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

Properties of some three-state, steady-state Ising systems, according to the Bragg-Williams approximation

We consider the steady-state properties of a lattice of three-state, cycling enzyme molecules, with nearest-neighbor interactions treated by the Bragg-Williams (mean field) approximation. Only a few particular cases are examined, but these illustrate the rich phase-transition possibilities of this class of systems. "Bifurcation" cases were treated in a previous paper; the present examples are of the nonbifurcation type. However, a few new theoretical properties of bifurcation cases are included.

Three-state, steady-state Ising systems: Monte Carlo and Bragg-Williams treatments

In two earlier papers, the steady-state critical and phase-transition properties of a lattice of three-state enzyme molecules were studied by using the "closed" Bragg-Williams (BW), or mean field, approximation. The "open" BW and Monte Carlo methods are applied to the same problem in this paper by using finite lattices. The open BW treatment provides a way of locating the cut across a van der Waals type of loop encountered in a phase transition in the closed BW system. Thermodynamic-like methods cannot be used for this purpose as they can with two-state, steady-state systems.

Maxwell-type constructions for multiple nonequilibrium steady states

A stochastic analysis is used to obtain Maxwell-type constructions for multiple nonequilibrium steady states in homogeneous systems. The stochastic theory shows that homogeneous molecular fluctuations can nucleate the Maxwell construction only on a time scale that is the order of a Poincaré recurrence time and indicates that hysteresis should be observed. The nonrelevance of homogeneous fluctuations for systems that undergo phase separation or in which spatial structures occur is discussed.

Steady-state phase or cooperative transitions between biochemical cycles

In a steady-state lattice of interacting enzyme molecules that have a multicycle kinetic diagram, a cooperative or phase transition may involve not only the conventional sudden change in the relative importance of the different states of a molecule but also a sudden change in the dominant cycles of the diagram. The latter effect implies a sudden switch in the dominant biochemistry (e.g., a sudden onset of active transport). An explicit example is discussed.

A theoretical analysis of binding to the Ca2+-specific sites on troponin incorporated into thin filaments

Recent data on the binding of Ca2+ to the specific sites on troponin, alone, in regulated actin, and in regulated actomyosin, as well as data on the Ca2+ activation of the actomyosin ATPase (Grabarek, Z., J. Grabarek, P.C. Leavis, and J. Gergely, 1983, J. Biol. Chem., 258:14098-14102.), are analyzed on the basis of a model used previously for qualitative theoretical studies of the Ca2+ activation of muscle contraction (Shiner and Solaro, 1982). The data allow and require an extension of the model to consider the effects of tropomyosin explicitly. Three major results of the analysis are at variance with previous investigations. A repulsive interaction between tropomyosins; and an attractive interaction between actins (or myosin heads attached to actin) are found, whereas others have found or assumed an attractive tropomyosin-tropomyosin interaction and no actin-actin interaction. The parameter values found here predict hysteresis under the conditions of the ATPase experiments; no other existing model for the interactions manifest in the Ca2+ activation of contraction can predict hysteresis. The prediction is of increased interest in light of experimental reports of hysteresis in the Ca2+ activation of isometric force (Ridgeway, E. B., A. M. Gordon, and D. A. Martyn, 1983, Science (Wash. DC), 219:1075-1077; Gordon, A. M., E. B. Ridgeway, and D. A. Martyn, 1984, Plenum Publishing Corp., New York, 553-563; Brandt, P. W., B. Gluck, M. Mini, and C. Cerri, 1985, J. Mus. Res. Cell Motil. 6:197-205.).

Mechanism of action of gentamicin components. Characteristics of their binding to Escherichia coli ribosomes

The binding of gentamicin (Gm) to Escherichia coli ribosomes and ribosomal subunits has been studied. By means of equilibrium dialysis and of statistical interpretation of the data it was found that [3H]gentamicin C2 and 6'-N-[3H]methylgentamicin C1a interact with three classes of sites on tight-coupled 70-S species: a first class concerning the tight and non-cooperative interaction with one drug molecule (Kd = 0.6 microM), a second class in which about five Gm molecules bind cooperatively (mean Kd = 10 microM), and a third class of very high capacity in which up to 70 drug molecules may interact. The extreme cooperativity of the third class of sites induces such an increase in the affinity for Gm that it may allow the shift of molecules already bound from high-affinity sites towards lower-affinity sites. The alteration of a ribosomal protein, L6, in a gentamicin-resistant mutant of E. coli abolished the multiclass and the cooperative aspects of ribosomes--gentamicin interaction. The large ribosomal subunits from E. coli MRE 600 strain interact cooperatively with Gm, whereas 50-S particles from the resistant mutant bind the drug in a diffuse way with high capacity and low affinity. The small subunits from both strains behave identically towards Gm. A good correlation is observed in comparing the gentamicin concentrations capable of saturating the different ribosomal classes of sites with concentrations inducing its multiphasic effects on protein synthesis.

