HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

J. Biol. Chem., Vol. 280, Issue 12, 11224-11232, March 25, 2005

This Article Abstract Full Text (PDF) Supplemental Data All Versions of this Article: 280/12/11224    most recent M411471200v1 Alert me when this article is cited Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Yang, C.-R. Articles by Hatfield, G. W. Search for Related Content PubMed PubMed Citation Articles by Yang, C.-R. Articles by Hatfield, G. W. Social Bookmarking                      What's this?

A Mathematical Model for the Branched Chain Amino Acid Biosynthetic Pathways of Escherichia coli K12^*

Chin-Rang Yang¶, Bruce E. Shapiro||, She-pin Hung, Eric D. Mjolsness**, and G. Wesley Hatfield

From the Department of Microbiology and Molecular Genetics, College of Medicine, the ^**School of Information and Computer Science, and the Institute for Genomics and Bioinformatics, University of California at Irvine, Irvine, California 92697 and the ^||Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

Received for publication, October 7, 2004 , and in revised form, January 18, 2005.

	ABSTRACT

TOP ABSTRACT INTRODUCTION EXPERIMENTAL PROCEDURES RESULTS DISCUSSION REFERENCES

 
As a first step toward the elucidation of the systems biology of the model organism Escherichia coli, it was our goal to mathematically model a metabolic system of intermediate complexity, namely the well studied end product-regulated pathways for the biosynthesis of the branched chain amino acids L-isoleucine, L-valine, and L-leucine. This has been accomplished with the use of kMech (Yang, C.-R., Shapiro, B. E., Mjolsness, E. D., and Hatfield, G. W. (2005) Bioinformatics 21, in press), a Cellerator (Shapiro, B. E., Levchenko, A., Meyerowitz, E. M., Wold, B. J., and Mjolsness, E. D. (2003) Bioinformatics 19, 677–678) language extension that describes a suite of enzyme reaction mechanisms. Each enzyme mechanism is parsed by kMech into a set of fundamental association-dissociation reactions that are translated by Cellerator into ordinary differential equations. These ordinary differential equations are numerically solved by Mathematica^TM. Any metabolic pathway can be simulated by stringing together appropriate kMech models and providing the physical and kinetic parameters for each enzyme in the pathway. Writing differential equations is not required. The mathematical model of branched chain amino acid biosynthesis in E. coli K12 presented here incorporates all of the forward and reverse enzyme reactions and regulatory circuits of the branched chain amino acid biosynthetic pathways, including single and multiple substrate (Ping Pong and Bi Bi) enzyme kinetic reactions, feedback inhibition (allosteric, competitive, and non-competitive) mechanisms, the channeling of metabolic flow through isozymes, the channeling of metabolic flow via transamination reactions, and active transport mechanisms. This model simulates the results of experimental measurements.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL PROCEDURES RESULTS DISCUSSION REFERENCES

 
Systems biology may be broadly defined as the integration of diverse data into useful biological models that allow scientists to easily observe complex cellular behaviors and predict the outcomes of metabolic and genetic perturbations. As a first step toward the elucidation of the systems biology of the model organism Escherichia coli, we have elected to limit our initial efforts to the development of a mathematical model for the complex but well studied metabolic pathways for the biosynthesis of the branched chain amino acids L-isoleucine, L-valine, and L-leucine.

