A reaction, A -> B with rate constant k is modeled using an ordinary differential equation and Gillespie's stochastic method.

Description

Reaction kinetics with a finite number of molecules.
This project file contains 3 separate models.

Model 1: ODE_Reaction computes the exponential decay of substance
A producing substance B use Ordinary Differential Equations.

Model 2: Stochastic_Reaction computes the decay of 30 molecules of A
producing 30 molecules of B. The Time between decay events is controlled
by a random exponential distribution ( see Gillespie, 1977). Either
Single Run or Loops Run to see envelope of stochastic solutions.

Model 3: Average_Stochastic_Time computes the average time for the
number of remaining molecules of A compared to the ODE_Reaction curves.
A thousand curves are generated and the average time as a function of the
molecules remaining is calculated. These curves are compared with the ODE
model.

Explanation of Gillespie's Stochastic Method:

Compute lambda1 as shown in the code.
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.

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

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.