Global dynamic optimization approach to predict activation in metabolic pathways

Background

During the last decade, a number of authors have shown that the genetic regulation of metabolic networks may follow optimality principles. Optimal control theory has been succesfully used to compute optimal enzyme profiles considering simple metabolic pathways. However, applying this optimal control framework to more general networks (e.g. branched networks, or networks incorporating enzyme production dynamics) yields problems that are analytically intractable and/or numerically very challenging. Further, these previous studies have only considered a single-objective framework.

Results

In this work we consider a more general multi-objective formulation and we present solutions based on recent developments in global dynamic optimization techniques. We illustrate the performance and capabilities of these techniques considering two sets of problems. First, we consider a set of single-objective examples of increasing complexity taken from the recent literature. We analyze the multimodal character of the associated non linear optimization problems, and we also evaluate different global optimization approaches in terms of numerical robustness, efficiency and scalability. Second, we consider generalized multi-objective formulations for several examples, and we show how this framework results in more biologically meaningful results.

Conclusions

The proposed strategy was used to solve a set of single-objective case studies related to unbranched and branched metabolic networks of different levels of complexity. All problems were successfully solved in reasonable computation times with our global dynamic optimization approach, reaching solutions which were comparable or better than those reported in previous literature. Further, we considered, for the first time, multi-objective formulations, illustrating how activation in metabolic pathways can be explained in terms of the best trade-offs between conflicting objectives. This new methodology can be applied to metabolic networks with arbitrary topologies, non-linear dynamics and constraints.     

A generalized computational framework to streamline thermodynamics and kinetics analysis of metabolic pathways

Metabolic engineering is a critical biotechnological approach in addressing global energy and environment challenges. Most engineering efforts, however, consist of laborious and inefficient trial-and-error of target pathways, due in part to the lack of methodologies that can comprehensively assess pathway properties in thermodynamics and kinetics. Metabolic engineering can benefit from computational tools that evaluate feasibility, expense and stability of non-natural metabolic pathways. Such tools can also help us understand natural pathways and their regulation at systems level. Here we introduce a computational toolbox, PathParser, which, for the first time, integrates multiple important functions for pathway analysis including thermodynamics analysis, kinetics-based protein cost optimization and robustness analysis. Specifically, PathParser enables optimization of the driving force of a pathway by minimizing the Gibbs free energy of least thermodynamically favorable reaction. In addition, based on reaction thermodynamics and enzyme kinetics, it can compute the minimal enzyme protein cost that supports metabolic flux, and evaluate pathway stability and flux in response to enzyme concentration perturbations. In a demo analysis of the Calvin–Benson–Bassham cycle and photorespiration pathway in the model cyanobacterium Synechocystis PCC 6803, the computation results are corroborated by experimental proteomics data as well as metabolic engineering outcomes. This toolbox may have broad application in metabolic engineering and systems biology in other microbial systems.     

Metabolic systems cost-benefit analysis for interpreting network structure and regulation

MOTIVATION: Interpretation of bioinformatics data in terms of cellular function is a major challenge facing systems biology. This question is complicated by robust metabolic networks filled with structural features like parallel pathways and isozymes. Under conditions of nutrient sufficiency, metabolic networks are well known to be regulated for thermodynamic efficiency however; efficient biochemical pathways are anabolically expensive to construct. While parameters like thermodynamic efficiency have been extensively studied, a systems-based analysis of anabolic proteome synthesis 'costs' and the cellular function implications of these costs has not been reported. RESULTS: A cost-benefit analysis of an in silico Escherichia coli network revealed the relationship between metabolic pathway proteome synthesis requirements, DNA-coding sequence length, thermodynamic efficiency and substrate affinity. The results highlight basic metabolic network design principles. Pathway proteome synthesis requirements appear to have shaped biochemical network structure and regulation. Under conditions of nutrient scarcity and other general stresses, E. coli expresses pathways with relatively inexpensive proteome synthesis requirements instead of more efficient but also anabolically more expensive pathways. This evolutionary strategy provides a cellular function-based explanation for common network motifs like isozymes and parallel pathways and possibly explains 'overflow' metabolisms observed during nutrient scarcity. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Monte-Carlo Modeling of the Central Carbon Metabolism of Lactococcus lactis: Insights into Metabolic Regulation

