Local Concentration Deviations in a Glycolytic Model of Nonequilibrium Thermodynamics: Emergence of Bistability from Thermodynamic Theories due to Violation of the Lyapunov Stability Theory

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 (sigma) 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, which however never exists in a living cell. In the unstable steady states in this model reaction, as for example, along a limit cycle trajectory not obeying any Lyapunov stability postulate, the two bistable e.p. branches characterized by two forms of deviation product function delta(s)delta(p) may be identified separated by a large delta(p) > 0 value (p being scaled concentration of the species P) for d sigma > 0 and d sigma < 0, respectively, which may be generalized as LCB principle for bistability/hysteresis in this model reaction. The method outlined here may be applied safely to other well-known two-variable models of chemistry/biology. For an appropriate three-variable model with one variable in a pseudo-steady-state approximation, this method may be applicable, albeit with some constraints.