**Morphogens**

The cell modelling system includes the facility to incorporate morphogenetic signalling. It is based on reacting and diffusing morphogens, first proposed by Turing as a mechanism for biological pattern formation. This allows for any system of chemical reactions coupled with passive transport modelled by the diffusion equation. Diffusion allows us to model long-range (high diffusion rate), short-range cell-to-cell (low diffusion rate), and cell-autonomous (zero diffusion rate) signal molecules within the same mathematical framework.

An arbitrary number of morphogens can be defined in the system. Each cell can be assigned particular rates which correspond to the production or degradation of each of the morphogens in the system.

**Genetic Script**

The genetic script is implemented via an embedded Tcl system. It may perform any sequence of Tcl instructions (e.g. logical conditions, resetting) on the cell parameters. In addition two procedures are defined. The divide procedure instigates cell division, instructing the spatial model to adjust itself accordingly. The script may also cause the cell to die via a kill procedure. Examples of the possible contents of the genetic script include:

Conditional division

if { $u0 > 1.0 } {

divide

}

A morphogen may induce its own production in a cell:

if { $u0 > 1.0 } {

set pu0 0.5

}

Morphogens may also control other parameters, perhaps via a direct relation:

if { $u0 > 1.0 } {

set growthrate [expr $u0*0.1]

}

Using the timer variable T it is possible to divide the cells activities into timed stages, e.g. an initialisation phase and a growth phase:

if { $T == 0.0 } {

set growthtype $parallel

set growthtype $parallel

}

if { $T < 100.0 } {

set growthrate 1.0

}

if { $T > 100.0 } {

set growthrate 0.0

}

set T [expr $T+1.0]

**Dynamic models for cellular patterning**

The dynamic model is based on a self-reproducing cellular automata, or cell, which has a dynamic state represented by a set of parameters. The cell-states are determined by a genetic script and a chemical signalling model.

The signalling model is an arbitrary set of reacting-diffusing morphogens, the concentrations of which partly define the cell-states. The genetic script is a sequence of logical operations which is repeatedly applied to the cell-state parameters, including the morphogen concentrations. Each cell runs an identical script directly mimicking real genetic logic.

Dynamic model

1. Execute genetic script for each cell

2. Iterate the reaction-diffusion system over Ns sub-time-steps

3. Adjust the mechanical properties of the walls

4. Solve the mechanical spring model

5. Repeat from step 1.

The cells are linked to a spatial/mechanical model which updates the shapes and arrangements of cells in response to their changing states. This then feeds-back on the signalling system by affecting the distributions and movement of morphogens.