GlycolyticOscillation

Systems / ODEs / Chemical & biological systems

Decroly and Goldbeter's multiply-regulated glycolytic oscillator — two product-activated enzymes in series, giving birhythmicity and chaos (the three-variable extension of the planar Selkov model).

continuous · ODE3 dimensionslimit cycle

time series

Definition

\[ \begin{aligned} \dot{a} &= - \frac{a s_{1} \left(a + 1\right) \left(b + 1\right)^{2}}{l_{1} + \left(a + 1\right)^{2} \left(b + 1\right)^{2}} + \nu \\ \dot{b} &= \frac{a q_{1} s_{1} \left(a + 1\right) \left(b + 1\right)^{2}}{l_{1} + \left(a + 1\right)^{2} \left(b + 1\right)^{2}} - \frac{b s_{2} \left(c + 1\right)^{2} \left(b d + 1\right)}{l_{2} + \left(c + 1\right)^{2} \left(b d + 1\right)^{2}} \\ \dot{c} &= \frac{b q_{2} s_{2} \left(c + 1\right)^{2} \left(b d + 1\right)}{l_{2} + \left(c + 1\right)^{2} \left(b d + 1\right)^{2}} - c k \end{aligned} \]

Parameters

Symbol	Default	Role
d	0	cooperativity modulation of the second enzyme
k	4.422	removal (sink) rate of the final product
l1	500000000	allosteric constant of the first enzyme
l2	100	allosteric constant of the second enzyme
nu	1	constant input rate of the substrate
q1	50	coupling ratio linking the two enzymatic stages
q2	0.02	coupling ratio linking the two enzymatic stages
s1	22.2222	maximum rate of the first allosteric enzyme
s2	22.2222	maximum rate of the second allosteric enzyme

State variables: a, b, c

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.GlycolyticOscillation()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Decroly & Goldbeter (1982), Proc. Natl. Acad. Sci. U.S.A. 79, 6917-6921
doi:10.1073/pnas.79.22.6917

BibTeX

@misc{glycolyticoscillation,
  title = {GlycolyticOscillation system},
  note = {Decroly & Goldbeter (1982), Proc. Natl. Acad. Sci. U.S.A. 79, 6917-6921},
  doi = {10.1073/pnas.79.22.6917}
}