Optimization of bioreactor using metabolic control analysis approach

A new methodology based on a metabolic control analysis (MCA) approach is developed for the optimization of continuous cascade bioreactor system. A general framework for representation of a cascade bioreactor system consisting of a large number of reactors as a single network is proposed. The kinetic and transport processes occurring in the system are represented as a reaction network with appropriate stoichiometry. Such representation of the bioreactor systems makes it amenable to the direct application of the MCA approach. The process sensitivity information is extracted using MCA methodology in the form of flux and concentration control coefficients. The process sensitivity information is shown to be a useful guide for determining the choice of decision variables for the purpose of optimization. A generalized problem of optimization of the bioreactor is formulated in which the decision variables are the operating conditions and kinetic parameters. The gradient of the objective function to be maximized with respect to all decision variables is obtained in the form of response coefficients. This gradient information can be used in any gradient-based optimization algorithm. The efficiency of the proposed technique is demonstrated with two examples taken from literature: biotransformation of crotonobetaine and alcohol fermentation in cascade bioreactor system.     

Design of large metabolic responses. Constraints and sensitivity analysis

Metabolic control analysis (Kacser & Burns (1973). Symp. Soc. Exp. Biol.27, 65-104; Heinrich & Rapoport (1974). Eur. J. Biochem.42, 89-95) has been extensively used to describe the response of metabolic concentrations and fluxes to small (infinitesimal) changes in enzyme concentrations and effectors. Similarly, metabolic control design (Acerenza (1993). J. theor. Biol.165, 63-85) has been proposed to design small metabolic responses. These approaches have the limitation that they were not devised to deal with large (non-infinitesimal) responses. Here we develop a strategy to design large changes in the metabolic variables. The only assumption made is that, for all the parameter values under consideration, the system has a unique stable steady state. The procedure renders the kinetic parameters of the rate equations that when embedded in the metabolic network produce the pattern of large changes in the steady-state variables that we aim to design. Structural and kinetic constraints impose restrictions on the type of responses that could be designed. We show that these conditions can be transformed into the language of mean-sensitivity coefficients and, as a consequence, a sensitivity analysis of large metabolic responses can be performed after the system has been designed. The mean-sensitivity coefficients fulfil conservation and summation relationships that in the limit reduce to the well-known theorems for infinitesimal changes. Finally, it is shown that the same procedure that was used to design metabolic responses and analyse their sensitivity properties can also be used to determine the values of kinetic parameters of the rate laws operating "in situ".     

Power-law modeling based on least-squares minimization criteria.

The power-law formalism has been successfully used as a modeling tool in many applications. The resulting models, either as Generalized Mass Action or as S-systems models, allow one to characterize the target system and to simulate its dynamical behavior in response to external perturbations and parameter changes. The power-law formalism was first derived as a Taylor series approximation in logarithmic space for kinetic rate-laws. The especial characteristics of this approximation produce an extremely useful systemic representation that allows a complete system characterization. Furthermore, their parameters have a precise interpretation as local sensitivities of each of the individual processes and as rate-constants. This facilitates a qualitative discussion and a quantitative estimation of their possible values in relation to the kinetic properties. Following this interpretation, parameter estimation is also possible by relating the systemic behavior to the underlying processes. Without leaving the general formalism, in this paper we suggest deriving the power-law representation in an alternative way that uses least-squares minimization. The resulting power-law mimics the target rate-law in a wider range of concentration values than the classical power-law. Although the implications of this alternative approach remain to be established, our results show that the predicted steady-state using the least-squares power-law is closest to the actual steady-state of the target system.     

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.     

The "independence principle" in the processes of water transport.

The processes of membrane transport exhibiting permeability coefficients depending on the species activities do not obey the "independence principle" and are assumed to take place by a mechanism of discrete nature, analyzable by a kinetic formalism. In this article, we study the dependence of the osmotic permeability coefficient on the water activities, from the steady-state analysis of a kinetic model of single-file water transport that simultaneously incorporates the vacancy-mediated and "knock-on" mechanisms into the state diagram. In particular, we study the relation between the near-equilibrium osmotic permeability (Pe) and the equilibrium water activity of the compartments (w). The analysis and numerical calculations performed for a simple case of the model show that, for values of the parameters consistent with experimental data, Pe exhibits only a small variation with w within the physiological range in the majority of the situations considered here. It is not possible to predict, from the study of these simple models, whether more complicated kinetic diagrams of water transport may be characterized by permeability coefficients with a more evident dependence on the water activities. Nevertheless, the results obtained here suggest that, for the case of physiological water pores, the analysis of the kinetic dependence of the permeability coefficients on the water activities may not yield evidence pointing to a discrete nature for the transport process.     

