Control Over the Contribution of the Mitochondrial Membrane Potential (ΔΨ) and Proton Gradient (ΔpH) to the Protonmotive Force (Δp)

The protonmotive force across the inner mitochondrial membrane (Δp) has two components: membrane potential (ΔΨ) and the gradient of proton concentration (ΔpH). The computer model of oxidative phosphorylation developed previously by Korzeniewski et al. (Korzeniewski, B., Noma, A., and Matsuoka, S. (2005) Biophys. Chem. 116, 145–157) was modified by including the K+ uniport, K+/H+ exchange across the inner mitochondrial membrane, and membrane capacitance to replace the fixed ΔΨ/ΔpH ratio used previously with a variable one determined mechanistically. The extended model gave good agreement with experimental results. Computer simulations showed that the contribution of ΔΨ and ΔpH to Δp is determined by the ratio of the rate constants of the K+ uniport and K+/H+ exchange and not by the absolute values of these constants. The value of Δp is mostly controlled by ATP usage. The metabolic control over the ΔΨ/ΔpH ratio is exerted mostly by K+ uniport and K+/H+ exchange in the presence of these processes, and by the ATP usage, ATP/ADP carrier, and phosphate carrier in the absence of them. The K+ circulation across the inner mitochondrial membrane is controlled mainly by K+ uniport and K+/H+ exchange, whereas H+ circulation by ATP usage. It is demonstrated that the secondary K+ ion transport is not necessary for maintaining the physiological ΔΨ/ΔpH ratio.

According to the Mitchell chemiosmotic theory (1), the intermediate that couples the electron flow through the respiratory chain with ATP synthesis by ATP synthase in the mitochondrial matrix is the protonmotive force (⌬p) across the inner mitochondrial membrane. This thermodynamic potential is composed of two components: electrical membrane potential (⌬⌿) and the difference between the cytosolic and matrix pH (⌬pH).
The contribution of ⌬⌿ and ⌬pH to ⌬p is very important from the kinetic point of view, because ⌬⌿ and ⌬pH exert a different influence on such elements of the system and processes as: complex III (C3), 2 complex IV (C4), ATP/ADP carrier, phosphate carrier (12), proton leak (13), and reactive oxygen species (ROS) production (8). The ATP/ADP carrier is driven by ⌬⌿, whereas the phosphate carrier is driven by ⌬pH. Complex III transfers four protons but only two positive charges, whereas complex IV transfers two protons and four positive charges (12); therefore complex III is relatively more sensitive to ⌬pH, whereas complex IV is to ⌬⌿. The proton leak seems to be more sensitive to ⌬⌿ (13), while ROS production is to ⌬pH (8). Therefore, the total value of ⌬p alone cannot define satisfactorily the kinetic properties of the oxidative phosphorylation system.
It is commonly accepted that the contribution of ⌬⌿ and ⌬pH to ⌬p is determined by secondary transport of ions, especially potassium ions (K ϩ ), driven by building up the proton gradient across the inner mitochondrial membrane (12). It is also accepted that without this secondary transport, ⌬p would be almost exclusively (about 99%) in the form of ⌬⌿, because, at the real electrical capacity of the inner mitochondrial membrane and the pH buffering capacity of the matrix, the transport of a very small amount of protons would build up the physiological value of ⌬p (12). It has been shown that ⌬p and its components do not depend very significantly on the extramitochondrial [K ϩ ] (5, 10), especially at higher physiological potassium ion concentrations. It has been observed in some studies (3,10) that an increase in [P i ] from 0 to physiological values (10 mM) causes a significant * The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. □ S The on-line version of this article (available at http://www.jbc.org) contains supplemental material. 1  Several trials of a quantitative description of the contribution of ⌬⌿ and ⌬pH to ⌬p have been undertaken (14 -16). These descriptions are based on different assumptions, give different predictions, and their verification by comparison with experimental data is limited. Generally, they are relatively simple and phenomenological. Recently Beard (17) developed a computer model of oxidative phosphorylation, based on the general structure of the model built by  that involves explicitly the K ϩ uniport, K ϩ /H ϩ exchange across the inner mitochondrial membrane, and membrane capacitance. This model predicts a very low value of ⌬pH (below 3 mV, 0.05 pH units), similar to that obtained in an experimental manner by Bose et al. (11).
