# The Essence of ATP Coupling

Authors: Nikolai Bazhin
Journal: ISRN Biochemistry
 (2012)

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## Abstract

The traditional explanation of ATP coupling is based on the raising of the equilibrium constants of the biochemical reactions. But in the frames of the detailed balance, no coupling occurs under thermodynamic equilibrium. The role of ATP in coupling is not that it provides an increase in the equilibrium constants of thermodynamically unfavorable reactions but that the unfavorable reactions are replaced by other reactions which kinetically are more favorable and give rise to the same products. The coupling with ATP hydrolysis results in the formation of quasistationary intermediate states.


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## Body

## 1. Introduction

The coupling to ATP hydrolysis is known to be favorable for various biochemical reactions [1]. It is usually explained in terms of equilibrium theory the principle argument of which is a substantial decrease in Gibbs function in the reaction of ATP hydrolysis [1]. No doubt that the latter is a necessary condition with, however, unavailable details of the mechanism. The present work is aimed to demonstrate that the process of coupling occurs in nonequilibrium conditions and that the reaction of ATP hydrolysis gives rise to the products of biochemical reaction of superequilibrium concentrations at the initial stages of the process. Under equilibrium conditions, no coupling is observed despite the presence of ATP.It is well known that in real systems, ATP concentration is kept almost constant by means of special synthetic systems. Therefore, it is rather difficult to perceive a detailed mechanism of ATP action in such systems. The present paper considers not only the real systems, but also the thermodynamic and the kinetic behavior of the systems prepared initially in the nonequilibrium state. The processes in these systems occur prior to thermodynamic equilibrium.The reaction of ATP hydrolysis, which results in a decrease in a standard value of Gibbs function, is often used as the exergonic reaction, participating in the processes of coupling [1, 2](1)ATP=ADP+Pi,ΔrG1′o<0,
where ΔrG1′o=-36.03 kJ/mol [3] for the standard state: pH = 7, I=0.25, and t=25°C. Under these conditions, the equilibrium constant is(2)K1′=[ADP]eq⋅[Pi]eq[ATP]eq=2⋅106≫1.It is assumed (see, e.g., [1, 2]) that if the equilibrium of the reaction (uncoupled system)
(3)A+B=AB,ΔrG3′o>0,
with
(4)K3′=[AB]eq[A]eq⋅[B]eq<1
is shifted towards reagents, its coupling with ATP hydrolysis provides displacement towards product AB. The summary reaction
(5)A+B+ATP=ADP+AB+Pi,ΔrG5′o=ΔrG1′o+ΔrG3′o<0
is taken as the simplest variant of coupling. In this case, the equilibrium constant of reaction (5)
(6)K5′=[ADP]eq⋅[Pi]eq[ATP]eq⋅[AB]eq[A]eq⋅[B]eq=K1′⋅K3′=exp⁡(-ΔrG5′oRT)>1
exceeds unity. It is concluded then [1, 2] that the reaction of ATP hydrolysis provides the energetically unfavorable conversion of reagents A and B into AB.However, in the equilibrium system, due to the principle of detailed balance, the concentration of the AB product is in equilibrium with the concentrations of reagents A and B and, thus, is as small as in the absence of ATP. In reaction (3), there is always equilibrium in the equilibrium system independent of other substances. The equilibrium constant of reaction (3) remains unchanged, because its value is determined by the structure of reagents and products and is calculated via a standard change in the Gibbs function. Thus, a thermodynamic coupling of reactions (1) and (3) is impossible in equilibrium conditions.The absence of coupling in equilibrium conditions can be assigned to the absence of the real chemical coupling between reactions (1) and (3) and the independence of both of the reactions. To provide the chemical coupling, intermediate APi (Scheme 1) is usually introduced [4].Scheme 1.
Consider the following:(7)ATP→K7ADP+Pi,(8)A+ATP⇆K8+K8-ADP+Pi,(9)APi+B⇆K9+K9-AB+Pi.
This, however, has no effect on the situation in the equilibrium system. According to the principle of detailed balance, all the feasible processes of the equilibrium system are in equilibrium. Thus, reactions (1) and (3), as well as (8) and (9), are in equilibrium. Due to the presence of the APi product, the amount of the AB product will be slightly less than that in the system with two reactions (1) and (3). It is impossible then to account for coupling in the framework of the equilibrium approach.
Nevertheless, the coupling does exist.In the real systems, the concentration of the AB product may substantially exceed the equilibrium concentration due to the coupling. What are the reasons for superequilibrium concentrations? One such reason could be the appearance of an additional local minimum of the Gibbs function because of the coupling with ATP hydrolysis. However, an ideal system has only one equilibrium state [5]. (The ideal system is the system in which a chemical potential of each substance is of the form μ=μo+RTlnC, where μo is the standard value of the chemical potential, and C is the concentration.) Therefore, there is no new equilibrium state which leads to superequilibrium concentrations. Since a thermodynamic system gradually tends to occupy a global minimum, the appearance of superequilibrium concentrations can be assigned to the appearance of quasiequilibrium states that slowly evolve towards the global minimum. We mention the quasiequilibrium state for the following reasons. The matter is that in biochemical systems, most of the chemical reactions proceed in the presence of enzymes. It is assumed then that in their absence, the coupling could be hardly observable despite the fact that enzymes have no effect on the thermodynamic parameters of the system. Enzymes affect only the reaction rate. In this case, this action equally concerns both the direct and inverse reactions. Thus, owing to enzymes, certain reactions are chosen from the variety of feasible reactions. The direct and inverse reactions, accelerated by enzymes, run faster than that of ATP hydrolysis and may be considered quasiequilibrium. During the process, the Gibbs function should decrease regularly despite the appearance of the superequilibrium AB concentrations.The goal of the present work is to demonstrate that the ATP coupling effect requires fairly fast reactions of the formation of intermediates resulting in the quasistationary state.


