=== 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. [[http://dx.doi.org/10.1016/S0022-5193(69)80016-0|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.  [[http://dx.doi.org/10.1038/nature08391|link]]

Morpheus models:

[[examples:miscellaneous#French Flag: Morphogen gradient|French Flag: Morphogen gradient]]

<code xml |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>

</code>

<code xml |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>
</code>