Optimization in integrated biochemical systems

Integrating systemic and molecular levels to infer key drivers sustaining metabolic adaptations

Metabolic adaptations to complex perturbations, like the response to pharmacological treatments in multifactorial diseases such as cancer, can be described through measurements of part of the fluxes and concentrations at the systemic level and individual transporter and enzyme activities at the molecular level. In the framework of Metabolic Control Analysis (MCA), ensembles of linear constraints can be built integrating these measurements at both systemic and molecular levels, which are expressed as relative differences or changes produced in the metabolic adaptation. Here, combining MCA with Linear Programming, an efficient computational strategy is developed to infer additional non-measured changes at the molecular level that are required to satisfy these constraints. An application of this strategy is illustrated by using a set of fluxes, concentrations, and differentially expressed genes that characterize the response to cyclin-dependent kinases 4 and 6 inhibition in colon cancer cells. Decreases and increases in transporter and enzyme individual activities required to reprogram the measured changes in fluxes and concentrations are compared with down-regulated and up-regulated metabolic genes to unveil those that are key molecular drivers of the metabolic response.

Deciphering the essential events in the reprogramming of metabolic networks subjected to complex perturbations, including the response to pharmacological treatments in multifactorial diseases like cancer, is crucial for the design of efficient therapies. Yet, tools to infer the molecular drivers sustaining such metabolic responses remain elusive for large metabolic networks. Here we develop an efficient computational strategy that integrates measured changes at systemic and molecular levels and combines metabolic control analysis with linear programming tools to infer key molecular drivers sustaining the metabolic adaptations to complex perturbations, such as an antitumoral drug therapy. The collective behavior is approximated using linear expressions where the adaptation of systemic concentrations and fluxes to a perturbation is described as a function of the molecular reprogramming of transport and enzyme activities. Starting from measured changes in fluxes and concentrations, we identify changes in the reprogramming of transporter and enzyme activities that are required to orchestrate the metabolic adaptation of colon cancer cells to a cell cycle inhibitor.

The three-legged stool of understanding metabolism: integrating metabolomics with biochemical genetics and computational modeling

Traditional biochemical research has resulted in a good understanding of many aspects of metabolism. However, this reductionist approach is time consuming and requires substantial resources, thus raising the question whether modern metabolomics and genomics should take over and replace the targeted experiments of old. We proffer that such a replacement is neither feasible not desirable and propose instead the tight integration of modern, system-wide omics with traditional experimental bench science and dedicated computational approaches. This integration is an important prerequisite toward the optimal acquisition of knowledge regarding metabolism and physiology in health and disease. The commentary describes advantages and drawbacks of current approaches to assessing metabolism and highlights the challenges to be overcome as we strive to achieve a deeper level of metabolic understanding in the future.

The best models of metabolism

Biochemical systems are among of the oldest application areas of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring a model and for formulating reactions has been constantly growing, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. In fact, the variety of options has become overwhelming and difficult to maneuver for novices and experts alike. This review outlines the metabolic model design process and discusses the numerous choices for modeling frameworks and mathematical representations. It tries to be inclusive, even though it cannot be complete, and introduces the various modeling options in a manner that is as unbiased as that is feasible. However, the review does end with personal recommendations for the choices of default models. WIREs Syst Biol Med 2017, 9:e1391. doi: 10.1002/wsbm.1391 For further resources related to this article, please visit the WIREs website.

The (Mathematical) Modeling Process in Biosciences

In this communication, we introduce a general framework and discussion on the role of models and the modeling process in the field of biosciences. The objective is to sum up the common procedures during the formalization and analysis of a biological problem from the perspective of Systems Biology, which approaches the study of biological systems as a whole. We begin by presenting the definitions of (biological) system and model. Particular attention is given to the meaning of mathematical model within the context of biology. Then, we present the process of modeling and analysis of biological systems. Three stages are described in detail: conceptualization of the biological system into a model, mathematical formalization of the previous conceptual model and optimization and system management derived from the analysis of the mathematical model. All along this work the main features and shortcomings of the process are analyzed and a set of rules that could help in the task of modeling any biological system are presented. Special regard is given to the formative requirements and the interdisciplinary nature of this approach. We conclude with some general considerations on the challenges that modeling is posing to current biology.

Redirector: designing cell factories by reconstructing the metabolic objective

Advances in computational metabolic optimization are required to realize the full potential of new in vivo metabolic engineering technologies by bridging the gap between computational design and strain development. We present Redirector, a new Flux Balance Analysis-based framework for identifying engineering targets to optimize metabolite production in complex pathways. Previous optimization frameworks have modeled metabolic alterations as directly controlling fluxes by setting particular flux bounds. Redirector develops a more biologically relevant approach, modeling metabolic alterations as changes in the balance of metabolic objectives in the system. This framework iteratively selects enzyme targets, adds the associated reaction fluxes to the metabolic objective, thereby incentivizing flux towards the production of a metabolite of interest. These adjustments to the objective act in competition with cellular growth and represent up-regulation and down-regulation of enzyme mediated reactions. Using the iAF1260 E. coli metabolic network model for optimization of fatty acid production as a test case, Redirector generates designs with as many as 39 simultaneous and 111 unique engineering targets. These designs discover proven in vivo targets, novel supporting pathways and relevant interdependencies, many of which cannot be predicted by other methods. Redirector is available as open and free software, scalable to computational resources, and powerful enough to find all known enzyme targets for fatty acid production.

