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

Increasing oil content in Brassica oilseed species

Brassica oilseed species are the third most important in the world, providing approximately 15 % of the total vegetable oils. Three species (Brassica rapa, B. juncea, B. napus) dominate with B. napus being the most common in Canada, China and Europe. Originally, B. napus was a crop producing seed with high erucic acid content, which still persists today, to some extent, and is used for industrial purposes. In contrast, cultivars which produce seed used for food and feed are low erucic acid cultivars which also have reduced glucosinolate content. Because of the limit to agricultural land, recent efforts have been made to increase productivity of oil crops, including Brassica oilseed species. In this article, we have detailed research in this regard. We have covered modern genetic, genomic and metabolic control analysis approaches to identifying potential targets for the manipulation of seed oil content. Details of work on the use of quantitative trait loci, genome-wide association and comparative functional genomics to highlight factors influencing seed oil accumulation are given and functional proteins which can affect this process are discussed. In summary, a wide variety of inputs are proving useful for the improvement of Brassica oilseed species, as major sources of global vegetable oil.

Optimal enzyme profiles in unbranched metabolic pathways

How to optimize the allocation of enzymes in metabolic pathways has been a topic of study for many decades. Although the general problem is complex and nonlinear, we have previously shown that it can be solved by convex optimization. In this paper, we focus on unbranched metabolic pathways with simplified enzymatic rate laws and derive analytic solutions to the optimization problem. We revisit existing solutions based on the limit of mass-action rate laws and present new solutions for other rate laws. Furthermore, we revisit a known relationship between flux control coefficients and enzyme abundances in optimal metabolic states. We generalize this relationship to models with density constraints on enzymes and metabolites, and present a new local relationship between optimal reaction elasticities and enzyme amounts. Finally, we apply our theory to derive simple kinetics-based formulae for protein allocation during bacterial growth.

On paradoxes between optimal growth, metabolic control analysis, and flux balance analysis

In Microbiology it is often assumed that growth rate is maximal. This may be taken to suggest that the dependence of the growth rate on every enzyme activity is at the top of an inverse-parabolic function, i.e. that all flux control coefficients should equal zero. This might seem to imply that the sum of these flux control coefficients equals zero. According to the summation law of Metabolic Control Analysis (MCA) the sum of flux control coefficients should equal 1 however. And in Flux Balance Analysis (FBA) catabolism is often limited by a hard bound, causing catabolism to fully control the fluxes, again in apparent contrast with a flux control coefficient of zero. Here we resolve these paradoxes (apparent contradictions) in an analysis that uses the 'Edinburgh pathway', the 'Amsterdam pathway', as well as a generic metabolic network providing the building blocks or Gibbs energy for microbial growth. We review and show that (i) optimization depends on so-called enzyme control coefficients rather than the 'catalytic control coefficients' of MCA's summation law, (ii) when optimization occurs at fixed total protein, the former differ from the latter to the extent that they may all become equal to zero in the optimum state, (iii) in more realistic scenarios of optimization where catalytically inert biomass is compensating or maintenance metabolism is taken into consideration, the optimum enzyme concentrations should not be expected to equal those that maximize the specific growth rate, (iv) optimization may be in terms of yield rather than specific growth rate, which resolves the paradox because the sum of catalytic control coefficients on yield equals 0, (v) FBA effectively maximizes growth yield, and for yield the summation law states 0 rather than 1, thereby removing the paradox, (vi) furthermore, FBA then comes more often to a 'hard optimum' defined by a maximum catabolic flux and a catabolic-enzyme control coefficient of 1. The trade-off between maintenance metabolism and growth is highlighted as worthy of further analysis.

