CaTwoPlusQuasiperiodic

Systems / ODEs / Chemical & biological systems

The Houart–Dupont–Goldbeter intracellular Ca²⁺ oscillator in its quasiperiodic/chaotic regime, driven by self-modulated InsP₃ feedback.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= V_{0} + V_{1} \beta - \frac{Vm_{2} y_{0}^{2}}{K_{2}^{2} + y_{0}^{2}} + \frac{Vm_{3} y_{0}^{m} y_{1}^{2} y_{2}^{4}}{\left(Ka^{4} + y_{2}^{4}\right) \left(Ky^{2} + y_{1}^{2}\right) \left(Kz^{m} + y_{0}^{m}\right)} - k y_{0} + kf y_{1} \\ \dot{y_{1}} &= \frac{Vm_{2} y_{0}^{2}}{K_{2}^{2} + y_{0}^{2}} - \frac{Vm_{3} y_{0}^{m} y_{1}^{2} y_{2}^{4}}{\left(Ka^{4} + y_{2}^{4}\right) \left(Ky^{2} + y_{1}^{2}\right) \left(Kz^{m} + y_{0}^{m}\right)} - kf y_{1} \\ \dot{y_{2}} &= V_{4} \beta - \frac{Vm_{5} y_{0}^{n} y_{2}^{p}}{\left(K_{5}^{p} + y_{2}^{p}\right) \left(Kd^{n} + y_{0}^{n}\right)} - eps y_{2} \end{aligned} \]

Parameters

Symbol	Default	Role
K2	0.1
K5	0.3
Ka	0.2
Kd	0.5
Ky	0.2
Kz	0.5
V0	2	constant Ca²⁺ influx
V1	2
V4	5
Vm2	6
Vm3	20
Vm5	30
beta	0.51	external stimulus level
eps	0.1	InsP₃ degradation rate
k	10	cytosolic Ca²⁺ removal
kf	1	store leak rate
m	2
n	4
p	2

Properties

Lyapunov spectrum

$+0.004669,\; -0.01023,\; -0.0183$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.457$

Divergence ∇·f

$\nabla\!\cdot f = - eps - k - kf$
constant part shown — state-dependent remainder omitted (too large)

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Houart, Dupont & Goldbeter (1999), Bull. Math. Biol. 61, 507-530

BibTeX

@misc{catwoplusquasiperiodic,
  title = {CaTwoPlusQuasiperiodic system},
  note = {Houart, Dupont & Goldbeter (1999), Bull. Math. Biol. 61, 507-530}
}