A deeper understanding of biological processes, along with methods in synthetic biology, is driving the frontier of metabolic engineering. In particular, a better representation of cell metabolism will enable the engineering of bacterial strains that can act as factories for valuable biochemical products, from medicines to biofuels. Models which predict the behavior of these complex biological systems enable better engineering design as well as a more comprehensive understanding of fundamental biological principles. Here we develop a new method, called Redirector, for modeling metabolic alterations, and their relationship to cell growth. This method optimizes genetic engineering changes to achieve metabolite production using a new representation of the metabolic impact of genetic manipulation, which is more biologically realistic than existing models. We discover proven and novel engineering targets to improve fatty acid production, correctly predicting how different combinations of genes build upon one another. This work demonstrates that Redirector is a powerful method for designing cell factories and improving our understanding of metabolic systems.

Biochemical Systems Theory: A Review

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

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 optimization of enzyme manipulations in metabolic networks considering resilience effects

BACKGROUND

Improving the synthesis rate of desired metabolites in metabolic systems is one of the main tasks in metabolic engineering. In the last decade, metabolic engineering approaches based on the mathematical optimization have been used extensively for the analysis and manipulation of metabolic networks. Experimental evidence shows that mutants reflect resilience phenomena against gene alterations. Although researchers have published many studies on the design of metabolic systems based on kinetic models and optimization strategies, almost no studies discuss the multi-objective optimization problem for enzyme manipulations in metabolic networks considering resilience phenomenon.

RESULTS

This study proposes a generalized fuzzy multi-objective optimization approach to formulate the enzyme intervention problem for metabolic networks considering resilience phenomena and cell viability. This approach is a general framework that can be applied to any metabolic networks to investigate the influence of resilience phenomena on gene intervention strategies and maximum target synthesis rates. This study evaluates the performance of the proposed approach by applying it to two metabolic systems: S. cerevisiae and E. coli. Results show that the maximum synthesis rates of target products by genetic interventions are always over-estimated in metabolic networks that do not consider the resilience effects.

CONCLUSIONS

Considering the resilience phenomena in metabolic networks can improve the predictions of gene intervention and maximum synthesis rates in metabolic engineering. The proposed generalized fuzzy multi-objective optimization approach has the potential to be a good and practical framework in the design of metabolic networks.

Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models

Background

Design of newly engineered microbial strains for biotechnological purposes would greatly benefit from the development of realistic mathematical models for the processes to be optimized. Such models can then be analyzed and, with the development and application of appropriate optimization techniques, one could identify the modifications that need to be made to the organism in order to achieve the desired biotechnological goal. As appropriate models to perform such an analysis are necessarily non-linear and typically non-convex, finding their global optimum is a challenging task. Canonical modeling techniques, such as Generalized Mass Action (GMA) models based on the power-law formalism, offer a possible solution to this problem because they have a mathematical structure that enables the development of specific algorithms for global optimization.

Results

Based on the GMA canonical representation, we have developed in previous works a highly efficient optimization algorithm and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolism. Here, we explore the possibility of recasting kinetic non-linear models into an equivalent GMA model, so that global optimization on the recast GMA model can be performed. With this technique, optimization is greatly facilitated and the results are transposable to the original non-linear problem. This procedure is straightforward for a particular class of non-linear models known as Saturable and Cooperative (SC) models that extend the power-law formalism to deal with saturation and cooperativity.

Conclusions

Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.

Analysis of operating principles with S-system models

Operating principles address general questions regarding the response dynamics of biological systems as we observe or hypothesize them, in comparison to a priori equally valid alternatives. In analogy to design principles, the question arises: Why are some operating strategies encountered more frequently than others and in what sense might they be superior? It is at this point impossible to study operation principles in complete generality, but the work here discusses the important situation where a biological system must shift operation from its normal steady state to a new steady state. This situation is quite common and includes many stress responses. We present two distinct methods for determining different solutions to this task of achieving a new target steady state. Both methods utilize the property of S-system models within Biochemical Systems Theory (BST) that steady states can be explicitly represented as systems of linear algebraic equations. The first method uses matrix inversion, a pseudo-inverse, or regression to characterize the entire admissible solution space. Operations on the basis of the solution space permit modest alterations of the transients toward the target steady state. The second method uses standard or mixed integer linear programming to determine admissible solutions that satisfy criteria of functional effectiveness, which are specified beforehand. As an illustration, we use both methods to characterize alternative response patterns of yeast subjected to heat stress, and compare them with observations from the literature.

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.

