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Analysis of Functional Coupling: Mitochondrial Creatine Kinase and Adenine Nucleotide Translocase

Marko Vendelin ^* , Maris Lemba  and Valdur A. Saks ^* 

^* Laboratory of Fundamental and Applied Bioenergetics, Institut National de la Santé et de la Recherche Médicale E0221, Joseph Fourier University, Grenoble, France; Institute of Cybernetics, Tallinn Technical University, Tallinn, Estonia; and Laboratory of Bioenergetics, National Institute of Chemical Physics and Biophysics, Tallinn, Estonia

Correspondence: Address reprint requests to Marko Vendelin at his present address: Laboratory of Fundamental and Applied Bioenergetics, INSERM E0221, Université J. Fourier, BP 53, F-38041, Grenoble, France. Tel.: 372-620-4151; Fax: 372-620-4169; E-mail: markov{at}ioc.ee.

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The mechanism of functional coupling between mitochondrial creatine kinase (MiCK) and adenine nucleotide translocase (ANT) in isolated heart mitochondria is analyzed. Two alternative mechanisms are studied: 1), dynamic compartmentation of ATP and ADP, which assumes the differences in concentrations of the substrates between intermembrane space and surrounding solution due to some diffusion restriction and 2), direct transfer of the substrates between MiCK and ANT. The mathematical models based on these possible mechanisms were composed and simulation results were compared with the available experimental data. The first model, based on a dynamic compartmentation mechanism, was not sufficient to reproduce the measured values of apparent dissociation constants of MiCK reaction coupled to oxidative phosphorylation. The second model, which assumes the direct transfer of substrates between MiCK and ANT, is shown to be in good agreement with experiments—i.e., the second model reproduced the measured constants and the estimated ADP flux, entering mitochondria after the MiCK reaction. This model is thermodynamically consistent, utilizing the free energy profiles of reactions. The analysis revealed the minimal changes in the free energy profile of the MiCK-ANT interaction required to reproduce the experimental data. A possible free energy profile of the coupled MiCK-ANT system is presented.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
In muscle and brain cells, phosphocreatine and adenylate kinase shuttles provide a link between ATP-producing and ATP-consuming sites (Bessman and Geiger, 1981; Wallimann et al., 1992; Saks and Ventura-Clapier, 1994; Joubert et al., 2002, 2004; Dzeja and Terzic, 2003). As a part of phosphocreatine shuttle, the functional coupling between mitochondrial creatine kinase (MiCK) and adenine nucleotide translocase (ANT) has been identified by stimulating oxidative phosphorylation with creatine (Cr) (Bessman and Fonyo, 1966) and has been further examined with kinetic and structural studies (Jacobus and Lehninger, 1973; Saks et al., 1975; Seppet, 1979; Gellerich and Saks, 1982; Barbour et al., 1984; Wallimann et al., 1992). Recently, it has been shown that the coupling plays an important role in preventing the opening of the permeability transition pore and, thus, is critical for cell life (Dolder et al., 2003). However, regardless of the large amount of experimental data on functional coupling between MiCK and ANT dating back to the 1970s, the intimate mechanism of the interaction between the proteins is still not clear.

There are two mechanisms suggested to explain the effective interaction between MiCK and ANT: 1), the dynamic compartmentation of ATP and ADP (Gellerich et al., 1987) and 2), the direct transfer of ATP and ADP between the proteins (Saks et al., 1975; Jacobus and Saks, 1982). According to the first mechanism, functional coupling between MiCK and ANT can be explained by differences between the concentrations of ATP and ADP in intermembrane space and those in the surrounding solution due to some limitation of their diffusion across the outer mitochondrial membrane (Gellerich et al., 1987). According to the second mechanism of coupling, ATP and ADP are directly transferred between MiCK and ANT without leaving the complex of proteins (Jacobus and Saks, 1982). Neither of the proposed mechanisms have been checked quantitatively against the experimental measurements by thermodynamically consistent models that incorporate all the basic types of available data. Dynamic compartmentation hypothesis was used either to fit a limited set of experimental data (Gellerich et al., 1987) or applied in the development of several simplified phenomenological models of MiCK-ANT coupling used as part of models of intracellular energy transfer (Aliev and Saks, 1997; Vendelin et al., 2000b; Saks et al., 2003). The direct channeling has been analyzed mathematically by Aliev and Saks (1993, 1994) using a probability approach. In the two latter works, to simulate the measured alterations in the kinetics of MiCK reaction brought about by oxidative phosphorylation, the dissociation of ATP from its ternary complex with MiCK and Cr (CK.ATP.Cr) was not allowed in the model, i.e., this complex was always utilized in the MiCK reaction. A thermodynamically consistent analysis of the mechanism was not included in Aliev and Saks (1993, 1994), and is still absent.

The aim of this work is to identify the simplest mechanism able to reproduce the available experimental data on functional coupling between MiCK and ANT. The following experimental results were analyzed by the mathematical models: 1), changes in the apparent kinetic properties of the MiCK reaction when coupled to oxidative phosphorylation (Jacobus and Saks, 1982; Saks et al., 1985); 2), competition between MiCK-activated mitochondrial respiration by the competitive ATP-regenerating system (Gellerich and Saks, 1982); and 3), studies of radioactively labeled adenine nucleotide uptake by mitochondria in the presence of MiCK activity (Barbour et al., 1984). The results show that the direct transfer of ATP and ADP between ANT and MiCK is involved in the phenomenon of functional coupling between the proteins. The possible free energy profile of the functionally coupled reactions is presented.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
To test the two hypotheses of functional coupling between mitochondrial creatine kinase (MiCK) and adenine nucleotide translocase (ANT)—i.e., dynamic compartmentation and direct transfer between the proteins—the corresponding mathematical models were composed. First, we describe a simple model of dynamic compartmentation. Next, the modeling strategy for simulating direct transfer between MiCK and ANT is given. Finally, the simulation protocol as well as numerical methods are described. Details of the model based on the direct transfer between MiCK and ANT are given in the Appendix.

