Author: Fabian Rost

• learn about ODE models (dynamics in morpheus, steady states analytically)
• develop first models

• get to know what students know about ODEs and adjust the module to the pre-knowledge
• give them very simple sketches of biomolecular models, which they should translate into ODEs, e.g.

• could be translated to the following ODEs:

\begin{align} \dot A &= k_1 \\ \dot A &= k_2 A \\ \dot A &= - k_3 A \\ \dot A &= k_4 - k_5 A \end{align}

• discuss those ODEs by
  • calculate steady state (do not calculate the stability, to complicated for biologists)
  • simulate in morpheus

• then discuss the delta-notch sketch with two species
  • start with the Collier model
  • let them simplify the Collier model sketch (remove the delta or notch species)
  • let them develop an ODE for this system (they should be able to do so from the above examples)
  • they could come up with something like:

\begin{align} \dot X_1 &= c \frac{\theta^n}{\theta^n + X_2^n} - k X_1 \\ \dot X_2 &= c \frac{\theta^n}{\theta^n + X_1^n} - k X_2 \end{align}

• this system is bistable for certain parameter ranges, if the students are advanced they might find this out themselves
• bistable e.g. for $\theta=0.5$, $n=4$, $c=k=1$
• if they have this system running in morpheus go spatial and let them simulate the system on a square and hexagonal grid
• then you could also move to shaped cpm cells or even moving cells

• students won't do so much on their own in this session, it is a lot teaching on ODEs (don't be theoretical here, not enough time!) and introducing morpheus

Paper:

• Joanne R. Collier, Nicholas A.M. Monk, Philip K. Maini, Julian H. Lewis, Pattern Formation by Lateral Inhibition with Feedback: a Mathematical Model of Delta-Notch Intercellular Signalling, Journal of Theoretical Biology, Volume 183, Issue 4, 21 December 1996, Pages 429-446, ISSN 0022-5193, http://dx.doi.org/10.1006/jtbi.1996.0233.

<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="3">
    <Description>
        <Details></Details>
        <Title></Title>
    </Description>
    <Space>
        <Lattice class="linear">
            <Neighborhood>
                <Order>1</Order>
            </Neighborhood>
            <Size symbol="size" value="100, 0, 0"/>
        </Lattice>
        <SpaceSymbol symbol="space"/>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="5"/>
        <TimeSymbol symbol="time"/>
    </Time>
    <Global>
        <Constant symbol="k1" value="1.0"/>
        <Variable symbol="A" value="1.0"/>
        <System solver="heun" time-step="0.01">
            <DiffEqn symbol-ref="A">
                <Expression>k1 * A</Expression>
            </DiffEqn>
        </System>
    </Global>
    <Analysis>
        <Logger time-step="0.01">
            <Input>
                <Symbol symbol-ref="A"/>
            </Input>
            <Output>
                <TextOutput/>
            </Output>
            <Plots>
                <Plot time-step="5">
                    <Style style="points"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="time"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="A"/>
                    </Y-axis>
                </Plot>
            </Plots>
        </Logger>
    </Analysis>
</MorpheusModel>