Well-mixed Actin Waves

L. Edelstein-Keshet (Author)

Y. Xiao (Contributor)

L. Edelstein-Keshet: Mathematical Models in Cell Biology

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ODEs for GTPase interacting with F-actin

Introduction

Here, we explore the time behavior of a well-mixed GTPase model with negative feedback from F-actin to the GTPase.

Description

We assume that GTPase ($G$) leads to actin assembly, and that F-actin, denoted by the variable ($F$), leads to GTPase inactivation:

$$\begin{align} \frac{\mathrm dG}{\mathrm dt} &= (b + \gamma \frac{G^n}{1 + G^n})G_i - G(\eta + sF) \\ \frac{\mathrm dF}{\mathrm dt} &= \epsilon(k_nG - k_sF) \\ \end{align}$$

Results

The model illustrates that cycling can ocurr in a bistable system with some negative feedback as shown in the figure below.

Model

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Download: wellmixedactinwaves.xml

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<MorpheusModel version="4">
    <Description>
        <Title>Actin Waves - well-mixed</Title>
        <Details>Full title:		Well-mixed 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/M2012
Reference:		L. Edelstein-Keshet: Mathematical Models in Cell Biology
Comment:		ODEs for GTPase interacting with F-actin. Illustrates that cycling can ocurr in a bistable system with some negative feedback. The GTPase equations in well-mixed form can be written in terms of the active GTPase and the amount (total - active) of the inactive GTPase. Here, a = active GTPase, F_a = F-actin concentration. Everything is well-mixed and two ODEs are solved for these two variables.</Details>
    </Description>
    <Global>
        <!--
Define the variables and indicate the method used to integrate the ODEs
-->
        <Variable symbol="a" value="2.2"/>
        <Variable symbol="F_a" value="1.3"/>
        <System time-step="0.1" solver="Runge-Kutta [fixed, O(4)]">
            <!--
Define a bunch of constants that appear in the ODEs
-->
            <Constant symbol="b" name="basal activation rate" value="0.067"/>
            <Constant symbol="gamma" name="feedback rate" value="2.0"/>
            <Constant symbol="n" name="Hill coefficient" value="3"/>
            <Constant symbol="k_n" name="Actin nucleation rate" value="11"/>
            <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="0.01"/>
            <Constant symbol="Tot" name="Total mean GTPase conc" value="4"/>
            <!--
For each variable, indicate that it satisfies a differential equation and
give the expression on the RHS of that ODE. The inactive GTPase is not
needed in the well-mixed system, so its ODE is commented out. It will be
needed once a PDE (spatially distributed) version of them odel is studied.
-->
            <DiffEqn symbol-ref="a">
                <!--              <Expression> (b+gamma*a^n/(1+a^n))*(Tot-a)- a*(eta+s*F_a/(F_0+F_a))  </Expression>
-->
                <Expression> (b+gamma*a^n/(1+a^n))*(Tot-a)- a*(eta+s*F_a)  </Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="F_a">
                <Expression> epsilon*(k_n*a-k_s*F_a)</Expression>
            </DiffEqn>
        </System>
    </Global>
    <!--
Define the spatial domain
-->
    <Space>
        <Lattice class="linear">
            <Size symbol="size" value="1, 0, 0"/>
            <Neighborhood>
                <Order>1</Order>
            </Neighborhood>
        </Lattice>
        <SpaceSymbol symbol="space"/>
    </Space>
    <!--
Set up the simulation StopTime
-->
    <Time>
        <StartTime value="0"/>
        <StopTime symbol="stoptime" value="500"/>
        <TimeSymbol symbol="time"/>
    </Time>
    <!--
Set up what is to be plotted and how
-->
    <Analysis>
        <Logger time-step="5">
            <Input>
                <Symbol symbol-ref="a"/>
                <Symbol symbol-ref="F_a"/>
            </Input>
            <Plots>
                <Plot time-step="-1">
                    <Style line-width="2.0" style="lines"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="time"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="a"/>
                        <Symbol symbol-ref="F_a"/>
                    </Y-axis>
                </Plot>
            </Plots>
            <Output>
                <TextOutput/>
            </Output>
        </Logger>
        <ModelGraph format="dot" reduced="false" include-tags="#untagged"/>
    </Analysis>
</MorpheusModel>

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Files associated with this model:

Actin Actin Wave Bistable System Contributed Examples F-actin Filament GTPase Microfilament Negative Feedback ODE Ordinary Differential Equation Oscillation Wave Well-mixed

Last updated on Tue, 25 Apr 2023