The biosynthetic pathways for the branched chain amino acids are shown in Fig. 1 (1–3). L-Threonine deaminase (TDA),¹ the first enzyme specific for the biosynthesis of L-isoleucine, is end product-inhibited by L-isoleucine, and -isopropylmalate synthase (IPMS), the first enzyme specific for the biosynthesis of L-leucine, is end product-inhibited by L-leucine. However, because the parallel pathways for L-valine and L-isoleucine biosynthesis are catalyzed by a set of bi-functional enzymes that bind substrates from either pathway, L-valine inhibition of the first enzyme specific for its biosynthesis catalyzed by a single -acetohydroxy acid synthase (AHAS) could compromise the cell for L-isoleucine biosynthesis or result in the accumulation of a toxic metabolic intermediate, -ketobutyrate (KB). This type of regulatory problem is often solved by using multiple isozymes with different substrate preferences that are differentially regulated by multiple end products of parallel pathways. In this case, there are three AHAS isozymes that catalyze the first step of the L-valine and the second step of the L-isoleucine pathway (4). AHAS I has substrate preferences for the condensation of two pyruvate molecules required for L-valine biosynthesis and is end product-inhibited by L-valine (4). AHAS III shows no preference for pyruvate or KB. Although this isozyme can produce intermediates for both L-valine and L-isoleucine, it is inhibited by L-valine (4). The AHAS II isozyme has substrate preferences for the condensation of pyruvate and the KB required for L-isoleucine biosynthesis, and it is not inhibited by any of the branched chain amino acids (4). However, AHAS II is not active in the E. coli. K12 strain (5). Consequently, this strain cannot grow in the presence of high levels of L-valine unless L-isoleucine is also added to the growth medium (6).

View larger version (17K): [in this window] [in a new window]   FIG. 1. Traditional metabolite conversion pathways for the biosynthesis of the branched chain amino acids L-isoleucine, L-valine, and L-leucine. The enzymes involved in the common pathway for branched chain amino acid biosynthesis are TDA (EC 4.3.1.19 [EC] ), AHAS (EC 4.1.3.18 [EC] ), acetohydroxy acid isomeroreductase (IR) (EC 1.1.1.86 [EC] ), dihydroxy acid dehydrase (DAD) (EC 4.2.1.9 [EC] ), TB (EC 2.6.1.42 [EC] ), transaminase C (TC) (EC 2.6.1.66 [EC] ), -isopropylmalate synthase (IPMS) (EC 4.1.3.12 [EC] ), -isopropylmalate isomerase (IPMI) (EC 4.2.1.33 [EC] ), -isopropylmalate dehydrogenase (IPMDH) (EC 1.1.1.85 [EC] ), the L-leucine, L-isoleucine, and L-valine transporter I (LIV-I), and the L-leucine-specific (LS) transporter. The metabolites used were L-threonine (Thr), L-isoleucine (Ile), L-valine (Val), L-leucine (Leu), L-glutamate (Glu), Ala, Pyr, KB, -acetolactate (AL), -acetohydroxybutyrate (AHB), ,-dihydroxy-isovalerate (DHIV), ,-dihydroxy--methylvalerate (DMV), -ketoisovalerate (KIV), -keto--methylvalerate (KMV), -ketoglutarate (KG), -isopropylmalate (IPM), -isopropylmalate (IPM), -ketoisocaproate (KIC), extracellular L-isoleucine (ex-Ile), extracellular L-valine (ex-Val), and extracellular L-leucine (ex-Leu). The italicized items in parentheses refer to abbreviations used in the figure that are not defined in the Abbreviations footnote. Gene names for each enzyme are italicized in the figure. Enzyme reactions are indicated by arrows. Feedback inhibition patterns are indicated by dashed lines. Activation is indicated by a plus sign, and inhibitions are indicated by vertical bars. The line through AHAS II, ilvGM, indicates that this isozyme is not active in E. coli strain K12.

TDA is an allosteric enzyme whose kinetic behavior can be described by the concerted allosteric transition mode of the Monod, Wyman, and Changeux (MWC) model (7, 8). According to the MWC model, TDA can exist in an active state (R) or an inactive state (T) (8, 9). The fraction of active enzyme in the R or T states is determined by the concentrations and relative affinities of the substrate (L-threonine), the inhibitor (L-isoleucine), and the activator (L-valine) for the R and T states.

In addition to these regulatory circuits, the intracellular levels of the branched chain amino acids are influenced by the reversible transamination reactions of each pathway. When the intracellular levels of any of the end product amino acids become high, reverse reactions to their cognate ketoacids are favored; for example, high concentrations of L-valine can be converted to -ketoisovalerate to supplement L-leucine production. In turn, intracellular amino acid levels can be affected by their active transport from the extracellular growth medium. Therefore, the enzyme reactions required for the active transport of the branched chain amino acids into the cell against a concentration gradient are included in our simulations.