Whole-cell metabolic control analysis

Since its conception some fifty years ago, metabolic control analysis (MCA) aims to understand how cells control their metabolism by adjusting the activity of their enzymes. Here we extend its scope to a whole-cell context. We consider metabolism in the evolutionary context of growth-rate maximisation by optimisation of protein concentrations. This framework allows for the prediction of flux control coefficients from proteomics data or stoichiometric modelling. Since genes compete for finite biosynthetic resources, we treat all protein concentrations as interdependent. We show that elementary flux modes (EFMs) emerge naturally as the optimal metabolic networks in the whole-cell context and we derive their control properties. In the evolutionary optimum, the number of expressed EFMs is determined by the number of protein-concentration constraints that limit growth rate. We use published glucose-limited chemostat data of S. cerevisiae to illustrate that it uses only two EFMs prior to the onset of fermentation and that it uses four EFMs during fermentation. We discuss published enzyme-titration data to show that S. cerevisiae and E. coli indeed can express proteins at growth-rate maximising concentrations. Accordingly, we extend MCA to elementary flux modes operating at an optimal state. We find that the expression of growth-unassociated proteins changes results from classical metabolic control analysis. Finally, we show how flux control coefficients can be estimated from proteomics and ribosome-profiling data. We analyse published proteomics data of E. coli to provide a whole-cell perspective of the control of metabolic enzymes on growth rate. We hope that this paper stimulates a renewed interest in metabolic control analysis, so that it can serve again the purpose it once had: to identify general principles that emerge from the biochemistry of the cell and are conserved across biological species.

The genotype-phenotype relationship and evolutionary genetics in the light of the Metabolic Control Analysis

Metabolic control analysis has long been used as a systemic model of the genotype-phenotype (GP) relationship. By considering kinetic parameters and enzyme concentrations as reflecting the genotype level and metabolic fluxes or pools as phenotypes related to fitness, MCA has given a biological basis to the relationship between these two levels. The non-linear and concave relationship between enzymes and fluxes can account for common genetic effects that reductionist approaches have been powerless to explain, such as the dominance of active alleles over less active alleles, the various types of epistasis and heterosis, and reveals the structural links between these genetic effects. The summation property of the flux control coefficients accounts for the L-shaped distribution of Quantitative Trait Locus (QTL) effects, irrespective of other possible causes. Metabolic models of response to selection results in evolutionary scenarios that are markedly different from those derived from the classical infinitesimal model of quantitative genetics. In particular, evolution towards selective neutrality appears to be a consequence of the diminishing return of the flux-enzyme relationship. In this paper, we survey the historical and recent achievements of MCA in genetics, quantitative genetics and evolution, focusing on epistasis and the evolution of flux in relation to enzyme concentrations.

Mathematical properties of optimal fluxes in cellular reaction networks at balanced growth

The physiology of biological cells evolved under physical and chemical constraints, such as mass conservation across the network of biochemical reactions, nonlinear reaction kinetics, and limits on cell density. For unicellular organisms, the fitness that governs this evolution is mainly determined by the balanced cellular growth rate. We previously introduced growth balance analysis (GBA) as a general framework to model and analyze such nonlinear systems, revealing important analytical properties of optimal balanced growth states. It has been shown that at optimality, only a minimal subset of reactions can have nonzero flux. However, no general principles have been established to determine if a specific reaction is active at optimality. Here, we extend the GBA framework to study the optimality of each biochemical reaction, and we identify the mathematical conditions determining whether a reaction is active or not at optimal growth in a given environment. We reformulate the mathematical problem in terms of a minimal number of dimensionless variables and use the Karush-Kuhn-Tucker (KKT) conditions to identify fundamental principles of optimal resource allocation in GBA models of any size and complexity. Our approach helps to identify from first principles the economic values of biochemical reactions, expressed as marginal changes in cellular growth rate; these economic values can be related to the costs and benefits of proteome allocation into the reactions' catalysts. Our formulation also generalizes the concepts of Metabolic Control Analysis to models of growing cells. We show how the extended GBA framework unifies and extends previous approaches of cellular modeling and analysis, putting forward a program to analyze cellular growth through the stationarity conditions of a Lagrangian function. GBA thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular growth.

