MATHEMATICAL MODELING OF ENERGY COUPLING IN MITOCHONDRIA IN FRAMES OF PROTONCHEMICAL HYPOTHESIS.
V.V.Lemeshko^{#},
A.G.Anishkin^{*}.^{
*}
Biology Research Institute of Kharkov State University, 310077,
Kharkov, Svobodi sq., 4, Ukraine.^{
#} National University of Colombia, Medellin, Colombia.
The dependence of ATPsynthesis and mitochondria respiration rates on membrane potential is analyzed in frames of protonchemical hypothesis of oxidative phosphorylation. The mathematical and electrical models of the unit of energy coupling are proposed. Computer analysis of the mathematical model allowed to obtain the theoretical dependences of the rates of ATP synthesis and of the respiration on membrane potential that are in good qualitative agreement with experimental data, contradicting chemiosmotic hypothesis.
Mitchell's chemiosmotic hypothesis [1] is commonly accepted for the description of the mechanism of energy coupling in biomembranes [2]. However, by present time the significant number of experimental data was accumulated that do not agree or obviously run counter to this hypothesis [3  9] and have caused the series of its modifications [10, 11, 12]. The chemical hypothesis of oxidative phosphorylation [4] was also improved in such a manner that though the availability of Mitchell's protonmotive force (Dm_{H}+) on coupling membrane was admitted, its minor role was ascribed.
Among main facts, which are difficult to explain in frames of chemiosmotic hypothesis, it is worth notice, in particular, the following:
1. Rate of oxidative phosphorylation as the function of membrane potential depends on the way of its change [3, 4, and 6].
2. Degree of mitochondrial respiration activation depends on way of reduction of protonmotive force magnitude on the same value [3, 9].
According to the chemiosmotic hypothesis, both the rate of oxidative phosphorylation and the intensity of respiration depend on the Dm_{H}+ magnitude, but not on the way of its change.
The satisfactory explanation of the above mentioned and number of other data, describing oxidative phosphorylation system, can be derived from the protonchemical hypothesis [13  16]. The latter includes in a consistent way both chemical hypothesis and the elements of chemiosmotic hypothesis.
The purpose of the present work was the development of a mathematical model for quantitative description of interrelation between the rate of ATP synthesis and magnitude of Dm_{H}+, that can be changed by uncoupler, by oxidation substrate or by inhibitor of respiration.
ELECTRICAL AND MATHEMATICAL MODELS OF PROTONCHEMICAL COUPLING
The mathematical model is based on equivalent electrical circuit of one protonchemical coupling unit in respiratory chain of mitochondria [16]. The simulation of energy coupling by means of equivalent electrical circuits was numerously used by different authors as the method of qualitative and quantitative analysis of oxidative phosphorylation hypotheses [11, 12, 16]. For simplification of description of one coupling unit equivalent electrical circuit, we emphasize two main theses, following from protonchemical hypotheses [1316]:
1. Electron flux in the coupling unit divided in two and can pass not only by Mitchell's way, when the energy of difference of redox potentials is spent on transmembrane carrying of protons, but also by way of ATP chemosynthesis.
2. In mitochondria, the act of synthesis of one ATP molecule requires the transmembrane transfer of proton through ATPsynthase in matrix, and also transfer through ATPsynthase of two atoms of hydrogen (two electrons with two protons) from donor to acceptor of electrons in respiratory chain. These two processes are coupled in such a manner that the mechanism of ATP synthesis has the chemical nature, but also includes use of Dm_{H}+ energy. Besides, the energy of transmembrane carrying of one more proton is spent (in account per one molecule of ATP synthesized) on transport of phosphorylation substrates through mitochondrial membrane.
Thus, synthesis of one ATP molecule requires the energy of transmembrane carrying of two protons and energy of carrying of two atoms of hydrogen between redoxcenters. Therefore, the summation of energy takes place, and this defines the magnitude of phosphate potential DG_{p}, which can be created in the system:
,
Where F  Faraday constant, Dm_{H}+  membrane electrochemical potential of hydrogen ions (Mitchell's protonmotive force), DE  difference of redox potentials in one coupling unit.
Let us consider mitochondrial protonchemical unit of coupling in detail (fig. 1). In the scheme, the unit of coupling is divided on two subunits  "a" and "b", according to [16]. In subunit "a", the transfer of electrons is coupled only with pumping out of protons from mitochondria. But in subunit "b", the bifurcation of electron flux takes place: on Mitchell's flux, coupled with pumping out of protons, and on chemical, coupled with ATP synthesis and transfer of two protons in matrix. The simple account gives the stoichiometry 4H^{+}/2e^{} per one coupling unit (subunit "a" plus subunit "b"), when there is no synthesis of ATP, and 1ATP/2e^{} per one coupling unit at completely coupled synthesis of ATP.
