The rationalization of high enzyme concentration in metabolic pathways such as glycolysis

The cellular concentration of enzymes of some major metabolic pathways, such as glycolysis, can approach millimolarity. This concentration of enzyme can catalyze in vitro rates which are 100-fold higher than maximum pathway flux. In an attempt to understand the need for such high enzyme concentration, an artificial metabolic pathway of five enzymes (apropos the central enzymes of glycolysis) has been modeled. Numerical methods were then used to determine the effect of enzyme concentration on: (1) the change in total free metabolite concentration as the pathway changes from low flux to high flux, (2) the time lag (transient time) in the rate of final product formation upon the transition from low flux to high flux. Both the changes in metabolite pool size and the transient time decreased with increased enzyme concentrations. When all enzymatic reactions were assigned Keq of unity, a concentration for each enzyme of 25 microM is sufficient to provide a transient time of 1 sec. When Keq different from unity are introduced, more enzyme is required to provide comparably short transient times. Under the latter condition, a pathway of sufficiently low transient time would require all the enzyme available in mammalian muscle. It is shown that there is little scope for further increases in either enzyme concentration or of catalytic efficiency of independent enzymes. Therefore, an alternative method of increasing efficiency is considered in which enzyme-bound metabolites can serve directly as substrates for subsequent enzymes in a metabolic pathway.     

Optimal temperature policy for immobilized enzyme packed bed reactor performing reversible Michaelis-Menten kinetics using the disjoint policy.

The optimal temperature policy that maximizes the time-averaged productivity of a continuous immobilized enzyme packed bed reactor is determined. This optimization study takes into consideration the enzyme thermal deactivation with substrate protection during the reactor operation. The general case of reversible Michaelis-Menten kinetics under constant reactor feed flow rate is assumed. The corresponding nonlinear optimization problem is solved using the calculus of variations by applying the disjoint policy. This policy reduces the optimization problem into a differential-algebraic system, DAE. This DAE system defines completely the optimal temperature-time profiles. These profiles depend on the kinetic parameters, feed substrate concentration, operating period, and the residence time and are characterized by increasing form with time. Also, general analytical expressions for the slopes of the temperature and residual enzyme activity profiles are derived. An efficient solution algorithm is developed to solve the DAE system, which results into a one-dimensional optimization problem with simple bounds on the initial feed temperature. The enzymatic isomerization of glucose into fructose is selected as a case study. The computed productivities are very close to that obtained by numerical nonlinear optimization with simpler problem to solve. Moreover, the computed conversion profiles are almost constant over 90% of the operating periods, thus producing a homogeneous product.     

Sequential Activation of Metabolic Pathways: a Dynamic Optimization Approach

The regulation of cellular metabolism facilitates robust cellular operation in the face of changing external conditions. The cellular response to this varying environment may include the activation or inactivation of appropriate metabolic pathways. Experimental and numerical observations of sequential timing in pathway activation have been reported in the literature. It has been argued that such patterns can be rationalized by means of an underlying optimal metabolic design. In this paper we pose a dynamic optimization problem that accounts for time-resource minimization in pathway activation under constrained total enzyme abundance. The optimized variables are time-dependent enzyme concentrations that drive the pathway to a steady state characterized by a prescribed metabolic flux. The problem formulation addresses unbranched pathways with irreversible kinetics. Neither specific reaction kinetics nor fixed pathway length are assumed. In the optimal solution, each enzyme follows a switching profile between zero and maximum concentration, following a temporal sequence that matches the pathway topology. This result provides an analytic justification of the sequential activation previously described in the literature. In contrast with the existent numerical approaches, the activation sequence is proven to be optimal for a generic class of monomolecular kinetics. This class includes, but is not limited to, Mass Action, Michaelis–Menten, Hill, and some Power-law models. This suggests that sequential enzyme expression may be a common feature of metabolic regulation, as it is a robust property of optimal pathway activation.     