## 2. Theoretical Description

Consider now the kinetic behavior of the system in terms of Scheme1. In the system, containing ATP, ADP, Pi, A, B, AB, and Pi, only three linearly independent reactions occur, and for convenience, we choose reactions (7), (8), and (9). It is assumed then that the rates of direct and inverse reactions (8) and (9) exceed much the rate of reaction (7). Reactions (8) and (9) proceed in a quasiequilibrium manner. Hence, we get
(10)k8+[A]real⋅[ATP]real≈k8-[APi]real⋅[ADP]real,k9+[APi]real⋅[B]real≈k9-[AB]real⋅[Pi]real.
These equations provide expressions for equilibrium constants under quasiequilibrium conditions
(11)K8,real′=k8+k8-=[APi]real⋅[ADP]real[A]real⋅[ATP]real,K9,real′=k9+k9-=[AB]real⋅[Pi]real[APi]real⋅[B]real,K8,real′⋅K9,real′≈K1′⋅K3′.In the quasiequilibrium state, we get(12)K1′⋅K3′≈K8,real′⋅K9,real′=[ADP]real⋅[Pi]real[ATP]real⋅[AB]real[A]real⋅[B]real,
which gives
(13)[AB]real[A]real⋅[B]real≈K1′⋅K3′⋅[ATP]real[ADP]real⋅[Pi]real.
Equation (13) verifies the existence of coupling, because in real conditions [6] [ATP]real/([ADP]real·[Pi]real)≥1, the value K1′·K3′>1, and hence, [AB]real/([A]real·[B]real)>1 as compared with the equilibrium ratio [AB]eq/([A]eq·[B]eq)<1. Thus, the appearance of quasistationary states results in the formation of intermediates in superequilibrium concentrations. The appearance of the superequilibrium concentrations does not contradict with the thermodynamics as the increasing in Gibbs function due to the superequilibrium concentrations is compensated by decreasing due to reaction of ATP hydrolysis.Thus, measuring reagent concentrations in the quasistationary conditions, the authors [4] calculated the K1′·K3′ product and, using one of the constants, calculated the second equilibrium constant.The approximate equation for a change in ATP concentration with time(14)d[ATP]dt≈-k8+[A][ATP]+k8-[APi][ADP]
shows that at the high A concentration, the time, at which the quasistationary state is reached, is estimated from the formula
(15)τst≈1(k8+[A]real).
It is worth noting that the K8,real′·K9,real′ product is calculated not only at times close to τst, but at longer times as well by realizing the quasistationary conditions.Thus, the coupling is reduced to the substitution of reactions (1) and (3) by reactions (7), (8), and (9) which gives rise to the quasiequilibrium state with the formation of the necessary products. This is determined by a favorable change in the Gibbs function upon ATP hydrolysis and the high rates of reactions (8) and (9) as compared with that of ATP hydrolysis. However, in the course of time, the system tends to true equilibrium, at which the concentrations of the products required are very low. This process is also driven by a favorable change in the Gibbs function in the reaction of ATP hydrolysis, as the equilibrium in reaction (7) must take the place.It is interesting to illustrate the aforementioned experimentally.


