# Example

Let’s go through an example. We’ll construct a model in which an

- intracellular cell cycle network (ODE) regulates
- the division of motile cells which (CPM)
- release a diffusive cytokine (PDE) which, in turn,
- controls the cell cycle (ODE).

Thus, there are 3 sub-models (ODE, CPM and PDE) that interact in a cyclic fashion:

## Step 1: Intracellular Model (ODE)

First, we define a ODE model of a cell cycle that we take directly from Ferrell *et al.* 2011.

Check out `CellCycle.xml`

to try yourself via:

**Morpheus Link**or**Morpheus GUI**:`Examples`

→`ODE`

→`CellCycle.xml`

or**Morpheus Model Repository**.

*et al.*Have in mind that you can take whatever model you want, e.g. by importing an SBML model file.

In Morpheus, we can describe the same system of equations together with the parameter values as below:

The `System`

is defined **within a CellType**. This makes sure that the ODE system is

**calculated separately for each member**of the population of this cell type. Furthermore the

**variables are defined as**. These are variables that are bound to a particular cell such that

`Properties`

**each cell can have a different value**for this variable.

If you run this model for a single cell with the parameters above, you will see oscillatory behavior:

## Step 2: Cell-Based Model (CPM)

Now, we would like to couple the above model to control the ‘physical’ cell division of a **spatial cell model**. This can be achieved easily by adding a few elements.

First, we change the space to be a `square`

`Lattice`

, sized 250x250. We initialize our cell centrally using the `Population`

s' `InitCellObjects`

component. We also use `Gnuplotter`

in the `Analysis`

section to check the results.

```
<InitCellObjects mode="order">
<Arrangement displacements="1, 1, 1" repetitions="1, 1, 1">
<Sphere center="125, 125, 0" radius="60"/>
</Arrangement>
</InitCellObjects>
```

Instead of adding one element after the other with the `+`

button or `right click`

/`Add`

in the GUI, you can select the above XML snippet and **drag and drop** it onto the `Population`

or **paste** it via the context menu.

Second, we add a `VolumeConstraint`

and `SurfaceConstraint`

that control the area and shape of a cell. And, obviously, we need to add a `CellDivision`

plugin that controls when and how a cell divides. Here, we specify that the cell should divide when `CDK1 > 0.5`

in the `Condition`

of cell division.

**multiscale coupling is automatically established**by the symbolic reference (outlined in red) between the condition for cell division and the variable in the intracellular ODE system.

One additional thing to specify is what happens when cell division occurs using the `Triggers`

. Here, we specify that the target volume `Vt`

of the two daughter cells is half the target volume of the mother cell (`Vt/2`

), i.e. to model cleavage without cell growth.

Lastly, we add the CPM section, providing the temporal simulation. Cells shall preferably adhere to each other, such that we set at negative interaction energy for cells.

```
<CPM>
<Interaction>
<Contact type1="sbml_cell" type2="sbml_cell" value="-12.0"/>
</Interaction>
<ShapeSurface scaling="norm">
<Neighborhood>
<Order>6</Order>
</Neighborhood>
</ShapeSurface>
<MonteCarloSampler stepper="edgelist">
<MCSDuration value="0.1"/>
<MetropolisKinetics temperature="5"/>
<Neighborhood>
<Order>1</Order>
</Neighborhood>
</MonteCarloSampler>
</CPM>
```

This generates a simulation like this:

Check out `CellCycle.xml`

to try yourself via:

**Morpheus Link**or**Morpheus GUI**:`Examples`

→`Multiscale`

→`CellCycle.xml`

or**Morpheus Model Repository**.

## Step 3: Intercellular Model (PDE)

The remaining two steps are:

- Allow cells to release a diffusive chemokine.
- Make this chemokine modulate the cell cycle.

### Adding a Diffusive Field

First, let’s define a cytokine $g$ that diffuses with coefficient $D_g$, is produced by cells proportionally to its concentration $\text{APC}$ and decays linearly with a certain rate ($0.05$):

$$\frac{\partial g}{\partial t} = D_g{\nabla}^2g + \text{APC} - 0.05 \cdot g$$

In Morpheus, we define a `Field`

`g`

in the `Global`

section and specify the diffusion coefficient $D_g$ in `Diffusion`

/`rate`

as `100`

. The reaction step of is defined in the `System`

where we provide a `DiffEqn`

with the production and decay terms in the expression `APC - 0.05*g`

:

`APC`

to be zero everywhere (`Constant`

: `APC`

=`0.0`

), but this value is overwritten wherever there are cells (i.e. in the `CellType`

scope).
This established the coupling between CPM and PDE: the movement and shape of CPM cells affect the production term in the PDE.

### Modulating the Cell Cycle

The final step is to let the local concentration of the cytokine modulate the cell cycle.

We first need to compute the local cytokine concentration $g_l$ for each cell (plugins highlighted in green below). Since there are different possibilities (i.e. we could take the sum, the mean or the maximum concentration ‘under’ the cell), we need to use a plugin called `Mapper`

which allows us to reduce the spatial granularity by selecting the mapping statistics we want. Here, we specify the `Mapper`

to take Field `g`

as an input and assign the `average`

value to the cell’s `Property`

`g_l`

– the local cytokine concentration.

Finally, we let the local concentration of $g$ affect the cell cycle dynamics. Here, we assume (quite artificially) that this concentration acts as an additional production term for $\text{CDK1}$ and add it to the `DiffEqn`

for `CDK1`

(see red outline). This implements the **coupling between PDE and ODE**: the extracellular cytokine concentration affects the intracellular cell cycle.

Here’s a video of a simulation of the full multiscale model we’ve constructed:

Check out `CellCycle_PDE.xml`

to try yourself via:

**Morpheus Link**or**Morpheus GUI**:`Examples`

→`Multiscale`

→`CellCycle_PDE.xml`