^{1}, Hong Qian

^{2,a)}and Yingfei Yi

^{3,b)}

### Abstract

We investigate the oscillatoryreactiondynamics in a closed isothermal chemical system: the reversible Lotka–Volterra model. The second law of thermodynamics dictates that the system ultimately reaches an equilibrium. Quasistationary oscillations are analyzed while the free energy of the system serves as a global Lyapunov function of the dissipative dynamics. A natural distinction between regions near and far from equilibrium in terms of the free energy can be established. The dynamics is analogous to a nonlinear mechanical system with time-dependent increasing damping. Near equilibrium, no oscillation is possible as dictated by Onsager’s reciprocal symmetry relation. We observe that while the free energy decreases in the closed system’s dynamics, it does not follow the steepest descending path.

We thank Ludovic Jullien for discussion, Richard Field, Robert Mazo, and Marc Roussel for helpful comments and Marc Roussel for carefully reading the manuscript. Y.Y. is partially supported by NSF Grant No. DMS0708331, NSFC Grant No. 10428101, and Changjiang Scholarship from Jilin University.

I. INTRODUCTION

II. REVERSIBLE LOTKA–VOLTERRA REACTIONSYSTEM

A. Standard Lotka–Volterra system

B. Quasisteady oscillation in a closed chemical reactionsystem

C. Driven reversible Lotka–Volterra reactions in an open chemical system

III. DISCUSSION

### Key Topics

- Free energy
- 24.0
- Chemical reactions
- 22.0
- Chemical dynamics
- 16.0
- Free oscillations
- 8.0
- Biochemical reactions
- 7.0

## Figures

Schematic comparison of qualitative dynamics between (a) standard damped oscillation and (b) reversible LV. The filled dots in (a) and (b) represent the equilibrium point of the system. The reversible LV system is 3D [Eq. (2)], with the vertical direction along the axis of the cone being intimately related to the total free energy of the system. The dynamics has two regimes: When far from equilibrium, chemical oscillations occur around the axis. (c) The motion is oscillatory, with a pair of complex eigenvalues and a third real eigenvalue. (e) When near the equilibrium, the motion is a monotonic, exponential relaxation with three real eigenvalues. The switch occurs when the pair of complex eigenvalues is equal and real (d). System in (b) has been termed either quasiplanar oscillation or “corkscrew oscillations” (Ref. 15), depending on the rate of free energy dissipation.

Schematic comparison of qualitative dynamics between (a) standard damped oscillation and (b) reversible LV. The filled dots in (a) and (b) represent the equilibrium point of the system. The reversible LV system is 3D [Eq. (2)], with the vertical direction along the axis of the cone being intimately related to the total free energy of the system. The dynamics has two regimes: When far from equilibrium, chemical oscillations occur around the axis. (c) The motion is oscillatory, with a pair of complex eigenvalues and a third real eigenvalue. (e) When near the equilibrium, the motion is a monotonic, exponential relaxation with three real eigenvalues. The switch occurs when the pair of complex eigenvalues is equal and real (d). System in (b) has been termed either quasiplanar oscillation or “corkscrew oscillations” (Ref. 15), depending on the rate of free energy dissipation.

Dynamics of reversible LV system (2) with a large total concentration of all molecules . Parameter used in the calculations: , , and . Initial value ; total run time . The “cone-shaped” dynamics is obligatory for quasiplanar oscillation in which the radial relaxation is faster than the vertical relaxation.

Dynamics of reversible LV system (2) with a large total concentration of all molecules . Parameter used in the calculations: , , and . Initial value ; total run time . The “cone-shaped” dynamics is obligatory for quasiplanar oscillation in which the radial relaxation is faster than the vertical relaxation.

Dynamics of reversible LV system (2) with a small total concentration of all molecules . Parameters used in the calculations: , , and initial values . All other parameters used are the same as in Fig. 2.

Dynamics of reversible LV system (2) with a small total concentration of all molecules . Parameters used in the calculations: , , and initial values . All other parameters used are the same as in Fig. 2.

Oscillation around the invariant curve (dashed). (a) is for initial point far from the invariant curve, . (b) is for initial point near the invariant curve, , which is much closer to the invariant curve. (c) and (d) are the projections of the 3D trajectory in (b) onto the and planes. Parameters used in the simulations are the same as in Fig. 2 except and .

Oscillation around the invariant curve (dashed). (a) is for initial point far from the invariant curve, . (b) is for initial point near the invariant curve, , which is much closer to the invariant curve. (c) and (d) are the projections of the 3D trajectory in (b) onto the and planes. Parameters used in the simulations are the same as in Fig. 2 except and .

(a) Perturbed system (solid line) vs unperturbed system (dotted lines). For the perturbed system, all the parameters used are the same as in Fig. 3, with initial point (0.5,0.1,0.045). The five closed orbits for the unperturbed system, from top to bottom, are with , 0.0361, 0.027, 0.0186, and 0.0105, respectively. (b) Chemical dynamics in a closed system does not follow the negative gradient of the free energy of the system. On the solution curve, all the arrows represent the direction of the negative gradient of the free energy. Note that the directions of the free energy gradient in (b), which should be perpendicular to the constant energy level curves, are approximately perpendicular to the closed orbits in (a).

(a) Perturbed system (solid line) vs unperturbed system (dotted lines). For the perturbed system, all the parameters used are the same as in Fig. 3, with initial point (0.5,0.1,0.045). The five closed orbits for the unperturbed system, from top to bottom, are with , 0.0361, 0.027, 0.0186, and 0.0105, respectively. (b) Chemical dynamics in a closed system does not follow the negative gradient of the free energy of the system. On the solution curve, all the arrows represent the direction of the negative gradient of the free energy. Note that the directions of the free energy gradient in (b), which should be perpendicular to the constant energy level curves, are approximately perpendicular to the closed orbits in (a).

The rate of free energy decreasing, , along a trajectory as a function of time . denotes the vector field of the 3D system and is the free energy function.

The rate of free energy decreasing, , along a trajectory as a function of time . denotes the vector field of the 3D system and is the free energy function.

Mechanical analog model according to Eq. (8) with and the initial values (5,5,45). This figure should be compared with Figs. 2(a) and 3(a).

Mechanical analog model according to Eq. (8) with and the initial values (5,5,45). This figure should be compared with Figs. 2(a) and 3(a).

Contour plot of with and is drawn against . The red dashed line indicates the critical driving force , and the white one is for .

Contour plot of with and is drawn against . The red dashed line indicates the critical driving force , and the white one is for .

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