## 3. Experimental Examples of Quasiequilibrium State Formation

### 3.1. Acetyl-CoA Formation

Consider now the production of Acetyl-CoA [4] as an experimental example of reaction kinetics in which the role of ATP is of essence. The reaction of Acetyl-CoA formation from acetate and CoA is impossible due to a small equilibrium constant
(16)acetate+CoA⟶acetyl-CoA+H2O,ΔrG16′o=+33.21kJ/mol.
However, in the presence of ATP, acetate kinase, and phosphate acetyltrassferase, the reaction
(17)acetate+CoA+ATP⟶acetyl-CoA+ADP+Pi,ΔrG17′o=-2.82kJ/mol
follows the mechanism shown by Scheme 2.Scheme 2.
Consider  the following:(18)ATP→K18ADP+Pi,ΔrG18′o=-36.03kJ/mol,(19)ATP+Acetate⇆K19+K19-APi+ADP,ΔrG19′o=+8.61kJ/molK9+,(20)APi+CoA⇆K20+K20-ACoA+Pi,ΔrG20′o=-11.43kJ/mol.Notation 1.
APi = acetyl phosphate and ACoA = acetyl-CoA. TheΔrG′o values were calculated according to [3]. Let us consider results from the second experiment [4, Table  2]. In the paper [4], the data are presented on the concentrations of reaction participants for times 15, 30, and 45 min. The initial concentrations and those at t=1800 s are listed in Table 1.Table 1
Initial concentrations, concentrations att=1800 s, and equilibrium concentrations, mM for the second experiment of Table 2 [4].
N1234567SubstanceAcetyl-phosphateATPADPAcetylCoACoAAcetatePhosphateInitial concentrations1.00.011.000.3864.9547.86Concentrations att=1800 s0.22 (calculated)0.6070.3620.1100.26865.5647.97Equilibrium concentrations att→∞, calculated this work48.8·10-924.17·10-91.0138.48·10-90.3865.9548.87Using the kinetic data on the second experiment [4, Table  2], we have chosen the values of rate constants (Table 2) and the calculated kinetic curves for all reaction participants by the standard Runge-Kutta method as described in the paper [7]. Figure 1 shows the kinetic curves for a short time interval of 0–5000 s, and Figure 2 shows for a longer one of 0–100000 s. The experimental data are well described by theoretical curves.Table 2
Reaction rate constants and reaction rates at 1800 s.
Constantk18 s−1k19+, (mM·s)−1k19-, (mM·s)−1k20+, (mM·s)−1k20-, (mM·s)−1Constant value0.000030.0000060.00350.0030.00004Reactions rate att=1800 s, mM/s1.6·10-52.3·10-42.8·10-41.8·10-42.0·10-4Figure 1
Kinetic curves over small time domain for the second experiment (Table2 [4]). Experimental data are denoted by the dots.Figure 2
Kinetic curves at long times for the second experiment (Table2 [4]).From the data of Table1 and Figure 1, the authors [4] concluded that they had reached the equilibrium state of the system. This statement, however, is erroneous for the following reasons. The authors describe the equilibrium state of the system, consisting of seven substances, that is, acetyl-phosphate, ATP, ADP, acetylCoA, CoA, acetate, and phosphate using two linearly independent reactions (19) and (20) of Scheme 2. However, a thermodynamically correct description of this system must include three linearly independent reactions, for example, those present in Scheme 2. The composition of the equilibrium system is easy to calculate from the equations of thermodynamic equilibrium
(21)K18′=(n30+ξ18+ξ19)⋅(n70+ξ18+ξ20)n20-ξ18-ξ19=2⋅106,K19′=(n10+ξ19-ξ20)⋅(n30+ξ18+ξ19)(n20-ξ18-ξ19)⋅(n60-ξ19)=0.031,K20′=(n40+ξ20)⋅(n70+ξ18+ξ20)(n10+ξ19-ξ20)⋅(n50-ξ20)=100.6.
In these reactions, the equilibrium constants K18′,K19′, and K20′ correspond to reactions (18), (19), and (20) of Scheme 2. For simplicity, we consider the system of volume 1 L. The ni0 value describes the number of moles of the i-th substance per one liter at the initial stage in moles. The values ξ18, ξ19, and ξ20 describe the chemical extents of reactions in Scheme 2. The chemical extents are given in moles [8] and introduced as follows:
(22)ξ=(ni-ni0)νi,
where ni is the amount of the ith substance in the system in any stage of the process, and νi is a stoichiometric coefficient of the reaction for the ith substance. Hence,
(23)ni=ni0+νiξ.
It is valid for the only reaction in the system. If there are several linearly independent reactions in the system, then
(24)ni=ni0+∑jνijξj,
where j is the number of the reactions, νij is the stoichiometric coefficient for the ith substance of the jth reaction, and ξj is the chemical extent of the jth reaction. Usually, in the initial stage of the process, it is convenient to assume that the ξj values are zero. The equilibrium constant of the j-th reaction is of the form
(25)Kj=∏iCjνij=∏i[(ni0+∑jνijξj)V]νij,