Optimization-based metabolic control analysis

In this work, a novel optimization-based metabolic control analysis (OMCA) method is introduced for reducing data requirement for metabolic control analysis (MCA). It is postulated that using the optimal control approach, the fluxes in a metabolic network are correlated to metabolite concentrations and enzyme activities as a state-feedback control system that is optimal with respect to a homeostasis objective. It is then shown that the optimal feedback gains are directly related to the elasticity coefficients (ECs) of MCA. This approach requires determination of the relative "importance" of metabolites and fluxes for the system, which is possible with significantly reduced experimental data, as compared with typical MCA requirements. The OMCA approach is applied to a top-down control model of glycolysis in hepatocytes. It is statistically demonstrated that the OMCA model is capable of predicting the ECs observed experimentally with few exceptions. Further, an OMCA-based model reconciliation study shows that the modification of four assumed stoichiometric coefficients in the model can explain most of the discrepancies, with the exception of elasticities with respect to the NADH/NAD ratio.     

Quantitative analysis of the fermentative metabolism of glycerol in Escherichia coli

Availability, low price, and high degree of reduction have made glycerol a highly attractive and exploited carbon source for the production of fuels and reduced chemicals. Here we report the quantitative analysis of the fermentative metabolism of glycerol in Escherichia coli through the use of kinetic modeling and metabolic control analysis (MCA) to gain a better understanding of glycerol fermentation and identify key targets for genetic manipulation that could enhance product synthesis. The kinetics of glycerol fermentation in a batch culture was simulated using a dynamic model consisting of mass balances for glycerol, ethanol, biomass, and 11 intracellular metabolites, along with the corresponding kinetic expressions for the metabolism of each species. The model was then used to calculate metabolic control coefficients and elucidate the control structure of the pathways involved in glycerol utilization and ethanol synthesis. The calculated flux control coefficients indicate that the glycolytic flux during glycerol fermentation is almost exclusively controlled by the enzymes glycerol dehydrogenase (encoded by gldA) and dihydroxyacetone kinase (DHAK) (encoded by dhaKLM). In agreement with the MCA findings, overexpression of gldA and dhaKLM led to significant increase in glycerol utilization and ethanol synthesis fluxes. Moreover, overexpression of other enzymes involved in the pathways that mediate glycerol utilization and its conversion to ethanol had no significant impact on glycerol utilization and ethanol synthesis, further validating the MCA predictions. These findings were then applied as a means of increasing the production of ethanol: overexpression of glycerol dehyrdogenase and DHAK enabled the production of 20 g/L ethanol from crude glycerol, a by-product of biodiesel production, indicating the potential for industrial scale conversion of waste glycerol to ethanol under anaerobic conditions.     

典范分析法在黄河三角洲菜州湾滨海区盐生植物群落研究中的应用

 典范变量分析能同时处理两种类型的变量集并给出它们之间的相关系数。本文采用该方法对黄河三角洲地区的10个植物群丛及其土壤的有关性质进行了分析。典范相关系数是0.9和1。文中同时求出了所有典范变量,保留了原数据的全部信息。结果表明该地区植被的动态变化与土壤水盐及有机质含量的动态变化显著相关。土壤的水盐动态是这一地区植被演替的制约因素。二维典范变量图较直观地表示出了群丛间的演替关系。初步判断出芦苇和白茅群丛是这一地区植被演替过程中相对稳定的阶段。该方法对研究植物群落的分布及植被演替等内容是很有帮助的。     

Dissecting the variance-covariance structure in insect physiology: the multivariate association between metabolism and morphology in the nymphs of the sand cricket (Gryllus firmus)

Energy metabolism in animals has been largely studied in relation to exogenous sources of variation. However, because they give insight into the relationship between whole metabolism and lower organizational levels such as organs and tissues, examination of endogenous determinants of metabolism other than body mass is itself very important. We studied the multivariate association of body parts and several aspects of energy metabolism in an insect, the nymphs of the sand cricket, Gryllus firmus. By using a variety of both univariate and multivariate techniques, we explored the resultant variance-covariance matrix to build a path diagram with latent variables. After controlling for body mass, we found a significant canonical correlation between metabolism and morphology. According to the factor loadings and path coefficients, the most important contributions of morphology to the correlation were thorax and abdomen size measures, whereas the most important metabolic contribution was resting metabolism. Activity metabolism was mostly explained by body mass rather than body parts, which could be a result of resting rates being chronic consequences of the functioning of the metabolic machinery that the insect must maintain.     

