INTRODUCTION

TOP ABSTRACT INTRODUCTION THE MODELS FOR METABOLICALLY... RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Mitochondria play a key role in the energy metabolism of aerobic cells and are formed by two membranes. The inner mitochondrial membrane (IMM) is known to be responsible for energy transformation, and its permeability is highly selective for metabolites and ions. Accumulated data suggest that the outer mitochondrial membrane (OMM) may be important in the regulation of metabolite fluxes and energy flow between the mitochondria and the cytoplasm (Liu and Colombini, 1992a; Sorgato and Moran, 1993; Saks et al., 1993, 1995; Brdiczka and Wallimann, 1994; Rostovtseva and Colombini, 1997), but the regulatory mechanisms remain to be elucidated.

In vitro, the rate of mitochondrial oxidative phosphorylation is significantly accelerated by external endergonic reactions utilizing ATP. Respiratory acceleration by ADP in the presence of inorganic phosphate (P_i) is known as the respiratory control phenomenon. In vivo, regulation of cellular energy metabolism is more complicated. Creatine kinase (CK), adenylate kinase, hexokinase, pyruvate kinase, and nucleoside diphosphokinase enzyme systems participate in energy channeling by specific metabolite compartmentation due to the highly organized structure of the cell and mitochondria (Wallimann et al., 1992; Saks et al., 1994; Brdiczka and Wallimann, 1994; Lipskaya et al., 1995). Various aspects of energy channeling are related to the voltage-dependent anion channel (VDAC), which was discovered more than 20 years ago (Schein et al., 1976; Colombini, 1979).

The VDAC forms large (up to 3 nm in diameter) aqueous pores in the OMM (Mannella et al., 1992; Colombini et al., 1996). The pore is composed of a 30-kDa peptide, called mitochondrial porin (Colombini et al., 1996), and sometimes may constitute more than 60% of the OMM total protein (Mannella, 1982). These pores have been shown to mediate the flux of charged metabolites, which are mostly organic anions, in a voltage-dependent manner (Colombini et al., 1996; Rostovtseva and Colombini, 1997; Rostovtseva and Bezrukov, 1998). VDAC is more permeable to anions in the open state, which normally occurs at membrane potentials with absolute values lower than 10-20 mV. At potentials with absolute values higher than 20 mV, VDAC undergoes transitions to multiple closed states and becomes more selectively permeable for cations (Schein et al., 1976; Colombini et al., 1996). Synthetic polyanions (Mangan and Colombini, 1987; Colombini et al., 1987; Zizi et al., 1995), reduced pyridine nucleotides (Lee et al., 1996), a protein localized in the mitochondrial intermembrane space (MIMS) (Liu and Colombini, 1992b; Holden and Colombini, 1993), and oncotic pressure (Zimmerberg and Parsegian, 1986) modulate the voltage dependence of the porin permeability. Synthetic polyanions greatly increase the slope of VDAC's permeability-voltage (PV) characteristic (Mangan and Colombini, 1987; Colombini et al., 1987). For instance, in the presence of dextran sulfate, a 10-fold decrease in VDAC's open probability has been observed with voltage changes from 0 mV to +5 mV or from 0 mV to 5 mV (Zizi et al., 1995). Modulation by various regulatory factors restricts VDAC's permeability to small organic anions, which may promote metabolite compartmentation. As a consequence, the rate of various mitochondrial enzymatic reactions may change.

The main question, which remains to be elucidated, is whether any mechanisms exist for electrical potential generation on the OMM, and if so, what values of the potential may be expected under physiological conditions. One possible mechanism of OMM potential (OMMP) generation may be Donnan potential (DP) (Liu and Colombini, 1992a, 1992b). Another proposed mechanism is capacitance coupling between the inner and outer membranes (Benz et al., 1990), but it has not been pursued in the literature. Some authors question the existence of a sufficiently high OMMP because many small ions, particularly potassium, are highly permeable through the OMM, and therefore they may counter any generated potential, with the exception of DP. Thus, several possible mechanisms exist which may generate the OMMP, but it has been difficult to obtain experimental evidence for any one of them.

