ~ ) ] 1 [

K¢[

(C-4)

(s-s)

(B-9)

k~c = rcD + s;

kcE = WCp

(C-5)

Application of the principle of detailed balance leads to the following relations (compare A p p e n dix A):

KE~

"4-kxEkEy~n( kcx + kcy~n ) ]

k~E = pCT + qCp;

(B-a0)

D 1 is given by Eqn. 1. Inserting Eqns. B-3, B-4 and B - 7 - B - 1 0 into Eqns. B-1 and B-2 and using Eqns. 8, 9, 13 and 14 yields t~H and q)¢ (Eqns. 15 and 16).

Appendix C: Flux equations for the four-state model with single binding site and barriers of arbitrary height (Fig. 2) In the following, we give the results for the p u m p mechanism described in subsection I I A (Fig. 2). In this case it is assumed that both barriers have finite height, so that transitions from the p r o t o n a t e d to the u n p r o t o n a t e d form occur by release of the p r o t o n either to the right or to the left side (Fig. 1). The corresponding rate constants

li~'ESW= li~htckECq lib;~'c li~X'E

c',, exp( u)=- v

(C-6)

(C-7)

Eqn. B-1 for the proton flux now has to be replaced by: ~ . = e.cli~ - PcX'~ + p.:~ - PEx'~

(c-8)

In a way analogous to that described in A p p e n d i x B, the flux equations are obtained as: 1 (kH = D0 [ Q " ( v - 1) + QH¢ ( Y -- 1)]

'h¢= D'~o[QcH(v -I)+Qc(Y-I)]

(C-9)

(C-IO)

338

Q._= x'bx'~ (.~/x'~) kc~k~¢ + X'~X'~(kc~k,~ + k¢~¢ + k ~ X c ) + X'CX'~(K~/)gE)kEck~, E

+ K .C.X. c. (.k E c k E c

+ kECX E

+ k~cXE)

(C-ll)

QHc=--rcOkcE( X'~/~'E)( ,'(;K'E -- K'cg~)

(C-12)

QcH=--rCDkcE( X'~/~e )( x~K ~ - Y~-K~)

(C-13)

If Qnc in Eqn. C-9 vanishes, the chemical driving force ( Y - 1 ) is no longer able to induce an ion flow. According to Eqn. C-12, this is the case for x~;r~ = r~r~. Under this condition, where the ratio x ' / r " of the dissociation rate constants of the ion from the binding site becomes equal in states H C / C and H E / E , there is no longer any asymmetry in the height of the barriers. In other words, coupling of the scalar chemical reaction to the vectorial transport process requires that r'c/x'(', ¢ t /

t/

E/KE •

e¢ =-rc o [ kce ( xcX'e+ XC:X'E+ KcX'~) + qce( kCEXE

+ kECXC+ XcXE)+kCEX'cX'~(~c/X'c)]

(C-14)

In the limiting case of ideal barrier asymmetry, the mechanism considered here becomes identical with the minimum model depicted in Fig. 4. In

fact,

l/

t

f o r KC,t X'~, r E , ~ktE ~ 0, KC, • C ,

t

it

r E , ~kE --'> O0,

x ~ / X ' c = K c / c ' , rE~A" ~ = K E / C " Eqns. C-9-C-15

Do---k~e [(X c + kcE)+ (rE + XE)+ kEC(Ke + Xc) ]

reduce to Eqns. 28-30 (with n = 1).

+ k c ~ ( , ~ + X~)+ k~*c[(X~ + k ~ ) ( ~ + x~) +kcE(K c+XE)]+kEcxE(xc+Xc) (C-15) Eqns. C-9 and C-10 have the same form as Eqns. 15 and 16. An essential property of the mechanism considered here (Fig. 2) is the assumption that all barriers have finite height. This means that the pump molecule still acts as a passive ionic channel when the conformational transitions are blocked. Indeed, in the absence of phosphorylation ( k c E = k~E = 0), Eqn. C-9 reduces to: ~t pt

trot

K~X'~ t --1)= rc c [ e x p ( a ~ n / R T ) - q (C-16) ¢#H Kc + Xc . v KC+Xc

Appendix D: Flux equations for light-driven ion pumps (a) Derivation o f Eqns. 117 and 118

For the reaction scheme of fig. 18 the proton flux, ~u, and the net photon absorption rate, ~r (both referred to a single pump molecule), in the steady state are given by ~bH = k E c P E -- k c E P C dPr = ° ( Js +

This is the relation for the ion flux through a passive channel with a single binding site [24]. According to Eqns. C-9 and C-10 the coupling ratio O=-~H/@~ is, in general, different from unity. Deviations from O = 1 result from 'leakiness' of the barriers which gives rise to back flow of ions, as well as from direct dephosphorylation (described by the rate constant s in Eqn. 5). The pump becomes completely coupled if in conformation C the right barrier and in conformation E the left barrier is infinitely high (r~ = A'~ = AtE = r ~ = 0) and if, in addition, direct phosphorylation and dephosphorylation are blocked (q = 0, s = 0). In this case, Eqns. C-9 and C-10 reduce to

J ) P H c -- [bf + o ( J s + J ) ] PHX

(D-l) (D-2)

(compare Eqns. 112 and 113). Pz is the probability of finding the pump molecule in state Z: PHC + PHX + PHE + PE + PC = 1

(D-3)

Since it has been assumed that the reaction H , X H , E and the two protonation-deprotonation reactions are always in equilibrium, the relations P H x / P H c = Q, P c / P H c = g c / c t, P E / P H E = K E / C " hold. This yields (together with Eqn. D-3)

four equations for the five unknown probabilities Pz- A fifth relation is given by the steady-state condition: d

~ n = ~c =

1 rCDIeC)kEkcE .... D~

( o Y--

1)

(C-17)

d~ (PHC + Pc ) = --( a + k~E)PH¢ + bPux + k ~cPHE -- k c E P c + k E e P E

= 0

(D-4)

339

In this way, the five probabilities Pz may be evaluated. Inserting the Pz into Eqns. D-1 and D-2 and using Eqns. 110-116 yields Eqns. 117 and 118.

(b) Flux equations at small driving forces If the radiation temperature, T~, of the light source (Fig. 19) is close to the ambient temperature, T, and if the electrochemical potential difference is small (A/~ H = 0), the system operates close to equilibrium. It can be shown that for the system depicted in Fig. 19 under the condition A/~ H --- 0, AT--T~ - T = 0, the dissipation function per mol of pump protein is given by: diS T - - ~ = ¢ku A Ftrt + tkr A X >10 A x = - L h I , ~~2 - AT= RT~(1-

(D-5)

Q)()'is - l)

(D-6)

The factor ~2/4~r in the expression for the 'radiation force', A X results from the fact that of the light quanta absorbed in the photosystem (temperature T ) only those which originate from the radiation source (temperature T~) contribute to free-energy dissipation. According to Eqn. D-5, the phenomenological equations have the form t~bH

=

LHA/~H + LHrAX

(D-7)

~r = LrHA/2H + LrA~(

(D-8)

In the vicinity of equilibrium, the quantity X (Eqn. 119) is given by:

),

X=7--l+

h~/kT AT 1-Q r

(D-9)

After insertion of Eqn. D-9 into Eqns. 117 and 118 (together with v" -- 1 + n A f t H / R T ), the phenomenological coefficients are obtained in the form: ,A3 L.=n'-~(a¢

+ k~E)

(D-10)

A3 81roQ LHr = L,. H = n ~ 4 ~ r o A p J . . ~ n.4 3 RTX 2

(D-11)

.4 3

. C'n keE) K c k c E

(D-12)

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