Synchronization of cells with activator-inhibitor pathways through adaptive environment-mediated coupling

In this paper, we report the synchronized dynamics of cells with activator-inhibitor pathways via an adaptive environment-mediated coupling scheme with feedbacks and control mechanisms. The adaptive character of the extracellular medium is modeled via its damping parameter as a physiological response aiming at annihilating the cellular differentiation existing between the chaotic biochemical pathways of the cells, in order to preserve homeostasis. We perform an investigation on the existence and stability of the synchronization manifold of the coupled system under the proposed coupling pattern. Both mathematical and computational tools suggest the accessibility of conducive prerequisites (conditions) for the emergence of a robust synchronous regime. The relevance of a phase-synchronized dynamics is appraised and several numerical indicators advocate for the prevalence of this fascinating phenomenon among the interacting cells in the phase space.

Figure 1

The spectrum of eigenvalues as a function of $\epsilon$: (a) Eigenvalue spectrum for which the synchronized state is unstable, $\epsilon \le 0.0771$ in dark line (blue online); Eigenvalue spectrum for which the synchronized state is stable, $\epsilon >0.0771$ in gray line (orange online). (b) Zoom of the small box in (a) as an evidence of the existence of at least one eigenvalue that exceeds zero, thereby yielding an unstable synchronization manifold when $\epsilon \le 0.0771$. The parameter values are $\mu =0.00612611845,\omega =1.0,\kappa =1.0$, and $\alpha =3.5001\times {10}^{-9}$.

Figure 2

The three largest Lyapunov exponents of the coupled system as a function of the coupling strength $\epsilon$. The parameter values are $q=0.1,k=0.003,L={10}^6,T=10$.

Figure 3

The Lyapunov diagrams of the coupled system Eq. (1), defined by the second-largest Lyapunov exponent of the spectrum, showing domains of stability of the synchronization manifold as function of the coupling strength $\epsilon$ and (a) the rate of degradation of the first substrate $k$ and (b) the maximum velocity of the enzyme $T$. The other parameters are fixed as $L={10}^6$ and $q=0.1$.

Figure 4

Complete synchronization in two indirectly coupled cells with activator-inhibitor pathways coupled through an adaptive environment with feedback control mechanism. Figures (a) $\epsilon =0$, no synchronization, and (b) $\epsilon =0.15$, synchronization, are the plots of the superposition of the time series of the two coupled cells. Figures (c) $\epsilon =0$, no synchronization, and (d) $\epsilon =0.15$, are the correlation graphs for different values of the coupling. For a suitable coupling strength, a complete synchronized dynamics is obtained.

Figure 5

Time series of the damping parameter of the environment $\kappa$ for (a) $\epsilon =0$ and (b) $\epsilon =0.15$.

Figure 6

Onset of phase synchronization in two indirectly coupled cells with activator-inhibitor pathways coupled through an adaptive environment with feedback control mechanism. (a) Occupation of the conditional observations with respect to the attractor, $\chi$. (b) End-product Normalized concentration $z_2^j$ of the second cell when the first cell makes the $j\text{th}$ crossing with the section $z=10$, for $\dot{z}>0$. (c) Mean value of the Kuramoto parameter $\langle R\rangle$. (d) The Kuramoto amplitude $\delta R$, as a function of the coupling strength ${\epsilon }_1={\epsilon }_2=\epsilon$, detecting the emergence of phase synchronization in the coupled biochemical system. The parameter values are $q=0.1,k=0.003,L={10}^6,T=10$.

Figure 7

Onset of phase synchronization in two indirectly coupled cells with activator-inhibitor pathways coupled through an adaptive environment with feedback control mechanism. Figures (a) $\epsilon =0$ and (b) $\epsilon =0.15$ are the plots of the attractor of the first cell in gray lines (pink online) and the stroboscopic projection of the attractor of the second cell in dark dots (blue online) on the cross section of the first cell, for different values of the coupling strength (the points are localized as the coupling increases, indicating the onset of phase synchronization in the system). Figures (c) $\epsilon =0$ and (d) $\epsilon =0.15$ are time series of the phase difference of the two coupled cells for different values of the coupling. As the coupling increases, the phase difference is bounded, confirming the onset of phase synchronization in the biochemical system. The parameter values are the same as described in the caption of of Fig. (6).

Figure 8

The Kuramoto diagrams of the coupled system Eq. (1), based on the mean value of the Kuramoto parameter, showing phase synchronization regions in black (blue) and regions of unsynchronized dynamics white (white) as a function of the coupling strength $\epsilon$ and (a) the rate of degradation of the first substrate $k$ and (b) the maximum velocity of the enzyme $T$. The other parameter being fixed as $L={10}^6$ and $q=0.1$.