The original model developed by  is able to reproduce, at least semiquantitatively, a broad range of different kinetic properties of the oxidative phosphorylation system in isolated mitochondria and intact tissues. However, this model uses a simplified, phenomenological description of the relationship between ⌬p, ⌬⌿, and ⌬pH: it assumes a fixed, constant contribution of ⌬⌿ to ⌬p, u ϭ 0.861. This assumption was based on experimental data concerning state 4, state 3, and intermediate states in isolated mitochondria (2), but is certainly oversimplified and unsatisfactory under many conditions. Metabolic Control Analysis (MCA) (23)(24)(25) has appeared to be an immensely useful quantitative tool for analyzing the dynamic behavior of biochemical systems. It was used in a great number of experimental and theoretical studies concerning the control of metabolic pathways (see Refs. 19, 26 -30 for a few examples).

THEORETICAL PROCEDURES
Computer Model-The computer dynamic model of oxidative phosphorylation in intact heart developed previously by Korzeniewski et al. (22) was extended to replace the fixed constant ⌬⌿/⌬p ratio (u ϭ 0.861) with a more mechanistic description of ion transport across the inner mitochondrial membrane, similarly as it was done by Beard (17). The K ϩ uniport, K ϩ /H ϩ exchange, and membrane capacitance were involved explicitly. The general scheme of the modeled system is presented in Fig. 1. The rates of K ϩ uniport (v Kuni ) and K ϩ /H ϩ exchange (v KHex ) are described by the following kinetic Equations 1 and 2, where k Kuni ϭ 1.0⅐10 Ϫ5 min Ϫ1 and k KHex ϭ 3.485⅐10 Ϫ3 min Ϫ1 ⅐M Ϫ1 are rate constants in the "reference point" (see below). R, T, and F have the typical meanings, subscript e means extramitochondrial and subscript i means intramitochondrial. The changes over time in intramitochondrial and extramitochondrial [H ϩ ] are described by the following differential Equations 3 and 4, where R cm ϭ 3.35 is the ratio of the cell volume to mitochondria volume in heart cells, buff Hi is the matrix proton-buffering coefficient, and buff He is the cytosol proton-buffering coefficient. Some fraction ( ])) of P i Ϫ transported by phosphate carrier (PiT) to the mitochondrial matrix dissociates to P i 2Ϫ and H ϩ , but they are quickly used by ATP synthase, and therefore no net protons are produced/consumed in the matrix. The opposite situation prevails in the cytosol. An analogical situation is the production/consumption of H ϩ (together with NADH) by the substrate dehydrogenation/complex I. All these processes are not related to proton and/or charge transfer across the inner mitochondrial membrane and therefore are not taken into account explicitly within the model. Complex I (C1) transports (for 2 electrons) 4 H ϩ from matrix to cytosol; complex III (C3) takes up (for 2 electrons) 2 H ϩ from the matrix, and releases 4 H ϩ to the cytosol; complex IV takes up (for 2 electrons or 1 oxygen atom) 4 H ϩ from the matrix (including 2 H ϩ for water molecule formation) and releases 2 H ϩ to the cytosol; ATP synthase (SN) transports n A ϭ 2.5 protons from cytosol to matrix for one ATP molecule synthesized; phosphate carrier (PiT), proton leak (LK), and K ϩ /H ϩ exchange (KHex) transport 1 H ϩ from cytosol to matrix. All these processes are taken into account explicitly in the model.
The changes over time in intramitochondrial [K ϩ ] are described by the following differential Equation 5.
The potassium ion concentration in heart is very well regulated. (There are huge potassium stores in other tissues, especially in skeletal muscle, which can buffer the potassium concentration in heart (31).) Therefore, we assumed that the FIGURE 1. Scheme of the oxidative phosphorylation system. The elements of the system taken into account explicitly within the model of oxidative phosphorylation used in the present study are presented. C m , membrane capacitance; ⌬⌿, membrane potential. extramitochondrial (cytosolic) [K ϩ ] is constant. (This assumption also corresponds well to the isolated mitochondria system.) All the simulations presented below were performed under this assumption. However, we also tested the possibility that there is a constant cellular (cytosolic plus mitochondrial) pool of potassium ions and therefore Equation 6 applies.
