In this work we propose a novel algorithmic strategy that allows for an efficient characterization of the whole set of stable fluxes compatible with the metabolic constraints. The algorithm, based on the well-known Bethe approximation, allows the computation in polynomial time of the volume of a non full-dimensional convex polytope in high dimensions. The result of our algorithm match closely the prediction of Monte Carlo based estimations of the flux distributions of the Red Blood Cell metabolic network but in incomparably shorter time. We also analyze the statistical properties of the average fluxes of the reactions in the E-Coli metabolic network and finally to test the effect of gene knock-outs on the size of the solution space of the E-Coli central metabolism.
Cellular metabolism is a complex biological problem. It can be viewed as a chemical engine that transforms available raw materials into energy or into the building blocks needed for the biological function of the cells. In more specific terms a metabolic network is indeed a processing system transforming input metabolites (nutrients), into output metabolites (amino acids, lipids, sugars etc.) according to very strict molecular proportions, often referred as stoichiometric coefficients of the reactions.
Although the general topological properties of these networks are well characterized, see for example [1][2][3], and non-trivial pathways are well known for many species [4] the cooperative role of these pathways is hard to comprehend. In fact, the large sizes of these networks, usually containing hundreds of metabolites and even more reactions, makes the comprehension of the principles that govern their global function a challenging task. Therefore, a necessary step to achieve this goal is the use of mathematical models and the development of novel statistical techniques to characterize and simulate these networks.
For example, it is well known that under evolutionary pressure, prokaryotes cells like E-Coli behave optimizing their growth performance [5]. Flux Balance Analysis (FBA) provides a powerful tool to predict, from the whole space of phenotipic states, which one will these cells acquire. In few words one may say that FBA maximizes a linear function (usually the growth rate of the cell) subject to biochemical and thermodynamic constraints [6]. On the other hand, cells with genetically engineered knockouts or bacterial strains that were not exposed to evolution pressures, need not to optimize their growth. In fact, the method of minimization metabolic adjustment (MOMA) [7] has shown that knockout metabolic fluxes undergo a minimal redistribution with respect to the flux configuration of the wild type. Yet, in more general situations, the results are unpredictable, therefore, a tool to characterize the shape and volume of the whole space of possible phenotipic solutions must be welcome.
Unfortunately, this characterization has remained an elusive task. As far as we know the attempts to obtain such a characterization were always based on the Monte Carlo sampling of the steadystate flux space [8]. But unfortunately, it appeared that this kind of sampling is a very expensive calculation and is unsuitable for very large networks. To address this problem we propose, using a technique derived from the fields of statistical physics and information theory, an algorithm that may efficiently characterize the whole set of stable fluxes compatibles with the stoichiometric constraints.
As already mentioned, a metabolic network is an engine that converts metabolites into other metabolites through a series of intra-cellular intermediate steps. The fundamental equation characterizing all functional states of a reconstructed biochemical reaction network is a mass conservation law that imposes simple linear constraints between the incoming and outcoming fluxes at any chemical reaction. It is call the dynamic mass balance equation:
where ρ is the vector of the M metabolite concentrations in the network. i (o) is the input (output) vector of fluxes, and ν are the reaction fluxes governed by the M ×N stoichiometric linear operator Ŝ (usually named stoichiometric matrix) encoding the coefficient of the M intra-cellular relations among the N fluxes.
As long as just steady-state cellular properties are concerned one can assume that a variation in the concentration of metabolites in a cell can be ignored and considered as constant. Therefore in case of fixed external conditions one can assume metabolites (quasi) stationarity and consequently the lhs of 1 can be set to zero. Under these hypotheses the problem of finding the metabolic fluxes compatible with flux-balance is mathematically described by the linear system of equations
where b is the net metabolite uptake by the cell.
Without loss of generality we can assume that the stoichiometric matrix Ŝ has full rows rank, i.e. that rank( Ŝ) = M , since linearly dependent equations can be easily identified and removed. Knowing that the number of metabolites M is lower than the number of fluxes N the subspace of solutions is a (N -M )-dimensional manifold embedded in the Ndimensional space of fluxes. In addition, the positivity of fluxes, together with the experimentally accessible values for the maximal fluxes, limit further the space of feasible solutions. This fact may be expressed by the following inequalities:
in such a way that together, 2 and 3, define the convex set of all the allowed time-independent phenotipic states of a given metabolic network.
Mathematically speaking, the space of feasible solutions consistent with the equations 2 constitutes an affine space V ⊂ R N of dimension N -M . The set of inequalities 3 then defines a convex polytope Π ⊂ V that, from
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