a simple nonequilibrium thermodynamic description of some inhibitors of oxidative phosphorylation

J. theor. Biol. (1985) l l7 , 471-488

A Simple Nonequilibrium Thermodynamic Description of some Inhibitors of Oxidative Phosphorylation

DAVID J o u AND FERNANDO FERRER

Departament de Termologia, Universitat Autbnorna de Barcelona, Bellaterra, Catalonia, Spain

(Received 18 March 1985, and in revised form 9 July 1985)

We propose a macroscopic description of some inhibitors of oxidative phosphorylation based on a simple modification of the phenomenological coefficients appearing in the constitutive equations of linear irreversible thermodynamics. In this theoretical model, we consider protonophores, some ATPase inactivators and some electron-chain inhibitors, and we provide quantitative expressions for their consequences on the protonmotive force, oxidation flux and phosphorylation flux as well as on heat generation.

1. Introduction

Biological processes are typical nonequilibrium situations. Their study may be undertaken from different points of view. The most fundamental one is that based on the kinetic mechanisms of the chemical reactions. However, in many circumstances there are many different chemical reactions involved in a phenomenon and, furthermore, they can be related to physical processes such as diffusion. This leads immediately to mathematical complications which, in spite of the possil~ility of their numerical analysis, may obscure the main trends of the global process. In these cases, it is of interest to dispose of a simplified, global instrument for the theoretical analysis of the main features of the phenomenon, at least as a first approximation. Such a f ramework of theoretical analysis is provided by nonequilibrium thermodynamics (Katchalski & Curran, 1965; Nicolis & Prigogine, 1977; Caplan & Essig, 1983), which may be able to describe the general course of some experimental situations and may provide some insight on some topics such as the restrictions of the second law of thermodynamics on the stoichiometry and on the energetic efficiency of the overall process. The situation is parallel to that of thermodynamics and statistical mechanics in physics: thermodynamics is able to lead to a set of very general and useful relations but, however, it cannot give any detail on the microscopic mechanisms. Though for a deep understanding of macroscopic phenomena statistical mechanics is indispensable, the thermodynamic analysis is not at all negligible, but

471

0022-5193/85/230471 + 18 $03.00/0 © 1985 Academic Press Inc. (London) Ltd

472 D. J O U A N D F. F E R R E R

may provide a useful test either for the experiments or for the theoretical microscopic models.

The linear irreversible thermodynamic equations describing oxidative phosphorylation have been stated by different authors (Caplan & Essig, 1969; Pietrobon et al., 1982; Rottenberg, 1979; Stucki et al., t983; Stucki, 1980; Van Dam et al., 1980; Walz, 1979; Westerhoff & Van Dam, 1979). However, such a set of equations has not been exhaustively used in the analysis of this phenomenon. Indeed, some specific topics have polarized the interest of the researchers, while some other ones have been left aside. Examples of the subjects most usually dealt with in this field are the efficiency of oxidative phosphorylation in energy conversion as a function of the degree of coupling (Caplan & Essig, 1969, 1983; Rottenberg, 1979), the suitability and the meaning of linear equations in such a highly nonlinear process (Van Dam et al., 1980; Stucki et aI., 1983) and the relation of different kinds of steady states to minimum entropy production (Stucki, 1980). On the other hand, some physically interesting topics have not yet been studied in sufficient depth. Examples of the latter are the dependence of the phenomenological coefficients on the different kinds of uncouplers.

These topics have been dealt with very thoroughly from an experimental point of view by many authors but, in our opinion, the considerable wealth of data has not been submitted to sufficient quantitative analysis, nor has it been synthesized in compact expressions for the phenomenological coefficients. Such a synthesis would lead to a wider potentiality of the existing nonequilibrium thermodynamic formalism and, on the other hand, it would provide a quantitative framework correlating the rich diversity of experimental results. This would contribute to the understanding of the action of uncouplers by linking some microscopic models at a molecular level with the corresponding macroscopic thermodynamic formulation.

Our aim in this paper is to provide a simple hypothesis on the dependence of the phenomenological coefficients of the thermodynamic equations on the concentration of several kinds of uncouplers, mainly protonophores, ATPase inactivators and electron-chain inactivators, so widely used in experimental research. Excluded from our present analysis are the uncouplers involving the flux of other ions than protons. We apply the formalism to calculate some quantitative expressions for the action of the uncouplers on protonmotive force, oxidation flux and phosphorytation flux, as well as on heat generation and the degree of coupling. We work in the usual linear thermodynamic framework and in the chemiosmotic hypothesis (Nicholls, 1982), our aim being only an increase in the usefulness of such equations and a deeper understanding of some of their aspects, rather than a consideration of some more fundamental problems in this field.