We propose that steady-state fluxes of charged metabolites through the OMM may be the source of the OMMP. Taking into account the complexity of the cellular energy distribution system, we propose a simplified model of energy transfer in the cell for computational study. Such an approach seems to be useful for the estimation of a possible range of MDP values that may be generated on the OMM under physiological steady-state energy demand rates in rodent heart.

Computational analysis of the model showed that generated MDP can be high enough to regulate metabolite fluxes, even in the presence of physiological concentrations of highly permeable free K⁺ and other small ions in the system. The model predicts electrodynamic compartmentation of charged metabolites, potassium, and other ions. The obtained results suggest that MDP on the OMM may be directly involved in the regulation of mitochondrial and cellular energy metabolism.

	THE MODELS FOR METABOLICALLY DERIVED MEMBRANE POTENTIAL GENERATION

TOP ABSTRACT INTRODUCTION THE MODELS FOR METABOLICALLY... RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Characteristics of the steady-state model for metabolically derived potential generation on a liposomal membrane

The main principle of MDP generation may be demonstrated with the model in Fig. 1. We assume that one liposome of volume V₂ is put in a medium of infinitely large volume V₁. The liposome contains an allosteric enzyme E catalyzing an essentially irreversible reaction converting organic anion A to another organic anion B. Both anions are able to permeate the liposomal membrane. Anions A and B are added to the system in the form of potassium salts, and their initial concentrations are equal in both the internal and external mediums. Additionally, initial concentrations of KCl are also equal inside and outside of the liposome. The liposomal membrane may be considered permeable or impermeable for K⁺ and Cl.

View larger version (12K): [in this window] [in a new window]   FIGURE 1   The liposomal model of steady-state MDP generation, described by Eqs. 1-15. The liposome contains KCl and enzyme E, catalyzing an essentially irreversible reaction of conversion organic anion A to B. The permeability of the liposomal membrane for A is higher than that for B. The membrane may be considered permeable or impermeable for K+ and Cl. The external medium, of infinitely large volume, contains A and B in equal concentrations, which corresponds to a nonequilibrium state of the reaction (assuming that at equilibrium state [A] [B]). Diffusional steady-state potential will be generated on the liposomal membrane, and its value will be determined by the rate of the enzymatic reaction inside the liposome, the relationship of membrane permeabilities for A and B, and the membrane permeability for K+ and Cl.

Now, an activator of the allosteric liposomal enzyme E is added to increase v_m,l, the maximum rate of enzymatic conversion of A to B in the liposome (Eq. 1), which was initially equal to 0. As a result, the initially equilibrated internal concentrations [A]₂ = [A]₁ and [B]₂ = [B]₁ change because of the internal enzymatic activity. In turn, the activated conversion of A to B in the liposome will cause the influx of A into the liposome and the efflux of B from the liposome into the external medium. Membrane electrical potential will be generated because of the difference in the permeability coefficients P_A and P_B. Potassium and chloride ion concentrations inside and outside the liposome will not change if the membrane is impermeable for these ions, but their distributions will reach equilibrium if the membrane is permeable for them. It is evident that membrane permeability for K⁺ and Cl will cause a significant decrease in the metabolically generated membrane potential. Nernstian redistribution of K⁺ and Cl, in the case of the permeable membrane, may affect steady-state A and B fluxes, the osmotic pressure difference, and electrodynamic compartmentation of the organic ions as demonstrated by computational study of this model.

Mathematical description of the steady-state model for metabolically derived potential generation on a liposomal membrane

The model in Fig. 1 may be described mathematically in the following way. Assuming that enzymatic conversion of A to B in the liposome is essentially irreversible and is characterized by a simple first-order Michaelis-Menten kinetics, the rate of this reaction may be described as

(1)

where v_m,l is the maximum rate and K_m,2 is the Michaelis-Menten constant.

The fluxes for A and B ions across the membrane due to the difference in their concentrations [A]₂, [A]₁, and [B]₂, [B]₁, caused by enzymatic activity inside the liposome, may be expressed by Goldman's equation for ionic flux at the approximation of constant electrical field across the membrane:

(2)

(3)

where P_A is the membrane permeability coefficient for A, P_B is the permeability coefficient for B, F is the Faraday constant, is the membrane potential, R is the gas constant, and T = 310 K is normal body temperature.

