Alternating Oscillations and Chaos in a Model of Two Coupled Biochemical Oscillators Driving Successive Phases of the Cell Cycle
Jeelean
Lim
Auckland Bioengineering Institute, The University of Auckland
Model Status
This CellML version of the model has been checked in COR and PCEnv and the model runs to replicate the original published results as depicted in figure 3 of the paper. The units have been checked and are consistent.
Model Structure
ABSTRACT: The animal cell cycle is controlled by the periodic variation of two cyclin-dependent protein kinases, cdk1 and cdk2, which govern the entry into the M (mitosis) and S (DNA replication) phases, respectively. The ordered progression between these phases is achieved thanks to the existence of checkpoint mechanisms based on mutual inhibition of these processes. Here we study a simple theoretical model for oscillations in cdk1 and cdk2 activity, involving mutual inhibition of the two oscillators. Each minimal oscillator is described by a three-variable cascade involving a cdk, together with the associated cyclin and cyclin-degrading enzyme. The dynamics of this skeleton model of coupled oscillators is determined as a function of the strength of their mutual inhibition. The most common mode of dynamic behavior, obtained under conditions of strong mutual inhibition, is that of alternating oscillations in cdk1 and cdk2, which correspond to the physiological situation of the ordered recurrence of the M and S phases. In addition, for weaker inhibition we obtain evidence for a variety of dynamic phenomena such as complex periodic oscillations, chaos, and the coexistence between multiple periodic or chaotic attractors. We discuss the conditions of occurrence of these various modes of oscillatory behavior, as well as their possible physiological significance.
The original paper reference is cited below:
Alternating Oscillations and Chaos in a Model of Two Coupled Biochemical Oscillators Driving Successive Phases of the Cell Cycle, Pierre-Charles Romond, Mauro Rustici, Didier Gonze, Albert GOldbeter, 1999, Annals of the New York Acedemy of Sciences, 879, 180-193. PubMed ID: 10415827
figure 1
Skeleton model of two coupled biochemical oscillators controlling the M and S phases of the cell cycle. Each oscillator consists of a three-variable cascade involving a cyclin (C1 or C2), a cyclin-dependent kinase (cdk) (M1 or M2), and a cdk-activated ubiquitin ligase (X1 or X2) that controls cyclin degradation. The + sign indicates the inactive form of the enzymes. The dashed lines ending with a horizontal bar represent the inhibition exerted by M1 and M2 on the synthesis of C1 and C2, respectively.
$\frac{d \mathrm{C\_1}}{d \mathrm{time}}=\frac{\mathrm{v\_i1}\mathrm{K\_im1}}{\mathrm{K\_im1}+\mathrm{M\_2}}-\frac{\mathrm{v\_d1}\mathrm{X\_1}\mathrm{C\_1}}{\mathrm{K\_d1}+\mathrm{C\_1}}-\mathrm{k\_d1}\mathrm{C\_1}$
$\frac{d \mathrm{M\_1}}{d \mathrm{time}}=\frac{\mathrm{V\_1}(1-\mathrm{M\_1})}{\mathrm{K\_1}+1-\mathrm{M\_1}}-\frac{\mathrm{V\_2}\mathrm{M\_1}}{\mathrm{K\_2}+\mathrm{M\_1}}$
$\frac{d \mathrm{X\_1}}{d \mathrm{time}}=\frac{\mathrm{V\_3}(1-\mathrm{X\_1})}{\mathrm{K\_3}+1-\mathrm{X\_1}}-\frac{\mathrm{V\_4}\mathrm{X\_1}}{\mathrm{K\_4}+\mathrm{X\_1}}$
$\frac{d \mathrm{C\_2}}{d \mathrm{time}}=\frac{\mathrm{v\_i2}\mathrm{K\_im2}}{\mathrm{K\_im2}+\mathrm{M\_1}}-\frac{\mathrm{v\_d2}\mathrm{X\_2}\mathrm{C\_2}}{\mathrm{K\_d2}+\mathrm{C\_2}}-\mathrm{k\_d2}\mathrm{C\_2}$
$\frac{d \mathrm{M\_2}}{d \mathrm{time}}=\frac{\mathrm{U\_1}(1-\mathrm{M\_2})}{\mathrm{H\_1}+1-\mathrm{M\_2}}-\frac{\mathrm{U\_2}\mathrm{M\_2}}{\mathrm{H\_2}+\mathrm{M\_2}}$
$\frac{d \mathrm{X\_2}}{d \mathrm{time}}=\frac{\mathrm{U\_3}(1-\mathrm{X\_2})}{\mathrm{H\_3}+1-\mathrm{X\_2}}-\frac{\mathrm{U\_4}\mathrm{X\_2}}{\mathrm{H\_4}+\mathrm{X\_2}}$
$\mathrm{V\_1}=\frac{\mathrm{C\_1}}{\mathrm{K\_c1}+\mathrm{C\_1}}\mathrm{V\_M1}$
$\mathrm{V\_3}=\mathrm{M\_1}\mathrm{V\_M3}$
$\mathrm{U\_1}=\frac{\mathrm{C\_2}}{\mathrm{K\_c2}+\mathrm{C\_2}}\mathrm{U\_M1}$
$\mathrm{U\_3}=\mathrm{M\_2}\mathrm{U\_M3}$
10415827
keyword
cell cycle
Added cmeta:id's to some variables
Alternating Oscillations and Chaos in a Model of Two Coupled Biochemical Oscillators Driving Successive Phases of the Cell Cycle
879
180
193
This is the CellML description of Romond et al's 1999 paper on alternating oscillations and chaos in a model of two coupled biochemical oscillators driving successive phases of the cell cycle
1999-06-00 00:00
Jeelean Lim
2009-02-04T00:00:00+00:00
2000
2009-02-12T10:17:30+13:00
Jeelean Lim
Didier
Gonze
This CellML version of the model has been checked in COR and PCEnv and the model runs to replicate the original published results as depicted in figure 3 of the paper. The units have been checked and are consistent. Initial concentrations and ratios were set with reference to the BioModels database.
Jeelean Lim
Jeelean
Lim
Pierre-Charles
Romond
Mauro
Rustici
Annals of the New York Academy of Sciences
Albert
Goldbeter
The University of Auckland
Auckland Bioengineering Institute
This CellML version of the model has been checked in COR and PCEnv and the model runs to replicate the original published results as depicted in figure 3 of the paper. The units have been checked and are consistent. Initial concentrations and ratios were set with reference to the BioModels database.
Jeelean
Lim
jlim063@aucklanduni.ac.nz