Molecular Synchronization Waves in Arrays of Allosterically Regulated Enzymes

Spatiotemporal pattern formation in a product-activated enzymic reaction at high enzyme concentrations is investigated. Stochastic simulations show that catalytic turnover cycles of individual enzymes can become coherent and that complex wave patterns of molecular synchronization can develop. The analysis based on the mean-field approximation indicates that the observed patterns result from the presence of Hopf and wave bifurcations in the considered system.

Figure 1

A sketch of the model.

Figure 2

Stochastic (a),(b) and mean-field (c),(d) simulations of 2D wave patterns: (a) ${\tau }_p=0.14$, $c=1$, and $\beta =300$; (b) ${\tau }_p=0.25$, $c=10$, and $\beta =10$; (c) ${\tau }_p=0.14$, $c=1$, and $\beta =300$; (d) ${\tau }_p=0.34$, $c=100$, and $\beta =1.42$. Other parameters are ${\alpha }_0=1$, ${\alpha }_1=1000$, $\kappa =10$, $\gamma =10$, $\sigma =0$, $D=100$. The linear size of the shown area is $L=40$ $l_{\mathrm{diff}}$ in all panels.

Figure 3

Phase diagram (${\alpha }_0=1$, ${\alpha }_1=1000$, $\kappa =10$, $\gamma =10$, $c=100$, $D=1000$). The Hopf bifurcation (solid line) and the wave bifurcation (dash-dotted line) boundaries are displayed. Gray lines show instability of the stationary state with respect to development of uniform oscillations with two (dashed) and three (dotted) groups in the well-mixed case. Lines separating parameter domains with different kinds of patterns are hand drawn, based on numerical simulations.

Figure 4

Spatiotemporal patterns in a 1D system (in each panel, the vertical axis is time, running down, and the horizontal axis is the coordinate). The upper two rows are stochastic simulations ($\sigma =0$) with concentrations $c=1$ and $c=10$, the bottom row shows mean-field simulations with $c=100$. (a) ${\tau }_p=0.3$, $\beta =95/c$; (b) ${\tau }_p=0.14$, $\beta =260/c$; (c) ${\tau }_p=0.22$, $\beta =600/c$; (d) ${\tau }_p=0.16$, $\beta =300/c$. Other parameters as in Fig. 3; the system size shown is $L=51$ $l_{\mathrm{diff}}$.