Metabolic Control Analysis of triacylglycerol accumulation in oilseed rape

The increasing global demand for vegetable oils will only be met if there are significant improvements in the productivity of the major oil crops, such as oilseed rape. Metabolic engineering offers the prospect of further gains in yield beyond that already achieved by breeding and selection but requires guidance as to the changes that need to be made. Metabolic Control Analysis, through measurement and estimation of flux control coefficients, can indicate which enzymes have the most influence on a desired flux. Some experiments have previously reported flux control coefficients for oil accumulation in the seeds of oilseed rape, and others have measured control coefficient distributions for multi-enzyme segments of oil synthesis in seed embryo metabolism measured in vitro. In addition, other reported manipulations of oil accumulation contain results that are exploited further here to calculate previously unknown flux control coefficients. These results are then assembled within a framework that allows an integrated interpretation of the controls on oil accumulation from the assimilation of CO₂ to deposition of oil in the seed. The analysis shows that the control is distributed to an extent that the gains from amplifying any single target are necessarily limited, but there are candidates for joint amplification that are likely to act synergistically to produce much more significant gains.

Structural Thermokinetic Modelling

To translate metabolic networks into dynamic models, the Structural Kinetic Modelling framework (SKM) assumes a given reference state and replaces the reaction elasticities in this state by random numbers. A new variant, called Structural Thermokinetic Modelling (STM), accounts for reversible reactions and thermodynamics. STM relies on a dependence schema in which some basic variables are sampled, fitted to data, or optimised, while all other variables can be easily computed. Correlated elasticities follow from enzyme saturation values and thermodynamic forces, which are physically independent. Probability distributions in the dependence schema define a model ensemble, which allows for probabilistic predictions even if data are scarce. STM highlights the importance of variabilities, dependencies, and covariances of biological variables. By varying network structure, fluxes, thermodynamic forces, regulation, or types of rate laws, the effects of these model features can be assessed. By choosing the basic variables, metabolic networks can be converted into kinetic models with consistent reversible rate laws. Metabolic control coefficients obtained from these models can tell us about metabolic dynamics, including responses and optimal adaptations to perturbations, enzyme synergies and metabolite correlations, as well as metabolic fluctuations arising from chemical noise. To showcase STM, I study metabolic control, metabolic fluctuations, and enzyme synergies, and how they are shaped by thermodynamic forces. Considering thermodynamics can improve predictions of flux control, enzyme synergies, correlated flux and metabolite variations, and the emergence and propagation of metabolic noise.

Evolution of enzyme levels in metabolic pathways: A theoretical approach

The central role of metabolism in cell functioning and adaptation has given rise to countless studies on the evolution of enzyme-coding genes and network topology. However, very few studies have addressed the question of how enzyme concentrations change in response to positive selective pressure on the flux, considered a proxy of fitness. In particular, the way cellular constraints, such as resource limitations and co-regulation, affect the adaptive landscape of a pathway under selection has never been analyzed theoretically. To fill this gap, we developed a model of the evolution of enzyme concentrations that combines metabolic control theory and an adaptive dynamics approach, and integrates possible dependencies between enzyme concentrations. We determined the evolutionary equilibria of enzyme concentrations and their range of neutral variation, and showed that they differ with the properties of the enzymes, the constraints applied to the system and the initial enzyme concentrations. Simulations of long-term evolution confirmed all analytical and numerical predictions, even though we relaxed the simplifying assumptions used in the analytical treatment.

Searching for principles of microbial physiology

Why do evolutionarily distinct microorganisms display similar physiological behaviours? Why are transitions from high-ATP yield to low(er)-ATP yield metabolisms so widespread across species? Why is fast growth generally accompanied with low stress tolerance? Do these regularities occur because most microbial species are subject to the same selective pressures and physicochemical constraints? If so, a broadly-applicable theory might be developed that predicts common microbiological behaviours. Microbial systems biologists have been working out the contours of this theory for the last two decades, guided by experimental data. At its foundations lie basic principles from evolutionary biology, enzyme biochemistry, metabolism, cell composition and steady-state growth. The theory makes predictions about fitness costs and benefits of protein expression, physicochemical constraints on cell growth and characteristics of optimal metabolisms that maximise growth rate. Comparisons of the theory with experimental data indicates that microorganisms often aim for maximisation of growth rate, also in the presence of stresses; they often express optimal metabolisms and metabolic proteins at optimal concentrations. This review explains the current status of the theory for microbiologists; its roots, predictions, experimental evidence and future directions.

