Kinetic modeling to optimize pentose fermentation in Zymomonas mobilis

Zymomonas mobilis engineered to express four heterologous enzymes required for xylose utilization ferments xylose along with glucose. A network of pentose phosphate (PP) pathway enzymatic reactions interacting with the native glycolytic Entner Doudoroff (ED) pathway has been hypothesized. We have investigated this putative reaction network by developing a kinetic model incorporating all of the enzymatic reactions of the PP and ED pathways, including those catalyzed by the heterologous enzymes. Starting with the experimental literature on in vitro characterization of each enzymatic reaction, we have developed a kinetic model to enable dynamic simulation of intracellular metabolite concentrations along the network of interacting PP and ED metabolic pathways. This kinetic model is useful for performing in silico simulations to predict how varying the different enzyme concentrations will affect intracellular metabolite concentrations and ethanol production rate during continuous fermentation of glucose and xylose mixtures. Among the five enzymes whose concentrations were varied as inputs to the model, ethanol production in the continuous fermentor was optimized when xylose isomerase (XI) was present at the highest level, followed by transaldolase (TAL). Predictions of the model that the interconnecting enzyme phosphoglucose isomerase (PGI) does not need to be overexpressed were recently confirmed through experimental investigations. Through such systematic analysis, we can develop efficient strategies for maximizing the fermentation of both glucose and xylose, while minimizing the expression of heterologous enzymes.     

Computer simulation of purine metabolism

A computer model of purine metabolism, including catabolism, salvage pathways and interconversion among nucleotides, is given. Steady-state rate equations corresponding to metabolic enzymes are written based on information from the literature about their kinetic behaviour. Numerical integration of this set of equations is performed employing selected parameters taken from the literature. After stabilization of purine compound concentrations is reached, simulation of enzyme deficit and enzyme overproduction is carried out. The latter is calculated by varying specified maximum velocities in the numerical integration. A pattern of intermediate metabolite concentrations is found. These results form a basis for the comparison of normal patterns or patterns reflecting the effects of inborn errors of metabolism. The aim of this paper is to demonstrate the usefulness of this computer simulation method in complex metabolism pathways.     

Control in channelled pathways. A matrix method calculating the enzyme control coefficients

The usual equations expressing the enzyme control coefficients (quantitative indicators of 'global' control properties of a pathway) via the elasticity coefficients (reflecting local kinetic properties of an enzyme reaction), cannot be applied to a variety of 'non-ideal' pathways, in particular to pathways with metabolic channelling. Here we show that the relationship between the control and elasticity coefficients can be obtained by considering such a metabolic pathway as a network of elemental chemical conversions (steps). To calculate the control coefficients of enzymes one should first determine the elasticity coefficients of such elemental steps and then take their appropriate combinations. Although the method is illustrated for a channelled pathway it can be used for any non-ideal pathway including those with high enzyme concentrations where the sequestration of metabolites by enzymes cannot be neglected.     

Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes

The metabolic control theory developed by Kacser, Burns, Heinrich, and Rapoport is briefly outlined, extended, and transformed so as optimally to address some biotechnological questions. The extensions include (i) a new theorem that relates the control of metabolite concentrations by enzyme activities to flux ratios at branches in metabolic pathways; (ii) a new theorem that does the same for the control of the distribution of the flux over two branches; (iii) a method that expresses these controls into properties (the so-called elasticity coefficients) of the enzymes in the pathway; and (iv) a theorem that relates the effects of changes in metabolite concentrations on reaction rates to the effects of changes in enzyme properties on the same rates. Matrix equations relating the flux control and concentration control coefficients to the elasticity coefficients of enzymes in simple linear and branched pathways incorporating feedback are given, together with their general solutions and a numerical example. These equations allow one to develop rigorous criteria by which to decide the optimal strategy for the improvement of a microbial process. We show how this could be used in deciding which property of which enzyme should be changed in order to obtain the maximal concentration of a metabolite or the maximal metabolic flux.     

