Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics

This paper examines the validity of the linlog approach, which was recently developed in our laboratory, by comparison of two different kinetic models for the metabolic network of Escherichia coli. The first model is a complete mechanistic model; the second is an approximative model in which linlog kinetics are applied. The parameters of the linlog model (elasticities) are derived from the mechanistic model. Three different optimization cases are examined. In all cases, the objective is to calculate the enzyme levels that maximize a certain flux while keeping the total amount of enzyme constant and preventing large changes of metabolite concentrations. For an average variation of metabolite levels of 10% and individual changes of a factor 2, the predicted enzyme levels, metabolite concentrations and fluxes of both models are highly similar. This similarity holds for changes in enzyme level of a factor 4-6 and for changes in fluxes up to a factor 6. In all three cases, the predicted optimal enzyme levels could neither have been found by intuition-based approaches, nor on basis of flux control coefficients. This demonstrates that kinetic models are essential tools in Metabolic Engineering. In this respect, the linlog approach is a valuable extension of MCA, since it allows construction of kinetic models, based on MCA parameters, that can be used for constrained optimization problems and are valid for large changes of metabolite and enzyme levels.     

Optimization of regulatory architectures in metabolic reaction networks

Successful biotechnological applications, such as amino acid production, have demonstrated significant improvement in bioprocess performance by genetic modifications of metabolic control architectures and enzyme expression levels. However, the stoichiometric complexity of metabolic pathways, along with their strongly nonlinear nature and regulatory coupling, necessitates the use of structured kinetic models to direct experimental applications and aid in quantitative understanding of cellular bioprocesses. A novel optimization problem is introduced here, the objective of which is to identify changes in the regulatory characteristics of pertinent enzymes and in their cellular content which should be implemented to optimize a particular metabolic process. The mathematical representation of the metabolic reaction networks used is the S-system representation, which at steady state is characterized by linear equations. Exploiting the linearity of the representation, we formulated the optimization problem as a mixed-integer linear programming (MILP) problem. This formulation allows the consideration of a regulatory superstructure that contains all alternative regulatory structures that can be considered for a given pathway. The proposed approach is developed and illustrated using a simple linear pathway. Application of the framework on a complicated pathway-namely, the xanthine monophosphate (XMP) and guanosine monophosphate (GMP) synthesis pathway-identified the modification of the regulatory architecture that, along with changes in enzyme expression levels, can increase the XMP and GMP concentration by over 114 times the reference value, which is 50 times more than could be achieved by changes in enzyme expression levels only. (c) 1996 John Wiley & Sons, Inc.     

Interrogating the effect of enzyme kinetics on metabolism using differentiable constraint-based models

Metabolic models are typically characterized by a large number of parameters. Traditionally, metabolic control analysis is applied to differential equation-based models to investigate the sensitivity of predictions to parameters. A corresponding theory for constraint-based models is lacking, due to their formulation as optimization problems. Here, we show that optimal solutions of optimization problems can be efficiently differentiated using constrained optimization duality and implicit differentiation. We use this to calculate the sensitivities of predicted reaction fluxes and enzyme concentrations to turnover numbers in an enzyme-constrained metabolic model of Escherichia coli. The sensitivities quantitatively identify rate limiting enzymes and are mathematically precise, unlike current finite difference based approaches used for sensitivity analysis. Further, efficient differentiation of constraint-based models unlocks the ability to use gradient information for parameter estimation. We demonstrate this by improving, genome-wide, the state-of-the-art turnover number estimates for E. coli. Finally, we show that this technique can be generalized to arbitrarily complex models. By differentiating the optimal solution of a model incorporating both thermodynamic and kinetic rate equations, the effect of metabolite concentrations on biomass growth can be elucidated. We benchmark these metabolite sensitivities against a large experimental gene knockdown study, and find good alignment between the predicted sensitivities and in vivo metabolome changes. In sum, we demonstrate several applications of differentiating optimal solutions of constraint-based metabolic models, and show how it connects to classic metabolic control analysis.     