I N H I B I T I O N O F O X I D A T I V E P H O S P H O R Y L A T I O N 473

Though the present model is limited to linear phenomena, it may be useful in the analysis of several experimental situations. The linearity assumed in linear nonequilibrium thermodynamics, concerns deviations around the equilibrium state, so that in the analysis of biological processes its use may be problematic. However, another kind of Iinearity is often encountered in practice, the so-called kinematic linearity, which implies a linearization around a (nonequilibrium) inflection point of a nonlinear flux-force relation. In such a point, the vanishing of the corresponding second derivative makes the linearization specially adequate. In this case, the laws describing the phenomenon are similar to those used in linear nonequilibrium thermodynamics, with the difference that a constant is added to the chemical affinities of oxidation and phosphorylation appearing in the corresponding equations (Westerhoff et al., 1982).

In this way, our results, besides their possible theoretical interest as an application of nonequilibrium thermodynamics to a basic biological situation, may be also useful for the analysis of some experimental situations of kinetic linearity, with the only modification of adding a constant to the chemical affinities. The plan of the paper is as follows; section 2 is a presentation of the usual linear constitutive equations of oxidative phosphorylation. Sections 3 and 4 deal respectively with protonophores, ATPase inactivators and electron-chain inactivators.

2. Linear Thermodynamic Equations of Oxidative Phosphorylation

Three kinds of processes participate in oxidative phosphorylation, at the simplest level of description (Nicholls, 1982): an oxidation-reduction reaction, a transport of protons through a closed membrane (inner mitochondrial membrane, sub-mitochondrial membrane or some bacterial membranes) and a phosphorylation reaction. The corresponding fluxes of such processes will be denoted as Jo (oxygen consumed per unit time and unit membrane area), Jp (ATP produced per unit time and unit membrane area) and JH(net influx proton flow through the unit area of membrane). Conjugated to such thermodynamic fluxes there are the corresponding thermodynamic forces, namely Ao(-AGo) chemical affinity of oxidation, Ap(-AGp), chemical affinity of phosphorylation, and A/~n, protonmotive force, i.e. the difference between outer and inner proton electrochemical potential (Caplan & Essig, 1983). The scalar processes, chemical reactions, are coupled to the vectorial process, proton transport, by means of the high anisotropy of the intervening membrane. The linear constitutive equations corresponding to such processes are, according to the usual theory of linear nonequilibrium thermodynamics, the following (Caplan & Essig, 1983; Rottenberg, 1979; Van Dam

474

et al., 1980)

D. J O U A N D F, F E R R E R

Jo = LoAo + LouA~H + LopAp (1)

JH = LoHAo + LHAI~. + L.pAp (2)

Jv = LopAo + L . p A t l . + LpA v (3)

where the L o are phenomenological coefficients which depend on the characteristics of the membrane and of the oxidation and phosphorylation pumps, as well as on the presence of uncouplers. The validity of such linear equations and of the supplementary Onsager's reciprocity relations is an open problem (Van Dam et al., 1980; Westerhoff & Van Dam, 1977), but it seems satisfactory enough at least as a first approximation to this problem.

The second law of thermodynamics implies that Lo, L , and Lp must be positive, as well as some other supplementary requirements as, for instance, LoLn >- Lone and LpLH >- LpH.z The chemiosmotic hypothesis implies, on the other hand, Lpo =0, i.e. that no direct coupling between oxidation and phosphorylation exists, the only one being due to proton transport across the membrane (Caplan & Essig, 1969, 1983). Finally, if one considers positive the influx of protons, one has Loll < 0 and L,p > 0, i.e. oxidation and proton influx are negatively coupled, while proton influx and phosphorylation are positively coupled (Caplan & Essig, 1969, 1983).

To have a further insight on the meaning of the coefficients, we may write the equation for the proton flux in the following alternative way

JH = -nolo + cuA f i . + np.lp (4)

where no and np (both positive) are the dynamical stoichiometries relating oxidation and proton efflux and phosphorylation and proton influx, respectively. On the other side, cH (positive) is the passive permeability of the membrane to protons. Equation (4) may be written in terms of Ao and Ap a s

JH = -noLoAo + ( c . - noLoH + npLHp)AI~H + npLpAp (5)

where we have adopted the chemiosmotic value Lop = 0. Comparison of (5) with (2) shows that indeed Lon = -noLo <0 and that LHp = npLp > 0, and it provides a physical meaning to the coefficient Ln. Note that Ln-- cn - noLo~ + npLn. cn + n2oLo + 2 = npLp>O, according to the requirement that Ln > 0. Furthermore, the requirements 2 LoL. >- Loll and LpLH >- L~p. are also fulfilled.

