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As discussed in Sec. VI of this reference, the analysis leading to the result can be regarded as a refined, corrected, and stochastically extended version of the analysis of F. Collins and G. Kimball, J. Colloid Sci. 4, 425 (1949). 10.1016/0095-8522(49)90023-9
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A derivation of Eq. that is further improved over the one given in the 2009 paper can be found in Secs. 3.7 and 4.8 of, (Oxford University Press).
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A derivation of Eq. that is further improved over the one given in the 2009 paper can be found in Secs. 3.7 and 4.8 of D. Gillespie and E. Seitaridou, Simple Brownian Diffusion (Oxford University Press, 2012).
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0(x) reveals that τ and j are statistically independent random variables with those respective PDFs. Integrating (summing) those PDFs over their respective arguments gives their CDFs, and Eqs. then follow from the inversion method. The generic forms of Eqs., being such straightforward consequences of applying the inversion Monte Carlo method to the aforementioned PDFs of τ and j, were known long before Gillespies 1976 paper; e.g., see page 36 of J. Hammersley and D. Handscomb, Monte Carlo Methods (Methuen, 1964).
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0(x) reveals that τ and j are statistically independent random variables with those respective PDFs. Integrating (summing) those PDFs over their respective arguments gives their CDFs, and Eqs. then follow from the inversion method. The generic forms of Eqs., being such straightforward consequences of applying the inversion Monte Carlo method to the aforementioned PDFs of τ and j, were known long before Gillespies 1976 paper; e.g., see page 36 of J. Hammersley and D. Handscomb, Monte Carlo Methods (Methuen, 1964). The main contributions of Gillespies 1976 paper were: (i) proving from simple kinetic theory that bimolecular reactions in a dilute gas, like unimolecular reactions, are describable by propensity functions as defined in; and (ii) proving that propensity functions as defined in imply that the time to next reaction and the index of next reaction are random variables distributed according to the joint PDF.
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An early conjecture was that the negativity problem in tau-leaping was caused by the unbounded Poisson random variables in the tau-leaping formula occasionally allowing too many reaction firings in a single lea That conjecture led to proposals to replace the Poisson random variables in Eq. with binomial random variables, which are bounded. But it was subsequently determined that the principal causes of the negativity problem lay elsewhere. First was the flawed implementation of the leacondition that was used in Ref.: the small-valued propensity functions that it failed to protect often have a reactant with a small molecular population which can easily be driven negative. Second, since the firing numbers of the individual reactions in the tau-leaping formula are generated independently of each other, two or more reaction channels that decrease the population of a common species could inadvertently collude to overdraw that species.
-
An early conjecture was that the negativity problem in tau-leaping was caused by the unbounded Poisson random variables in the tau-leaping formula occasionally allowing too many reaction firings in a single leap. That conjecture led to proposals to replace the Poisson random variables in Eq. with binomial random variables, which are bounded. But it was subsequently determined that the principal causes of the negativity problem lay elsewhere. First was the flawed implementation of the leap condition that was used in Ref.: the small-valued propensity functions that it failed to protect often have a reactant with a small molecular population which can easily be driven negative. Second, since the firing numbers of the individual reactions in the tau-leaping formula are generated independently of each other, two or more reaction channels that decrease the population of a common species could inadvertently collude to overdraw that species. Neither of these two problems is fixed by the ad hoc substitution of binomial random variables for the Poisson random variables in Eq.. But both problems are effectively dealt with in the heavily revised tau-leaping procedure described in Ref., which uses the theoretically appropriate Poisson random variables.
-
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in, edited by M. Bernardo, P. Degano, and G. Zavattaro (Springer), p, Sec. 3 gives a tutorial on tau-leaping, Secs. 4 and 5 give a tutorial on the ssSSA.
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D. Gillespie, Simulation methods in systems biology., in Formal Methods for Computational Systems Biology, edited by, M. Bernardo, P. Degano, and, G. Zavattaro, (Springer, 2008), pp. 125-167, Sec. 3 gives a tutorial on tau-leaping, Secs. 4 and 5 give a tutorial on the ssSSA.
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The CLE can be shown to be mathematically equivalent to the Fokker-Planck equation that is obtained by first making a formal Taylor series expansion of the right side of the CME, which yields the so-called Kramers-Moyal equation, and then truncating that expansion after the second derivative term. However, that way of obtaining the CLE, which was known well before this 2000 paper, does not qualify as a derivation, because it does not make clear under what conditions the truncation will be accurate. In contrast, the derivation of the CLE given here provides a clear and testable criterion for the accuracy of its approximations, namely, the extent to which both leaconditions are satisfied. But see also the proviso in Ref.. 10.1063/1.481811
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D. Gillespie, J. Chem. Phys. 113, 297 (2000). The CLE can be shown to be mathematically equivalent to the Fokker-Planck equation that is obtained by first making a formal Taylor series expansion of the right side of the CME, which yields the so-called Kramers-Moyal equation, and then truncating that expansion after the second derivative term. However, that way of obtaining the CLE, which was known well before this 2000 paper, does not qualify as a derivation, because it does not make clear under what conditions the truncation will be accurate. In contrast, the derivation of the CLE given here provides a clear and testable criterion for the accuracy of its approximations, namely, the extent to which both leap conditions are satisfied. But see also the proviso in Ref.. 10.1063/1.481811
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The approximation P(m)≈N(m,m) that was made in deriving the CLE from the tau-leaping formula, while accurate for likely values of those two random variables, is very inaccurate in the tails of their probability densities, even when m 1. Although both tails are very near 0, they differ by many orders of magnitude. Since rare events arise from the unlikely firing numbers under those tails, it follows that the CLE will not accurately describe the atypical behavior of a chemical system, even if both leaconditions are well satisfied.