Metabolic design: how to engineer a living cell to desired metabolite concentrations and fluxes

A biotechnological aim of genetic engineering is to increase the intracellular concentration or secretion of valuable compounds, while making the other concentrations and fluxes optimal for viability and productivity. Efforts to accomplish this based on over-expression of the enzyme, catalyzing the so-called "rate-limiting step," have not been successful. Here we develop a method to determine the enzyme concentrations that are required to achieve such an aim. This method is called Metabolic Design Analysis and is based on the perturbation method and the modular ("top-down") approach-formalisms that were first developed for the analysis of biochemical regulation such as, Metabolic Control Analysis. Contrary to earlier methods, the desired alterations of cellular metabolism need not be small or confined to a single metabolite or flux. The limits to the alterations of fluxes and metabolite concentrations are identified. To employ Metabolic Design Analysis, only limited kinetic information concerning the pathway enzymes is needed.     

Towards kinetic modeling of genome-scale metabolic networks without sacrificing stoichiometric,thermodynamic and physiological constraints

Mathematical modeling is an essential tool for the comprehensive understanding of cell metabolism and its interactions with the environmental and process conditions. Recent developments in the construction and analysis of stoichiometric models made it possible to define limits on steady-state metabolic behavior using flux balance analysis. However, detailed information on enzyme kinetics and enzyme regulation is needed to formulate kinetic models that can accurately capture the dynamic metabolic responses. The use of mechanistic enzyme kinetics is a difficult task due to uncertainty in the kinetic properties of enzymes. Therefore, the majority of recent works considered only mass action kinetics for reactions in metabolic networks. Herein, we applied the optimization and risk analysis of complex living entities (ORACLE) framework and constructed a large-scale mechanistic kinetic model of optimally grown Escherichia coli. We investigated the complex interplay between stoichiometry, thermodynamics, and kinetics in determining the flexibility and capabilities of metabolism. Our results indicate that enzyme saturation is a necessary consideration in modeling metabolic networks and it extends the feasible ranges of metabolic fluxes and metabolite concentrations. Our results further suggest that enzymes in metabolic networks have evolved to function at different saturation states to ensure greater flexibility and robustness of cellular metabolism.     

Mathematical modelling of dynamics and control in metabolic networks. V. Static bifurcations in single biochemical control loops

Here we expand an earlier study of feedback activation in simple linear reaction sequences by searching the parameter space of biologically realistic rate laws for multiple stable steady states. The impetus for this work is to seek the origin of decision making strategies at the metabolic level, with particular emphasis on the switching between the operating conditions needed to meet changing substrate availability and organism requirements. The control loop considered herein is a linear reaction chain in which the end product of the reaction sequence feedback activates the first reaction in the sequence to produce feedback control. It has been found that the criteria for the existence of multiple steady state solutions in such loops involve only the kinetics of the regulatory enzyme controlling the first reaction and that of end product removal. The effects of these kinetics are examined here using two representative models for the regulatory enzyme: the lumped controller, based on Hill-type kinetics, and the symmetry model. The behavior of these two models is qualitatively similar, and both show the characteristics needed for switching between low and high substrate utilization. The removal rate is assumed to be of the Michaelis-Menten type. Judicious scaling of the governing equations permits separation of genetically determined kinetic parameters from concentration dependent ones. This allows us to conclude that, for a fixed set of kinetic parameters, the steady state flux through the loop can be switched between stable steady states by merely varying metabolite or enzyme concentrations. In particular, when the initial substrate exceeds a certain critical level, the loop can be "switched on" (by a discontinuous increase in the flux through the chain), and similarly, when it falls below a critical level, the pathway is shut down. Similar effects can be realized by varying the ratios of enzyme concentrations. It is proposed that by identifying these critical points one can gain significant insight into the objectives of decision making at the metabolic level.     