In this model, oxidative phosphorylation is characterized by five independent parameters, namely, L. , Lo, Lp, Lo. and L.p or, alternatively by c., Lo, Lp, no and np. In the usual circumstances, Jo, Ao and Jp are positive,

I N H I B I T I O N O F O X I D A T I V E P H O S P H O R Y L A T I O N 4 7 5

while Ap remains negative. The latter fact is associated with the potential of ATP being hydrolysed releasing the accumulated energy or, in other words, the oxidation process is spontaneous while the phosphorylation process proceeds in a nonspontaneous way. Some indicative values for the coefficients for rat liver mitochondria (Van Dam et al., 1980) at pH = 7.0, T= 25°C, with a substrate of sucrose and tris-succinate are as follows

c , = 3.16± 1.0 nmol H+/(mg protein min mV)

L , = 7-9+ 2-0 nmol ATP/(mg protein min mV)

Lo = 1-9+0.1 natoms O/(mg protein min mV)

The use of several kinds of inhibitors is of first importance in the experimental and theoretical investigation of oxidative phosphorylation. There- fore, one of the first questions that arises in the thermodynamic context is how to describe the action of such inhibitors by means of constitutive equations. Obviously, they manifest their influence through the phenomenological coefficients, which depend in principle on the inhibitor concentration. Here, we will state an explicit and simple model to account for the action of some inhibitors.

In this paper, we will emphasize the following special aspects of the inhibitors action of experimental significance (1) the relation Jp as a function of Jo and Ao, which has been studied in some cases previously (Van Dam et al., 1980); (2) the degree of coupling of oxidation and phosphorylation; (3) the consequences on A//H, Jo and Jp at constant Ao and Ap; (4) the heat generation per unit area and unit time. We will restrict ourselves to the steady state of vanishing net proton flux.

(a) Jp as a function of Jo and Ao. At the steady state JH = 0, we may eliminate one of the thermodynamic forces in terms of the other two.

Following Van Dam et al. (1980) we choose to eliminate A/~m which is the most difficult to determine experimentally. Note that Ao and Ap refer to the values measured outside the mitochondria, which are more easily accessible to experiments. We have, after some trivial calculations

LpLH -- L2Hp Jo + LpLHL° -- LoL~p - L2on Lp Ao. (6) JP = LoI.ILHp LoHLHp

This may also be written in terms of cH, no and tip in the simpler form

j =Fno+ cn ] jo_ c , Ao. (7) L np nonpLo J nonp

The coefficients of the straight lines Jp = XJo-yAo coincide with those obtained by Van Dam et al. (1980), except for the sign of Jp and the

476 D . J O U A N D F. F E R R E R

introduction of an 'activity factor Yo". In differing with these authors, we have assumed no independent constant terms in the phenomenological equations, so that we do not have, in equation (7), any independent term corresponding to such constants. Note that both x and y are positive according to equation (7).

(b) Oxidation-phosphorylation degree o f coupling. Kedem & Caplan (1965) some years ago introduced the concept and quantitative expression of the degree of coupling of two processes. Such degree of coupling is defined as the coupling coefficient divided by the geometric mean of the corresponding diagonal or direct coefficients. In the case of two coupled processes 1 and 2 described by the equations

Jl = LIA1 + L12A2 and J2 = L12Al + L2A2 (8)

the degree of coupling of both processes is defined as q = L12(L~L2) -~/2. Such a coefficient is especially useful and significant in the analysis of the efficiency of energy conversion and of the stoichiometry of the coupled processes. In the steady state, when J . = 0, the three phenomenological equations (1)-(3) reduce to

• lo = [ Lo - ( L 2 o . / L . )]Ao - ( L o n L n p / L . )A t, (9)

J,, = - ( L o n L n p / L . )Ao + [ Lp - ( L2p/ L . )lAp.

Therefore, the oxidation-phosphorylation degree of coupling is given by

q = LonLt#/x / (LoLH - L2oH)(LpLH - L~ . ) . (10)

(c) Entropy production and restrictions on proton pump stoichiometries, Some restrictions on the stoichiometries no and np of the redox and ATPase proton pumps may be reached from a detailed consideration of entropy production. In the situation considered here, we will have for the entropy production o- the following expression

To" = Joao + JHA/.~n + JvAp. (1 l)

There is a more convenient way of writing equation (11), which manifests explicitly the origin of the dissipation. Indeed, introduction into equation (11) of equation (4) for the proton flux leads to

Q = (SoAo - noJoA/2n) + c.(AlSn)2+(n~JpAlYH +JpAt,). (12)

The second of these three terms corresponds to ohmic dissipation due to the flow of protons through passive leaks. The former one corresponds to the heat dissipated in the oxidation protonmotive pump (energy left by the chemical reaction minus energy gained by protons). The latter term gives

I N H I B I T I O N O F O X I D A T I V E P H O S P H O R Y L A T I O N 4 7 7

account of heat dissipated in the ATPase pump (energy left by ingoing protons minus energy accumulated in nonequilibrium ATP). In this way equation (12) is much more illustrative than equation (11). In particular, if we carry a step further the presumptions of the second law, we could state for the particular efficiency of each pump the following expressions

~?o=noAl~H/Ao<-I and ~p=lAvl/npA~.<_l (13)

independently of the values of Jo or Jp. These inequalities set some limits on the stoichiometric coefficients in terms of only static quantities as A~H, Ao and Ap. Indeed, one has no <- Ao/AlJ, and np> [Ap[/A~H. In the limiting case of a reversible process (Ao, Ap and A//H very small), such inequalities would reduce to equalities, and we would recover the results of previous authors concerning no and np (Rottenberg et aL, 1970; Van Dam et aL, 1980). In differing from those expressions, our relations are valid for relatively high values of the thermodynamic forces or fluxes. This is of interest because it is in principle conceivable that no and np vary with the thermodynamic forces. The static criterion (13) may supplement the usual dynamical measurements of no and np.