Metabolic pathways are complex dynamic systems whose response to perturbations and environmental challenges are governed by multiple interdependencies between enzyme properties, reactions rates, and substrate levels. Understanding the dynamics arising from such a network can be greatly enhanced by the construction of a computational model that embodies the properties of the respective system. Such models aim to incorporate mechanistic details of cellular interactions to mimic the temporal behavior of the biochemical reaction system and usually require substantial knowledge of kinetic parameters to allow meaningful conclusions. Several approaches have been suggested to overcome the severe data requirements of kinetic modeling, including the use of approximative kinetics and Monte-Carlo sampling of reaction parameters. In this work, we employ a probabilistic approach to study the response of a complex metabolic system, the central metabolism of the lactic acid bacterium Lactococcus lactis, subject to perturbations and brief periods of starvation. Supplementing existing methodologies, we show that it is possible to acquire a detailed understanding of the control properties of a corresponding metabolic pathway model that is directly based on experimental observations. In particular, we delineate the role of enzymatic regulation to maintain metabolic stability and metabolic recovery after periods of starvation. It is shown that the feedforward activation of the pyruvate kinase by fructose-1,6-bisphosphate qualitatively alters the bifurcation structure of the corresponding pathway model, indicating a crucial role of enzymatic regulation to prevent metabolic collapse for low external concentrations of glucose. We argue that similar probabilistic methodologies will help our understanding of dynamic properties of small-, medium- and large-scale metabolic networks models.     

Elementary mode analysis: a useful metabolic pathway analysis tool for characterizing cellular metabolism

Elementary mode analysis is a useful metabolic pathway analysis tool to identify the structure of a metabolic network that links the cellular phenotype to the corresponding genotype. The analysis can decompose the intricate metabolic network comprised of highly interconnected reactions into uniquely organized pathways. These pathways consisting of a minimal set of enzymes that can support steady state operation of cellular metabolism represent independent cellular physiological states. Such pathway definition provides a rigorous basis to systematically characterize cellular phenotypes, metabolic network regulation, robustness, and fragility that facilitate understanding of cell physiology and implementation of metabolic engineering strategies. This mini-review aims to overview the development and application of elementary mode analysis as a metabolic pathway analysis tool in studying cell physiology and as a basis of metabolic engineering.     

Kinetic-model-based pathway optimization with application to reverse glycolysis in mammalian cells

Over the last two decades, model-based metabolic pathway optimization tools have been developed for the design of microorganisms to produce desired metabolites. However, few have considered more complex cellular systems such as mammalian cells, which requires the use of nonlinear kinetic models to capture the effects of concentration changes and cross-regulatory interactions. In this study, we develop a new two-stage pathway optimization framework based on kinetic models that incorporate detailed kinetics and regulation information. In Stage 1, a set of optimization problems are solved to identify and rank the enzymes that contribute the most to achieving the metabolic objective. Stage 2 then determines the optimal enzyme interventions for specified desired numbers of enzyme adjustments. It also incorporates multi-scenario optimization, which allows the simultaneous consideration of multiple physiological conditions. We apply the proposed framework to find enzyme adjustments that enable a reverse glucose flow in cultured mammalian cells, thereby eliminating the need for glucose feed in the late culture stage and enhancing process robustness. The computational results demonstrate the efficacy of the proposed approach; it not only captures the important regulations and key enzymes for reverse glycolysis but also identifies differences and commonalities in the metabolic requirements for different carbon sources.     

ROS and pentose phosphate pathway: mathematical modelling of the metabolic regulation in response to xenobiotic-induced oxidative stress and the proposed Impact of the gluconate shunt

Abstract

Elevated intracellular levels of reactive oxygen species (ROS), e.g. resulting from exposure to xenobiotics, can cause severe damages. Antioxidant defence mechanisms, which involve regulation of enzyme activities, protect cells to a certain extent. Nevertheless, continuous or increased exposure can overwhelm this system resulting in an adverse cellular state. To simulate exposure scenarios and to investigate the transition to an adverse cellular state, a mathematical model for the dynamics of ROS in response to xenobiotic-induced oxidative stress has been developed. It is based on exposure experiments of human urothelial cells (RT4) to the nitrated polycyclic aromatic hydrocarbon 3-nitrobenzanthrone (3-NBA), a component of diesel engine exhaust, and takes into account the following metabolic pathways of the antioxidant defence system: glutathione redox cycle scavenging directly ROS, the pentose phosphate pathway and the gluconate shunt as NADPH supplier and the beginning of glycolysis. In addition, ROS generation due to the bioactivation of 3-NBA has been implemented. The regulation of enzyme activities plays an important role in the presented mathematical model. The in silico model consists of ordinary differential equations on the basis of enzyme kinetics and mass action for the metabolism of 3-NBA. Parameters are either estimated from performed in vitro experiments via least-squares fitting or obtained from the literature. The results underline the importance of the pentose phosphate pathway to cope with oxidative stress and suggest an important role of the gluconate shunt during low-dose exposure.     