In both cases the theoretical results were similar (not shown), and therefore the above assumption is not particularly important for the properties of the system.
The changes over time in ⌬⌿ are described by Equation 7, where the inner mitochondrial membrane capacitance, C m , is 1 M/mV, as in Refs. 17 and 32. C1 transfers (for 2 electrons) 4 positive charges form matrix to cytosol; C3 transfers (per 2 electrons) 2 positive charges from matrix to cytosol (4 protons and 2 electrons); C4 transfers (for 2 electrons or 1 oxygen atom) 4 positive charges from matrix to cytosol (2 protons plus 2 electrons from cytosol to matrix); ATP synthase transfers n A ϭ 2.5 positive charges from cytosol to matrix; ATP/ADP carrier, proton leak, and K ϩ uniport transport 1 positive charge from cytosol to matrix. The reference point for the computer simulation carried out in the present study corresponded to a slowly beating intact heart (22). In this point, the following variable values were recorded: VO 2  The rate constants of K ϩ uniport and K ϩ /H ϩ antiport were adjusted to be relatively quick and to give the u ϭ ⌬⌿/⌬p value of 0.861 used in the original model (22) The complete description of the model of oxidative phosphorylation with a mechanistic description of the relationship between ⌬⌿ and ⌬pH in intact heart is located on the web site as supplemental material.
Metabolic Control Analysis-The most fundamental idea of MCA is based on the ratio of the relative change (dY/Y) in some variable Y caused by a small relative change (dX/X) in some parameter/variable X, to the latter change in Equation 8.
In principle, these changes are infinitesimal, but in practice sufficiently small finite changes can be considered. In particular, control coefficients defined within MCA characterize the control exerted by particular components of a given metabolic system (enzymes, carriers, processes, and metabolic blocks) over different macroscopic properties (variables) of the system. Control coefficients are defined in Equation 9, where U is some property (variable) of the system and E i is the activity/concentration of the i-th component of the system. Several control coefficients have been defined within MCA, concerning the control over the flux, metabolite concentrations, ⌬p etc. (see e.g. Refs. 7, 25, 28). The metabolic control of particular elements of the oxidative phosphorylation system over ⌬p, u ϭ ⌬⌿/⌬p, K ϩ cycling, and H ϩ cycling was calculated according to Equation 9 (U represented ⌬p; u, K ϩ cycling rate or H ϩ cycling rate). In subsequent computer simulations, the original rate constants of particular steps were increased by a relative factor of 0.01 (by 1%) and the relative changes in particular U-s between the original and the new steady state were recorded.

THEORETICAL RESULTS
First of all, our model predicts that in the reference point (that reflects the physiological conditions) there is a very small difference between the cytosolic and matrix potassium ion concentration (120.0 mM versus 118.0 mM, respectively) and ⌬⌿ is the dominant component of ⌬p. These properties of the model agree excellently with recent conclusions drawn by Nicholls (33): "In an intact cell, or with isolated mitochondria in a physiologically relevant high K ϩ medium, there is essentially no gradient of K ϩ across the inner mitochondrial membrane, the concentration being close to 120 mM in each compartment. Under these conditions, ⌬pH is usually small . . . ".
To simulate the dependence of ⌬p, ⌬⌿, and ⌬pH on k Kuni , the value of this constant was changed by six orders of magnitude, while k KHex was kept constant. An increase in k Kuni slightly decreases ⌬p, as it can be seen in Fig. 2A. However, ⌬⌿ and ⌬pH change significantly with the increase in k Kuni : ⌬⌿ strongly decreases and ⌬pH strongly increases. At very low k Kuni values, almost the entire ⌬p is in the form of ⌬⌿. An increase in k Kuni elevates the mitochondrial [K ϩ ], but these changes are rather moderate: in the whole range of k Kuni an about 2-fold increase in [K ϩ ] i is observed.
The dependence of ⌬p, ⌬⌿, and ⌬pH on k KHex is opposite to the dependence on k Kuni , as it can be seen in the simulations shown in Fig. 2B. In these simulations, k KHex was changed by six orders of magnitude, while k Kuni was kept constant. An increase in k KHex slightly increases ⌬p, strongly increases ⌬pH, and strongly decreases ⌬⌿. At very high k KHex values, almost the entire ⌬p is in the form of ⌬⌿. [K ϩ ] i drops about twice with the increase in k KHex .