P_B was set at 0.2P_A to model a fivefold difference in the liposomal membrane permeabilities for A and B ions. The P_A value was varied in some range. The liposomal volume V₂ was held constant and equal to 1 µl. To consider the external volume V₁ infinitely large relative to V₂, V₁ was set at 1 ml, and the relationship between concentrations [A]₁ and [B]₁ was defined as constant and independent of the rate of A to B conversion in the liposome. The initial concentrations of A and B in the two media were set at 10 mM each. The average concentration of A together with B (the sum of the molecules A and B in the system is constant because of the 1:1 stoichiometry of the reaction in the liposome), as well as the average concentrations of added K⁺ (100 mM) and Cl (80 mM), may be expressed by the following equations, respectively:

(4)

(5)

(6)

In addition, the space-charge neutrality principle should be maintained; that is,

(7)

(8)

At steady state, the rate of A to B conversion and the fluxes of A and B across the membrane must be equal:

(9)

(10)

In the case where the membrane is impermeable to K⁺ and Cl, the following equations should be satisfied:

(11)

(12)

When the membrane is permeable for these ions, Nernstian distribution has to be observed (when the membrane potential is generated by the steady-state fluxes of A and B):

(13)

(14)

The osmotic pressure difference between the liposomal matrix and the external medium may be expressed by the following equation:

(15)

The system of Eqs. 1-12 and 15, or 1-10 and 13-15, may be solved computationally, using standard software that uses numerical methods.

General characteristics of the simplified cell model

Energy transfer through the OMM of muscle cells may be represented in general as shown in Fig. 2 A. It includes oxidative phosphorylation, creatine kinase, adenylate kinase, and cytoplasmic ATPase systems. CK in the MIMS utilizes creatine (Cr) and ATP to produce phosphocreatine (PCr) and ADP. PCr molecules diffuse into the cytoplasm, where cytoplasmic CK utilizes them and ADP to produce ATP and Cr. Cr diffuses back into the MIMS, while ATP is used in endergonic processes by myofibrils, producing P_i, ADP, and work. In parallel, ATP may diffuse directly from the MIMS into the cytoplasm to be utilized by myofibrils. ADP, formed by hydrolysis of ATP in the cytoplasm, can be utilized by cytoplasmic CK or by adenylate kinase (AK), or diffuse into the MIMS. AMP, formed by the cytoplasmic AK reaction, diffuses into the MIMS, where the MIMS AK uses AMP and ATP to produce ADP for oxidative phosphorylation.

View larger version (23K): [in this window] [in a new window]   FIGURE 2   Cell model of metabolite flux between mitochondria and the cytoplasm. (A) Complete scheme reflecting ATP, ADP, AMP, PCr, Cr, and Pi diffusion through the OMM. (B) Simplified scheme, assuming that only PCr2, Pi2, Pi, and Cr are permeable through the OMM. D3 is ADP3, T4 is ATP4. The CK-ATPase system includes the cytoplasmic CK and ATPase. The OMM is permeable to small inorganic ions.

The general model of metabolite fluxes shown in Fig. 2 A could be described mathematically if the literature contained all experimental data for the components in this model. On the other hand, a simplified model may illustrate the basic principle of OMMP generation, by reducing the general model in Fig. 2 A to the model in Fig. 2 B, which considers only the CK system-mediated energy transfer. This system has been shown to play a key role in tissues with high and fluctuating energy demands, and many of its kinetic characteristics have been determined experimentally (Wallimann et al., 1992; Saks et al., 1993, 1994, 1995; Brdiczka and Wallimann, 1994; O'Gorman et al., 1996).

