Periodic forcing of a mathematical model of the eukaryotic cell cycle

In a differential equation model of the molecular network governing cell growth and division, cell cycle phases and transitions through checkpoints are associated with certain bifurcations of the underlying vector field. If the cell cycle is driven by another rhythmic process, interactions between forcing and bifurcations lead to emergent orbits and oscillations. In this paper, by varying the amplitude and frequency of forcing of the synthesis rates of regulatory proteins and the mass growth rate in a minimal model of the eukaryotic cell cycle, we study changes of the probability distributions of interdivision time and mass at division. By computing numerically the Lyapunov exponent of the model, we show that the splitting of probability distributions is associated with mode-locked solutions. We also introduce a simple, integrate-and-fire model to analyze mode locking in the cell cycle.

Figure 1

Nullclines and birhythmicity in Eqs. (7, 8). Dashed line; $X$ nullcline, solid line; $Z$ nullcline. Two dotted circles, marked by ${\mathrm{LC}}_1$ and ${\mathrm{LC}}_2$, show two stable limit cycle oscillations. Mass is fixed at $m=4.2$.

Figure 2

A bifurcation diagram of Eqs. (7, 8). Solid line; stable steady state, dashed line; unstable steady state, open circles; unstable oscillations, filled circles; stable limit cycles. The dotted curve with arrows shows the trajectory of motion when Eqs. (7, 8) are supplemented by the mass growth equation, $\dot{m}=\mu m$. The cell divides $(m\to \tau m)$ when $X$ drops below $X_{\mathit{thr}}=0.05$; hence the trajectory undergoes an abrupt horizontal jump to the left at $X=X_{\mathit{thr}}$.

Figure 3

Two parameter bifurcation diagram. The solid line is the continuation of the SNIC bifurcation. Dotted lines separate regions with different oscillation periods (T). Also indicated is a range of perturbation of $k_{10}$.

Figure 4

Emergent orbits. Forcing may induce small amplitude, slow oscillations near the steady state. The inset magnifies the region of the small amplitude oscillations near the steady state. The parameters are: $m=3.2$, $A=0.4$, and $f=0.003$.

Figure 5

Cyclicfold bifurcations at $k_2=0.0689$ (left) and $k_2=0.0795$ (right). Open circles indicate unstable oscillations, filled circles indicate stable oscillations. In the interval between the two vertical solid lines, the system is monorhythmic but oscillates on either a small or a large amplitude orbit depending on $k_2$.

Figure 6

A typical case of probability distributions of mass doubling time and mass at division at weak forcing.

Figure 7

Emergent orbits in Eqs. (12, 13, 14, 15). (a) Period doubling. (b) Toruslike motion. (c) A combination of bifurcations shown in (a) and (b). (d) Cell cycle arrest.

Figure 8

Cell division under periodic forcing. Solid line shows periodically modulated mass dynamics. Other lines show the oscillations of $X$, $Y$, and $Z$, respectively. Parameters are as in Table , except $A=1$, $f=0.07$ (4:3 resonance), $X_{\mathit{trh}}=0.05$, $k_{10}=0.0025$, and $B=0$.

Figure 9

Splitting of the probability distributions. Parameters are the same as in Fig. 8.

Figure 10

A bifurcation diagram of $T^n$ and $m_d$, and the corresponding Lyapunov exponents for Eqs. (12, 13, 14, 15). The parameters are the same as in Fig. 9 except $A=B$, $f=0.035$ (2:3 resonance), and $k_2=0.0795$.

Figure 11

The bifurcation diagrams of $T^n$ and $m_d$, and the corresponding Lyapunov exponents of Eqs. (16, 17, 18, 19). Parameters are $f\approx 0.0209$, $m^{*}=2.5$, ${\psi }_0=0$, and $A=B$.