Optimizing the sensitivity of enzyme systems to perturbations from equilibrium

Algebraic derivations and numerical examples illustrate how metabolite pool sizes and enzyme rate constants influence the rate at which a multireactant enzyme system, initially poised in a near-equilibrium steady state, responds to small perturbations in the concentrations of the reactants. Certain enzymes, such as those employing the ordered bi bi catalytic mechanism, become relatively insensitive to perturbations when the reactants are all present at high concentrations. Other enzymes, such as those employing the ping-pong bi bi mechanism, are most sensitive to perturbations at high reactant concentrations. The ratio of the reactant concentrations to one another significantly alters sensitivity to perturbations; equations are presented for calculation of the reactant concentrations yielding maximal sensitivity to perturbations. Natural selection could choose metabolite pool sizes and enzyme rate constants which would optimize the performance of these systems, but changing metabolic loads (naturally or experimentally imposed) constantly alter the sensitivity of these systems to perturbations, changing the relative strengths of various connections in metabolic control networks.     

Mechanism of activation by anions of phosphoglycolate phosphatases from spinach and human red blood cells

Phosphoglycolate phosphatases from spinach and human red blood cells show a number of common features not often found in enzymes. Both enzymes are activated more than 50-fold by millimolar concentrations of Cl-. Other inorganic anions and a number of carboxylic acids also activate. Each enzyme has limited substrate specificity yet each hydrolyzes P-glycolate and ethyl-P with the same maximal velocity. L-P-lactate is only a good substrate for the red cell enzyme. With both enzymes initial rate data obtained by varying both the P-glycolate and Cl- give parallel line double reciprocal plots. Similar experiments with ethyl-P as substrate give intersecting lines with both enzymes. The likelihood that both classes of substrates are acting at the same site is strengthened by the results of inhibition studies with alternative substrates and the constancy of inhibition constants for glycolate with all substrates for a given enzyme. For each substrate the experimentally observed variation in V/Km with different activators is small, suggesting that the enzyme has an ordered mechanism with the phosphorylated substrate reacting first. A mechanism that is consistent with all of the data is presented.     

Control of the rate of photosynthetic carbon dioxide fixation

A simplified model of the reductive pentose phosphate pathway of photosynthesis is analysed in order to quantify the degree to which each of the constituent reactions controls the rate of CO₂ fixation (given by the control coefficient). The analysis focuses on the four largely irreversible reactions of the cycle together with the first irreversible reaction in the sucrose and starch synthetic pathways. The model assumes that the other reactions are at equilibrium. The photorespiratory and electron transport systems are not included in the model. The analysis demonstrates that: (1) an analytical approach can be used to investigate the distribution of flux control in autocatalytic and moiety-conserved cycles; (2) measurements of enzyme kinetic parameters and certain fluxes and substrate concentrations can be used to solve the equations defining the enzyme control coefficients; (3) the conservation of total stromal phosphate and the intricate regulatory mechanisms of the photosynthetic system result in a relationship between the control coefficients that is complex and may defy any intuitive assessment of ‘rate limitation’; (4) ribulose-1,5-bisphosphate carboxylase / oxygenase may, under certain conditions, be a major controller of the rate of CO₂ fixation and, by regulating the concentration of ribulose 1,5-bisphosphate, may be important in governing the ratio of organic to inorganic phosphate in the stroma; (5) the other enzymes may also serve an important role in determining the distribution of phosphate between organic and inorganic species because they catalyze reactions at the branch points between starch and sucrose synthesis and ribulose 1,5-bisphosphate regeneration; (6) these enzymes that catalyze ‘branch-pint’ reactions may have negative control coefficients because of their ability to reduce the total concentration of cycle intermediates; (7) an approach combining the use of the equations presented in this paper and flux and substrate concentration measurements may be adequate for determining the control coefficients of several enzymes of the reductive pentose phosphate pathway.     

Mathematical modelling of glycolysis and adenine nucleotide metabolism of human erythrocytes. I. Reaction-kinetic statements, analysis of in vivo state and determination of starting conditions for in vitro experiments

A mathematical model of energy metabolism of human red cells is presented, which includes besides the glycolytic reactions the adenine nucleotide metabolism. The model is based on the network of chemical reactions, the thermodynamic equilibrium constants of fast reversible reactions and on the kinetic equations for irreversible enzyme reactions. The model consists of a system of 16 differential equations and allows the mathematical evaluation of metabolic levels in the steady state of energy metabolism corresponding to the in vivo state erythrocytes with the kinetic data for the enzymes derived from in vitro experiments. The dependence of the levels of metabolites in the steady state on the activity of some enzymes is analysed to characterize the regulatory properties of the system. The comparison of the steady state levels of the model with experimental data makes it possible to estimate values of some controversial enzyme parameters. Estimates of the kinetic parameters of the following intracellular processes are presented: 1) rate constant of AMP-phosphatase, 2) maximum rate of adenylate deaminase, 3) activity of adenine phosphoribosylpyrophosphate transferase and 4) adenosine transport through the cell membrane. The simulation of the preparatory phase before incubation of erythrocytes indicates, that the model also permits to compute the time course of changes of levels of metabolites. To solve the initial problem the stiff differential equation system is integrated numerically by an efficient program without the application of the quasi-steady-state approximation.     