(d) Heat generation. One of the subsidiary roles of mitochondria besides phosphorylation is thermogenesis, especially in some systems such as brown adipose tissue of hamsters and guinea pigs. Therefore, it is of interest the analysis of mitochondrial thermogenesis. This is an experimentally accessible quantity which has been used in the study of oxidative phosphorylation. Let AHo and AHp be the corresponding reaction enthalpies (heat of reaction) of oxidation and of phosphorylation reactions, respectively. Then, the global heat generated per unit time and unit membrane area (or unit protein mass) will be given by

Q( t) = Jo( t)AHo + Jp( t)AHp. (14)

Here, we have indicated time t as a variable to emphasize that AHo and AHp are constant, which depend only on temperature and on the substrate, but not on the degree or the rate of reaction. In the analysis of thermograms, heat is recorded as a function of time. The temporal dependence of Q will be due in this case to the temporal dependence of the reaction rates Jo and Jp. Often, expression (14) is simplified and the heat generated is directly related to the rate of oxygen consumption, by writing a global heat of reaction AH"

O(t) ~- L ( t ) A H ' . (15)

In many cases this is a useful simplification and, according to experimenters, it works quite well in practice. Nevertheless, the approximate character and


the limitations of expression (15) must be kept in mind: the global heat of reaction AH" depends now on the degree of coupling between oxidation and phosphorylation, so that, in principle, it may depend on inhibitor concentrations and therefore on time, in nonstationary experiments.

3. Protonophores

One of the simplest uncoupling agents is provided by some substances which are able to increase the permeability of the membrane to protons. As examples we may quote dinitrophenol and FCCP, as the most well known (Nicholls, 1982; Terada, 1981). Though the mechanistic translocating behaviour of these agents may be quite different at a molecular level, their thermodynamic description is essentially identical, because such a description is more general than the microscopic one, and many of the particular microscopic details are lost in it.

The simplest way to account for the action of such uncouplers is, according to the views widely held, an increase of the passive permeability of the membrane to protons, cn. We will assume therefore that

OI/2 cH(n) = cn(O) + - - (16)

1 +/3n"

Here, cH(O) is the native conductance of the membrane, n the external concentration of protonophore and a and/3 are parameters which depend on the properties of both the protonophore and the membrane. The parameter o~ will be given by the conductance per particle of protonophore times the distribution coefficient characterizing the ratio of adsorbed protonophores over the external concentration n, while /3 is related to the saturation concentration N of adsorbed protonophores as/3 = 1/N. There- fore O/ and/3 are given by

1 a = A e x p ( - W / R T ) /3=--l~exp(-W/RT). (17)

Here, A is a parameter related to the conductance per particle, and W is the activation energy per mole of protonophore, i.e. the energy necessary to adsorb one mole of protonophore on the inner mitochondrial membrane. This factor depends on the charge and radius of the ionophore, as well as on the properties of the membrane. In the case of channel-forming protonophores A will be given by A = zra2D/l, with a the effective radius, D the diffusion coefficient of protons in water and I the length of protonophore or the membrane thickness. In some cases, a may be much greater than the physical dimensions of the particle, because of co-operative factors,

I N H I B I T I O N OF OXIDATIVE PHOSPHORYLATION 4 7 9

especially in the case of highly active uncouplers, e.g. SF 6847 (Terada, 1981). This approximation is not expected to be always valid. The interaction between protonophores may give to c , ( n ) a sigmoidal character. However, a behaviour like expression (16) is indeed observed for many uncouplers in some range of concentrations of experimental interest. Some indicative values of c~ and /3 are, in rat liver mitochondria (Terada, 1981), cx = 3-9 x 10 -3 nmol H÷/(min mg protein mV riM) and /3 = 1 × 10 -6 nM -1 for dinitrophenol, and ce = 11.2 nmol H+/(min mg protein mV nM) and /3 = 1.4 × 10 -2 nM -~ for SF 6847.

Now, we will restrict to a linear approximation in expression (16), to evaluate the first-order modifications produced by the uncoupler on physio- logical parameters. The consequence of expression (16) is that L H ( n ) =

L H ( O ) + oen. In this way, we are led to the following results. The parameters x and y of the straight lines Jp = XJo- y A o will increase

with concentration, since oe is positive for uncouplers. This is in agreement with the results of Van Dam et al. (1980). The expressions for x and y as a function of protonophore concentration n are

a 6g x = xo + n and y = yo + n. (18)

nonpZo nonp

The corresponding behaviour of the lines is shown in Fig. l(a). The significance of the possible single intersection point of all these lines in relation

(o) 0 nl nz

/ (b) O~ nl,o 2

/ Jo

(c) nz nl 0 /// Jo

FIG. 1. The lines Jp = xJ o - yA o are shown at different inhibitor concentrations (0 < n~ < n2). (a) refers to protonophores, (b) to ATPase inactivators and (c) to electron-chain inactivators.


with the stoichiometries has been examined carefully by Van Dam et al. (1980).