Defining control coefficients in non-ideal metabolic pathways

The extent to which an enzyme controls a flux has been defined as the effect on that flux of a small modulation of the activity of that enzyme divided by the magnitude of the modulation. We here show that in pathways with metabolic channelling or high enzyme concentrations and conserved moieties involving both enzymic and non-enzymic species, this definition is ambiguous; the magnitude of the corresponding flux control coefficient depends on how the enzyme activity is modulated. This is illustrated with two models of biochemically relevant pathways, one in which dynamic metabolite channelling plays a role, and one with a moiety-conserved cycle. To avoid such ambiguity, we view biochemical pathways in a more detailed manner, i.e., as a network of elemental steps. We define 'elemental control coefficients' in terms of the effect on a flux of an equal modulation of the forward and reverse rate constant of any such elemental step (which may correspond to transitions between enzyme states). This elemental control coefficient is independent of the method of modulation. We show how metabolic control analysis can proceed when formulated in terms of the elemental control coefficients and how the traditional control coefficients are related to these elemental control coefficients. An 'impact' control coefficient is defined which quantifies the effect of an activation of all elemental processes in which an enzyme is involved. It equals the sum of the corresponding elemental control coefficients. In ideal metabolic pathways this impact control coefficient reduces to the traditional flux control coefficient. Differences between the traditional control coefficients are indicative of non-ideality of a metabolic pathway, i.e. of channelling or high enzyme concentrations.     

An integrated modeling-experimental strategy for the analysis of metabolic pathways

An inherent problem in studying the behavior of a metabolic pathway is the impossibility of developing a complete, detailed model that includes all the cellular processes that have an impact on the set of fluxes in such a pathway. Lacking this, one requires some means of modeling the interactions between a metabolic pathway and other cellular processes for the purpose of analyzing pathway characteristics within the cell (e.g., determining sensitivity coefficients for various steps in the pathway) with a minimal amount of time and effort. A general framework is developed for studying these issues in a rigorous manner. Using this framework, detailed knowledge about a metabolic pathway (i.e., a set of rate expressions for steps in the pathway) can be combined with the results from a relatively simple set of experiments in order to obtain estimates for the sensitivity of the pathway to enzyme activities, inhibition constants, and other parameters that determine the pathway's behavior, while accounting for the pathway's interaction with the rest of the cellular metabolism. A model system representing amino acid production is used to illustrate the problem and to provide results based on computational experiments. The modeling strategy described here should be useful in genetic design to improve pathway fluxes and metabolic network selectivity.     

基于图论和约束优化的异源代谢途径设计方法及应用

构建高产高附加值产品的微生物细胞工厂是代谢工程的研究目标之一,设计高效的产品合成途径是实现这一目标的重要方式.不同微生物底盘因其代谢能力有所差异,故而可以利用的底物和生产的产品范围有限.为了扩大其生产能力,需要进行代谢途径从无到有的设计.传统代谢工程基于经验进行异源途径设计的方式低效且无法确保结果的全面性,而系统生物学...     

An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

The intracellular equilibrium thermodynamic and steady-state concentrations of metabolites

A new model for the organization and flow of metabolites through a metabolic pathway is presented. The model is based on four major findings. (1) The intracellular concentrations of enzyme sites exceed the concentrations of intermediary metabolites that bind specifically to these sites. (2) The concentration of the excessive enzyme sites in the cell is sufficiently high so that nearly all the cellular intermediary metabolites are enzyme-bound. (3) Enzyme conformations are perturbed by the interactions with substrates and products; the conformations of enzyme-substrate and enzyme-product complexes are different. (4) Two enzymes, catalyzing reactions that are sequential in a metabolic pathway, transfer the common metabolite back and forth via an enzyme-enzyme complex without the intervention of the solvent environment. The model proposes that the enzyme-enzyme recognition is ligand-induced. Conversion of E₂S and E₂P results in the loss of recognition of E₂ by E₁ and the concomitant recognition of E₂ by E₃. This model substantially alters existent views of the bioenergetics and the kinetics of intracellular metabolism. The rates of direct transfer of metabolite from enzyme to enzyme are comparable to the rates of interconversion between substrate and product within an individual enzyme. Consequently, intermediary metabolites are nearly equipartitioned among their high-affinity enzyme sites within a metabolic pathway. Metabolic flux involves the direct transfer of metabolite from enzyme to enzyme via a set of low and nearly equal energy barriers.     

