Minimization of intermediate concentrations as a suggested optimality principle for biochemical networks. II. Time hierarchy, enzymatic rate laws, and erythrocyte metabolism

Resource allocation in mammalian systems

Cells execute biological functions to support phenotypes such as growth, migration, and secretion. Complementarily, each function of a cell has resource costs that constrain phenotype. Resource allocation by a cell allows it to manage these costs and optimize their phenotypes. In fact, the management of resource constraints (e.g., nutrient availability, bioenergetic capacity, and macromolecular machinery production) shape activity and ultimately impact phenotype. In mammalian systems, quantification of resource allocation provides important insights into higher-order multicellular functions; it shapes intercellular interactions and relays environmental cues for tissues to coordinate individual cells to overcome resource constraints and achieve population-level behavior. Furthermore, these constraints, objectives, and phenotypes are context-dependent, with cells adapting their behavior according to their microenvironment, resulting in distinct steady-states. This review will highlight the biological insights gained from probing resource allocation in mammalian cells and tissues.

On the optimality of the enzyme-substrate relationship in bacteria

Much recent progress has been made to understand the impact of proteome allocation on bacterial growth; much less is known about the relationship between the abundances of the enzymes and their substrates, which jointly determine metabolic fluxes. Here, we report a correlation between the concentrations of enzymes and their substrates in Escherichia coli. We suggest this relationship to be a consequence of optimal resource allocation, subject to an overall constraint on the biomass density: For a cellular reaction network composed of effectively irreversible reactions, maximal reaction flux is achieved when the dry mass allocated to each substrate is equal to the dry mass of the unsaturated (or “free”) enzymes waiting to consume it. Calculations based on this optimality principle successfully predict the quantitative relationship between the observed enzyme and metabolite abundances, parameterized only by molecular masses and enzyme–substrate dissociation constants (K_m). The corresponding organizing principle provides a fundamental rationale for cellular investment into different types of molecules, which may aid in the design of more efficient synthetic cellular systems.

This study shows that in E. coli, the cellular mass of each metabolite approximately equals the combined mass of the free enzymes waiting to consume it; this simple relationship arises from the optimal utilization of cellular dry mass, and quantitatively describes available experimental data.

Using optimal control to understand complex metabolic pathways

BACKGROUND

Optimality principles have been used to explain the structure and behavior of living matter at different levels of organization, from basic phenomena at the molecular level, up to complex dynamics in whole populations. Most of these studies have assumed a single-criteria approach. Such optimality principles have been justified from an evolutionary perspective. In the context of the cell, previous studies have shown how dynamics of gene expression in small metabolic models can be explained assuming that cells have developed optimal adaptation strategies. Most of these works have considered rather simplified representations, such as small linear pathways, or reduced networks with a single branching point, and a single objective for the optimality criteria.

RESULTS

Here we consider the extension of this approach to more realistic scenarios, i.e. biochemical pathways of arbitrary size and structure. We first show that exploiting optimality principles for these networks poses great challenges due to the complexity of the associated optimal control problems. Second, in order to surmount such challenges, we present a computational framework which has been designed with scalability and efficiency in mind, including mechanisms to avoid the most common pitfalls. Third, we illustrate its performance with several case studies considering the central carbon metabolism of S. cerevisiae and B. subtilis. In particular, we consider metabolic dynamics during nutrient shift experiments.

CONCLUSIONS

We show how multi-objective optimal control can be used to predict temporal profiles of enzyme activation and metabolite concentrations in complex metabolic pathways. Further, we also show how to consider general cost/benefit trade-offs. In this study we have considered metabolic pathways, but this computational framework can also be applied to analyze the dynamics of other complex pathways, such as signal transduction or gene regulatory networks.

