Strategies for representing metabolic pathways within biochemical systems theory: reversible pathways

The search for systematic methods to deal with the integrated behavior of complex biochemical systems has over the past two decades led to the proposal of several theories of biochemical systems. Among the most promising is biochemical systems theory (BST). Recent comparisons of this theory with several others that have recently been proposed have demonstrated that all are variants of BST and share a common underlying formalism. Hence, the different variants can be precisely related and ranked according to their completeness and operational utility. The original and most fruitful variant within BST is based on a particular representation, called an S-system (for synergistic and saturable systems), that exhibits many advantages not found among alternative representations. Even within the preferred S-system representation there are options, depending on the method of aggregating fluxes, that become especially apparent when one considers reversible pathways. In this paper we focus on the paradigm situation and clearly distinguish the two most common strategies for generating an S-system representation. The first is called the "reversible" strategy because it involves aggregating incoming fluxes separately from outgoing fluxes for each metabolite to define a net flux that can be positive, negative, or zero. The second is the "irreversible" strategy, which involves aggregating forward and reverse fluxes through each reaction to define a net flux that is always positive. This second strategy has been used almost exclusively in all variants of BST. The principal results of detailed analyses are the following: (1) All S-system representations predict the same changes in dependent concentrations for a given change in an independent concentration. (2) The reversible strategy is superior to the irreversible on the basis of several criteria, including accuracy in predicting steady-state flux, accuracy in predicting transient responses, and robustness of representation. (3) Only the reversible strategy yields a representation that is able to capture the characteristic feature of amphibolic pathways, namely, the reversal of nets flux under physiological conditions. Finally, the results document the wide range of variation over which the S-system representation can accurately predict the behavior of intact biochemical systems and confirm similar results of earlier studies [Voit and Savageau, Biochemistry 26: 6869-6880 (1987)].     

A comparison of variant theories of intact biochemical systems. I. Enzyme-enzyme interactions and biochemical systems theory

The need for a well-structured theory of intact biochemical systems becomes increasingly evident as one attempts to integrate the vast knowledge of individual molecular constituents, which has been expanding for several decades. In recent years, several apparently different approaches to the development of such a theory have been proposed. Unfortunately, the resulting theories have not been distinguished from each other, and this has led to considerable confusion with numerous duplications and rediscoveries. Detailed comparisons and critical tests of alternative theories are badly needed to reverse these unfortunate developments. In this paper we (1) characterize a specific system involving enzyme-enzyme interactions for reference in comparing alternative theories, and (2) analyze the reference system by applying the explicit S-system variant within biochemical systems theory (BST), which represents a fundamental framework based upon the power-law formalism and includes several variants. The results provide the first complete and rigorous numerical analysis within the power-law formalism of a specific biochemical system and further evidence for the accuracy of the explicit S-system variant within BST. This theory is shown to represent enzyme-enzyme interactions in a systematically structured fashion that facilitates analysis of complex biochemical systems in which these interactions play a prominent role. This representation also captures the essential character of the underlying nonlinear processes over a wide range of variation (on average 20-fold) in the independent variables of the system. In the companion paper in this issue the same reference system is analyzed by other variants within BST as well as by two additional theories within the same power-law formalism--flux-oriented and metabolic control theories. The results show how all these theories are related to one another.     

A comparison of variant theories of intact biochemical systems. II. Flux-oriented and metabolic control theories

In the past two decades, several theories, all ultimately based upon the same power-law formalism, have been proposed to relate the behavior of intact biochemical systems to the properties of their underlying determinants. Confusion concerning the relatedness of these alternatives has become acute because the implications of these theories have never been compared. In the preceding paper we characterized a specific system involving enzyme-enzyme interactions for reference in comparing alternative theories. We also analyzed the reference system by using an explicit variant that involves the S-system representation within biochemical systems theory (BST). We now analyze the same reference system according to two other variants within BST. First, we carry out the analysis by using an explicit variant that involves the generalized mass action representation, which includes the flux-oriented theory of Crabtree and Newsholme as a special case. Second, we carry out the analysis by using an implicit variant that involves the generalized mass action representation, which includes the metabolic control theory of Kacser and his colleagues as a special case. The explicit variants are found to provide a more complete characterization of the reference system than the implicit variants. Within each of these variant classes, the S-system representation is shown to be more mathematically tractable and accurate than the generalized mass action representation. The results allow one to make clear distinctions among the variant theories.     

Biochemical systems theory and metabolic control theory: 2. the role of summation and connectivity relationships

Perhaps the major obstacle to recognizing the relatedness of Biochemical Systems Theory (BST) and a subsequently developed approach some have called Metabolic Control Theory (MCT) is the summation and connectivity relationships. These are the most visible and central features of the MCT approach to the understanding of intact biochemical systems, whereas in the BST approach they appear to be invisible and peripheral. Generalized versions of these relationships are shown to be inherent to BST, and it is shown how their role differs from that within MCT. The significance of summation and connectivity relationships is shown to be historical and secondary in the sense that one can understand fully the integrated behavior of complex biochemical systems in steady state with BST and never explicitly invoke these relationships. It also is shown that the summation and connectivity relationships in MCT have inherent limitations that make them inadequate as the basis for a general theory of biochemical systems. The results in this paper, together with those in the previous paper, clearly demonstrate that MCT is a special case of BST.     

