In this article, the steady state condition for the multi-compartment models for cellular metabolism is considered. The problem is to estimate the reaction and transport fluxes, as well as the concentrations in venous blood when the stoichiometry and bound constraints for the fluxes and the concentrations are given. The problem has been addressed previously by a number of authors, and optimization based approaches as well as extreme pathway analysis have been proposed. These approaches are briefly discussed here. The main emphasis of this work is a Bayesian statistical approach to the flux balance analysis (FBA). We show how the bound constraints and optimality conditions such as maximizing the oxidative phosphorylation flux can be incorporated into the model in the Bayesian framework by proper construction of the prior densities. We propose an effective Markov Chain Monte Carlo (MCMC) scheme to explore the posterior densities, and compare the results with those obtained via the previously studied Linear Programming (LP) approach. The proposed methodology, which is applied here to a two-compartment model for skeletal muscle metabolism, can be extended to more complex models.

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In this paper we present a partially orthogonal decomposition for a matrix A. Using this decomposition the linear least squares problem is reduced to solving two linear systems. The matrix of the first system is symmetric and positive definite, and the matrix of the second system is nonsingular upper triangular. We show that this approach can provide computational savings.

The steady state condition of two-compartment skeletal muscle model is considered. The inverse problem is to estimate the reaction and transport fluxes, as well as the concentrations in venous blood when the stoichiometry and bound constraints for the fluxes and the concentrations are given. A new methodology based on a Bayesian statistical approach to the flux balance analysis (FBA) is proposed. We show how the bound constraints and optimality condition can be incorporated into the model in the Bayesian framework by proper construction of the prior densities. An effective Markov Chain Monte Carlo (MCMC) scheme to explore the posterior densities is proposed.

In [25] it was proposed a parametric linear transformation, which is a "convex" combination of the Gauss transformation of elimination method and the Gram-Schmidt transformation of modified orthogonalization process. Using this transformation, in particular, elimination methods were generalized, Dantzig's optimality criterion and simplex method were developed [26]. The essence of the simplex method development is the following. At each sth step the pivot (positive) vector of length Ks is selected, that allows us to move to improved feasible solution after the step of the generalized Gauss-Jordan complete elimination method. In this method the movement to the optimal point takes place over pseudobases, i.e., over interior points. This method is parametric and finite. Since the method is parametric there are various variants to choose the pivot vectors (rules), in the sense of their lengths and indices. In this article we propose three rules, which are the development of Dantzig's first rule. These rules are investigated on the Klee-Minty cube (problem) [14, 31]. It is shown that for two rules the number of steps necessary equals to 2n, and for third rule we obtain the standard simplex method with the largest coefficient rule, i.e., the number of steps for solving this problem is 2n - 1.