Abstract
Reaction rates (fluxes) in a metabolic network can be analyzed using constraint-based modeling which imposes a steady state assumption on the system. 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. One can relax the steady state constraint, and also include experimental noise in the model, through a stochastic formulation of the problem. 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. 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. We apply the ACHR, OPTGP, CHRR and Gibbs sampling algorithms to ten metabolic networks and evaluate their convergence, consistency and efficiency. The coordinate hit-and-run with rounding (CHRR) is found to perform best among the algorithms suitable for the deterministic formulation. A desirable property of CHRR is its guaranteed distributional convergence. Among the three other algorithms, ACHR has the largest consistency with CHRR for genome scale models. For the stochastic formulation, the Gibbs sampler is the only method appropriate for sampling at genome scale. However, our analysis ranks it as less efficient than the samplers used for the deterministic formulation.
Highlights
Cell metabolism involves many chemical reactions, catalyzed by thousands of enzymes, and is often represented as metabolic networks [1]
In this study we have evaluated ACHR, optimized general parallel sampler (OPTGP) and coordinate hit-and-run with rounding (CHRR) algorithms which are appropriate for the deterministic formulation
To get a visual impression of what this amounts to in a density plot, Fig 5 shows flux densities and Kullback-Leibler divergence (KLD) values for the Fumarase mitochondrial reaction (v553) of the iAT_PLT_636 model. According to this KLD scale ACHR has a good similarity to CHRR (KLD = 0.01 < 0.05), and OPTGP has a medium similarity to CHRR (0.05 < KLD = 0.43 < 0.5) while the Gibbs algorithm has a poor similarity to CHRR (KLD = 0.82 > 0.5)
Summary
Cell metabolism involves many chemical reactions, catalyzed by thousands of enzymes, and is often represented as metabolic networks [1]. Since the CHRR uses CHR Markov chain for sampling purpose, its convergence to the target distribution is guaranteed in contrast to ACHR based algorithms [28] In this part we review the studies of Van den Meersche et al [20] and Heinonen et al [22] in which statistical frameworks have been proposed to analyze metabolic fluxes while integrating flux measurements with their noise in the formulation and relaxing the steady state assumption in Eq (2). To our knowledge, these two studies are the only studies presenting sampling algorithms applicable at genome scale. A summary of the sampling algorithms and their main characteristics are presented in the Table 1
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