Nonequilibrium thermodynamics of glycolytic traveling wave: Benjamin-Feir instability

Evolution of the nonequilibrium thermodynamic entities corresponding to dynamics of the Hopf instabilities and traveling waves at a nonequilibrium steady state of a spatially extended glycolysis model is assessed here by implementing an analytically tractable scheme incorporating a complex Ginzburg-Landau equation (CGLE). In the presence of self and cross diffusion, a more general amplitude equation exploiting the multiscale Krylov-Bogoliubov averaging method serves as an essential tool to reveal the various dynamical instability criteria, especially Benjamin-Feir (BF) instability, to estimate the corresponding nonlinear dispersion relation of the traveling wave pattern. The critical control parameter, wave-number selection criteria, and magnitude of the complex amplitude for traveling waves are modified by self- and cross-diffusion coefficients within the oscillatory regime, and their variabilities are exhibited against the amplitude equation. Unlike the traveling waves, a low-amplitude broad region appears for the Hopf instability in the concentration dynamics as the system phase passes through minima during its variation with the control parameter. The total entropy production rate of the uniform Hopf oscillation and glycolysis wave not only qualitatively reflects the global dynamics of concentrations of intermediate species but almost quantitatively. Despite the crucial role of diffusion in generating and shaping the traveling waves, the diffusive part of the entropy production rate has a negligible contribution to the system's total entropy production rate. The Hopf instability shows a more complex and colossal change in the energy profile of the open nonlinear system than in the traveling waves. A detailed analysis of BF instability shows a contrary nature of the semigrand Gibbs free energy with discrete and continuous wave numbers for the traveling wave. We hope the Hopf and traveling wave pattern around the BF instability in terms of energetics and dissipation will open up new applications of such dynamical phenomena.

Figure 1

Wave number vs the amplitude equation coefficient $\alpha$ is shown in Fig. 1. Figure 1 illustrates variation of the magnitude of the complex amplitude with respect to the same coefficient, $\alpha$. Both figures are obtained by varying the cross-diffusion coefficient $D_{12}$ from $-0.00001$ to $-0.0005$ and $D_{21}$ from $0.00001$ to 0.0005 while other parameters are fixed, i.e., $D_{11}=D_{22}=0.00051$, $\nu =2.45$ and $w=2$.

Figure 2

The system's phase as a function of the control parameter $\nu$ is illustrated in Figs. 2 and 2 for Hopf instability and traveling waves, respectively, for finite domain of length $l=200$. The phase change in the limit of infinite size is illustrated in Fig. 2. The normalized real part of the Hopf amplitude denoted by Re $A_H$ comprised of magnitude and phase is obtained analytically as a function of control parameter $\nu$ at time $t=400$ in Fig. 2. The corresponding normalized real part of amplitude fields for traveling waves for discrete and continuous wave number are shown in Figs. 2 and 2, respectively. These amplitude figures will provide a better understanding of the local concentration profile in the 1D Selkov model in the parameter space of Hopf instability and traveling waves. Here the diffusion coefficients are $D_{11}=D_{22}=0.00051,D_{12}=-0.0002,D_{21}=0.0002,$ and the value of the parameter $\omega$ is set as 2.

Figure 3

In Fig. 3, the 3D concentration field of $X$ for the Hopf instability in the Selkov model of length $l=200$ as a function of the externally controlled parameter $\nu$ is presented (plot of $Y$ is similar) at time $t=400$ and temperature $T=300\text{K}$ for a fixed value of parameter $\omega =2$. The “jet” color map is used to show contrast in concentration field. Figure 3 illustrates the corresponding image of the concentration field of $X$. The 3D concentration field of $X$ for the same system in the case of traveling waves with finite system consideration is illustrated in Fig. 3 and the corresponding image is shown in Fig. 3. The extended spatial dimension is considered along the vertical axis. The 3D concentration field of $X$ and corresponding image for traveling waves with a continuous family of wave numbers in the limit of infinite size are demonstrated in Figs. 3 and 3, respectively. Diffusion coefficients are $D_{11}=D_{22}=0.00051,D_{12}=-0.0002,D_{21}=0.0002,$ and all the reactions are weakly reversible, i.e., $K_{-\rho }={10}^{-4}$.

Figure 4

The Benjamin-Feir (BF) instability and phase-reversal defining conditions corresponding to the amplitude equation are shown in Fig. 4. Figure 4 illustrates a plot similar to a phase portrait in the case of Hopf instability. Here $X$ and $Y$ concentrations dynamics are obtained by varying $\nu$ but for a fixed time $t=400$. The red dotted line in Fig. 4 corresponds to $\alpha +\beta$ and the solid blue line represents $1-\alpha \beta$. As the $1-\alpha \beta$ line crosses the zero line (the dotted black line), we enter the BF instability regime. Diffusion coefficients are $D_{11}=D_{22}=0.00051,D_{12}=-0.0002,D_{21}=0.0002,$ and all the reactions are weakly reversible, i.e., $K_{-\rho }={10}^{-4}$.

Figure 5

The wave number near the onset of Hopf instability. For the finite system of length $l=200$, we have shown discrete allowed values of wave number below the continuous wave number. The continuous wave number in the limit of infinite size is illustrated by the dashed line. Here the diffusion coefficients are $D_{11}=D_{22}=0.00051,D_{12}=-0.0002,D_{21}=0.0002,$ and the value of the parameter $\omega$ is set as 2.

Figure 6

Total entropy production of Hopf instability with respect to control parameter $\nu$ is obtained analytically for a 1D Selkov model of length $l=200$ at time $t=400$ and absolute temperature, $T=300\text{K}$ for $\omega =2$. Total entropy production rate is comprised of entropy production rates due to the homogeneous part and reaction part. Global concentration fields of intermediate species $X$ and $Y$ as a function of $\nu$ are presented in Fig. 6. Total entropy production of traveling waves of the same system with discrete wave numbers is presented in Fig. 6. In the case of traveling waves, global concentration fields of intermediate species $X$ and $Y$ vs control parameter are shown in Fig. 6. In the limit of infinite size, the total entropy production rate and global concentration fields of intermediate species of traveling waves are presented in Figs. 6 and 6. It is very apparent from Figs. 6 and 6 and Figs. 6 and 6 that entropy production rate is proportional to global concentration of $Y$ in terms of both magnitude and phase. For all the cases the diffusion coefficients are $D_{11}=D_{22}=0.00051,D_{12}=-0.0002,D_{21}=0.0002,$ and elementary chemical reactions are weakly reversible, i.e., $K_{-\rho }={10}^{-4}.$

Figure 7

The semigrand Gibbs free energy and corresponding slope of the Hopf instability are illustrated in Figs. 7 and 7, respectively, as functions of the control parameter $\nu$ at $t=400$ and $T=300\text{K}$ for fixed parameter $\omega =2$. The same entities for traveling waves of discrete wave numbers are presented in Figs. 7 and 7, respectively. In the limit of infinite size, the semigrand Gibbs free energy and corresponding slope profile of traveling waves are shown in Figs. 7 and 7, respectively. The dotted lines are for the unstable homogeneous state of the system in both cases. For all the cases diffusion coefficients are $D_{11}=D_{22}=0.00051,D_{12}=-0.0002,D_{21}=-0.0002$.