Is the catalytic activity of triosephosphate isomerase fully optimized? An investigation based on maximization of entropy production

Triosephosphate isomerase (TIM) is often described as a fully evolved housekeeping enzyme with near-maximal possible reaction rate. The assumption that an enzyme is perfectly evolved has not been easy to confirm or refute. In this paper, we use maximization of entropy production within known constraints to examine this assumption by calculating steady-state cyclic flux, corresponding entropy production, and catalytic activity in a reversible four-state scheme of TIM functional states. The maximal entropy production (MaxEP) requirement for any of the first three transitions between TIM functional states leads to decreased total entropy production. Only the MaxEP requirement for the product (R-glyceraldehyde-3-phosphate) release step led to a 30% increase in enzyme activity, specificity constant kcat/KM, and overall entropy production. The product release step, due to the TIM molecular machine working in the physiological direction of glycolysis, has not been identified before as the rate-limiting step by using irreversible thermodynamics. Together with structural studies, our results open the possibility for finding amino acid substitutions leading to an increased frequency of loop six opening and product release.

The Protein Cost of Metabolic Fluxes: Prediction from Enzymatic Rate Laws and Cost Minimization

Bacterial growth depends crucially on metabolic fluxes, which are limited by the cell’s capacity to maintain metabolic enzymes. The necessary enzyme amount per unit flux is a major determinant of metabolic strategies both in evolution and bioengineering. It depends on enzyme parameters (such as k_cat and K_M constants), but also on metabolite concentrations. Moreover, similar amounts of different enzymes might incur different costs for the cell, depending on enzyme-specific properties such as protein size and half-life. Here, we developed enzyme cost minimization (ECM), a scalable method for computing enzyme amounts that support a given metabolic flux at a minimal protein cost. The complex interplay of enzyme and metabolite concentrations, e.g. through thermodynamic driving forces and enzyme saturation, would make it hard to solve this optimization problem directly. By treating enzyme cost as a function of metabolite levels, we formulated ECM as a numerically tractable, convex optimization problem. Its tiered approach allows for building models at different levels of detail, depending on the amount of available data. Validating our method with measured metabolite and protein levels in E. coli central metabolism, we found typical prediction fold errors of 4.1 and 2.6, respectively, for the two kinds of data. This result from the cost-optimized metabolic state is significantly better than randomly sampled metabolite profiles, supporting the hypothesis that enzyme cost is important for the fitness of E. coli. ECM can be used to predict enzyme levels and protein cost in natural and engineered pathways, and could be a valuable computational tool to assist metabolic engineering projects. Furthermore, it establishes a direct connection between protein cost and thermodynamics, and provides a physically plausible and computationally tractable way to include enzyme kinetics into constraint-based metabolic models, where kinetics have usually been ignored or oversimplified.

“Enzyme cost”, the amount of protein needed for a given metabolic flux, is crucial for the metabolic choices cells have to make. However, due to the technical limitations of linear optimization methods, this cost has traditionally been ignored by constraint-based metabolic models such as Flux Balance Analysis. On the other hand, more detailed kinetic models which use ordinary differential equations to simulate fluxes for different choices of enzyme allocation, are computationally demanding and not scalable enough. In this work, we developed a method which utilizes the full kinetic model to predict steady-state enzyme costs, using a scalable and robust algorithm based on convex optimization. We show that the minimization of enzyme cost is a meaningful optimality principle by comparing our predictions to measured enzyme and metabolite levels in exponentially growing E. coli. This method could be used to quantify the enzyme cost of many other pathways and explain why evolution has selected some low-yield metabolic strategies, including aerobic fermentation in yeast and cancer cells. Furthermore, future metabolic engineering projects could benefit from our method by choosing pathways that reduce the total amount of enzyme required for the synthesis of a value-added product.