OptForce: an optimization procedure for identifying all genetic manipulations leading to targeted overproductions

Computational procedures for predicting metabolic interventions leading to the overproduction of biochemicals in microbial strains are widely in use. However, these methods rely on surrogate biological objectives (e.g., maximize growth rate or minimize metabolic adjustments) and do not make use of flux measurements often available for the wild-type strain. In this work, we introduce the OptForce procedure that identifies all possible engineering interventions by classifying reactions in the metabolic model depending upon whether their flux values must increase, decrease or become equal to zero to meet a pre-specified overproduction target. We hierarchically apply this classification rule for pairs, triples, quadruples, etc. of reactions. This leads to the identification of a sufficient and non-redundant set of fluxes that must change (i.e., MUST set) to meet a pre-specified overproduction target. Starting with this set we subsequently extract a minimal set of fluxes that must actively be forced through genetic manipulations (i.e., FORCE set) to ensure that all fluxes in the network are consistent with the overproduction objective. We demonstrate our OptForce framework for succinate production in Escherichia coli using the most recent in silico E. coli model, iAF1260. The method not only recapitulates existing engineering strategies but also reveals non-intuitive ones that boost succinate production by performing coordinated changes on pathways distant from the last steps of succinate synthesis.

Over the past few years, there has been an unprecedented increase in the use of microorganisms for the production of biofuels, industrial chemicals and pharmaceutical precursors. In this regard, biotechnologists are confronted with the challenge to efficiently convert biomass and other renewable resources into useful biochemicals. With the advent of organism-specific mathematical models of metabolism, scientists have used computations to identify genetic modifications that maximize the yield of a desired product. In this paper, we introduce OptForce, an algorithm that identifies all possible metabolic interventions that lead to the overproduction of a biochemical of interest. Unlike existing techniques, OptForce does not rely on the maximization of a fitness function to predict metabolic fluxes. Instead, OptForce contrasts the metabolic flux patterns observed in an initial strain and a strain overproducing the chemical at the target yield. The essence of this procedure is the identification of all coordinated reaction modifications that force the network towards the overproduction target. We used OptForce to predict metabolic interventions for succinate overproduction in Escherichia coli. The results described in this paper not only uncover existing strain designs for succinate production but also elucidate new ones that can be experimentally explored.

Identifying quantitative operation principles in metabolic pathways: a systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses

BACKGROUND

Optimization methods allow designing changes in a system so that specific goals are attained. These techniques are fundamental for metabolic engineering. However, they are not directly applicable for investigating the evolution of metabolic adaptation to environmental changes. Although biological systems have evolved by natural selection and result in well-adapted systems, we can hardly expect that actual metabolic processes are at the theoretical optimum that could result from an optimization analysis. More likely, natural systems are to be found in a feasible region compatible with global physiological requirements.

RESULTS

We first present a new method for globally optimizing nonlinear models of metabolic pathways that are based on the Generalized Mass Action (GMA) representation. The optimization task is posed as a nonconvex nonlinear programming (NLP) problem that is solved by an outer-approximation algorithm. This method relies on solving iteratively reduced NLP slave subproblems and mixed-integer linear programming (MILP) master problems that provide valid upper and lower bounds, respectively, on the global solution to the original NLP. The capabilities of this method are illustrated through its application to the anaerobic fermentation pathway in Saccharomyces cerevisiae. We next introduce a method to identify the feasibility parametric regions that allow a system to meet a set of physiological constraints that can be represented in mathematical terms through algebraic equations. This technique is based on applying the outer-approximation based algorithm iteratively over a reduced search space in order to identify regions that contain feasible solutions to the problem and discard others in which no feasible solution exists. As an example, we characterize the feasible enzyme activity changes that are compatible with an appropriate adaptive response of yeast Saccharomyces cerevisiae to heat shock

CONCLUSION

Our results show the utility of the suggested approach for investigating the evolution of adaptive responses to environmental changes. The proposed method can be used in other important applications such as the evaluation of parameter changes that are compatible with health and disease states.

Recent developments in parameter estimation and structure identification of biochemical and genomic systems

The organization, regulation and dynamical responses of biological systems are in many cases too complex to allow intuitive predictions and require the support of mathematical modeling for quantitative assessments and a reliable understanding of system functioning. All steps of constructing mathematical models for biological systems are challenging, but arguably the most difficult task among them is the estimation of model parameters and the identification of the structure and regulation of the underlying biological networks. Recent advancements in modern high-throughput techniques have been allowing the generation of time series data that characterize the dynamics of genomic, proteomic, metabolic, and physiological responses and enable us, at least in principle, to tackle estimation and identification tasks using 'top-down' or 'inverse' approaches. While the rewards of a successful inverse estimation or identification are great, the process of extracting structural and regulatory information is technically difficult. The challenges can generally be categorized into four areas, namely, issues related to the data, the model, the mathematical structure of the system, and the optimization and support algorithms. Many recent articles have addressed inverse problems within the modeling framework of Biochemical Systems Theory (BST). BST was chosen for these tasks because of its unique structural flexibility and the fact that the structure and regulation of a biological system are mapped essentially one-to-one onto the parameters of the describing model. The proposed methods mainly focused on various optimization algorithms, but also on support techniques, including methods for circumventing the time consuming numerical integration of systems of differential equations, smoothing overly noisy data, estimating slopes of time series, reducing the complexity of the inference task, and constraining the parameter search space. Other methods targeted issues of data preprocessing, detection and amelioration of model redundancy, and model-free or model-based structure identification. The total number of proposed methods and their applications has by now exceeded one hundred, which makes it difficult for the newcomer, as well as the expert, to gain a comprehensive overview of available algorithmic options and limitations. To facilitate the entry into the field of inverse modeling within BST and related modeling areas, the article presented here reviews the field and proposes an operational 'work-flow' that guides the user through the estimation process, identifies possibly problematic steps, and suggests corresponding solutions based on the specific characteristics of the various available algorithms. The article concludes with a discussion of the present state of the art and with a description of open questions.