Model of dynamic compartmentation
According to the dynamic compartmentation hypothesis, the functional coupling between MiCK and ANT occurs through high ATP and ADP concentrations either in an intermembrane space (Gellerich et al., 1987) or in a narrow-space microcompartment (i.e., a gap) between the proteins (Aliev and Saks, 1997). In this study, our model assumes that there is a difference in the concentrations between the compartment and surrounding solution. Basic principles underlying the model composition were as follows (Fig. 1 A):

1. ANT was assumed to translocate adenine nucleotides between the matrix space and the compartment.
   
2. MiCK was linked to ATP and ADP in the compartment, while interacting with Cr and PCr from the solution.
   
3. Diffusion between the compartment and solution was considered to be restricted.
   

View larger version (26K): [in this window] [in a new window]   FIGURE 1  Scheme of interaction between mitochondrial creatine kinase (MiCK) and adenine nucleotide translocase (ANT). The interaction between the proteins is considered as a sum of two interaction modes: ATP and ADP are transferred through solution (subplot A) or directly channeled between the proteins (subplot B). In the model based on dynamic compartmentation hypothesis, only the interaction through solution is considered. When direct channeling between MiCK and ANT is assumed, both modes are taken into account. (Subplot A) In the first mode of MiCK-ANT interaction, ATP after transport from matrix to intermembrane space by ANT (link between ANTx.ATP and ANTi.ATP in the scheme) is released to solution. Next, the released ATP participates in MiCK reaction according to the random Bi-Bi mechanism with fast equilibrium between all states of MiCK. ADP produced by MiCK reaction is released into solution by MiCK and picked up by ANT for transport to mitochondrial matrix (link between ANTi.ADP and ANTx.ATP in the scheme). (Subplot B) In direct transfer mode, ATP is transferred from ANT to MiCK without leaving two-protein complex to solution. Since MiCK has only one binding site for ATP and ADP, such transfer is possible only if this site is free, i.e., MiCK is either free (CK) or has only Cr or PCr bound (CK.Cr and CK.PCr). In the scheme, we grouped the states of MiCK according to whether ATP or ADP is bound to enzyme or not (open boxes in the scheme with three states of MiCK in each group). During direct transfer of ATP from ANT to MiCK, MiCK is transferred from states CK, CK.Cr, and CK.PCr to states CK.MgATP, CK.Cr.MgATP, and CK.PCr.MgATP, respectively. In the scheme, this transfer is shown as a link between ANTi.ATP and two corresponding groups of MiCK states. Next, after MiCK reaction (link between states CK.Cr.MgATP and CK.PCr.MgADP in the scheme), ADP is transferred directly to ANT. Note, that MiCK operates with Mg-bound ATP and ADP and ANT requires Mg-free ATP and ADP forms. Thus, during direct transfer between MiCK and ANT, Mg is either bound or released, as shown in the scheme.

(1)

(2)

(3)

(4)

where the factor F_volume denotes the ratio of the volume of the solution to that of the compartment and _CK, _ANT, _difATP, _difADP, _ATPase, and _PK correspond to the rate of MiCK reaction, ATP export from the matrix to the compartment by ANT, diffusion of ATP and ADP between compartment and solution, residual ATPase in solution, and PK reaction, respectively. Diffusion of nucleotides was governed by

(5)

(6)

where D_ATP and D_ADP are exchange coefficients. The concentrations of Cr, PCr, Mg²⁺, Pi, and phosphoenolpyruvate (if present) were fixed at the onset of the simulations and assumed to be essentially constant during the experiment. The differential equations governing the system were solved until steady-state values of the variables were obtained.

Adenine nucleotides are known to form complexes with magnesium ions Mg²⁺, which are subjected to creatine kinase reaction, whereas only Mg²⁺-free nucleotides are translocated by ANT. The concentrations of the Mg²⁺-bound nucleotides (MgATP_s, MgADP_s, MgATP_c, and MgADP_c) and Mg²⁺ free forms (fATP_s, fADP_s, fATP_c, and fADP_c) were related to the total amount of nucleotides (ATP_c and ADP_s) as

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

where [Mg²⁺] is the concentration of free Mg²⁺, and K_DT and K_DD are Mg²⁺ dissociation constants from MgATP and MgADP, respectively. Concentration [Mg²⁺] was assumed to depend only on the amount of total Mg²⁺, Pi, and nucleotides in the solution.

The rate equation of the MiCK reaction was derived according to the random-order Bi-Bi mechanism, assuming rapid equilibrium of substrate binding and product release (Jacobs and Kuby, 1970; Jacobus and Saks, 1982, Morrison and James, 1965),

(15)

where

(16)

and K_aK_icr = K_iaK_cr, K_dK_icp = K_idK_cp. The constants V_1CK, V_–1CK, K_ia, K_a, K_icr, K_id, K_icp, K_cp, K_Icp, and K_Icr in the equations are the maximal forward and reverse rates of the MiCK reaction, and dissociation constants, respectively. According to the hypothesis of dynamic compartmentation, the constants were taken equal to those of the soluble MiCK. The indexes a, cr, d, and cp mark the dissociation of ATP, Cr, ADP, and PCr, respectively; these indexes correspond to the dissociation of a ternary complex and with an additional index i to the dissociation of the complex composed of MiCK and only one substrate. The parameters K_Icp and K_Icr are the dissociation constants of dead-end complexes.

Description of ANT kinetics was based on the phenomenological model by Korzeniewski (1998). The net rate of ATP export by ANT from the matrix to the compartment is described by the kinetic equations (Vendelin et al., 2000b)

(17)

where [fATP_x] and [fADP_x] are the concentrations of Mg²⁺-free nucleotides in the matrix, K_g is the ANT reaction dissociation constant, is the membrane potential, and V_ANT is the maximal relative nucleotide transportation rate. Parameter Z is equal to RT/F, where R is the gas constant, T is the absolute temperature, and F is the Faraday number.