	EXPERIMENTAL PROCEDURES

TOP ABSTRACT INTRODUCTION EXPERIMENTAL PROCEDURES RESULTS DISCUSSION REFERENCES

 
The Mathematical Model—Here we use kMech (10), a Cellerator (11) language extension that describes a suite of enzyme reaction mechanisms. Each enzyme mechanism is parsed by kMech into a set of fundamental association-dissociation reactions that are translated by Cellerator into ordinary differential equations (ODEs) that are numerically solved by Mathematica^TM (10). To build a model for a metabolic pathway, users need only to string together appropriate kMech enzyme mechanism models and provide the physical and kinetic parameters for each enzyme. The development of approximation methods for estimating unavailable model parameters such as forward and reverse rate constants (k_f and k_r) from kinetic measurements (K_m and k_cat) is described elsewhere (10).

The detailed kMech models for each of the pathway enzymes (Fig. 1), a Mathematica^TM-executable kMech.m file, and a Mathematica^TM notebook file with detailed kMech inputs and the corresponding ODEs, kinetic rate constants, and initial conditions for solving the ODEs (or its Systems Biology Markup Language (SBML) version) are available at the University of California, Irvine, Institute for Genomics and the Bioinformatics web site (www.igb.uci.edu/servers/sb.html). The PDF version of the Mathematica^TM notebook and a list of reported and optimized enzyme kinetic and physical parameters used in simulations are available in the supplemental data in the on-line version of this article. Cellerator, available at www.aig.jpl.nasa.gov/public/mls/cellerator/feedback.html, is free of charge to academic, United States government, and other nonprofit organizations.

Carbon Flow Channeling—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 focus on conversion between metabolites (metabolic flux) rather than enzyme mechanisms. Although metabolic flux provides valuable information about biomass conversions (12), it cannot simulate, for example, the pathway-specific regulation patterns that control carbon flow channeling through the three AHAS isozymes of the parallel L-isoleucine and L-valine pathways and the final transamination reactions. This level of mathematical modeling requires a detailed understanding of enzyme kinetic mechanisms and regulatory circuits (Fig. 2) as described below.

View larger version (28K): [in this window] [in a new window]   FIG. 2. Enzyme-centric, metabolic pathways for the biosynthesis of the branched chain amino acids L-isoleucine, L-valine, and L-leucine. The abbreviations of enzymes and metabolites are the same as those in Fig. 1. Ovals represent enzyme molecules. White ovals indicate free enzyme states, and shaded ovals indicate intermediate enzyme states with a function group attached to enzymes. Enzyme reactions are indicated by lines with arrowheads. Reversible reactions are indicated by gray lines with arrowheads. Switching between free and intermediate enzyme states are indicated by dashed lines with double arrowheads.

Carbon flow through these isozymes is controlled by the affinities (K_m) of the enzyme intermediates for their second substrates as shown in Fig. 2 (13). For example, the AHAS II enzyme reactions shown in Fig. 1 are described by the two reaction sets designated Reaction 1 and Reaction 2.

View larger version (9K): [in this window] [in a new window]   REACTION 1

View larger version (10K): [in this window] [in a new window]   REACTION 2

View larger version (14K): [in this window] [in a new window]   REACTION 3

View larger version (15K): [in this window] [in a new window]   REACTION 4

View larger version (20K): [in this window] [in a new window]   REACTION 5

View larger version (47K): [in this window] [in a new window]   REACTION 6

Reversible Transamination Mechanism—The pyridoxal 5'-phosphate-dependent transaminase B (TB) enzyme catalyzes the final, reversible step of the biosynthetic pathways of all three of the branched chain amino acids (Figs. 1 and 2). The first step of each of these Ping Pong Bi Bi transamination reactions uses glutamate as an amino donor to form a pyridoxamine-bound enzyme intermediate (TBNH₂, shaded oval in Fig. 2) for the transamination of the three different -ketoacids of each pathway. Carbon flow through TB is controlled by the affinities (K_m) of the enzyme intermediates for their second -ketoacid substrates as shown in Fig. 2. The TB enzyme reactions of Fig. 1 are described by the chemical equations shown in Reactions 7, 8, 9. Because the first substrate reaction with glutamate is the same for all three of the branched chain -ketoacid second substrates, the Mathematica^TM Union operator is once again used to eliminate this redundancy as shown in Reaction 10. Because transamination is reversible, kMech models must be entered in both reaction directions for each of the three branched chain amino acid transaminations and, again, the Union operator is used to eliminate the duplicated second substrate reactions (TBNH₂ + aKG TBNH₂·aKG TB + Glu; TBNH₂ is underlined to indicate that the TB enzyme is in the intermediate state after reacting with the first substrate) of each transamination (long gray arrows in Fig. 2) as shown in Reaction 11.

