Dynamics and stochastic resonance in a mathematical model of bistable phosphorylation and nuclear size control

Abstract

Robust oscillations play crucial roles in a wide variety of biological processes and are often generated by deterministic mechanisms. However, stochastic fluctuations often generate complex perturbations of these deterministic oscillations, potentially strengthening or weakening their robustness. In this paper, we study bistable phosphorylation as a mechanism for robust oscillation. We present a simple nucleocytoplasmic transport and cell growth model where cargo proteins undergo bistable phosphorylation prior to nuclear import. We perform a detailed bifurcation analysis to examine the system's dynamical behavior. We then introduce additive noise into the model and study the stochastic resonance behavior and robustness of oscillations under noise. Our results show that, depending on the phosphorylation threshold, time-scale parameters, and nucleocytoplasmic transport rate, bistable phosphorylation may generate oscillations via Hopf bifurcations; moreover, stochastic resonance and Bautin bifurcations enhance the robustness of the oscillations.

Figures (11)

• Figure 1: Effect of $K_c$ on the dynamical behaviors of the system. (a), phase plane. Solid lines, trajectories; dashed lines, $c_{no}$-nullclines; dotted lines, $c_{nop}$-nullclines. For three values of $K_c$: the nullclines with leftmost inflection point and bordeaux trajectories correspond to $K_c=1$, the nullclines with the middle inflection point and blue periodic trajectory correspond to $K_c=2.75$, while the nullclines with the rightmost inflection point and red trajectory correspond to $K_c=4.2$. In each case, there was a single fixed point, marked with a red asterisk. (b), corresponding trajectories of the $c_{no}$ fraction as a function of time. The colors match those used in panel (a): bordeaux for $K_c=1$, blue for $K_c=2.75$ and red for $K_C=4.2$. In all simulations, $\tau=0.01$ and $k_{nt}=0.1$.
• Figure 2: Effect of $\tau$ on the dynamical behaviors of the system. (a), phase plane. Solid lines, trajectories; dashed lines, $c_{no}$-nullclines; dotted lines, $c_{nop}$-nullclines. For five values of $\tau$: the purple trajectory corresponds to $\tau=0.01$, the light blue trajectory corresponds to $\tau=1$, the green trajectory corresponds to $\tau=10$, the bordeaux trajectory corresponds to $\tau=40$, and the blue trajectory corresponds to $\tau=70$. (b), corresponding trajectories of $c_{no}$ fraction as a function of time. The colors match those used in panel (a): light blue for $\tau=1$, green for $\tau=10$, and bordeaux for $\tau=40$. In all simulations, $K_c=2.75$ and $k_{nt}=0.1$.
• Figure 3: Effect of $k_{nt}$ on the dynamical behaviors of the system. Solid lines, trajectories; dashed lines, $c_{no}$-nullclines; dotted lines, $c_{nop}$-nullclines. For three values of $k_{nt}$: the $c_{nop}$-nullcline with leftmost inflection point and bordeaux trajectories correspond to $k_{nt}=0.005$, the $c_{nop}$-nullcline with the middle inflection point and blue periodic trajectory correspond to $k_{nt}=0.02$, while the $c_{nop}$-nullcline with the rightmost inflection point and red trajectory correspond to $k_{nt}=0.1$. In each case, there was a single fixed point, marked with a red asterisk. In all simulations, $K_c=6.56$ and $\tau=14.5$.
• Figure 4: Example of bistability of equilibria. Solid lines, trajectories; dashed lines, $c_{no}$-nullclines; dotted lines, $c_{nop}$-nullclines. Each fixed point is marked with a red asterisk. In all simulations, $K_c=14.2$, $\tau=5$, and $k_{nt}=0.00397$.
• Figure 5: Two-parameter bifurcation analysis with respect to $(\tau,K_c)$ with $k_{nt}=0.1$. (a), $(\tau, K_c)$ plane partitioned by Hopf bifurcation curve of equilibria (gray) and fold bifurcation curve of limit cycles (orange). Blue (Region I), one stable equilibrium. Red (Region II), one stable limit cycle. Green (Region III), two limit cycles of opposite stability. GH, Bautin (generalized Hopf) bifurcation points (labeled by asterisks). Oscillations occur in red and green regions. Diagram computed using the Matcont package on Matlab dhooge2003matcont. (b), bifurcation with respect to $K_c$ under various $\tau$ values. Stable and unstable limit cycles are shown by filled and hollow circles, respectively. Diagram computed using XPPAUT.