The residual ATPase activity _ATPase was characterized by the simple Michaelis-Menten equation of

(18)

The formula for the pyruvate kinase reaction was obtained from Saks et al. (1984).

All values of the model parameters pertinent to the description of coupling between MiCK and ANT via dynamic compartmentation are given in Table 1.

For simplicity, the following assumptions were made:

1. The concentrations of the substrates near ANT and MiCK were taken to be the same as the surrounding solution.
   
2. The total number of binding sites for ATP (ADP) on all MiCK and all ANT molecules is taken to be the same, with binding sites on MiCK molecules organized in pairs with the binding sites on ANT molecules. If one considers MiCK as an octamer, this may be taken to interact with the cluster of eight ANT dimers (Wallimann et al., 1992), with each ANT dimer having one binding site (Klingenberg, 1985).
   

The interaction between MiCK and ANT is considered as a sum of two interaction modes: 1), ATP and ADP are liberated into intermembrane space and then bound to MiCK or ANT or 2), directly channeled between the proteins. The first interaction mode is similar to that of the dynamic compartmentation, but without any diffusion restrictions after ATP or ADP release. Thus, when the proteins interact through ATP and ADP release and uptake from solution, ANT and MiCK reaction schemes are exactly the same as for the uncoupled system (see Fig. 1 A). In this case, ATP as well as all other substrates are in fast equilibrium with MiCK and the reaction follows the random Bi-Bi type mechanism. However, MiCK with no bound ATP or ADP molecule can also accept ATP (ADP) from ANT, which is transferred directly (see Fig. 1 B). In this case, we assume that when MiCK accepts ATP (ADP) from ANT directly, bound ATP and ADP cannot be in fast equilibrium with the surrounding solution. Thus, the equilibration of the MiCK binding site for ATP and ADP with the surrounding solution is prevented. For simplicity, we assumed that, in the case of direct transfer, MiCK is not exchanging ATP and ADP with solution at all, but returns ATP in the form of ADP or ATP back to ANT (Fig. 1 B). To distinguish the states of MiCK that are so tightly coupled to ANT from other MiCK states, an index c would be used in notations. For example, CK_c.ATP would indicate the MiCK state with attached ATP received from ANT directly.

A thermodynamically consistent model was derived, taking into account the free energies of transition during each reaction as well as the principle of microscopic reversibility. This approach, based on the transition state theory of enzymatic reactions, allows one to study the free energy profile of the coupled system. In generalized form, the rates of monomolecular transition between two states of MiCK-ANT complex, noted as A and B, are governed by

(19)

(20)

where ₊ and _– are the rates of transitions between states A and B in forward and backward directions, respectively; [A] and [B] are normalized concentrations; is a factor that depends on the nature of the transition; G is a free energy in the transition state; G_A and G_B were the free energies of the states A and B, respectively; R is the gas constant; and T is the absolute temperature. Since in our further analysis we seek the overall rate constants and not the exact value of G (see below), it is possible to predefine the value of in the calculations. Using the free energy change G_AB = G_B–G_A and the free energy of transition G = G–G_A, the rates ₊ and v_– would be

(21)

(22)

The binding of the substrates to MiCK-ANT complex is modeled using similar equations, which are modified to take into account the bimolecular nature of the process. (For examples of these bimolecular reactions, see the equations following as well as the equations in the Appendix.)

Let us consider direct transfer of ATP from ANT to coupled MiCK, with the nucleotide binding site of the carrier directed toward intermembrane space and MiCK having no substrates bound. The process of ATP (ADP) transfer between ANT and MiCK is considered to be a conformational change within the MiCK-ANT complex. During this conformational change, a Mg²⁺ ion has to be attached, since MiCK reacts with the Mg-bound form of ATP only. The rate ₊ of the transfer of ATP from ANT to MiCK and the rate for reverse transfer would be determined by the free energy of transition G and the free energies of the participating substrates, which, in addition to the free energies of ANT and MiCK complexes, includes the free energy of Mg²⁺ (G_Mg), as

(23)

(24)

(25)

where [ANTi.ATP-CK] is the relative concentration of ANT and MiCK complex, with ANT directed toward intermembrane space and binding ATP, and substrate-free MiCK; [ANTi-CK.ATP] is the relative concentration of ANT and MiCK complex, with substrate-free ANT directed toward intermembrane space and MiCK forming a binary complex with ATP; and G_ANTi.ATP-CK and G_ANTi-CK.ATP are free energies of MiCK-ANT complex in states ANTi.ATP-CK and ANTi-CK.ATP, respectively.

To keep the number of model parameters as small as possible, we assumed that all transformations changing only a state of MiCK or ANT, depend on the participating states of this protein only. For example, according to this assumption, the rate of ATP binding from matrix to ANT does not depend on the state of MiCK. Thus, the respective rate of ATP binding should be specified in the model with only one G and the free energies of ANT states involved without any dependence on MiCK state. The latter is achieved in the model by assuming that the free energy of the MiCK-ANT complex is a sum of the free energies of the ANT and MiCK states. For example, the free energies used in Eqs. 23–25 are G_ANTi.ATP-CK = G_ANTi.ATP + G_CK and G_ANTi-CK.ATP = G_ANTi + G_CK.ATP.

In terms of the composed mathematical model, our aim is to find the values of the free energy of transition G for every reaction between the states of the MiCK-ANT complex and the free energies of all states of the complex. In search for the simplest possible kinetic scheme of the coupling, we use the free energies of the enzyme and the carrier states as well as the G estimated for an uncoupled system as much as possible. In the simplest possible model, only the G for ATP and ADP binding with ANT would be changed, in addition to introducing the G for direct transfer of ATP and ADP between the proteins. If this model fails to reproduce the measured data, then more parameters would be altered, until the minimal combination of changed parameters sufficient to reproduce the data is identified. This minimal combination of parameters would be the main result of our analysis. Naturally, this requires the parameters to be varied in a large range, before ruling a combination out as not sufficient to reproduce the experiment.