Evolutionary pressures on microbial metabolic strategies in the chemostat

Protein expression is shaped by evolutionary processes that tune microbial fitness. The limited biosynthetic capacity of a cell constrains protein expression and forces the cell to carefully manage its protein economy. In a chemostat, the physiology of the cell feeds back on the growth conditions, hindering intuitive understanding of how changes in protein concentration affect fitness. Here, we aim to provide a theoretical framework that addresses the selective pressures and optimal evolutionary-strategies in the chemostat. We show that the optimal enzyme levels are the result of a trade-off between the cost of their production and the benefit of their catalytic function. We also show that deviations from optimal enzyme levels are directly related to selection coefficients. The maximal fitness strategy for an organism in the chemostat is to express a well-defined metabolic subsystem known as an elementary flux mode. Using a coarse-grained, kinetic model of Saccharomyces cerevisiae’s metabolism and growth, we illustrate that the dynamics and outcome of evolution in a chemostat can be very counter-intuitive: Strictly-respiring and strictly-fermenting strains can evolve from a common ancestor. This work provides a theoretical framework that relates a kinetic, mechanistic view on metabolism with cellular physiology and evolutionary dynamics in the chemostat.

Visual account of protein investment in cellular functions

Proteomics techniques generate an avalanche of data and promise to satisfy biologists' long-held desire to measure absolute protein abundances on a genome-wide scale. However, can this knowledge be translated into a clearer picture of how cells invest their protein resources? This article aims to give a broad perspective on the composition of proteomes as gleaned from recent quantitative proteomics studies. We describe proteomaps, an approach for visualizing the composition of proteomes with a focus on protein abundances and functions. In proteomaps, each protein is shown as a polygon-shaped tile, with an area representing protein abundance. Functionally related proteins appear in adjacent regions. General trends in proteomes, such as the dominance of metabolism and protein production, become easily visible. We make interactive visualizations of published proteome datasets accessible at www.proteomaps.net. We suggest that evaluating the way protein resources are allocated by various organisms and cell types in different conditions will sharpen our understanding of how and why cells regulate the composition of their proteomes.

Kinetic modeling of riboflavin biosynthesis in Bacillus subtilis under production conditions

To study the network dynamics of the riboflavin biosynthesis pathway and to identify potential bottlenecks in the system, an ordinary differential equation-based model was constructed using available literature data for production strains. The results confirmed that the RibA protein is rate limiting in the pathway. Under the conditions investigated, we determined a potential limiting order of the remaining enzymes under increased RibA concentration (>0.102 mM) and therefore higher riboflavin production (>0.045 mmol g(CDW)(-1) h(-1) and 0.0035 mM s(-1), respectively). The reductase activity of RibG and lumazine synthase (RibH) might be the next most limiting steps. The computational minimization of the enzyme concentrations of the pathway suggested the need for a greater RibH concentration (0.251 mM) compared with the other enzymes (RibG: 0.188 mM, RibB: 0.023 mM).

Elucidating the adaptation and temporal coordination of metabolic pathways using in-silico evolution

Cellular metabolism, the interconversion of small molecules by chemical reactions, is a tightly coordinated process that requires integration of diverse environmental and intracellular cues. While for many organisms the topology of the network of metabolic reactions is increasingly known, the regulatory principles that shape the network's adaptation to diverse and changing environments remain largely elusive. To investigate the principles of metabolic adaptation and regulation in metabolic pathways, we propose a computational approach based on in-silico evolution. Rather than analyzing existing regulatory schemes, we let a population of minimal, prototypical metabolic cells evolve rate constants and appropriate regulatory schemes that allow for optimal growth in static and fluctuating environments. Applying our approach to a small, but already sufficiently complex, minimal system reveals intricate transitions between metabolic modes. These results have implications for trade-offs in resource allocation. Going from static to varying environments, we show that for fluctuating nutrient availability, active metabolic regulation results in a significantly increased overall rate of metabolism.