In the simplified model (Fig. 2 B), the charges of metabolites were included for computational estimation of the OMMP generated by the steady-state fluxes of those charged metabolites. According to this model, ATP⁴, formed in the matrix of mitochondria from P_i and ADP³, is transported into the MIMS in exchange for ADP³ through the adenine nucleotide translocatior (ANT). In the MIMS, CK produces PCr² and ADP³ from Cr and ATP⁴. ADP³ is transported into the matrix through ANT to bring a new molecule of ATP⁴ into the MIMS. PCr² diffuses into the cytoplasm and is utilized together with cytoplasmic ADP³ by cytoplasmic CK to produce ATP⁴ and Cr. Formed ATP⁴ is utilized by myofibrils to produce P_i, ADP³, and work. Therefore, if we call cytoplasmic CK and ATPase a "CK-ATPase system," we may say that PCr², diffusing into the cytoplasm from the MIMS, is "hydrolyzed" by the cytoplasmic CK-ATPase system. To close the cycle, Cr, P_i, and P_i² diffuse back into the MIMS, where P_i is transported into the mitochondrial matrix and P_i² is protonated for transport. Finally, Cr is utilized in the MIMS CK reaction for PCr² synthesis (Fig. 2 B).

The fluxes of ATP⁴, ADP³, and AMP² through the OMM are omitted in the model in Fig. 2 B for the purpose of simplification. This is based on the data showing that PCr² flux through the OMM is ~10 times higher than the ATP⁴ flux in working rat heart (Saks et al., 1994). In addition, it has been shown that VDAC can significantly limit the flow of adenine nucleotides between the MIMS and the cytoplasm (Saks et al., 1993; Rostovtseva and Colombini, 1997; Hodge and Colombini, 1997). While data for PCr² permeability through VDAC are to the best of our knowledge absent from the literature, we used the experimental data of Colombini (personal communication) for open and closed states of VDAC. The data for P_i and P_i² permeabilities are available in the literature (Hodge and Colombini, 1997). Free potassium, chloride, and magnesium ions were included in the model at their physiological concentrations.

The creatine molecule has a net zero charge at physiological conditions. Although we did not find any experimental data about Cr permeation through VDAC or the OMM, we assumed that the OMM permeability for Cr is not a limiting factor in energy transfer between mitochondria and the cytoplasm. This assumption was based on the fact that VDAC, being a large water-filled pore, restricts its permeability for ions mostly by electrostatic mechanisms (Colombini et al., 1996). As a result, among the metabolites in Fig. 2 A, only the exchange of P_i, P_i², and PCr² ions and the flux of noncharged Cr through the OMM were considered.

Mathematical description of the simplified cell model

The process of energy utilization, by means of the CK-ATPase system (Fig. 2 B) in the cytoplasm, can be described in general by two reactions,

(16)

(17)

the sum of which yields

(18)

Resultant PCr² "hydrolysis" (Eq. 18) by the cytoplasmic CK-ATPase system is a practically irreversible reaction under physiological conditions. If we assume that reaction 18 follows simple first-order Michaelis-Menten kinetics, the rate of PCr hydrolysis, v_o, can be described as

(19)

where [PCr²]_o is the PCr² concentration in the cytoplasm, and v_max,o is the maximum rate of PCr utilization by the CK-ATPase system in the cytoplasm. v_max,o was scanned in a wide range of values for computational study of the model. v_max,o is known to be modulated by Ca²⁺ during the muscle contraction cycle. The rate of PCr² hydrolysis at steady state depends on v_max,o and on the steady-state concentration of PCr² in the cytoplasm, [PCr²]_o. In turn, [PCr²]_o depends on the rate of Cr rephosphorylation in the MIMS and on PCr² flux through the OMM. Using Goldman's constant field approximation, PCr² flux through the OMM, J_PCr², is described as

(20)

where

(21)

In these equations, P_PCr² is the OMM permeability for PCr²; [PCr²]_i and [PCr²]_o are PCr² concentrations in the MIMS and the cytoplasm, respectively; F is the Faraday constant; is the OMMP; R is the gas constant; and T = 310 K is normal body temperature.

PCr production in the MIMS is described by the following reaction:

(22)

ATP is produced in the matrix of mitochondria from ADP and P_i and transported into the MIMS in exchange for ADP through ANT in the IMM. For simplicity, the processes of P_i and ADP transport from the MIMS into the matrix, ATP production in the matrix from P_i and ADP, and ATP transport from the matrix into the MIMS may be described as a single reaction in the MIMS:

(23)

because there are known approximate relationships between concentrations of ATP, ADP, and P_i in the matrix and the MIMS (see Appendix A). Combining reactions 22 and 23 written for the MIMS compartment yields

(24)