The system of equations for modeling the direct transfer between MiCK and ANT and values of the model parameters are described in the Appendix.

Protocol of simulations
There were two types of simulations performed in this study: 1), scanning of parameter space by changing specified model parameters independently from each other and 2), fitting the experimental data by minimization of residual function by variation of parameter values. Although analysis of the dynamic compartmentation comprised only the first type of simulation, both were employed in the direct transfer model.

In simulations with the dynamic compartmentation model, the values of maximal ATPase activity in the solution, ATP and ADP exchange constants D_ATP and D_ADP, were varied as follows: ATPase activity was varied from 0 to 17% of maximal MiCK activity in accordance with estimations of Jacobus and Saks (1982); exchange constants were varied from 10^–10 s^–1 to 10¹ s^–1. During a scan in the model of direct transfer between MiCK and ANT, the free energies of the MiCK and ANT states were varied in a wide range from –15 kJ mol^–1 to +10 kJ mol^–1 and the free energies of transition were varied from 20 kJ mol^–1 to 50 kJ mol^–1 (corresponding to rate constants variation from 0.002 s^–1 to 300 s^–1). The step sizes of these variations are specified for every simulation performed in the section Results.

When fitting the experimental data by the model, the residual function, subject to minimization, consisted of a sum of terms, each of which corresponded to one measurement: ((f_exp–f_cal)/)² with measured (f_exp) and computed (f_cal) rate (or apparent kinetic constant) and standard deviation () of the measurements. To specify the dependence of the direction of MiCK reaction on oxidative phosphorylation (Saks et al., 1985), the following penalty term was added to all residual functions used in this study,

(26)

where _CK is the MiCK reaction rate in the presence of oxidative phosphorylation and 4 mM PCr, 0.12 mM ATP, 0.05 mM ADP, and 40 mM Cr; was taken equal to 0.01 s^–1; and a large value = 10³ was used to ensure that _CK remained larger than during optimizations.

In some experiments used, standard deviation (SD) was not reported, and the following approximations were used in this study:

1. MiCK reaction rates estimated from kinetic constants were assumed to have SD = 15% of the maximal rate of the MiCK reaction in the direction of PCr production.
   
2. Measurements of respiration rate after inhibition by the competitive ATP-regenerating system (Gellerich and Saks, 1982) were assumed to have SD = 10% of the respiration rate recorded without the competitive ATP-regenerating system in solution.
   
3. The estimation of the amount of ADP retransported to mitochondria after the MiCK reaction, without mixing with the surrounding solution (Barbour et al., 1984), was assumed to have the same SD at all ADP concentrations and to be equal to 10%.
   

The analysis of the direct transfer between MiCK and ANT was organized as follows:

1. The calculated solution was checked against the experimental data that required the smallest amount of simulations—i.e., dependence of MiCK reaction direction on the presence of oxidative phosphorylation in certain conditions (Saks et al., 1985).
   
2. All parameter values combinations that satisfied the first test were used to compute kinetic constants of MiCK reaction in the presence of oxidative phosphorylation.
   

Numerical methods
The system of ordinary differential equations was solved by the backward differentiation formula that is able to treat stiff equations (Brown et al., 1989). The accuracy of the solution was tested by varying the tolerance of the ordinary differential equation solver. The required optimization was performed using the Levenberg-Marquardt algorithm (Moré et al., 1984). The models were implemented using C++ and FORTRAN programming languages. The parts of C++ code for the model based on direct transfer hypothesis were generated by a Python script. Source code is available on request.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Dynamic compartmentation of ATP and ADP
According to the hypothesis of dynamic compartmentation, the concentrations of ADP and ATP in the compartment are different from those in the solution. This is due to the restricted diffusion of ATP and ADP between the solution and the compartment. The analysis of Eqs. 1–4 reveals that, in steady-state conditions, the sum of ATP and ADP concentrations in the compartment can differ from that in the solution only if the ATPase activity _ATPase is assigned a non-zero value. The latter can be demonstrated by taking the right side of the time-derivatives of Eqs. 1–4 equal to zero, reflecting the steady-state conditions. In that case, we obtain from Eqs. 3–6

(27)

(28)

assuming that there is no pyruvate kinase in the solution (_PK = 0). Adding Eqs. 27 and 28 results in

(29)

It becomes clear from Eq. 29 that the sum of ATP and ADP concentrations in the compartment can differ from the concentration in the solutions only if both of the following conditions are satisfied: 1), residual ATPase activity is present in the solution and 2), the values of ATP and ADP exchange constants are not equal to each other. Equation 29 also demonstrates that, to increase the total concentrations in the compartment, the term (1/D_ADP–1/D_ATP) should be negative and as large as possible by its absolute value, i.e., the exchange is fast for ADP and slow for ATP.

In Fig. 2, the apparent dissociation constants of ATP and Cr in MiCK reaction were computed as a function of _ATPase activity in the solution. In accordance with the analysis presented above, the exchange between solution and compartment was assumed to be fast for ADP and slow for ATP in these simulations. As Fig. 2, A and B, clearly demonstrates, increase of ATPase activity in the solution leads to a drop in computed apparent dissociation constants of ATP both from ternary and binary complexes with MiCK, K_a, and K_ia, respectively. However, the K_a and K_ia values that have been measured by Jacobus and Saks (1982) are reached by the model at different _ATPase activities (Fig. 2). Within the model, it is not possible to reproduce the experimentally observed 10-fold and 2.5-fold decreases in the values of soluble MiCK dissociation constants simultaneously with one parameter set. This can be demonstrated by computing K_a and K_ia at different combinations of _ATPase, D_ATP, and D_ADP values (Fig. 3). Indeed, the computed line, characterizing the relationship between K_a and K_ia values, does not pass the point coordinates, which are the measured dissociation constant values in the presence of oxidative phosphorylation. This holds true regardless of the ATPase activity used and the values of the exchange constants in the wide range of values used. The increase of model ANT activity leads to the shift of the K_a–K_ia relationship toward the measured values to a certain limit. This limit is still adrift from the measurements (Fig. 3). We conclude from the results that the model based on dynamic compartmentation hypothesis is unable to reproduce the measurements. Due to such a strict relationship between computed values of K_a and K_ia there is no need to use minimization for fitting the experimental data to check this conclusion—altering parameter values is neither going to change the slope of the lines nor cross the limit obtained with the high ANT activities, as is evident from scanning the parameter space (Fig. 3).

