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.     

Power-law modeling based on least-squares minimization criteria.

The power-law formalism has been successfully used as a modeling tool in many applications. The resulting models, either as Generalized Mass Action or as S-systems models, allow one to characterize the target system and to simulate its dynamical behavior in response to external perturbations and parameter changes. The power-law formalism was first derived as a Taylor series approximation in logarithmic space for kinetic rate-laws. The especial characteristics of this approximation produce an extremely useful systemic representation that allows a complete system characterization. Furthermore, their parameters have a precise interpretation as local sensitivities of each of the individual processes and as rate-constants. This facilitates a qualitative discussion and a quantitative estimation of their possible values in relation to the kinetic properties. Following this interpretation, parameter estimation is also possible by relating the systemic behavior to the underlying processes. Without leaving the general formalism, in this paper we suggest deriving the power-law representation in an alternative way that uses least-squares minimization. The resulting power-law mimics the target rate-law in a wider range of concentration values than the classical power-law. Although the implications of this alternative approach remain to be established, our results show that the predicted steady-state using the least-squares power-law is closest to the actual steady-state of the target system.     

Determining important regulatory relations of amino acids from dynamic network analysis of plasma amino acids

The changes in the concentrations of plasma amino acids do not always follow the flow-based metabolic pathway network. We have previously shown that there is a control-based network structure among plasma amino acids besides the metabolic pathway map. Based on this network structure, in this study, we performed dynamic analysis using time-course data of the plasma samples of rats fed single essential amino acid deficient diet. Using S-system model (conceptual mathematical model represented by power-law formalism), we inferred the dynamic network structure which reproduces the actual time-courses within the error allowance of 13.17%. By performing sensitivity analysis, three of the most dominant relations in this network were selected; the control paths from leucine to valine, from methionine to threonine, and from leucine to isoleucine. This result is in good agreement with the biological knowledge regarding branched-chain amino acids, and suggests the biological importance of the effect from methionine to threonine.     

Biochemical systems theory: operational differences among variant representations and their significance

An appropriate language or formalism for the analysis of complex biochemical systems has been sought for several decades. The necessity for such a formalism results from the large number of interacting components in biochemical systems and the complex non-linear character of these interactions. The Power-Law Formalism, an example of such a language, underlies several recent attempts to develop an understanding of integrated biochemical systems. It is the simplest representation of integrated biochemical systems that has been shown to be consistent with well-known growth laws and allometric relationships--the most regular, quantitative features that have been observed among the systemic variables of complex biochemical systems. The Power-Law Formalism provides the basis for Biochemical Systems Theory, which includes several different strategies of representation. Among these, the synergistic-system (S-system) representation is the most useful, as judged by a variety of objective criteria. This paper first describes the predominant features of the S-system representation. It then presents detailed comparisons between the S-system representation and other variants within Biochemical Systems Theory. These comparisons are made on the basis of objective criteria that characterize the efficiency, power, clarity and scope of each representation. Two of the variants within Biochemical Systems Theory are intimately related to other approaches for analyzing biochemical systems, namely Metabolic Control Theory and Flux-Oriented Theory. It is hoped that the comparisons presented here will result in a deeper understanding of the relationships among these variants. Finally, some recent developments are described that demonstrate the potential for further growth of Biochemical Systems Theory and the underlying Power-Law Formalism on which it is based.     

A simplified method for power-law modelling of metabolic pathways from time-course data and steady-state flux profiles

Background  

In order to improve understanding of metabolic systems there have been attempts to construct S-system models from time courses. Conventionally, non-linear curve-fitting algorithms have been used for modelling, because of the non-linear properties of parameter estimation from time series. However, the huge iterative calculations required have hindered the development of large-scale metabolic pathway models. To solve this problem we propose a novel method involving power-law modelling of metabolic pathways from the Jacobian of the targeted system and the steady-state flux profiles by linearization of S-systems.     