Continuous modeling of metabolic networks with gene regulation in yeast and in vivo determination of rate parameters

A continuous model of a metabolic network including gene regulation to simulate metabolic fluxes during batch cultivation of yeast Saccharomyces cerevisiae was developed. The metabolic network includes reactions of glycolysis, gluconeogenesis, glycerol and ethanol synthesis and consumption, the tricarboxylic acid cycle, and protein synthesis. Carbon sources considered were glucose and then ethanol synthesized during growth on glucose. The metabolic network has 39 fluxes, which represent the action of 50 enzymes and 64 genes and it is coupled with a gene regulation network which defines enzyme synthesis (activities) and incorporates regulation by glucose (enzyme induction and repression), modeled using ordinary differential equations. The model includes enzyme kinetics, equations that follow both mass-action law and transport as well as inducible, repressible, and constitutive enzymes of metabolism. The model was able to simulate a fermentation of S. cerevisiae during the exponential growth phase on glucose and the exponential growth phase on ethanol using only one set of kinetic parameters. All fluxes in the continuous model followed the behavior shown by the metabolic flux analysis (MFA) obtained from experimental results. The differences obtained between the fluxes given by the model and the fluxes determined by the MFA do not exceed 25% in 75% of the cases during exponential growth on glucose, and 20% in 90% of the cases during exponential growth on ethanol. Furthermore, the adjustment of the fermentation profiles of biomass, glucose, and ethanol were 95%, 95%, and 79%, respectively. With these results the simulation was considered successful. A comparison between the simulation of the continuous model and the experimental data of the diauxic yeast fermentation for glucose, biomass, and ethanol, shows an extremely good match using the parameters found. The small discrepancies between the fluxes obtained through MFA and those predicted by the differential equations, as well as the good match between the profiles of glucose, biomass, and ethanol, and our simulation, show that this simple model, that does not rely on complex kinetic expressions, is able to capture the global behavior of the experimental data. Also, the determination of parameters using a straightforward minimization technique using data at only two points in time was sufficient to produce a relatively accurate model. Thus, even with a small amount of experimental data (rates and not concentrations) it was possible to estimate the parameters minimizing a simple objective function. The method proposed allows the obtention of reasonable parameters and concentrations in a system with a much larger number of unknowns than equations. Hence a contribution of this study is to present a convenient way to find in vivo rate parameters to model metabolic and genetic networks under different conditions.     

Evolutionary optimization of the catalytic efficiency of enzymes.

1. The rate equation for a generalized Michaelian type of enzymic reaction mechanism has been analyzed in order to establish how the mechanism should be kinetically designed in order to optimize the catalytic efficiency of the enzyme for a given average magnitude of true and apparent first-order rate constants in the mechanism at given concentrations of enzyme, substrate and product. 2. As long as on-velocity constants for substrate and product binding to the enzyme have not reached the limiting value for a diffusion-controlled association process, the optimal state of enzyme operation will be characterized by forward (true and apparent) first-order rate constants of equal magnitude and reverse rate constants of equal magnitude. The drop in free energy driving the catalysed reaction will occur to an equal extent for each reaction step in the mechanism. All internal equilibrium constants will be of equal magnitude and reflect only the closeness of the catalysed reaction to equilibrium conditions. 3. When magnitudes of on-velocity constants for substrate and product binding have reached their upper limits, the optimal kinetic design of the reaction mechanism becomes more complex and has to be established by numerical methods. Numerical solutions, calculated for triosephosphate isomerase, indicate that this particular enzyme may or may not be considered to exhibit close to maximal efficiency, depending on what value is assigned to the upper limit for a ligand association rate constant. 4. Arguments are presented to show that no useful information on the evolutionary optimization of the catalytic efficiency of enzymes can be obtained by previously taken approaches that are based on the application of linear free-energy relationships for rate and equilibrium constants in the reaction mechanism.     