Multi-criteria optimization of regulation in metabolic networks

Determining the regulation of metabolic networks at genome scale is a hard task. It has been hypothesized that biochemical pathways and metabolic networks might have undergone an evolutionary process of optimization with respect to several criteria over time. In this contribution, a multi-criteria approach has been used to optimize parameters for the allosteric regulation of enzymes in a model of a metabolic substrate-cycle. This has been carried out by calculating the Pareto set of optimal solutions according to two objectives: the proper direction of flux in a metabolic cycle and the energetic cost of applying the set of parameters. Different Pareto fronts have been calculated for eight different "environments" (specific time courses of end product concentrations). For each resulting front the so-called knee point is identified, which can be considered a preferred trade-off solution. Interestingly, the optimal control parameters corresponding to each of these points also lead to optimal behaviour in all the other environments. By calculating the average of the different parameter sets for the knee solutions more frequently found, a final and optimal consensus set of parameters can be obtained, which is an indication on the existence of a universal regulation mechanism for this system.The implications from such a universal regulatory switch are discussed in the framework of large metabolic networks.     

Prediction of temporal gene expression. Metabolic opimization by re-distribution of enzyme activities.

A computational approach is used to analyse temporal gene expression in the context of metabolic regulation. It is based on the assumption that cells developed optimal adaptation strategies to changing environmental conditions. Time-dependent enzyme profiles are calculated which optimize the function of a metabolic pathway under the constraint of limited total enzyme amount. For linear model pathways it is shown that wave-like enzyme profiles are optimal for a rapid substrate turnover. For the central metabolism of yeast cells enzyme profiles are calculated which ensure long-term homeostasis of key metabolites under conditions of a diauxic shift. These enzyme profiles are in close correlation with observed gene expression data. Our results demonstrate that optimality principles help to rationalize observed gene expression profiles.     

Analysis of metabolic networks using a pathway distance metric through linear programming

The solution of the shortest path problem in biochemical systems constitutes an important step for studies of their evolution. In this paper, a linear programming (LP) algorithm for calculating minimal pathway distances in metabolic networks is studied. Minimal pathway distances are identified as the smallest number of metabolic steps separating two enzymes in metabolic pathways. The algorithm deals effectively with circularity and reaction directionality. The applicability of the algorithm is illustrated by calculating the minimal pathway distances for Escherichia coli small molecule metabolism enzymes, and then considering their correlations with genome distance (distance separating two genes on a chromosome) and enzyme function (as characterised by enzyme commission number). The results illustrate the effectiveness of the LP model. In addition, the data confirm that propinquity of genes on the genome implies similarity in function (as determined by co-involvement in the same region of the metabolic network), but suggest that no correlation exists between pathway distance and enzyme function. These findings offer insight into the probable mechanism of pathway evolution.     

Kinetic analysis of the general modifier mechanism of Botts and Morales involving a suicide substrate

