CircadianRhythm

Systems / ODEs / Chemical & biological systems

A molecular circadian clock — entrainable biochemical oscillation.

continuous · ODE5 dimensionsoscillatory

time series

Definition

\[ \begin{aligned} \dot{y_{0}} &= \frac{2.5 Ki^{n} \left(vmax - vmin\right) \left(vmin + 0.5 \cos{\left(y_{4} \right)} + 0.5\right)}{Ki^{n} + y_{3}^{n}} - \frac{vm y_{0}}{km + y_{0}} \\ \dot{y_{1}} &= - k y_{1} - k_{1} y_{1} + k_{2} y_{3} + ks y_{0} \\ \dot{y_{2}} &= k y_{1} - \frac{vd y_{2}}{kd + y_{2}} \\ \dot{y_{3}} &= k_{1} y_{1} - k_{2} y_{3} - \frac{vdn y_{3}}{kdn + y_{3}} \\ \dot{y_{4}} &= \frac{\pi}{12} \end{aligned} \]

Parameters

Symbol	Default	Role
Ki	1	threshold for repression by nuclear PER
k	0.5	phosphorylation / turnover rate of cytosolic PER
k1	0.3	transport rate of PER into the nucleus
k2	0.15	transport rate of PER out of the nucleus
kd	1.4	Michaelis constant of cytosolic PER degradation
kdn	0.4	Michaelis constant of nuclear PER degradation
km	0.4	Michaelis constant of mRNA degradation
ks	1	rate of PER synthesis from mRNA
n	4	Hill coefficient of transcriptional repression
vd	6	maximum degradation rate of cytosolic PER
vdn	1.5	maximum degradation rate of nuclear PER
vm	0.7	maximum rate of mRNA degradation
vmax	4.7	upper bound of the phase-modulated transcription rate
vmin	1	lower bound of the phase-modulated transcription rate
vs	6	maximum rate of mRNA synthesis

Properties

Lyapunov spectrum

$-0.02716,\; -0.02647,\; -0.06307,\; -1.037,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1$

Divergence ∇·f

$\nabla\!\cdot f = - k - k_{1} - k_{2} + \frac{vd y_{2}}{kd^{2} + 2 kd y_{2} + y_{2}^{2}} - \frac{vd}{kd + y_{2}} + \frac{vdn y_{3}}{kdn^{2} + 2 kdn y_{3} + y_{3}^{2}} - \frac{vdn}{kdn + y_{3}} + \frac{vm y_{0}}{km^{2} + 2 km y_{0} + y_{0}^{2}} - \frac{vm}{km + y_{0}}$
state-dependent

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Leloup, Gonze & Goldbeter (1999); Gonze, Leloup & Goldbeter (2000)

BibTeX

@misc{circadianrhythm,
  title = {CircadianRhythm system},
  note = {Leloup, Gonze & Goldbeter (1999); Gonze, Leloup & Goldbeter (2000)}
}