where Cj is the equilibrium concentration, and V is the system volume. The νij values are positive for products and negative for reagents. When the system volume is 1 L, the equilibrium constant may be given in a simpler form
(26)Kj=∏i(ni0+∑jνijξj)νij.
We use chemical extents because the method of chemical extents is a simple and powerful means of describing the equilibrium chemical systems as compared with the method of concentrations. This method is particularly suitable for complex chemical systems, involving several linearly independent reactions in which one and the same substance can serve both the product and the reagent. When equilibrium equations (21) are written using concentrations, they are sure to contain seven unknown values. If we write them in terms of chemical extents, these will involve three unknowns, because the mass conservation laws are taken into account which favors further calculations. When the system volume is 1 L, the number of moles is numerically equal to the concentration, which is also convenient. The scale of changes in chemical extents is determined by the initial amount of reacting substances. The chemical extent may be both positive (reaction is directed to the right) and negative (reaction is directed to the left). It is worth noting that we use the values of chemical extents rather than the ε ones, that vary only from 0 to 1. The ε values are used but rarely [8].Equations (21) are solved as follows:(1)
the number of phosphate moles in reaction varies but slightly. The range of variations inξj does not exceed 1.01·10-3 mol. It is assumed then that n70+ξ18+ξ20≈n70. As a result, from equation for K1′, we find(27)ξ18+ξ19≈10-3-24.17⋅10-12mol,(2)
the number of acetate moles in reaction varies but little. Therefore, we assume thatn60-ξ19≈n60. Thus, from equation for K19′, we find(28)ξ19-ξ20≈-10-3+48.18⋅10-12mol,(3)
in equation forK20′, we assume that n70 varies but slightly and substitute the value for ξ19-ξ20 to determine ξ20(29)ξ20≈38.48⋅10-12mol.Hence,(30)ξ18≈1.01⋅10-3-110.83⋅10-12mol,ξ19≈-10-3+86.66⋅10-12mol,ξ20≈38.48⋅10-12mol.
The equilibrium constants in (21) are given to within 2% using the values ξ18,ξ19, and ξ20. The values of equilibrium concentrations are listed in Table 1 which shows substantial difference in the values of equilibrium concentrations and those in the quasiequilibrium state.Figures1 and 2 demonstrate the curves for the change in the Gibbs function (ΔG+71) kJ of the reacting system with time. The value of the Gibbs function is observed to decrease monotonically with time.The time dependence of the Gibbs function was calculated for a solution of volume 1 L from the expression(31)G(t)=∑ini(t)[μio+RTln⁡Ci(t)].
The standard μio values were taken from [3], where ni(t) is the amount of the ith reagent in the system of volume 1 L at time t, and Ci(t) is the concentration of the i-th reagent at time t.As follows from Figure1, the quasistationary regime is realized at times exceeding 1000 s, which is in accord with the calculations performed by (15). The reaction rates at 1800 s are summarized in Table 2. The rates of direct and inverse reactions (19) and (20) are observed to be close to each other and exceed the ATP hydrolysis rate by about order of magnitude. Thus, the quasistationary regime is satisfied. Figures 1 and 2 show that the product of the equilibrium constants of reactions (19) and (20) is held constant within a time domain of 3000–100000 s.As follows from Figure2, in the course of time, the system tends to true equilibrium at which the concentration of the ACoA product is very low, and there is no point in discussing coupling in the case of thermodynamic equilibrium. Thus, the experiment is in fair agreement with the theoretical concepts.It is readily seen that in the ATP coupling of essence are not only thermodynamic factors but also the kinetic ones. For example, let us decrease the rate constants of reactions (20), k20+ and k20- by a factor of 10. The equilibrium constant and the ΔrG20′o values remain unchanged. However, the maximum amount of ACoA and the time during which a maximum is reached vary substantially (Table 3). Thus, ATP is sure to produce desired products under quasistationary conditions without varying the equilibrium constant of unfavorable reactions.Table 3
The effect of rate constant on the maximum amount of ACoA.
Rate constants of reaction (20)k20+, (mM·s)−1k20-, (mM·s)−1The time at which the maximum ACoA concentration is reached, s[ACoA]max, mMRate constants are the same0.0030.000046000.13Rate constants are decreased 10 times0.00030.00000477000.082


