Rac-Rho-ECM Spatial Model with Catch-Slip Bond Integrin Biophysics
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Introduction
This paper describes the regimes of behaviour of a 1D spatial (PDE) model for the mutually antagonistic Rac-Rho GTPases, with feedback to and from the extracellular matrix (ECM).
Description
The Rac-Rho submodels are bistable, and ECM enhances Rho activation. Rac and Rho contribute positive and negative feedback, respectively, to the ECM. The full model has regimes of uniform, polar, random, and oscillatory dynamics.
Results
The file listed below was used to produce Figure 6, Model III: the catch-slip bond model in 1D, showing regimes of no pattern, polarized patterns, and some oscillatory patterns.
 showing Model III regimes ([CC BY 4.0](https://creativecommons.org/licenses/by/4.0/): [**Rens *et al.***](#reference))](/media/model/m2063/RensKeshetModel3image_hu26eebdc51820b9c3d4c6353fdfcfab39_159019_4bc31e09aa99ca958819a31f0f47df81.jpg)
Reference
This model is the original used in the publication, up to technical updates:
E. G. Rens, L. Edelstein-Keshet: Cellular Tango: how extracellular matrix adhesion choreographs Rac-Rho signaling and cell movement. Phys. Biol. 18 (6): 066005, 2021.
Model
RacRhoECM3-catchslip.xml
XML Preview
<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="4">
<Description>
<Details>Spatially distributed Model III
R= Rac, P = Rho, E = ECM
Based on: Model 2, with ECM represented as catch-slip bonds
Same Rac-Rho Equations.
ECM Equation given by
dE/dt=epsilon*(K*(Et-E) - E*min(k0*exp(force/(p*(E+En)))+k0c*exp(-force/(p*(E+En)))
where the force depends on Rac and Rho</Details>
<Title>Model3CatchSlipBondRacRhoECMPDEsIn1D</Title>
</Description>
<Space>
<Lattice class="linear">
<Neighborhood>
<Order>1</Order>
</Neighborhood>
<Size symbol="size" value="60, 0, 0"/>
<NodeLength symbol="dx" value="0.05"/>
<BoundaryConditions>
<Condition boundary="x" type="noflux"/>
</BoundaryConditions>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="1000"/>
<TimeSymbol symbol="time"/>
</Time>
<Global>
<Constant symbol="bR" value="3" name="Rac activation rate"/>
<Constant symbol="delta" value="1.0" name="Rac decay rate"/>
<Constant symbol="kE" value="2" name="Rho basal activation rate"/>
<Constant symbol="GammaE" value="4" name="Rho activation due to ECM feedback"/>
<Constant symbol="epsilon" value="0.001" name="1/( ECM timescale)"/>
<Constant symbol="K" value="10" name="ECM basal rate of increase"/>
<Constant symbol="E0" value="300" name="ECM level for half-max Rho activation"/>
<Constant symbol="n" value="3" name="Hill coefficient"/>
<Constant symbol="k0" value="exp(-7.78)" name="ECM decay rate due to slip bond"/>
<Constant symbol="En" value="100" name="small ECM contact"/>
<Constant symbol="p" value="0.08" name="Force per bond reference value"/>
<Constant symbol="betaR" value="1000" name="Maximal Rac-dependent force magnitude"/>
<Constant symbol="betaP" value="1600" name="Maximal Rho-dependent force magnitude"/>
<Constant symbol="Et0" value="1000" name="Maximal adhesion size at zero force"/>
<Constant symbol="k0c" value="exp(4.02)" name="catch bond part of ECM decay rate"/>
<Field symbol="P" value="0.0" name="Rho">
<Diffusion rate="0.1"/>
</Field>
<Field symbol="R" value="if(x<=0.3, 4, 0)" name="Rac">
<Diffusion rate="0.1"/>
</Field>
<Field symbol="E" value="50
" name="ECM contact">
<Diffusion rate="0"/>
</Field>
<Field symbol="RI" value="1.5" name="Inactive Rac">
<Diffusion rate="1"/>
</Field>
<Field symbol="PI" value="1.5" name="Inactive Rho">
<Diffusion rate="1"/>
</Field>
<Field symbol="ECM" value="0.0"/>
<Field symbol="forcefield" value="0.0"/>
<Field symbol="plotR" value="0.0" name="Rac for plotting"/>
<Function symbol="x">
<Expression>dx*space.x</Expression>
</Function>
<System solver="adaptive45" time-step="0.1">
<DiffEqn symbol-ref="R">
<Expression>(bR/(1+P^n))*RI-1*delta*R</Expression>
</DiffEqn>
<DiffEqn symbol-ref="P">
<Expression>(kE+GammaE*f_E1())*PI/(1+R^n)-1*P</Expression>
</DiffEqn>
<DiffEqn symbol-ref="E">
<Expression>epsilon*(K*(Et-E) - E*min(k0*exp(force/(p*(E+En)))+k0c*exp(-force/(p*(E+En))),1000000))</Expression>
</DiffEqn>
<Function symbol="f_E1">
<Expression>E^3/(E0^3+E^3)</Expression>
</Function>
<DiffEqn symbol-ref="RI">
<Expression>(-bR/(1+P^n))*RI+1*delta*R</Expression>
</DiffEqn>
<DiffEqn symbol-ref="PI">
<Expression>-(kE+GammaE*f_E1())*PI/(1+R^n)+1*P</Expression>
</DiffEqn>
<Function symbol="force">
<Expression>max(0,-betaR*R/(1+R)+betaP*P/(1+P))</Expression>
</Function>
<Rule symbol-ref="ECM">
<Expression>E/100</Expression>
</Rule>
<Rule symbol-ref="forcefield">
<Expression>force/200</Expression>
</Rule>
<Function symbol="Et">
<Expression>Et0</Expression>
</Function>
<Rule symbol-ref="plotR">
<Expression>min(R,2)</Expression>
</Rule>
</System>
</Global>
<Analysis>
<Logger time-step="1" name="chemical profiles">
<Input>
<Symbol symbol-ref="R"/>
<Symbol symbol-ref="P"/>
<Symbol symbol-ref="E"/>
<Symbol symbol-ref="PI"/>
<Symbol symbol-ref="RI"/>
</Input>
<Output>
<TextOutput/>
</Output>
<Plots>
<Plot time-step="-1">
<Style point-size="2.0" style="points" decorate="false"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="x"/>
</X-axis>
<Y-axis>
<Symbol symbol-ref="time"/>
</Y-axis>
<Color-bar minimum="0.0" maximum="2">
<Symbol symbol-ref="plotR"/>
</Color-bar>
</Plot>
<Plot time-step="10">
<Style style="lines" line-width="2.0"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="x"/>
</X-axis>
<Y-axis minimum="0.0" maximum="7">
<Symbol symbol-ref="P"/>
<Symbol symbol-ref="ECM"/>
<Symbol symbol-ref="forcefield"/>
<Symbol symbol-ref="R"/>
</Y-axis>
<Range>
<Time mode="current" history="1.0"/>
</Range>
</Plot>
</Plots>
</Logger>
</Analysis>
</MorpheusModel>
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