Mathematical Problems in Engineering

Volume 2015, Article ID 673120, 9 pages

http://dx.doi.org/10.1155/2015/673120

## Dynamics Analysis and Prediction of Genetic Regulation in Glycerol Metabolic Network via Structural Kinetic Modelling

^{1}School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350108, China^{2}School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China^{3}Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia^{4}School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China^{5}College of Life Science, Fujian Normal University, Fuzhou, Fujian 350108, China^{6}School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China

Received 16 August 2014; Accepted 22 December 2014

Academic Editor: Carlo Cosentino

Copyright © 2015 Jianxiong Ye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Glycerol can be biologically converted to 1,3-propanediol (1,3-PD) by *Klebsiella pneumoniae*. In the synthesis pathway of 1,3-PD, the accumulation of an intermediary metabolite 3-hydroxypropionaldehyde (3-HPA) would cause an irreversible cessation of the dynamic system. Genetic manipulation on the key enzymes which control the formation rate and consumption rate of 3-HPA would decrease the accumulation of 3-HPA, resulting in nonlinear regulation on the dynamic system. The interest of this work is to focus on analyzing the influence of 3-HPA inhibition on the stability of the dynamic system. Due to the lack of intracellular knowledge, structural kinetic modelling is applied. On the basis of statistical account of the dynamical capabilities of the system in the parameter space, we conclude that, under weak or no inhibition to the reaction of 3-HPA consumption, the system is much easier to obtain a stable state, whereas strong inhibition to its formation is in favor of stabilizing the system. In addition, the existence of Hopf bifurcation in this system is also verified. The obtained results are helpful for deeply understanding the metabolic and genetic regulations of glycerol fermentation by *Klebsiella pneumoniae*.

#### 1. Introduction

1,3-Propanediol (1,3-PD) has wide applications for a variety of markets, especially as a monomer for polyesters, polyethers, and polyurethanes [1]. Microbial production of 1,3-PD, a socially beneficial route to obtain chemicals from renewable resources, has been widely investigated and considered as a competitor to the traditional petrochemical routes. Over the past years, some microorganisms such as* Klebsiella pneumoniae*,* Clostridium butyricum,* and* Citrobacter freundii* have been used to synthesize 1,3-PD from glycerol [1–3], among which* Klebsiella pneumoniae* (*K. pneumoniae*) was most popularly investigated due to its high productivity [4].

Over the past years, modelling, stability, and optimal control of glycerol fermentation have been extensively investigated. For details, see [5–10] and the references therein. However, most of the previous researches were based on explicit models for the extracellular substance concentrations instead of a consideration of the metabolic process in the intracellular environment which, in essence, is the origin of multistationarity and oscillation. Up to 2008, the intracellular dynamics of this bioprocess was firstly studied by Sun et al. [11]. Thereafter, we introduced several quantitative measures of biological robustness to estimate the kinetic parameters of Sun’s model in the context where the intracellular data were limited and some of the metabolic mechanisms were not completely known [12–14]. The problem is that huge computing workload will be faced in the evaluation of the biological robustness index, even though the scale of the metabolic network is not too large.

In the reductive pathway of glycerol conversion to 1,3-PD, the accumulation of an intermediary metabolite, 3-hydroxypropion-aldehyde (3-HPA), would greatly affect the productivity of the fermentation. There have been many researches on the genetic engineering technology about decreasing 3-HPA accumulation by overexpressing the related genes of the strains [15–17]. However, some questions would emerge on the engineered strains: what kind of changes may happen to the new strains or whether the new strains can inherit “good” properties from the host strains after genetic manipulation. Therefore, it is necessary to reevaluate the dynamics of the engineered strains. In particular, due to the presence of the negative feedback in the 3-HPA accumulation, it is worthwhile to analyze the role of 3-HPA inhibition in the metabolic system of glycerol fermentation.

Over the past decades, detailed kinetic models have been widely applied to investigate the dynamic properties of metabolic system, in which differential equations are employed to describe the temporal behavior of the system. Unfortunately, there is a disproportion between the high number of parameters contained in the kinetic models and the relatively incomplete data available [18]. To address these problems, researchers have devoted great effort to exploit other methods to quantitatively evaluate dynamic properties of metabolic processes in the recent years [19–24]. It is worth mentioning that Steuer et al. [19] proposed a structural kinetic modelling approach, which has an advantage in drawing quantitative conclusions about the possible dynamics of the system without assuming detailed knowledge of the underlying enzyme-kinetic rate equations and parameters. One other advantage of this approach is that the classical Michaelis-Menten kinetic model can be transformed to this model scheme with saturation parameters well defined [19].

In this paper, our interest is to focus on investigating how 3-HPA inhibition would influence the stability of the glycerol metabolic system of the engineered* K. pneumoniae*, in which the genes encoding two key enzymes glycerol dehydratase (GDHt) and 1,3-PD oxydoreductase (PDOR) are overexpressed. A structural kinetic model (SKM) is developed, which is presented in a parametric form with the ranges of all associated parameters well defined. A statistical exploration on the proposed model is performed on the comprehensive parameter space to investigate the system’s capability to obtain a stable state. Additionally, by varying the strength of 3-HPA inhibitions upon its upstream and downstream reactions, we verify the existence of Hopf bifurcation. Finally, we analyze how the ratio of GDHt activity to PDOR activity may influence the stability of the system.

This paper is organized as follows. In Section 2, we briefly introduce the metabolic process of glycerol in* K. pneumoniae* and present the explicit rate equations for the considered metabolites. In Section 3, a SKM is developed for the reductive pathway of this process. In Section 4, the physiologically feasible ranges of the associated parameters in the SKM are specified. In Section 5, the stability of the system is statistically investigated on the comprehensive parameter space. Conclusions and discussions of the computational results are presented at the end of this paper.

#### 2. Glycerol Metabolism in* K. pneumoniae* and Its Explicit Mathematical Model

During glycerol metabolism by* K. pneumoniae* under anaerobic condition, glycerol is first transported across cell membrane from the extracellular environment to the intracellular environment. In the intracellular environment, glycerol is dissimilated through coupled oxidative and reductive pathways as shown in Figure 1. The goal product 1,3-PD is produced by the reductive branch in two successive enzymatic reactions [25]: glycerol is first dehydrated to 3-HPA by the enzyme GDHt; 3-HPA is then converted to 1,3-PD by the enzyme PDOR. In the oxidative pathway, glycerol oxidation is catalyzed by the enzyme glycerol dehydrogenase (GDH), leading to the formation of dihydroxyacetone (DHA); DHA is further phosphorylated by two dihydroxyacetone kinases and is channelled into glycolysis, yielding the same fermentation products as in glucose fermentation (acetate, ethanol) and to the generation of energy and reducing power. Finally, the products are transported across cell membrane from the intracellular environment to the extracellular environment.