Model of tryptophan metabolism, readily scalable using tissue-specific gene expression data

Tryptophan is utilized in various metabolic routes including protein synthesis, serotonin, and melatonin synthesis and the kynurenine pathway. Perturbations in these pathways have been associated with neurodegenerative diseases and cancer. Here we present a comprehensive kinetic model of the complex network of human tryptophan metabolism based upon existing kinetic data for all enzymatic conversions and transporters. By integrating tissue-specific expression data, modeling tryptophan metabolism in liver and brain returned intermediate metabolite concentrations in the physiological range. Sensitivity and metabolic control analyses identified expected key enzymes to govern fluxes in the branches of the network. Combining tissue-specific models revealed a considerable impact of the kynurenine pathway in liver on the concentrations of neuroactive derivatives in the brain. Moreover, using expression data from a cancer study predicted metabolite changes that resembled the experimental observations. We conclude that the combination of the kinetic model with expression data represents a powerful diagnostic tool to predict alterations in tryptophan metabolism. The model is readily scalable to include more tissues, thereby enabling assessment of organismal tryptophan metabolism in health and disease.

How biochemical constraints of cellular growth shape evolutionary adaptations in metabolism

Evolutionary adaptations in metabolic networks are fundamental to evolution of microbial growth. Studies on unneeded-protein synthesis indicate reductions in fitness upon nonfunctional protein synthesis, showing that cell growth is limited by constraints acting on cellular protein content. Here, we present a theory for optimal metabolic enzyme activity when cells are selected for maximal growth rate given such growth-limiting biochemical constraints. We show how optimal enzyme levels can be understood to result from an enzyme benefit minus cost optimization. The constraints we consider originate from different biochemical aspects of microbial growth, such as competition for limiting amounts of ribosomes or RNA polymerases, or limitations in available energy. Enzyme benefit is related to its kinetics and its importance for fitness, while enzyme cost expresses to what extent resource consumption reduces fitness through constraint-induced reductions of other enzyme levels. A metabolic fitness landscape is introduced to define the fitness potential of an enzyme. This concept is related to the selection coefficient of the enzyme and can be expressed in terms of its fitness benefit and cost.

Systems biology of lactic acid bacteria: a critical review

Understanding the properties of a system as emerging from the interaction of well described parts is the most important goal of Systems Biology. Although in the practice of Lactic Acid Bacteria (LAB) physiology we most often think of the parts as the proteins and metabolites, a wider interpretation of what a part is can be useful. For example, different strains or species can be the parts of a community, or we could study only the chemical reactions as the parts of metabolism (and forgetting about the enzymes that catalyze them), as is done in flux balance analysis. As long as we have some understanding of the properties of these parts, we can investigate whether their interaction leads to novel or unanticipated behaviour of the system that they constitute.

There has been a tendency in the Systems Biology community to think that the collection and integration of data should continue ad infinitum, or that we will otherwise not be able to understand the systems that we study in their details. However, it may sometimes be useful to take a step back and consider whether the knowledge that we already have may not explain the system behaviour that we find so intriguing. Reasoning about systems can be difficult, and may require the application of mathematical techniques. The reward is sometimes the realization of unexpected conclusions, or in the worst case, that we still do not know enough details of the parts, or of the interactions between them.

We will discuss a number of cases, with a focus on LAB-related work, where a typical systems approach has brought new knowledge or perspective, often counterintuitive, and clashing with conclusions from simpler approaches. Also novel types of testable hypotheses may be generated by the systems approach, which we will illustrate. Finally we will give an outlook on the fields of research where the systems approach may point the way for the near future.

Molecular-level tradeoffs and metabolic adaptation to simultaneous stressors

Life is a dynamic process driven by the complex interplay between physical constraints and selection pressures, ranging from nutrient limitation to inhibitory substances to predators. These stressors are not mutually exclusive; microbes have faced concurrent challenges for eons. Genome-enabled systems biology approaches are adapting economic and ecological concepts like tradeoff curves and strategic resource allocation theory to analyze metabolic adaptations to simultaneous stressors. These methodologies can accurately describe and predict metabolic adaptations to concurrent stresses by considering the tradeoff between investment of limiting resources into enzymatic machinery and the resulting cellular function. The approaches represent promising links between computational biology and well-established economic and ecological methodologies for analyzing the interplay between physical constraints and microbial fitness.

