Linear thermodynamic analysis of the reversible Selkov model: An interpretation of the Chatelier-like principle for local concentration fluctuations near thermodynamic equilibrium
01 Mar 1990-Journal of Chemical Physics (American Institute of Physics)-Vol. 92, Iss: 5, pp 3058-3061
TL;DR: In this article, it was shown that at thermodynamic equilibrium, the product δS δP is always a positive quantity which appears to be a Chatelier-like principle for local concentration fluctuation applicable to the autocatalytic step S ⇄ P of this model.
Abstract: Thermodynamic analysis of the reversible Selkov model (a simple kinetic model describing glycolytic oscillations) has been done by an entropy production technique of Prigogine and it is shown that only the autocatalytic step can destabilize the steady state in this model. It is derived that at thermodynamic equilibrium, the product δS δP is always a positive quantity which appears to be a Chatelier‐like principle for local concentration fluctuation applicable to the autocatalytic step S ⇄ P of this model.
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TL;DR: In this paper, an amplitude equation was derived for a model glycolytic diffusion-reaction system, and linear stability analysis of this amplitude equation interprets the structural transitions and stability of various forms of Turing structures.
Abstract: For a model glycolytic diffusion-reaction system, an amplitude equation has been derived in the framework of a weakly nonlinear theory. The linear stability analysis of this amplitude equation interprets the structural transitions and stability of various forms of Turing structures. This amplitude equation also conforms to the expectation that time-invariant amplitudes in Turing structures are independent of complexing reaction with the activator species, whereas complexing reaction strongly influences Hopf-wave bifurcation.
26 citations
TL;DR: For a reaction-diffusion system of glycolytic oscillations containing analytical steady state solution in complicated algebraic form, Turing instability condition and the critical wavenumber at the Turing bifurcation point, have been derived by a linear stability analysis as discussed by the authors.
Abstract: For a reaction-diffusion system of glycolytic oscillations containing analytical steady state solution in complicated algebraic form, Turing instability condition and the critical wavenumber at the Turing bifurcation point, have been derived by a linear stability analysis. In the framework of a weakly nonlinear theory, these relations have been subsequently used to derive an amplitude equation, which interprets the structural transitions and stability of various forms of Turing structures. Amplitude equation also conforms to the expectation that time-invariant amplitudes are independent of complexing reaction with the activator species.
19 citations
TL;DR: In this article, the authors analyze the features of oscillation in a well-studied biochemical model of glycolysis with autocatalytic kinetic step and explore the sensitivity analysis technique to assess degree of susceptibility of the system with respect to different input parameters.
Abstract: Oscillation is ubiquitous and important in both bio-engineering and biosystems. We here analyze the features of oscillation in a well-studied biochemical model of glycolysis with autocatalytic kinetic step. This Selkov model is followed deterministically using mass action kinetics. We explore the sensitivity analysis technique to assess degree of susceptibility of the system with respect to different input parameters for a thorough understanding of the merit of individual parameters controlling the core dynamics. Emphasis is concentrated on the system’s biological response via oscillation, bracketing the range of perturbation allowed to the kinetic parameters. An interesting observation is the switchover of the dynamics from non-oscillating to damping followed by stable sustained undulation with one or two controlling kinetic parameters which are quite appealing. The result may be illuminating in understanding the dynamics behind oscillations in biochemical systems in general. A key outcome of the study is the prioritization of most sensitive parameter for the network model controlling the dynamical features via scatterplot analysis.
2 citations
TL;DR: This theory interprets the appropriate limit of validity of the generalized Le Chatelier-Braun (LCB) principle for stable nonequilibrium and equilibrium steady states obeying the Lyapunov stability postulate and its implications in terms of the local concentration deviations of the reacting intermediate species S(ATP) and P(ADP) in response to external excitations.
Abstract: The thermodynamics of nonequilibrium states in the reversible Sel’kov model, a mathematical model of glycolytic oscillations, is reported in terms of the Lyapunov properties of the second differential of its local equilibrium entropy (S) and that of its entropy production (e.p.) function (σ) per unit volume. This theory interprets the appropriate limit of validity of the generalized Le Chatelier–Braun (LCB) principle for stable nonequilibrium and equilibrium steady states obeying the Lyapunov stability postulate and its implications in terms of the local concentration deviations of the reacting intermediate species S(ATP) and P(ADP) in response to external excitations. The local concentration deviations of the reacting intermediates are reported to be asymmetric for stable steady states in this model system obeying the Lyapunov stability postulate out of equilibrium (both linear and nonlinear domains), whereas symmetrical local concentration deviations prevail at the state of thermodynamic equilibrium, wh...
