Table of Contents
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
- typical models for biochemistry/systems biology use hill kinetics:
\begin{align} \dot A &= \frac{\theta^n}{\theta^n + B^n} - k A \end{align}
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 A_1 &= \frac{\theta^n}{\theta^n + A_2^n} - A_1 \\ \dot A_2 &= \frac{\theta^n}{\theta^n + A_1^n} - A_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 $\theta=0.5$, $n=4$,
- 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
<?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>
LateralInhibition_2cell.xml
<?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="8"/>
<TimeSymbol symbol="time"/>
</Time>
<Global>
<Variable symbol="A" value="0.51"/>
<Variable symbol="A_neighbour" value="0.5"/>
<System solver="heun" time-step="0.01">
<DiffEqn symbol-ref="A">
<Expression>theta^n / (theta^n + A_neighbour^n)
- k * A</Expression>
</DiffEqn>
<DiffEqn symbol-ref="A_neighbour">
<Expression>theta^n / (theta^n + A^n)
- k * A_neighbour</Expression>
</DiffEqn>
</System>
<Constant symbol="theta" value="0.1"/>
<Constant symbol="n" value="6"/>
<Constant symbol="k" value="1.0"/>
</Global>
<Analysis>
<Logger time-step="0.1">
<Input>
<Symbol symbol-ref="A"/>
</Input>
<Output>
<TextOutput/>
</Output>
<Plots>
<Plot time-step="8">
<Style style="points"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="time"/>
</X-axis>
<Y-axis>
<Symbol symbol-ref="A"/>
<Symbol symbol-ref="A_neighbour"/>
</Y-axis>
</Plot>
</Plots>
</Logger>
</Analysis>
</MorpheusModel>
LateralInhibition_space.xml
<?xml version='1.0' encoding='UTF-8'?>
<MorpheusModel version="3">
<Description>
<Details></Details>
<Title></Title>
</Description>
<Space>
<Lattice class="hexagonal">
<Neighborhood>
<Order>1</Order>
</Neighborhood>
<Size symbol="size" value="10,10,0"/>
<BoundaryConditions>
<Condition boundary="x" type="periodic"/>
<Condition boundary="-x" type="periodic"/>
<Condition boundary="x" type="periodic"/>
<Condition boundary="-y" type="periodic"/>
</BoundaryConditions>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="50"/>
<TimeSymbol symbol="time"/>
<RandomSeed value="42"/>
</Time>
<CellTypes>
<CellType class="biological" name="cells">
<Property symbol="X" value="0.0"/>
<Constant symbol="theta" value="0.1"/>
<Constant symbol="n" value="6"/>
<Property symbol="X_neighbour" value="0.0"/>
<NeighborhoodReporter>
<Input scaling="cell" value="X"/>
<Output symbol-ref="X_neighbour" mapping="average"/>
</NeighborhoodReporter>
<System solver="heun" time-step="0.01">
<DiffEqn symbol-ref="X">
<Expression>theta^n/(theta^n+X_neighbour^2)
- X</Expression>
</DiffEqn>
</System>
</CellType>
</CellTypes>
<CellPopulations>
<Population size="1" type="cells">
<InitCellLattice/>
<InitProperty symbol-ref="X">
<!-- <Disabled>
<Expression>if(cell.id == 1, 1, 0)</Expression>
</Disabled>
-->
<Expression>rand_uni(0, 0.01)</Expression>
</InitProperty>
</Population>
</CellPopulations>
<Analysis>
<Gnuplotter time-step="50">
<Plot>
<Cells value="X"/>
</Plot>
<Terminal name="png"/>
</Gnuplotter>
</Analysis>
<Global>
<Constant symbol="X" value="0.0"/>
</Global>
</MorpheusModel>