Systematic Construction of Kinetic Models from Genome-Scale Metabolic Networks

The quantitative effects of environmental and genetic perturbations on metabolism can be studied in silico using kinetic models. We present a strategy for large-scale model construction based on a logical layering of data such as reaction fluxes, metabolite concentrations, and kinetic constants. The resulting models contain realistic standard rate laws and plausible parameters, adhere to the laws of thermodynamics, and reproduce a predefined steady state. These features have not been simultaneously achieved by previous workflows. We demonstrate the advantages and limitations of the workflow by translating the yeast consensus metabolic network into a kinetic model. Despite crudely selected data, the model shows realistic control behaviour, a stable dynamic, and realistic response to perturbations in extracellular glucose concentrations. The paper concludes by outlining how new data can continuously be fed into the workflow and how iterative model building can assist in directing experiments.     

Kinetic study of a substrate cycle involving a chemical step: highly amplified determination of phenolic compounds

A kinetic analysis of a substrate cycle in which one of the two steps was substituted by a chemical reaction has been made. The model is illustrated by the amplified determination, in a continuous assay, of phenolic compounds at low concentrations using the enzyme tyrosinase and β-NADH to reduce the o-quinone product of catalytic activity. Progress curves corresponding to β-NADH disappearance were not linear and followed first-order kinetics. Knowledge of the kinetics of the reaction has allowed us to achieve detection limits as low as 50 nM in a simple 10-min assay. There is no analytical solution to the non-linear differential equation system that describes the kinetics of the reaction, therefore, computer simulations of its dynamic behaviour are also presented, good agreement with the experimental results being obtained. The method is applicable to the measurement of any other metabolite, and its amplification capacity as well as the simplicity of determining kinetic parameters enable it to be implemented in a bioreactor for automation purposes.     

Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics

This paper presents a new mathematical framework for modeling of in vivo dynamics and for metabolic re-design: the linlog approach. This approach is an extension of metabolic control analysis (MCA), valid for large changes of enzyme and metabolite levels. Furthermore, the presented framework combines MCA with kinetic modeling, thereby also combining the merits of both approaches. The linlog framework includes general expressions giving the steady-state fluxes and metabolite concentrations as a function of enzyme levels and extracellular concentrations, and a metabolic design equation that allows direct calculation of required enzyme levels for a desired steady state when control and response coefficients are available. Expressions giving control coefficients as a function of the enzyme levels are also derived. The validity of the linlog approximation in metabolic modeling is demonstrated by application of linlog kinetics to a branched pathway with moiety conservation, reversible reactions and allosteric interactions. Results show that the linlog approximation is able to describe the non-linear dynamics of this pathway very well for concentration changes up to a factor 20. Also the metabolic design equation was tested successfully.     

Effect of enzyme organization on the stability of Yates-Pardee pathways

Yates-Pardee-type metabolic pathways in a heterogenous cell milieu are modeled as a system of coupled non-linear partial differential equations. A numerical solution to this systmm is described and some properties of such a physiological system are studied. Confinement with and without a membrane is considered and it is shown how confinement results in an increase in the stability of the metabolite concentrations. These results suggest that the enzyme organization may contribute to the stability of the cellular metabolism.     

Kinetic evidence of the existence of a regulatory phosphoenolpyruvate binding site in maize leaf phosphoenolpyruvate carboxylase

Phenylphosphate, a structural analog of phosphoenolpyruvate (PEP), was found to be an activator of phosphoenolpyruvate carboxylase (PEP carboxylase) purified from maize leaves. This finding suggested the presence in the enzyme of a regulatory site, to which PEP could bind. We carried out kinetic studies on this enzyme using controlled concentrations of free PEP and of Mg-PEP complex and developed a theoretical kinetic model of the reaction. In summary, the main conclusions drawn from our results, and taken as assumptions of the model, were the following: (i) The affinity of the active site for the complex Mg-PEP is much higher than that for free PEP and Mg2+ ions, and therefore it can be considered that the preferential substrate of the PEP-catalyzed reaction is Mg-PEP. (ii) The enzyme has a regulatory site specific for free PEP, to which Mg2+ ions can not bind. (iii) The binding of free PEP, or an analog molecule, to this regulatory site yields a modified enzyme that has much lower apparent Km values and apparent Vmax values than the unmodified enzyme. So, free PEP behaves as an excellent activator of the reaction at subsaturating substrate concentrations, and as an inhibitor at saturating substrate concentrations. These findings may have important physiological implications on the regulation of the PEP carboxylase in vivo activity and, consequently, of the C4 pathway, since increased reaction rates would be obtained when the concentration of PEP rises, even at limiting Mg2+ concentrations.     