### 3.2. Phosphorylation of Glucose

Consider now the process of glucose phosphorylation. A change in the Gibbs function in the direct process of phosphorylation using phosphate amounts to 11.57 kJ/mol. It is assumed then that the equilibrium constant of glucose phosphorylation increases by2·105 due to ATP hydrolysis [6]. However, the equilibrium system, consisting of ATP, ADP, Pi, glucose, and glucose-6-phosphate, can be described in terms of two linearly independent reactions, either uncoupled
(32)ATP=ADP+Pi,ΔrG32′o=-36.03kJ/mol,K32′=2.0⋅106,(33)glucose+Pi=glucose-6-phosphate,ΔrG33′o=+11.57kJ/mol,K33′=9.4⋅10-3
or coupled(34)ATP=ADP+Pi,K34′=K32′,(35)glucose+ATP=glucose-6-phosphate+ADP,ΔrG35′o=-24.41kJ/mol,K35′=1.9⋅104.
It is worth noting that the equilibrium state is independent of the choice of linearly independent reactions. As follows from (32) and (33) or (34) and (35), in the equilibrium state, the system mainly contains ADP, Pi, glucose, and the small amounts of ATP and glucose-6-phosphate. Indeed, let us consider the system with initial concentrations:
(36)[ATP]=a0=10-3M,[glucose]=g0=10-3M,[ADP][Pi]=[glucose-6-phosphate]=0M.
The equilibrium constants are
(37)K32′=ξ32⋅(ξ32-ξ33)(a0-ξ32),K33′=ξ33[(ξ32-ξ33)⋅(g0-ξ33)],K34′=ξ34⋅(ξ34+ξ35)(a0-ξ34-ξ35),K35′=ξ35⋅(ξ34+ξ35)[(a0-ξ34-ξ35)⋅(g0-ξ35)],
where ξ32,ξ33,ξ34, and ξ35 are the extents of reactions (32), (33), (34), and (35), accordingly, at equilibrium. The extents of the reactions at equilibrium
(38)ξ32=(10-3-0.5·10-12)mol,ξ33=9.4·10-9mol,ξ34=(10-3-9.4·10-9-0.5·10-12)mol,ξ35=9.4·10-9mol
satisfy (37). In this case, the chemical extents were calculated in the same manner as for the system of (21).The extents of reactions (33) and (35) of glucose-6-phosphate formation in the equilibrium conditions are very small and equal to each other. Thus, no coupling is observed in the equilibrium system. The coupling manifests itself at fairly short times due to reaction (35). As the rate constants of the direct and inverse reactions are relatively high, the quasistationary state arises which gives a quantity of glucose-6-phosphate. Further, the quasistationary equilibrium in (35) shifts gradually to the left due to the ATP hydrolysis in reaction (1) which is unobservable for real biochemical systems, because glucose-6-phosphate enters rather quickly into other reactions, and the ATP concentration is kept constant. In real biochemical systems, the ATP concentration is much higher than the equilibrium one, which favors the fast reactions of phosphorylation.


