Thermodynamics of the reactions of carbamoyl phosphate

Two measurements of equilibrium constants by Marshall and Cohen make it possible to calculate standard Gibbs energies of formation of the species of carbamate and carbamoyl phosphate. Carbamate formation from carbon dioxide and ammonia does not require an enzyme, and the equilibrium concentrations of carbamate in ammonium bicarbonate are calculated. Knowing the values of standard Gibbs energies of formation of species of carbamate and carbamoyl phosphate make it possible to calculate the dependencies of the standard transformed Gibbs energies of formation of these reactants on pH and ionic strength and to calculate apparent equilibrium constants for several enzyme-catalyzed reactions and several chemical reactions. These calculations are sufficiently complicated that computer programs in Mathematica are used to make tables and plots. The dependences of apparent equilibrium constants on pH are consequences of the production or consumption of hydrogen ions, which are shown in plots. As usual the increase in the number of enzyme-catalyzed reactions for which apparent equilibrium constants can be calculated is larger than the number of reactions required to obtain the thermodynamic properties of the species involved.     

Thermodynamics and kinetics of the glyoxylate cycle

Because the standard Gibbs energies of formation of all the species of reactants in the glyoxylate cycle are known at 298.15 K, it is possible to calculate the apparent equilibrium constants of the five reactions in the cycle in the pH range 5-9 and ionic strengths from 0 to approximately 0.35 M. In making calculations on such a system, it is convenient to specify concentrations of coenzymes like NADox and NADred because they are involved in many reactions and may be in steady states. Calculations are given for [NADox] = 1000[NADred] and [NADox] = 10[NADred]. Equilibrium compositions are calculated using computer programs when all the reactants are present initially and when only glyoxylate and CoA are present initially. The kinetics of the reactions in the glyoxylate cycle at specified concentrations of NADox and NADred are calculated by numerical solution of the steady-state rate equations for the case where the reactant concentrations are below their Michaelis constants and only glyoxylate and CoA are present initially.     

Levels of thermodynamic treatment of biochemical reaction systems.

Equilibrium calculations on biochemical reaction systems can be made at three levels. Level 1 is the usual chemical calculation with species at specified temperature and pressure using standard Gibbs energies of formation of species or equilibrium constants K. Level 2 utilizes reactants such as ATP (a sum of species) at specified T, P, pH, and pMg with standard transformed Gibbs energies of formation of reactants or apparent equilibrium constants K'. Calculations at this level can also be made on the enzymatic mechanism for a biochemical reaction. Level 3 utilizes reactants at specified T, P, pH, and pMg, but the equilibrium concentrations of certain reactants are also specified. The fundamental equation of thermodynamics is derived here for Level 3. Equilibrium calculations at this level use standard transformed Gibbs energies of formation of reactants at specified concentrations of certain reactants or apparent equilibrium constants K". Level 3 is useful in calculating equilibrium concentrations of reactants that can be reached in a living cell when some of the reactants are available at steady-state concentrations. Calculations at all three levels are facilitated by the use of conservation matrices and stoichiometric number matrices for systems. Three cases involving glucokinase, glucose-6-phosphatase, and ATPase are discussed.     

Thermodynamics of the purine nucleotide cycle

Since the standard Gibbs energies of formation are known for all the species in the purine nucleotide cycle at 298.15 K, the functions of pH and ionic strength that yield the standard transformed Gibbs energies of formation of the ten reactants can be calculated. This makes it possible to calculate the standard transformed Gibbs energies of reaction, apparent equilibrium constants, and changes in the binding of hydrogen ions for the three reactions at desired pHs and ionic strengths. These calculations are also made for the net reaction and a reaction that is related to it. The equilibrium concentrations for the cycle are calculated when all the reactants are initially present or only some are present initially. Since the concentrations of GTP, GDP, and P(i) may be in steady states, the equilibrium concentrations are also calculated for the system at specified steady-state concentrations.     

Quantifying the inactivation rate constants for the molecular species comprising the catalytic cycle of Leuconostoc mesenteroides glucose 6-phosphate dehydrogenase

When an unstable enzyme is incubated with its substrate(s), catalysis may cease before chemical equilibrium is attained. The residual substrate concentrations depend on their initial concentrations, the initial enzymic activity, and the inactivation rate constants for each molecular species that comprise the catalytic cycle. The underlying theory has been elaborated previously for single-substrate reactions and here it is extended to bi-substrate reactions. The theory is illustrated by application to glucose 6-phosphate dehydrogenase, which is unstable when exposed to a low concentration of sodium dodecyl sulphate. It is shown that the ternary complex containing both substrates is resistant to inactivation while each of the remaining complexes undergoes first-order decay. Rate constants for the inactivation of each complex are calculated.     