Scaffold proteins in mammalian MAP kinase cascades

The mitogen-activated protein kinase (MAPK) signaling pathway, which is conserved from yeast to humans, is activated in response to a variety of extra- and intracellular stimuli, and plays key roles in multiple cellular processes, including proliferation, differentiation, and apoptosis. The MAPK pathway transmits its signal through the sequential phosphorylation of MAPK kinase kinase to MAPK kinase to MAPK. Specific and efficient activation of the MAPK cascades is crucial for proper cellular responses to stimuli. As shown in yeast, the mammalian MAPK signaling system may also employ scaffold proteins, in part, to organize the MAPK signaling components into functional MAPK modules, thereby enabling the efficient activation of specific MAPK pathways. This review article describes recent advances in the study of potential mammalian scaffold proteins that may help us understand the complex regulation, including the spatial and temporal control, of the mammalian MAPK signaling pathways.     

Modelling the optimal timing in metabolic pathway activation—Use of Pontryagin's Maximum Principle and role of the Golden section

The time course of enzyme concentrations in metabolic pathways can be predicted on the basis of the optimality criterion of minimizing the time period in which an essential product is generated. This criterion is in line with the widely accepted view that high fitness requires high pathway flux. Here, based on Pontryagin's Maximum Principle, a method is developed to solve the corresponding constrained optimal control problem in an almost exclusively analytical way and, thus, to calculate optimal enzyme profiles, when linear, irreversible rate laws are assumed. Three different problem formulations are considered and the corresponding optimization results are derived. Besides the minimization of transition time, we consider an operation time in which 90% of the substrate has been converted into product. In that case, only the enzyme at the lower end of the pathway rather than all enzymes are active in the last phase. In all cases, biphasic or multiphasic time courses are obtained. The biological meaning of the results in terms of a consecutive just-in-time expression of metabolic genes is discussed. For the special case of two-enzyme systems, the role of the Golden section in the solution is outlined.     

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.     

MetaCyc and AraCyc. Metabolic pathway databases for plant research

MetaCyc (http://metacyc.org) contains experimentally determined biochemical pathways to be used as a reference database for metabolism. In conjunction with the Pathway Tools software, MetaCyc can be used to computationally predict the metabolic pathway complement of an annotated genome. To increase the breadth of pathways and enzymes, more than 60 plant-specific pathways have been added or updated in MetaCyc recently. In contrast to MetaCyc, which contains metabolic data for a wide range of organisms, AraCyc is a species-specific database containing only enzymes and pathways found in the model plant Arabidopsis (Arabidopsis thaliana). AraCyc (http://arabidopsis.org/tools/aracyc/) was the first computationally predicted plant metabolism database derived from MetaCyc. Since its initial computational build, AraCyc has been under continued curation to enhance data quality and to increase breadth of pathway coverage. Twenty-eight pathways have been manually curated from the literature recently. Pathway predictions in AraCyc have also been recently updated with the latest functional annotations of Arabidopsis genes that use controlled vocabulary and literature evidence. AraCyc currently features 1,418 unique genes mapped onto 204 pathways with 1,156 literature citations. The Omics Viewer, a user data visualization and analysis tool, allows a list of genes, enzymes, or metabolites with experimental values to be painted on a diagram of the full pathway map of AraCyc. Other recent enhancements to both MetaCyc and AraCyc include implementation of an evidence ontology, which has been used to provide information on data quality, expansion of the secondary metabolism node of the pathway ontology to accommodate curation of secondary metabolic pathways, and enhancement of the cellular component ontology for storing and displaying enzyme and pathway locations within subcellular compartments.     

Alteration of Mitochondrial Proteome Due to Activation of Notch1 Signaling Pathway

The Notch signaling pathway, a known regulator of cell fate decisions, proliferation, and apoptosis, has recently been implicated in the regulation of glycolysis, which affects tumor progression. However, the impact of Notch on other metabolic pathways remains to be elucidated. To gain more insights into the Notch signaling and its role in regulation of metabolism, we studied the mitochondrial proteome in Notch1-activated K562 cells using a comparative proteomics approach. The proteomic study led to the identification of 10 unique proteins that were altered due to Notch1 activation. Eight of these proteins belonged to mitochondria-localized metabolic pathways like oxidative phosphorylation, glutamine metabolism, Krebs cycle, and fatty acid oxidation. Validation of some of these findings showed that constitutive activation of Notch1 deregulated glutamine metabolism and Complex 1 of the respiratory chain. Furthermore, the deregulation of glutamine metabolism involved the canonical Notch signaling and its downstream effectors. The study also reports the effect of Notch signaling on mitochondrial function and status of high energy intermediates ATP, NADH, and NADPH. Thus our study shows the effect of Notch signaling on mitochondrial proteome, which in turn affects the functioning of key metabolic pathways, thereby connecting an important signaling pathway to the regulation of cellular metabolism.     