## 3.1. Acetyl-CoA Formation

Consider now the production of Acetyl-CoA [4] as an experimental example of reaction kinetics in which the role of ATP is of essence. The reaction of Acetyl-CoA formation from acetate and CoA is impossible due to a small equilibrium constant
(16)acetate+CoA⟶acetyl-CoA+H2O,ΔrG16′o=+33.21kJ/mol.
However, in the presence of ATP, acetate kinase, and phosphate acetyltrassferase, the reaction
(17)acetate+CoA+ATP⟶acetyl-CoA+ADP+Pi,ΔrG17′o=-2.82kJ/mol
follows the mechanism shown by Scheme 2.Scheme 2.
Consider  the following:(18)ATP→K18ADP+Pi,ΔrG18′o=-36.03kJ/mol,(19)ATP+Acetate⇆K19+K19-APi+ADP,ΔrG19′o=+8.61kJ/molK9+,(20)APi+CoA⇆K20+K20-ACoA+Pi,ΔrG20′o=-11.43kJ/mol.Notation 1.
APi = acetyl phosphate and ACoA = acetyl-CoA. TheΔrG′o values were calculated according to [3]. Let us consider results from the second experiment [4, Table  2]. In the paper [4], the data are presented on the concentrations of reaction participants for times 15, 30, and 45 min. The initial concentrations and those at t=1800 s are listed in Table 1.Table 1
Initial concentrations, concentrations att=1800 s, and equilibrium concentrations, mM for the second experiment of Table 2 [4].
N1234567SubstanceAcetyl-phosphateATPADPAcetylCoACoAAcetatePhosphateInitial concentrations1.00.011.000.3864.9547.86Concentrations att=1800 s0.22 (calculated)0.6070.3620.1100.26865.5647.97Equilibrium concentrations att→∞, calculated this work48.8·10-924.17·10-91.0138.48·10-90.3865.9548.87Using the kinetic data on the second experiment [4, Table  2], we have chosen the values of rate constants (Table 2) and the calculated kinetic curves for all reaction participants by the standard Runge-Kutta method as described in the paper [7]. Figure 1 shows the kinetic curves for a short time interval of 0–5000 s, and Figure 2 shows for a longer one of 0–100000 s. The experimental data are well described by theoretical curves.Table 2
Reaction rate constants and reaction rates at 1800 s.
Constantk18 s−1k19+, (mM·s)−1k19-, (mM·s)−1k20+, (mM·s)−1k20-, (mM·s)−1Constant value0.000030.0000060.00350.0030.00004Reactions rate att=1800 s, mM/s1.6·10-52.3·10-42.8·10-41.8·10-42.0·10-4Figure 1
Kinetic curves over small time domain for the second experiment (Table2 [4]). Experimental data are denoted by the dots.Figure 2
Kinetic curves at long times for the second experiment (Table2 [4]).From the data of Table1 and Figure 1, the authors [4] concluded that they had reached the equilibrium state of the system. This statement, however, is erroneous for the following reasons. The authors describe the equilibrium state of the system, consisting of seven substances, that is, acetyl-phosphate, ATP, ADP, acetylCoA, CoA, acetate, and phosphate using two linearly independent reactions (19) and (20) of Scheme 2. However, a thermodynamically correct description of this system must include three linearly independent reactions, for example, those present in Scheme 2. The composition of the equilibrium system is easy to calculate from the equations of thermodynamic equilibrium
(21)K18′=(n30+ξ18+ξ19)⋅(n70+ξ18+ξ20)n20-ξ18-ξ19=2⋅106,K19′=(n10+ξ19-ξ20)⋅(n30+ξ18+ξ19)(n20-ξ18-ξ19)⋅(n60-ξ19)=0.031,K20′=(n40+ξ20)⋅(n70+ξ18+ξ20)(n10+ξ19-ξ20)⋅(n50-ξ20)=100.6.
In these reactions, the equilibrium constants K18′,K19′, and K20′ correspond to reactions (18), (19), and (20) of Scheme 2. For simplicity, we consider the system of volume 1 L. The ni0 value describes the number of moles of the i-th substance per one liter at the initial stage in moles. The values ξ18, ξ19, and ξ20 describe the chemical extents of reactions in Scheme 2. The chemical extents are given in moles [8] and introduced as follows:
(22)ξ=(ni-ni0)νi,
where ni is the amount of the ith substance in the system in any stage of the process, and νi is a stoichiometric coefficient of the reaction for the ith substance. Hence,
(23)ni=ni0+νiξ.
It is valid for the only reaction in the system. If there are several linearly independent reactions in the system, then
(24)ni=ni0+∑jνijξj,
where j is the number of the reactions, νij is the stoichiometric coefficient for the ith substance of the jth reaction, and ξj is the chemical extent of the jth reaction. Usually, in the initial stage of the process, it is convenient to assume that the ξj values are zero. The equilibrium constant of the j-th reaction is of the form
(25)Kj=∏iCjνij=∏i[(ni0+∑jνijξj)V]νij,