View larger version (29K): [in this window] [in a new window]   FIGURE 2  Calculated apparent dissociation constants of MiCK reaction as functions of ATPase activity (ATPase) in the solution in the presence of oxidative phosphorylation. Coupling between MiCK and oxidative phosphorylation was modeled according to the dynamic compartmentation hypothesis. ATPase activity is presented relative to maximal MiCK activity, in percents. Subplots A, B, C, and D correspond to apparent Ka, Kia, Kcr, and Kicr, respectively. The simulation results (solid lines) obtained with the model are compared with the measured values of apparent dissociation constants, indicated by the average measured value (dashed line) and shaded area corresponding to the measured average ± standard deviation (SD) (data from Jacobus and Saks, 1982). In the calculations, maximal activity of ANT was taken equal to that of MiCK; ATP and ADP exchange constants DATP and DADP were 10–9 s–1 and 10–1 s–1, respectively. Note that with the increase of ATPase activity in the solution, apparent dissociation constants of ATP from ternary and binary complexes with MiCK (Ka and Kia, respectively) are reduced considerably.

View larger version (15K): [in this window] [in a new window]   FIGURE 3  Calculated apparent dissociation constants Ka and Kia of the MiCK reaction in the presence of oxidative phosphorylation. Coupling between MiCK and oxidative phosphorylation was modeled according to the dynamic compartmentation hypothesis. Here, apparent dissociation constants Ka and Kia (represented by small dots in the figure) were computed in case of different combinations of the values of ATPase activity in the solution ATPase and exchange constants DATP and DADP. In the figure, the measured values are shown (open circles) in the right upper corner (no oxidative phosphorylation) and in the left lower corner (with oxidative phosphorylation). When using the kinetic constants of ANT transport as given in Table 1, all combinations of computed Ka and Kia are aligned along the line with index 1 (indexes are shown within larger open circles in the figure). By increasing the maximal activity of ANT by 10 or 100 times, this line can be shifted to the left (lines with indexes 2 and 3, respectively). When instead of increasing the maximal activity of ANT, the apparent dissociation constant Kg is increased, the line shifts to the right (line with index 4). Note that regardless of the values used of ANT kinetic constants, all computed combinations of Ka and Kia were considerably adrift from the measured values of these constants in the presence of oxidative phosphorylation. Experimental data from Jacobus and Saks (1982).

View larger version (35K): [in this window] [in a new window]   FIGURE 4  Analysis of the direction of the MiCK reaction in the presence of oxidative phosphorylation. In this contour plot, the rates of the MiCK reaction, relative to the maximal rate of the MiCK reaction in the direction of PCr synthesis, are depicted by different shades and iso-lines. Numbers on the iso-lines show the value of reaction rates; positive values correspond to PCr synthesis. The shades correspond to the values of MiCK reaction rate, changing gradually from white (maximal rate, PCr synthesis) to dark gray (minimal rate, ATP synthesis). Concentrations in solution were kept at the following values: 2 mM PCr, 0.12 mM ATP, 0.05 mM ADP, and 5 mM Pi. According to Saks et al. (1985), the direction of MiCK reaction depends on whether MiCK is coupled to oxidative respiration or not (Saks et al., 1975). Namely, whereas coupling of MiCK to oxidative phosphorylation favors the synthesis of PCr by MiCK (positive reaction rates on the figure), noncoupled MiCK reaction is driven toward the breakdown of PCr (negative reaction rates on the figure). DATP and DADP are exchange constants that characterize the exchange of ATP and ADP between the compartment and the solution (both constants are given in s–1). Note that at smaller values of DATP and DADP, the MiCK reaction direction predicted by the model is in correspondence with the measurements (reaction rate is positive).

View larger version (33K): [in this window] [in a new window]   FIGURE 5  In this contour plot, the relative decrease in respiration rate after addition of the external ADP-trapping system into the solution containing isolated respiring mitochondria in the presence of oxidative phosphorylation is shown. The decrease v/v0, where v0 and v are the respiration rate before and after pyruvate kinase and phosphoenolpyruvate (PK+PEP) addition, is depicted by different shades and iso-lines. Numbers on the iso-lines show the value of v/v0, shades change gradually from white (v/v0 = 1, no drop in respiration) to dark gray (v/v0 = 0, respiration completely inhibited). Initial conditions in the solution: 0.33 mM ATP, 0 mM ADP and PCr, 33 mM Cr, 5 mM Mg2+, 10 mM Pi, and 2.5 mM PEP. In the calculations, the activity of PK+PEP system was taken to be infinite compared to that of MiCK. According to Gellerich and Saks (1982), PK+PEP system is able to decrease the rate of respiration by 60% at maximum, i.e., the decrease v/v0 is 0.4. DATP and DADP are exchange constants that characterize the exchange of ATP and ADP between the compartment and the solution (both constants are given in s–1). In these simulations, ATPase activity in the solution was taken to be equal to 0.1% of the maximal activity of MiCK. Varying ATPase activity did not alter the results qualitatively. Note that the region in DATP–DADP plane corresponding to the measured drop in respiration rate is rather limited (see area around v/v0 = 0.4 in the figure).

Direct transfer of ATP and ADP between creatine kinase and adenine nucleotide translocase
The model of direct transfer of substrates between MiCK and ANT contains two important modifications if compared with the model based on the dynamic compartmentation hypothesis. First, the concentrations of the metabolites near MiCK and ANT are the same as in solution; i.e., there is no diffusion restriction separating the compartment from the surrounding solution. Second, ATP and ADP can be transferred directly between MiCK and ANT, in addition to transfer through solution.