典范分析法在黄河三角洲莱州湾滨海区盐生植物群落研究中的应用

典范变量分析能同时处理两种类型的变量集并给出它们之间的相关系数。本文采用该方法对黄河三角洲地区的10个植物群丛及其土壤的有关性质进行了分析。典范相关系数是0.9和1。文中同时求出了所有典范变量,保留了原数据的全部信息。结果表明该地区植被的动态变化与土壤水盐及有机质含量的动态变化显著相关。土壤的水盐动态是这一地区植被演替的制约因素。二维典范变量图较直观地表示出了群丛间的演替关系。初步判断出芦苇和白茅群丛是这一地区植被演替过程中相对稳定的阶段。该方法对研究植物群落的分布及植被演替等内容是很有帮助的。     

Automated oncogene detection in complex protein networks with applications to the MAPK signal transduction pathway

Activation of the extracellular signal-regulated kinases (ERK1/2; p42/p44 mitogen-activated protein kinase (MAPK)) is one of the most extensively studied signaling pathways not least because it occurs downstream of oncogenic RAS. Here, we take advantage of the wealth of experimental data available on the canonical RAS/RAF/MEK/ERK pathway of Bhalla et al. to test the utility of a newly developed nonlinear analysis algorithm designed to predict likelihood of cellular transformation. By using ERK phosphorylation as an "output signal", the method analyzes experimentally determined kinetic data and predicts putative oncogenes and tumor suppressor gene products impacting the RAS/MAPK module using a purely theoretical approach. This analysis identified several modifiers of ERK/MAPK activation described previously. In addition, several novel enzymes are identified which are not previously described to affect ERK/MAPK phosphorylation. Importantly, the nonlinear analysis enables a ranking of modifiers of MAPK activation predicting their relative importance in RAS-dependent oncogenesis. The results are compared with a linearized analysis based on sensitivity analysis about the steady state or metabolic control analysis (MCA). The results are favorable, pointing to the utility of first-order sensitivity analysis and MCA in the analysis of complex signaling networks for oncogenes.     

Response to temporal parameter fluctuations in biochemical networks

Metabolic response coefficients describe how variables in metabolic systems, like steady state concentrations, respond to small changes of kinetic parameters. To extend this concept to temporal parameter fluctuations, we define spectral response coefficients that relate Fourier components of concentrations and fluxes to Fourier components of the underlying parameters. It is also straightforward to generalize other concepts from metabolic control theory, such as control coefficients with their summation and connectivity theorems. The first-order response coefficients describe forced oscillations caused by small harmonic oscillations of single parameters: they depend on the driving frequency and comprise the phases and amplitudes of the concentrations and fluxes. Close to a Hopf bifurcation, resonance can occur: as an example, we study the spectral densities of concentration fluctuations arising from the stochastic nature of chemical reactions. Second-order response coefficients describe how perturbations of different frequencies interact by mode coupling, yielding higher harmonics in the metabolic response. The temporal response to small parameter fluctuations can be computed by Fourier synthesis. For a model of glycolysis, this approximation remains fairly accurate even for large relative fluctuations of the parameters.     

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.     

Summation theorems for flux and concentration control coefficients of dynamic systems

Metabolic control analysis (MCA) was developed to quantify how system variables are affected by parameter variations in a system. In addition, MCA can express the global properties of a system in terms of the individual catalytic steps, using connectivity and summation theorems to link the control coefficients to the elasticity coefficients. MCA was originally developed for steady-state analysis and not all summation theorems have been derived for dynamic systems. A method to determine time-dependent flux and concentration control coefficients for dynamic systems by expressing the time domain as a function of percentage progression through any arbitrary fixed interval of time is reported. Time-dependent flux and concentration control coefficients of dynamic systems, provided that they are evaluated in this novel way, obey the same summation theorems as steady-state flux and concentration control coefficients, respectively.     

