ODE model: Cell cycle

Introduction

This model is a simple three-species ODE model of the Xenopus embryonic cell cycle (Ferrell et al., 2011). It exhibits sustained limit cycle oscillations.

Time plots of ODE model of *Xenopus* embryonic cell cycle
Time plots of ODE model of Xenopus embryonic cell cycle.

Model description

One CellType is created that has three variables of Properties representing the concentrations of $\mathrm{APC}$, $\mathrm{Plk1}$, and $\mathrm{CDK1}$. These variables are coupled in a System of DiffEqns.

In the System, a number of Constants are defined, whose symbols are used in the DiffEqn. Note that the equations are entered in simple plain text.

The System uses the runga-kutta (4th order) solver for the differential equations and speficies a particular time step (here $ht = 10^{-2}$) which is interpreted in global time steps.

The global time is defined in the Time element and runs from StartTime to StopTime ($0$ to $25$). In this non-spatial model, Space defines a Lattice of size $(x,y,z)=(1,1,0)$.

Results are written to a file using the Analysis plugin Logger. The Logger also visualizes the time plot to screen (in interactive mode) or to PNG files (in local mode).

Things to try

  • Change the dynamics by altering System/time-scaling.

Model

In Morpheus GUI: Examples 🠒 ODE 🠒 CellCycle.xml.

<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="3">
    <Description>
        <Title>Example-CellCycle</Title>
        <Details>James Ferrell, Tony Yu-Chen Tsai and Qiong Yang (2011) Modeling the Cell Cycle: Why Do Certain Circuits Oscillate?, Cell 144, p874-885. http://dx.doi.org/10.1016/j.cell.2011.03.006</Details>
    </Description>
    <Global/>
    <Space>
        <Lattice class="square">
            <Size symbol="size" value="1,1,0"/>
            <Neighborhood>
                <Order>1</Order>
            </Neighborhood>
        </Lattice>
        <SpaceSymbol symbol="space"/>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="25"/>
        <TimeSymbol symbol="time"/>
    </Time>
    <CellTypes>
        <CellType class="biological" name="cells">
            <Property symbol="APC" value="0"/>
            <Property symbol="Plk1" value="0"/>
            <Property symbol="CDK1" value="0"/>
            <System solver="runge-kutta" time-scaling="1" time-step="1e-2">
                <Constant symbol="n" value="8"/>
                <Constant symbol="K" value="0.5"/>
                <Constant symbol="α1" value="0.1"/>
                <Constant symbol="α2" value="3.0"/>
                <Constant symbol="α3" value="3.0"/>
                <Constant symbol="β1" value="3.0"/>
                <Constant symbol="β2" value="1.0"/>
                <Constant symbol="β3" value="1.0"/>
                <DiffEqn symbol-ref="CDK1">
                    <Expression>α1 - β1 * CDK1 * (APC^n) / (K^n + APC^n)</Expression>
                </DiffEqn>
                <DiffEqn symbol-ref="Plk1">
                    <Expression>α2*(1-Plk1) * ((CDK1^n) / (K^n + CDK1^n)) - β2*Plk1</Expression>
                </DiffEqn>
                <DiffEqn symbol-ref="APC">
                    <Expression>α3*(1- APC) * ((Plk1^n) / (K^n + Plk1^n)) - β3*APC</Expression>
                </DiffEqn>
            </System>
        </CellType>
    </CellTypes>
    <CellPopulations>
        <Population size="0" type="cells">
            <InitCellLattice/>
        </Population>
    </CellPopulations>
    <Analysis>
        <Logger time-step="1e-2">
            <Restriction>
                <Celltype celltype="cells"/>
            </Restriction>
            <Input>
                <Symbol symbol-ref="APC"/>
                <Symbol symbol-ref="CDK1"/>
                <Symbol symbol-ref="Plk1"/>
            </Input>
            <Output>
                <TextOutput file-format="csv"/>
            </Output>
            <Plots>
                <Plot time-step="-1">
                    <Style style="lines" line-width="4.0"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="time"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="APC"/>
                        <Symbol symbol-ref="CDK1"/>
                        <Symbol symbol-ref="Plk1"/>
                    </Y-axis>
                </Plot>
                <Plot time-step="-1">
                    <Style style="lines" line-width="4.0"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="CDK1"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="APC"/>
                    </Y-axis>
                    <Color-bar>
                        <Symbol symbol-ref="Plk1"/>
                    </Color-bar>
                </Plot>
            </Plots>
        </Logger>
    </Analysis>
</MorpheusModel>

Reference

J. E. Ferrell Jr., T. Y. Tsai, Q. Yang. Modeling the Cell Cycle: Why Do Certain Circuits Oscillate? Cell, 144 (6): 874-885, 2011.

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