Regulatory functions of allosteric enzymes in far-from-equilibrium systems

A theoretical study is made on catalytic activities of allosteric enzymes in non-equilibrium systems. It is demonstrated that the amount of chemical flow catalyzed by allosteric enzymes in systems maintained far-from-equilibrium can be qualitatively different from that familiar in the near-equilibrium situations. To be more specific, a study is made of a system containing two chemical species, S and P, and an allosteric enzyme, E, which catalyzes the reaction of the interconversion between them. This system interacts with its environment in a way characterized by a set of controlled parameters. By this interaction the system is maintained far-from-equilibrium. More than one steady state is possible for a certain range of controlled parameters. For continuous changes of the controlled parameters, discontinuous transitions between the multiple steady states can occur. This new aspect of the enzyme kinetics of allosteric proteins may play a role in the regulation of metabolic flows within living cells.     

Simple chemical reaction systems with limit cycle behaviour

It is shown that for exhibiting limit cycle behaviour a two-component chemical reaction system has to involve at least three reactions among which one must be autocatalytic of the type 2X + … ? 3X + … Under this condition, possible candidates for chemical limit cycle systems are selected by postulating their steady state to be an instable focus. This procedure can be reduced to the selection of appropriate stoichiometric coefficients and is readily performed by a medium size computer. The result is quite a lot of limit cycle systems which are altogether simpler than, for example, the “Brusselator” with its number of four reactions. One of the results is briefly discussed.     

Steroid hormone antagonism and a cyclic model of receptor kinetics

Steady state agonist-antagonist relations have been derived for a general version of a cyclic model of glucocorticoid-receptor kinetics. The model was previously shown to account quantitatively for the transient and steady state distribution of hormone-receptor complexes formed in thymus cells by several glucocorticoids. Agonist-antagonist properties of a steroid in the model are expressed quantitatively by its "agonist activity" A, the steady state ratio of nuclear-bound to total complexes it forms. For a pure agonist A = 1, for a pure antagonist A = 0. This ratio is found to be independent of steroid concentration and a function only of the rate constants of reactions involving complexes formed by the steroid. Analysis of the dependence of A on each rate constant reveals how each reaction in the cyclic model--activation, nuclear binding, dissociation of activated and nuclear-bound complexes--influences antagonist properties. The steady state interaction of an antagonist with an agonist is shown to be governed by relations that are indistinguishable from competition relations for the simplest equilibrium system, and to yield dose-response curves that are very similar to those produced by two-state allosteric models of steroid hormone antagonism, despite the fact that the cyclic model includes no allosteric mechanisms. With steroids for which relevant rate constants can be measured, the model is directly testable. Limitations of the model arise from lack of information about the nuclear events that lead to biological activity following binding of activated complexes to the nucleus.     

Non-equilibrium thermodynamics of marginal- and conditional-open chemical systems

Every open chemical system treated in this paper is restricted to the case involving a sequence of monomolecular reactions. Various kinds of probability distribution governing it are introduced according to the situations in which it is placed. The chemical system subject to marginal distribution is given the term marginal-open system MO. The open chemical system ō discussed by Nicolis and Babloyantz can be regarded as the limiting system of MO. For an open chemical system, itself in contact with an external reservoir of finite volume, the probability distribution conditioned on the marginal distribution for the external reservoir in an arbitrarily fixed state is more appropriate. Such an open chemical system is called a conditional-open system CO. However, in the case of the external reservoir of infinite volume, although it is not certainly trivial, another conditional probability distribution has to be proposed; it is derived on the hypothesis that the probability distribution for an arbitrary total number of molecules in the open chemical system is known. The open chemical system so specified is called conditional-open system $\text{C}\text{O}\text{?}$. It is shown that for each system $\text{MO, CO}\text{and}\text{C}\text{O}\text{?}$ the change of entropy starting from the steady state provides a Liapunov function under some conditions and that the steady state is asymptotically stable. The relation of the entropy change to non-equilibrium fluctuations of chemical components in each system is discussed in comparison with that in the corresponding open chemical system ō, for which the steady state surely exists and is always stable. It is shown that the concept of $\text{C}\text{O}\text{?}$ is useful for investigating the phenomenon of steady-state coupling.     

Implications of Network Topology on Stability

In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network’s “influence topology”, a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network’s steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network’s stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.     

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

On the role of enzyme cooperativity in metabolic oscillations: analysis of the Hill coefficient in a model for glycolytic periodicities.

The role of enzyme cooperativity in the mechanism of metabolic oscillations is analyzed in a concerted allosteric model for the phosphofructokinase reaction. This model of a dimer enzyme activated by the reaction product accounts quantitatively for glycolytic periodicities observed in yeast and muscle. The Hill coefficient characteristic of enzyme-substrate interactions is determined in the model, both at the steady state and in the course of sustained oscillations. Positive cooperativity is a prerequisite for periodic behavior. A necessary condition for oscillation in a dimer K system is a Hill coefficient larger than 1.6 at the unstable stationary state. The analysis suggests that positive as well as negative effectors of phosphofructokinase inhibit glycolytic oscillations by inducing a decrease in enzyme cooperativity. The results are discussed with respect to glycolytic and other metabolic periodicities.     

