documentation:course:module2

Module 2: Morphogen gradients

Module 2: Morphogen gradients

Author: Lutz Brusch

Aim:

Adapt an established theoretical model to a fundamental developmental patterning process.
Thereby going through the typical modeling workflow seen in Module 1.

Description:

Read abstract of Yu et al.
Derive diffusion equation from 1st and 2nd Fick’s law
Morphogen interpretation entails morphogen binding thereby eliminating it from (observed) diffusible pool → derive Eq. (1)
Run Gradient.xml to see gradient building up
Solve steady state of Eq. (1) analytically, c(x)=c0*exp(-x/lambda) lambda=sqrt(D/k)
Find parameter value for D in the paper (Fig.3, why multiple?, relation between molecule weight and D?, fluorescent tag is source of artefacts – let’s see how big they are)
Calculate k from lambda given in the text above Fig.4 (which D corresponds to the data?)
Discuss role of Dynamin and Rab5 for ligand-receptor uptake
Using Morpheus, reproduce the results of Fig.4B and document in protocol, verify role of Dynamin and Rab5
Discuss role of right boundary condition, vary system size to explore changing steady state conc. at same position for no flux bc. in small vs. large system size
Estimate D for naked Fgf8
Simulate true morphogen profile and discuss difference to measured profile, just control case → this is a 50% tagging artefact

Study Wolpert’s French Flag model, Discuss idea qualitatively
Change gradient decay length and observe positions of tissue boundaries
Change threshold values and observe positions of tissue boundaries
Change cell size (node length) and observe positions of tissue boundaries

Extend the morphogen PDE-model by a cell differentiation model = bistable reporter gene that controls differential gene expression downstream and is activated by the morphogen (Hill+additive morphogen-degradation), Which diffusion constant should the gene equation get? (D=0)
Compare your model extension to Gradient_Interpreter.xml
Simulate for different morphogen degradation rate and diffusion constant, observe position of emergent differentiation boundary

Paper:

Wolpert L (1969). “Positional information and the spatial pattern of cellular differentiation”. J. Theor. Biol. 25 (1): 1–47. doi:10.1016/S0022-5193(69)80016-0. PMID 4390734. link
Yu S R et al. (2009). “Fgf8 morphogen gradient forms by a source-sink mechanism with freely diffusing molecules” Nature 461, 533-536. doi:10.1038/nature08391. link

Morpheus models:

French Flag: Morphogen gradient

h Gradient.xml |h

<MorpheusModel version="1">
    <Description>
        <Title>Example-MorphogenGradient</Title>
    </Description>
    <Space>
        <Lattice class="linear">
            <Size value="100 0 0"/>
            <BoundaryConditions>
                <Condition boundary="x" type="noflux"/>
                <Condition boundary="-x" type="constant"/>
            </BoundaryConditions>
            <NodeLength unit="micron" value="1"/>
        </Lattice>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="500"/>
        <SaveInterval value="0"/>
        <RandomSeed value="1"/>
    </Time>
    <PDE>
        <Layer symbol="A" name="activator">
            <Diffusion rate="1" unit="µm²/s"/>
            <Initial>
                <InitPDEExpression>
                    <Expression>rand_uni(0,0.01)</Expression>
                </InitPDEExpression>
            </Initial>
            <BoundaryConditions>
                <Condition boundary="-x" value="1.0"/>
            </BoundaryConditions>
        </Layer>
        <System solver="runge-kutta" time-step="1">
            <DiffEqn symbol-ref="A">
                <Expression>- k * A</Expression>
            </DiffEqn>
            <Constant symbol="k" value="0.01"/>
        </System>
    </PDE>
    <Analysis>
        <SpaceTimeLogger interval="20">
            <Layer symbol-ref="A"/>
            <Plot interval="500" every="0" terminal="png" persist="true"/>
        </SpaceTimeLogger>
        <Logger interval="10">
            <Format string="A"/>
            <Input>
                <PDE mapping="all"/>
            </Input>
            <Plot terminal="png" persist="true">
                <X-axis column="2"/>
                <Y-axis columns="5"/>
                <color-bar column="1"/>
            </Plot>
        </Logger>
        <!--    <Disabled>
        <Gnuplotter interval="500">
            <Terminal name="png"/>
            <PDE symbol-ref="A"/>
        </Gnuplotter>
    </Disabled>
-->
    </Analysis>
</MorpheusModel>

h Gradient_Interpreter.xml |h

<MorpheusModel version="1">
    <Description>
        <Title>Example-ActivatorInhibitor1D</Title>
    </Description>
    <Space>
        <Lattice class="linear">
            <Size value="100 0 0"/>
            <BoundaryConditions>
                <Condition boundary="x" type="noflux"/>
                <Condition boundary="-x" type="constant"/>
            </BoundaryConditions>
            <NodeLength unit="micron" value="1"/>
        </Lattice>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="500"/>
        <SaveInterval value="0"/>
        <RandomSeed value="1"/>
    </Time>
    <PDE>
        <Layer symbol="A" name="activator">
            <Diffusion rate="1" unit="µm²/s"/>
            <Initial>
                <InitPDEExpression>
                    <Expression>rand_uni(0,0.01)</Expression>
                </InitPDEExpression>
            </Initial>
            <BoundaryConditions>
                <Condition boundary="-x" value="1.0"/>
            </BoundaryConditions>
        </Layer>
        <Layer symbol="G" name="Gene">
            <Diffusion rate="0"/>
            <Initial>
                <InitPDEExpression>
                    <Expression>0</Expression>
                </InitPDEExpression>
            </Initial>
        </Layer>
        <System solver="runge-kutta" time-step="0.01">
            <DiffEqn symbol-ref="A">
                <Expression>- k * A</Expression>
            </DiffEqn>
            <Constant symbol="k" value="0.01"/>
            <DiffEqn symbol-ref="G" name="threshold activation">
                <Expression>A+G^2/(0.2^2+G^2)-1*G</Expression>
            </DiffEqn>
        </System>
    </PDE>
    <Analysis>
        <SpaceTimeLogger interval="20">
            <Layer symbol-ref="A"/>
            <Plot interval="100" every="0" terminal="png" persist="true"/>
        </SpaceTimeLogger>
        <Logger interval="10">
            <Format string="A"/>
            <Input>
                <PDE mapping="all"/>
            </Input>
            <Plot terminal="png" persist="true">
                <X-axis column="2"/>
                <Y-axis columns="5"/>
                <color-bar column="1"/>
            </Plot>
        </Logger>
        <!--    <Disabled>
        <Gnuplotter interval="500">
            <Terminal name="png"/>
            <PDE symbol-ref="A"/>
        </Gnuplotter>
    </Disabled>
-->
        <SpaceTimeLogger interval="5">
            <Layer symbol-ref="G"/>
            <Plot interval="100" every="0" terminal="png" persist="true"/>
        </SpaceTimeLogger>
    </Analysis>
</MorpheusModel>

Table of Contents

Module 2: Morphogen gradients