Abstract
The uniform sampling of convex polytopes is an interesting computational problem with many applications in inference from linear constraints, but the performances of sampling algorithms can be affected by ill-conditioning. This is the case of inferring the feasible steady states in models of metabolic networks, since they can show heterogeneous time scales. In this work we focus on rounding procedures based on building an ellipsoid that closely matches the sampling space, that can be used to define an efficient hit-and-run (HR) Markov Chain Monte Carlo. In this way the uniformity of the sampling of the convex space of interest is rigorously guaranteed, at odds with non markovian methods. We analyze and compare three rounding methods in order to sample the feasible steady states of metabolic networks of three models of growing size up to genomic scale. The first is based on principal component analysis (PCA), the second on linear programming (LP) and finally we employ the Lovazs ellipsoid method (LEM). Our results show that a rounding procedure dramatically improves the performances of the HR in these inference problems and suggest that a combination of LEM or LP with a subsequent PCA perform the best. We finally compare the distributions of the HR with that of two heuristics based on the Artificially Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good agreement with the results of the HR for the small network, while on genome scale models present inconsistencies.
Highlights
The metabolism of cells is based on a complex metabolic network of chemical reactions performed by enzymes, which are able to degrade nutrients in order to produce biomass and generate the energy needed to sustain all other tasks the cell has to perform [1]
In this article we have proposed rounding methods in order to reduce the condition number for the application of the hit–and–run (HR) Markov Chain Monte Carlo to the problem of the uniform sampling of steady states in metabolic network models
Such ellipsoids were built by applying principle component analysis to previous sampling, by solving a set of linear programming problems— to a technique called Flux Variability Analysis in the field of metabolic network analysis, and by the Lovazs ellipsoid method
Summary
The metabolism of cells is based on a complex metabolic network of chemical reactions performed by enzymes, which are able to degrade nutrients in order to produce biomass and generate the energy needed to sustain all other tasks the cell has to perform [1]. The highthroughput data coming from genome sequencing of single organisms can be used to reconstruct the complete set of enzymes devoted to metabolic functions, leading to models of metabolism at the scale of the whole genome, whose analysis is computationally challenging [2]. If we want to model a metabolic system in terms of the dynamics of the concentration levels, even upon assuming well-mixing (no space) and neglecting noise (continuum limit), we have a very. For a chemical reaction network in which M metabolites participate in N reactions (where N,M ’ O(102–3) in genome-scale models) with the stoichiometry encoded in a matrix S = {Sμr}, the concentrations cμ change in time according to mass-balance equations c_ 1⁄4 S Á v ð1Þ where vi is the flux of the reaction i that in turn is a possibly unknown function of the concentration levels vi(c), with possibly unknown parameters.
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