Quantifying the inactivation rate constants for the molecular species comprising the catalytic cycle of Leuconostoc mesenteroides glucose 6-phosphate dehydrogenase

When an unstable enzyme is incubated with its substrate(s), catalysis may cease before chemical equilibrium is attained. The residual substrate concentrations depend on their initial concentrations, the initial enzymic activity, and the inactivation rate constants for each molecular species that comprise the catalytic cycle. The underlying theory has been elaborated previously for single-substrate reactions and here it is extended to bi-substrate reactions. The theory is illustrated by application to glucose 6-phosphate dehydrogenase, which is unstable when exposed to a low concentration of sodium dodecyl sulphate. It is shown that the ternary complex containing both substrates is resistant to inactivation while each of the remaining complexes undergoes first-order decay. Rate constants for the inactivation of each complex are calculated.     

Standard transformed Gibbs energies of coenzyme A derivatives as functions of pH and ionic strength

The best way to store data on apparent equilibrium constants for enzyme-catalyzed reactions is to calculate the standard Gibbs energies of formation of the species involved at 298.15 K and zero ionic strength so that equilibrium constants can be calculated at the desired pH and ionic strength. These calculations are described for CoA, acetyl-CoA, oxalyl-CoA, succinyl-CoA, methylmalonyl-CoA, malyl-CoA and CoA-glutathione. The species properties are then used to calculate standard transformed Gibbs energies of formation for these reactants as functions of pH at ionic strength 0.25 M. The species data also make it possible to calculate apparent equilibrium constants of 23 enzyme-catalyzed reactions as a function of pH, including some that cannot be determined directly because they are so large.     

The role of water in the thermodynamics of dilute aqueous solutions

Water plays a role in the thermodynamics of dilute aqueous solutions that is unusual in two ways. First, knowledge of hydration equilibrium constants of species is not required in calculations of thermodynamic properties of biochemical reactants and reactions at specified pH. Second, since solvent provides an essentially infinite source of oxygen atoms in a reaction system where water is a reactant, oxygen atoms are not conserved in the reaction system in dilute aqueous solutions. This is related to the fact that H₂O is omitted in equilibrium expressions for dilute aqueous solutions. Calculations of the standard transformed Gibbs energies of formation of total carbon dioxide and total ammonia at specified pH are discussed, and the average bindings of hydrogen ions by these reactants are calculated by differentiation. Since both of these reactants are involved in the urease reaction, the apparent equilibrium constants and changes in the numbers of hydrogen ions bound are calculated for this reaction as functions of pH.     

Recommendations for terminology and databases for biochemical thermodynamics

Chemical equations are normally written in terms of specific ionic and elemental species and balance atoms of elements and electric charge. However, in a biochemical context it is usually better to write them with ionic reactants expressed as totals of species in equilibrium with each other. This implies that atoms of elements assumed to be at fixed concentrations, such as hydrogen at a specified pH, should not be balanced in a biochemical equation used for thermodynamic analysis. However, both kinds of equations are needed in biochemistry. The apparent equilibrium constant K' for a biochemical reaction is written in terms of such sums of species and can be used to calculate standard transformed Gibbs energies of reaction Δ(r)G'°. This property for a biochemical reaction can be calculated from the standard transformed Gibbs energies of formation Δ(f)G(i)'° of reactants, which can be calculated from the standard Gibbs energies of formation of species Δ(f)G(j)° and measured apparent equilibrium constants of enzyme-catalyzed reactions. Tables of Δ(r)G'° of reactions and Δ(f)G(i)'° of reactants as functions of pH and temperature are available on the web, as are functions for calculating these properties. Biochemical thermodynamics is also important in enzyme kinetics because apparent equilibrium constant K' can be calculated from experimentally determined kinetic parameters when initial velocities have been determined for both forward and reverse reactions. Specific recommendations are made for reporting experimental results in the literature.     

Kinetic parameters of enzymatic reactions in states of maximal activity; an evolutionary approach

A theoretical investigation is presented which allows the calculation of states of maximal reaction rates for single enzymes and for unbranched enzymatic chains. As an extension to previous papers (Heinrich & Holzhütter, 1985, Biomed. biochim. Acta 44, 959-969; Heinrich et al., 1987, Bull. math. Biol. 49, 539-595) a detailed enzymatic mechanism was taken into consideration. Conclusions are drawn for the optimal values of the microscopic rate constants as well as of the maximal activities and Michaelis constants. Ten solutions are found which depend on the equilibrium constant as well as on the concentrations of substrates and products. It is shown that for high equilibrium constants one of the solutions applies to a very large range of the concentrations of the outer reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. In contrast to previous assumptions (Albery & Knowles, 1976b, Biochemistry 15, 5631-5640; Burbaum et al., 1989, Biochemistry 28, 9293-9305) states of maximal reaction rate are not always characterized by the highest possible values of the second-order rate constants which are related to the diffusion of the substrate and the product to the active site of the enzyme. Predictions are made concerning the ratios of maximal activities in optimal states as well as for the adaptation of the Michaelis constants to the concentrations of the outer reactants. Using metabolic control analysis it is shown that the solutions obtained for single enzymes may also be applied in multi-enzyme systems.     

