GlycolyticOscillation

Systems / ODEs / Chemical & biological systems

Sel'kov-type glycolytic oscillations — metabolic limit cycles.

continuous · ODE3 dimensionslimit cycle

time series

Definition

\[ \begin{aligned} \dot{y_{0}} &= \nu - \frac{s_{1} y_{0} \left(y_{0} + 1\right) \left(y_{1} + 1\right)^{2}}{l_{1} + \left(y_{0} + 1\right)^{2} \left(y_{1} + 1\right)^{2}} \\ \dot{y_{1}} &= \frac{q_{1} s_{1} y_{0} \left(y_{0} + 1\right) \left(y_{1} + 1\right)^{2}}{l_{1} + \left(y_{0} + 1\right)^{2} \left(y_{1} + 1\right)^{2}} - \frac{s_{2} y_{1} \left(y_{2} + 1\right)^{2} \left(d y_{1} + 1\right)}{l_{2} + \left(y_{2} + 1\right)^{2} \left(d y_{1} + 1\right)^{2}} \\ \dot{y_{2}} &= - k y_{2} + \frac{q_{2} s_{2} y_{1} \left(y_{2} + 1\right)^{2} \left(d y_{1} + 1\right)}{l_{2} + \left(y_{2} + 1\right)^{2} \left(d y_{1} + 1\right)^{2}} \end{aligned} \]

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

Properties

Lyapunov spectrum

$-2.06e-05,\; -0.1578,\; -4.421$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 0$

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Decroly & Goldbeter (1982), Proc. Natl. Acad. Sci. U.S.A. 79, 6917-6921

BibTeX

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