Metabolic Engineering with power-law and linear-logarithmic systems

Metabolic Engineering aims to improve the performance of biotechnological processes through rational manipulation rather than random mutagenesis of the organisms involved. Such a strategy can only succeed when a mathematical model of the target process is available. Simplifying assumptions are often needed to cope with the complexity of such models in an efficient way, and the choice of such assumptions often leads to models that fall within a certain structural template or formalism. The most popular formalisms can be grouped in two categories: power-law and linear-logarithmic. As optimization and analysis of a model strongly depends on its structure, most methods in Metabolic Engineering have been defined within a given formalism and never used in any other. In this work, the four most commonly used formalisms (two power-law and two linear-logarithmic) are placed in a common framework defined within Biochemical Systems Theory. This framework defines every model as matrix equations in terms of the same parameters, enabling the formulation of a common steady state analysis and providing means for translating models and methods from one formalism to another. Several Metabolic Engineering methods are analysed here and shown to be variants of a single equation. Particularly, two problem solving philosophies are compared: the application of the design equation and the solution of constrained optimization problems. Generalizing the design equation to all the formalisms shows it to be interchangeable with the direct solution of the rate law in matrix form. Furthermore, optimization approaches are concluded to be preferable since they speed the exploration of the feasible space, implement a better specification of the problem and exclude unrealistic results. Beyond consolidating existing knowledge and enabling comparison, the systematic approach adopted here can fill the gaps between the different methods and combine their strengths.

Optimization of biotechnological systems through geometric programming

Background

In the past, tasks of model based yield optimization in metabolic engineering were either approached with stoichiometric models or with structured nonlinear models such as S-systems or linear-logarithmic representations. These models stand out among most others, because they allow the optimization task to be converted into a linear program, for which efficient solution methods are widely available. For pathway models not in one of these formats, an Indirect Optimization Method (IOM) was developed where the original model is sequentially represented as an S-system model, optimized in this format with linear programming methods, reinterpreted in the initial model form, and further optimized as necessary.

Results

A new method is proposed for this task. We show here that the model format of a Generalized Mass Action (GMA) system may be optimized very efficiently with techniques of geometric programming. We briefly review the basics of GMA systems and of geometric programming, demonstrate how the latter may be applied to the former, and illustrate the combined method with a didactic problem and two examples based on models of real systems. The first is a relatively small yet representative model of the anaerobic fermentation pathway in S. cerevisiae, while the second describes the dynamics of the tryptophan operon in E. coli. Both models have previously been used for benchmarking purposes, thus facilitating comparisons with the proposed new method. In these comparisons, the geometric programming method was found to be equal or better than the earlier methods in terms of successful identification of optima and efficiency.

Conclusion

GMA systems are of importance, because they contain stoichiometric, mass action and S-systems as special cases, along with many other models. Furthermore, it was previously shown that algebraic equivalence transformations of variables are sufficient to convert virtually any types of dynamical models into the GMA form. Thus, efficient methods for optimizing GMA systems have multifold appeal.

On the local optimal solutions of metabolic regulatory networks using information guided genetic algorithm approach and clustering analysis

Biological information generated by high-throughput technology has made systems approach feasible for many biological problems. By this approach, optimization of metabolic pathway has been successfully applied in the amino acid production. However, in this technique, gene modifications of metabolic control architecture as well as enzyme expression levels are coupled and result in a mixed integer nonlinear programming problem. Furthermore, the stoichiometric complexity of metabolic pathway, along with strong nonlinear behaviour of the regulatory kinetic models, directs a highly rugged contour in the whole optimization problem. There may exist local optimal solutions wherein the same level of production through different flux distributions compared with global optimum. The purpose of this work is to develop a novel stochastic optimization approach-information guided genetic algorithm (IGA) to discover the local optima with different levels of modification of the regulatory loop and production rates. The novelties of this work include the information theory, local search, and clustering analysis to discover the local optima which have physical meaning among the qualified solutions.

