# Brusselator

Model number
0329

The Brusselator (combination of Brussels and Oscillator) equations of I. Prigogine and R. Lefever are solved both continuously and stochastically.

## Description

``` The Brusselator equations form an oscillating system that
approaches a limit cycle. For an example of oscillating systems watch a video
of the Briggs-Rauscher Oscillating Reaction (several videos available on the
internet).

The same equations are solved stochastically using the Gillespie method.
A and B are fixed, X and Y vary, C and D are not calculated.

NOTE: to increase the number of molecules by a factor
of 10, divide k1 by 10, k2 by 1000, k3 by 100, and k4 by 10.
Be sure to make the corresponding change in the ODE model.

Compute lambda1, lambda2, ... lambda4 as shown in the code
for model StochasticBrusselator.

Sum the lambda's to calculate lambda.
Create a random exponential distribution by taking the natural
log of uniform random numbers.
When multiplied by -1/lambda it becomes the time step when
another reaction takes place. The uneven time steps are
accumulated in tau.
A second uniform random number for which reaction takes place
is determined by the probability of the reaction which is the
ratio of its lambda to the sum of all the other lambdas.
We Calculate P1=lambda1/lambda, P2=(lambda1+lambda2)/lambda, ...
For the reaction selected, the appropriate pools are
incremented or decremented in whole units.```

## Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

References
```I. Prigogine and R. Lefever, Symmetry Breaking Instabilities in Dissipative Systems. II,
J. Chem. Phys. 48, 1695 (1968),

D.T. Gillespie, Exact stochastic simulation of coupled chemical reactions,
J. Phys. Chem., 1977, 81 (25), pp 2340–2361.```
Key terms
Brusselator
stochastic
Gillespie
oscillating reaction
equilibria
limit cycle
Acknowledgements

Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.