View larger version (10K): [in this window] [in a new window]   REACTION 7

View larger version (10K): [in this window] [in a new window]   REACTION 8

View larger version (10K): [in this window] [in a new window]   REACTION 9

View larger version (19K): [in this window] [in a new window]   REACTION 10

View larger version (17K): [in this window] [in a new window]   REACTION 11

Allosteric Regulation—Threonine deaminase is an allosteric enzyme whose kinetic behavior can be described by the concerted allosteric transition model of Monod, Wyman, Changeux known as the MWC model (7, 8). According to the MWC model, TDA can exist in an active state (R) or an inactive state (T) (8, 9). The fraction of enzyme in the R or T state is determined by the concentrations and relative affinities of the substrate (L-threonine), the inhibitor (L-isoleucine), and the activator (L-valine) for each state. This model is described by two equations, designated Equations 1 and 2,

(Eq. 1)

(Eq. 2)

in which S, I, and A are substrate, inhibitor, and activator concentrations, respectively, K_m, K_i, and K_a are their respective dissociation constants, n is the number of substrate and effector ligand binding sites, c is the ratio of the affinity of the substrate for the catalytically active R state and the inhibited T state, L₀ is the equilibrium constant (allosteric constant) for the R and T states in the absence of ligands, v_o is the initial reaction velocity, and V_max is the maximal reaction velocity.

Equation 1 describes the fraction of the enzyme in the catalytically active state (R) as a function of substrate and effector concentrations. Equation 2 describes the fractional saturation (Y_f = v_o/V_max) of the enzyme occupied by substrate as a function of substrate and effector concentrations (7).

We have recently described implementation of the MWC model in Cellerator (10). Experimental values of the kinetic parameters and ligand concentrations listed above are most often available in the literature. However, values of c and L₀ are often not available. These values can be calculated by fitting substrate saturation curves in the presence and absence of several inhibitor concentrations (10, 14, 15).

Approximation of Intracellular Enzyme Concentrations—With few exceptions, intracellular enzyme concentrations are not available. However, with careful experimental documentation, these concentrations can be approximated from the yields and specific activities of purified enzymes. For example, calculations based on purification tables in the literature suggest that the intracellular concentration of TDA is 4 µM (16). Furthermore, recent experiments have shown a positive correlation between mRNA levels measured with DNA microarrays and protein abundance in both E. coli (17) and yeast cells (18, 19). Thus, the intracellular levels of the remaining enzymes of the branched chain amino acid biosynthetic pathway can be inferred from the calculated intracellular level of TDA and the relative mRNA levels of the other branched chain amino acid biosynthetic enzymes using DNA microarray data (20). The data in supplemental Table I in the on-line version of this article demonstrate that this is a reasonable method. Indeed, simulations using intracellular enzyme concentrations inferred in this manner produce experimentally observed steady-state pathway intermediate and end product levels (21, 22), usually within 2-fold to one-half adjustments of these inferred values.

Optimization of Model Parameters—A list of reported enzyme kinetic and physical parameters needed to solve the differential equations for the simulations reported here, and their literature sources are available in supplemental Table I in the on-line version of this article. The optimized values to simulate known steady-state intracellular levels of pathway substrates, intermediates, and end products are also listed for comparison. In brief, for each enzyme there are at least three parameters needed, namely the total enzyme concentration (E_T), the K_m for each substrate, and the k_cat for each enzyme reaction. For enzymes with additional regulatory mechanism, extra parameters such as the K_i for each inhibitor and the K_a for each activator also are required.

