Simulation and optimization of metabolic pathways

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.

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.

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.

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.

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.

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.

Analysis of metabolic networks using a pathway distance metric through linear programming

The solution of the shortest path problem in biochemical systems constitutes an important step for studies of their evolution. In this paper, a linear programming (LP) algorithm for calculating minimal pathway distances in metabolic networks is studied. Minimal pathway distances are identified as the smallest number of metabolic steps separating two enzymes in metabolic pathways. The algorithm deals effectively with circularity and reaction directionality. The applicability of the algorithm is illustrated by calculating the minimal pathway distances for Escherichia coli small molecule metabolism enzymes, and then considering their correlations with genome distance (distance separating two genes on a chromosome) and enzyme function (as characterised by enzyme commission number). The results illustrate the effectiveness of the LP model. In addition, the data confirm that propinquity of genes on the genome implies similarity in function (as determined by co-involvement in the same region of the metabolic network), but suggest that no correlation exists between pathway distance and enzyme function. These findings offer insight into the probable mechanism of pathway evolution.

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.

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.

Rational engineering of the TOL meta-cleavage pathway

The meta-cleavage pathway of Pseudomonas putida mt-2 was simulated using a biochemical systems simulation developed by Regan (1996). A non-competitive inhibition term for catechol-2,3-dioxygenase (C23O) by 2-OH-pent-2,4-dienoate (Ki = 150 μM) was incorporated into the model. The simulation predicted steady state accumulation levels in the μM range for metabolites pre-meta-cleavage, and in the mM range for metabolites post-meta-cleavage. The logarithmic gains L[V-i, Xj] and L[X-i, Xj] clearly indicated that the pathway was most sensitive to the concentration of the starting substrate, benzoate, and the first enzyme of the pathway, toluate-1, 2-dioxygenase (TO). The simulation was validated experimentally; it was found that the amplification of TO increased the steady state flux from 0.024 to 0.091 (mmol/g cell dwt)/h. This resulted in an increased accumulation of a number of the pathway metabolites (intra- and extracellularly), especially cis-diol, 4-OH-2-oxovalerate, and 4-oxalocrotonate. Metabolic control analysis indicated that C23O was, in fact, the major controling enzymic step of the pathway with a scaled control coefficient of 0.83. The amplification of TO resulted in a shift of some of the control away from C23O. Catechol-2,3-dioxygenase, however, remained as the major controling element of the pathway. Copyright 1998 John Wiley & Sons, Inc.

Flux analysis of microbial metabolic pathways using a visual programming environment

This paper describes the use of a visual programming environment (LabVIEW) for the flux analysis of metabolic pathways. Representations of metabolic pathways are constructed in software from individual reaction elements (icons) which are linked together to indicate potential flux routes. Off-line bioprocess data are then used to supply the inputs and outputs to the metabolic pathway and the pathway fluxes are calculated. The metabolic system can be modelled at different levels of complexity and new pathways can be inserted into existing models. To illustrate this, flux analyses are performed on three Escherichia coli mutants with metabolic pathway deletions and insertions. The first analysis looks at organic acid production and the second at the effect of the presence in E. coli of an engineered pathway for toluene degradation.