The above simulations clearly show that the contribution of ⌬⌿ and ⌬pH to ⌬p strongly depends on the absolute values of the rate constants of the K ϩ uniport and K ϩ /H ϩ antiport: k Kuni and k KHex . However, when the values of these constants are changed in parallel in relation to the reference point, by the same factor, and thus the ratio of the rate constants is kept constant (k Kuni /k KHex ϭ 2.896⅐10 Ϫ3 in the reference point), the value of ⌬p and the ⌬⌿/⌬pH ratio are essentially unaffected. This is shown in Fig. 2C. Therefore, the contribution of ⌬⌿ and ⌬pH to ⌬p is determined by the relative values (ratio) of k Kuni and k KHex . It is also worth noticing that [K ϩ ] i remains constant at the constant k Kuni /k KHex ratio. At very high values of k Kuni and k KHex (for the constant k Kuni /k KHex ratio), the values of ⌬⌿, ⌬pH, and ⌬p slightly decrease, because the very quick K ϩ circulation and thus K ϩ /H ϩ exchange prevailing under such conditions leads to a significant dissipation of ⌬p. Whereas the K ϩ /H ϩ exchange contributes in 0.044% to the overall H ϩ cycling in the reference point, this contribution increases to 36% at the right-most point in Fig. 2C. , ⌬pH slightly increases in the first case and slightly decreases in the second. An even more interesting situation takes place in the case of an increase in [P i ], here at higher P i concentrations ⌬⌿ slightly increases (after an initial decrease), at the cost of a significant decrease in ⌬pH. Therefore, the contribution of ⌬⌿ to ⌬p increases of course at higher [P i ]. The simulated dependence of ⌬p, ⌬⌿, and ⌬pH on [P i ] is compared in Fig. 3C with experimental data by Kunz et al. (3) and Nicholls (10). Generally, a good agreement can be observed (see "Discussion" for details). In our simulations, ⌬p, ⌬⌿, and ⌬pH are essentially independent of the extramitochondrial K ϩ concentration (Fig. 3D) (in this theoretical analysis, different constant [K ϩ ] e in the range between 0 and 120 mM were fixed in subsequent simulations) although at low [K ϩ ] the system approaches the steady state very slowly (starting from the reference point) because the K ϩ uniport and K ϩ /H ϩ exchange are very slow (not shown). Our simulations are compared in Fig. 3D with experimental data by Czyż et al. (5). Again, a good agreement can be observed (see also "Discussion").
The increase in [P i ] causes an increase in the rate of oxygen consumption and ATP synthesis (not shown) and an increase in the reduction level of cytochrome c (see Fig. 4). This effect is not related to the activation of complex III by P i , because such activation is not included in our model.
The absolute value of ⌬p is controlled to the greatest extent by the ATP demand, although the control is distributed among essentially all components of the system. This is shown in Fig. 5. Generally, the processes that participate (directly or indirectly) in the production of ⌬p (substrate dehydrogenation, complex I, complex III, and complex IV) have a positive control, while the processes dissipating ⌬p (ATP usage, proton leak) have a negative control over ⌬p. The sum of the concentration control coefficients over ⌬p equals zero (negative controls counter-balance positive controls), in agreement with the so-called connectivity property (25).
In the presence of K ϩ uniport and K ϩ /H ϩ antiport, just these two processes control to the greatest extent the contribution of ⌬⌿ and ⌬pH to ⌬p. As can be seen in Fig. 6A, K ϩ uniport has a great negative control over u ϭ ⌬⌿/⌬p, while K ϩ /H ϩ antiport has a great positive control over this variable. ATP usage has a moderate positive control over u, and the contribution of the remaining steps to the control is minor.
In the absence of K ϩ uniport and K ϩ /H ϩ antiport (k Kuni and k KHex set to 0) the metabolic control over u is exerted mainly by ATP usage and phosphate carrier (positive control) and ATP/ ADP carrier (negative control) (Fig. 6B). Other components control this variable to a smaller extent. Under these conditions u ϭ 0.847, which is similar to u ϭ 0.861 in the presence of K ϩ uniport and K ϩ /H ϩ antiport. Therefore, the system is able to regulate effectively the contribution of ⌬⌿ and ⌬pH to ⌬⌿ even without the secondary K ϩ transport.
