Cooperative effects in models of steady-state transport across membranes. I

Cooperative denaturation kinetics of homogeneous polymers

The kinetics of denaturation of a homogeneous, helical biopolymer with nearest neighbor interactions are described, using a kinetic Ising model in which the configuration of its neighbors dictates the transition probability for a single residue in the chain. The actual kinetics that are simulated using Monte Carlo techniques are compared with the results of analytical kinetic equations for the fraction of helix, (s), generated using the mean-field approximation. This mean-field rate equation is expanded as a hierarchy of terms that characterize the nature of rate constants for interacting systems. The first term in the expansion is first order in (s) and varies linearly with the interaction energy. Subsequent rate terms involve higher powers of (s) and demonstrate the need for nonlinear equations in systems with larger interaction energies. Both the simulations and the mean-field approximation show an intrinsic induction period for the single-step kinetic process. They also yield an apparent first-order rate constant that changes as the reaction proceeds. However, only the simulated kinetics yield ordered regions of chain and a nonzero, nearest-neighbor correlation function.

Cooperativity in viral fusion

A simple model for membrane fusion mediated by vial spike glycoproteins is presented. The viral proteins are considered to be allosteric proteins that undergo concerted conformational transitions when they bind the ligand. The ligand in this case is H+. The effect of the conformational transition is to bring membranes together and induce their fusion. An equation is derived for the dependence of fusion rates on ligand concentration, for a given dissociation constant (Kd), equilibrium constant for the conformational change (L), and number of cooperating subunits (n). Curves generated by this equation provide a reasonable fit to data on the rates of fusion of Vesicular Stomatitis virus with cells for a pKd of 6.3, L = 1000 and n = 6.

Cole-Moore effect in the frog node

Potassium currents were recorded from the voltage-clamped frog node (Rana esculenta) during various test pulses that followed hyperpolarizing prepulses of different amplitudes and durations. Both the delay in potassium current onset and the shape of the current trace as a function of time were found to be a function of prepulse parameters. This finding is different from the current trace superposition described by Cole and Moore for a specific test pulse, sodium equilibrium potential in the squid giant axon. The Cole-Moore effect, which was found here only under a specific set of conditions, thus may be a special case rather than the general property of the membrane. The implication of these findings to the various excitable membrane potassium channel models, which are based on the Cole-Moore effect, is discussed.

Further study of the effect of enzyme-enzyme interactions on steady-state enzyme kinetics

This paper continues an earlier one [Hill, T.L. (1977) Proc. Natl . Acad. Sci. USA 74, 3632-3632] and presents further introductory examples. Most attention is devoted to a closed linear chain of two-state enzyme molecules with nearest-neighbor interactions. The one-dimensional Ising theory can be used here. The Bragg-Williams (mean field) approximation is introduced to deal with a one-, two-, or three-dimensional lattice of enzyme molecules, at steady state, with an arbitrary kinetic diagram. The behavior of the flux in a phase transition is noted. Finally, a treatment is given for the first effect (second "viral" coefficient) of interactions on the flux in a dilute solution of two-state enzyme molecules.

Theoretical study of the effect of enzyme-enzyme interactions on steady-state enzyme kinetics

Equilibrium statistical mechanics is much concerned with problems involving intermolecularinteractions, either in lattices or in pure fluids or solutions. The possibility of enzyme-enzyme interactions suggests that the same problems might be studied profitably at steady state as well as at equilibrium. In the systems we consider, each of the identical enzyme molecules of the system undergoes steady-state stochastic cycling among states i equal 1,....,n. But the molecules do not cycle independently. Two neghboring molecules, in states i and j, interact with a free energy wij (a function of the distance r in the solution case). The instantaneous transition probabilities between states for a given molecule will depend on the instantaneous interactions between the molecule in question and its neighbors. The primary question of interest is how the enzyme flux is influenced by the interactions. The general problem is outlined here and some simple special cases are treated. The discussion will be continued in a following paper [Hill, T. L. (1977) Proc. Natl. Acad. Sci. USA 74, in press]

The dipole model and phase transitions in biological membranes

Assuming the dipole model for a membrane, approximate calculations are made which employ a dipole-dipole interaction energy. The calculations are based upon the assumption of cooperative coupling of membrane polar molecules and make use of the Bragg-Williams approximation. A theoretical estimate is made of the critical temperature at which phase changes might occur in certain biological membranes. Proposals are presented which explain how the dipole transition might relate to the sometimes observed thermal phase transitions in biological membranes.

Cooperative effects in models of steady-state transport across membranes. IV. One-site, two-site, and multisite models

Several different one-site, two-site, and multisite models of steady-state ion transport across a membrane are investigated. The basic features, including cooperative interactions between channels, are the same as in earlier papers in this series. In particular, the present paper represents a considerable elaboration of part III. The models might apply to artificial or possibly to biological membranes, but particular applications must await further elucidation of the molecular structure and operation of these membranes.

On the theory of ion transport across the nerve membrane. II. Potassium ion kinetics and cooperativity (with x = 4)

We use a tetrahedral model of four interacting protein subunits to represent the K(+) channel or gate in the squid nerve membrane. The kinetic predictions, with varying degrees of cooperativity, are compared with experimental observations, especially those of Hodgkin and Huxley (J. Physiol. 117, 500, 1952) and of Cole and Moore (Biophys. J. 1, 1, 1960). The tentative conclusion reached is that if there is any cooperativity present it must be rather weak. There is no indication here that cooperativity improves the Hodgkin-Huxley assumption of independent "subunits". Other related models will be discussed in Part III. We also find evidence against the suggestion that there is cooperativity between K(+) channels arranged in patches of a two-dimensional lattice.

Cooperative effects in models of steady-state transport across membranes. II. Oscillating phase transition

A simple model for steady-state transport of a neutral molecule across a membrane is investigated in a preliminary way. In this model, there is a possible conformation change in each membrane unit which alters the access of the binding site for the transported molecule from one bath to the other. Thus, transport cannot be accomplished without a conformation change. Furthermore, we assume a cooperative interaction between nearest-neighbor membrane units in the same conformation. Then, with suitable rate constants and bath concentrations, and if the interaction energy is large enough, the membrane will oscillate back and forth between the two conformational phases, producing a surge of flux in each cycle. The period of the cycle depends on the times necessary to nucleate the two phase transitions.