In the initial simulation, E_T values were inferred from microarray data as described above, and the K_m and k_cat values were obtained from in vitro enzyme kinetic data of purified enzymes with the exception of transaminase C and -isopropylmalate isomerase, where empirical values were used because of a lack of experimental data. These values were manually adjusted to match the published in vivo steady-state levels of intermediate and end product metabolites (21, 22). Interestingly, the inferred E_T and in vitro K_m values work quit well, because the adjustments are usually within 2-fold to one-half of the initial values. However, because many variables can influence in vitro measurements, including the relative activities of purified enzymes, larger corrections were sometimes necessary for the estimation of k_cat values (5 of 9 enzymes). Once the mathematical model was optimized with the parameters reported in supplemental Table I in the on-line version of this article, it was used without further adjustment for the simulations of the metabolic and genetic perturbations reported below.

	RESULTS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL PROCEDURES RESULTS DISCUSSION REFERENCES

 
Computational Modeling of the Dynamics of Carbon Flow through the Branched Chain Amino Acid Biosynthetic Pathways of E. coli K12—The three interacting metabolic pathways simulated here consist of 11 enzymes, 18 metabolic intermediates, and three enzyme cofactors. The mathematical model for this metabolic system consists of 105 ODEs with 110 association and dissociation rate constants and 52 catalytic rate constants. The enzymes of these interacting pathways employ three distinct enzyme mechanisms (simple catalytic, Bi Bi, and Ping Pong Bi Bi) that are regulated by allosteric, competitive, or noncompetitive inhibition mechanisms. As described under "Experimental Procedures," the physical parameters for these models have been obtained directly from the literature, calculated from data in the literature, or estimated by fitting experimental data (supplemental Table I in the on-line version of this article). Relative intracellular enzyme levels have been inferred from enzyme purification and DNA microarray data (20).

The steady-state levels for the thirteen pathway intermediates and end products are shown in Fig. 3. Steady-state enzyme activity levels were optimized to properly channel the steady-state flow of intermediates through these pathways to match reported in vivo levels of pathway intermediates and end products (21, 22). The detailed kMech inputs, corresponding ODEs, kinetic rate constants, and initial conditions for solving the ODEs are presented in supplemental Fig. 1, available in the on-line version of this article.

View larger version (20K): [in this window] [in a new window]   FIG. 3. Simulated flow of carbon through the branched chain amino acid biosynthetic pathways of E. coli K12. The graphical insets show the approach (minutes) to steady-state (µM) synthesis and utilization of the substrates, intermediates, and end products of the pathways. The intermediates are abbreviated as described in the legend of Fig. 1. The starting substrates L-threonine and pyruvate are supplied at rates to maintain constant levels of 520 and 1000 µM, respectively. L-Glutamate (Glu) and Ala for the transamination reactions are supplied at a rate to maintain constant levels of 2000 µM each. For the acetohydroxy acid isomeroreductase reaction, NADPH is supplied at a rate to maintain a constant level of 1000 µM. For the -isopropylmalate synthase (IPMS) reaction, acetyl-CoA is supplied at a rate to maintain a constant level of 1000 µM. The beginning substrates (L-threonine and pyruvate) levels, as well as the end product (L-isoleucine, L-valine, and L-leucine) levels, agree with measured intracellular values (21, 22). Where available, the ranges of reported values for pathway intermediate and end product levels in cells growing in a glucose minimal salts medium are shown in parentheses (µM) in the inset graphs.

View larger version (20K): [in this window] [in a new window]   FIG. 4. Allosteric regulation of L TDA. A, the fraction of TDA in the active R state. At t = 0 and an initial L-threonine concentration of 520 µM, 65% of the TDA enzyme is in the active R state. As L-isoleucine accumulates, TDA is rapidly end product-inhibited and, as L-valine accumulates, this inhibition is slowly countered until, at steady state, only 5.5% of the total enzyme is in the active R state. B, the fractional saturation of TDA with L-threonine (vo/Vmax). At t = 0 and an initial L-threonine (Yf) concentration of 520 µM, 8% of the total enzyme is saturated with L-threonine. At a final steady-state level of end product synthesis, it is only 1.2% saturated with L-threonine.

