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Module 1: Delta-Notch (ODE systems, ODEs on a grid)

Author: Fabian Rost

Aim

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

Description

Basic ODEs

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

Delta-Notch

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.

Morpheus models

h ExponentialGrowth.xml |h

<?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>

Table of Contents

Module 1: Delta-Notch (ODE systems, ODEs on a grid)

Aim

Description

Basic ODEs

Delta-Notch

Morpheus models