Redesigning metabolism based on orthogonality principles

Modifications made during metabolic engineering for overproduction of chemicals have network-wide effects on cellular function due to ubiquitous metabolic interactions. These interactions, that make metabolic network structures robust and optimized for cell growth, act to constrain the capability of the cell factory. To overcome these challenges, we explore the idea of an orthogonal network structure that is designed to operate with minimal interaction between chemical production pathways and the components of the network that produce biomass. We show that this orthogonal pathway design approach has significant advantages over contemporary growth-coupled approaches using a case study on succinate production. We find that natural pathways, fundamentally linked to biomass synthesis, are less orthogonal in comparison to synthetic pathways. We suggest that the use of such orthogonal pathways can be highly amenable for dynamic control of metabolism and have other implications for metabolic engineering.

How important is thermodynamics for identifying elementary flux modes?

We present a method for computing thermodynamically feasible elementary flux modes (tEFMs) using equilibrium constants without need of internal metabolite concentrations. The method is compared with the method based on a binary distinction between reversible and irreversible reactions. When all reactions are reversible, adding the constraints based on equilibrium constants reduces the number of elementary flux modes (EFMs) by a factor of two. Declaring in advance some reactions as irreversible, based on reliable biochemical expertise, can in general reduce the number of EFMs by a greater factor. But, even in this case, computing tEFMs can rule out some EFMs which are biochemically irrelevant. We applied our method to two published models described with binary distinction: the monosaccharide metabolism and the central carbon metabolism of Chinese hamster ovary cells. The results show that the binary distinction is in good agreement with biochemical observations. Moreover, the suppression of the EFMs that are not consistent with the equilibrium constants appears to be biologically relevant.

Optimality principles reveal a complex interplay of intermediate toxicity and kinetic efficiency in the regulation of prokaryotic metabolism

A precise and rapid adjustment of fluxes through metabolic pathways is crucial for organisms to prevail in changing environmental conditions. Based on this reasoning, many guiding principles that govern the evolution of metabolic networks and their regulation have been uncovered. To this end, methods from dynamic optimization are ideally suited since they allow to uncover optimality principles behind the regulation of metabolic networks. We used dynamic optimization to investigate the influence of toxic intermediates in connection with the efficiency of enzymes on the regulation of a linear metabolic pathway. Our results predict that transcriptional regulation favors the control of highly efficient enzymes with less toxic upstream intermediates to reduce accumulation of toxic downstream intermediates. We show that the derived optimality principles hold by the analysis of the interplay between intermediate toxicity and pathway regulation in the metabolic pathways of over 5000 sequenced prokaryotes. Moreover, using the lipopolysaccharide biosynthesis in Escherichia coli as an example, we show how knowledge about the relation of regulation, kinetic efficiency and intermediate toxicity can be used to identify drug targets, which control endogenous toxic metabolites and prevent microbial growth. Beyond prokaryotes, we discuss the potential of our findings for the development of antifungal drugs.

Identification of Conserved Moieties in Metabolic Networks by Graph Theoretical Analysis of Atom Transition Networks

Conserved moieties are groups of atoms that remain intact in all reactions of a metabolic network. Identification of conserved moieties gives insight into the structure and function of metabolic networks and facilitates metabolic modelling. All moiety conservation relations can be represented as nonnegative integer vectors in the left null space of the stoichiometric matrix corresponding to a biochemical network. Algorithms exist to compute such vectors based only on reaction stoichiometry but their computational complexity has limited their application to relatively small metabolic networks. Moreover, the vectors returned by existing algorithms do not, in general, represent conservation of a specific moiety with a defined atomic structure. Here, we show that identification of conserved moieties requires data on reaction atom mappings in addition to stoichiometry. We present a novel method to identify conserved moieties in metabolic networks by graph theoretical analysis of their underlying atom transition networks. Our method returns the exact group of atoms belonging to each conserved moiety as well as the corresponding vector in the left null space of the stoichiometric matrix. It can be implemented as a pipeline of polynomial time algorithms. Our implementation completes in under five minutes on a metabolic network with more than 4,000 mass balanced reactions. The scalability of the method enables extension of existing applications for moiety conservation relations to genome-scale metabolic networks. We also give examples of new applications made possible by elucidating the atomic structure of conserved moieties.