Responses to Metabolic and Genetic Perturbations
L-Valine Growth Inhibition of E. coli K12 Is Due to KB Accumulation, Not L-Isoleucine Starvation—It is well known that adding L-valine at a final concentration of 1 mM to a culture of E. coli K12 cells growing in a glucose minimal salts medium inhibits their growth and that this L-valine inhibition can be reversed by L-isoleucine addition (6). Because the AHAS I and AHAS III isozymes of E. coli K12 strains are inhibited by L-valine and because the ilvG gene for AHAS II in E. coli K12 strains contains a frameshift mutation that destroys AHAS II activity (5), it was assumed that L-valine inhibition of AHAS I and AHAS III might inhibit growth by interfering with L-isoleucine biosynthesis. However, later studies demonstrated that the intracellular L-isoleucine level is not suppressed by L-valine because its biosynthesis is sustained, even at saturating L-valine concentrations, by AHAS III that remains 15–20% active (13, 23) and by the L-valine activation of TDA that shuttles more substrate into the L-isoleucine pathway. Indeed, the simulation in Fig. 5A shows that, in the presence of extracellular L-valine, the intracellular L-isoleucine level in fact accumulates nearly 9-fold (Fig. 5A). At the same time, the pathway precursor of L-isoleucine, KB, increases 4-fold (Fig. 5B). This build-up of KB is caused by L-valine activation of TDA (Fig. 5C), which increases its production, and by L-valine inhibition of the AHAS I and AHAS III isozymes, which reduces its consumption. It is now known that this KB accumulation is toxic to cells because of its ability to inhibit the glucose PTS transport system (24, 25). Thus, as reproduced by our simulations, L-valine growth inhibition of E. coli K12 is not a consequence of L-isoleucine starvation.

View larger version (19K): [in this window] [in a new window]   FIG. 5. Simulated effects of excess L-valine on branched chain amino acid biosynthesis in E. coli K12. Conditions described in Fig. 2 were used for the simulations presented here, except that excess extra-cellular L-valine was added at a rate sufficient to be maintained at a concentration of 1 mM. The data in panel A show that, as described under "Results," excess L-valine increases rather than inhibits L-isoleucine biosynthesis. The data in panel B show that excess L-valine also causes a 4-fold increase in the intracellular accumulation of KB, which is restored to control levels by the extracellular addition of 500 µM L-isoleucine. The data in panel C show that the accumulation of KB observed in the presence of excess L-valine coincides with the conversion of nearly 18% of the cellular TDA to a catalytically active R state and that the subsequent extracellular addition of 500 µM L-isoleucine reverses this transition to the control level (Fig. 3A).

Simulating the Metabolic Engineering of an L-Isoleucine Overproducing E. coli K12 Strain—An obvious goal of modeling biological systems is to facilitate metabolic engineering for the commercial production of specialty chemicals such as amino acids. In the past, this has been largely accomplished by genetic manipulation and selection methods. For example, a common strategy used to overproduce an amino acid has been to isolate a strain with a feedback-resistant mutation in the gene for the first enzyme for the biosynthesis of that amino acid. Here we use our model to determine the effects of a feedback-resistant TDA for the overproduction of L-isoleucine. We can simulate a TDA feedback-resistant mutant strain (TDA^R) by increasing the K_i for L-isoleucine to a large number, (e.g. 100,000 µM). The simulation in Fig. 6A shows that in the absence of L-isoleucine inhibition, the activator and substrate ligands drive nearly 100% of cellular feedback-resistant TDA^R to the active R state compared with the wild type enzyme that is only 6% present in the active R state. However, despite this increased amount of enzyme in the active state, the data in Fig. 6B show that AHAS III is able to support only a 5 to 6-fold increase in L-isoleucine production in a feedback-resistant E. coli K12 compared with a wild type strain. At the same time, the steady-state level of the AHAS III substrate, KB, is increased 40-fold (Fig. 6C). This is because E. coli K12 does not have an active AHAS II isozyme that favors the condensation of pyruvate and KB for L-isoleucine production; thus, KB would accumulate to toxic levels. These simulation results suggest that in order to overproduce L-isoleucine, the KB accumulation must be reduced. The results in Fig. 6D show that restoring a wild type AHAS II isozyme and simulating an attenuator mutation that elevates the levels of all of the enzymes of the L-isoleucine and L-valine parallel pathways 11-fold (26) both avoids buildup of KB and subsequent pathway intermediates (Fig. 6C) and results in a 40-fold increase in L-isoleucine production. These simulated results, which show that high level overproduction of L-isoleucine in E. coli requires a functional AHAS II isozyme and a de-attenuated genetic background (ilvGMEDA-att^–), agree with experiments performed by Hashiguchi et al. of the Ajinomoto Co., Tokyo, Japan (27).

