Steady state far from equilibrium

Turner, J. S. (1974). Finite fluctuations and multiple steady states far from equilibrium. Bull. Math. Biol., 36 (2), 205-13. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

In vivo, under steady-state conditions, there is a net flux from left to right because there is a continuous supply of A and removal of D. In practice, there are invariably one or more nonequilibrium reactions in a metabolic pathway, where the reactants are present in concentrations that are far from equilibrium. In attempting to reach equilibrium, large losses of free energy occur as heat, making this type of reaction essentially irreversible, eg. 

Steady states may also arise under conditions that are far from equilibrium. If the deviation becomes larger than a critical value, and the system is fed by a steady inflow that keeps the free energy high (and the entropy low), it may become unstable and start to oscillate, or switch chaotically and unpredictably between steady state levels. 

King et a/.54,138,155 applied the steady state approximation to both systems. In the case of Fe(CO)5, as shown in Scheme 7b, both C02 and H2 production rates should be the same, and so 2[Fe(CO)5][OH-] = fc4[H2Fe(CO)4]. Reactant concentrations are far from equilibrium, and the reactions are assumed to be driven to the right. 

A chemical reaction can be designated as oscillatory, if repeated maxima and minima in the concentration of the intermediates can occur with respect to time (temporal oscillation) or space (spatial oscillation). A chemical system at constant temperature and pressure will approach equilibrium monotonically without overshooting and coming back. In such a chemical system the concentrations of intermediate must either pass through a single maximum or minimum rapidly to reach some steady state value during the course of reaction and oscillations about a final equilibrium state will not be observed. However, if mechanism is sufficiently complex and system is far from equilibrium, repeated maxima and minima in concentrations of intermediate can occur and chemical oscillations may become possible. 

TNC. 18. R. Lefever, G. Nicolis and I. Prigogine, On the occurrence of oscillations around the steady state in systems of chemical reactions far from equilibrium, J. Chem. Phys. 47, 1045-1047 (1967). 

In the course of time open systems that exchange matter and energy with then-environment generally reach a stable steady state. However, as shown by Glansdorff and Prigogine, once the system operates sufficiently far from equilibrium and when its kinetics acquire a nonlinear nature, the steady state may become unstable [15, 18]. Feedback regulatory processes and cooperativity are two major sources of nonlinearity that favor the occurrence of instabilities in biological systems. 

A commonly held belief is that, for an enzyme reaction within a metabolic pathway, a large excess of catalytic capacity relative to a pathway s metabolic flux ensures that a given step is at or near thermodynamic equilibrium. Brooks recently treated the kinetic behavior of reaction schemes one might judge to be at equilibrium, and he showed that individual steps can remain far from equilibrium, even at a high ratio of an enzyme s flux to a pathway s steady-state flux. His calculations indicate that whether a reaction is near equilibrium depends on (a) the overall flux through the enzyme locus and (b) the kinetic parameters of the other enzymes in the pathway. S. P. Brooks (1996) Biochem. Cell Biol. 74, 411. 

Control of polymerizing conditions that allows steady-state production far from equilibrium... 

Other reactions are far from equilibrium in the cell. For example, for the phosphofructokinase-1 (PFK-1) reaction in glycolysis is about 1,000, but Q ([fructose 1,6-bisphosphate] [ADP] / [fructose 6-phosphate] [ATP]) in a typical cell in the steady state is about 0.1 (Table 15-2). It is because the reaction is so far from equilibrium that the process is exergonic under cellular con-... 

In multistep processes such as glycolysis, certain reactions are essentially at equilibrium in the steady state the rates of these substrate-limited reactions rise and fall with substrate concentration. Other reactions are far from equilibrium their rates are too slow to produce instant equilibration of substrate and product. These enzyme-limited reactions are... 

As shown notably by the thermodynamic school of Brussels,9,10 systems maintained far from equilibrium and endowed with appropriate non-linearities and feedback interactions may display such nontrivial behaviors as sustained oscillations and multiple steady states (see papers by I. Prigogine and G. Nicolis in this volume). Even though the structures studied are much simpler than biological systems, this type of work provides a firm fundamental basis, not sufficient but absolutely necessary, for the future understanding of such processes as cell differentiation. Clearly, the sustained oscillations and multiple steady states displayed by simpler systems are related, respectively, with the two types of biological regulation described above. 

In this model (Table 3), substrate, A, is transformed to product, B, by an enzyme, E. The supply of A is large, ensuring far-from-equilibrium conditions. An intermediate, X, is produced autocatalytically, and degraded by the enzyme. (This feature of the model makes it unrealistic, as few autocatalytic processes arise this way.) The steady-state equation for X is cubic, and has three roots, or solutions for certain values of the parameters. One of the solutions is unstable a real system cannot maintain a steady-state concentration, [X]ss, with a value corresponding to this solution. Therefore, before [X]ss approaches such a value too closely, it jumps to a different value, corresponding to one of the stable solutions. This behavior leads, to hysteresis, as shown in Fig. 1. 

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