Specific features in realization of the principle of minimum energy dissipation during individual development

Realization of the principle of minimum energy dissipation (Prigogine??s theorem) during individual development has been analyzed. This analysis has suggested the following reformulation of this principle for living objects: when environmental conditions are constant, the living system evolves to a current steady state in such a way that the difference between entropy production and entropy flow (?? _u function) is positive and constantly decreases near the steady state, approaching zero. In turn, the current steady state tends to a final steady state in such a way that the difference between the specific entropy productions in an organism and its environment tends to be minimal. In general, individual development completely agrees with the law of entropy increase (second law of thermodynamics).     

Properties of solution of the diffusion-reaction equation

An arbitrary set of chemical reactions is considered to occur among chemical speciesX _i . In a closed uniform reaction system certain linear combinations of the concentrations of theX _i are constants. The general construction of all such linear combinations with non-negative coefficients is given in terms of the molecular formulae for theX _i . It is shown that to each such linear combination there corresponds another which is a harmonic function when the reactions take place in an open spatially distributed stationary reaction system of arbitrary shape. Under the usual boundary conditions these harmonic functions are constants. With some restrictions upon the diffusion and permeability coefficients these constants are evaluated. This evaluation is the basis for relations between the total concentration of a given chemical group (e.g., the sum of the concentrations of a free molecule, or ion, and its various bound forms) in the reaction system, and in the surrounding medium. The bearing of these relations on apparent active transport is noted and illustrated.     

A transfer-function representation for regulatory responses of a controlled metabolic pathway

A transfer-function representation for the response of a controlled metabolic pathway to the changes in influx and efflux rates of metabolites is formulated to describe analytically and approximately the regulatory behavior of the pathway around a steady state. The pathway model analysed is an open and homogeneous system which consists of two consecutive enzymatic reactions catalyzed by an allosteric enzyme of Monod-Wyman-Changeux (MWC) dimeric model and a Michaelis-Menten-type enzyme, respectively, and undergoes the feedback inhibition by the end product. The rate equation for the system (a system of ordinary differential equations) is linearized about a steady state, so that the responses of the reaction rates to the changes in influx rate of the substrate and efflux rate of the end product are expressed in a form of transfer function. The formulation leads to the transfer function for the response of production rate of the end product to the change in its efflux rate to clarify the regulatory response of feedback mechanism in controlled metabolic pathways. The relationship among the chemical species in the system at steady stete also supports a reasonable assumption that the regulatory mechanisms in metabolic pathways are to control the production of end product against the change in its demand from the cellular environments.     

TRH influences the pterin cofactor- and time-dependent instabilities of rat raphe tryptophan hydroxylase activity assessed under far-from-equilibrium conditions

Systematic effects on the dynamics of rat raphé tryptophan hydroxylase were observed in the presence of thyrotropin-releasing hormone under the far-from-equilibrium conditions of a kinetic scattering paradigm. The peptide reduced the amplitude of velocity fluctuations across fine-grained increases in cofactor or time and induced rare, high-amplitude irregularities suggesting phase transitions at relatively long wavelengths. Respectively, these two effects of the neuropeptide resemble the statistical changes observed in the same enzyme system in the presence of the lithium ion and the tricyclic antidepressant chlorimipramine under comparable assay conditions. Neither the subtleties of control dynamics nor their responses to the peptide were demonstrable under the saturating conditions of Michaelis-Menten or allosteric kinetics. These findings have possible implications for neurotransmitter regulation in view of the most current information about dynamical interactions among proteins, peptides, and ion ligands in aqueous environment.     

Concordant chemical reaction networks and the Species-Reaction Graph

In a recent paper it was shown that, for chemical reaction networks possessing a subtle structural property called concordance, dynamical behavior of a very circumscribed (and largely stable) kind is enforced, so long as the kinetics lies within the very broad and natural weakly monotonic class. In particular, multiple equilibria are precluded, as are degenerate positive equilibria. Moreover, under certain circumstances, also related to concordance, all real eigenvalues associated with a positive equilibrium are negative. Although concordance of a reaction network can be decided by readily available computational means, we show here that, when a nondegenerate network’s Species-Reaction Graph satisfies certain mild conditions, concordance and its dynamical consequences are ensured. These conditions are weaker than earlier ones invoked to establish kinetic system injectivity, which, in turn, is just one ramification of network concordance. Because the Species-Reaction Graph resembles pathway depictions often drawn by biochemists, results here expand the possibility of inferring significant dynamical information directly from standard biochemical reaction diagrams.     

Chaotic oscillations in a simple collapsible-tube model.

A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.     