View larger version (22K): [in this window] [in a new window]   FIG. 6. Simulation of an E. coli K12 strain that overproduces L-isoleucine. The simulation conditions described in the Fig. 2 legend were used for the simulations presented here, except that a L-threonine deaminase feedback-resistant mutant, TDAR, was simulated by increasing its Ki for L-isoleucine to 100,000 µM, and the ilvGMEDA operon attenuator mutant (ilvGMEDA-att–) was simulated by increasing TDA, AHAS II, acetohydroxy acid isomeroreductase, dihydroxy acid dehydrase, and TB total enzyme levels 11-fold (26) (3). The simulation in panel A shows that the effect of the feedback-resistant TDA mutant TDAR is to allow the positive effector ligands L-threonine and L-valine to transition nearly 100% of the TDA enzyme to the active R state. The simulation results in panel B show that L-isoleucine production in the TDAR mutant is 5–6-fold increased. The simulation in panel C demonstrates that in the TDAR K12 mutant, the intermediate, KB, accumulates to a level 40-fold higher than in a wild type K12 strain; however, when the AHAS II isozyme is restored and the bi-functional enzymes of the L-isoleucine and L-valine pathways are genetically de-repressed 11-fold (ilvGMEDA-att–), KB accumulation is relieved (panel C) and L-isoleucine synthesis is increased more than 40-fold over the wild-type K12 level (panel D).

View larger version (16K): [in this window] [in a new window]   FIG. 7. An acetohydroxy acid isomeroreductase mutant (ilvC) E. coli K12 strain is auxotrophic for L-isoleucine and L-valine but not for L-leucine. The simulation conditions described in Fig. 2 were used for the simulations presented here, except that the initial concentration of acetohydroxy acid isomeroreductase was set to 0 to simulate an ilvC mutation, and extracellular L-valine and L-isoleucine were supplied at a level of 500 µM each. The results show that -ketoisovalerate (KIV) (panel A) and L-leucine (panel B) are produced in an ilvC strain in the presence of extracellular L-valine and L-isoleucine.

	DISCUSSION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL PROCEDURES RESULTS DISCUSSION REFERENCES

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{at}uci.edu. To whom correspondence on biology questions should be addressed. E-mail: gwhatfie{at}uci.edu.

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

	ACKNOWLEDGMENTS

 
We are appreciative of helpful advice from Donald F. Senear.

	REFERENCES

TOP ABSTRACT INTRODUCTION EXPERIMENTAL PROCEDURES RESULTS DISCUSSION REFERENCES

 

