CellCycle

Systems / ODEs / Chemical & biological systems

A model of the eukaryotic cell-division oscillator.

continuous · ODE6 dimensionsoscillatory

time series

Definition

\[ \begin{aligned} \dot{y_{0}} &= \frac{Kim vi}{Kim + y_{4}} - kd_{1} y_{0} - \frac{vd y_{0} y_{2}}{Kd_{1} + y_{0}} \\ \dot{y_{1}} &= - \frac{V_{2} y_{1}}{K + y_{1}} + \frac{Vm_{1} y_{0} \left(1 - y_{1}\right)}{\left(Kc + y_{0}\right) \left(K - y_{1} + 1\right)} \\ \dot{y_{2}} &= - \frac{V_{4} y_{2}}{K + y_{2}} + \frac{Vm_{3} y_{1} \left(1 - y_{2}\right)}{K - y_{2} + 1} \\ \dot{y_{3}} &= \frac{Kim vi}{Kim + y_{1}} - kd_{1} y_{3} - \frac{vd y_{3} y_{5}}{Kd_{1} + y_{3}} \\ \dot{y_{4}} &= - \frac{V_{2} y_{4}}{K + y_{4}} + \frac{Vm_{1} y_{3} \left(1 - y_{4}\right)}{\left(Kc + y_{3}\right) \left(K - y_{4} + 1\right)} \\ \dot{y_{5}} &= - \frac{V_{4} y_{5}}{K + y_{5}} + \frac{Vm_{3} y_{4} \left(1 - y_{5}\right)}{K - y_{5} + 1} \end{aligned} \]

Parameters

Symbol	Default	Role
K	0.01	common Michaelis constant of the modification cycles
Kc	0.5	threshold for cdk activation by cyclin
Kd1	0.02	Michaelis constant of cyclin degradation
Kim	0.65	inhibition constant of the mutual coupling
V2	0.15	maximum rate of cdk inactivation
V4	0.05	maximum rate of protease inactivation
Vm1	0.3	maximum rate of cdk activation
Vm3	0.1	maximum rate of protease activation
kd1	0.001	basal rate of cyclin degradation
vd	0.025	maximum rate of cyclin degradation
vi	0.05	cyclin synthesis rate

Properties

Lyapunov spectrum

$-0.003656,\; -0.01692,\; -0.6235,\; -1.765,\; -2.105,\; -2.348$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 0$

Divergence ∇·f

$\nabla\!\cdot f = - 2 kd_{1}$
constant part shown — state-dependent remainder omitted (too large)

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.CellCycle()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Romond, Rustici, Gonze & Goldbeter (1999), Ann. N.Y. Acad. Sci. 879, 180-193

BibTeX

@misc{cellcycle,
  title = {CellCycle system},
  note = {Romond, Rustici, Gonze & Goldbeter (1999), Ann. N.Y. Acad. Sci. 879, 180-193}
}