For the dependence of the degree of coupling we have, in the first-order approximation obtained by developing expression (10)

q(n) q ( O ) - ½ [ ( L , ( O ) ~ -1 , -1 = -noL o) + ( L n ( O ) - n p L p ) ]an. (19)

This shows that, as was expected, the degree of coupling decreases with increasing protonophore concentration.

The previous equations are able to deal with a variety of experimental situations. For instance, constant Jp = 0 corresponds to State 4 of respiration. Here, we assume a situation in which Ao and Ap--i .e. the concentrations of reactants and products--remain constant. This is a possible experimental situation because these parameters may be in principle controlled (Van Dam et al., 1980). The curves represented in the figures correspond to such a situation. Other cases may also be examined: the data of Table 1 correspond to Jo = 0, Ap constant and Jp = 0 and Ao constant. In some cases,

TABLE 1

Uncoupling activities for some uncouplers in rat liver mitochondria

Uncoupler Uncoupl ing activity

Respiration ATPase

SF 6847 10 nM 3 nM

S 13 20 nM 7 nM

S 6 150 nM 100 nM

FCCP 70 nM 15 nM

CCCP 110 nM 35 nM 2,4-Dinitrophenol 24 I.~M 8 ~M

however, it is difficult to associate the actual experimental conditions with direct requirements on the thermodynamic fluxes (rate of variation of concentrations) and thermodynamic forces (concentrations), the general case being a mixture of both conditions (constant rate of variation for some substances and constant concentration for other substances), whose specification may be rather complicated and, in some cases, may require a more fundamental, kinetic approach. At constant Ao and Ap, the illustrative example dealt with here, the influence of the protonophore concentration


on A/2H, Jo and Jp is given by

A fin ( n ) = A fin (O) -- L-H'( O )A f i . ( O )an (20)

Jo(n) = Jo( O)+ L-H~( O)AI2n( O)noLoan (21)

Jp( n ) = Jp( O ) - L-n~( O )h f in( O )npLpan. (22)

The main trends indicated in these expressions do correspond to the observed experimental behaviour: addition of protonophores causes a decrease in AfiH while stimulating respiration (increase of Jo) and decreasing the phosphorylation rate Jp. An interesting conclusion may be reached from these expressions: the ratio of the slopes of the curves A/~.H(n) and Jo(n) and of A/2H(n) and Jp(n) is given by the expression (23)

slope of A/2H(n) _ _(noLo)_ l slope of A/~H(n ) = (npLp)_l. (23) slope of Jo(n) slope of Jp(n)

It is remarkable that, at least at low concentrations, the previous ratios do not depend on the kind of uncoupler, but only on native properties of the membrane. This result may be useful in correlating experimental data, which should confirm this prediction. In this case, the measurement of only one of the three variables A/2H(n), Jo(n) or Jp(n) would be sufficient to find the remaining ones. The interest of this result is increased by the fact that it is more general than the linear hypothesis of expression (16). In fact, it may be obtained in a straightforward way by differentiation of expressions (1) and (3) with respect to n, by assuming that Ao and Ap are constant and the only coefficient which depends on n is L , . Finally, the heat generation per unit area and unit time in the steady state is

Q(n) = Q(O)+AIZH(O)anL-H~(O)[noLoAHo - npLpAHp]. (24)

As a final application we will consider with detail some experimental data in order to see some implications of irreversible thermodynamics in their analysis. In a study of the interaction of highly active uncouplers with mitochondria, Terada (1981) characterized their interaction on the basis of two parameters: the uncoupling activities for respiration and for ATPase stimulation. Such uncoupling activities are defined as the concentration of the uncoupler required for a 50% of the maximum stimulation of state 4 respiration and ATPase activity. The experimental results for these parameters are listed in Table 1. Examination of the data shows that for four of the six uncouplers mentioned, the ratio of respiration/ATPase uncoupling activities is approximately three. Is this fact a mere coincidence or is there a general explanation? In this case, what is the meaning of the discrepancy of other two values?