Detection of potential enzyme targets by metabolic modelling and optimization: Application to a simple enzymopathy

MOTIVATION

A very promising approach in drug discovery involves the integration of available biomedical data through mathematical modelling and data mining. We have developed a method called optimization program for drug discovery (OPDD) that allows new enzyme targets to be identified in enzymopathies through the integration of metabolic models and biomedical data in a mathematical optimization program. The method involves four steps: (i) collection of the necessary information about the metabolic system and disease; (ii) translation of the information into mathematical terms; (iii) computation of the optimization programs prioritizing the solutions that propose the inhibition of a reduced number of enzymes and (iv) application of additional biomedical criteria to select and classify the solutions. Each solution consists of a set of predicted values for metabolites, initial substrates and enzyme activities, which describe a biologically acceptable steady state of the system that shifts the pathologic state towards a healthy state.

RESULTS

The OPDD was used to detect target enzymes in an enzymopathy, the human hyperuricemia. An existing S-system model and bibliographic information about the disease were used. The method detected six single-target enzyme solutions involving dietary modification, one of them coinciding with the conventional clinical treatment using allopurinol. The OPDD detected a large number of possible solutions involving two enzyme targets. All except one contained one of the previously detected six enzyme targets. The purpose of this work was not to obtain solutions for direct clinical implementation but to illustrate how increasing levels of biomedical information can be integrated together with mathematical models in drug discovery.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Cooperativity and saturation in biochemical networks: A saturable formalism using Taylor series approximations

Cooperative and saturable systems are common in molecular biology. Nevertheless, common canonical formalisms for kinetic modeling that are theoretically well justified do not have a saturable form. Modeling and fitting data from saturable systems are widely done using Hill-like equations. In practice, there is no theoretical justification for the generalized use of these equations, other than their ability to fit experimental data. Thus it is important to find a canonical formalism that is (a) theoretically well supported, (b) has a saturable functional form, and (c) can be justifiably applicable to any biochemical network. Here we derive such a formalism using Taylor approximations in a special transformation space defined by power-inverses and logarithms of power-inverses. This formalism is generalized for processes with n-variables, leading to a useful mathematical representation for molecular biology: the Saturable and Cooperative Formalism (SC formalism). This formalism provides an appropriate representation that can be used for modeling processes with cooperativity and saturation. We also show that the Hill equation can be seen as a special case within this formalism. Parameter estimation for the SC formalism requires information that is also necessary to build Power-Law models, Metabolic Control Analysis descriptions or (log)linear and Lin-log models. In addition, the saturation fraction of the relevant processes at the operating point needs to be considered. The practical use of the SC formalism for modeling is illustrated with a few examples. Similar models are built using different formalisms and compared to emphasize advantages and limitations of the different approaches.

Identification of metabolic system parameters using global optimization methods

BACKGROUND

The problem of estimating the parameters of dynamic models of complex biological systems from time series data is becoming increasingly important.

METHODS AND RESULTS

Particular consideration is given to metabolic systems that are formulated as Generalized Mass Action (GMA) models. The estimation problem is posed as a global optimization task, for which novel techniques can be applied to determine the best set of parameter values given the measured responses of the biological system. The challenge is that this task is nonconvex. Nonetheless, deterministic optimization techniques can be used to find a global solution that best reconciles the model parameters and measurements. Specifically, the paper employs branch-and-bound principles to identify the best set of model parameters from observed time course data and illustrates this method with an existing model of the fermentation pathway in Saccharomyces cerevisiae. This is a relatively simple yet representative system with five dependent states and a total of 19 unknown parameters of which the values are to be determined.

CONCLUSION

The efficacy of the branch-and-reduce algorithm is illustrated by the S. cerevisiae example. The method described in this paper is likely to be widely applicable in the dynamic modeling of metabolic networks.

Kinetic modeling to optimize pentose fermentation inZymomonas mobilis

Zymomonas mobilis engineered to express four heterologous enzymes required for xylose utilization ferments xylose along with glucose. A network of pentose phosphate (PP) pathway enzymatic reactions interacting with the native glycolytic Entner Doudoroff (ED) pathway has been hypothesized. We have investigated this putative reaction network by developing a kinetic model incorporating all of the enzymatic reactions of the PP and ED pathways, including those catalyzed by the heterologous enzymes. Starting with the experimental literature on in vitro characterization of each enzymatic reaction, we have developed a kinetic model to enable dynamic simulation of intracellular metabolite concentrations along the network of interacting PP and ED metabolic pathways. This kinetic model is useful for performing in silico simulations to predict how varying the different enzyme concentrations will affect intracellular metabolite concentrations and ethanol production rate during continuous fermentation of glucose and xylose mixtures. Among the five enzymes whose concentrations were varied as inputs to the model, ethanol production in the continuous fermentor was optimized when xylose isomerase (XI) was present at the highest level, followed by transaldolase (TAL). Predictions of the model that the interconnecting enzyme phosphoglucose isomerase (PGI) does not need to be overexpressed were recently confirmed through experimental investigations. Through such systematic analysis, we can develop efficient strategies for maximizing the fermentation of both glucose and xylose, while minimizing the expression of heterologous enzymes.

Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.

Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics

This paper examines the validity of the linlog approach, which was recently developed in our laboratory, by comparison of two different kinetic models for the metabolic network of Escherichia coli. The first model is a complete mechanistic model; the second is an approximative model in which linlog kinetics are applied. The parameters of the linlog model (elasticities) are derived from the mechanistic model. Three different optimization cases are examined. In all cases, the objective is to calculate the enzyme levels that maximize a certain flux while keeping the total amount of enzyme constant and preventing large changes of metabolite concentrations. For an average variation of metabolite levels of 10% and individual changes of a factor 2, the predicted enzyme levels, metabolite concentrations and fluxes of both models are highly similar. This similarity holds for changes in enzyme level of a factor 4-6 and for changes in fluxes up to a factor 6. In all three cases, the predicted optimal enzyme levels could neither have been found by intuition-based approaches, nor on basis of flux control coefficients. This demonstrates that kinetic models are essential tools in Metabolic Engineering. In this respect, the linlog approach is a valuable extension of MCA, since it allows construction of kinetic models, based on MCA parameters, that can be used for constrained optimization problems and are valid for large changes of metabolite and enzyme levels.

Multicriteria optimization of biochemical systems by linear programming: application to production of ethanol by Saccharomyces cerevisiae

In this study we present a method for simultaneous optimization of several metabolic responses of biochemical pathways. The method, based on the use of the power law formalism to obtain a linear system in logarithmic coordinates, is applied to ethanol production by Saccharomyces cerevisiae. Starting from an experimentally based kinetic model, we translated it to its power law equivalent. With this new model representation, we then applied the multiobjective optimization method. Our intent was to maximize ethanol production and minimize each of the internal metabolite concentrations. To ensure cell viability, all optimizations were carried out under imposed constraints. The different solutions obtained, which correspond to alternative patterns of enzyme overexpression, were implemented in the original model. We discovered few discrepancies between the S-system-optimized steady state and the corresponding optimized state in the original kinetic model, thus demonstrating the suitability of the S-system representation as the basis for the optimization procedure. In all optimized solutions, the ATP level reached its maximum and any increase in its activity positively affected the optimization process. This work illustrates that in any optimization study no single criteria is of general application being the multiobjective and constrained task the proper way to address it. It is concluded that the proposed multiobjective method can serve to carry out, in a single study, the general pattern of behavior of a given metabolic system with regard to its control and optimization.

Optimization of biochemical systems by linear programming and general mass action model representations

A new method is proposed for the optimization of biochemical systems. The method, based on the separation of the stoichiometric and kinetic aspects of the system, follows the general approach used in the previously presented indirect optimization method (IOM) developed within biochemical systems theory. It is called GMA-IOM because it makes use of the generalized mass action (GMA) as the model system representation form. The GMA representation avoids flux aggregation and thus prevents possible stoichiometric errors. The optimization of a system is used to illustrate and compare the features, advantages and shortcomings of both versions of the IOM method as a general strategy for designing improved microbial strains of biotechnological interest. Special attention has been paid to practical problems for the actual implementation of the new proposed strategy, such as the total protein content of the engineered strain or the deviation from the original steady state and its influence on cell viability.

Design principles and operating principles: the yin and yang of optimal functioning

Metabolic engineering has as a goal the improvement of yield of desired products from microorganisms and cell lines. This goal has traditionally been approached with experimental biotechnological methods, but it is becoming increasingly popular to precede the experimental phase by a mathematical modeling step that allows objective pre-screening of possible improvement strategies. The models are either linear and represent the stoichiometry and flux distribution in pathways or they are non-linear and account for the full kinetic behavior of the pathway, which is often significantly effected by regulatory signals. Linear flux analysis is simpler and requires less input information than a full kinetic analysis, and the question arises whether the consideration of non-linearities is really necessary for devising optimal strategies for yield improvements. The article analyzes this question with a generic, representative pathway. It shows that flux split ratios, which are the key criterion for linear flux analysis, are essentially sufficient for unregulated, but not for regulated branch points. The interrelationships between regulatory design on one hand and optimal patterns of operation on the other suggest the investigation of operating principles that complement design principles, like a user's manual complements the hardwiring of electronic equipment.

Modeling, optimization and experimental assessment of continuous L-(-)-carnitine production by Escherichia coli cultures

In a previous paper Cánovas et al. (Biotechnol Bioeng 2002;77:764-775) presented a model for L-(-)-carnitine production using Escherichia coli O44 K74, in a cell-recycle bioreactor for the biotransformation of crotonobetaine into L-carnitine. In this work we optimize this biotechnological setup and experimentally verify the predicted optimal parameter profiles. Provided with a reliable and robust S-system description of the cell-bioreactor combined system, we applied the Indirect Optimization Method described by Torres et al. (Biotechnol Bioeng 1997;55(5):758-772; Food Technol Biotechnol 1998;36(3):177-184). This optimization approach provides different parameter value profiles, all of which are compatible with the cell physiology and the bioreactor operating conditions, that yield increased rates of L-(-)-carnitine production. Three parameters were seen to be of critical importance for maximizing L-(-)-carnitine production: the dilution rate, the initial crotonobetaine concentration, and the carnitine dehydratase activity. When the first two were changed in the experimental setup, there was a 74% increase in the L-(-)-carnitine production rate, performance that was in close agreement with the predictions of the model. In accordance with the optimized solution, a further improvement (90% increase in the L-(-)-carnitine production rate) could be attained by over-expressing up to 5 times the carnitine dehydratase basal activity. Thus the optimization approach shown herein provides experimental evidence of a new strategy which demonstrates the possible variables that can be subjected to modifications compatible with the cell physiology and bioreactor operating conditions, and which are able to yield increased rates of L-(-)-carnitine production.