The computerized derivation of steady-state rate equations for enzyme kinetics.

A FORTRAN 77 program is described for the derivation of steady-state rate equations for enzyme kinetics. Input is very simple and consists of the two enzyme forms and the two rate constants for each step in the mechanism. The program may be run interactively or off-line. The results are produced after collecting together the algebraic coefficients of like concentration terms, taking account of sign. A fully interactive BASIC version running on a BBC Microcomputer is also available. Details of the programs have been deposited as Supplementary Publication SUP 50126 (45 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies may be obtained as indicated in Biochem. J. (1984) 217, 5.     

An optimized algorithm for flux estimation from isotopomer distribution in glucose metabolites

MOTIVATION: Analysis of the conversion of (13)C glucose within the metabolic network allows the evaluation of the biochemical fluxes in interconnecting metabolic pathways. Such analyses require solving hundreds of equations with respect to individual isotopomer concentrations, and this assumes applying special software even for constructing the equations. The algorithm, proposed by others could be improved. METHOD: A C-code linked to the program written in Mathematica (Wolfram Research Inc.), constructs and solves differential equations for all isotopomer concentrations, using the general enzyme characteristics (K(m), equilibrium constant, etc.). This code uses innovative algorithm of determination for the isotopomers-products, thus essentially decreasing the computation time. Feasible metabolic fluxes are provided by the parameters of enzyme kinetics found from the data fitting. RESULTS: The software effectively evaluates metabolic fluxes based on the measured isotopomer distribution, as was illustrated by the analysis of glycolysis and pentose phosphate cycle. The mechanism of transketolase and transaldolase catalysis was shown to induce a specific kind of isotopomer re-distribution, which, despite the significance of its effect, usually is not taken into account. AVAILABILITY: The software could be freely downloaded from the site: http://bq.ub.es/bioqint/label_distribution/.     

Measuring in vivo elasticities of Calvin cycle enzymes: network structure and patterns of modulations

To measure the kinetics of enzymes, the proteins are usually assayed in vitro after isolation from their parent organisms. We make an attempt to show how one might determine enzyme elasticities in an intact system by a multiple modulation approach. Certain target enzymes are modulated in their activities and the changes in metabolite concentrations and flux rates upon the modulations are used to calculate the enzyme elasticities. Central to this approach is that the modulations must be independent of each other, and an algorithm is developed for finding all independent modulations that allow determining the elasticities of a given enzyme. This approach is applied to a mass-action model of the Calvin cycle. The goal is to determine the elasticities of as many enzymes as possible by modulating the activities of as few of them as possible. It is shown that the elasticities of 20 (out of 22) Calvin cycle enzymes can be determined by modulating just five reactions. Moreover, visualization of independence of modulations may be used to decompose the Calvin cycle into several sections that are independent of each other regarding flow of matter and information.     

Substrate-containing gel electrophoresis: sensitive detection of amylolytic, nucleolytic, and proteolytic enzymes

Electrophoresis of hydrolytic enzymes under nondenaturing conditions on acrylamide gels containing the appropriate high-molecular-weight substrates entrapped on the gel has been explored as a general method for sensitive enzyme resolution and detection. Under electrophoresis conditions of optimal enzyme activity, the enzymes may bind tightly to the fixed substrate and can only migrate in the electrophoretic field as the substrate is hydrolyzed. When the gels after electrophoresis in this “binding mode” are stained with substrate-detecting reagents, clear tracks of enzyme migration are observed, and the length of each track is a function of the amount of enzyme present in that track. Multiple forms of a given enzyme activity have not been and are not likely to be observed under these conditions. Under electrophoresis conditions of minimal (or suboptimal) enzyme activity, the enzymes do not bind to the fixed substrate and their mobility in the electrophoretic field does not appear to be significantly affected by the presence of substrate. After electrophoresis in this “nonbinding mode” the gels are incubated under conditions of optimal enzyme activity to allow substrate hydrolysis to take place before they are stained with substrate-detecting reagents, and active enzymes are detected as clear bands. Multiple forms of a given activity which were resolved during electrophoresis in the nonbinding mode are reflected by the presence of individual bands. The substrate-containing gel electrophoresis technique does not appear to be amenable to precise quantification of enzymes. By comparing the length of the clear tracks or the degree of staining of the activity bands for a range of enzyme concentrations, however, it is possible to establish the smallest amount of enzyme that can unequivocally be detected under a given set of conditions; from such studies we estimate that the sensitivity of detection with the substrate-containing gel electrophoresis technique can be orders of magnitude better than that obtained with other methods. The levels of detection observed in the work presented here were about 50 pg for α-amylase run on starch-containing gels, 1 pg to 1 ng for nucleases run on DNA- or RNA-containing gels, and 100 pg to 10 ng for 11 different pure and crude protease preparations run on gels containing heat-denatured bovine serum albumin.     