where Cj is the equilibrium concentration, and V is the system volume. The νij values are positive for products and negative for reagents. When the system volume is 1 L, the equilibrium constant may be given in a simpler form
(26)Kj=∏i(ni0+∑jνijξj)νij.
We use chemical extents because the method of chemical extents is a simple and powerful means of describing the equilibrium chemical systems as compared with the method of concentrations. This method is particularly suitable for complex chemical systems, involving several linearly independent reactions in which one and the same substance can serve both the product and the reagent. When equilibrium equations (21) are written using concentrations, they are sure to contain seven unknown values. If we write them in terms of chemical extents, these will involve three unknowns, because the mass conservation laws are taken into account which favors further calculations. When the system volume is 1 L, the number of moles is numerically equal to the concentration, which is also convenient. The scale of changes in chemical extents is determined by the initial amount of reacting substances. The chemical extent may be both positive (reaction is directed to the right) and negative (reaction is directed to the left). It is worth noting that we use the values of chemical extents rather than the ε ones, that vary only from 0 to 1. The ε values are used but rarely [8].Equations (21) are solved as follows:(1)
the number of phosphate moles in reaction varies but slightly. The range of variations inξj does not exceed 1.01·10-3 mol. It is assumed then that n70+ξ18+ξ20≈n70. As a result, from equation for K1′, we find(27)ξ18+ξ19≈10-3-24.17⋅10-12mol,(2)
the number of acetate moles in reaction varies but little. Therefore, we assume thatn60-ξ19≈n60. Thus, from equation for K19′, we find(28)ξ19-ξ20≈-10-3+48.18⋅10-12mol,(3)
in equation forK20′, we assume that n70 varies but slightly and substitute the value for ξ19-ξ20 to determine ξ20(29)ξ20≈38.48⋅10-12mol.Hence,(30)ξ18≈1.01⋅10-3-110.83⋅10-12mol,ξ19≈-10-3+86.66⋅10-12mol,ξ20≈38.48⋅10-12mol.
The equilibrium constants in (21) are given to within 2% using the values ξ18,ξ19, and ξ20. The values of equilibrium concentrations are listed in Table 1 which shows substantial difference in the values of equilibrium concentrations and those in the quasiequilibrium state.Figures1 and 2 demonstrate the curves for the change in the Gibbs function (ΔG+71) kJ of the reacting system with time. The value of the Gibbs function is observed to decrease monotonically with time.The time dependence of the Gibbs function was calculated for a solution of volume 1 L from the expression(31)G(t)=∑ini(t)[μio+RTln⁡Ci(t)].
The standard μio values were taken from [3], where ni(t) is the amount of the ith reagent in the system of volume 1 L at time t, and Ci(t) is the concentration of the i-th reagent at time t.As follows from Figure1, the quasistationary regime is realized at times exceeding 1000 s, which is in accord with the calculations performed by (15). The reaction rates at 1800 s are summarized in Table 2. The rates of direct and inverse reactions (19) and (20) are observed to be close to each other and exceed the ATP hydrolysis rate by about order of magnitude. Thus, the quasistationary regime is satisfied. Figures 1 and 2 show that the product of the equilibrium constants of reactions (19) and (20) is held constant within a time domain of 3000–100000 s.As follows from Figure2, in the course of time, the system tends to true equilibrium at which the concentration of the ACoA product is very low, and there is no point in discussing coupling in the case of thermodynamic equilibrium. Thus, the experiment is in fair agreement with the theoretical concepts.It is readily seen that in the ATP coupling of essence are not only thermodynamic factors but also the kinetic ones. For example, let us decrease the rate constants of reactions (20), k20+ and k20- by a factor of 10. The equilibrium constant and the ΔrG20′o values remain unchanged. However, the maximum amount of ACoA and the time during which a maximum is reached vary substantially (Table 3). Thus, ATP is sure to produce desired products under quasistationary conditions without varying the equilibrium constant of unfavorable reactions.Table 3
The effect of rate constant on the maximum amount of ACoA.
Rate constants of reaction (20)k20+, (mM·s)−1k20-, (mM·s)−1The time at which the maximum ACoA concentration is reached, s[ACoA]max, mMRate constants are the same0.0030.000046000.13Rate constants are decreased 10 times0.00030.00000477000.082