Biochemical systems theory: increasing predictive power by using second-order derivatives measurements

Models based on the power-law formalism provide a useful tool for analyzing metabolic systems. Within this methodology, the S-system variant furnishes the best strategy. In this paper we explore an extension of this formalism by considering second-order derivative terms of the Taylor series which the power-law is based upon. Results show that the S-system equations which include second-order Taylor coefficients give better accuracy in predicting the response of the system to a perturbation. Hence, models based on this new approach could provide a useful tool for quantitative purposes if one is able to measure the required derivatives experimentally. In particular we show the utility of this approach when it comes to discriminating between two mechanisms that are equivalent in the S-system a representation based on first-order coefficients. However, the loss of analytical tractability is a serious disadvantage for using this approach as a general tool for studying metabolic systems.     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

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.     

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.     

Enzyme organization and stability of metabolic pathways: A comparison of various approaches

The aim of this paper is to compare various methods for the quantification of metabolic pathways dynamics. A Yates-Pardee metabolic pathway with enzyme organization, i.e. with spatial localization of the enzymes in a specific cellular compartment, was studied using: (i) the classical Henri-Michaelis-Menten (HMM) equations, (ii) linearization of the HMM equations in the vicinity of a steady state (linearized formalism), and (iii) Biochemical Systems Theory formalism (BST formalism). It is shown that transient solutions computed via either the linearized formalism or the BST formalism can greatly differ from transient solutions computed with the HMM equations. However, in the studied example, results remain qualitatively the same for the three formalisms. This suggests that the study of the topology of the system may give useful insights into the metabolic pathways dynamics.     

Inference of biochemical network models in S-system using multiobjective optimization approach

MOTIVATION: The inference of biochemical networks, such as gene regulatory networks, protein-protein interaction networks, and metabolic pathway networks, from time-course data is one of the main challenges in systems biology. The ultimate goal of inferred modeling is to obtain expressions that quantitatively understand every detail and principle of biological systems. To infer a realizable S-system structure, most articles have applied sums of magnitude of kinetic orders as a penalty term in the fitness evaluation. How to tune a penalty weight to yield a realizable model structure is the main issue for the inverse problem. No guideline has been published for tuning a suitable penalty weight to infer a suitable model structure of biochemical networks. RESULTS: We introduce an interactive inference algorithm to infer a realizable S-system structure for biochemical networks. The inference problem is formulated as a multiobjective optimization problem to minimize simultaneously the concentration error, slope error and interaction measure in order to find a suitable S-system model structure and its corresponding model parameters. The multiobjective optimization problem is solved by the epsilon-constraint method to minimize the interaction measure subject to the expectation constraints for the concentration and slope error criteria. The theorems serve to guarantee the minimum solution for the epsilon-constrained problem to achieve the minimum interaction network for the inference problem. The approach could avoid assigning a penalty weight for sums of magnitude of kinetic orders.     

Modelization and experimental studies on the control of the glycolytic-glycogenolytic pathway in rat liver

In this paper we construct a model of the glycolytic-glycogenolytic converging pathway in rat liver, by integrating experimental data obtained in anin vitro system and information available from the literature. The model takes the mathematical expression of an S-system representation within the power law formalism (Savageau, 1976. Biochemical System Analysis: A study of function and design in Molecular Biology. Addison-Wesley, Reading, Mass.). By using this theoretical framework a model analysis was carried out that allowed us a) the assessment of the quality of the model in terms of its consistency and robustness, b) the steady state analysis and control characterization of the system, and c) the study of the dynamics of the system after changes in the level of two magnitudes of biological significance: the glucose concentration and the phosphofructokinase enzyme activity. Model predictions are compared with experimental measurements referred to Logarithmic Gains through fluxes and substrates concentrations showing that there is a good correlation between the model predictions and the experimentally determined values.     

From systems biology to systems biomedicine

Systems Biology is about combining theory, technology, and targeted experiments in a way that drives not only data accumulation but knowledge as well. The challenge in Systems Biomedicine is to furthermore translate mechanistic insights in biological systems to clinical application, with the central aim of improving patients' quality of life. The challenge is to find theoretically well-chosen models for the contextually correct and intelligible representation of multi-scale biological systems. In this review, we discuss the current state of Systems Biology, highlight the emergence of Systems Biomedicine, and highlight some of the topics and views that we think are important for the efficient application of Systems Theory in Biomedicine.     