Fig. 1. General scheme of coupling unit. _{Symbols: } _{D,B,A}_{  redoxcenters of respiratory chain; } _{  electron flux in respiratory chain; } _{ electron flux by the chemical way; } _{  transmembrane flux of protons;} _{  ATP synthesis flux;} _{  fluxes coupling.}

Therefore, the synthesis of one ATP molecule in the coupling unit consumes energy, released at the transfer of two electrons by chemical way, and energy of transmembrane transfer of two protons.
Phosphitethyole mechanism of oxidative phosphorylation was previously offered as one of possible variants of protonchemical coupling [13, 14, 16]. From this mechanism follows, that the rate of ATP synthesis is exponentially reduced at the reduction of membrane potential and difference of redox potentials between appropriate donor and acceptor of electrons in coupling unit [16]:
Where and  concentration of phosphite and phosphate, respectively.
Certainly, total reduction of phosphate to phosphite is considerably energetically unprofitable reaction [18] in view of very low redoxpotential of this redoxpair [16], that makes the probability of participation of phosphite in mechanism of transformation of energy rather small, but does not exclude completely. At the same time, just based of hypothetical phosphite mechanism the general principle of protonchemical coupling [16] is the most simply entered. Therefore, in analogy with formula (1), the exponential nature of ATP synthesis rate dependence on Dm_{H}+ and DE_{BA} is also the result of any other redox mechanism of ATP synthesis in protonchemical coupling unit, including oneelectron and freeradical mechanisms [15].
In addition, "radical hypothesis" was offered later by other authors [19]. According to this hypothesis, the role of highenergy intermediate can be performed by cardiolipin, which is converted to cardiolipin enolphosphate in course of the redox reaction.
In the most general kind in frames of protonchemical principle of coupling the distribution of electron flow between chemiosmotic and chemical branches of subunit "b", that determinates the rate of ATP synthesis, it is possible to describe as follows: when the coupling membrane have been maximal charged on chemiosmotic way, current through membrane (that is the sort of condenser) stops to proceed, and difference of redoxpotentials DE_{BA} in subunit "b" becomes also maximal one. All this is the reason of activation of electron transfer on chemical way. Thus, the mechanism of switching is thermodynamic one. In frames of this general mechanism some particular molecular mechanism of switching can exists, for example one of the mechanisms of regulatory effect of transmembrane electrical potential on functional activity of biomembranes, described by Konev and Kaler [17].
The equivalent electrical circuit (fig. 2), corresponding to one unit of protonchemical coupling (in accordance to [16] (fig. 1)), can be presented as five circuits (fig. 2). The flux of electrons, in the form of a hydrogen atoms flux, from donor D to acceptor B (see fig. 1), is caused by appropriate difference of redox potentials (DE_{BD}), which is the driving force of the oxidative phosphorylation process as a whole. On the equivalent electrical circuit (fig. 2) current in circuit I corresponds to the electron flux through the whole coupling unit, and the source of voltage E_{o}, with internal resistance r_{o}, reflects the stationary difference of redox potentials DE_{BD} between D and B. The stationary difference of redox potentials (DE_{AD}), is established in subunit "a" and is reflected in circuit I as the source of voltage e_{a}.
Fig. 2. Equivalent electric circuit of general coupling point. See commentary in the text. 
Appropriate difference of redox potentials in subunit "b" (DE_{BA}) is reflected as the source of voltage e_{b}. The value of current J in circuit I is equal to the rate of the transfer of electrons through unit of coupling, and this value is determined by ratio between E_{o}, e_{a} and e_{b}:
(2). 
The transfer of one electron from donor D to acceptor A in subunit "a", is coupled with transmembrane carrying of one H^{+} and generation of hydrogen ions membrane electrochemical potential. The latter on the electrical circuit (fig. 2) corresponds to the source of voltage e_{m} in circuit II, in which the resistance R_{ah} of H^{+}  pump is also included. The current, flowing in circuit II under action of voltage (e_{a}  e_{m}), is equal to the rate of transport of protons by H^{+} pump of subunit “a”, and also is equal to current J, flowing through whole unit of coupling:
(3) 
The current through subunit "b" is determined as the sum of currents, flowing by chemiosmotic way and in parallel by way of ATP chemosynthesis (fig. 1). In circuit III of electrical circuit (fig. 2) it is shown, that the current J_{1} on right, chemiosmotic branch (circuit III.1) and current J_{2} on left, chemical branch (circuit III.2) flow under the action of voltage e_{b}. The sum of currents J_{1} and J_{2} through subunit "b" is equal to general current J through whole unit of coupling, i.e., to the current in circuit I:
(4) 
The current J_{1} is coupled with active transport of protons by H^{+ } pump of subunit "b" and is determined by voltage (e_{b}  e_{m}) and resistance of pump R_{bh}. According to the method of mesh currents, it is possible to write down:
(5) 
According to considered twoelectron variant of protonchemical coupling, synthesis of one ATP molecule is coupled with transfer of two electrons and, simultaneously, with transfer of two protons by gradient of their membrane electrochemical potential. So, value of current J_{2} in circuit III.2 is determined by sum of voltages (e_{b} +e_{m}) and by magnitude of ATPsynthase resistance, R_{A}. Beside, ATP chemosynthesis rate and magnitude of current J_{2} depend on magnitude of phosphate potential e_{p}. This dependence can be reflected by introduction in circuit III.2 of additional source of voltage, e_{p}/2, directed opposite to e_{b} and e_{m}. The divider 2 at e_{p} was added due to stoichiometry 1ATP / 2e^{} per one unit of protonchemical coupling. In this case, the phosphate potential in bulk phase out of mitochondrion was taken as phosphate potential magnitude.
