MembraneProperties: Cell Polarization and Chemotaxis

Y. Mori, A. Jilkine, L. Edelstein-Keshet (Authors)

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Introduction

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

A cellular Potts model with DirectedMotion that responds to cell polarity (but not directly to the external signal gradient),
a co-moving PDE model on the membrane of the cell (to let cell polarity emerge from distributed receptor-signal interactions) and
an external signal gradient (implemented as a global field with random fluctuations).

The cell membrane polarizes in response to the external signal U that is sensed as a pattern (of receptor activation) s along the cell membrane.

Cell dynamically repolarizes in response to switching external gradient.

Description

This example implements two models of cell polarity formation: Meinhardt’s substrate-depletion model (ASDM) and Edelstein-Keshet’s wave-pinning (WP) model. The user can switch the polarity model by disabling/enabling the relevant System under CellType. Equations and parameter values are copied from Mori et al..

Each model defines a two-component reaction-diffusion system (of MembraneProperty A and B) representing membrane-bound molecules, and is mapped to and co-moves with a cell. Cell motility and cell shape dynamics are given by the cellular Potts model. An external gradient with random fluctuations, specified in Global, provides a signal for the polarization of the cell. In turn, the pattern of the MembraneProperty A gives the polarity vector of the cell. The polarity vector then controls the DirectedMotion behavior of the cell. A TiffPlotter exports the full 3D plus time data set.

After a switch in direction of the gradient, the cell re-polarizes in the new direction and starts to move in the new direction, if the wave pinning model has been selected (see video below). A GnuPlotter visualizes the gradient, cell shape and MembraneProperty A at a horizontal slice through 3D space (shown below) and a Logger visualizes the (phi, theta)-dependency of the MembraneProperty A (not shown).

The alternative ASDM maintains its previous polarity for much longer and does not re-polarize during the runtime (see video below). On a longer time scale, the ASDM may eventually re-polarize the 3D cell by rotating the activator maximum over the cell membrane. The cell would take a U-turn with a large radius. The same behaviours are observed for any other strength of the gradient’s fluctuations as well as for a smooth gradient.

Reference

This model reproduces a published result, originally obtained with a different simulator:

Y. Mori, A. Jilkine, L. Edelstein-Keshet: Wave-Pinning and Cell Polarity from a Bistable Reaction-Diffusion System.. Biophysical Journal 94(9): 3684-3697, 2008.

Model

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Download: CellPolarity.xml

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<MorpheusModel version="4">
    <Description>
        <Title>Example-CellPolarity</Title>
        <Details>Chemotaxis of polarized cell
----------------------------
Show feedback between cell motility, cell polarization and external gradient.
 
Two cell polarization models are taken from the original publication:
Y. Mori, A. Jilkine and L. Edelstein-Keshet (2008). Wave-Pinning and Cell Polarity from a Bistable Reaction-Diffusion System. Biophysical Journal 94(9), 3684-3697. DOI:10.1529/biophysj.107.120824

and can be enabled/disabled for a cell in 3D through the respective System in the Celltype 'cells':
- Substrate-depletion model: no repolarization
- Wave-pinning model: repolarization (as studied in detail in the above publication)
 