Environmental selection of the feed-forward loop circuit in gene-regulation networks

Gene-regulation networks contain recurring elementary circuits termed network motifs. It is of interest to understand under which environmental conditions each motif might be selected. To address this, we study one of the most significant network motifs, a three-gene circuit called the coherent feed-forward loop (FFL). The FFL has been demonstrated theoretically and experimentally to perform a basic information-processing function: it shows a delay following ON steps of an input inducer, but not after OFF steps. Here, we ask under what environmental conditions might the FFL be selected over simpler gene circuits, based on this function. We employ a theoretical cost-benefit analysis for the selection of gene circuits in a given environment. We find conditions that the environment must satisfy in order for the FFL to be selected over simpler circuits: the FFL is selected in environments where the distribution of the input pulse duration is sufficiently broad and contains both long and short pulses. Optimal values of the biochemical parameters of the FFL circuit are determined as a function of the environment such that the delay in the FFL blocks deleterious short pulses of induction. This approach can be generally used to study the evolutionary selection of other network motifs.

Mathematical modeling of living cell metabolism using the method of steady-state stoichiometric flux balance

This approach uses a set of algebraic linear equations for reaction rates (the method of steady-state stoichiometric flux balance) to model the purposeful metabolism of the living self-reproducing biochemical system (i.e. cell), which persists in steady-state growth. Linear programming (SIMPLEX method) is used to derive the solution for the model equations set (determining reaction rates which provide flux balance at given conditions). Here, we demonstrate the approach through the mathematical modeling of steady-state metabolism in Saccharomyces cerevisiae mitochondria.

Use of physiological constraints to identify quantitative design principles for gene expression in yeast adaptation to heat shock

Background

Understanding the relationship between gene expression changes, enzyme activity shifts, and the corresponding physiological adaptive response of organisms to environmental cues is crucial in explaining how cells cope with stress. For example, adaptation of yeast to heat shock involves a characteristic profile of changes to the expression levels of genes coding for enzymes of the glycolytic pathway and some of its branches. The experimental determination of changes in gene expression profiles provides a descriptive picture of the adaptive response to stress. However, it does not explain why a particular profile is selected for any given response.

Results

We used mathematical models and analysis of in silico gene expression profiles (GEPs) to understand how changes in gene expression correlate to an efficient response of yeast cells to heat shock. An exhaustive set of GEPs, matched with the corresponding set of enzyme activities, was simulated and analyzed. The effectiveness of each profile in the response to heat shock was evaluated according to relevant physiological and functional criteria. The small subset of GEPs that lead to effective physiological responses after heat shock was identified as the result of the tuning of several evolutionary criteria. The experimentally observed transcriptional changes in response to heat shock belong to this set and can be explained by quantitative design principles at the physiological level that ultimately constrain changes in gene expression.

Conclusion

Our theoretical approach suggests a method for understanding the combined effect of changes in the expression of multiple genes on the activity of metabolic pathways, and consequently on the adaptation of cellular metabolism to heat shock. This method identifies quantitative design principles that facilitate understating the response of the cell to stress.

Evolution of enzymes in a series is driven by dissimilar functional demands

That distinct enzyme activities in an unbranched metabolic pathway are evolutionarily tuned to a single functional requirement is a pervasive assumption. Here we test this assumption by examining the activities of two consecutively acting enzymes in human erythrocytes with an approach to quantitative evolutionary design that avoids the above-mentioned assumption. We previously found that avoidance of NADPH depletion during the pulses of oxidative load to which erythrocytes are normally exposed is the main functional requirement mediating selection for high glucose-6-phosphate dehydrogenase activity. In the present study, we find that, in contrast, the maintenance of oxidized glutathione at low concentrations is the main functional requirement mediating selection for high glutathione reductase activity. The results in this case show that, contrary to the assumption of a single functional requirement, natural selection for the normal activities of the distinct enzymes in the pathway is mediated by different requirements. On the other hand, the results agree with the more general principles that underlie our approach. Namely, that (i) the values of biochemical parameters evolve so as to fulfill the various performance requirements that are relevant to achieve high fitness, and (ii) these performance requirements can be inferred from quantitative systems theory considerations, informed by knowledge of specific aspects of the biochemistry, physiology, genetics, and ecology of the organism.

