Discussion
- A Mathematical Model for the Branched Chain Amino Acid Biosynthetic Pathways of Escherichia coli K12

DISCUSSION

In this report, we describe a mathematical simulation of branched chain amino acid biosynthesis and regulation in E. coli. This approach involves the following steps. (i) Step 1 is the identification of all of the molecular participants, including enzymes, metabolites, and coenzymes as well as the enzyme kinetic and regulatory mechanism of each enzyme (defined in Fig. 1 and supplemental Table I in the on-line version of this article). For well studied model organisms such as E. coli, these types of information are often available in 50 years of scientific literature and several on-line databases (28–32). (ii) Step 2 is the development of approximation methods for unavailable model parameters. Examples include the approximation of rate constants (k_f and k_r) from kinetic measurements (K_m and k_cat) described by Yang et al. (10), the approximation of k_cat from the activity of purified enzymes, and the approximation of intracellular enzyme concentrations (E_T) from enzyme purification and DNA microarray data. (iii) Step 3 is the use of the information obtained in steps 1 and 2 to create, as accurately as possible, calculation-independent models that describe the catalytic and regulatory mechanisms of each enzymatic step (kMech). (iv) Step 4 is the stringing together of appropriate kMech models and providing the physical and kinetic parameters for each enzyme in the pathway. (v) Step 5 is the generation of ordinary differential equations to describe each enzyme mechanism in terms of fundamental molecular interactions (Cellerator). (vi) Step 6 is the optimization of model parameters to simulate known steady-state intracellular levels of pathway substrates, intermediates, and end products. (vii) Step 7 is the comparison of simulated and observed results of biochemical and genetic perturbation experiments.

This type of deterministic continuous modeling of metabolic systems can provide valuable information such as predicted steady-state levels of metabolic substrates, intermediates, and end products and can predict the outcomes of biochemical and genetic perturbations that require detailed enzyme kinetic and regulatory mechanisms. Traditional modeling approaches use the Michaelis-Menten kinetic equation for one substrate/one product reactions and the King-Altman method to derive equations for more complex multiple reactant reactions. These types of equations are called steady-state velocity equations, because the derivatives of concentration of each reactant in the model over time are set to 0 in order to simplify a set of non-linear differential equations to linear algebra equations (33). Therefore, the kinetic model based on this approach has embedded the steady-state hypothesis. In contrast, the model generated by kMech/Cellerator consists of non-simplified, non-linear differential equations that describe the rates of change of each reactant in the model over time. To build a pathway model, users need only to call upon kMech models for the enzyme mechanisms of a pathway without writing any differential equations. Because of this simple user input and the integration of kMech, Cellerator, and Mathematica^TM, human errors are greatly reduced (10). To allow kMech/Cellerator to be utilized by an audience with little or no programming experience, a Java-based graphical user interface (GUI) is under development. This graphical editor is designed to help users construct pathways, select enzyme mechanisms, and enter required physical and kinetic parameters with simple point and click methods.

In contrast to "top down" metabolic flux balance analysis methods (34), which provide valuable information about biomass conversions without knowing individual enzyme mechanism and pathway-specific regulation patterns (12), the kMech/Cellerator models described here represent a "bottom up" approach to an understanding of complex metabolic networks. The model presented here is incomplete for many reasons, primarily, because it does not exist in the context of the bacteria cell. In addition to the metabolic regulatory mechanisms considered here, carbon flow through metabolic pathways is affected by a hierarchy of additional controls of gene expression levels that affect pathway enzyme activities and amounts. These hierarchical levels of control, from the most general to the most specific, are as follows: (i) global control of gene activity mediated by chromosome structure (3); (ii) global control of the genes of stimulons and regulons (35); and (iii) operon-specific controls. The first or highest level of control is exemplified by DNA topology-dependent mechanisms that coordinate basal level expression of all of the genes of the cell (independent of operon-specific controls). This level is mediated by DNA architectural proteins and the actions of topoisomerases in response to nutritional and environmental growth conditions (3). The second level of control is mediated by site-specific DNA-binding proteins, which, in cooperation with operon-specific controls, regulate often overlapping groups of metabolically related operons in response to environmental or metabolic transitions or stress conditions (35). The third level of control is mediated by less abundant regulatory proteins that respond to operon-specific signals and bind in a site-specific manner to one or a few DNA sites to regulate single operons. Each of these levels of control impacts metabolic regulation by influencing enzyme levels. Thus, a complete model of branched chain amino acid biosynthesis in E. coli must include these higher levels of gene regulation. To incorporate these higher levels of regulation, we are currently developing a set of models that describe the genetic regulatory mechanisms that control the operons of the ilv regulon. To these ends, we face new challenges. For example, whereas the ordinary differential equations we are using for metabolic pathways are a deterministic and continuous approximation for an average representation of interactions between large numbers of discrete molecules (e.g. enzymes and metabolites), McAdams and Arkin point out that because each cell contains only one gene/operon there can be large differences in the time between successive events in regulatory cascades across a cell population that can produce probabilistic outcomes (36). To address this and other issues, we are currently working on another software package for genetic regulatory mechanisms, gMech, that implements stochastic simulation algorithms such as Gillespie's algorithm (37) and the Langevin equation (38), which accommodate stochastic noise. This gMech software package will contain models for genetic regulatory mechanisms such as attenuation, activation, and repression as well as DNA topological controls. Therefore, the work presented here should be considered as a first step of a bottom up approach that integrates biochemical information from the literature and bioinformatics databases and relative gene expression data from DNA microarrays to build a self-regulated metabolic pathway in E. coli. As high throughput technologies for genomics, proteomics, and metabolomics grow, we expect that a similar approach will soon be feasible in higher organisms.

FOOTNOTES

^* This work was supported in part by National Institutes of Health Grants GM55073 and GM68903 (to G. W. H.). The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

The on-line version of this article (available at http://www.jbc.org) contains further mathematical modeling data in the form of supplemental Fig. 1 and supplemental Table I.

^¶ A trainee of the Biomedical Informatics Training (BIT) Program of the University of California at Irvine Institute for Genomics and Bioinformatics and the recipient of National Library of Medicine Postdoctoral Fellowship T15 LM-07443.

To whom correspondence on computation questions should be addressed. E-mail: emj@uci.edu . $§$ $§$ To whom correspondence on biology questions should be addressed. E-mail: gwhatfie@uci.edu .

¹ The abbreviations used are: TDA, L-threonine deaminase; AHAS, ${alpha}$ -acetohydroxy acid synthase; aKB, ${alpha}$ -ketobutyrate; MWC, Monod, Wyman, and Changeux (model); ODE, ordinary differential equation; Pyr, pyruvate; TB, transaminase B.