CaTwoPlus

Systems / ODEs / Chemical & biological systems

A calcium-induced calcium-release oscillator with self-modulated InsP₃ signalling that produces spikes, bursting, chaos, and birhythmicity.

continuous · ODE3 dimensionsbursting

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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	Michaelis constant of Ca²⁺ pumping
K5	0.3194	Michaelis constant of the kinase flux
Ka	0.1	Michaelis constant of InsP₃ degradation
Kd	1	Michaelis constant (InsP₃ degradation)
Ky	0.3	Michaelis constant (store Ca²⁺)
Kz	0.6	Michaelis constant (cytosolic Ca²⁺)
V0	2	constant Ca²⁺ influx into the cytosol
V1	2	stimulus-dependent Ca²⁺ influx
V4	3	maximum stimulus-induced InsP₃ synthesis rate
Vm2	6	maximum Ca²⁺ pumping rate into the store
Vm3	30	maximum CICR-mediated release rate
Vm5	50	maximum InsP₃-degrading kinase rate
beta	0.65	external stimulus saturation
eps	13	linear InsP₃ degradation rate
k	10	linear cytosolic Ca²⁺ removal rate
kf	1	passive Ca²⁺ leak from the store
m	2	Hill coefficient of the CICR term
n	4	Hill coefficient of the CICR term
p	1	Hill coefficient of the InsP₃-degradation term

Properties

Lyapunov spectrum

$+0.365,\; -0.01787,\; -9.11$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.038$

Divergence ∇·f

$\nabla\!\cdot f = - eps - k - kf$
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.CaTwoPlus()
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{catwoplus,
  title = {CaTwoPlus system},
  note = {Houart, Dupont & Goldbeter (1999), Bull. Math. Biol. 61, 507-530}
}