### Contributors

Otto-von-Guericke Universität Magdeburg, Germany: Mary Ann Blätke, Jan Wegener

### Background

We consider the following scenario:

• One species (here cAMP) diffuses continuously in space; i.e., it evolves simultaneously over time and space.
• The state-dependent diffusion rate follows mass/action kinetics, i.e., the rate is defined by the product of the species involved times some constant, summing up all dependencies on pressure, temperature, etc.
• The observation starts with a high concentration (e.g., 100) in the middle of the space, with all other space positions initially set to 0.

### Modelling

Tip: Click on image of a plain Petri net opens animation window. Animation does not consider (stochastic or continuous ) rates.

1-dimensional grid Reading this Petri net as a continuous Petri net, and assuming the diffusion rates to follow mass/action kinetics with the common rate parameter k, we obtain the following ODEs; for sake of readability we abbreviate cAMP_i by ci. We obtain a general model pattern for an arbitrary, but static size of a discrete, 1-dimensional space by folding this plain (continuous) Petri net into a coloured (continuous) Petri net.

```constant
D1 = int with 5;	       // grid size
MIDDLE = int with (D1+1)/2; // middle position

colorset
CS = int with 1−D1;         // addresses of grid positions

variable
x,y : CS;

function
neighbour1D (CS x,a) bool:
// a is neighbour of x
( a=x−1 | a=x+1) & (1<=a) & (a<=D1);``` Changing constant D1 adapts the model pattern to a specific 1-dimensional grid size.

2-dimensional grid with four neighbours `   ` ```colorset
CD1 = 1−D1;
CD2 = 1−D2;
Grid2D = CD1 x CD2;

var
x,a : CD1;
y,b : CD2;

function
neighbour2D4 (CD1 x , CD2 y , CD1 a , CD2 b) bool :
// (a,b) is one of the up to four neighbours of (x,y)
(a=x & b=y−1) | (a=x & b=y+1) | (b=y & a=x−1) | (b=y & a=x+1);```

Unfolding the coloured Petri net with constants D1=D2=5 yields the following plain Petri net.

2-dimensional grid with eight neighbours `   ` ```function
neighbour2D8 (CD1 x, CD2 y, CD1 a, CD2 b) bool :
// (a,b) is one of the up to eight neighbours of (x,y)
(a=x | a=x+1 | a=x−1) & (b=y | b=y+1 | b = y−1)
& (!(a=x & b=y))
& (1<=a & a<=D1) & (1<=b & b<=D2);```

Unfolding the coloured Petri net with constants D1=D2=5 yields the following plain Petri net.

### Analysis

All computational experiments are done on the automatically unfolded Petri nets. With other words, the coloured continuous Petri nets serve as high-level description of the continuous systems under study.

Some statistics

 grid size no. places no. transitions 15x15 225 1,624 30x30 900 3,480 60x60 3,600 14,160 120x120 14,400 57,120

For simulation purposes, a continuous Petri net is represented by a system of Ordinary Differential Equations (ODEs), with one equation for each place. Thus, coloured continuous Petri nets can be seen as a very high-level description of ODEs.

2D representation of a continuous run

Converting the continuous simulation traces into 2D representations (heat map style) allows us to observe the development of a gradient in time and space. We give six snapshots (time points: 0, 5, 10, 15, 20, 30) for the 15x15 grid.      Increasing resolution

We give the snapshots at time point 50 for the grids 15x15, 30x30, 60x60, and 120x120.    ### References

latest update: December 12, 2011, at 04:28 PM