Use of a multiple-enzyme/multiple-reagent assay system to quantify activity levels in samples containing mixtures of matrix metalloproteinases

Matrix metalloproteinases (MMPs) are a family of enzymes that are up-regulated in many diseases, including osteoarthritis (OA) and rheumatoid arthritis (RA). Here we report on a novel technique that can be used to simultaneously measure activity levels for a panel of enzymes, such as the MMPs. The technique, termed the multiple-enzyme/multiple-reagent assay system (MEMRAS), relies on the use of reagents such as substrates with varying selectivity profiles against a group of enzymes. When reaction rates are measured by following a change in fluorescence with time, for mixtures of enzymes, an equation with unknown concentrations for each activity is generated for each reagent used. Simultaneously solving the set of equations leads to a solution for the unknown concentrations. We have applied this mathematical technique to measure activity levels for mixtures of MMPs such as collagenase 3 and gelatinase A. In addition, because we were most interested in determining collagenase 3 levels as a potential biological marker for OA, we developed highly selective substrates for this enzyme by using results found in previous bacteriophage substrate-mapping experiments. Some of the best substrates tested have specific activities for collagenase 3 that are 37,000-, 17,000-, 90-, and 200-fold selective over stromelysin 1, collagenase 1, and gelatinases A and B, respectively.     

A generalized theory of the transition time for sequential enzyme reactions.

In a sequence of coupled enzyme reactions the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. Provided that a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaelis-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Application of the theory to coupled enzyme assays allows a definition of the minimum requirements for successful operation of the assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration.     

A microcomputer method for designing optimal experiments for estimating enzyme kinetic parameters

A computer program, written in BASIC, for designing optimal experiments with the aim of evaluating estimates of the parameters for any enzyme kinetic model is given. This computer program can be run on any microcomputer with less than 32 Kbytes of random access memory. The program uses the termed D-optimization design criterion, which minimizes the determinant of the variance-covariance matrix. The user only supplies the rate equation, the maximum and minimum concentrations of substrates and inhibitors, the weighting pattern, and the best possible values of the parameters. The computer supplies the optimal substrate and inhibitor concentrations (one for each parameters), for estimating the parameter values, and the determinant of the variance-covariance matrix. Likewise, the microcomputer supplies the eigenvalues and eigenvectors of information and redundancy matrices, the sensitivity and the global redundancy.     

Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. I. The basis for model construction.

A steady-state model of the tricarboxylic acid cycle was constructed using a dynamic systems analysis computer program, METASIM. The model was based on radioactive tracer analyses which provided flux relationships and compartmented metabolite concentrations. Ten of the enzymes modeled were purified and characterized from Dictyostelium discoideum. Although experimentally determined enzyme mechanisms and constants were used in the model, Vmax values were found to be unreliable, i.e. they did not reflect enzyme activity in vivo. This value was therefore calculated as the only unknown in each enzyme kinetic equation and called Vvivo, to distinguish it from Vmax determined in vitro.     

'Slave' metabolites and enzymes. A rapid way of delineating metabolic control.

Although control of fluxes and concentrations tends to be distributed rather than confined to a single rate-limiting enzyme, the extent of control can differ widely between enzymes in a metabolic network. In some cases, there are enzymes that lack control completely. This paper identifies one surprising origin of such lack of control: If, in a metabolic system, there is a metabolite that affects the catalytic rate of only one enzyme, the corresponding enzyme cannot control any metabolic variable other than the concentration of that metabolite. We call such enzymes 'slave enzymes', and the corresponding metabolites 'slave metabolites'. Implications of the existence of slave enzymes for the control properties of enzymes further down the metabolic pathway are discussed and examined for the glycolytic pathway of yeast. Inadvertent assumptions in metabolic models may cause the latter incorrectly to calculate absence of metabolic control. The phenomenon of slave enzymes may well be important in enhancing metabolic signal transduction.