Conserved moieties are transferred between metabolites in internal reactions of a metabolic network but are not synthesised, degraded or exchanged with the environment. The total amount of a conserved moiety in the metabolic network is therefore constant over time. Metabolites that share a conserved moiety have interdependent concentrations because their total amount is constant. Identification of conserved moieties results in a concise description of all concentration dependencies in a metabolic network. The problem of identifying conserved moieties has previously been formulated in terms of the stoichiometry of metabolic reactions. Methods based on this formulation are computationally intractable for large networks. We show that reaction stoichiometry alone gives insufficient information to identify conserved moieties. By first incorporating additional data on the fate of atoms in metabolic reactions, we developed and implemented a computationally tractable algorithm to identify conserved moieties and their atomic structure.

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.

Which sets of elementary flux modes form thermodynamically feasible flux distributions?

Elementary flux modes (EFMs) are non-decomposable steady-state fluxes through metabolic networks. Every possible flux through a network can be described as a superposition of EFMs. The definition of EFMs is based on the stoichiometry of the network, and it has been shown previously that not all EFMs are thermodynamically feasible. These infeasible EFMs cannot contribute to a biologically meaningful flux distribution. In this work, we show that a set of thermodynamically feasible EFMs need not be thermodynamically consistent. We use first principles of thermodynamics to define the feasibility of a flux distribution and present a method to compute the largest thermodynamically consistent sets (LTCSs) of EFMs. An LTCS contains the maximum number of EFMs that can be combined to form a thermodynamically feasible flux distribution. As a case study we analyze all LTCSs found in Escherichia coli when grown on glucose and show that only one LTCS shows the required phenotypical properties. Using our method, we find that in our E. coli model < 10% of all EFMs are thermodynamically relevant.

Steady-state metabolite concentrations reflect a balance between maximizing enzyme efficiency and minimizing total metabolite load

Steady-state metabolite concentrations in a microorganism typically span several orders of magnitude. The underlying principles governing these concentrations remain poorly understood. Here, we hypothesize that observed variation can be explained in terms of a compromise between factors that favor minimizing metabolite pool sizes (e.g. limited solvent capacity) and the need to effectively utilize existing enzymes. The latter requires adequate thermodynamic driving force in metabolic reactions so that forward flux substantially exceeds reverse flux. To test this hypothesis, we developed a method, metabolic tug-of-war (mTOW), which computes steady-state metabolite concentrations in microorganisms on a genome-scale. mTOW is shown to explain up to 55% of the observed variation in measured metabolite concentrations in E. coli and C. acetobutylicum across various growth media. Our approach, based strictly on first thermodynamic principles, is the first method that successfully predicts high-throughput metabolite concentration data in bacteria across conditions.

Identifying the preferred subset of enzymatic profiles in nonlinear kinetic metabolic models via multiobjective global optimization and Pareto filters

Optimization models in metabolic engineering and systems biology focus typically on optimizing a unique criterion, usually the synthesis rate of a metabolite of interest or the rate of growth. Connectivity and non-linear regulatory effects, however, make it necessary to consider multiple objectives in order to identify useful strategies that balance out different metabolic issues. This is a fundamental aspect, as optimization of maximum yield in a given condition may involve unrealistic values in other key processes. Due to the difficulties associated with detailed non-linear models, analysis using stoichiometric descriptions and linear optimization methods have become rather popular in systems biology. However, despite being useful, these approaches fail in capturing the intrinsic nonlinear nature of the underlying metabolic systems and the regulatory signals involved. Targeting more complex biological systems requires the application of global optimization methods to non-linear representations. In this work we address the multi-objective global optimization of metabolic networks that are described by a special class of models based on the power-law formalism: the generalized mass action (GMA) representation. Our goal is to develop global optimization methods capable of efficiently dealing with several biological criteria simultaneously. In order to overcome the numerical difficulties of dealing with multiple criteria in the optimization, we propose a heuristic approach based on the epsilon constraint method that reduces the computational burden of generating a set of Pareto optimal alternatives, each achieving a unique combination of objectives values. To facilitate the post-optimal analysis of these solutions and narrow down their number prior to being tested in the laboratory, we explore the use of Pareto filters that identify the preferred subset of enzymatic profiles. We demonstrate the usefulness of our approach by means of a case study that optimizes the ethanol production in the fermentation of Saccharomyces cerevisiae.

