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...ons required by the algorithm is seven for starting up, and six or seven during the simulation. The step method Λ for these ODE modules are implemented with a standard adaptive step sizing mechanism =-=(Press et al., 2002). In additi-=-on, time step sizes are constrained by the following condition to ensure that the Stepper does not change variable values too much at once in a single step: ∀j ∈ ˆ Ri : xjɛrel + ɛabs ≥ � δ...

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...s θ be the expected time for the reactant of the current Process with the smallest population to change its value by 10 %. If precision precedes efficiency, set ξ = 0. The Mersenne-Twister algorithm=-= (Matsumoto and Nishimura, 1998-=-) has been employed as a pseudo-random number generator. 5 4 Results Two relatively simple models have been constructed to examine the performance of the meta-algorithm. The first is a model of the he...

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..., 2002). In addition to those continuous representations, discrete formulation is often used. Stochastic discrete event simulation of coupled chemical reactions was pioneered by Gillespie and others (=-=Gillespie, 1976-=-), and improved by Gibson (Gibson and Bruck, 2000) to work more efficiently. StochSim is a discrete-time stochastic simulation algorithm originally devised as a means of investigating the bacterial ch...

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...epresentations, discrete formulation is often used. Stochastic discrete event simulation of coupled chemical reactions was pioneered by Gillespie and others (Gillespie, 1976), and improved by Gibson (=-=Gibson and Bruck, 2000-=-) to work more efficiently. StochSim is a discrete-time stochastic simulation algorithm originally devised as a means of investigating the bacterial chemotaxis signaling pathway (Morton-Firth and Bray...

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...on of cellular processes. By far the most widely accepted scheme of describing continuous dynamical systems is representation by differential equations. Apart from standard generic solver algorithms (=-=Gear, 1971-=-), specialized methods are often used for S-System and Generalized Mass-Action (GMA) canonical forms of power-law differential systems in simulations of metabolic and gene regulations (Irvine and Sava...

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...ave been implemented: second-order Fehlberg with a third-order error estimation (Fehlberg 2(3)) (Fehlberg, 1969), and fourth-order Dormand-Prince with a fifth-order error term (DormandPrince 5(4)7M) (=-=Dormand and Prince, 1980). For -=-some integration algorithms including these two, it is possible to derive interpolants by using Hermite polynomial interpolation, the general form of which is: g(x) = � � (j) i j αi,j(x)f i , �...

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... factor in the behavior of the system (Singer, 1953). ODE models isolate the biochemical system into a group of deterministic and continuous reactions, and tacitly ignore fluctuations in the pathway (=-=Rao et al., 2002-=-). With identical parameter settings, the stochastic and deterministic models can produce different results, and stochastic models are generally believed to be more accurate (Marion et. al., 1998; Sri...

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...ibson and Bruck, 2000) to work more efficiently. StochSim is a discrete-time stochastic simulation algorithm originally devised as a means of investigating the bacterial chemotaxis signaling pathway (=-=Morton-Firth and Bray, 1998-=-). Simulation methods based on cellular automaton have many applications from reactive gas physics to sociology, and have been found to be very useful for biological simulations (for example, see Weim...

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...002). With identical parameter settings, the stochastic and deterministic models can produce different results, and stochastic models are generally believed to be more accurate (Marion et. al., 1998; =-=Srivastava et al., 2002).-=- Despite its advantages, however, Gillespie’s scheme of exact stochastic simulation has limited utility due to its computational cost which is proportional to the number of molecules. The composite ...

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...e algorithm. Two instances of embedded explicit Runge-Kutta algorithms with dense output capabilities have been implemented: second-order Fehlberg with a third-order error estimation (Fehlberg 2(3)) (=-=Fehlberg, 1969-=-), and fourth-order Dormand-Prince with a fifth-order error term (DormandPrince 5(4)7M) (Dormand and Prince, 1980). For some integration algorithms including these two, it is possible to derive interp...

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...e of a composite simulation scheme, we set up a pure stochastic model, a pure ODE model and a stochastic/ODE composite model, based on Srivastava’s stochastic petri net model of the E.coli heatshock=-= (Srivastava et al., 2001-=-). Protein folding/unfolding processes have been added to the model, and modeled as differential equations in the ODE and composite models, and using the Gillespie algorithm in the stochastic model (F...

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...hms (Gear, 1971), specialized methods are often used for S-System and Generalized Mass-Action (GMA) canonical forms of power-law differential systems in simulations of metabolic and gene regulations (=-=Irvine and Savageau, 1990-=-). To complement a lack of quantitative kinetic data, an attempt to marry dynamic rate-equation based continuous models and static flux balance from stoichiometry is ongoing (Yugi et al., 2002). In ad...

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...down because the steady-state fluctuations in the number of molecules (which is proportional to the square root of the number of molecules) becomes a significant factor in the behavior of the system (=-=Singer, 1953-=-). ODE models isolate the biochemical system into a group of deterministic and continuous reactions, and tacitly ignore fluctuations in the pathway (Rao et al., 2002). With identical parameter setting...

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... achieved almost an order of magnitude better performance than the original. 5 Discussion In the limit of large numbers of reactant molecules, stochastic and deterministic simulations are equivalent (=-=Gillespie, 1977-=-). In contrast, if the system has low copy numbers of species, the deterministic law of mass action breaks down because the steady-state fluctuations in the number of molecules (which is proportional ...

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...1998). Simulation methods based on cellular automaton have many applications from reactive gas physics to sociology, and have been found to be very useful for biological simulations (for example, see =-=Weimar, 2002-=-). Brownian dynamics has proven to be an useful framework of interactions between ligands and binding sites on receptors, enzymes, and transporters (Bartol et al., 1996). Different algorithms have dif...

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...Irvine and Savageau, 1990). To complement a lack of quantitative kinetic data, an attempt to marry dynamic rate-equation based continuous models and static flux balance from stoichiometry is ongoing (=-=Yugi et al., 2002-=-). In addition to those continuous representations, discrete formulation is often used. Stochastic discrete event simulation of coupled chemical reactions was pioneered by Gillespie and others (Gilles...

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...computational challenges that are distinct from those encountered in problems of other disciplines such as molecular dynamics, or computational physics, and even conventional biochemical simulations (=-=Takahashi et al., 2002-=-). A vast array of molecular processes occur simultaneously in the cell. The involved physical and chemical processes include molecular diffusion, molecular binding, enzymatic catalysis and higher ord...

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...ological simulations (for example, see Weimar, 2002). Brownian dynamics has proven to be an useful framework of interactions between ligands and binding sites on receptors, enzymes, and transporters (=-=Bartol et al., 1996-=-). Different algorithms have different strengths, and are often suited to different spatial and temporal scales. When constructing models that span multiple scales, one might choose between two princi...