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2 Compute candidate step sizes (ex) , (im) for explicit and implicit tau-leaping; 3 If (ex) < na / 0 (x) (im) < na / 0 (x) then simulate nb single reactions, update t and x, and goto 1;

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4 Compute candidate step size as expected time to next critical reaction: Generate Exponential

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5 If (ex) min( (im) /nd ; ) then use explicit tau-leaping with := min( (ex) ; ); else use implicit tau-leaping with := min( (im) ; ); 6 If x + m km vm has negative components then reduce (ex) and (im) , and goto 3; Note that the last step is required because there is still a positive probability of generating negative population sizes, though this probability should be small for appropriately chosen parameters Hence, altogether the step size selection procedure can be interpreted as an acceptance rejection method The inventors more speci cally reduce (ex) and (im) by half, but this seems rather arbitrary and may be subject to changes It remains to precisely specify Step 2, which has been subject to various improvements that we shall brie y outline In order to formalize the leap condition of approximately constant propensity functions, an error control parameter > 0 is required In early versions the goal was to bound for every reaction the expected change in its propensity function during a time step of size by 0 (x), hence by times the sum of all propensity functions evaluated at state x The original tau-selection procedure in [37] does not always yield an appropriate step size that satis es this condition, but later in [38] it was shown that the largest value of that indeed satis es it can be obtained by bounding the mean and the standard deviation of the expected change in the propensity function of each reaction by 0 (x) It was also recognized that instead of bounding the change in the propensity function for all reactions by 0 (x) it is more appropriate to bound the change in the propensity function individually for every reaction Rm by m (x), which corresponds to bounding the relative changes in each propensity function by Strictly applied, this implies that becomes zero and the simulation does not advance at all if any of the propensity functions evaluated at state x is very small But, as noted in [18], if m changes at all, then according to equation (114) it changes by at least cm such that a change of less than cm does not make sense, and consequently the change in m can be bounded by the maximum of m (x) and cm Furthermore, [18] presented a procedure that approximately enforces this bound, which is much faster than estimating the mean and the standard deviation according to [38] The essential underlying rationale is, instead of directly considering propensity functions, to bound the relative changes in populations of certain molecular species such that the relative changes in the propensity functions will be all approximately bounded by It differs for explicit and implicit tau-leaping only in the species that are taken into account and is the current state of the art in step size selection for

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tau-leaping Here, we present it in a compact form The details of the derivation can be found in [18] In either case, it suf ces to consider reactant species Denote by R the set of indices of all reactant species and de ne, for all i R, i,M (x) :=

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where M denotes a set of indices of reactions Then the candidate step size dependent on M is M = min

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