Rac-Rho-ECM Spatial Model with Slip-Bond Integrin Dynamics

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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 supplementary Figure 2. Model II regimes: As in SI Fig. 1, but for the slip-bond integrin model in the ECM dynamics PDE.

[Suppl. Fig. 2](https://iopscience.iop.org/article/10.1088/1478-3975/ac2888/pdf) showing Model II regimes ([CC BY 4.0](https://creativecommons.org/licenses/by/4.0/): [**Rens *et al.***](#reference))
Suppl. Fig. 2 showing Model II regimes (CC BY 4.0: Rens et al.)

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

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    <?xml version='1.0' encoding='UTF-8'?>
    <MorpheusModel version="4">
        <Description>
            <Details>Spatially distributed Model II
    
    
    R= Rac, P = Rho, E = ECM
    
    Based on: Model 1,  with ECM represented as slip-bonds
    Same Rac-Rho Equations.
    ECM Equation given by
    
    dE/dt=epsilon*(K*(Et-E) - E*min(k0*exp(force/(p*(E+En))),100000))
    
    where the force depends on Rac and Rho</Details>
            <Title>Model2SlipBondRacRhoECMPDEsIn1D</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="5" name="Rac activation rate"/>
            <Constant symbol="RT" value="3.0" name="Total Rac"/>
            <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="PT" value="6.0" name="Total Rho"/>
            <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"/>
            <Constant symbol="n" value="3"/>
            <Constant symbol="k0" value="5" name="ECM decay"/>
            <Constant symbol="En" value="100" name="small ECM contact"/>
            <Constant symbol="p" value="0.35" name="force scaling factor"/>
            <Constant symbol="Et0" value="1000"/>
            <Constant symbol="betaR" value="1200" name="rac force"/>
            <Constant symbol="betaP" value="1600" name="rho force"/>
            <Function symbol="x">
                <Expression>dx*space.x</Expression>
            </Function>
            <Field symbol="P" value="0.0" name="Rho">
                <Diffusion rate="0.1"/>
            </Field>
            <Field symbol="R" value="if(x&lt;=0.3, 4, 0)" name="Rac">
                <Diffusion rate="0.1"/>
            </Field>
            <Field symbol="E" value="50&#xa;" 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"/>
            <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))),100000))</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>
            </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="R"/>
                        </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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