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...modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm =-=[30, 31, 32]-=-. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1, 2, . . . }) specifying the integral molecular counts of the species. A reaction α is a tuple 〈l,r, k〉 ∈ N p ...

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...ut rather the coagulation process is simulated stochastically by assuming that coagulation events are Poisson distributed with a Brownian kernel. [14] Applying such a Monte Carlo approach for simulating the evolution of particle distributions dates back to Gillespie [1975], who developed the exact Stochastic Simulation Algorithm [see also Gillespie, 1976, 1977, 1992] to treat the stochastic collision-coalescence process in clouds. Variants of Gillespie’s algorithm are widely used in different fields, including simulations of gene regulatory networks [El Samad et al., 2005], chemical kinetics [Gillespie, 2007], and sintering in flames [Wells et al., 2006]. [15] Since Gillespie [1975], particle-resolved methods have been used to study aerosols by many authors. We do not attempt to give a comprehensive literature survey here. Babovsky [1999] and Eibeck and Wagner [2001] developed the Mass Flow Algorithm with variable computational/ physical particle ratios, Kolodko and Sabelfeld [2003] gave relevant error estimates, and Debry et al. [2003] coupled it to evaporation and condensation. Somewhat similarly, Laurenzi et al. [2002] and Alfonso et al. [2008] (on the basis of ideas from Spouge [1985]) stored...

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...les that evolve according to given rules, which in turn represent biochemical interactions. Rules are applied according to an extension of Gillespie’s well known Stochastic Simulation Algorithm (SSA) =-=[7]-=- to the multicompartmental structure of P system models [10]. Biological functions are the results of the interactions between modules made up of many molecular species. By definition, a module is a d...

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...or binding analysis in biochemical signaling [19], although it is valid only where the random perturbations in the chemical reactions are negligible due to the large number of molecules, as stated in =-=[20]-=-. The ligand-receptor kinetics model in this paper is an extension of the classical chemical kinetics formulation that allows us to have a simulation model for the random perturbations in the ligand-r...

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...used, ODEs suffer from several limitations. First, the passage from a finite number of molecules to real-valued concentration is not always justified, especially when the number of molecules is small =-=[22]-=-. Secondly, many biological phenomena, for example gene activation, are more naturally modeled as transitions between discrete states. Pure ODEs cannot easily accommodate this mixture of continuous ev...

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...aporation is handled deterministically, emission, dilution and coagulation are treated with a stochastic Monte Carlo approach. Applying such a Monte Carlo approach for simulating the evolution of particle distributions dates back to Gillespie (1975), who developed the exact Stochastic Simulation Algorithm (see also Gillespie, 1976, 1977, 1992) to treat the stochastic collisioncoalescence process in clouds. Variants of Gillespie's algorithm are widely used in different fields, including simulations of gene regulatory networks (El Samad, Khammash, Petzold, & Gillespie, 2005), chemical kinetics (Gillespie, 2007), and sintering in flames (Wells, Morgan, Kraft, & Wagner, 2006). Coagulation between aerosol particles is simulated in PartMC by generating a realization of a Poisson process with a Brownian coagulation kernel. For the large number of particles used here we cannot simply implement the Stochastic Simulation Algorithm by Gillespie (1975). Instead it is necessary to employ an efficient approximate simulation method. We developed a binned sampling method to efficiently sample from the highly multi-scale coagulation kernel (in our case the Brownian kernel) in the presence of a very non-uniform par...

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...rch for some probability distributions and small networks. A variety of enhancements to the SSA have also been developed to enable its use for different problems. For example, much work has been done =-=[14,31,32,66,67,86]-=- to address the issue of stiffness of the dynamical systems embodied in biological networks, and extend the timescale over which the SSA can be used, while bounding the errors induced by multi-timesca...

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...? As an example application, chemical reactions can be conveniently modeled by systems of nonlinear stochastic differential equations, whose variables encode the densities of various chemical species =-=[1, 2]-=-. Complex biological networks might involve hundreds of such species [3], and learning stochastic models from data is an important (and challenging) computational task [4]. Considering one such chemic...