## 3.2. Phosphorylation of Glucose

Consider now the process of glucose phosphorylation. A change in the Gibbs function in the direct process of phosphorylation using phosphate amounts to 11.57 kJ/mol. It is assumed then that the equilibrium constant of glucose phosphorylation increases by2·105 due to ATP hydrolysis [6]. However, the equilibrium system, consisting of ATP, ADP, Pi, glucose, and glucose-6-phosphate, can be described in terms of two linearly independent reactions, either uncoupled
(32)ATP=ADP+Pi,ΔrG32′o=-36.03kJ/mol,K32′=2.0⋅106,(33)glucose+Pi=glucose-6-phosphate,ΔrG33′o=+11.57kJ/mol,K33′=9.4⋅10-3
or coupled(34)ATP=ADP+Pi,K34′=K32′,(35)glucose+ATP=glucose-6-phosphate+ADP,ΔrG35′o=-24.41kJ/mol,K35′=1.9⋅104.
It is worth noting that the equilibrium state is independent of the choice of linearly independent reactions. As follows from (32) and (33) or (34) and (35), in the equilibrium state, the system mainly contains ADP, Pi, glucose, and the small amounts of ATP and glucose-6-phosphate. Indeed, let us consider the system with initial concentrations:
(36)[ATP]=a0=10-3M,[glucose]=g0=10-3M,[ADP][Pi]=[glucose-6-phosphate]=0M.
The equilibrium constants are
(37)K32′=ξ32⋅(ξ32-ξ33)(a0-ξ32),K33′=ξ33[(ξ32-ξ33)⋅(g0-ξ33)],K34′=ξ34⋅(ξ34+ξ35)(a0-ξ34-ξ35),K35′=ξ35⋅(ξ34+ξ35)[(a0-ξ34-ξ35)⋅(g0-ξ35)],
where ξ32,ξ33,ξ34, and ξ35 are the extents of reactions (32), (33), (34), and (35), accordingly, at equilibrium. The extents of the reactions at equilibrium
(38)ξ32=(10-3-0.5·10-12)mol,ξ33=9.4·10-9mol,ξ34=(10-3-9.4·10-9-0.5·10-12)mol,ξ35=9.4·10-9mol
satisfy (37). In this case, the chemical extents were calculated in the same manner as for the system of (21).The extents of reactions (33) and (35) of glucose-6-phosphate formation in the equilibrium conditions are very small and equal to each other. Thus, no coupling is observed in the equilibrium system. The coupling manifests itself at fairly short times due to reaction (35). As the rate constants of the direct and inverse reactions are relatively high, the quasistationary state arises which gives a quantity of glucose-6-phosphate. Further, the quasistationary equilibrium in (35) shifts gradually to the left due to the ATP hydrolysis in reaction (1) which is unobservable for real biochemical systems, because glucose-6-phosphate enters rather quickly into other reactions, and the ATP concentration is kept constant. In real biochemical systems, the ATP concentration is much higher than the equilibrium one, which favors the fast reactions of phosphorylation.


## 4. Conclusions

ATP is an effective phosphorylating agent which, due to favorable change in Gibbs function and in the presence of suitable enzymes, provides fast phosphorylation resulting in superequilibrium concentrations of phosphorylation products in quasistationary states. A quasistationary system is a point at the Gibbs function surface which slowly tends to a global minimum in the course of ATP hydrolysis. The closer to the global minimum, the lower the concentration of the phosphorylation products. In real biochemical systems, the quasistationary phenomena are unobservable, because the ATP concentration is kept almost constant at the level which substantially exceeds the equilibrium one which provides (with the help of enzymes) the fast processes of phosphorylation. In equilibrium systems, no coupling is observed. This coupling phenomenon is attained by combining thermodynamic and kinetic factors.Problems of using ATP as energy carrier in biochemical systems have been discussed in the Appendix.



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*Source: 827604-2012-05-10.xml*