The rate of K ϩ cycling across the inner mitochondrial membrane is controlled mostly by the processes directly participating in this cycling: K ϩ uniport and K ϩ /H ϩ exchange. This is shown in Fig. 7. Both processes exert positive control of comparable size (the control by the H ϩ /K ϩ exchange is somewhat higher). The H ϩ cycling across the inner mitochondrial membrane is controlled to the greatest extent by ATP usage (not shown).

DISCUSSION
One of the most important findings of the present study is that the contribution of ⌬⌿ and ⌬pH to ⌬p is determined by the relative activity of the K ϩ uniport and K ϩ /H ϩ antiport. The simulations presented in Fig. 2 clearly demonstrate that it is not the absolute values of the rate constants of K ϩ uniport and K ϩ /H ϩ antiport (k Kuni and k KHex , respectively), but the ratio of these rate constants that determines the value of parameter u ϭ ⌬⌿/⌬p. Significant variations in either k Kuni or k KHex , when the other rate constant remains unchanged, significantly affect the contribution of ⌬⌿ and ⌬pH to ⌬p (Fig. 2, A and B). On the other hand, an increase in both rate constants by six orders of magnitude has only an insignificant effect on ⌬p, ⌬⌿, and ⌬pH when the k Kuni /k KHex ratio is fixed at 2.896⅐10 Ϫ3 .
Extramitochondrial ADP decreases and extramitochondrial ATP increases ⌬p and ⌬⌿, whereas a small opposite effect on ⌬pH is observed (Fig. 3, A and B). Such a behavior of the system is not surprising, because a high external [ADP] activates the ATP/ADP carrier, accelerates the dissipation of ⌬⌿ and thus increases the contribution of ⌬pH to ⌬p, while a high external [ATP] has the opposite effect. However, one must bear in mind that the potential effect of [ATP] on the mitoK ATP channel is not included in our simple model. On the other hand the half-inhibiting [ATP] for this channel is very low, in the micromolar range (34).
An increase in the extramitochondrial [P i ] decreases ⌬p and ⌬pH, but slightly increases (at higher concentrations) ⌬⌿. This simulated behavior of the system is generally similar to the experimental results obtained by Nicholls (10) and Kunz et al. (3), as shown in Fig. 3C. The contribution of ⌬pH to ⌬p was probably somewhat overestimated in Ref. 10, because small amounts of valinomycin were used in that study to allow Rb ϩ flow across the inner mitochondrial membrane; valinomycin dissipates ⌬⌿ and thus builds up ⌬pH. In the more recent study (3), DDA ϩ that does not need valinomycin was used to determine ⌬⌿.
On the other hand, Bose et al. (11) observed a significant (much higher than in our simulations, but the qualitative tendency is similar) P i -induced increase in ⌬⌿, but they measured a very low value of ⌬pH, below 3 mV (0.05 pH units), which is not what was usually observed by other investigators (2-10). It is not clear if these differences are due to different experimental systems and conditions used (resulting in e.g. different values of k Kuni and/or k KHex ) or different methods of ⌬⌿ and (especially) ⌬pH determination.
Our simulations predict that ⌬p, ⌬⌿, and ⌬pH are essentially independent of extramitochondrial K ϩ concentration (Fig. 3D). These theoretical predictions are generally similar to the experimental data obtained by Czyż et al. (5), although the authors observed that ⌬⌿ slightly increases and ⌬pH slightly increases at low, unphysiological [K ϩ ] e . A similar behavior of the system was encountered by Nicholls (10). Of course this (rather insig-   nificant, considering low [K ϩ ]) difference between model predictions and experimental data can be caused by the fact that within the model, the kinetic description of K ϩ uniport and K ϩ /H ϩ antiport is oversimplified and does not work properly at very low [K ϩ ]. Another possibility is that, as discussed above, the system approaches very slowly a steady-state at low [K ϩ ], and therefore the steady-state had not been reached in the discussed experimental studies. Indeed, computer simulations demonstrated that a dependence of ⌬⌿ and ⌬pH on [K ϩ ] similar to that seen in Refs. 5, 10 can be easily obtained if the values of ⌬⌿ and ⌬pH from the transient state preceding the steady state are plotted in the diagram (not shown).
