Cooperative effects in models of steady-state transport across membranes. 3. Simulation of potassium ion transport in nerve

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.

The effects of modifiers on enzyme catalysis: a non-classical nearest neighbor approach

We present a nearest neighbor lattice model of the effects of modifiers on two-state enzyme catalysis of the reaction S in equilibrium with p. We do not in general make the assumptions of the classical approach to cooperative catalysis that yield (1) adsorption isotherms of the same form as those for the corresponding equilibrium system and (2) a rate of the catalyzed reaction proportional to the number of occupied catalytic sites. Closed form results are obtained for two approximations, the Bragg-Williams and the quasi-chemical. The latter requires (1), but is exact for several simple cases, including the concerted model, under this condition. Under (1) it is found that an interaction between modifier and catalytic sites, whether attractive or repulsive, increases the magnitudes of the slopes of the adsorption isotherms but that interactions between identical sites (catalytic or modifier) increase these magnitudes if attractive and decrease them if repulsive. Thus, the former interaction allows for phase transitions if sufficiently attractive or repulsive, but the latter only if sufficiently attractive. Herein also lies the explanation for why the concerted model displays only "positive cooperativity". It is further seen that it is not possible to classify a modifier as an activator or inhibitor of the catalyzed reaction solely on the basis of the sign of the interaction energy between catalytic and modifier sites. For a given energy, the rate of the reaction may increase or decrease in response to the modifier, or it may respond biphasically. Similarly, the rate may respond biphasically to the activities of s or p, leading to instabilities. Thus, possibilities of multiple nonequilibrium stationary states or spatio-temporal patterns are raised.

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]

On the theory of ion transport across the nerve membrane. VI. Free energy and activation free energies of conformational change

Empirical functions, such as n(infinity)(V) and tau(n)(V) (of the Hodgkin-Huxley type), can be recast in terms of more fundamental functions F(V) (related to a conformational free energy change) and theta(V) (related to the corresponding free energies of activation). Examples of F(V) and theta(V) are given, for squid and frog node. F(V) is essentially a quadratic function of V. The possible molecular origin, for protein-like subunits, of the linear (e.g., net charge) and quadratic (e.g., polarizability) terms in F(V) is discussed. The F(V), theta(V) kind of analysis leads rather automatically to a simple explanation of the well-known approximate coincidence in location (V value) of the maximum in tau(n)(V) (time constant) and the steeply rising part of n(infinity)(V) (also m, 1 - h).

On the theory of ion transport across the nerve membrane. 3. Potassium ion kinetics and cooperativity (with x=4,6,9)

The calculations of Part II of this series have been extended to square (x = 4) and octahedral (x = 6) arrangements of subunits in a potassium channel (or gate). The conclusion is the same as before: experimental induction and superposition properties of gK(t), on depolarization, seem to rule out any significant degree of interaction or cooperativity between the (protein?) subunits of K(+) channel. Calculations for x = 4, 6, and 9 have also been made for a square lattice of interacting channels (periodic boundary conditions). Because of apparent rapid convergence with x, it seems fairly safe to conclude that this model is unsatisfactory. There is some difficulty with superposition but the principal shortcoming is a failure to produce induction behavior. Aggregation models for the K(+) channel are also discussed briefly here. They, too, appear rather unpromising (for the same reasons as seem to exclude conformational cooperativity within a channel).

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.