Computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae

Abstract

Flux balance analysis (FBA) is an effective tool in the analysis of metabolic network. It can predict the flux distribution of engineered cells, whereas the accurate prediction depends on the reasonable objective function. In this work, we propose two nonlinear bilevel programming models on anaerobic glycerol metabolism in Klebsiella pneumoniae (K. pneumoniae) for 1,3-propanediol (1,3-PD) production. One intends to infer the metabolic objective function, and the other is to analyze the robustness of the objective function. In view of the models’ characteristic an improved genetic algorithm is constructed to solve them, where some techniques are adopted to guarantee all chromosomes are feasible and move quickly towards the global optimal solution. Numerical results reveal some interesting conclusions, e.g., biomass production is the main force to drive K. pneumoniae metabolism, and the objective functions, which are obtained in term of several different groups of flux distributions, are similar.

Introduction

1,3-Propanediol (1,3-PD) possesses potential applications on a large commercial scale, especially as a monomer of polyesters or polyurethanes, its biosynthesis has attracted worldwide interests Bibel et al., 1999, Nakamura and Whited, 2003, etc. Among all kinds of microbial production of 1,3-PD, dissimilation of glycerol by Klebsiella pneumoniae (K. pneumoniae) has been widely investigated due to its high productivity since 1980s Zeng and Biebl, 2002, Menzel et al., 1997. However, compared with the competing chemical process, the microbial production is difficult to obtain a high 1,3-PD concentration in the fermentation broth. Its an area of interest to develop an improved technique to improve the productivity of 1,3-PD.

The knowledge of cell physiology and metabolic regulation is helpful to improve productivity of 1,3-PD by a metabolic engineering approach on the strain (Stephanopoulos et al., 1998). Moreover, computational methods for cellular metabolism play an important role in understanding the complex biochemical interactions within the cells. Among those methods, FBA has been proved to be an effectively computational tool for understanding cell physiology and regulation of metabolism, and has been successfully applied to different microorganisms Maczek et al., 2006, Ozkan et al., 2005, Sanchez et al., 2006, Shirai et al., 2005. FBA can predict the metabolic fluxes by using a reduced set of measured fluxes and mass balance equations around intracellular metabolites, and thereby provide a better characterization of cellular phenotypes. However, the accurate prediction depends on a reasonable objective function. Objective functions in practice can take on a linear form, i.e., $f=cv$, where c denotes the vector defining the coefficients or weights for each flux in $v$ (Beard et al., 2002). The elements of c enable the formulation of a number of diverse objectives. Common objective functions include maximizing biomass or cell growth, maximizing ATP production or maximizing the rate of synthesis of a particular product (Kauffman et al., 2003). Other objective functions include minimizing ATP production in order to determine conditions of optimal metabolic energy efficiency, and minimizing nutrient uptake in order to evaluate the conditions under which a cell will perform its metabolic functions while consuming the minimum amount of nutrients (Lee et al., 2006), etc.

Although many hypotheses have been put forward as surrogates for cellular objective functions, substantially less work has been conducted toward systematically validating them with experimentally derived flux distributions of metabolic networks. An optimization-based framework is proved to be effective for inferring the objective function of E. coli metabolism (Burgard and Maranas, 2003). For solving the bilevel problem, the authors transform it into an equivalent nonlinear programming making use of the duality theory. However, up to date, no attempt has been made to infer the objective function of anaerobic glycerol metabolism in K. pneumoniae for 1,3-PD production.

In this work, we introduce a rigorous mathematical model (NBP1) to infer whether a weighted sum of fluxes can explain a set of experimental data observed from anaerobic glycerol metabolism in K. pneumoniae for 1,3-PD production. The model is a nonlinear bilevel programming, whose lower level is a conventional FBA model with undetermined parameters and upper level is a quadratic programming for evaluating the consistency with observed fluxes. Bard (1991) proves that the bilevel linear programming is NP-hard. So it is more difficult to solve NBP1. To obtain the weight coefficients in the model, a fast convergent genetic algorithm is constructed by adopting some techniques, which greatly reduce the searching space and avoid the difficulty to deal with the infeasible chromosomes. In this work, we also propose a model (NBP2) to investigate the effect of deviations between the flux distribution and the experimental ones on the weight coefficients. Some meaningful conclusions are drawn by several groups of data observed at different dilution rates and initial glycerol concentrations.

The remainder of this paper is organized as follows. In Section 2, optimization models are formulated to determine the objective function and analyze the robustness of the objective function. In Section 3, we construct an improved genetic algorithm to solve NBP1(2). Section 4 gives the main results. Finally, Section 5 concludes some remarks and future research directions.