-
The approximation P(m)≈N(m,m) that was made in deriving the CLE from the tau-leaping formula, while accurate for likely values of those two random variables, is very inaccurate in the tails of their probability densities, even when m 1. Although both tails are very near 0, they differ by many orders of magnitude. Since rare events arise from the unlikely firing numbers under those tails, it follows that the CLE will not accurately describe the atypical behavior of a chemical system, even if both leap conditions are well satisfied.
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The foregoing chain of reasoning can be summarized from the perspective of computational mathematics as follows: The definition of the propensity function implies, for any time steτ that is small enough to satisfy the first leacondition, the tau-leaping formula. If the second leacondition is also satisfied, the tau-leaping formula becomes Eq., which is the forward Euler formula for a stochastic differential equation. And that formula becomes in the thermodynamic limit, where its diffusion term will be negligibly small compared to its drift term, the forward Euler formula for an ordinary differential equation.
-
The foregoing chain of reasoning can be summarized from the perspective of computational mathematics as follows: The definition of the propensity function implies, for any time step τ that is small enough to satisfy the first leap condition, the tau-leaping formula. If the second leap condition is also satisfied, the tau-leaping formula becomes Eq., which is the forward Euler formula for a stochastic differential equation. And that formula becomes in the thermodynamic limit, where its diffusion term will be negligibly small compared to its drift term, the forward Euler formula for an ordinary differential equation.
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49
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84877776005
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A common misconception is that, while the molecular population X is obviously a discrete variable, the molecular concentration Z XΩ is a continuous variable. The error in that view becomes apparent when one realizes that simply by adopting a unit of length that gives Ω the value 1, Z becomes numerically equal to X. But even if that is not done, a sudden change in X, say from 10 to 9, will always result a discontinuous 10 decrease in Z. The molecular concentration Z is no less discrete, and no more continuous, than the molecular population X.
-
A common misconception is that, while the molecular population X is obviously a discrete variable, the molecular concentration Z XΩ is a continuous variable. The error in that view becomes apparent when one realizes that simply by adopting a unit of length that gives Ω the value 1, Z becomes numerically equal to X. But even if that is not done, a sudden change in X, say from 10 to 9, will always result a discontinuous 10 decrease in Z. The molecular concentration Z is no less discrete, and no more continuous, than the molecular population X.
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jX(t′)dt′, see Refs. and. Note that all the integrals here can be written as finite algebraic sums, since X(t′)
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jX(t′)dt′, see Refs. and. Note that all the integrals here can be written as finite algebraic sums, since X(t′) stays constant between successive reactions.
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A more concise tutorial presentation of the ssSSA, featuring illustrative applications to reactions and the classical enzyme-substrate reactions E S ES → E P, is given in Ref.. A more thorough examination of the relation between the ssSSAs treatment of the enzyme-substrate system and the classical Michaelis-Menten approximation is given in, 10.1049/iet-syb.2009.0057
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A more concise tutorial presentation of the ssSSA, featuring illustrative applications to reactions and the classical enzyme-substrate reactions E S ES → E P, is given in Ref.. A more thorough examination of the relation between the ssSSAs treatment of the enzyme-substrate system and the classical Michaelis-Menten approximation is given in K. Sanft, D. Gillespie, and L. Petzold, IET Syst. Biol. 5, 58 (2011). 10.1049/iet-syb.2009.0057
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1 must be also. This illustrates the important fact that fast and slow reaction channels cannot always be identified solely on the basis of the magnitudes of their rate constants.
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1 must be also. This illustrates the important fact that fast and slow reaction channels cannot always be identified solely on the basis of the magnitudes of their rate constants.
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See, for example, (Oxford University Press), Cha 5. That chapters revised Sec. 5.6 (which can be downloaded from the books webpage on the publishers website) shows that below a certain value of h, no propensity function can give a physically accurate modeling of diffusive transfers of molecules between voxels.
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See, for example, D. Gillespie and E. Seitaridou, Simple Brownian Diffusion (Oxford University Press, 2012), Chap. 5. That chapters revised Sec. 5.6 (which can be downloaded from the books webpage on the publishers website) shows that below a certain value of h, no propensity function can give a physically accurate modeling of diffusive transfers of molecules between voxels.
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