Use of maximum entropy principle with Lagrange multipliers extends the feasibility of elementary mode analysis

Elementary mode (EM) analysis is potentially effective in integrating transcriptome or proteome data into metabolic network analyses and in exploring the mechanism of how phenotypic or metabolic flux distribution is changed with respect to environmental and genetic perturbations. The EM coefficients (EMCs) indicate the quantitative contribution of their associated EMs and can be estimated by maximizing Shannon's entropy as a general objective function in our previous study, but the use of EMCs is still restricted to a relatively small-scale networks. We propose a fast and universal method that optimizes hundreds of thousands of EMCs under the constraint of the Maximum entropy principle (MEP). Lagrange multipliers (LMs) are applied to maximize the Shannon's entropy-based objective function, analytically solving each EMC as the function of LMs. Consequently, the number of such search variables, the EMC number, is dramatically reduced to the reaction number. To demonstrate the feasibility of the MEP with Lagrange multipliers (MEPLM), it is coupled with enzyme control flux (ECF) to predict the flux distributions of Escherichia coli and Saccharomycescerevisiae for different conditions (gene deletion, adaptive evolution, temperature, and dilution rate) and to provide a quantitative understanding of how metabolic or physiological states are changed in response to these genetic or environmental perturbations at the elementary mode level. It is shown that the ECF-based method is a feasible framework for the prediction of metabolic flux distribution by integrating enzyme activity data into EMs to genetic and environmental perturbations.