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4 Determine reaction type Rm : m = min {k : 1 (x) + + k (x) > u2 0 (x)}; 5 Set t := t + ; x := x + vm ; Gillespie [34] referred to this method of trajectory generation as the direct method and also discussed an equivalent method that he called the rst reaction method which is an equivalent interpretation of the CTMC dynamics often referred to as a race, for example in computer performance evaluation, particularly common in the context of stochastic Petri nets Given any state, the tentative times until entering a particular state are all exponentially distributed with mean the reciprocal of the according transition rate Hence, the next state will be entered according to the fastest transition Since the minimum of exponential distributions is again exponentially distributed with mean the reciprocal of the sum of the parameters (rates) of the exponential distributions involved, the equivalence to the former interpretation easily follows The pseudo-code above does not provide implementation details but expresses the properties to be computed mathematically Basic skills and fundamental knowledge of data structures and algorithms immediately imply a binary search in Step 4 Astonishingly, for quite a long time much of the work and publications on stochastic simulation for biochemically reacting systems was concerned with improvements by reducing the computation costs of determining the reaction type in the direct method, until nally the binary search version was formulated Several mathematically equivalent implementations of the direct method all bear their own name; see [20, 45, 48] For obvious reasons, we avoid the details here Applying the race version of the CTMC dynamics, Gibson and Bruck [33] presented the next reaction method , now commonly referred to as the Gibson Bruck algorithm Their improvement is simply a better implementation than a naive one of the direct method Roughly speaking, only properties that change are recalculated (others are reused) after a reaction is simulated, and some standard advanced data structures that can be found in, for example, [24] are used The algorithm was celebrated as more ef cient than the direct method, but a simple count of the number of necessary operations leads to the conclusion that this cannot be true Not surprisingly, [20] showed that a proper implementation of the direct method is indeed more ef cient, though this seems hitherto to have gone largely unrecognized The Gibson Bruck algorithm is still often considered the best available method for statistically exact trajectory generation and it is therefore in widespread use in Monte Carlo studies of biological and genetic networks In contrast to the efforts expended on implementations of trajectory generation, hardly any investigation of statistical accuracy or robustness of estimators is available A notable exception is [22] where distance measures for probability distributions are considered. java pdf 417 reader Topic: pdf417 · GitHub
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