ODE Model: Cell Cycle
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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.
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 DiffEqn
s.
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
.
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.
Model
Examples
→ ODE
→ CellCycle.xml
or
CellCycle.xml
XML Preview
<?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>
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