4 8 2 D. J O U A N D F. F E R R E R

According to equation (7), Jo the oxidation rate, in state 4 (Jp = 0), is given by

CH J° - Lon2o+ cn AoLo. (25)

Since the original values of the respiration rate is given by (25) with cn = cn(O) , the native value of the proton permeability, the 50% of the maximum stimulation of state 4 respiration will be reached when

Jo.~/2 =½LoAo L o n - - - - - a x ) q . (26)

Combination of expressions (26) with (25) and (16) leads for the respiration 50% stimulation to n~/2 (respiration 50% stimulation)

Lon2o + cn( O ) (27) n , / 2 - a + fl( Lon2 + on(O))"

The apparent ATPase stimulation is determined by measuring the amount of P~ liberated from ATP. Assume that we are originally in a state 4 (Jp = 0), so that P~ is neither absorbed nor liberated from ATP. In this case, Jp = 0 = Lp(Ap + npA~n). Assume now that uncoupler is added, at constant Ap and constant respiration rate Jo. The situation is in this way parallel to the previous one, where we have calculated Jo stimulation at constant Jp and Ao, whereas now we calculate Jp negative stimulation at constant Jo and A~. The maximum rate of liberation of P~ from ATP, proportional to Jp, will be reached when A~H gets its minimum value, so that the 50% ATP stimulation will be reached when

Jp.1/2 =½Lp(Ap+ npAt~u min). (28)

Taking into account that, from expressions (1)-(3) we have

AI~n no.Io - npLvA p - cn + n2Lp (29)

then, at constant Jo and Ap, combination of expressions (28), (29) and (16) leads to a concentration n'~/2 corresponding to ATPase 50% stimulation

Lpn2 + cn( O) n~/2(ATPase 50% stimulation) = a +fl(Lpn2+ cn(O))" (30)

Therefore, the relation of uncoupling activities becomes

n~/2(respiration 50% stimulation)

n'l/2(ATPase 50% stimulation)

_ cH(O)+n2Lo a + f l ( c n ( O ) + n 2 L p ) (3t) cn (O)+n2Lp a+fl(cn(O)+n2oLo)"


Therefore, in the present situation, i.e. when the uncoupling effects are due exclusively to an increase in the proton permeability of the membrane, and when saturation effects are negligible (/3 = 0), the ratio of the aforemen- tioned uncoupling activities does not depend on the specific uncoupler, but only on the membrane properties. According to the values of the coefficients for rat liver mitochondria (see Table 1), we get for this ratio the values 2.2±0.9. Independently of the degree of agreement of this theoretical estimate with the experimental data we stress the fact that this simple model allows to expect that, as far as saturation affects can be neglected, the ratio is indeed independent of the specific uncoupler.

When saturation effects are taken into account, the ratio becomes depen- dent on the saturation value, a/fl, of the uncoupler induced permeability. Such a ratio decreases with increasing/3 (lower saturation values). In this way, one may understand the behaviour of $6, for which the ratio is almost the half of the one corresponding to the other substances. Nevertheless, the ratio of FCCP uncoupling activities turns out to be higher than that of the other substances. This is not understandable in the present simplified model. Some possibilities of explaining this discrepancy would be the existence of nonlinear effects leading to an apparent negative value of/3 in a range of concentrations, or that, besides increasing the proton permeability, this substance acted on the oxidation or phosphorylation sites by increasing n2oLo or decreasing n~Lp. The latter behaviour would be more complicated than the simple one which is usually assumed as considering it only as a proton translocator.

4. ATPase Inactivators and Electron-chain Inactivators

To present a less trivial example of the consequences of the inhibitors on the phenomenological coefficients, we deal with the case of some ATPase inactivators, such as oligomycin and DCCD (Kozlov & Skulachev, 1977; Nicholls, 1982), along the line of development set forth in the preceding section. Such antibiotics inhibit both the synthesis and the uncoupler- stimulated hydrolysis of ATP. They are widely used in experiments on oxidative phosphorylation and they are known to interact with the Fo part of ATPase, so that they inactivate the corresponding pump by obstructing the inflow of protons through the Fo channels. The simplest hypothesis in describing this behaviour by means of a modification of the phenomenological coefficients is to assume that the coefficient Lp is modified as

Lp(n)=Lp(O)(1 1 +/3'~/ (32)

4 8 4 D. J O U A N D F. F E R R E R

where Lp(O) is the native value of the corresponding coefficient and a ' and fl' depend on the properties of both the mitochondrial membrane and the inactivator, while n is its external concentration. Expression (32) reflects the idea that these inactivators interact with ATPase by reducing the useful ATPase sites. If one assumes that initially there are N ATPase sites per unit area in the membrane, the coefficient fl will be simply the distribution coefficient, giving the ratio of inactivator adsorbed on the membrane over the external concentration, over the original N number of ATPase sites. Therefore

c~'= N -1 exp ( - W / R T ) (33)

W being the activation energy and R the constant of ideal gases. This choice of a ' represents the fraction of ATPase sites inactivated, by assuming that each molecule of inactivator acts on one ATPase site.

Some co-operative effects are possible, however, by means of which one molecule of inactivator could act on several ATPase sites. In order to have values for a and/3 we have evaluated it for DCCD in E. coli with AN 180 strain (Fillingame, 1975) in the linear region between 1 I~M-10 txM, and we have found c~'=0-10(tXM) -~, /3'=0"11(1~M)-1; while for DPBP~ Fe 2- (Carlsson, 1981) we have o¢'=7,65 IXM -~ and /3'=6"98 }xM -~. In terms of the phenomenological coefficients of equations (1)-(3) and in the first order approximation, the modifications subsequent to equation (32) are

LpH = L n p ( O ) ( 1 - a ' n ) and LH(n) = L,(O)-npLHp(O)a'n.