Metabolic modeling: a tool of drug discovery in the post-genomic era

The Human Genome Project, DNA microarrays, proteome chips and metabolic profiles, among others, are generating unprecedented amounts of valuable information about the genetic and metabolic responses of organisms to stimuli. This wealth of information poses a great scientific challenge, namely the development of novel, effective methods for functional analysis and interpretation. It is proposed here that biomathematical systems models must accompany data generation and management. A particularly effective framework for this purpose is canonical modeling based on Biochemical Systems Theory. The key concept of this theory is the formulation of biological phenomena as systems of differential equations, in which all processes are represented as products of power-law functions.

Stoichiometric model for evaluating the metabolic capabilities of the facultative methylotroph Methylobacterium extorquens AM1, with application to reconstruction of C(3) and C(4) metabolism

A stoichiometric model of central metabolism was developed based on new information regarding metabolism in this bacterium to evaluate the steady-state growth capabilities of the serine cycle facultative methylotroph Methylobacterium extorquens AM1 during growth on methanol, succinate, and pyruvate. The model incorporates 20 reversible and 47 irreversible reactions, 65 intracellular metabolites, and experimentally-determined biomass composition. The flux space for this underdetermined system of equations was defined by finding the elementary modes, and constraints based on experimental observations were applied to determine which of these elementary modes give a reasonable description of the flux distribution for each growth substrate. The predicted biomass yield, on a carbon atom basis, is 49.8%, which agrees well with the range of published experimental yield measurements (37-50%). The model predicts the cell to be limited by reduced pyridine nucleotide availability during methylotrophic growth, but energy-limited when growing on multicarbon substrates. Mutation and phenotypic analysis was used to explore a previously unknown region of the metabolic map and to confirm the stoichiometry of the pathways in this region used in the metabolic model. Based on genome sequence data and simulation results, three enzymes involved in C(3)-C(4) interconversion pathways were predicted to be mutually redundant: malic enzyme, phosphoenolpyruvate carboxykinase, and phosphoenolpyruvate synthase. Insertion mutations in the genes predicted to encode these enzymes were made and these mutants were capable of growing on all substrates tested, confirming the redundancy of these pathways. Likewise, pathway analysis suggests that the TCA cycle enzymes citrate synthase and succinate dehydrogenase are essential for all growth substrates. In keeping with these predictions, null mutants could not be obtained in these genes. Finally, a similar model was developed for the ribulose monophosphate pathway obligate methylotroph Methylobacillus flagellatum KT to compare the efficiency of carbon utilization in the two types of methylotrophic carbon utilization pathways. The predicted yield for this organism on methanol is 65.9%.

Probing the performance limits of the Escherichia coli metabolic network subject to gene additions or deletions

An optimization-based procedure for studying the response of metabolic networks after gene knockouts or additions is introduced and applied to a linear flux balance analysis (FBA) Escherichia coli model. Both the gene addition problem of optimally selecting which foreign genes to recombine into E. coli, as well as the gene deletion problem of removing a given number of existing ones, are formulated as mixed-integer optimization problems using binary 0-1 variables. The developed modeling and optimization framework is tested by investigating the effect of gene deletions on biomass production and addressing the maximum theoretical production of the 20 amino acids for aerobic growth on glucose and acetate substrates. In the gene deletion study, the smallest gene set necessary to achieve maximum biomass production in E. coli is determined for aerobic growth on glucose. The subsequent gene knockout analysis indicates that biomass production decreases monotonically, rendering the metabolic network incapable of growth after only 18 gene deletions. In the gene addition study, the E. coli flux balance model is augmented with 3,400 non-E. coli reactions from the KEGG database to form a multispecies model. This model is referred to as the Universal model. This study reveals that the maximum theoretical production of six amino acids could be improved by the addition of only one or two genes to the native amino acid production pathway of E. coli, even though the model could choose from 3,400 foreign reaction candidates. Specifically, manipulation of the arginine production pathway showed the most promise with 8.75% and 9.05% predicted increases with the addition of genes for growth on glucose and acetate, respectively. The mechanism of all suggested enhancements is either by: 1) improving the energy efficiency and/or 2) increasing the carbon conversion efficiency of the production route.