Two elementary models for the regulation of skeletal muscle contraction by calcium

It is shown by use of an extremely simple explicit two-state model that two basic ideas may be sufficient to understand at least qualitatively the sensitive activation of isometric muscle contraction by Ca2+. (a) Ca2+ binds much more strongly on troponin if myosin is already attached to actin. The steady state analogue of this is that the single rate constant (in the two-state model) for myosin attachment plus Pi release is much larger if Ca2+ is bound to troponin. (b) End-to-end tropomyosin interactions are responsible for positive cooperativity. Although these ideas seem to be sufficient, this of course does not mean that they are necessary. These same ingredients were used in two previous, more elaborate models for the cooperative equilibrium binding of myosin subfragment-1 on actin-tropomyosin-troponin, with and without Ca2+, and for a study of the steady state ATPase activity of the same system. Essentially as an appendix, the above-mentioned simple treatment is extended to a somewhat more realistic and complicated model of isometric contraction.

Calcium and proton dependence of sarcoplasmic reticulum ATPase

The influence of Ca2+ and H+ concentrations on the sequential reactions of the ATPase cycle was studied by a series of pre-steady state and steady state experiments with sarcoplasmic reticulum vesicles. It is shown that H+ competition with calcium binding results in a reduced population of activated enzyme, which is manifested by a lower level of phosphorylated enzyme intermediate following addition of ATP. Further effects of Ca2+ and H+ are demonstrated on the progression of the phosphoenzyme through the reaction cycle and on the final hydrolytic cleavage of Pi. The overall dependence of steady state ATP flux on Ca2+ and H+ concentrations in leaky vesicles is expressed by a series of curves showing that as the H+ concentration is raised higher Ca2+ concentrations are required to obtain half-maximal ATP fluxes. At saturating Ca2+, maximal ATP fluxes are observed at an intermediate H+ concentration (pH 7.2), while lower levels are obtained as the H+ concentration is reduced (to pH 8) or increased (to pH 6). A preliminary model is then proposed based on the presence of two interacting domains permitting competitive binding of Ca2+ or H+, per each catalytic site undergoing phosphorylation by ATP. The model considers three main states and thirteen substates (depending on the occupancy of the binding sites in each state by Ca2+, H+, or neither) in the progression of the ATP cycle, coupled to transport of Ca2+ and counter transport of H+ in leaky vesicles. Considering the preliminary nature of the model and the experimental scatter, a rather satisfactory agreement is noted between a family of curves generated by theoretical analysis and the ATP flux curves obtained experimentally.

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.

Theoretical models for cooperative steady-state ATPase activity of myosin subfragment-1 on regulated actin

Recent theoretical work on the cooperative equilibrium binding of myosin subfragment-1-ADP to regulated actin, as influenced by Ca2+, is extended here to the cooperative steady-state ATPase activity of myosin subfragment-1 on regulated actin. Exact solution of the general steady-state problem will require Monte Carlo calculations. Three interrelated special cases are discussed in some detail and sample computer (not Monte Carlo) solutions are given. The eventual objective is to apply these considerations to in vitro experimental data and to in vivo muscle models.

Steady-state head-to-tail polymerization of actin or microtubules. II. Two-state and three-state kinetic cycles

In a previous paper, bioenergetic aspects of head-to-tail polymerization for a two-state actin ATPase cycle were discussed. In section 2, here, the steady-state polymer length distribution for this case is derived. The distribution has the same mathematical form as at equilibrium, but the parameters are different. In section 3, both bioenergetic topics and the polymer length distribution are considered for the more general and realistic case of a three-state actin ATPase cycle. Again, the mathematical form of the steady-state distribution is the same as at equilibrium, but the parameters are more complicated. In section 4, the question is examined of how much the mean and variance of a polymer length distribution, obtained from a finite experimental sample of polymer (aggregate) molecules, would be expected to deviate from the true mean and variance (from an infinite sample). Also considered briefly in section 4 is the effect of hard polymer-polymer interactions on the equilibrium polymer length distribution, at finite polymer concentrations.