Monte Carlo simulation of diffusion and reaction in water radiolysis – a study of reactant `jump through' and jump distances

In Monte Carlo simulations of water radiolysis, the diffusion of reactants can be approximated by “jumping” all species randomly, to represent the passage of a short period of time, and then checking their separations. If, at the end of a jump, two reactant species are within a distance equal to the reaction radius for the pair, they are allowed to react in the model. In principle, the possibility exists that two reactants could “jump through” one another and end up with a separation larger than the reaction radius with no reaction being scored. Ignoring this possibility would thus reduce the rate of reaction below that intended by such a model. By making the jump times and jump distances shorter, any error introduced by `jump through' is made smaller. This paper reports numerical results of a systematic study of `jump through' in Monte Carlo simulations of water radiolysis. With a nominal jump time of 3 ps, it is found that more than 40% of the reactions of the hydrated electron with itself and of the H atom with itself occur when reactions during `jump through' are allowed. For all other reactions, for which the effect is smaller, the contributions of `jump through' lie in the range l%–16% of the total. Corrections to computed rate constants for two reactions are evaluated for jump times between 0.1 and 30 ps. It is concluded that jump-through corrections are desirable in such models for jump times that exceed about 1 ps or even less. In a separate study, we find that giving all species of a given type the same size jump in a random direction yields results that are indistinguishable from those when the jump sizes are selected from a Gaussian distribution. In this comparison, the constant jump size is taken to be the root-mean-square jump size from the Gaussian distribution. Received: 8 September 1997 / Accepted in revised form: 27 October 1997     

Thermodynamics of systems of biochemical reactions

When a reaction system described in terms of species is in a certain state, the Gibbs energy G provides the means for determining whether each reaction will go to the right or the left, and the equilibrium composition of the whole system can be calculated using G. When the pH is specified, a system of biochemical reactions is described in terms of reactants, like ATP (a sum of species), and the transformed Gibbs energy G' provides the means for determining whether each reaction will go to the right or the left. The equilibrium composition of the whole system can be calculated using G'. Since metabolism is complicated, the thermodynamics of systems of reactions like glycolysis and the citric acid cycle can also be considered at specified concentrations of coenzymes like ATP, ADP, NAD(ox), and NAD(red). This is of interest because coenzymes tend to be in steady states because they are involved in many reactions. When the concentrations of coenzymes are constant, the further transformed Gibbs energy G" provides the means for calculating whether each reaction will go to the right or the left, and the equilibrium composition of the whole system can be calculated using G". Under these conditions, a metabolic reaction system can be reconceptualized in terms of sums of reactants; for example, glycolysis can be represented by C(6)=2C(3), where C(6) is the sum of the reactants with six carbon atoms and C(3) is the sum of the reactants with three carbon atoms. These calculations can also be described by use of semigrand partition functions. Semigrand partition functions have the advantage of containing all the thermodynamic information on a series of reactions at specified pH or at specified pH and specified concentrations of coenzymes.     

Thermodynamic properties of enzyme-catalyzed reactions involving cytosine, uracil, thymine, and their nucleosides and nucleotides

The standard Gibbs energies of formation of species in the cytidine triphosphate series, uridine triphosphate series, and thymidine triphosphate series have been calculated on the basis of the convention that Delta(f)G=0 for the neutral form of cytidine in aqueous solution at 298.15 K at zero ionic strength. This makes it possible to calculate apparent equilibrium constants for a number of reactions for which apparent equilibrium constants have not been measured or cannot be measured because they are too large. This paper adds fifteen reactants to the database BasicBiochemData3 at MathSource that includes 199 reactants. The standard transformed Gibbs energies of formation of these fifteen reactants are used to calculate apparent equilibrium constants at 298.15 K, ionic strength 0.25 M, and pHs 5, 6, 7, 8, and 9 for thirty two reactions. The pKs, standard Gibbs energies of hydrolysis, and standard Gibbs energies of deamination are given for these fifteen reactants.     