Assessment of effects of experimental imprecision on optimized biochemical systems

Metabolic pathways may be optimized with S-system models that prescribe profiles of control variables leading to optimal output while keeping metabolites and enzyme activities within predefined ranges. Monte Carlo simulations show how much the yield and the corresponding metabolite concentrations would be affected by inaccuracies in the experimental implementation of the prescribed profiles. For a recent model of citric acid production in Aspergillus niger, the yield is roughly normally distributed, whereas the distributions of metabolite concentrations differ greatly in shape and statistical characteristics. Even moderate inaccuracies may lead to constraint violations, which appear to be correlated with high logarithmic gains.

Metabolism of citric acid production by Aspergillus niger: model definition, steady-state analysis and constrained optimization of citric acid production rate

In an attempt to provide a rational basis for the optimization of citric acid production by A. niger, we developed a mathematical model of the metabolism of this filamentous fungus when in conditions of citric acid accumulation. The present model is based in a previous one, but extended with the inclusion of new metabolic processes and updated with currently available kinetic data. Among the different alternatives to represent the system behavior we have chosen the S-system representation within power-law formalism. This type of representation allows us to verify not only the ability of the model to exhibit a stable steady state of the integrated system but also the robustness and quality of the representation. The model analysis is shown to be self-consistent, with a stable steady state, and in good agreement with experimental evidence. Moreover, the model representation is sufficiently robust, as indicated by sensitivity and steady-state and dynamic analyses. From the steady-state results we concluded that the range of accuracy of the S-system representation is wide enough to model realistic deviations from the nominal steady state. The dynamic analysis indicated a reasonable response time, which provided further indication that the model is adequate. The extensive assessment of the reliability and quality of the model put us in a position to address questions of optimization of the system with respect to increased citrate production. We carried out the constrained optimization of A. niger metabolism with the goal of predicting an enzyme activity profile yielding the maximum rate of citrate production, while, at the same time, keeping all enzyme activities within predetermined, physiologically acceptable ranges. The optimization is based on a method described and tested elsewhere that utilizes the fact that the S-system representation of a metabolic system becomes linear at steady state, which allows application of linear programming techniques. Our results show that: (i) while the present profile of enzyme activities in A. niger at idiophase steady state yields high rates of citric acid production, it still leaves room for changes and suggests possible optimization of the activity profile to over five times the basal rate synthesis; (ii) when the total enzyme concentration is allowed to double its basal value, the citric acid production rate can be increased by more than 12-fold, and even larger values can be attained if the total enzyme concentration is allowed to increase even more (up to 50-fold when the total enzyme concentration may rise up to 10-fold the basal value); and (iii) the systematic search of the best combination of subsets of enzymes shows that, under all conditions assayed, a minimum of 13 enzymes need be modified if significant increases in citric acid are to be obtained. This implies that improvements by single enzyme modulation are unlikely, which is in agreement with the findings of some investigators in this and other fields.

Engineering a living cell to desired metabolite concentrations and fluxes: pathways with multifunctional enzymes

With molecular genetics enabling modulation of the concentrations of cellular enzymes, metabolic engineering becomes limited by the question of which modulations of the enzyme concentrations are required to bring about a desired pattern of cellular metabolism. In an earlier paper (Kholodenko et al. (1998). Biotechnol. Bioeng. 59, 239-247) we derived a method to determine the required modulations. This method, however, cannot be immediately applied to cellular pathways with enzymes catalyzing more than one step in metabolism (multifunctional enzymes). In the present paper we show to which extent the presence of multifunctional enzymes limits biotechological ambitions, which one might otherwise pursue in vain. In particular, it is impossible to change the concentration of a single intermediate and leave the rest of metabolism unperturbed if that intermediate interacts directly with a multifunctional enzyme. The analytical machinery of Metabolic Control Analysis is used to relate the desired and ensuing changes in the metabolic pattern. An explicit solution to this problem of engineering metabolism is then given in the form of a single matrix equation.

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.

Mathematical models of purine metabolism in man

Experimental and clinical data on purine metabolism are collated and analyzed with three mathematical models. The first model is the result of an attempt to construct a traditional kinetic model based on Michaelis-Menten rate laws. This attempt is only partially successful, since kinetic information, while extensive, is not complete, and since qualitative information is difficult to incorporate into this type of model. The data gaps necessitate the complementation of the Michaelis-Menten model with other functional forms that can incorporate different types of data. The most convenient and established representations for this purpose are rate laws formulated as power-law functions, and these are used to construct a Complemented Michaelis-Menten (CMM) model. The other two models are pure power-law-representations, one in the form of a Generalized Mass Action (GMA) system, and the other one in the form of an S-system. The first part of the paper contains a compendium of experimental data necessary for any model of purine metabolism. This is followed by the formulation of the three models and a comparative analysis. For physiological and moderately pathological perturbations in metabolites or enzymes, the results of the three models are very similar and consistent with clinical findings. This is an encouraging result since the three models have different structures and data requirements and are based on different mathematical assumptions. Significant enzyme deficiencies are not so well modeled by the S-system model. The CMM model captures the dynamics better, but judging by comparisons with clinical observations, the best model in this case is the GMA model. The model results are discussed in some detail, along with advantages and disadvantages of each modeling strategy.