Events in Global:
- Switch on/off and change gradients (noise through global constant)</Details>
    </Description>
    <Global>
        <Field symbol="U" value="noisy(l.x / lattice.x)">
            <Diffusion rate="0.0"/>
        </Field>
        <Event trigger="on-change" time-step="100">
            <Condition>time>=1000</Condition>
            <Rule symbol-ref="U">
                <Expression>noisy(0.5)</Expression>
            </Rule>
        </Event>
        <Event trigger="on-change" time-step="100">
            <Condition>time>=1500</Condition>
            <Rule symbol-ref="U">
                <Expression>noisy((lattice.x-l.x) / lattice.x)</Expression>
            </Rule>
        </Event>
        <Constant symbol="l_U" value="0.0"/>
        <Constant symbol="lengthscale" value="3"/>
        <Constant symbol="noise" value="0.2"/>
        <Function symbol="noisy">
            <Parameter symbol="v" name="value"/>
            <Expression>max(0,(v+rand_norm(0,noise)))</Expression>
        </Function>
    </Global>
    <Space>
        <Lattice class="cubic">
            <Size symbol="lattice" value="150, 75, 30"/>
            <BoundaryConditions>
                <Condition type="noflux" boundary="x"/>
                <Condition type="periodic" boundary="y"/>
                <Condition type="constant" boundary="z"/>
            </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="2500"/>
        <SaveInterval value="0"/>
        <RandomSeed value="0"/>
        <TimeSymbol symbol="time"/>
    </Time>
    <CellTypes>
        <CellType name="cells" class="biological">
            <VolumeConstraint target="5000" strength="0.1"/>
            <SurfaceConstraint target="1" strength="0.02" mode="aspherity"/>
            <MembraneProperty symbol="A" name="A" value="1.5 + rand_uni(-0.1,0.1)">
                <Diffusion rate="0.05*lengthscale^2"/>
            </MembraneProperty>
            <MembraneProperty symbol="B" name="B" value="0.8">
                <Diffusion rate="10*lengthscale^2"/>
            </MembraneProperty>
            <MembraneProperty symbol="c" name="chemotactic strength" value="0">
                <Diffusion rate="0"/>
            </MembraneProperty>
            <MembraneProperty symbol="s" name="signal" value="0">
                <Diffusion rate="0"/>
            </MembraneProperty>
            <System time-step="0.05" name="WavePinning" solver="Dormand-Prince [adaptive, O(5)]">
                <Constant symbol="k_0" value="0.067"/>
                <Constant symbol="gamma" value="1"/>
                <Constant symbol="delta" value="1"/>
                <Constant symbol="K" value="1"/>
                <!--    <Disabled>
        <Constant symbol="sigma" name="spatial-length-signal" value="5.0"/>
    </Disabled>
-->
                <Constant symbol="n" name="Hill coefficient" value="2"/>
                <Intermediate symbol="F" value="B*(k_0+ 0.2*(s-U.mean) + (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 name="Sensing the field">
                <Input value="U"/>
                <Output symbol-ref="s" mapping="average"/>
            </Mapper>
            <PropertyVector symbol="p" name="polarity" value="0.0, 0.0, 0.0"/>
            <Mapper name="Polarity extraction from membrane">
                <Input value="A"/>
                <Polarity symbol-ref="p"/>
            </Mapper>
            <DirectedMotion strength="0.04" direction="p"/>
            <!--    <Disabled>
        <System time-step="0.05" name="Substrate-Depletion(ASDM)" solver="Dormand-Prince [adaptive, O(5)]">
            <Rule symbol-ref="c">
                <Expression>A^2*5e1</Expression>
            </Rule>
            <DiffEqn symbol-ref="A">
                <Expression>B*(s.norm*A^2/(1+s_a*A^2)) - r_a * A</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="B">
                <Expression>b_b - B*(s.norm*A^2/(1+s_a*A^2)) - r_b*B</Expression>
            </DiffEqn>
            <Constant symbol="s_a" value="0.1"/>
            <Constant symbol="r_a" value="0.1"/>
            <Constant symbol="b_b" value="0.15"/>
            <Constant symbol="r_b" value="0.0"/>
            <Function symbol="s.norm">
                <Expression>0.1+0.01*(s-U.mean)/U.mean</Expression>
            </Function>
            <Disabled>
                <Constant symbol="s.norm" value="0.1"/>
            </Disabled>
        </System>
    </Disabled>
-->
            <Property symbol="U.mean" value="0.0"/>
            <Mapper name="scalar signal average over whole cell">
                <Input value="U"/>
                <Output symbol-ref="U.mean" mapping="average"/>
            </Mapper>
        </CellType>
    </CellTypes>
    <CPM>
        <Interaction default="0.0"/>
        <MonteCarloSampler stepper="edgelist">
            <MCSDuration value="1"/>
            <Neighborhood>
                <Order>2</Order>
            </Neighborhood>
            <MetropolisKinetics yield="0.05" temperature="1"/>
        </MonteCarloSampler>
        <ShapeSurface scaling="norm">
            <Neighborhood>
                <Order>6</Order>
            </Neighborhood>
        </ShapeSurface>
    </CPM>
    <CellPopulations>
        <Population type="cells" size="0">
            <InitCellObjects mode="distance">
                <Arrangement repetitions="1, 1, 1" displacements="1, 1, 1">
                    <Sphere center="25 37 15" radius="10"/>
                </Arrangement>
            </InitCellObjects>
        </Population>
    </CellPopulations>
    <Analysis>
        <Gnuplotter time-step="50">
            <Terminal name="png"/>
            <Plot slice="15">
                <Field symbol-ref="U"/>
                <Cells min="0.2" value="A"/>
                <CellArrows orientation="p * 50"/>
            </Plot>
        </Gnuplotter>
        <TiffPlotter time-step="100" format="8bit" OME-header="false">
            <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>
        <ModelGraph include-tags="#untagged" format="svg" reduced="false"/>
    </Analysis>
</MorpheusModel>

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Last updated on Fri, 02 Dec 2022