MembraneProperties: Cell Polarization and Chemotaxis

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Introduction

This model of cell polarity shows the coupling of three model formalisms:

  • A cellular Potts model,
  • a PDE model, solved on the membrane of the cell and
  • an external gradient.

The cell membrane polarizes in response to the external gradient. Chemotactic cell motility depends on the polarity of the cell and the external gradient.

Cell dynamically repolarizes in response to switching external gradient.
Cell dynamically repolarizes in response to switching external gradient.

Description

This example implements two models of cell polarity: Meinhardt’s substrate-depletion model and Edelstein-Keshet’s wave-pinning model. The user can switch polarity model by Disabling/Enabling the relevant System.

The model defines a one-dimensional reaction-diffusion system (MembraneProperty) representing membrane-bound molecules, and is mapped to a cellular Potts model defining a discrete shaped cell. An external gradient, specified in a PDE, provides a signal for the polarization of the cell. In turn, the polarity of the cell influences its chemotaxic behavior.

After a switch in direction of the gradient, the cell re-polarizes in the new direction and starts to move up the gradient, if the wave pinning model has been selected.

Model

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<MorpheusModel version="3">
    <Description>
        <Title>Example-CellPolarity</Title>
        <Details>Chemotaxis of polarized cell
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Show feedback between cell motility, cell polarization and external gradient.

Two simple polarization models can be enabled/disabled:

  • Substrate-depletion model: no repolarization
  • Wave-pinning model: repolarization

PDE Equation:

  • Switch on/off and change gradients </Details> </Description> <Global> <Field symbol="U" value="l.x / lattice.x"> <Diffusion rate="0.0"/> </Field> <!– <Disabled> <Equation symbol-ref="U"> <Expression>l.x / lattice.x + 0if( t > 500 and t &lt; 1500, l.x / lattice.x, 0)</Expression> </Equation> </Disabled> –> <Constant symbol="c" value="0.0"/> <!– <Disabled> <Constant symbol="A" value="0.0"/> </Disabled> –> </Global> <Space> <Lattice class="cubic"> <Size symbol="lattice" value="150 75 50"/> <BoundaryConditions> <Condition boundary="x" type="noflux"/> <Condition boundary="y" type="periodic"/> </BoundaryConditions> <NodeLength value="1.0"/> <Neighborhood> <Distance>2.0</Distance> </Neighborhood> </Lattice> <SpaceSymbol symbol="l"/> <MembraneLattice> <Resolution symbol="memsize" value="50"/> <SpaceSymbol symbol="m"/> </MembraneLattice> </Space> <Time> <StartTime value="0"/> <StopTime value="2000"/> <SaveInterval value="0"/> <RandomSeed value="0"/> <TimeSymbol symbol="t"/> </Time> <CellTypes> <CellType class="biological" name="cells"> <Chemotaxis field="U" strength="100(A)" retraction="true"/> <VolumeConstraint target="5000" strength="0.1"/> <SurfaceConstraint target="1" mode="aspherity" strength="0.05"/> <MembraneProperty symbol="A" value="0.5" name="A"> <Diffusion rate="0.05"/> </MembraneProperty> <MembraneProperty symbol="B" value="0.5" name="B"> <Diffusion rate="1"/> </MembraneProperty> <MembraneProperty symbol="c" value="0" name="chemotactic strength"> <Diffusion rate="0"/> </MembraneProperty> <MembraneProperty symbol="s" value="0" name="signal"> <Diffusion rate="0"/> </MembraneProperty> <Property symbol="l_U" value="0"/> <!– <Disabled> <System solver="runge-kutta" time-step="1.0" name="Substrate-Depletion"> <Rule symbol-ref="c"> <Expression>A^25e1</Expression> </Rule> <DiffEqn symbol-ref="A"> <Expression>(rhoA^2) / B - mu_a * A + rho_a + 0.001s</Expression> </DiffEqn> <DiffEqn symbol-ref="B"> <Expression>(rhoA^2) - mu_i * B</Expression> </DiffEqn> <Constant symbol="rho_a" value="0.01"/> <Constant symbol="mu_i" value="0.03"/> <Constant symbol="mu_a" value="0.02"/> <Constant symbol="rho" value="0.001"/> </System> </Disabled> –> <System solver="runge-kutta" time-step="1.0" name="WavePinning"> <Constant symbol="k_0" value="0.067"/> <Constant symbol="gamma" value="1"/> <Constant symbol="delta" value="0.25"/> <Constant symbol="K" value="1"/> <Constant symbol="sigma" value="5.0" name="spatial-length-signal"/> <Constant symbol="n" value="4" name="Hill coefficient"/> <Intermediate symbol="F" value="B*(k_0+ s + (gamma*A^n) / (K^n + A^n) ) - delta*A"/> <DiffEqn symbol-ref="A"> <Expression>F</Expression> </DiffEqn> <DiffEqn symbol-ref="B"> <Expression>-F</Expression> </DiffEqn> <Rule symbol-ref="c"> <Expression>A^2*1000</Expression> </Rule> </System> <Mapper> <Input value="U"/> <Output symbol-ref="s" mapping="average"/> </Mapper> </CellType> <CellType class="medium" name="medium"> <Property symbol="l_U" value="0"/> </CellType> </CellTypes> <CPM> <Interaction default="0.0"> <Contact type1="cells" type2="medium" value="-10"/> <Contact type1="cells" type2="cells" value="-20"/> </Interaction> <MonteCarloSampler stepper="edgelist"> <MCSDuration value="1"/> <Neighborhood> <Order>2</Order> </Neighborhood> <MetropolisKinetics temperature="1" yield="0.05"/> </MonteCarloSampler> <ShapeSurface scaling="norm"> <Neighborhood> <Order>6</Order> </Neighborhood> </ShapeSurface> </CPM> <CellPopulations> <Population size="0" type="cells"> <!– <Disabled> <InitRectangle mode="regular" number-of-cells="1"> <Dimensions size="150, 75, 50" origin="0.0, 0.0, 0.0"/> </InitRectangle> </Disabled> –> <Cell id="1" name="1"> <Nodes>25 37 25</Nodes> </Cell> </Population> </CellPopulations> <Analysis> <Gnuplotter time-step="50"> <Terminal name="png"/> <Plot> <Field symbol-ref="U"/> <Cells value="A" slice="25"/> </Plot> </Gnuplotter> <TiffPlotter format="8bit" OME-header="false" time-step="100"> <Channel symbol-ref="cell.id" celltype="cells"/> <Channel symbol-ref="c" celltype="cells"/> <Channel symbol-ref="U"/> </TiffPlotter> <Logger time-step="50"> <Input> <Symbol symbol-ref="A"/> </Input> <Output> <TextOutput file-format="matrix"/> </Output> <Plots> <SurfacePlot time-step="50"> <Color-bar> <Symbol symbol-ref="A"/> </Color-bar> <Terminal terminal="png"/> </SurfacePlot> </Plots> <Restriction> <Celltype celltype="cells"/> </Restriction> </Logger> </Analysis> </MorpheusModel>

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