Subunit neighbor interactions in enzyme kinetics: half-of-the-sites reactivity in a dimer

We consider an isologous enzyme dimer in which the subunits, if operating independently, would obey Michaelis-Menten kinetics. However, because of neighbor interactions, the rate constants of the kinetic cycle in either subunit depend on the state (E or ES) of the other subunit. The steady-state behavior of this dimer system, with interactions, is investigated. In what is probably the most important special case, ES x ES is destabilized considerably by the neighbor interaction compared to E x ES. This leads to half-of-the-sites reactivity (one subunit is in state ES; the other subunit cycles between E and ES), negative cooperativity, and a considerable enhancement of enzyme activity relative to the activity of independent subunits.

Theoretical model for the cooperative equilibrium binding of myosin subfragment 1 to the actin-troponin-tropomyosin complex

Recent experimental data on the equilibrium binding of myosin subfragment 1 (S-1) to regulated actin filaments in the presence and in the absence of Ca(2+) are analyzed by using a linear Ising model. In the model, each tropomyosin-troponin unit (including seven sites on the actin filament) can be in one of two possible states, which have different intrinsic free energies and different binding constants for S-1. Bound S-1 molecules do not interact with each other. There are nearest-neighbor (pair) interactions between these units that depend on the state of each member of the pair and on the number of Ca(2+) bound to one member of the pair. There are two sources of positive cooperativity in this system: the fact that seven actin sites change state together as part of a single unit; and the existence of attractive nearest-neighbor interactions between units. Parameters in the model are evaluated by fitting the data, both in the presence and in the absence of Ca(2+). Several extensions of this model are discussed.

Approximate steady-state properties of lattices of interacting three-state enzyme molecules: a novel phase transition

Previous work on the cooperative behavior of lattices of interacting two-state enzyme molecules at steady state is extended here to interacting three-state enzyme molecules with a one-way cycle. The Bragg-Williams (mean field) approximation is used. A phase-transition example with a bifurcation point is discussed. Compared to conventional phase transitions (with a van der Waals loop), several new and complicated features appear. A second paper on this subject will contain a number of other examples of three-state systems.

Interacting enzyme systems at steady state: location of the phase transition in approximations of the mean field type

We consider a phase transition "loop," obtained from a mean field type of approximate treatment of a closed steady-state Ising system. Where is the cut (stable path) across the loop located? The general procedure, in answering this question, is to pass to an open version of the same system and use the cut that appears automatically in this case (no loop is possible in an open system). This is equivalent to finding the point at which the two phases have equal total probability in the open system. It is shown here that this procedure, when applied to a system of two-state enzyme molecules, is formally equivalent to well-known thermodynamic methods (Maxwell's theorem, etc.). These can be applied directly to the closed system without considering the open system explicitly. However, for enzyme molecules with more than two states, the "thermodynamic" method generally fails and one must fall back on the open system procedure mentioned above. Practical implementation of this procedure is not easy.

Unsymmetrical and concerted examples of the effect of enzyme--enzyme interactions on steady-state enzyme kinetics

In previous papers of this series, emphasis has been placed on the steady-state phase transition and critical properties of large lattices of interacting, symmetrical, and identical enzyme molecules. The present paper is concerned with a number of examples of enzyme--enzyme interactions that do not belong to the class of models of the earlier papers. These are more biochemically oriented and include heterologous dimers, a linear chain with unsymmetrical interactions, and concerted isologous dimers (half-the-sites reactivity).

"Viral" expansion of enzyme flux and use of quasi-chemical approximation for two-state enzymes with enzyme-enzyme interactions

Two examples of enzyme systems with interactions, at steady state, are treated here. In both cases, the enzyme cycle has two states and quasi-equilibrium in spatial distributions obtains at steady state (because f alpha + f beta = 1). The first example is a dilute solution of enzyme molecules in a solvent. The flux (turnover) per molecule is expanded in powers of the enzyme concentration (a "viral" expansion). Aggregation of the enzyme molecules in solution is considered as a special case. In the second example, we treat an arbitrary lattice of enzyme molecules, with nearest-neighbor interactions, using the well-known quasi-chemical approximation. The flux per molecule is obtained. Critical behavior and hysteresis are illustrated.