The decrease in the contribution of ⌬pH to ⌬p with the increase in [P i ] from 0 to physiological values can be explained as follows. The phosphate carrier carries out the transport of P i Ϫ from cytosol to the mitochondrial matrix coupled with the transport of H ϩ to the matrix. Therefore, fast action of this carrier causes intensive ⌬pH dissipation. At low P i concentrations, this carrier is inhibited, and therefore the dissipation of ⌬pH is restricted, which leads to a relatively higher contribution of ⌬pH to ⌬p. An increase in [P i ] abolishes this effect and causes a decrease of the ⌬pH contribution and thus an increase in the ⌬⌿ contribution to ⌬p. Fig. 4 shows that an increase in [P i ] from 0 to physiological values results in an increase in the reduction level of cytochrome c. It has been postulated by Bose et al. (11) that the P i -induced increase in ⌬⌿ and cytochrome c reduction level constitutes evidence for a direct activation of complex III of the respiratory chain by extramitochondrial inorganic phosphate. However, the present studies do not confirm this proposition, because both effects can be accomplished by a computer model of oxidative phosphorylation that does not involve the possible activation of complex III by P i (although we have to admit that the simulated increase in ⌬⌿ is very small).
The model developed by Beard (17) predicts that almost the entire ⌬p is in the form of ⌬⌿, and therefore ⌬pH is very low, in accordance with the experimental results obtained in Ref. 11. However, this prediction is simply a consequence of the assumption that there is no passive potassium flux (k Kuni ϭ 0) and that the K ϩ /H ϩ exchange is very active (17). If a lower rate constant of the K ϩ /H ϩ exchange is assumed, along with a nonzero rate constant of K ϩ uniport, as is done in the present study, a much higher value of ⌬pH of about 30 mV is obtained, in agreement with the great majority of experimental data coming from different laboratories (2)(3)(4)(5)(6)(7)(8)(9)(10).
It is not surprising that ATP usage has the greatest (negative) control over the absolute value of ⌬p (Fig. 5), because ATP usage is the main process determining ATP turnover. An increase in the intensity of this process leads to a decrease in the cytosolic phosphorylation potential. This decrease is transferred (through the ATP/ADP carrier and phosphate carrier) to the mitochondrial phosphorylation potential and then (through ATP synthase) to the protonmotive force (⌬p). The proton leak dissipates ⌬p, and therefore it also has a negative control. However, the fraction of protons returning to the matrix by this process is relatively small (17% in a slowly beating heart within the model), which results in a relatively small control coefficient.
The presented theoretical results agree very well with the experimental determination of the control of oxidative subsystem (substrate dehydrogenation, complex I, complex III, and complex IV), phosphorylation subsystem (ATP synthase, ATP/ ADP carrier, phosphate carrier, ATP usage), and proton leak subsystem over ⌬p carried out within the frame of the 'topdown approach' to MCA (7,35).
In the presence of K ϩ uniport and K ϩ /H ϩ antiport, the contribution of ⌬⌿ to ⌬p (parameter u) is mainly controlled just by these processes (Fig. 6A). The magnitude of the control is identical for both processes, but has the opposite sign. The K ϩ uniport (leak) dissipates ⌬⌿ and therefore exerts a negative control over u ϭ ⌬⌿/⌬p, whereas the K ϩ /H ϩ uniport dissipates ⌬pH and therefore its control over u is positive. The equal FIGURE 6. Simulated control coefficients of particular components of the system over u ‫؍‬ ⌬⌿/⌬p. A, in the presence of K ϩ uniport and K ϩ /H ϩ exchange; B, in the absence of K ϩ uniport and K ϩ /H ϩ exchange. Control over ⌬⌿/⌬pH NOVEMBER 28, 2008 • VOLUME 283 • NUMBER 48 in size but oppositely-directed control exerted by the K ϩ uniport and K ϩ /H ϩ explains why the contribution of ⌬⌿ and ⌬pH to ⌬p remains essentially unaffected when the activities (rate constants) of both processes are changed in parallel by the same relative factor (Fig. 2C).