This cumulative reaction is driven by mitochondrial oxidative phosphorylation and is essentially reversible, requiring the use of the next equation (derived in Appendix A) for the rate of PCr_i production in the MIMS, v_i:

(25)

where v_max,m = 0.0067 fmol/s, the maximum rate of ATP production by an average rat heart mitochondrion in the coupled "oxidative phosphorylation-ANT-MIMS creatine kinase" system. [P_i]_i is the sum of P_i and P_i² concentrations in the MIMS, and v_max,i,r = 0.0133 fmol/s is the maximum rate of the reverse CK reaction in an average rat heart mitochondrion (see Appendix A), i.e., the reaction of Cr and P_i production in the MIMS from PCr. v_max,i,r is evidently overestimated because it only corresponds to the reversed CK reaction. It may be smaller for the cumulative reaction (Eq. 24), which will lead to a higher MDP.

The flux of P_i² through the OMM may be described by an equation similar to Eq. 20, using Goldman's approximation. The P_i² flux is dependent on the OMM permeability for P_i² (P_Pi²), on P_i² concentrations in the MIMS ([P_i²]_i) and the cytoplasm ([P_i²]_o), and on the steady-state OMMP according to the following equation:

(26)

where y is defined in Eq. 21.

The flux of P_i through the OMM may be described in the same manner:

(27)

At steady state, the rates of PCr² production, PCr² efflux from the MIMS into the cytoplasm, PCr² utilization by the CK-ATPase system in the cytoplasm, and the flux of P_i through the OMM from the cytoplasm into the MIMS should all be equal. The steady-state boundary conditions can be set in the following three equations:

(28)

(29)

(30)

where

(31)

The minus sign in Eq. 28 appears because P_i and PCr² flow in opposite directions.

From Eqs. 28-30, the steady-state metabolite flux through the OMM of a single mitochondrion, J, can be defined as

(32)

In addition to PCr², P_i², and P_i (see Appendix A for pK_a2 and pH), physiological concentrations of free K⁺, Cl, and Mg²⁺ ions were included. To have a sufficient concentration of counterions for physiological concentrations of K⁺ and Mg²⁺, arbitrary anions W were included in the system (see details below), as well as impermeable macromolecules Z²⁰ with the arbitrary charge 20. Z²⁰ anions represent some equivalent of the negatively charged surface of membranes and impermeable negatively charged macromolecules. The Donnan potential was modeled by setting different concentrations of nonpermeating macromolecules Z²⁰ in the MIMS and the cytoplasm.

The Nernst equation is applied to describe the distribution of freely permeating ions K⁺, Cl, Mg²⁺, and H⁺ between the cytoplasm and the MIMS:

(33)

(34)

(35)

(36)

The model is closed; i.e., it does not lose or gain any ions from the outside. Thus average ion concentrations were defined in the system using the following equations:

(37)

(38)

(39)

(40)

(41)

where V_i and V_o are the MIMS and the cytoplasm volumes, respectively, and

(42)

According to the space-charge neutrality principle, the total charge of cations is equal to the total charge of anions in a given volume (if the membrane electric capacity is negligibly small):

(43)

(44)

Making v_max,o = 0 fmol/s for the equilibrium state and using the average concentrations (see also Appendix A) [K⁺] = 150 mM, [Mg²⁺] = 1 mM, [Cl] = 5 mM, [PCr²] = 19 mM, [P_i²] = 0.34 mM and [P_i] = 0.21 mM (for pK_a2 = 7.2 and pH = 7.0), [ADP³] = 0.04 mM, and [ATP⁴] = 10 mM, [W] was found to be 8 mM by utilizing Eqs. 43 and 44 to maintain the space-charge neutrality principle, when concentrations of [Z²⁰] in the MIMS and in the cytoplasm were 3 mM to model DP = 0 mV. To model a nonzero DP, [Z²⁰]_i was set at 5 mM and [Z²⁰]_o was set at 3 mM. In this case, [W] was calculated to be around 5 mM (at v_max,o = 0).