Fluorescence correlation spectroscopy. II. An experimental realization

This paper describes the first experimental application of fluorescence correlation spectroscopy, a new method for determining chemical kinetic constants and diffusion coefficients. These quantities are measured by observing the time behaviour of the tiny concentration fluctuations which occur spontaneously in the reaction system even when it is in equilibrium. The equilibrium of the system is not disturbed during the experiment. The diffusion coefficients and chemical rate constants which determine the average time behaviour of these spontaneous fluctuations are the same as those sought by more conventional methods including temperature-jump or other perturbation techniques. The experiment consists essentially in measuring the variation with time of the number of molecules of specified reactants in a defined open volume of solution. The concentration of a reactant is measured by its fluorescence; the sample volume is defined by a focused laser beam which excites the fluorescence. The fluorescent emission fluctuates in proportion with the changes in the number of fluorescent molecules as they diffuse into and out of the sample volume and as they are created or eliminated by the chemical reactions. The number of these reactant molecules must be small to permit detection of the concentration fluctuations. Hence the sample volume is small (10^?8 ml) and the concentration of the solutes is low (～ 10^?9 M). We have applied this technique to the study of two prototype systems: the simple example of pure diffusion of a single fluorescent species, rhodamine 6G, and the more interesting but more challenging example of the reaction of macromolecular DNA with the drug ethidium bromide to form a fluorescent complex. The increase of the fluorescence of the ethidium bromide upon formation of the complex permits the observation of the decay of concentration fluctuations via the chemical reaction and consequently the determination of chemical rate constants.     

Equilibrium concentrations for pyruvate dehydrogenase and the citric acid cycle at specified concentrations of certain coenzymes

It is of interest to calculate equilibrium compositions of systems of biochemical reactions at specified concentrations of coenzymes because these reactants tend to be in steady states. Thermodynamic calculations under these conditions require the definition of a further transformed Gibbs energy G" by use of a Legendre transform. These calculations are applied to the pyruvate dehydrogenase reaction plus the citric acid cycle, but steady-state concentrations of CoA, acetyl-CoA and succinyl-CoA cannot be specified because they are involved in the conservation of carbon atoms. These calculations require the use of linear algebra to obtain further transformed Gibbs energies of formation of reactants and computer programs to calculate equilibrium compositions. At specified temperature, pH, ionic strength and specified concentrations of several coenzymes, the equilibrium composition depends on the specified concentrations of the coenzymes and the initial amounts of reactants.     

An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

Thermodynamic properties of enzyme-catalyzed reactions involving guanine, xanthine, and their nucleosides and nucleotides

The standard Gibbs energies of formation of species in the guanosine triphosphate and the xanthosine triphosphate series have been calculated on the basis of the convention that the standard Gibbs energy of formation for the neutral form of guanosine is equal to zero in aqueous solution at 298.15 K and zero ionic strength. This makes it possible to calculate apparent equilibrium constants for a number of enzyme-catalyzed reactions for which apparent equilibrium constants have not been measured or cannot be measured directly because they are too large. The eventual elimination of this convention is discussed. This adds ten reactants to the database BasicBiochemData3 that has 199 reactants. The standard transformed Gibbs energies of formation of these ten reactants are used to calculate apparent equilibrium constants at 298.15 K, 0.25 M ionic strength, and pHs 5, 6, 7, 8, and 9. The pKs, standard Gibbs energies of hydrolysis, and standard Gibbs energies of deamination are given for the reactants in the ATP, IMP, GTP, and XTP series.     

Energy barriers and rates of tautomeric transitions in DNA bases: ab initio quantum chemical study

Tautomeric transitions of DNA bases are proton transfer reactions, which are important in biology. These reactions are involved in spontaneous point mutations of the genetic material. In the present study, intrinsic reaction coordinates (IRC) analyses through ab initio quantum chemical calculations have been carried out for the individual DNA bases A, T, G, C and also A:T and G:C base pairs to estimate the kinetic and thermodynamic barriers using MP2/6-31G** method for tautomeric transitions. Relatively higher values of kinetic barriers (about 50-60 kcal/mol) have been observed for the single bases, indicating that tautomeric alterations of isolated single bases are quite unlikely. On the other hand, relatively lower values of the kinetic barriers (about 20-25 kcal/mol) for the DNA base pairs A:T and G:C clearly suggest that the tautomeric shifts are much more favorable in DNA base pairs than in isolated single bases. The unusual base pairing A':C, T':G, C':A or G':T in the daughter DNA molecule, resulting from a parent DNA molecule with tautomeric shifts, is found to be stable enough to result in a mutation. The transition rate constants for the single DNA bases in addition to the base pairs are also calculated by computing the free energy differences between the transition states and the reactants.     

A Fast Method to Estimate Kinetic Constants for Enzyme Inhibitors

We present a method to determine the reaction type and kinetic constants for enzyme inhibitors that decreases the number of experimental assays by at least a factor of five. It is based on a new theoretical formalism in terms of concentrations that dismisses the requirement of estimating initial velocities. Expressions for the time evolution of the concentrations of all the reactants are also given.