## File: SimulationTau.htd

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 `12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152` `````` Tau-Leaping

Tau-Leaping

With tau-leaping we take steps forward in time. For each reaction we calculate a predicted average propensity. We then generate Poisson deviates to determine how many times each reaction will fire during the step. The advantage of tau-leaping is that it can jump over many reactions and thus may be much more efficient than exact methods. The disadvantage is that it is not an exact method.

There are several options for the tau-leaping solver. By default it will use an adaptive step size and will correct negative populations. You can also choose to not correct negative populations, the simulation will fail if a species is overdrawn. There is also a fixed time step option. This option is only useful for studying the tau-leaping method. With a fixed step size it is difficult to gauge the accuracy of the simulation.

In tau-leaping, one uses an expected value of the propensities in advancing the solution. The propensities are assumed to be constant over the time step. There are several ways of selecting the expected propensity values. The simplest is forward stepping; The expected propensities are the values at the beginning of the step. One can also use midpoint stepping. In this case one advances to the midpoint of the interval with a deterministic step. Then one uses the midpoint propensity values to take a stochastic step and fire the reactions. Midpoint stepping is analogous to a second order Runge-Kutta method for ODE's. One can also use higher order approximations to determine the expected propensities. You can use a fourth order Runge-Kutta scheme with deterministic steps to choose the expected propensities and then take a stochastic step with these values. Note that regardless of how you choose the expected propensities, the tau-leaping solver is still a first-order accurate stochastic method. That is, you can choose a first, second, or fourth order method for calculating the expected propensities, but you still assume that the propensities are constant when taking the stochastic step. Thus it is a first-order stochastic method. However, using higher order formulas for the expected propensities is typically more accurate.

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