Kinetics of creatine kinase reaction coupled to oxidative phosphorylation
In the direct transfer model, ADP and ATP release by MiCK-ANT complex into the solution was reduced and a link between MiCK and ANT was established. For that, free energies of transition G corresponding to ATP (ADP) release to the solution and to direct transfer of ATP (ADP) between MiCK and ANT had to be specified. To investigate whether such changes would be sufficient for reproduction of the measured data on MiCK-ANT coupling, we varied the values of these G in a wide range independently from each other and computed the rate of MiCK reaction coupled to oxidative phosphorylation, with 4 mM phosphocreatine (PCr), 0.12 mM ATP, 0.05 mM ADP, and 40 mM creatine (Cr) in solution. According to Saks et al. (1985), MiCK reaction is directed toward PCr synthesis (positive direction) in these conditions. However, regardless of the G combinations used, the computed MiCK rate was always in the opposite direction. The computed MiCK reaction was positive when, in addition to the parameters mentioned above, at least one of the following parameters was varied: 1), the free energy of transition G of the MiCK reaction coupled with ANT; 2), the free energy of ANT directed toward the intermembrane space with ATP attached; or 3), the free energies of the MiCK states coupled with ANT. In the last case, the free energy of CK_c.ATP was varied and the free energies of all other coupled MiCK states with bound ATP or ADP (Fig. 1 B) were computed to keep differences between the free energies of MiCK states the same as for similar states in isolated MiCK.

Next, we computed the apparent kinetic constants of the MiCK reaction in the presence of oxidative phosphorylation and compared these with the measured data (Jacobus and Saks, 1982). The apparent kinetic constants were computed only for such combinations of parameter values as would reproduce the measured direction of the MiCK reaction as described above (i.e., that would satisfy the criteria in the following text and tables). To check whether direct transfer of metabolites between MiCK and ANT allows us to overcome the difficulties of modeling the MiCK reaction kinetics encountered using the dynamic compartmentation hypothesis (Fig. 3), we plotted all computed apparent dissociation constants K_a and K_ia against each other in a diagram (Fig. 6). As made clear from the figure, the region with the measured values of K_a and K_ia is covered by the model solution, and it is possible to find a combination of model parameters that would lead to the measured combination of K_a and K_ia values. The best fits obtained by the model during such independent variation (denoted by scan) for different varied parameter sets as well as the results of model fitting simulations (see below) are summarized in Table 2. Note that in some cases the computed kinetic constants were negative. This is due to the procedure used to calculate kinetic constants from the computed MiCK reaction rate (the same as in Jacobus and Saks, 1982), and clearly indicates that the model cannot reproduce the measured data with the corresponding set of parameters.

View larger version (16K): [in this window] [in a new window]   FIGURE 6  Calculated apparent dissociation constants Ka and Kia of the MiCK reaction in the presence of oxidative phosphorylation. Coupling between MiCK and oxidative phosphorylation was modeled assuming direct transfer of metabolites between ANT and MiCK. Apparent dissociation constants Ka and Kia (represented by small dots in the figure) are computed for different combinations of the free energies of the MiCK-ANT complex states and the free energies of transition. In the figure, the measured values are shown by open circles in the right upper corner (no oxidative phosphorylation) and in the left lower corner (with oxidative phosphorylation). Note that the range of computed Ka–Kia combinations covers the area near the measured values of these constants in the presence of oxidative phosphorylation. Experimental data from Jacobus and Saks (1982).

ADP flux between creatine kinase and adenine nucleotide translocase
We analyzed further those parameter combinations able to reproduce the MiCK reaction rate kinetics relatively well. For this analysis, in addition to the experimental data on the kinetics of the MiCK reaction coupled to oxidative phosphorylation, we analyzed ADP flux between MiCK and ANT. In our simulations, the model was used to reproduce the following experimental data:

1. Rate of MiCK reaction coupled to oxidative phosphorylation, estimated from apparent kinetic constants (Jacobus and Saks, 1982).
   
2. Inhibition of MiCK-activated mitochondrial respiration by the competitive ATP-regenerating system (Gellerich and Saks, 1982).
   
3. Studies of radioactively labeled adenine nucleotide uptake by mitochondria in presence of MiCK activity (Barbour et al., 1984).
   

The best fits, obtained using the same set of initial estimates as in the simulations OKm and OVel in Table 2, are presented in Table 3 and Figs. 7–9. These results are analyzed below.

View larger version (27K): [in this window] [in a new window]   FIGURE 7  Computed MiCK reaction rate as a function of ATP with (subplot A) and without (subplot B) oxidative phosphorylation. Coupling between MiCK and oxidative phosphorylation was modeled assuming direct transfer of metabolites between ANT and MiCK. (Subplot A) Experimental data (black dots) was reproduced by model with five different parameter sets. The parameter sets are indexed (see Table 3); correspondence between line-type and the index is shown in the legend. In this subplot, the MiCK reaction rate is shown for ATP titrations in the presence of three different combinations of Cr and PCr concentrations, which are indicated in the figure in mM. Note that all used parameter sets fit experimental data quite well, with parameter set 3 somewhat underestimating the reaction rate. (Subplot B) Computed MiCK rate with inhibited oxidative phosphorylation is in correspondence with the experimental data (black dots). The same parameter sets were used as in subplot A. Experimental MiCK rate used in both subplots was estimated on the basis of apparent MiCK reaction kinetic constants reported in Jacobus and Saks (1982).

View larger version (17K): [in this window] [in a new window]   FIGURE 8  Inhibition of respiration rate by a concurrent enzyme system, 60 IU/ml pyruvate kinase and phosphoenolpyruvate (PEP). The respiration was stimulated by the addition of 0.33 mM of ADP in the presence of 33 mM Cr. In the figure, the respiration rate after addition of PK+PEP (V) is normalized by the respiration rate obtained before PK+PEP addition (VADP), as shown in the y-axis label (V0 corresponds to the respiration rate without added ADP). The simulation results (bars) are compared with the measured drop in VO2, indicated by the average measured VO2 (dashed line). The parameter sets are indexed (Table 3). Note that results obtained with all parameter sets used are close to the measurements. Experimental data is from Gellerich and Saks (1982).

