MODEL NUMBER: // MODEL NUMBER: 0329 MODEL NAME: Brusselator SHORT DESCRIPTION: The Brusselator (combination of Brussels and Oscillator) equations of I. Prigogine and R. Lefever are solved both continuously and stochastically. DIAGRAM: Four chemical reactions are being modeled: Reaction Rate Constant A -> X (k1) 2X + Y -> 3X (k2) B + X -> Y + C (k3) X -> D (k4) DETAILED 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. KEY WORDS: Brusselator, stochastic, Gillespie, oscillating reaction, equilibria, limit cycle 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. REVISION HISTORY: JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE: JSim software was developed with support from NIH grants HL088516, and HL073598. Please cite these grants in any publication for which this software is used and send one reprint of published abstracts or articles to the address given below. Academic use is unrestricted. Software may be copied so long as this copyright notice is included. Copyright (C) 1999-2011 University of Washington. Contact Information: The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle, WA 98195-5061