Robert A. Alberty

Biochemical thermodynamics and rapid-equilibrium enzyme kinetics.

Biochemical thermodynamics is based on the chemical thermodynamics of aqueous solutions, but it is quite different because pH is used as an independent variable. A transformed Gibbs energy G is used, and that leads to transformed enthalpies H and transformed entropies S. Equilibrium constants for enzyme-catalyzed reactions are referred to as apparent equilibrium constants K to indicate that they are functions of pH in addition to temperature and ionic strength. Despite this, the most useful way to store basic thermodynamic data on enzyme-catalyzed reactions is to give standard Gibbs energies of formation, standard enthalpies of formation, electric charges, and numbers of hydrogen atoms in species of biochemical reactants like ATP. This makes it possible to calculate standard transformed Gibbs energies of formation, standard transformed enthalpies of formation of reactants (sums of species), and apparent equilibrium constants at desired temperatures, pHs, and ionic strengths. These calculations are complicated, and therefore, a mathematical application in a computer is needed. Rapid-equilibrium enzyme kinetics is based on biochemical thermodynamics because all reactions in the mechanism prior to the rate-determining reaction are at equilibrium. The expression for the equilibrium concentration of the enzyme-substrate complex that yields products can be derived by applying Solve in a computer to the expressions for the equilibrium constants in the mechanism and the conservation equation for enzymatic sites. In 1979, Duggleby pointed out that the minimum number of velocities of enzyme-catalyzed reactions required to estimate the values of the kinetic parameters is equal to the number of kinetic parameters. Solve can be used to do this with steady-state rate equations as well as rapid-equilibrium rate equations, provided that the rate equation is a polynomial. Rapid-equilibrium rate equations can be derived for complicated mechanisms that involve several reactants and various types of inhibitors, activators, and moderators.

Determination of Rapid-Equilibrium Kinetic Parameters of Ordered and Random Enzyme-Catalyzed Reaction A + B = P + Q

This article deals with the rapid-equilibrium kinetics of the forward and reverse reactions together for the ordered and random enzyme-catalyzed A + B = P + Q and emphasizes the importance of reporting the values of the full set of equilibrium constants. Equilibrium constants that are not in the rate equation can be calculated for random mechanisms using thermodynamic cycles. This treatment is based on the use of a computer to derive rate equations for three mechanisms and to estimate the kinetic parameters with the minimum number of velocity measurements. The most general of these three programs is the one to use first when the mechanism for A + B = P + Q is studied for the first time. This article shows the effects of experimental errors in velocity measurements on the values of the kinetic parameters and on the apparent equilibrium constant calculated using the Haldane relation.

Determination of Kinetic Parameters of Enzyme-Catalyzed Reaction A + B + C → Products with the Minimum Number of Velocity Measurements

Rapid-equilibrium rate equations are derived for the five different mechanisms for the enzymatic catalysis of A + B + C → products using a computer. These rate equations are used to determine the minimum number of velocities required to estimate the values of the kinetic parameters. The rate equation for the completely ordered mechanism involves four kinetic parameters, and the rate equation for the completely random mechanism involves eight kinetic parameters. Therefore, the four to eight kinetic parameters can be estimated by determining four to eight velocities and solving four to eight simultaneous equations. General recommendations are made as to the choices of triplets of substrate concentrations {[A], [B], [C]} to be used to determine the velocities. The effects of 5% errors in the measured velocities, one at a time, are calculated and are summarized in tables. Calculations of effects of experimental errors are useful in choosing the triplets of substrate concentrations to be used to obtain the most accurate values of the kinetic parameters. When the kinetic parameters for A + B + C → products are to be determined for the first time, it is recommended that the program for the completely random mechanism be used because it can identify the mechanism and determine the kinetic parameters in one operation.

Consumption of hydrogen ions in rapid-equilibrium enzyme kinetics.

Enzyme-catalyzed reductase reactions in particular are characterized by large changes in the binding of hydrogen ions Δ(r)N(H). This is a thermodynamic property of the reaction that is catalyzed. For example, in the ferredoxin-nitrite reductase reaction, there is an increase of eight in the binding of hydrogen ions for every molecule of nitrite reduced to ammonia H(2)O. If these hydrogen ions are consumed in the rate-determining reaction, the limiting velocity is proportional to [H(+)](8). This would make it practically impossible to determine the kinetic parameters. This article shows that when n hydrogen ions are consumed in reactions preceding the rate-determining reaction the limiting velocity is not proportional to [H(+)](n) and may only vary with pH according to the pKs of the enzyme-substrate complex that produces products. Rapid-equilibrium rate equations for ordered A + B → products are derived for two mechanisms in which a single hydrogen ion is consumed prior to the rate-determining reaction. Rate equations are tested by calculating velocities for the minimum number of velocity measurements required to estimate the kinetic parameters and using these velocities to estimate the kinetic parameters.