MetMAP: an integrated Matlab package for analysis and optimization of metabolic systems

In previous works we have presented and applied a method to predict the parameter profile that optimizes biochemical systems regarding either a single or a set of metabolic responses within physiological constraints [Vera et al., 2003a]. This optimization technique requires a previous model definition and a translation to S-system form and the use of widely available linear programming packages. However, in dealing with these issues the interested researcher has to confront additional difficulties because of a lack of connectivity among available software packages or routines specifically designed to perform different tasks. In addition to this difficulty is the unavailability of any automated package which is capable of performing such optimizations and the previous required analysis. This situation prompted us to develop an integrated software package able to deal with these tasks in a single program environment. In this paper we present a software package for the model definition, analysis and optimization of a biochemical system. It starts with a given model definition that is directly translated to its equivalent S-system form. Once the model quality assessment is performed (stability and sensitivity analysis) the program determines the parameter profile that yields the optimized response compatible with a predefined set of constraints. Moreover the package finds the set of solutions obtained when more than one system's responses are to be optimized (multiobjective optimization).     

Efficient Inference of Parsimonious Phenomenological Models of Cellular Dynamics Using S-Systems and Alternating Regression

The nonlinearity of dynamics in systems biology makes it hard to infer them from experimental data. Simple linear models are computationally efficient, but cannot incorporate these important nonlinearities. An adaptive method based on the S-system formalism, which is a sensible representation of nonlinear mass-action kinetics typically found in cellular dynamics, maintains the efficiency of linear regression. We combine this approach with adaptive model selection to obtain efficient and parsimonious representations of cellular dynamics. The approach is tested by inferring the dynamics of yeast glycolysis from simulated data. With little computing time, it produces dynamical models with high predictive power and with structural complexity adapted to the difficulty of the inference problem.     

Control, regulation and thermodynamics of free-energy transduction

The quantitative formalism called Metabolic Control Theory makes it possible to be precise in discussions of metabolic control. To illustrate this, I will mention 2 experimental systems where free energy is converted from one form to another, i.e., bacteriorhodopsin liposomes and mitochondrial oxidative phosphorylation. More specifically I shall discuss how the distribution of the control of fluxes, concentrations and potentials, among the various enzymes (catalysts) in these systems has been measured and how this distribution can be understood in terms of the enzyme properties. From the outset, Metabolic Control Theory was valid for branched metabolic pathways with non-linear kinetics. Yet, it seemed to be limited to metabolic pathways without enzyme-enzyme interactions and to steady states. It is now clear that these limitations were apparent only and recent extensions to Metabolic Control Theory deal explicitly with enzyme-enzyme interaction and with transient-time analysis. Other limitations are inherent. For instance, Metabolic Control Theory pays for its clarity and exactness by being limited to small modulations. Mosaic Non Equilibrium Thermodynamics and Biochemical System Analysis are formalisms that deal with larger changes, at the cost of accuracy and exactness.     

Optimization of nonlinear biotechnological processes with linear programming: Application to citric acid production by Aspergillus niger

The metabolic pathway and the properties of many of the enzymes involved in the citric acid biosynthesis in the mold Aspergillus niger are well known. This fact, together with the availability of new theoretical frameworks aimed at quantitative analyses of control and dynamics in metabolic systems, has allowed us to construct a mathematical model of the carbohydrate metabolism in Aspergillus niger under conditions of citric acid accumulation. The model makes use of the S-system representation of biochemical systems, which renders it possible to use linear programming to optimize the process. It was found that maintaining the metabolite pools within narrow physiological limits (20% around the basal steady-state level) and allowing the enzyme concentrations to vary within a range of 0.1 to 50 times their basal values it is possible to triple the glycolytic flux while maintaining 100% yield of substrate transformation. To achieve these improvements it is necessary to modulate seven or more enzymes simultaneously. Although this seems difficult to implement at present, the results are useful because they indicate what the theoretical limits are and because they suggest several alternative strategies. (c) 1996 John Wiley & Sons, Inc.     

Advantages and disadvantages of aggregating fluxes into synthetic and degradative fluxes when modelling metabolic pathways.

It is now widely accepted that mathematical models are needed to predict the behaviour of complex metabolic networks in the cell, in order to have a rational basis for planning metabolic engineering with biotechnological or therapeutical purposes. The great complexity of metabolic networks makes it crucial to simplify them for analysis, but without violating key principles of stoichiometry or thermodynamics. We show here, however, that models for branched complex systems are sometimes obtained that violate the stoichiometry of fluxes at branch points and as a result give unrealistic metabolite concentrations at the steady state. This problem is especially important when models are constructed with the S-system form of biochemical systems theory. However, the same violation of stoichiometry can occur in metabolic control analysis if control coefficients are assumed to be constant when trying to predict the effects of large changes. We derive the appropriate matrix equations to analyse this type of problem systematically and to assess its extent in any given model.     

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.