Well-mixed Actin Waves

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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.

In the well-mixed single GTPase model, when actin negative feedback is included, the system can begin to oscillate. In the Morpheus model, $a$ represents active GTPase (denoted by $G$ in the [ODEs above](#description)) and $F_a$ (denoted by $F$ [above](#description)) is the filamentous actin.
In the well-mixed single GTPase model, when actin negative feedback is included, the system can begin to oscillate. In the Morpheus model, $a$ represents active GTPase (denoted by $G$ in the ODEs above) and $F_a$ (denoted by $F$ above) is the filamentous actin.

Model

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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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