Multi-objective mixed integer strategy for the optimisation of biological networks

In this contribution, the authors consider multi-criteria optimisation problems arising from the field of systems biology when both continuous and integer decision variables are involved. Mathematically, they are formulated as mixed-integer non-linear programming problems. The authors present a novel solution strategy based on a global optimisation approach for dealing with this class of problems. Its usefulness and capabilities are illustrated with two metabolic engineering case studies. For these problems, the authors show how the set of optimal solutions (the so-called Pareto front) is successfully and efficiently obtained, providing further insight into the systems under consideration regarding their optimal manipulation.

An indirect optimization method for biochemical systems: description of method and application to the maximization of the rate of ethanol, glycerol, and carbohydrate production in Saccharomyces cerevisiae

Three metabolic models for the production of ethanol, glycerol, and carbohydrates in yeast are optimized with respect to different production rates. While originally nonlinear, all three optimization problems are reduced in such a way that methods of linear programming can be used. The optimizations lead to profiles of enzyme activities that are compatible with the physiology of the cells, which guarantees their viability and fitness, and yield higher rates of the desired final end products than the original systems. In order to increase ethanol rate production at least three times, six enzymes must be modulated. By contrast, when the production of glycerol or carbohydrates is optimized, modulation of just one enzyme (in the case of glycerol) or two enzymes (in the case of carbohydrates) is necessary to yield significant increases in product flux rate. Comparisons of our results with those obtained from other methods show great similarities and demonstrate that both are valid methods. The choice of one or the other method depends on the question of interest. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 758-772, 1997.

Use of game-theoretical methods in biochemistry and biophysics

Evolutionary game theory can be considered as an extension of the theory of evolutionary optimisation in that two or more organisms (or more generally, units of replication) tend to optimise their properties in an interdependent way. Thus, the outcome of the strategy adopted by one species (e.g., as a result of mutation and selection) depends on the strategy adopted by the other species. In this review, the use of evolutionary game theory for analysing biochemical and biophysical systems is discussed. The presentation is illustrated by a number of instructive examples such as the competition between microorganisms using different metabolic pathways for adenosine triphosphate production, the secretion of extracellular enzymes, the growth of trees and photosynthesis. These examples show that, due to conflicts of interest, the global optimum (in the sense of being the best solution for the whole system) is not always obtained. For example, some yeast species use metabolic pathways that waste nutrients, and in a dense tree canopy, trees grow taller than would be optimal for biomass productivity. From the viewpoint of game theory, the examples considered can be described by the Prisoner's Dilemma, snowdrift game, Tragedy of the Commons and rock-scissors-paper game.

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.

The convex basis of the left null space of the stoichiometric matrix leads to the definition of metabolically meaningful pools

The stoichiometric matrix, S, represents a mapping of reaction rate vectors into a space of concentration time derivatives. The left null space of the stoichiometric matrix contains the dynamic invariants: a combination of concentration variables, referred to as metabolic pools, whose total concentration does not change over time. By analogy to the traditional reaction map formed by S, a compound map can be derived from -S(T). The analogy to flux analysis of the (right) null space of S enables us to classify the metabolic pools into three categories: Type A that contains chemical elements and their combinations in the form of certain moieties, Type B that contains such moieties in addition to cofactors carrying such moieties that are internal to the network, and Type C that contains only the cofactors. A convex formulation of the basis for the left null space allows us to directly classify the metabolic pools into these three categories. Type B metabolic pools include conservation pools that form conjugates of moiety-occupied and moiety-vacant concentration states of metabolites and cofactors. Type B metabolic pools thus describe the various states of moiety exchange between the primary substrates and the cofactors that capture properties like energy and redox potential. The convex basis gives clear insight into this exchange for glycolytic pathway in human red blood cell, including the identification of high and low energy pools that form conjugates. Examples suggest that pool maps may be more appropriate for signaling pathways than flux maps. The analysis of the left null space of the stoichiometric matrix allows us to define the achievable states of the cell and their physiological relevance.

