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decroly1

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decroly1

The SBML for this model was obtained from the BioModels database (BioModels ID: BIOMD0000000319) Biomodels notes: The upper panel shows the time course of concentration of alpha for the chaotic regime as in upper panel of fig 3B, the lower panel a bifurcation diagram of alpha against ks similar to the upper panel of fig 1 of the reference publication. The time course was calculated using Copasi v 4.6.33, the bifurcation diagram using the Oscill8 v 2.0.11 and SBW 2.7.10. In the bifurcation diagram thick lines indicate stable, thin instable solutions. The orange line stands for the steady state of alpha, the purple line for its maximal value in oscillations. JWS Online curation: This model was curated by reproducing the figures as described in the BioModels Notes. No additional changes were made.

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Manuscript information

Birhythmicity, chaos, and other patterns of temporal self-organization in a multiply regulated biochemical system.

  • O Decroly
  • Albert Goldbeter
Proc. Natl. Acad. Sci. U.S.A. 1982; 79 (22): 6917-6921
Abstract
We analyze on a model biochemical system the effect of a coupling between two instability-generating mechanisms. The system considered is that of two allosteric enzymes coupled in series and activated by their respective products. In addition to simple periodic oscillations, the system can exhibit a variety of new modes of dynamic behavior; coexistence between two stable periodic regimes (birhythmicity), random oscillations (chaos), and coexistence of a stable periodic regime with a stable steady state (hard excitation) or with chaos. The relationship between these patterns of temporal self-organization is analyzed as a function of the control parameters of the model. Chaos and birhythmicity appear to be rare events in comparison with simple periodic behavior. We discuss the relevance of these results with respect to the regularity of most biological rhythms.

Unit definitions 4

Unit definitions have no effect on the numerical analysis of the model. It remains the responsibility of the modeler to ensure the internal numerical consistency of the model. If units are provided, however, the consistency of the model units will be checked.

Name Definition
— 1.0 dimensionless
— 1.0 dimensionless
— 1.0 second^(-1.0)
— 1.0 dimensionless

Compartments 1

Id Name Spatial dimensions Size
cell cell 3.0 1.0

Species 3

Id Name Initial quantity Compartment Fixed
alpha alpha 29.19988 cell (cell) ✘
beta beta 188.8 cell (cell) ✘
gamma gamma 0.3367 cell (cell) ✘

Initial assignments 0

Initial assignments are expressions that are evaluated at time=0. It is not recommended to create initial assignments for all model entities. Restrict the use of initial assignments to cases where a value is expressed in terms of values or sizes of other model entities. Note that it is not permitted to have both an initial assignment and an assignment rule for a single model entity.

Definition
Reactions 4
Id Name Objective coefficient Reaction Equation and Kinetic Law Flux bounds
r1 — ∅ > alpha

v_Km1
r2 — alpha > {50.0}beta

sigma1 * alpha * (1 + alpha) * pow(1 + beta, 2) / (L1 + pow(1 + alpha, 2) * pow(1 + beta, 2))
r3 — beta > {0.02}gamma

sigma2 * beta * (1 + d * beta) * pow(1 + gamma, 2) / (L2 + pow(1 + d * beta, 2) * pow(1 + gamma, 2))
r4 — gamma > ∅

ks * gamma

Parameters 7   0

Global parameters

Id Value

Local parameters

Id Value Reaction
v_Km1 0.45 per sec r1
L1 500000000.0 dimensionless r2
sigma1 10.0 per sec r2
L2 100.0 dimensionless r3
d 0.0 dimensionless r3
sigma2 10.0 per sec r3
ks 1.99 per sec r4

Rules 0   0   0

Assignment rules

Definition

Rate rules

Definition

Algebraic rules

Definition

Events 0

Trigger Assignments