Pathway kinetics and metabolic control analysis of a high-yielding strain of Penicillium chrysogenum during fed batch cultivations

A kinetic model representing the pathway for the biosynthesis of penicillin by P. chrysogenum has been developed. The model is capable of describing the flux through the biosynthetic pathway, and model simulations correspond well with measurements of intermediates and end products. One feature of the present model structure is that it assumes the kinetics of the enzyme isopenicillin N synthetase (IPNS) to be first order with respect to the dissolved oxygen concentration in the range of 0.070 to 0.18 mM (25% to 70% saturation with air). Thus, it indicates the importance that molecular oxygen has on the rate of the reaction catalyzed by this enzyme, and consequently as an enhancer of the specific rate of penicillin production. Using the kinetic model, metabolic control analysis (MCA) of the pathway was performed. The determined flux control coefficients suggested that, during the production phase, the flux is controlled by IPNS as this enzyme becomes saturated with tripeptide delta-(L-alpha-amino-adipyl)-L-cysteinyl-D-valine (LLD-ACV). In the simulations, oxygen was shown to be a bottleneck alleviator by stimulating the rate of IPNS which prevents the accumulation of LLD-ACV. As a consequence of this stimulation, the rate-controlling step was moved to another place in the pathway. (c) 1996 John Wiley & Sons, Inc.     

Sticking coefficient-orchestrated selection in a kinetic model for the first origin

The mathematical formalism of the steady-state Poisson equation is applied to a variant of Freeman Dyson's "Toy Model" for a first origin. Our kinetic approach allows for an examination of the requisite conditions under which metabolism is quantized into discrete eigenstates (e.g. Dyson's disordered, saddle point, and metabolically active toy cell states). The surface reaction machinery additionally allows for more realistic modeling, whence the crucial role of sticking coefficients (catalyst precursors) as prebiotic selectors emerges. In our interior source model, a steady influx of vent nutrients fuels the intracellular synthesis of (impermeable) monomers within a rock-encradled cavity. Random adsorptions and desorptions occur at inactive "cell" wall sites (where the inert monomers remain impermeable until their eventual return to the intracellular metabolite pool). Occasionally, metabolizing reactions also occur due to endogeneous source monomers adsorbing at their "active" sites. Dyson's mean field approach is used to simplify the species-specific sticking coefficients at empty active (substrate) sites to functions of the fraction x of sites occupied by (catalytically) active monomers. In short, our work suggests that disorder-order transition models based on random drift between discrete metabolic eigenstates (Dyson's Toy Model) do not, in general, extend to more realistic metabolisms. From a perspective based on quasi-random feedback kinetics, the contraindication for discretization (spontaneous generation) in non-autocatalytic metabolisms is consistent with the emergence of ordered metabolism under hydrothermal driving forces, a provisio the occurrence of each period of vent dormancy coincides with a discrete zero-source (dormant) metabolic state. Cell drift to higher order is induced by the random reactions which happen to enhance the substrate specificity (chemical selectivity) of the sticking coefficients for active monomers. The result is stronger sink effects for metabolizing species, whence active adsorptions are promoted in favor of inactive adsorptions at substrate sites. Positive feedback plays a crucial role in preserving ("propagating") order in the cell wall reaction kinetics and is held in check by negative feedback inhibition of excessive cell growth. Finally, the eventual desorption of randomly growing dysfunctional proteins is postulated as a deterrent to deterioration catastrophes.     

Metabolic control and its analysis. Additional relationships between elasticities and control coefficients

Existing theorems from the analysis of metabolic control have been taken and embedded in a simple matrix algebra procedure for calculating the flux control coefficients of enzymes (formerly known as sensitivities) in a metabolic pathway from their kinetic properties (their elasticities). New theorems governing the flux control coefficients of branched pathways and substrate cycles have been derived to allow the procedure to be applied to complex pathway configurations. Modifications to the elasticity terms used in the equations have been theoretically justified so that the method remains valid for pathways with conserved metabolites (for example, the adenine nucleotide pool or the intermediates of a catalytic cycle such as the tricarboxylic acid cycle) or with pools of metabolites kept very near to equilibrium by very rapid reactions. The matrix equations generated using these theorems and relationships may be solved algebraically or numerically. Algebraic solutions have been used to determine the factors responsible for the degree of amplification of flux control coefficients by substrate cycles and to show that it is possible to derive expressions for the elasticities of a group of enzymes.