Our model predicts that the system seems to be able to regulate efficiently the contribution of ⌬⌿ and ⌬pH to ⌬p even without the secondary K ϩ transport. The simulations shown in Fig. 6B explain the basis of this surprising system property. Namely, in the absence of K ϩ circulation, the control over u is taken over by ATP usage, ATP/ADP carrier, and phosphate carrier. ATP usage had already a significant control in the presence of the K ϩ uniport and K ϩ /H ϩ antiport, because a decrease in ⌬p causes an increase in u. On the other hand, in the absence of the secondary K ϩ transport, the ATP/ADP carrier and phosphate carrier seem to take over the role of K ϩ uniport and K ϩ /H ϩ antiport in determining the ⌬⌿/⌬pH ratio. This is logical, because the ATP/ADP carrier dissipates ⌬⌿ (as the K ϩ uniport) and the phosphate carrier dissipates ⌬pH (as the K ϩ /H ϩ antiport).
It is also noteworthy that the obtained simulation results do not depend qualitatively on the value of membrane capacitance C m , because changing this parameter within several orders of magnitude has only a minor influence on the distribution of ⌬p between ⌬⌿ and ⌬pH (1% relative change in the value of parameter u) without a significant change in ⌬p (results not shown).
The rate of K ϩ circulation across the inner mitochondrial membrane is mostly controlled by K ϩ uniport and K ϩ /H ϩ antiport (Fig. 7). The control exerted by the latter is somewhat greater, because the former is very sensitive to the dominant component of ⌬p: ⌬⌿, appearing in the power expression (Equation 1). According to the Metabolic Control Analysis paradigm, the greater the sensitivity to metabolite concentrations (or thermodynamic forces), the lower the control over the flux, and the inverse (25).
Of course, the model developed in the present study contains several simplifications. A simple kinetics (linear dependence of rates on ion concentrations) of the K ϩ uniport and K ϩ /H ϩ exchange was assumed. Constant cytosolic and mitochondrial free [Mg 2ϩ ] was assumed; this assumption is justified by the presence of a significant magnesium binding pool in both compartments (36) and by the essentially constant [ATP] (changes in [ATP] caused by AMP deamination are potentially the greatest source of short-term variations in [Mg 2ϩ ]). It is also widely accepted, that the matrix [Mg 2ϩ ] modulates K ϩ /H ϩ antiport activity. Nevertheless, under physiological magnesium concentrations, exchanger inhibition is reported to be already saturated, and moderate [Mg 2ϩ ] changes should not significantly affect exchanger turnover rate (37). It was assumed that the mitochondrial matrix volume is substantially constant in intact tissues. (This is certainly not the case in isolated mitochondria.) Nevertheless, it seems unlikely that the general theoretical results obtained in the present study depend significantly on these assumptions.
The present theoretical study is important for understanding the functioning of the system. As mentioned in the Introduction, various oxidative phosphorylation complexes, proton leak, and free radical production are differentially regulated by ⌬⌿ and ⌬pH. The distribution of control over u ϭ ⌬⌿/⌬pH among particular oxidative phosphorylation complexes may be particularly important in pathological conditions, especially in the case of inborn enzyme deficiencies (causing mitochondrial diseases, Ref. 38), where an elevated free radical production is one of the most important pathogenic factors.
Summing up, in the present study the computer model of oxidative phosphorylation developed previously by Korzeniewski et al. was extended by including explicitly the K ϩ uniport, K ϩ /H ϩ exchange, and capacitance of the inner mitochondrial membrane. Computer simulations demonstrated that the ⌬⌿/⌬pH ratio is mainly determined by the relative activities (rate constants) of the K ϩ uniport and K ϩ /H ϩ exchange rather than by the absolute values of these activities. It was shown that the absolute value of ⌬p is mainly controlled by ATP usage, although the control is distributed among essentially all components of the system. The model predicted that in the presence of the K ϩ uniport and K ϩ /H ϩ antiport, the metabolic control (defined according to the Metabolic Control Analysis paradigm) over the contribution of ⌬⌿ to ⌬p (u ϭ ⌬⌿/⌬p) is exerted mainly by these processes, while in the absence of them, by ATP usage, ATP/ADP carrier, and P i carrier. It is shown that K ϩ circulation across the inner mitochondrial membrane is controlled by K ϩ uniport and K ϩ /H ϩ antiport. Finally, it was postulated that the system is able to control effectively the physiological ⌬⌿/⌬pH ratio even without the secondary K ϩ transport system.