An immediate analysis of the effects of addition of inactivator may be carried out on the lines stated previously. The parameters x and y characterizing the lines Jp = xJo -yAo are not modified (Fig. l(b)), as may be seen from expression (7), so that the inactivator concentration has no influence on them, at least in the present model. This provides a conspicuous ditIerence with the case of protonophores (Fig. l(a)). On the other side, the degree of coupling will be modified as follows

q(n)=q(O)[1 ~ , cH ] - ~ a n +--~2 ° . (34) CH to

Assuming, as before, that Ao and Ap remain constant, we have for the corresponding modifications of A/J.H, Jo and Jp

A/2. (n) = A/ / . ( O ) + L-H](O)nJp(O)o~'n (35)

Jo( n ) = Jo( O ) - L-.~( O )nrlp( O )noLoa'n (36)

Jp(n)=Jp(O)-L-H~(O)Jp(O)(c.+n2oLo)a'n. (37)


This shows that the presence of this kind of inactivator increases the protonmotive force and that decreases both Jo and Je. Note that, as well as in the preceding section, the ratio of the slopes of At2rt (n), Jo (n) and Jp(n) does not depend on the specific inactivator, but only on native properties of the membrane (Fig. 2).

" 'jo l i

#3

FIG. 2. In our model, the ratios of the slopes of the curves A/2H(n), Jo(n) and Jp(n) turn out to be constant, depending on the native properties of the membrane but not on the particular inhibitor. The values of such constant are different for protonophores, ATPase inactivators and electron-chain inactivators.

The relation Jo(n) may be used to obtain the so-called control strength of the inhibitors on the respiration rate, defined as

nm~,,~ dJo(n) z - - - - ( 3 8 )

Jo(o) dn

With nma x the concentration of inhibitor necessary for total inhibition. In this way we obtain for the control strength as a function of Jo

L . ( O) Jo - LoAo " [ L . (O) ] (39)

Z(Jo) = n-pLp(O) Jo l + a - /3 n ~ ) J -LoAo

In practice, the function Z(Jo) is more complicated than this expression, which describes the behaviour at low Jo but not at high Jo, where nonlinear effects become important. The definition of Z is rather general and could have been applied also with reference to protonophores, and to other kinds of inhibitors. Here, we have shown the expression for the control strength only as an example of possible applications of the theory. Finally, we have for heat dissipation the result

Q(n)=Q(O)-a'n[npnoLoAHo+(C,+n2oLo)AHp]. (40)

There may be other inactivators that affect only the Fi part of ATPase sites,


in such a way they become simple proton leaks, through the Fo part. Such inactivators would decrease the number of active ATPase sites, as in the case dealt with in this section, but simultaneously they would increase the membrane proton permeability, as in the case of protonophores. In consequence, the quantitative action of such uncouplers could be obtained by addition of the results of this section with those of the precedent one, in the present linear approximation. Furthermore, the coefficient Lp depends also on the transport properties of the membrane, so that in the present simplified formalism, it could account also for the action of the inhibitors of ADP-ATP translocases.

The results in the preceding paragraphs suggest examining the consequences of a modification of Lo rather than Lp.

This would be the case of some electron-chain inactivators, as for instance antimycin or funiculosin. In this case, the physical situation is more involved since the electron chain is more complex than the ATPase sites. Some inhibitors may produce only a partial but not a total inhibition of each redox site, depending on the kind of substrate used. We will consider here only the simple case of total inhibition of each redox site affected by the inhibitor. However, from a thermodynamic point of view the description would be quite similar to that of the previous case. Here, a molecule of inactivator does not inactivate the full electron chain, but only some of its steps. Similarly to expression (32) we assume that

( L o ( n ) = L o ( O ) 1 l + 3 " n / " (41)

As in the previous definitions, n is the external inhibitor concentration and a" and /3" parameters characteristic of both the inactivator and the membrane, which express the fractional inactivation of the electron chain. Obviously, the values of ~" and fl" will depend on the specific step at which such chain is inactivated. Though in this case there is not so clear a molecular model as in the previous cases, we will analyse it for the sake of a wider perspective on the formal consequences of our development.

The parameters of expressions (1)-(3) which will be modified as a consequence of expression (41) are now Loll and Lm analogously to expression (34). The parameters x and y of the lines Jp = xJo -yAo are modified according to

CH x = Xo + a"n Y = Yo. (42)

npnoto

The behaviour of such lines is shown in Fig. 1(c), where it may be compared with the rather different behaviours of Fig. l(a) and Fig. l(b). In this case,


addition of inactivator in state 4 mitochondria leads to a decrease in respiration, in contrast with the case of protonophores. Such a result is quite logical and evident.