Suicide substrates are widely used in enzymology for studying enzyme mechanisms and designing potential drugs. The presence of a reversible modifier decreases or increases the rate of substrate-induced inactivation, with evident physiological and experimental consequences. To date, only the action of a competitive or uncompetitive inhibitor of an enzyme system involving suicide substrate has been reported. In this paper, we analyse the kinetics of enzyme-catalysed reactions which evolve in accordance with the general modifier mechanisms of Botts and Morales in which enzyme inactivation is induced by suicide substrate. Rapid equilibrium of all of the reversible reaction steps involved is assumed and the time course equations for the residual enzyme activity, the inactive enzyme forms and the reaction product are derived. Partition ratios giving the relative weight of the product and inactive enzyme concentrations, and the relative contribution to the product formation of each of the unmodified and modified catalytic routes, are studied. New indices pointing to the conditions under which the modifier acts as inhibitor or as activator are suggested. The goodness of the analytical solutions is tested by comparison with the simulated curves obtained by numerical integration. An experimental design and kinetic data analysis to evaluate the kinetic parameters from the time progress curves of the product are proposed. From these results, those corresponding to several reaction mechanisms involving both a suicide substrate and a modifier, and which can be regarded as particular cases of the general case analysed here, can be directly and easily derived.     

Estimating optimal profiles of genetic alterations using constraint-based models

Metabolic engineering involves application of recombinant DNA methods to manipulate metabolic networks to improve cellular properties. It is critical that the genetic alterations be performed in an optimal manner to maximize profit. In addition to the product yield, productivity consideration is also critical, especially for the production of bulk chemicals such as 1,3-propanediol. In this work, we demonstrate that it is suboptimal from the standpoint of productivity to induce genetic alteration at the start of the production process. A bi-level optimization scheme is formulated to determine the optimal temporal flux profile for the manipulated reaction. In the first case study, an optimal flux in the reaction catalyzed by glycerol kinase is determined to maximize the glycerol production at the end of a 6-h batch cultivation of Escherichia coli under aerobic conditions. The final glycerol concentration is 30% higher for the optimal flux profile compared with having an active flux during the entire batch. The effect of the mass transfer coefficient on the optimal profile and the glycerol concentration is also determined. In the second case study, the anaerobic batch fermentation of the ldh(-) strain of Escherichia coli is considered. The optimal flux in the acetate pathway is determined to maximize the final ethanol concentration. The optimal flux results in higher ethanol concentration (11.92 mmol L(-1)) compared to strains with no acetate flux (8.36 mmol L(-1)) and fully active acetate flux (6.22 mmol L(-1)). We also examine the effects of growth inhibition due to high ethanol concentrations and variations in final batch time on ethanol production.     

Optimal choice of species and size class for transplanting coral community

Transplantation of sessile organisms living in a planned destruction site to a safe site is an important means of restoration to mitigate biodiversity loss following anthropogenic developments. In particular, corals, which play fundamental roles in the coral reef ecosystem and contribute to biodiversity, are good candidates for transplantation. In this study, we investigate the optimal choice of species and size class to be used for coral transplantation. We first studied a case in which the objective function to evaluate the success of transplantation is the maximum total coverage. The optimal strategy is to choose the species and size class with higher net coverage gain per unit handling effort. It is often recommended to transplant only one or a few species and neglect others, even if the original community consists of many species. This may achieve high coverage in the restored coral community but cause loss of species diversity. To overcome this problem, we next study a case in which the objective of the transplantation operation is to maximize the “prosperity index”, defined as the product of total coverage and species diversity. In this case, the optimal strategy depends on the species property, population size, and the limitation of total cost allowed for transplantation, but it tends to recommend more species to be transplanted than what is recommended by the coverage maximization criterion. We conclude that maximization of the prosperity index is a better criterion for transplantation than simple coverage maximization.     

Metabolic control analysis. The effects of high enzyme concentrations

Differing views have been given in the literature as to whether the presence in a pathway of an enzyme at a concentration comparable to that of its substrate affects the values of control coefficients and the theorems of metabolic control analysis. Here we argue in favour of one of those views: that there is no effect unless the enzyme sequesters a substrate that contains a conserved moiety. In this particular case, we derive both a general criterion for estimating whether such an effect will be of a significant magnitude, and equations for determining the changes in the flux control coefficients. The nature of the phenomenom and the application of the equations are illustrated with a numerical simulation.     

Minimal time control of fed-batch bioreactor with product inhibition

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green’s theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.