Competition for enzymes in metabolic pathways: implications for optimal distributions of enzyme concentrations and for the distribution of flux control

The structures of biochemical pathways are assumed to be determined by evolutionary optimization processes. In the framework of mathematical models, these structures should be explained by the formulation of optimization principles. In the present work, the principle of minimal total enzyme concentration at fixed steady state fluxes is applied to metabolic networks. According to this principle there exists a competition of the reactions for the available amount of enzymes such that all biological functions are maintained. In states which fulfil these optimization criteria the enzyme concentrations are distributed in a non-uniform manner among the reactions. This result has consequences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix epsilon, and the vector e of enzyme concentrations fulfil in optimal states the relations c(T)e = e and epsilon(T)e = 0. Starting from a well-balanced distribution of enzymes the minimization of total enzyme concentration leads to a lowering of the SD of the flux control coefficients.     

A model-based optimization framework for the inference on gene regulatory networks from DNA array data

MOTIVATION: Identification of the regulatory structures in genetic networks and the formulation of mechanistic models in the form of wiring diagrams is one of the significant objectives of expression profiling using DNA microarray technologies and it requires the development and application of identification frameworks. RESULTS: We have developed a novel optimization framework for identifying regulation in a genetic network using the S-system modeling formalism. We show that balance equations on both mRNA and protein species led to a formulation suitable for analyzing DNA-microarray data whereby protein concentrations have been eliminated and only mRNA relative concentrations are retained. Using this formulation, we examined if it is possible to infer a set of possible genetic regulatory networks consistent with observed mRNA expression patterns. Two origins of changes in mRNA expression patterns were considered. One derives from changes in the biophysical properties of the system that alter the molecular-interaction kinetics and/or message stability. The second is due to gene knock-outs. We reduced the identification problem to an optimization problem (of the so-called mixed-integer non-linear programming class) and we developed an algorithmic procedure for solving this optimization problem. Using simulated data generated by our mathematical model, we show that our method can actually find the regulatory network from which the data were generated. We also show that the number of possible alternate genetic regulatory networks depends on the size of the dataset (i.e. number of experiments), but this dependence is different for each of the two types of problems considered, and that a unique solution requires fewer datasets than previously estimated in the literature. This is the first method that also allows the identification of every possible regulatory network that could explain the data, when the number of experiments does not allow identification of unique regulatory structure.     

On the stability of metabolic cycles

We investigate the stability properties of two different classes of metabolic cycles using a combination of analytical and computational methods. Using principles from structural kinetic modeling (SKM), we show that the stability of metabolic networks with certain structural regularities can be studied using a combination of analytical and computational techniques. We then apply these techniques to a class of single input, single output metabolic cycles, and find that the cycles are stable under all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical methods can be used to understand the relationship between kinetic parameters and stability, and that results from these analytical methods can be confirmed with computational experiments. In addition, our results suggest that elevated metabolite concentrations and certain crucial saturation parameters can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic probability of being stable. Furthermore, our conclusions support the hypothesis that certain types of metabolic cycles may have played a role in the development of primitive metabolism despite the absence of regulatory mechanisms.     