Characterization of the interactions of NADH with the dimeric and tetrameric states of methaemoglobin

The binding of NADH to the dimeric (αβ) and tetrameric (α₂β₂) states of human aquomethaemoglobin has been characterized by sedimentation equilibrium studies of the effect of the concentration of free ligand on the macromolecular state of the haemoprotein. Both macromolecular states of aquomethaemoglobin exhibit a single binding site for NADH, which interacts approximately tenfold more strongly (6000 cf. 700 M⁻¹) with the tetramer under the conditions studied (pH 6.0, I 0.10). Because the structure of aquomethaemoglobin resembles that of the deoxy state of haemoglobin, there is a high probability that organic phosphates also bind to dimeric deoxyhaemoglobin, a phenomenon which is not considered in thermodynamic treatments of the interplay between oxygen binding and its allosteric inhibition by 2,3-bisphosphoglycerate. Fortunately, the equilibrium constant for deoxyhaemoglobin self-association is so large that neglect of the interaction between allosteric inhibitor and dimeric haemoglobin is an oversight that should have no deleterious implications in the resultant thermodynamic analysis of the interplay between the preferential interactions of oxygen and organic phosphate with the various macromolecular states of deoxyhaemoglobin.     

New method for calculating the dissipation parameters in ultrafast biochemical reactions from protein crystal structure data

A new method of calculating the spectral function of system-bath interaction during an elementary biochemical reaction is proposed. This method was applied to the primary electron transfer in the photosynthetic reaction center of purple bacteria Rhodobacter sphaeroides. The calculated spectral functions differ significantly from the commonly used ohmic function. It is shown that the unidirectionality of the electron transfer along the A-branch in the reaction center of Rh. sphaeroides can be caused by the asymmetry of the reaction system interaction with the protein environment.     

1,10‐Phenanthroline–H2O2–KSCN–CuSO4–NaOH oscillating chemiluminescence system

In this paper, oscillating chemiluminescence (CL), 1,10‐phenanthroline H₂O₂–KSCN–CuSO₄–NaOH system, was studied in a batch reactor. The system described is a novel, slowly damped oscillating CL system, generated by coupling the well‐known Epstein–Orban, H₂O₂–KSCN–CuSO₄–NaOH chemical oscillator reaction with the CL reaction involving the oxidation of 1,10‐phenanthroline by hydrogen peroxide, catalyzed by copper(II) in alkaline medium. In this system, the CL reaction acts as a detector or indicator system of the far‐from‐equilibrium dynamic system. Narrow and slightly asymmetric light pulses of 1.2 s half‐width are emitted at 440 nm with an emitted light time of 200–1000 s, induction period of 3.5–357 s and oscillation period of 28–304 s depending on the reagent concentrations. In this report the effect of the concentration variation of components involved in the oscillating CL system on the induction period, the oscillation period and amplitude was investigated and the parameters were plotted with respect to reagent concentrations. Copper concentration showed a significant effect on the oscillation period. The possible mechanism for the oscillating CL reaction was also discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

The synergetic enzyme theory of muscular contraction: II Relation between Hill's equations and functions of two-headed myosin.

Based on the assumption of nonidentical two heads of myosin it is pointed out that a strong motive force is generated in actomyosin pair only when ATP-decomposition occurs co-operatively at the both heads and that the tension-independent part of shortening heat is liberated when an ATP molecule is decomposed only at the burst head. These two actions of actomyosin pair are related to the two states of force-generator in Huxley-Simmons' model. Elementary cycles at different positions in a sarcomere are interacted each other through feedback loop via sliding motion of muscular filaments. Due to this synergetic interaction the rate constant for the rate-determining step of elementary cycle has a dependence on velocity v of shortening such as k = k ° + κv. From these functions and properties of actomyosin system in vivo, the following properties of muscle are explained consistently in a quantitative manner: (1) Hill's equation on the relationship between tension and velocity of shortening, (2) damped oscillations in tension and in muscular length around steady state, (3) Hill's energy equation improved in 1964, (4) the chemical equivalence of shortening heat, (5) the influence of tension on the incorporation of radio-active phosphate into ATP and (6) the asymmetric activation by actomyosin system only for the forward reaction, the decomposition of ATP.     

Prolyl Isomerization as a Molecular Memory in the Allosteric Regulation of the Signal Adapter Protein c-CrkII

c-CrkII is a central signal adapter protein. A domain opening/closing reaction between its N- and C-terminal Src homology 3 domains (SH3N and SH3C, respectively) controls signal propagation from upstream tyrosine kinases to downstream targets. In chicken but not in human c-CrkII, opening/closing is coupled with cis/trans isomerization at Pro-238 in SH3C. Here, we used advanced double-mixing experiments and kinetic simulations to uncover dynamic domain interactions in c-CrkII and to elucidate how they are linked with cis/trans isomerization and how this regulates substrate binding to SH3N. Pro-238 trans → cis isomerization is not a simple on/off switch but converts chicken c-CrkII from a high affinity to a low affinity form. We present a double-box model that describes c-CrkII as an allosteric system consisting of an open, high affinity R state and a closed, low affinity T state. Coupling of the T-R transition with an intrinsically slow prolyl isomerization provides c-CrkII with a kinetic memory and possibly functions as a molecular attenuator during signal transduction.