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TL;DR: In this paper, the thermodynamics of nonequilibrium states in the reversible Selkov model are reported and the reaction velocity and entropy production as a function of reaction affinity are computed.
Abstract: The thermodynamics of nonequilibrium states in the reversible Selkov model is reported. The reaction velocity and entropy production as a function of reaction affinity are computed.
2 citations
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TL;DR: It is shown that nonequilibrium may become a source of order and that irreversible processes may lead to a new type of dynamic states of matter called "dissipative structures" and the thermodynamic theory of such structures is outlined.
Abstract: Fundamental conceptual problems that arise from the macroscopic and microscopic aspects of the second law of thermodynamics are considered. It is shown that nonequilibrium may become a source of order and that irreversible processes may lead to a new type of dynamic states of matter called "dissipative structures." The thermodynamic theory of such structures is outlined. A microscopic definition of irreversible processes is given, and a transformation theory is developed that allows one to introduce nonunitary equations of motion that explicitly display irreversibility and approach to thermodynamic equilibrium. The work of the group at the University of Brussels in these fields is briefly reviewed. In this new development of theoretical chemistry and physics, it is likely that thermodynamic concepts will play an ever-increasing role.
864 citations
TL;DR: A comparison between the model and the phosphofructokinase reaction shows a close resemblance between their dynamical properties, which makes it possible to explain qualitatively most experimental data on single-frequency oscillations in glycolysis.
Abstract: The paper describes a simple kinetic model of an open monosubstrate enzyme reaction with substrate inhibition and product activation. A comparison between the model and the phosphofructokinase reaction shows a close resemblance between their dynamical properties. This makes it possible to explain qualitatively most experimental data on single-frequency oscillations in glycolysis. A mathematical analysis of the model has shown the following.
1In the model, at a definite relationship between the parameters, self-oscillations arise.
2The condition of self-excitation is satisfied more readily with a lower source rate, larger product sink rate constants, lower product-enzyme affinity and higher enzyme activity.
3Self-oscillations exist only in a certain range of values of the parameter determining the degree of substrate inhibition. This range increases with decreasing source rate. Too strong or, conversely, too weak substrate inhibition leads to damped oscillations.
4The period of self-oscillations depends on the degree of substrate inhibition, the source rate, the sink rate constant, the enzyme activity, the affinity of the substrate and the product for the enzyme; it decreases with an increase in these values.
5With an increase in the relative sink rate constant the steady state amplitude of self-oscillations initially increases until a definite maximum is reached and then drops to zero.
6A self-oscillatory state in the phosphofructokinase reaction exists only when the maximum rate of this reaction is essentially higher than the source rate, and lower than the maximum rate of the reactions controlling the sink of the products.
7An experimental investigation of self-oscillations in the phosphofructokinase reaction may be considerably simplified by using a reconstituted system consisting of a small number of reactions with an irreversible sink of the products and artificial substrate supply. In this case the above relationship (section 6) should hold.
675 citations
TL;DR: In this article, the Langevin equation was used to derive the Navier-Stokes equations for the Brownian motion of a particle of arbitrary shape, and these terms and their correlation properties are presented, and then used to obtain the Lagrangian Lagrangians for linearized hydrodynamical equations, which were first proposed by Landau and Lifshitz.
Abstract: The velocity of a particle in Brownian motion as described by the Langevin equation is a stationary Gaussian–Markov process. Similarly, in the formulation of the laws of non‐equilibrium thermodynamics by Onsager and Machlup, the macroscopic variables describing the state of a system lead to an n‐component stationary Gaussian–Markov process, which, in addition, these authors assumed to be even in time. By dropping this assumption, the most general stationary Gaussian–Markov process is discussed. As a consequence, the theory becomes applicable to the linearized hydrodynamical equations and suggests that the Navier–Stokes equations require additional fluctuation terms which were first proposed by Landau and Lifshitz. Such terms and their correlation properties are presented, and these equations are then used to derive the Langevin equation for the Brownian motion of a particle of arbitrary shape.
295 citations
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185 citations