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Stochastic simulation algorithms for computational systems biology: Exact, approximate, and hybrid methods

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Abstract Nowadays, mathematical modeling is playing a key role in many different research fields. In the context of system biology, mathematical models and their associated computer simulations constitute essential tools of investigation. Among the others, they provide a way to systematically analyze systems perturbations, develop hypotheses to guide the design of new experimental tests, and ultimately assess the suitability of specific molecules as novel therapeutic targets. To these purposes, stochastic simulation algorithms (SSAs) have been introduced for numerically simulating the time evolution of a well‐stirred chemically reacting system by taking proper account of the randomness inherent in such a system. In this work, we review the main SSAs that have been introduced in the context of exact, approximate, and hybrid stochastic simulation. Specifically, we will introduce the direct method (DM), the first reaction method (FRM), the next reaction method (NRM) and the rejection‐based SSA (RSSA) in the area of exact stochastic simulation. We will then present the τ‐leaping method and the chemical Langevin method in the area of approximate stochastic simulation and an implementation of the hybrid RSSA (HRSSA) in the context of hybrid stochastic‐deterministic simulation. Finally, we will consider the model of the sphingolipid metabolism to provide an example of application of SSA to computational system biology by exemplifying how different simulation strategies may unveil different insights into the investigated biological phenomenon. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Analytical and Computational Methods > Computational Methods
A graphical representation of the simulation algorithms introduced in the review. Starting from a common root node representing a generic stochastic simulation algorithm, the methodologies differentiate in terms of accuracy and runtime according to exact and approximate methods. We depict with rectangles the algorithm classes and with circles the specific methods. Hybrid methods are here represented as a part of the approximate methods, however, they are often referred as a class of simulation algorithms itself. In the diagram, the τ‐leaping and the chemical Langevin methods are connected since, following the Gillespie's approach, the latter can be derived as an approximation of the former. Analogously, the deterministic methods are connected with the chemical Langevin method. Deterministic methods are indicated with dashed lines since they are not described in detail in this review
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Distribution of the number of molecules for the species ceramide‐1‐phosphate (Cer1P) and sphingosine‐1‐phosphate (S1P) at the end of the simulation (t = 24 hr) using rejection‐based SSA (RSSA) and the RSSA version that accounts for extrinsic noise (10,000 simulation runs)
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The comparison of the simulation results of Reali et al. model using a deterministic approach and rejection‐based SSA (RSSA) in combination with a scaling factor of 1e‐03 to reduce the population abundance. The results highlight how the stochastic noise is affecting the less abundant species ceramide‐1‐phosphate (Cer1P) and sphingosine‐1‐phosphate (S1P), suggesting the use of exact or hybrid stochastic simulation algorithms
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Schematic overview of a hybrid simulation strategy in which the set of reactions is divided into two subsets of slow and fast reactions. The two subsets are simulated with different methods and then are synchronized in order to define the new system state
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Two outcomes of the simulation of the Lotka‐Volterra model with different simulation algorithms. Figure a Shows the results of a deterministic simulation dynamics. Figure b Shows the dynamics of the same model using a stochastic algorithm. x‐axis: Time in absolute units. y‐axis: Abundance of the species
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Analytical and Computational Methods > Computational Methods
Models of Systems Properties and Processes > Mechanistic Models

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