Terrell L. Hill

Monte Carlo calculations on critical behavior in two‐state, steady‐state Ising systems

This is the first part of a study, by the Monte Carlo method, of one‐dimensional and two‐dimensional steady‐state Ising systems. We examine phase transitions and critical behavior at equilibrium (for reference) and steady state. The primary model is a 10×10 or 100×1 periodic square lattice of M=100 interacting, two‐state, cycling enzyme molecules. We use a flat top on P (N2) (the probability N2 out of M molecules are in state 2) as a criterion to locate the approximate ’’critical point’’ in these finite systems. At equilibrium, cases M=10×10, 5×5, and 12 have been examined. Reducing M increases the nearest‐neighbor correlation function c1 at fixed T and increases the critical temperature. At steady state, with M=100 in either one or two dimensions, the cooperativity of the system may be either enhanced or reduced compared to equilibrium, depending on the choice of certain kinetic parameters. With the rather conservative choices of parameters used here, phase transition behavior at steady state is qualitat...

Critical behavior of two‐state, steady‐state Ising systems, according to the Bragg–Williams approximation

The Bragg–Williams (mean field) approximation is applied to an infinite arbitrary lattice of two‐state enzymes, with nearest‐neighbor interactions, cycling at a stable steady‐state arbitrarily far from equilibrium. General equations are given for the fraction of enzymes in state 2 and for the flux. A simple numerical procedure is introduced for the determination of critical constants. A considerable sampling of results, especially on critical properties, is given. Because ’’van der Waals loops’’ are often obtained, some quite complicated, hysteresis is of course possible in the conventional way. The equations are simple enough so that the interested reader can easily generate further examples of his own, if desired. Of particular interest are special cases (a) in which the phase transition occurs in two steps and (b) in which either attractive or repulsive interactions will produce a phase transition.

On the one‐dimensional steady‐state Ising problem

The exact linear flux–force relation, near equilibrium, is found for a one‐dimensional, steady‐state, two‐state Ising system with nearest‐neighbor interactions that affect the rate constants of the two‐state cycle. The physical interpretation of the linear flux–force coefficient is shown to be a one‐way cycle flux at equilibrium. A new approximation is introduced for the same system at an arbitrary steady state.

Effect of enzyme-enzyme interactions on steady-state enzyme kinetics. IV. "Strictly steady-state" examples.

Abstract A number of two-state and three-state examples of enzyme interaction effects at steady state are considered. In contrast to most examples in parts I, II, and III, these are not of the quasi-equilibrium type. The Bragg—Williams approximation is used here for lattices of both two-state and three-state enzymes. In addition, several examples of small (oligomeric) systems are treated. Diffusion in lattice-fluid models is introduced in a simple way in the text and commented on further in the appendix (together with diffusion in solution).

Steady-state kinetics of models of respiratory chain enzymes with isopotential pools and conformational site enzymes

Abstract The formal kinetics of the respiratory chain enzymes is a very complicated mathematical problem. Our main objective here is to introduce methodology that should be useful in approaching this problem, especially at steady state. Most of the work reported here is on models that involve one or two isopotential pools of enzymes and also possibly a site enzyme between two pools that may or may not undergo a conformation change. Preliminary topics treated are the “cross-over” phenomenon and the “mass action” approximation.

Effective coupling of four independent membrane systems in steady-state oxidative phosphorylation

Alexandre et al. have proposed a four-system model of oxidative phosphorylation. The thermodynamic consequences of this model are explored, assuming as a first approximation that these four systems can be treated as a self-contained group. The method can be generalized, in higher approximations, to include further systems and other complications. Respiratory control is considered from the point of view of the model. Self-consistent numerical examples are given to represent mitochondrial activity in state 3 and in state 4.