Propagation of the slow subsystem-jump equations

On the other hand, the time evolution of the subset of slow reactions is propagated in time using a slight modification of the Next Reaction variant of SS A, developed by Gibson and Bruck. A system of differential jump equations is used to calculate the next jump of any slow reaction. The jump equations are defined as follows. 

Equations 18.7 are also ltd differential equations even though they do not contain any Wiener process, because the propensities of the slow reactions depend on the state of the system, which in turn depends on the system of CLEs. Due to the coupling between the system of CLEs and the differential jump equations, a simultaneous numerical integration is necessary. If there is no coupling between fast and slow subsets or there are only slow reactions the system of differential jump equations simplifies to the Next Reaction variant. 

The method can be further sped up by allowing more than one zero crossing, i.e., more than one slow reaction, to occur in the time it takes the system of CLEs to advance by At. Though this is an additional approximation contributing to the error introduced by the approximation of the fast reactions as continuous Markov processes, it results in a significant decrease in simulation times. The accuracy depends on the number of slow reactions allowed within At and decreases as the number increases. 

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