A theory of optimal differential gene expression

We investigate a model of optimal regulation, intended to describe large-scale differential gene expression. Relations between the optimal expression patterns and the function of genes are deduced from an optimality principle: the regulators have to maximise a fitness function which they influence directly via a cost term, and indirectly via their control on important cell variables, such as metabolic fluxes. According to the model, the optimal linear response to small perturbations reflects the regulators' functions, namely their linear influences on the cell variables. The optimal behaviour can be realised by a linear feedback mechanism. Known or assumed properties of response coefficients lead to predictions about regulation patterns. A symmetry relation predicted for deletion experiments is verified with gene expression data. Where the optimality assumption is valid, our results justify the use of expression data for functional annotation and for pathway reconstruction and suggest the use of linear factor models for the analysis of gene expression data.

An extension to the metabolic control theory taking into account correlations between enzyme concentrations

The classical metabolic control theory [Kacser, H. & Burns, J.A. (1973) Symp. Soc. Exp. Biol.27, 65-104; Heinrich, R. & Rapoport, T. (1974) Eur. J. Biochem.42, 89-95.] does not take into account experimental evidence for correlations between enzyme concentrations in the cell. We investigated the implications of two causes of linear correlations: competition between enzymes, which is a mere physical adaptation of the cell to the limitation of resources and space, and regulatory correlations, which result from the existence of regulatory networks. These correlations generate redistribution of enzyme concentrations when the concentration of an enzyme varies; this may dramatically alter the flux and metabolite concentration curves. In particular, negative correlations cause the flux to have a maximum value for a defined distribution of enzyme concentrations. Redistribution coefficients of enzyme concentrations allowed us to calculate the 'combined response coefficient' that quantifies the response of flux or metabolite concentration to a perturbation of enzyme concentration.

The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks

Cellular functions are ultimately linked to metabolic fluxes brought about by thousands of chemical reactions and transport processes. The synthesis of the underlying enzymes and membrane transporters causes the cell a certain 'effort' of energy and external resources. Considering that those cells should have had a selection advantage during natural evolution that enabled them to fulfil vital functions (such as growth, defence against toxic compounds, repair of DNA alterations, etc.) with minimal effort, one may postulate the principle of flux minimization, as follows: given the available external substrates and given a set of functionally important 'target' fluxes required to accomplish a specific pattern of cellular functions, the stationary metabolic fluxes have to become a minimum. To convert this principle into a mathematical method enabling the prediction of stationary metabolic fluxes, the total flux in the network is measured by a weighted linear combination of all individual fluxes whereby the thermodynamic equilibrium constants are used as weighting factors, i.e. the more the thermodynamic equilibrium lies on the right-hand side of the reaction, the larger the weighting factor for the backward reaction. A linear programming technique is applied to minimize the total flux at fixed values of the target fluxes and under the constraint of flux balance (= steady-state conditions) with respect to all metabolites. The theoretical concept is applied to two metabolic schemes: the energy and redox metabolism of erythrocytes, and the central metabolism of Methylobacterium extorquens AM1. The flux rates predicted by the flux-minimization method exhibit significant correlations with flux rates obtained by either kinetic modelling or direct experimental determination. Larger deviations occur for segments of the network composed of redundant branches where the flux-minimization method always attributes the total flux to the thermodynamically most favourable branch. Nevertheless, compared with existing methods of structural modelling, the principle of flux minimization appears to be a promising theoretical approach to assess stationary flux rates in metabolic systems in cases where a detailed kinetic model is not yet available.

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.

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.