View larger version (14K): [in this window] [in a new window]   FIGURE 9  Competition between ADP supplied by MiCK reaction and ADP from solution. The amount of ADP retransported to mitochondria after MiCK reaction, without mixing with ADP in the intermembrane space, is shown at different levels of ADP concentration in surrounding solution. Simulation results obtained for the five parameter sets (see legend at upper right for parameter set index) are close to the measured data (black dots). Experimental data from Barbour et al. (1984).

According to our simulations (Fig. 8), the computed inhibition of oxidative phosphorylation by the competitive ATP-regenerating system (PK+PEP) was close to the results of measurements by Gellerich and Saks (1982). Additionally, the model was able to reproduce the competition between ADP supplied by the coupled MiCK reaction and ADP from solution estimated from radioactively labeled adenine nucleotide uptake by mitochondria (Fig. 9). Namely, the computed amount of adenine nucleotides that were retransported to mitochondria without mixing with surrounding solution is close to the estimations by Barbour et al. (1984). In sum, the model was able to reproduce an estimated competition between ADP sources and inhibition of respiration by pyruvate kinase with all combinations of varied parameters that reproduced kinetic measurements of the MiCK reaction (Jacobus and Saks, 1982; Saks et al., 1985).

Free energy profile of coupled MiCK-ANT reaction
The free energy profiles of the MiCK reaction estimated for the uncoupled case and that coupled to ANT are presented in Fig. 10, parts A and B, respectively. The following predictions can be drawn from our analysis of MiCK reaction coupled to ANT.

View larger version (29K): [in this window] [in a new window]   FIGURE 10  The free energy profile of the MiCK reaction in the absence (subplot A) or presence (subplot B) of oxidative phosphorylation. Note that the free energies of the states are shown relative to different initial states in these schemes: MiCK without any substrates bound and with MgATP2– and Cr in solution (uncoupled case); MiCK-ANT complex without any bound substrates, ANT directed toward the intermembrane space, and with Mg2+, Mg-free ATP4–, and Cr in solution (coupled case). Such difference in selection of initial states of reactions is due to the difference in the forms of ATP binding to MiCK and ANT: MiCK binds MgATP2– and ANT binds Mg-free ATP4–. (Subplot A) The relative free energies of MiCK states are estimated from measured kinetic constants. Here, MgATP and MgADP are denoted as T and D, respectively. Note that the free energy change during reaction is clear from the difference in free energies after (CK+D+PCr) and before (CK+T+Cr) the reaction. The differences in maximal rates of forward and backward MiCK reactions can be explained by the differences in the free energies of CK.T.Cr and CK.D.PCr (Eqs. 21–22). (Subplot B) Partial free energy profile of the MiCK reaction coupled to ANT. The free energies of the coupled system (boxes with solid border) are compared with the free energies of uncoupled MiCK and ANT (boxes with dashed border). In the first column, three states of two-protein complexes are shown with ATP (T in the scheme) attached to ANT directed to intermembrane space (Ni). The total free energy of the MiCK-ANT complex depends on the state of the coupled MiCK, as shown in the first column. ATP, attached to ANT, is either released to solution (second column) or transferred from ANT to MiCK (third column). After the MiCK reaction, ADP (D in the scheme) is transferred to ANT (last column) and then either released to solution (fourth column) or transported to mitochondrial matrix (not shown). This profile corresponds to the parameter set 4 (see Table 3). In the scheme, all reactions that are in the pathway leading to synthesis of PCr after the transfer of ATP from ANT to MiCK are shown by thick lines. Note that there are several simplifications made to keep the profile as simple as possible. First, the free energies changes indicated in the profile are induced by differences of the free energies of the complex states as well as changes in solution due to binding and release of the substrates and magnesium. However, in the scheme, only release and binding of ATP and ADP are indicated. Second, possible binding of ATP and ATP by ANT during the MiCK reaction is not indicated in this profile.

Second, in line with the changes of the free energy of the coupled MiCK-ANT complex with ATP^4– attached to ANT_i (left column in the scheme), the free energy of the complex with ADP^3– attached to ANT_i is slightly lower than in the uncoupled system (right column in the scheme). This is clear from inspecting the difference between states CK+D and CK-Ni.D in the scheme (compare the free energies for coupled and uncoupled cases in Fig. 10 B).

Thus, the free energies of MiCK-ANT complex before the transfer of ATP^4– to MiCK are considerably elevated and the free energies after transport of ADP^3– to ANT are slightly dropped. Due to such changes, the synthesis of PCr from ATP that is transferred from mitochondrial matrix by ANT becomes energetically advantageous. The net free energy change during the transfer of ATP^4– from ANT to MiCK, MiCK reaction, and the transfer of ADP^3– from MiCK back to ANT, is negative and ranges from –3.7 kJ mol^–1 to –20.7 kJ mol^–1, depending on the states of MiCK-ANT complex at the beginning and end of the coupled reaction along the main pathway (thick lines in the scheme). Note that if the free energies of MiCK-ANT states would be kept the same as in the uncoupled case, then the corresponding net free energy difference would be from –0.3 kJ mol^–1 to +16.7 kJ mol^–1 (see boxes with dashed borders in Fig. 10 B).