1. Umbarger, H. E. (1978) Annu. Rev. Biochem. 47, 532–606[Medline] [Order article via Infotrieve]
2. Umbarger, H. E. (1996) in Escherichia coli and Salmonella: Cellular and Molecular Biology (Neidhardt, F. C., Curtiss, R., III, Ingraham, J. L., Lin, E. C. C., Low, K. B., Magasanik, B., Reznikoff, W. S., Riley, M., Schaechter, M., and Umbarger, H. E., eds) 2nd Ed., pp. 442–457, American Society for Microbiology Press, Washington, D. C.
3. Hatfield, G. W., and Benham, C. J. (2002) Annu. Rev. Genet. 36, 175–203[CrossRef][Medline] [Order article via Infotrieve]
4. Barak, Z., Chipman, D. M., and Gollop, N. (1987) J. Bacteriol. 169, 3750–3756[Abstract/Free Full Text]
5. Lawther, R. P., Calhoun, D. H., Gray, J., Adams, C. W., Hauser, C. A., and Hatfield, G. W. (1982) J. Bacteriol. 149, 294–298[Abstract/Free Full Text]
6. Umbarger, H. E., and Brown, B. (1955) J. Bacteriol. 70, 241–248[Free Full Text]
7. Monod, J., Wyman, J., and Changeux, J. P. (1965) J. Mol. Biol. 12, 88–118[Medline] [Order article via Infotrieve]
8. Hatfield, G. W., and Umbarger, H. E. (1970) J. Biol. Chem. 245, 1742–1747[Abstract/Free Full Text]
9. Changeux, J. P. (1963) Cold Spring Harbor Symp. Quant. Biol. 28, 497–504
10. Yang, C.-R., Shapiro, B. E., Mjolsness, E. D., and Hatfield, G. W. (2005) Bioinformatics 21, in press
11. Shapiro, B. E., Levchenko, A., Meyerowitz, E. M., Wold, B. J., and Mjolsness, E. D. (2003) Bioinformatics 19, 677–678[Abstract/Free Full Text]
12. Edwards, J. S., Ibarra, R. U., and Palsson, B. O. (2001) Nat. Biotechnol. 19, 125–130[CrossRef][Medline] [Order article via Infotrieve]
13. Herring, P. A., McKnight, B. L., and Jackson, J. H. (1995) Biochem. Biophys. Res. Commun. 207, 48–54[CrossRef][Medline] [Order article via Infotrieve]
14. Segel, I. H. (1993) Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady State Enzyme Systems, p. 427, John Wiley & Sons, Inc., New York
15. Hatfield, G. W. (1970) The Regulation of L-Threonine Deaminase in Bacillus subtilis by Repression and Endproduct Inhibition. Ph.D. thesis, pp. 103–112, Purdue University
16. Calhoun, D. H., Rimerman, R. A., and Hatfield, G. W. (1973) J. Biol. Chem. 248, 3511–3516[Abstract/Free Full Text]
17. Arfin, S. M., Long, A. D., Ito, E. T., Tolleri, L., Riehle, M. M., Paegle, E. S., and Hatfield, G. W. (2000) J. Biol. Chem. 275, 29672–29684[Abstract/Free Full Text]
18. Futcher, B., Latter, G. I., Monardo, P., McLaughlin, C. S., and Garrels, J. I. (1999) Mol. Cell. Biol. 19, 7357–7368[Abstract/Free Full Text]
19. Ideker, T., Thorsson, V., Ranish, J. A., Christmas, R., Buhler, J., Eng, J. K., Bumgarner, R., Goodlett, D. R., Aebersold, R., and Hood, L. (2001) Science 292, 929–934[Abstract/Free Full Text]
20. Hung, S. P., Baldi, P., and Hatfield, G. W. (2002) J. Biol. Chem. 277, 40309–40323[Abstract/Free Full Text]
21. Epelbaum, S., LaRossa, R. A., VanDyk, T. K., Elkayam, T., Chipman, D. M., and Barak, Z. (1998) J. Bacteriol. 180, 4056–4067[Abstract/Free Full Text]
22. Quay, S. C., Dick, T. E., and Oxender, D. L. (1977) J. Bacteriol. 129, 1257–1265[Abstract/Free Full Text]
23. Jackson, J. H., Herring, P. A., Patterson, E. B., and Blatt, J. M. (1993) Biochimie (Paris) 75, 759–765
24. Daniel, J., Dondon, L., and Danchin, A. (1983) Mol. Gen. Genet. 190, 452–458[CrossRef][Medline] [Order article via Infotrieve]
25. Danchin, A., Dondon, L., and Daniel, J. (1984) Mol. Gen. Genet. 193, 473–478[CrossRef][Medline] [Order article via Infotrieve]
26. Adams, C. W., Rosenberg, M., and Hatfield, G. W. (1985) J. Biol. Chem. 260, 8538–8544[Abstract/