Actin Waves

L. Edelstein-Keshet (Author)

Y. Xiao (Contributor)

L. Edelstein-Keshet: Mathematical Models in Cell Biology

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Spatial relaxation oscillator produces spatial dynamics and spiral waves

Introduction

We consider a full spatial variant of the model of a F-actin negative feedback to GTPase system.

Description

To do so, we add diffusion terms to well-mixed GTPase equations. The PDEs are:

$$\begin{align} \frac{\partial G}{\partial t} &= D\frac{\partial^2 G}{\partial x^2} + (b+\gamma \frac{G^n}{1+G^n})G_{\mathrm i} - G(\eta + sF) \\ \frac{\partial G_{\mathrm i}}{\partial t} &= D_{\mathrm i}\frac{\partial^2 G_{\mathrm i}}{\partial x^2} - (b+\gamma \frac{G^n}{1+G^n})G_{\mathrm i} + G(\eta + sF) \\ \frac{\partial F}{\partial t} &= \epsilon(k_{\mathrm n}G-k_{\mathrm s}F) \\ \end{align}$$

We explore the equation system first in a single spatial dimension in ActinWavesPDE_main.xml and continue in two spatial dimensions using a square lattice in place of the linear lattice in ActinWavesPDE_2D.xml, with the actin reaction rate $\epsilon$ changed from $0.001$ to $0.01$ to speed up the dynamics.

Results

Consider the actin waves PDE system in 1D spatial coordinates (top figures) and in 2D (lower panels):

Simulation of the actin waves PDE system in 1D, showing $G(x, t)$ in the top row and $F(x, t)$ in the bottom row for several values of the feedback parameter $s$. Produced with [`ActinWavesPDE_main.xml`](#model). — Simulation of the actin waves PDE system in 1D, showing $G(x, t)$ in the top row and $F(x, t)$ in the bottom row for several values of the feedback parameter $s$. Produced with `ActinWavesPDE_main.xml`.

Simulation video of ActinWavesPDE_main.xml with the profile of all three variables active GTPase $G(x, t)$, inactive GTPase $G_{\mathrm i}(x, t)$ and F-actin $F(x, t)$ changing over time.

Simulation of the actin waves PDE system in 2D, showing (right, left) $G(x,y,t)$ and $F(x,y,t)$ (side by side) over time. Produced with [`ActinWavesPDE_2D.xml`](#downloads). — Simulation of the actin waves PDE system in 2D, showing (right, left) $G(x,y,t)$ and $F(x,y,t)$ (side by side) over time. Produced with `ActinWavesPDE_2D.xml`.

Simulation video of ActinWavesPDE_2D.xml

Model

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Download: ActinWavesPDE_main.xml

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<MorpheusModel version="4">
    <Description>
        <Title>Actin Waves (PDE) in 1D</Title>
        <Details>Full title:		Actin Waves
Authors:		L. Edelstein-Keshet
Contributors:	Y. Xiao
Date:		23.06.2022
Software:		Morpheus (open-source). Download from https://morpheus.gitlab.io
Model ID:		https://identifiers.org/morpheus/M2013
File type:		Main model
Reference:		L. Edelstein-Keshet: Mathematical Models in Cell Biology
Comment:		Simulations of F-actin negative feedback to the wave-pinning GTPase model produced the actin waves PDE system. Here we simulate this in 1D, showing profiles of the variables over space and time, as well as kymographs of the active GTPase and the F-actin summarizing the entire dynamics.</Details>
    </Description>
    <Global>
        <Field symbol="a" name="activeGTPase" value="1.5+0.7*(0.5-rand_norm(1,0.5))">
            <Diffusion rate="0.01"/>
        </Field>
        <Field symbol="i" name="Inactive" value="2.0">
            <Diffusion rate="1"/>
        </Field>
        <Field symbol="F_a" name="F-actin" value="1.3">
            <Diffusion rate="0.0"/>
        </Field>
        <System time-step="0.05" solver="Runge-Kutta [fixed, O(4)]">
            <DiffEqn symbol-ref="a">
                <Expression> i*(b+gamma*a^n/(1+a^n))- a*(eta+s*F_a) </Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="i">
                <Expression> -i*(b+gamma*a^n/(1+a^n))+ a*(eta+s*F_a)</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="F_a">
                <Expression> epsilon*(k_n*a-k_s*F_a)</Expression>
            </DiffEqn>
            <Constant symbol="b" name="basal activation rate" value="0.067"/>
            <Constant symbol="gamma" name="feedback rate" value="4.5"/>
            <Constant symbol="n" name="Hill coefficient" value="3"/>
            <Constant symbol="k_n" name="Actin nucleation rate" value="20"/>
            <Constant symbol="k_s" name="Actin disassembly rate" value="2"/>
            <Constant symbol="eta" name="basal GTPase decay rate" value="0.5"/>
            <Constant symbol="s" name="actin-dependent GTPase decay rate" value="0.5"/>
            <Constant symbol="F_0" name="actin set point" value="1"/>
            <Constant symbol="epsilon" name="actin reaction rate" value="0.001"/>
        </System>
    </Global>
    <Space>
        <Lattice class="linear">
            <Size symbol="size" value="100, 0, 0"/>
            <BoundaryConditions>
                <Condition type="noflux" boundary="x"/>
            </BoundaryConditions>
            <NodeLength value="0.02"/>
            <Neighborhood>
                <Order>1</Order>
            </Neighborhood>
        </Lattice>
        <SpaceSymbol symbol="space"/>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="6000"/>
        <SaveInterval value="0"/>
        <!--    <Disabled>
        <RandomSeed value="1"/>
    </Disabled>
-->
        <TimeSymbol symbol="time"/>
    </Time>
    <Analysis>
        <Logger time-step="50">
            <Input>
                <Symbol symbol-ref="a"/>
                <Symbol symbol-ref="i"/>
                <Symbol symbol-ref="F_a"/>
            </Input>
            <Output>
                <TextOutput/>
            </Output>
            <Plots>
                <Plot time-step="10" title="space plot">
                    <Style line-width="3.0" style="lines"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="space.x"/>
                    </X-axis>
                    <Y-axis minimum="0" maximum="10">
                        <Symbol symbol-ref="a"/>
                        <Symbol symbol-ref="i"/>
                        <Symbol symbol-ref="F_a"/>
                    </Y-axis>
                    <Range>
                        <Time mode="current"/>
                    </Range>
                </Plot>
                <Plot time-step="-1" title="time-space plot">
                    <Style decorate="true" point-size="2" style="points"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="space.x"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="time"/>
                    </Y-axis>
                    <Color-bar>
                        <Symbol symbol-ref="a"/>
                    </Color-bar>
                </Plot>
                <Plot time-step="-1" title="time-space plot">
                    <Style decorate="true" point-size="2" style="points"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="space.x"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="time"/>
                    </Y-axis>
                    <Color-bar>
                        <Symbol symbol-ref="F_a"/>
                    </Color-bar>
                </Plot>
            </Plots>
        </Logger>
        <ModelGraph format="dot" reduced="false" include-tags="#untagged"/>
    </Analysis>
</MorpheusModel>

Downloads

Files associated with this model:

1D 2D Actin Actin Wave Contributed Examples F-actin Filament Lattice Linear Lattice Microfilament Partial Differential Equation PDE Reaction Rate Square Lattice Wave Well-mixed

Last updated on Mon, 13 May 2024