Dynamic model of CHO cell metabolism

Fed-batch cultures are extensively used for the production of therapeutic proteins. However, process optimization is hampered by lack of quantitative models of mammalian cellular metabolism in these cultures. This paper presents a new kinetic model of CHO cell metabolism and a novel framework for simulating the dynamics of metabolic and biosynthetic pathways of these cells grown in fed-batch culture. The model defines a subset of the intracellular reactions with kinetic rate expressions based on extracellular metabolite concentrations and temperature- and redox-dependent regulatory variables. The simulation uses the rate expressions to calculate pseudo-steady state flux distributions and extracellular metabolite concentrations at discrete time points. Experimental data collected in this study for several different CHO cell fed-batch cultures are used to derive the rate expressions, fit the parameters, and validate the model. The simulations accurately predicted the effects of process variables, including temperature shift, seed density, specific productivity, and nutrient concentrations.     

iSCHRUNK – In Silico Approach to Characterization and Reduction of Uncertainty in the Kinetic Models of Genome-scale Metabolic Networks

Accurate determination of physiological states of cellular metabolism requires detailed information about metabolic fluxes, metabolite concentrations and distribution of enzyme states. Integration of fluxomics and metabolomics data, and thermodynamics-based metabolic flux analysis contribute to improved understanding of steady-state properties of metabolism. However, knowledge about kinetics and enzyme activities though essential for quantitative understanding of metabolic dynamics remains scarce and involves uncertainty. Here, we present a computational methodology that allow us to determine and quantify the kinetic parameters that correspond to a certain physiology as it is described by a given metabolic flux profile and a given metabolite concentration vector. Though we initially determine kinetic parameters that involve a high degree of uncertainty, through the use of kinetic modeling and machine learning principles we are able to obtain more accurate ranges of kinetic parameters, and hence we are able to reduce the uncertainty in the model analysis. We computed the distribution of kinetic parameters for glucose-fed E. coli producing 1,4-butanediol and we discovered that the observed physiological state corresponds to a narrow range of kinetic parameters of only a few enzymes, whereas the kinetic parameters of other enzymes can vary widely. Furthermore, this analysis suggests which are the enzymes that should be manipulated in order to engineer the reference state of the cell in a desired way. The proposed approach also sets up the foundations of a novel type of approaches for efficient, non-asymptotic, uniform sampling of solution spaces.     

A mathematical model of the Calvin photosynthesis cycle

1. A mathematical model is presented for photosynthetic carbohydrate formation in C3 plants under conditions of light and carbon dioxide saturation. The model considers reactions of the Calvin cycle with triose phosphate export and starch production as main output processes, and treats concentrations of NADPH, NAD+, CO2, and H+ as fixed parameters of the system. Using equilibrium approximations for all reaction steps close to equilibrium steady-state and transient-state relationships are derived which may be used for calculation of reaction fluxes and concentrations of the 13 carbohydrate cycle intermediates, glucose 6-phosphate, glucose 1-phosphate, ATP, ADP, and inorganic (ortho)phosphate. 2. Predictions of the model were examined with the assumption that photosynthate export from the chloroplast occurs to a medium containing orthophosphate as the only exchangeable metabolite. The results indicate that the Calvin cycle may operate in a single dynamically stable steady state when the external concentration of orthophosphate does not exceed 1.9 mM. At higher concentrations of the external metabolite, the reaction system exhibits overload breakdown; the excessive rate of photosynthate export deprives the system of cycle intermediates such that the cycle activity progressively approaches zero. 3. Reactant concentrations calculated for the stable steady state that may obtain are in satisfactory agreement with those observed experimentally, and the model accounts with surprising accuracy for experimentally observed effects of external orthophosphate on the steady-state cycle activity and rate of starch production. 4. Control analyses are reported which show that most of the non-equilibrium enzymes in the system have a strong regulatory influence on the steady-state level of all of the cycle intermediates. Substrate concentration control coefficients for cycle enzymes may be positive, such that an increase in activity of an enzyme may raise the steady-state concentration of the substrate is consumes. 5. Under optimal external conditions (0.15-0.5 mM orthophosphate), reaction flux in the Calvin cycle is controlled mainly by ATP synthetase and sedoheptulose bisphosphatase; the cycle activity approaches the maximum velocity that can be supported by the latter enzyme. At lower concentrations of external orthophosphate the cycle activity is controlled almost exclusively by the phosphate translocator.(ABSTRACT TRUNCATED AT 400 WORDS)     

Towards Kinetic Modeling of Global Metabolic Networks: Methylobacterium extorquens AM1 Growth as Validation