The degree of coupling is affected as follows

q(n) q(O)[1 ~a"n cH 1 . . . . ~ . (43) CH + n-pLp

Assuming again a situation in which Ao and Ap remain constant, the perturbations in A/2n(n); Jo(n) and Jp(n) are

At~H(n) = At~H( O ) -- L-nl( O)n,ffo( O)cd' n (44)

Jo(n)=Jo(O)_L~, (O)(cH + 2 ,, npLp)L(O)a n (45)

Jp(n) = Jp( O ) - t-n'( O)npn,,tflo( O)a"n. (46)

All results in this section indicate, as expected, a global reduction of the activity of all processes involved in oxidative phosphorylation. Finally, heat dissipation is given by

Q( n ) = Q( O ) - L ~t( O )Jo( O )a" n[ ( cH + n~Lp)A H,, + npnoLp A Hp]. (47)

5. Conclusions

In this paper we have proposed an irreversible thermodynamic description of the action of some uncouplers of oxidative phosphorylation. We have assumed a simple model in which the influence of several kinds of uncouplers has been included in the suitable phenomenological coefficients of the thermodynamic constitutive equations. In the case of protonophofic uncouplers, the coefficient to be changed is evidently Cn, the membrane permeability to protons. In the case of some ATPase inactivators (oligomycin, DCCD), the adequate parameter seems to be Lp, while for some electron-chain inhibitors, Lo appears as the most suitable one in describing the corresponding action. In this way, the several microscopic models of uncoupling mechanisms find a macroscopic formulation, less detailed than the microscopic one but closer to the usual experimental situations. This provides a quantitative framework for the detailed analysis, comparison and correlation of experimental results.

In our model, we have studied theoretically the consequences of inhibitors on the lines Jp = xJ o - y A o , on the oxidation-phosphorytation degree of coupling, on AfiH(n), Jo(n) and Jp(n) at constant A,, and A m and on heat dissipation. All such parameters have a sufficiently clear experimental meaning which justifies the theoretical analysis from different points of

488 D, JOU A N D F. F E R R E R

v iew. In o u r m o d e l , e a c h u n c o u p l e r is d e s c r i b e d b y t w o p a r a m e t e r s , r e l a t e d

to t h e m o d i f i c a t i o n o f t he c o r r e s p o n d i n g p h e n o m e n o l o g i c a l coe f f i c i en t .

R E F E R E N C E S

AZZONE, G. F., POZZAN, T. & MASSARI, S. (1978). Biochim. biophys. Acta 501, 307. CAPLAN, S. R. • ESSIG, A. (1969). Proc. natn. Acad. Sci. U.S.A. 64, 211. CAPLAN, S. R. t~ ESSIG, A. (1983). Bioenergetics and linear nonequilibrium thermodynamics.

The steady state. Cambridge, Massachussets: Harvard University Pres. FILLINGAME, R. H. (1975). J. Baeter. 124, 870. KATCHALSKI, A. & CURRAN, P. F. (1965). Nonequilibrium thermodynamics in biophysics.

Cambridge, Massachussets: Harvard University Press. KEDEM, O. & CAPLAN, S. R. (1965). Trans. Faraday Soc. 61, 1897. KOZLOV, I. A. & SKULACHEV, V. P. (1977). Biochim. biophys. Acta 463, 29. NICOLIS, G. & PRIGOGINE, I. (1977). Self-organization in nonequiIibrium systems. New York:

Wiley. NICHOLLS, D. G. (1974). Ear. J. Biochem. 49, 573. NICHOLLS, D. G. (1982). Bioenergetics. An introduction to the chemiosmotic Theory. London:

Academic Press. PIETROBON, D., ZORAT'F1, M., AZZONE, G. F., STUCK1, J. W. & WALZ, D. (1982). Eur. J.

Biochem. 127, 483. POOLE, R. K. & HADDOCK, B. A. (1975). FEBS Lett. 58, 249. ROTTENBERG, G. H. (1979). Biochim. biophys. Acta 549, 225. RO'I'rENBERO, H., CAPLAN, S. R. & ESSIG, A. (1970). In: Membranes and Ion Transport

(Bittar, E. E., ed.). vol. 1, pp. 165-191. London: Wiley Interscience. STUCKI, J. W., COMPIANI, M. & CAPLAN, S. R. (1983). Biophys. Chem. 18, 101 STUCK1, J. W. (1980). Eur. J. Biochem. 109, 269. TERADA, H. (1981). Biochim. biophys. Acta 639, 225. VAN DAM, K., WESTERHOFF, H. V., KRAB, K., VAN DER MEER, R. & ARENTS, J. C. (1980).

Biochim. biophys. Acta 591, 240. WALZ, D. (1979). Biochim. biophys. Acta 505, 279. WESTERHOFF, H. V. & VAN DAM, K. (1979). Curr. Top. Bioenergetics 9, 1.

a simple nonequilibrium thermodynamic description of some inhibitors of oxidative phosphorylation

Documents