Delay Differential Equations: Cell Cycle
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Introduction
This model is a two-species version of the Xenopus embryonic cell cycle shown above that uses delay differential equations (Ferrell et al., 2011). It exhibits sustained limit cycle oscillations.
![Time plots of ODE model of *Xenopus* embryonic cell cycle, modeled with delay differential equations.](/media/model/m0003/cellcycledelay_hu249487c71ba864985ce3f469b9600f3f_13032_30a76faf39584a707966c86a10c6946f.png)
Description
This model uses two Properties
($\mathrm{CDK1}$ and $\mathrm{APC}$) and two DelayProperties
($\text{CDK1}_d$ and $\text{APC}_d$) with delay $\tau$. The latter are properties that return the value that has been assigned at time $t-\tau$.
The updated values of $\mathrm{CDK1}$ and $\mathrm{APC}$ are assigned to (the back of) $\text{CDK1}_d$ and $\text{APC}_d$ using Equations
. When these properties used in the DiffEqn
, they return the value assigned in the past.
The two variables are logged and both a time plot and a phase plot are drawn.
Things to try
- Explore the effect of delays by altering the
DelayProperty/delay
.
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
→ CellCycleDelay.xml
or
CellCycleDelay.xml
XML Preview
<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="3">
<Description>
<Title>Example-CellCycleDelay</Title>
<Details>Example of delay differential equations.
Implements equation 23 and 24 and reproduces figure 7 from:
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="linear">
<Size symbol="size" value="1 0 0"/>
<Neighborhood>
<Order>1</Order>
</Neighborhood>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="25"/>
<SaveInterval value="0"/>
<TimeSymbol symbol="time"/>
</Time>
<CellTypes>
<CellType class="biological" name="cells">
<Property symbol="APC" value="0"/>
<Property symbol="CDK1" value="0"/>
<DelayProperty symbol="APC_d" value="0" delay="0.5"/>
<DelayProperty symbol="CDK1_d" value="0" delay="0.5"/>
<Equation symbol-ref="APC_d">
<Expression>APC</Expression>
</Equation>
<Equation symbol-ref="CDK1_d">
<Expression>CDK1</Expression>
</Equation>
<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="β1" value="3.0"/>
<Constant symbol="β2" value="1.0"/>
<DiffEqn symbol-ref="CDK1">
<Expression>α1 - β1 * CDK1 * (APC_d^n) / (K^n + APC_d^n)</Expression>
</DiffEqn>
<DiffEqn symbol-ref="APC">
<Expression>α2*(1- APC) * ((CDK1_d^n) / (K^n + CDK1_d^n)) - β2*APC</Expression>
</DiffEqn>
</System>
</CellType>
</CellTypes>
<CellPopulations>
<Population size="1" type="cells"/>
</CellPopulations>
<Analysis>
<Logger time-step="1e-2">
<Restriction>
<Celltype celltype="cells"/>
</Restriction>
<Input>
<Symbol symbol-ref="APC"/>
<Symbol symbol-ref="APC_d"/>
<Symbol symbol-ref="CDK1"/>
<Symbol symbol-ref="CDK1_d"/>
</Input>
<Output>
<TextOutput/>
</Output>
<Plots>
<Plot time-step="-1">
<Style style="lines" line-width="3.0"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="time"/>
</X-axis>
<Y-axis>
<Symbol symbol-ref="CDK1"/>
<Symbol symbol-ref="CDK1_d"/>
<Symbol symbol-ref="APC"/>
<Symbol symbol-ref="APC_d"/>
</Y-axis>
</Plot>
</Plots>
</Logger>
</Analysis>
</MorpheusModel>
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