To satisfy Eqs. 43 and 44, the floating concentrations of W, originating from the dissociation of a week acid WH, were included in the system. The dissociation constant of WH was assumed to be the same in the MIMS and the cytoplasm:
If WH is infinitely permeable across the OMM, then [WH]_i = [WH]_o = [WH], which yields the following equation, regardless of whether W ion is permeable or not:
Or, taking into account Eq. 36, it yields

(45)

Thus, W ions will be distributed across the OMM according to the Nernst equation if electrochemical equilibrium exists for protons across the OMM. The pH_o was set and the system of equations was allowed to find the concentrations of W to satisfy the space-charge neutrality principle of each medium for various steady states of the model. The space-charge neutrality principle must be satisfied because of the extremely low electrical capacity of the outer membrane.

The floating concentration [W] depends on the steady state and on the value of the external pH (pH_o), which is taken to be constant. Higher energy demand in the cytoplasm will lead to higher steady-state concentrations of P_i and P_i² in the system. P_i² is supplied from PCr² "hydrolysis," and P_i appears to be due to the protonation of some of the P_i², according to pK_a2, fixed pH_o, and settled pH_i, by utilizing H⁺ ions originating from week acid WH dissociation, which increases the concentration of W in the system.

Values for v_max,o were varied from 0 to 8 × 10³ fmol/s, covering and exceeding the physiological range of workloads related to one average mitochondrion in rat heart (see Appendix A). The average MIMS volume of rat heart single mitochondrion was taken as V_i = 0.03 fl (see Appendix B). The sum of cytoplasmic and myofibrillar compartment volumes, V_o = 0.36 fl, was chosen to be in proportion to the MIMS volume of 12:1 (see Saks and Aliev, 1996). The cytoplasmic volume V_o was set as a constant (V_o = 12V_i), even in the case when an osmotic pressure difference appeared between the MIMS and the cytoplasm compartments.

Critical parameters for MDP generation are the OMM permeabilities for P_i, P_i², and P_PCr². Experimentally measured ion fluxes of P_i, P_i², and P_PCr² (in 10⁶ ions/s) through a single VDAC reconstituted into the planar phospholipid membrane under standard conditions are 14.0, 6.30, and 3.90, respectively, in the open state and 1.60, 0.29, and 0.24, respectively, in the closed state. The VDAC ion flux values for P_i and P_i² were taken from the literature (Hodge and Colombini, 1997); and the data for PCr² were kindly provided by Dr. Colombini (personal communication). These data were used to obtain the relationship of the OMM permeabilities for the metabolites above, taking all of them relative to the OMM permeability for P_i in the VDAC open state (Table 1).

View this table: [in this window] [in a new window]   TABLE 1   The OMM relative permeabilities for Pi, Pi2, and PCr2 in the open (Po) and closed (Pc) states of the VDAC, corresponding to the relationship between experimentally determined fluxes of Pi, Pi2 (Hodge and Colombini, 1997), and PCr2 (Colombini's unpublished data) through a single VDAC reconstituted in planar phospholipid membranes

To model the form of the experimentally determined permeability-voltage (PV) characteristics of the VDAC (Zizi et al., 1998), the following mathematical approximation was used:

(46)

where P is the OMM absolute permeability of a single average mitochondrion for an ion, and P_o and P_c are the OMM relative permeabilities corresponding to the VDAC's open and closed states, respectively (Table 1). The absolute permeability coefficient a₀ in Eq. 46 was set at 3.6 fl/s for all calculations (Appendix B). The voltage-sensitive part of the absolute permeability function will lie between a₀P_c and a₀P_o. Higher values of a₀ would lead to a dramatic decrease in MDP, while lower values of a₀ would lead to a nonphysiological restriction of metabolite flux through the OMM (see Appendix B for more details) and to a higher value of MDP. The value of coefficient a in Eq. 46 allows an adequate bell-shaped and symmetrical PV characteristic of the VDAC. The higher the a, the steeper the slope of VDAC's permeability dependence on the OMMP. Constant  shifts the PV curve to the left or to the right under the influence of various factors, but it was set at 0 in all calculations. In Eq. 46,  and are expressed in volts, where is the OMMP (i.e., MDP in the case where DP = 0 or a combination of MDP and DP). Fig. 3 shows three symmetrical PV characteristics of the VDAC, modeled by Eq. 46, where a₀ = 3.6 fl/s, a = 300 V¹,  = 0. Fig. 3 demonstrates how DP may change permeabilities and how they can be modulated by MDP.