Non-linear optimization of biotechnological processes by stochastic algorithms: application to the maximization of the production rate of ethanol, glycerol and carbohydrates by Saccharomyces cerevisiae

A non-linear optimization, based on an stochastic multi-start search algorithm, has been applied to the maximization of the production rates of ethanol, glycerol and carbohydrates by Saccharomyces cerevisiae. This optimization is applied to two alternative (non-linear) model representations of the same system, namely the Michaelis-Menten and the generalized mass action forms. We find a complete agreement between the results obtained using both representations. This is, maximization of the ethanol production rate requires modulation of up to six enzymes, while modification of only one enzyme is sufficient to obtain a significant improvement in the production rate of glycerol and carbohydrates. When the results are compared with those previously obtained using an indirect linear optimization method (Torres, N.V., Voit, E.O., González-Alcón, C., Rodríguez, F. 1997. An integrated optimization method for biochemical systems. Description of method and application to ethanol, glycerol and carbohydrate production in S. cerevisiae. Biotechnol. Bioeng. 55(5), 758-772.), we find close agreement between both optimization techniques. Qualitatively, both optimization approaches render the same profile of enzymes to be modulated, while quantitatively, discrepancies arise when the objective function is the maximization of the ethanol production rate. Reasons for such discrepancies and an evaluation of the advantages of each method (linear vs non-linear) are given.

Channelling can decrease pool size

It is widely considered that a possible advantage of metabolite channelling, in which a product of an enzyme is transferred to the next enzyme in a metabolic pathway without being released to the 'bulk' solution, is that channelling can decrease the steady-state concentrations of 'pool' intermediates. This then spares the limited solvent capacity of the cell, and reduces the loss of pathway flux due to leakage or instability of the free intermediate. Recently, however, based on simulations of a particular model of a 'dynamic' channel, Cornish-Bowden ["Failure of channelling to maintain low concentrations of metabolic intermediates" (1991) Eur. J. Biochem. 195, 103-108] has argued that this is not in fact the case; his simulations indicated that the channel was rather ineffective at decreasing the concentration of the pool intermediate, and in some cases actually increased it. However, although his simulations were restricted to very specific thermodynamic and kinetic parameters, he generalised his conclusions, arguing that "channelling has no effect on the free concentration of a channelled intermediate in a pathway". By showing that, for a number of kinetic cases, the concentration of the pool intermediate did decrease substantially with increased channelling, we demonstrate here that the conclusion of Cornish-Bowden is not correct. In particular, if the reaction catalysed by the enzymes forming the channel has an equilibrium constant K higher than 1, and if the enzyme removing the product of the channel reaction is kinetically competent, channelling in the model system studied by Cornish-Bowden (1991) can decrease the steady-state concentration of the pool by a factor of 1000, independently of the mechanism of the terminal reaction and under conditions of essentially constant overall flux. If the channel is a 'static' channel, the decrease in the pool can be to arbitrarily low levels. This conclusion also holds for a system in which other reactions may consume the pool intermediate. Thus, channelling can maintain metabolite concentrations at low levels.

Minimization of intermediate concentrations as a suggested optimality principle for biochemical networks. II. Time hierarchy, enzymatic rate laws, and erythrocyte metabolism

The multiobjective problem of minimizing all intermediate concentrations is solved for a model of glycolysis, the pentose monophosphate shunt and the glutathione system in human erythrocytes. It turns out that one solution out of four obtained corresponds qualitatively to the real system. Furthermore, it is shown that for any reaction system, the mentioned optimality principle implies distinct time hierarchy in that some reactions are infinitely fast and subsist in quasi-equilibrium. Finally, the relationships to the standard method of deriving enzymatic rate laws are discussed.