Using the method of mesh currents for circuit III.2, it is possible to write down:
(6) 
On the other hand, the value of current J_{2} in circuit III.2 is equal to 2 of ATP chemosynthesis rate and at the same time,  to the rate of influx in matrix of protons coupled with ATP synthesis.
According to previously published work [16] simple accounts, in case of hypothetical phosphitethyole mechanism of protonchemical coupling, rate of ATP synthesis exponentially decreases at Dm_{H}+ and DE_{BA} decrease [16], as it is visible from formula (1). In electrical circuit (fig. 2, circuit III) this exponential dependence is reflected by nonlinear nature of ATPsynthase resistance R_{A}, which depends on e_{b} and e_{m} in the following way:
(7) 
Where R_{A0}  normalizing multiplier.
The magnitude of Dm_{H}^{+}, which is presented in electron equivalent circuits as the source of voltage e_{m}, depends on the ratio between rate of pumping out of protons by H^{+}pumps and proton conductivity of coupling membrane. The circuit IV (fig. 2) reflects that fact, that the rate of pumping out of protons is equal to sum of rates of pumping out through the H^{+} pump of subunit “a” (current J in circuit II, fig. 2) and subunit "b" (current J_{1} in circuit III.1, fig. 2), i.e., J +J_{1}. The proton conductivity, in general case, is conditioned by following processes (see the circuit IV, fig. 2):
The resulting current of protons, coupled with ATP syntheses and phosphorylation substrates transport, is numerically equal to appropriate electron flux J_{2}, according to twoelectron variant of protonchemical coupling. Thus, it is possible to write down, that the summary current, flowing in circuit IV through resistances R_{U} and R_{L}, is equal (J +J_{1}J_{2}) or, in view of equation (3), is equal 2J_{1}. On the other hand, it is possible to express summary current, flowing through resistance R_{U} and R_{L}, through e_{m}:
(8) 
The magnitude of phosphate potential, simulated in scheme by source of voltage e_{p}, is determined by ratio between rate of ATP synthesis in coupling unit and rate of ATP hydrolysis in endergonic processes. Let us assume, that all these endergonic processes proceed in extramitochondrial space, where the magnitude of phosphate potential in considered case is determined. The processes of ATP hydrolysis are modeled in electrical scheme by resistance R_{g} in circuit V. Then in stationary conditions the rate of ATP hydrolysis is equal to the rate of ATP synthesis, i.e. J_{2}/2, and can be expressed as the ratio of phosphate potential magnitude to value of resistance R_{g} :
(9) 
Thereby, we have the system of eight nonidentical equations (1) (8), in which seven unknown variables (J, J_{1}, J_{2}, e_{a}, e_{b}, e_{m}, e_{p} and R_{A}) are present. We transform this system of equations to more convenient kind:
(10) 

(11)  
(12)  
(13)  
(14)  
(15)  
(16)  
(17) 
Where 
The resistances r_{o}, R_{ah}, R_{bh}, R_{U}, R_{L}, R_{g} and R_{A0} reflect the separate parameters of simulated system of energy coupling in real conditions of experiment. Therefore, inhibition of respiration by malonate at succinate oxidation by mitochondria is possible to be modeled by varying of internal resistance r_{o} of voltage source E_{o}. The addition of uncoupler in incubation media of mitochondria is reflected by varying of resistance R_{U} in definite limits.
The activation of ATP synthesis at decrease of phosphate potential (for instance, owing to addition of ADP or hexokinase in incubation media of mitochondria) is modeled by reduction of resistance R_{g}.
