# Example reaction–diffusion models

The `demo/models.yaml`

contains a number of simple reaction system and reaction–diffusion system models from the literature.

Note that the units of the quantities expressed in the models file are in counts, metres, and seconds; this leads to some very large or small quantities for typical cell volumes, such as the 25 µm^3 cells used in the Turing example below. This problem will be addressed in a future revision that includes general unit support. For single well-mixed volume simulations, a ‘unit’ volume allows rate constants to be considered to be in s^{-1}.

## Schnakenberg

An example demonstrating limit cycle behaviour, first raised in (Schnakenberg, 1979, p. 398). The rate constants and examples come from (Erban, Chapman, & Maini, 2007, p. 30), and have been chosen to demonstrate stochastic behaviour that differs qualitatively from that of the corresponding deterministic system of ODEs.

**Reaction system**

reaction | rate constants |
---|---|

2A + B → 3A |
k_{1}=4×10^{-5} s^{-1} |

∅ ⇄ A |
k_{2}=40 s^{-1}, k_{3}=10 s^{-1} |

∅ → B |
k_{4}=25 s^{-1} |

**Initial conditions**

species | concentration |
---|---|

A: |
10 |

B: |
10 |

## Schlögl

Single species model reaction model exhibiting bistable behaviour, as introduced by Schlögl in (Schlögl, 1972, p. 150) and discussed in (Erban et al., 2007, p. 29). The system exhibits two stable steady states with concentrations 100 and 400; stochastic simulation demonstrates spontaneous transition between these two states.

**Reaction system**

reaction | rate constants |
---|---|

2A ⇄ 3A |
k_{1}=0.18 s^{-1}, k_{2}=2.5×10^{-4} s^{-1} |

∅ ⇄ A |
k_{3}=2200 s^{-1}, k_{4}=37.5 s^{-1} |

Note that this description has rescaled the reaction constants from per-minute quantities to per-second quantities.

**Initial conditions**

species | concentration |
---|---|

A: |
0 |

## Turing

A reaction–diffusion model exhibiting spontaneous emergence of spatial inhomogeneity, based on Turing patterns (Turing, 1952). The example constructed in (Erban et al., 2007, p. 32) uses the Schnakenberg reaction system with rate constants described below, on a 1-D mesh of length 1 mm, divided into 40 equal sized compartments.

Note that the description in the paper expresses concentrations as counts per mesh element; the model implementation has 40 cells each of volume 25 µm^{3}, and the reaction rates below have been scaled accordingly.

**Reaction system**

reaction | rate constants |
---|---|

2A + B → 3A |
k_{1}=6.25×10^{-4} µm^{6}s^{-1} |

∅ ⇄ A |
k_{2}=0.04 µm^{3}·s^{-1}, k_{3}=0.02 s^{-1} |

∅ → B |
k_{4}=0.12 µm^{3}·s^{-1} |

**Diffusion coefficients**

species | diffusivity |
---|---|

A: |
10^{-5} mm^{2}·s^{-1} |

B: |
10^{-3} mm^{2}·s^{-1} |

**Initial conditions**

species | concentration |
---|---|

A: |
8 µm^{-3} |

B: |
0.12 µm^{-3} |

## References

Erban, R., Chapman, J., & Maini, P. (2007). A practical guide to stochastic simulations of reaction–diffusion processes. Retrieved from http://arxiv.org/abs/0704.1908v2

Schlögl, F. (1972). Chemical reaction models for non-equilibrium phase transitions. *Zeitscrift für Physik*, *253*(2), 147–161.

Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour. *Journal of Theoretical Biology*, *81*, 389–400.

Turing, A. M. (1952). The chemical basis of morphogenesis, *237*(541), 37–72.