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
According to our analysis, the simplest kinetic scheme that can reproduce the experimental measurements on functional coupling between mitochondrial creatine kinase (MiCK) and adenine nucleotide translocase (ANT) involves the direct transfer of ATP and ADP between the proteins. The model composed on the basis of the dynamic compartmentation mechanism of functional coupling of MiCK and ANT was not sufficient to reproduce the measured values of apparent dissociation constants of the MiCK reaction (Fig. 3). However, when one assumes the direct transfer of ATP and ADP between MiCK and ANT, it is possible to reproduce the measured kinetic properties of MiCK (Fig. 7) and the estimated direct flux of ADP between MiCK and ANT (Figs. 8 and 9). Direct transfer of ATP and ADP between MiCK and ANT was analyzed by composing free energy profiles of the reactions using a thermodynamically consistent model. To our knowledge, it is the first time that this approach was used to study interactions between MiCK and ANT. Earlier, the analysis based on free energy profiles of reactions was successfully applied in the studies of several biological systems such as mitochondrial inner membrane carriers (Kramer, 1994) and actomyosin cross-bridges in skeletal and heart muscles (Cooke et al., 1994; Eisenberg et al., 1980; Hill, 1974; Vendelin et al., 2000a). Our analysis of the direct transfer mechanism revealed that:

1. The mere establishment of direct transfer between ANT and MiCK, as well as limitation of ATP and ADP release by the proteins, was not sufficient to reproduce the measurements.
   
2. The measurements can be reproduced if, at least, the free energy of the ANT state with the ANT binding site directed toward the intermembrane space with ATP attached (state ANT_i.ATP) is changed (Table 2).
   

The mechanism of functional interaction between MiCK and ANT, as suggested by our results, is in accord with the measurements of MiCK kinetics on intact mitochondria in isotonic KCl solution (Kuznetsov et al., 1989). Kuznetsov and co-workers (1989) showed that, after detachment of MiCK from the mitochondrial inner membrane, the influence of oxidative phosphorylation on MiCK reaction kinetics disappears—despite the presence of MiCK in the intermembrane space and the intactness of the mitochondrial outer membrane.

Dynamic compartmentation hypothesis
Gellerich et al. (1987) was able to reproduce, with a simple mathematical model, the inhibition of MiCK-activated mitochondrial respiration by the concurrent ATP-regenerating system. The authors assumed that the diffusion of ADP and ATP between the compartment and the surrounding solution is limited, and that this limitation is the same for both metabolites. This result was confirmed by our model: it is possible to find from such exchange coefficients for ADP and ATP that the exogenously added ATP-regenerating system is inhibiting mitochondrial respiration by 60% (Fig. 5). Additionally, the model based on the dynamic compartmentation hypothesis was able to reproduce reversal of the MiCK reaction after coupling to oxidative phosphorylation in certain conditions (Fig. 4), even if the exchange coefficients for ADP and ATP are the same as considered by Gellerich et al. (1987).

To check whether the dynamic compartmentation mechanism can reproduce changes in the apparent kinetic constants of the MiCK reaction when coupled to oxidative phosphorylation, we had to extend the original model of Gellerich et al. (1987) by adding more degrees of freedom to the model. Namely, we added ATPase activity into the surrounding solution as well as considering ADP and ATP exchange coefficients independent from each other. These changes were introduced into the model to create large concentration differences between the compartment and the surrounding solution (see Results, Eq. 29). Without these changes, assuming that ADP and ATP exchange coefficients are, for example, the same, neither of the computed apparent coefficients K_a and K_ia were reduced by >1% from the values measured in the uncoupled case, regardless of the exchange coefficients (D_ATP = D_ADP) used, the ATPase activities in the solution, and the ANT activity used (results not shown). Thus, these extensions of the original model of Gellerich et al. (1987) were the required ones if the reduction of apparent kinetic constants in the model of the coupled MiCK-ANT interaction is desired (Fig. 2). However, even with these extensions, it is impossible to fit the measured dissociation constants K_a and K_ia with the model simultaneously (Fig. 3). Thus, the mechanism of dynamic compartmentation is not sufficient to reproduce the measured changes in the apparent kinetic constants of the MiCK reaction coupled to oxidative phosphorylation in isolated heart mitochondria (Jacobus and Saks, 1982; Kuznetsov et al. (1989) and in inner membrane-matrix preparation (Saks et al., 1985).

Direct transfer hypothesis
When the direct transfer between MiCK and ANT is assumed as a basic mechanism of the coupling, it is possible to reproduce the measured kinetic properties of MiCK as well as the estimated direct flux of ADP between MiCK and ANT. Our model was able to reproduce the measurements only if, in addition to the limitation of ATP and ADP release from the MiCK-ANT complex, the free energy of the ANT state with the ANT binding site directed toward the intermembrane space with ATP attached (state ANT_i.ATP) was changed (Table 2). Thus, the mere establishment of direct transfer between ANT and MiCK, as well as limitation of ATP and ADP release by the proteins, was not sufficient to reproduce the measurements, and the free energy profile of MiCK-ANT interaction had to be modified further. The elevation of the free energy of the ANT_i.ATP state proposed by the model is the main result obtained from the analysis of the free energy profile of the coupled MiCK reaction. As it is clear from the free energy profile (Fig. 10 B), the MiCK reaction is driven toward synthesis of PCr by altering the free energy of ANT_i.ATP. Note that such increase of the free energy of intermediate state ANT_i.ATP does not inhibit transport of ATP^4– from the matrix to the inner membrane space by ANT due to the large differences in the free energies of ATP^4– in the intermembrane space and the mitochondrial matrix induced by mitochondrial potential (Klingenberg, 1985). Indeed, we assumed in our model that during a transport of ATP^4– from matrix to inner membrane space a net negative charge is transported, close to the estimation by Gropp et al. (1999). With the membrane potential taken equal to –180 mV, the membrane potential contributes 17 kJ mol^–1 of the free energy change during a transport of ATP^4– from the matrix solution to the inner membrane space solution. In the model, this was accounted for by increasing the free energy of ATP^4– in the matrix as well as the free energy of the ANT_x.ATP state by the contribution of membrane potential. Thus, even after increase of the free energy of the intermediate state of ATP^4– transport (ANT_i.ATP) by 13.5 kJ mol^–1, the transition from ANT_x.ATP to ANT_i.ATP was still energetically advantageous, and ATP^4– transport was not greatly inhibited in the model.

Aliev and Saks (1993