Here we report a systematic method for constructing a large scale kinetic metabolic model and its initial application to the modeling of central metabolism of Methylobacterium extorquens AM1, a methylotrophic and environmental important bacterium. Its central metabolic network includes formaldehyde metabolism, serine cycle, citric acid cycle, pentose phosphate pathway, gluconeogensis, PHB synthesis and acetyl-CoA conversion pathway, respiration and energy metabolism. Through a systematic and consistent procedure of finding a set of parameters in the physiological range we overcome an outstanding difficulty in large scale kinetic modeling: the requirement for a massive number of enzymatic reaction parameters. We are able to construct the kinetic model based on general biological considerations and incomplete experimental kinetic parameters. Our method consists of the following major steps: 1) using a generic enzymatic rate equation to reduce the number of enzymatic parameters to a minimum set while still preserving their characteristics; 2) using a set of steady state fluxes and metabolite concentrations in the physiological range as the expected output steady state fluxes and metabolite concentrations for the kinetic model to restrict the parametric space of enzymatic reactions; 3) choosing enzyme constants K’s and K’eqs optimized for reactions under physiological concentrations, if their experimental values are unknown; 4) for models which do not cover the entire metabolic network of the organisms, designing a dynamical exchange for the coupling between the metabolism represented in the model and the rest not included.     

K-FIT: An accelerated kinetic parameterization algorithm using steady-state fluxomic data

Kinetic models predict the metabolic flows by directly linking metabolite concentrations and enzyme levels to reaction fluxes. Robust parameterization of organism-level kinetic models that faithfully reproduce the effect of different genetic or environmental perturbations remains an open challenge due to the intractability of existing algorithms. This paper introduces Kinetics-based Fluxomics Integration Tool (K-FIT), a robust kinetic parameterization workflow that leverages a novel decomposition approach to identify steady-state fluxes in response to genetic perturbations followed by a gradient-based update of kinetic parameters until predictions simultaneously agree with the fluxomic data in all perturbed metabolic networks. The applicability of K-FIT to large-scale models is demonstrated by parameterizing an expanded kinetic model for E. coli (307 reactions and 258 metabolites) using fluxomic data from six mutants. The achieved thousand-fold speed-up afforded by K-FIT over meta-heuristic approaches is transformational enabling follow-up robustness of inference analyses and optimal design of experiments to inform metabolic engineering strategies.     

A generalized theoretical treatment of the kinetics of an enzyme-catalysed reaction in the presence of an unstable irreversible modifier

A generalized theoretical treatment of the kinetics of an enzyme-catalysed reaction in the presence of an unstable irreversible inhibitor (or activator) is presented. Analytical expressions describing the time-dependence of product formation have been derived in coefficient form amenable to non-linear regression analysis for two operationally distinct types of reaction mechanism dependent on whether the reaction of the unstable modifier (X) with either or both the free enzyme (E) and enzyme-substrate complex (ES) occurs as a simple bimolecular process, or proceeds through the intermediacy of either or both adsorptive enzyme-modifier (EX) and enzyme-modifier-substrate (EXS) complexes in what may be considered as an extension of the Botts-Morales general modifier mechanism for (stable) reversible enzyme inhibitors and activators. Special cases of both models are classified in an analogous way to the traditional naming of reversible enzyme modifications, and guidelines concerning tests of mechanism and determination of kinetic parameters are given. In particular, it has been shown that kinetic constants describing enzyme inactivation by an unstable site-specific inhibitor forming a reversible EX complex prior to covalent modification step may be determined from a single progress curve. Kinetic analysis of the extended Botts-Morales mechanism describing irreversible enzyme inactivation has demonstrated that analytical expressions describing the time-course of product formation may be derived for a stable modifier by retaining the usual steady-state assumptions regarding the fluxes around ES and EXS provided quasi-equilibrium modifier binding to E and ES is assumed, but for unstable modifiers all of the binding steps must be assumed to be at quasi-equilibrium in the steady-state, except under restrictive circumstances.