View larger version (25K): [in this window] [in a new window]   FIGURE 3   Permeability-voltage characteristics of VDAC (in fl/s) modeled by Eq. 46 for Pi (), Pi2 (- - -), and PCr2 (· · · ·). In all plots a0 = 3.6 fl/s, a = 300 V1, and  = 0. Relative permeability coefficients of the OMM for Pi, Pi2, and PCr2 in open (Po) and closed (Pc) states of the VDAC are presented in Table 1. Permeabilities may be changed by the DP and further modulated by MDP. The DP and MDP form the OMMP.

The difference in osmotic pressure between the MIMS and the cytoplasm needs to be considered as an additional parameter X, which is calculated using the following equation:

(47)

Parameter X is important for morphological changes in mitochondria accompanying their metabolic state variations. The system of equations with the parameters described above was solved numerically using Mathcad 8.0 software (MathSoft, Cambridge, MA).

	RESULTS

TOP ABSTRACT INTRODUCTION THE MODELS FOR METABOLICALLY... RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Computational analysis of the liposomal model (Fig. 1) showed that enzymatic conversion of metabolite A to metabolite B inside the liposome led to the generation of membrane potential (Fig. 4 A), under the following conditions: 1) when permeability coefficients of the membrane were different for these two metabolites and 2) concentrations [A]₁ and [B]₁ (in the external medium) were maintained to be essentially constant at steady state ([A]₁ = [B]₁). The value of MDP depended on the rate of the enzymatic reaction modulated by changing v_m,l in Eq. 1. In a real system, an allosteric enzyme may be modulated by its allosteric activator, for example, by Ca²⁺. MDP was significantly diminished because of Nernstian redistribution of the other permeable ions (K⁺ and Cl in the considered liposomal model). An increase in metabolite flux J (Fig. 4 B) and the effect of electrodynamic compartmentation of metabolites (Fig. 4 C) are observed as well, when the membrane is considered permeable for K⁺ and Cl. MDP led to an osmotic pressure difference between the liposomal matrix and the external medium through K⁺ and Cl Nernstian redistribution and because of electrodynamic compartmentation of the metabolites (Fig. 4 D). Not only the ratio of membrane permeabilities for A and B, but their absolute permeabilities as well, were essential for MDP (Fig. 4 E) and metabolite flux (Fig. 4 F) dependence on the rate of liposomal enzymatic reaction.

View larger version (31K): [in this window] [in a new window]   FIGURE 4   Solutions of the liposomal model for steady-state MDP generation (Fig. 1) described by Eqs. 1-15. (A) MDP dependence on the maximum rate of A to B conversion in the liposome, vm,l. (B) Flux J of A or B dependence on vm,l (the fluxes of A and B are equal at steady state). (C) Dependence of A and B concentrations in the liposome on vm,l. (D) Dependence of osmotic pressure difference between the liposome and external medium on vm,l. (E) MDP dependence on the permeability coefficient a, where PA = a and PB = 0.2a. (F) Flux J of A or B dependence on permeability coefficient PA, where PB = 0.2PA. For A-D: PA = 1.0 µl/s and PB = 0.2 µl/s. For E and F: vm,l = 100 nmol/s (a, c) and vm,l = 10 nmol/s (b, d). The liposomal membrane is permeable ( , Eqs. 13 and 14 are used) or impermeable (- - -, Eqs. 11 and 12 are used) for K+ and Cl ions.

In computational analysis of the simplified cell model (Fig. 2 B), the OMM permeability was assumed to be the main limiting step in metabolite flux, as long as the rate of energy demand in the cytoplasm or ATP production by mitochondria were not limiting. Dependence of the generated OMMP on the rate of PCr "hydrolysis" in the cytoplasm at pH_o 7.0 was calculated under conditions where the OMM was permeable for all ions, except nonpermeating macromolecules Z²⁰ (Fig. 5). Zero values were assigned to a in Eq. 46 when the voltage dependence of VDAC permeability was not considered. The "basic relationship" of the OMM permeabilities for P_i, P_i², PCr² (Table 1) and a₀ = 3.6 fl/s were used in Eq. 46.

View larger version (30K):