Basic Rac-Rho-ECM Spatial Model

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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 1 showing Model I regimes: Behaviour of Rac for the full 1D spatially distributed version of Model I, Eqs. 2.3a and 2.3b in the paper. These 1D PDEs are a spatial variant of a basic model previously studied by Holmes et al. Here the ECM dynamics is phenomenological, with Rac enhancing and Rho damping the ECM signaling.

[Suppl. Fig. 1](https://iopscience.iop.org/article/10.1088/1478-3975/ac2888/pdf) showing Model I regimes ([CC BY 4.0](https://creativecommons.org/licenses/by/4.0/): [**Rens *et al.***](#reference))
Suppl. Fig. 1 showing Model I 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 I (Related to the Model 3 in Holmes et al (2017)

R= Rac, P = Rho, E = ECM

Rac-Rho equations d R/dt= {b_R }/{1+\rho_a^3}R_I - \delta R +D_R Laplacian( R), d \rho/dt ={b_\rho }{1+R_a^3}\rho_I - \delta \rho +D_\rho Laplacian( Rho)

ECM equation

dE/dt= epsilon*((K+GammaRf_R1())-E(kP+GammaPf_P1())) where the f’s are Hill functions of Rac or Rho.</Details> <Title>Model1RacRhoECMPDEsIn1D</Title> </Description> <Space> <Lattice class="linear"> <Neighborhood> <Order>1</Order> </Neighborhood> <Size symbol="size" value="30, 0, 0"/> <NodeLength symbol="dx" value="0.1"/> <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="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="GammaP" value="10" name="Rho feedback to ECM reduction"/> <Constant symbol="GammaR" value="5.0" name="Rac-dependent ECM rate of increase"/> <Constant symbol="R0" value="1.0" name="Rac level for half-max ECM activation"/> <Constant symbol="P0" value="2.4" name="Rho level for half-max ECM inhibition"/> <Constant symbol="epsilon" value="0.001" name="1/( ECM timescale)"/> <Constant symbol="K" value="0.1" name="ECM basal rate of increase"/> <Constant symbol="kP" value="0.45" name="Rho-induced ECM downregulation"/> <Constant symbol="E0" value="1.5" name="ECM level for half-max Rho activation"/> <Constant symbol="n" value="3" name="Hill coefficient"/> <Function symbol="x"> <Expression>dxspace.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="0&#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="plotR" value="0.0" name="Rac for plotting"/> <System solver="adaptive45" time-step="0.1"> <DiffEqn symbol-ref="R" name="PDE for active Rac"> <Expression>(bR/(1+P^n))RI-1deltaR</Expression> </DiffEqn> <DiffEqn symbol-ref="P" name="PDE for active Rho"> <Expression>(kE+GammaEf_E1())PI/(1+R^n)-1P</Expression> </DiffEqn> <Function symbol="f_E1" name="function for ECM-dependent rate of Rho activation "> <Expression>E^3/(E0^3+E^3)</Expression> </Function> <DiffEqn symbol-ref="E" name="PDE for ECM variable"> <Expression>epsilon*((K+GammaRf_R1())-E(kP+GammaPf_P1()))</Expression> </DiffEqn> <Function symbol="f_R1" name="Formula for Rac-dependent ECM activation"> <Expression>R^n/(R0^n+R^n)</Expression> </Function> <Function symbol="f_P1" name="formula for Rho-dependent ECM supression"> <Expression>P^n/(P0^n+P^n)</Expression> </Function> <DiffEqn symbol-ref="RI" name="PDE for inactive Rac"> <Expression>(-bR/(1+P^n))RI+1deltaR</Expression> </DiffEqn> <DiffEqn symbol-ref="PI" name="PDE for inactive Rho"> <Expression>-(kE+GammaE*f_E1())PI/(1+R^n)+1P</Expression> </DiffEqn> <Rule symbol-ref="ECM" name="ECM"> <Expression>E</Expression> </Rule> <Rule symbol-ref="plotR"> <Expression>min(R,2)</Expression> </Rule> </System> <Field symbol="ECM" value="0"/> </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="4.0" 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="R"/> </Y-axis> <Range> <Time mode="current" history="1.0"/> </Range> </Plot> </Plots> </Logger> </Analysis> </MorpheusModel>

Model Graph
Model Graph

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