The system might have many feasible solutions in a closed convex polytope, the n-dimensional analogue to the three dimensional polyhedron, formed by the intersection of the kernel of S and the linear inequalities in Eq (3). m < n, the system in Eqs (2) and (3) is in general undetermined. However, since a typical metabolic network has fewer metabolites than reactions, i.e. Often the flux rates are constrained with upper and lower bounds the problem goes from being a set of differential equations in x to become an algebraic problem, with the flux rates v as unknowns. In Flux Balance Analysis (FBA) the model system is assumed to be in a steady state The non-linearity of the ODE system also makes the system susceptible to chaotic behavior, bifurcation and sensitivity to parameter values. A challenge is to establish models of the different flux rates, in general nonlinear in x, and to estimate the k parameters in α through in-vivo and in-vitro experiments. The stoichiometric matrix S is constructed so that element S ij is positive (negative) if metabolite i is created (consumed) by reaction j, represented by the flux rate v j, and is assumed constant. a matrix representation of the network, and are the flux rates in the n reactions. Here, is a vector containing of metabolite concentrations, is a vector of parameters, is the stoichiometric matrix, i.e. The dynamics of a metabolic network, consisting of m metabolites and n reactions, can be mathematically modelled by a system of Ordinary Differential Equations (ODEs) written in short form as The funding body did not play any role in the design of the study and in the 705 writing of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist.Ĭell metabolism involves many chemical reactions, catalyzed by thousands of enzymes, and is often represented as metabolic networks. įunding: This research was supported by the Research Council of Norway through grant 248840, 704 dCod 1.0. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.ĭata Availability: All data supporting the findings of this study are available within the paper and can also be accessed on figshare or generated using the publicly available source code via the following URLs: and. Received: OctoAccepted: JPublished: July 1, 2020Ĭopyright: © 2020 Fallahi et al. However, our analysis ranks it as less efficient than the samplers used for the deterministic formulation.Ĭitation: Fallahi S, Skaug HJ, Alendal G (2020) A comparison of Monte Carlo sampling methods for metabolic network models. For the stochastic formulation, the Gibbs sampler is the only method appropriate for sampling at genome scale. Among the three other algorithms, ACHR has the largest consistency with CHRR for genome scale models. A desirable property of CHRR is its guaranteed distributional convergence. The coordinate hit-and-run with rounding (CHRR) is found to perform best among the algorithms suitable for the deterministic formulation. We apply the ACHR, OPTGP, CHRR and Gibbs sampling algorithms to ten metabolic networks and evaluate their convergence, consistency and efficiency. In this study we give an overview of both the deterministic and stochastic formulation of the problem, and of available Monte Carlo sampling methods for sampling the corresponding solution space. Uniform sampling of fluxes, feasible in both the deterministic and stochastic formulation, can provide us with statistical properties of the metabolic network, such as marginal flux probability distributions. One can relax the steady state constraint, and also include experimental noise in the model, through a stochastic formulation of the problem. In a deterministic formulation of the problem the steady state assumption has to be fulfilled exactly, and the observed fluxes are included in the model without accounting for experimental noise. Reaction rates (fluxes) in a metabolic network can be analyzed using constraint-based modeling which imposes a steady state assumption on the system.
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