Applications of genetically-encoded biosensors for the construction and control of biosynthetic pathways

Cells are filled with biosensors, molecular systems that measure the state of the cell and respond by regulating host processes. In much the same way that an engineer would monitor a chemical reactor, the cell uses these sensors to monitor changing intracellular environments and produce consistent behavior despite the variable environment. While natural systems derive a clear benefit from pathway regulation, past research efforts in engineering cellular metabolism have focused on introducing new pathways and removing existing pathway regulation. Synthetic biology is a rapidly growing field that focuses on the development of new tools that support the design, construction, and optimization of biological systems. Recent advances have been made in the design of genetically-encoded biosensors and the application of this class of molecular tools for optimizing and regulating heterologous pathways. Biosensors to cellular metabolites can be taken directly from natural systems, engineered from natural sensors, or constructed entirely in vitro. When linked to reporters, such as antibiotic resistance markers, these metabolite sensors can be used to report on pathway productivity, allowing high-throughput screening for pathway optimization. Future directions will focus on the application of biosensors to introduce feedback control into metabolic pathways, providing dynamic control strategies to increase the efficient use of cellular resources and pathway reliability.     

Engineering the spatial organization of metabolic enzymes: mimicking nature's synergy

A growing body of evidence indicates that many cellular reactions within metabolic pathways are catalyzed not by free-floating 'soluble' enzymes, but via one or more membrane-associated multienzyme complexes. This type of macromolecular organization has important implications for the overall efficiency, specificity, and regulation of metabolic pathways. An ever-increasing number of biochemical and genetic studies on primary and secondary metabolism have laid a solid foundation for this model, providing compelling evidence in favor of the so-called channeling of intermediates between enzyme active sites and colocalization of enzymes inside a cell. In this review, we discuss several of nature's most notable multifunctional enzyme systems including the AROM complex and tryptophan synthase, each of which provides new fundamental insights into the structural organization of metabolic machinery within living cells. We then focus on the growing body of literature related to engineering strategies using protein chimeras and post-translational assembly mechanisms. Common among these techniques is the desire to mimic natural enzyme organization for optimizing the production of valuable metabolites with industrial and medical importance.     

Using isotopomer path tracing to quantify metabolic fluxes in pathway models containing reversible reactions

As a more complete picture of the genetic and enzymatic composition of cells becomes available, there is a growing need to describe how cellular regulatory elements interact with the cellular environment to affect cell physiology. One means for describing intracellular regulatory mechanisms is concurrent measurement of multiple metabolic pathways and their interactions by metabolic flux analysis. Flux of carbon through a metabolic pathway responds to all cellular regulatory systems, including changes in enzyme and substrate concentrations, enzyme activation or inhibition, and ultimately genetic control. The extent to which metabolic flux analysis can describe cellular physiology depends on the number of pathways in the model and the quality of the data. Intracellular information is obtainable from isotopic tracer experiments, the most extensive being the determination of the isotopomer distribution, or specific labeling pattern, of intracellular metabolites. We present a rapid and novel solution method that determines the flux of carbon through complex pathway models using isotopomer data. This time-consuming problem was solved with the introduction of isotopomer path tracing, which drastically reduces the number of isotopomer variables to the number of isotopomers observed experimentally. We propose a partitioned solution method that takes advantage of the nearly linear relationship between fluxes and isotopomers. Whereas the stoichiometric matrix and the isotopomer matrix are invertible, simulated annealing and the Newton-Raphson method are used for the nonlinear components. Reversible reactions are described by a new parameter, the association factor, which scales hyperbolically with the rate of metabolite exchange. Automating the solution method permits a variety of models to be compared, thus enhancing the accuracy of results. A simplified example that contains all of the complexities of a comprehensive pathway model is presented. Copyright John Wiley & Sons, Inc.     

大肠杆菌苏氨酸合成途径动力学模型的构建与分析

动力学模型分析有利于理解生物系统的调控机制,从而为高效细胞工厂的理性设计提供指导。基于以往发表的相关途径动力学模型和测量的酶动力学数据,开发了大肠杆菌苏氨酸合成途径的动力学模型。模型包含从天冬氨酸至苏氨酸的合成途径及葡萄糖开始的为合成途径提供前体以及能量的代谢途径。与以往模型不同的是新模型中考虑了能量和还原力的平衡,从而使模型模拟的系统自身成为一个不需要从外界提供能量和还原力的自洽系统。模型稳态分析的结果表明PTS、G6PDH和HDH等反应对苏氨酸合成反应的通量控制系数较大,通过过表达这些反应的酶可以有效增加苏氨酸合成反应的通量。     

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.     

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.