THE CHOOSING OF THE PARAMETERS OF PROTONCHEMICAL COUPLING UNIT MODEL
Let us analyze the rates of ATP synthesis and respiration in one protonchemical coupling unit in dependence on Dm_{H}^{+} magnitude, which is changed by two different ways. To set the parameters of system, we take the voltage E_{o} equal 400 mV, i.e., approximately 1/3 general differences of redoxpotentials between oxygen and hydrogen electrodes.
The magnitudes of resistances were chosen taking into account following criteria: range of respiratory control value, maximal degree of activation of respiration by uncoupler, low proton conductivity of coupling membrane and other, including the empirical selection with subsequent computer analysis of model behavior. Resistances chosen below are normalized on one coupling unit and have not absolute, but relative nature, though they are expressed in ohm.
We accept the resistances r_{o}, R_{ah} and R_{bh} equal 1 ohm. The resistance of natural outflow of protons through membrane, R_{L}, we take equal 100 ohm, i.e., considerably more than resistances of proton pumps. Let's take the ATPsynthase resistance normalizing multiplier R_{A0} equal 2*10^{11} ohm (in conditions of ATP synthesis the real resistance R_{A} thus will take order of 10 ohm). We take initial resistance of outflow through uncoupler R_{U} equal 10000 ohm, that approximately can be considered as infinitely large resistance R_{U} in absence of uncoupler. 10000 ohm was taken as initial value of resistance R_{g} that simulates the respiration state 4 (by Chance).
THE COMPUTER ANALYSIS OF THE CHOSEN MODEL OF PROTONCHEMICAL COUPLING
The system of equations (10  17) cannot be resolved in obvious kind because of existence of nonlinear equation (17), in which variable R_{A} depends on its own magnitude, entering in exponent expression. So the solution was found using computer by method of consecutive approximations on variable R_{A} at the above chosen values of assigned parameters of equivalent electrical circuit. At these conditions, the calculation of necessary parameters  rate of ATP syntheses and rate of mitochondria respiration in dependence on membrane potential  was conducted for two ways of Dm_{H}^{+} change.
The dependence of ATP synthesis rate on Dm_{H}^{+} changed by uncoupler addition was investigated by varying of resistance R_{U} from initial chosen value 10000 ohm to 1 ohm. The lower level of resistance (1 ohm) is chosen two order less than resistance R_{L} of natural proton outflow through coupling membrane. Resistance R_{g} was taken 1 ohm, that simulates D G_{p} = 0.
The same dependence on Dm_{H}^{+} changed by addition of competitive inhibitor of substrate SH_{2} oxidation, was investigated by varying of internal resistance r_{o} in range from 1 ohm (initial value) to 1000 ohm.
Dependence of ATP synthesis rate on Dm_{H}^{+} magnitude for the two considered cases received by computer solution is presented on fig. 3. It is obvious, that the calculated curves exponentially decrease at reduction of membrane potential and are divergent, i.e., the same magnitude Dm_{H}^{+} corresponds to the different magnitudes of ATP synthesis rate. The received data of the computer experiment are on principle agreed with experimental data for mitochondria [6].
Fig. 3. Theoretical ATP synthesis
rate (Jp) dependence on membrane potential. Symbols: 1  titration by the uncoupler; 2 – titration by respiratory inhibitor 
Fig. 4. Theoretical
respiration rate (Jr) dependence on membrane potential. Symbols: 1  titration by the uncoupler; 2 – titration by hexokinase 
It is impossible to explain in the frames of chemiosmotic hypothesis the abrupt fall of ATP synthesis rate at insignificant decrease of membrane potential. Even if we admit, that it is consequence of specific kinetic characteristics of ATPsynthase, any case, it isn't possible in frames of chemiosmotic hypothesis to explain the existence of two various magnitudes of ATP synthesis rate at the same magnitude of membrane potential, changed by different ways. However, this effect mathematically strictly follows from the protonchemical model of energy coupling.
At modeling of dependence of mitochondria respiration rate on magnitude of membrane potential, the change of Dm_{H}^{+} was considered at titration by uncouplerprotonophore, or at titration by hexokinase. Titration by uncoupler, as previously, was modeled by varying of resistance R_{U} from initial value 10000 ohm to 1 ohm. Titration by hexokinase was modeled by varying of ATP hydrolysis resistance R_{g} from initial value 10000 ohm to 1 ohm.
As the result of modeling two divergent curves were received (fig. 4), that is agreed qualitatively with experimental data [3, 9], which run counter to Mitchell's hypothesis.
The mathematical and electrical models of protonchemical coupling, considered in present work, permits to receive number of interrelations between other thermodynamic parameters of oxidative phosphorylation system and to explain from single position the large totality of experimental facts, including those, which are in contradiction to Mitchell's hypothesis. The distinctive property of model is the opportunity to receive by calculation and coordination with experimental data the series of such characteristics of energy coupling system, which defy direct experimental measurement.