Canonical sensitivities: a useful tool to deal with large perturbations in metabolic network modeling

The dynamic range of metabolic models can be extended to deal with large perturbations by introducing the related concepts of "generalized" kinetic order and "canonical" sensitivities. Generalized kinetic orders are built as a well-defined non linear combination of the canonical sensitivities coefficients, which in turn are obtained by a least-squares regression on central composite factorial design data. In a such way, the whole domain of the operating variables is mapped without need to determine locally neither the first nor the second order model derivatives. The method was validated through numerical simulations, its predictions being compared with those coming from a Michaelis-Menten formalism taken as reference. In parallel, two variants of the Power-law formalism (S-system, least-squares GMA) also were tested. The canonical sensitivities method produced the widest range to predict metabolite concentrations and metabolic fluxes at the steady states. In addition, the variation pattern for the logarithmic gains and for the characteristic eigenvalues have been accurately determined from a unique overall model, being both required to make realistic analysis in metabolic engineering. The achieved information also can be expressed in terms of those typical coefficients derived from the Metabolic Control Analysis (MCA). Even if current first order Power-law or MCA formalisms were used, the canonical sensitivities approach provides a significant advantage, since complete sets of homologous, accurate, locally valid metabolic coefficients can be simultaneously recovered from the array proposed, being representative of the whole range of the operating variables instead of a unique nominal condition as is usual.     

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.     

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.     

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)].     

Design of metabolic engineering strategies for maximizing L-(-)-carnitine production by Escherichia coli. Integration of the metabolic and bioreactor levels

In this work metabolic engineering strategies for maximizing L-(-)-carnitine production by Escherichia coli based on the Biochemical System Theory and the Indirect Optimization Method are presented. The model integrates the metabolic and the bioreactor levels using power-law formalism. Based on the S-system model, the Indirect Optimization Method was applied, leading to profiles of parameter values that are compatible with both the physiology of the cells and the bioreactor system operating conditions. This guarantees their viability and fitness and yields higher rates of L-(-)-carnitine production. Experimental results using a high cell density reactor were compared with optimized predictions from the Indirect Optimization Method. When two parameters (the dilution rate and the initial crotonobetaine concentration) were directly changed in the real experimental system to the prescribed optimum values, the system showed better performance in L-(-)-carnitine production (74% increase in production rate), in close agreement with the model's predictions. The model shows control points at macroscopic (reactor operation) and microscopic (molecular) levels where conversion and productivity can be increased. In accordance with the optimized solution, the next logical step to improve the L-(-)-carnitine production rate will involve metabolic engineering of the E. coli strain by overexpressing the carnitine transferase, CaiB, activity and the protein carrier, CaiT, responsible for substrate and product transport in and out of the cell. By this means it is predicted production may be enhanced by up to three times the original value.     

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.     

Synergism analysis of biochemical systems. I. Conceptual framework

The detection of synergisms--deviations from additive or linear behaviour--is often an important step in uncovering mechanisms of biochemical processes. Yet, a theoretical background for systemic analysis of synergisms in metabolic networks is lacking. Based on suitable mathematical models, such a theoretical approach should allow predicting synergisms and analysing what mechanistic features contribute to specific synergisms. This work presents a conceptual framework and formalism that fulfil these purposes. The synergism between perturbations of a pair of parameters is quantified as the difference between the response to the simultaneous perturbation of both parameters and the sum of the individual responses to the perturbations of each parameter. A generalisation measures deviations from multiplicative or power-law behaviour. These deviations were called log-synergisms, as in logarithmic coordinates they are quantified in the same way as the synergisms are in Cartesian coordinates. For small perturbations, synergisms and log-synergisms are approximately proportional to the second derivatives (in Cartesian and logarithmic coordinates, respectively) of the observable to the perturbed parameter(s). These derivatives, here called synergism or log-synergism coefficients, measure how steeply the responses diverge from linearity/additivity or power-law/multiplicativity. The formalism now presented allows evaluating (log-)synergism coefficients for systemic steady-state responses, and relates these coefficients to intrinsic kinetic properties of the underlying processes. A robust homeostasis of metabolite concentrations requires that these have moderate systemic log- and relative-synergism coefficients.     

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.     

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.     

Symmetries of S-systems.

An S-system is a set of first-order nonlinear differential equations that all have the same structure: The derivative of a variable is equal to the difference of two products of power-law functions. S-systems have been used as models for a variety of problems, primarily in biology. In addition, S-systems possess the interesting property that large classes of differential equations can be recast exactly as S-systems, a feature that has been proven useful in statistics and numerical analysis. Here, simple criteria are introduced that determine whether an S-system possesses certain types of symmetries and how the underlying transformation groups can be constructed. If a transformation group exists, families of solutions can be characterized, the number of S-system equations necessary for solution can be reduced, and some boundary value problems can be reduced to initial value problems.     

An S-System Parameter Estimation Method (SPEM) for biological networks

Advances in experimental biology, coupled with advances in computational power, bring new challenges to the interdisciplinary field of computational biology. One such broad challenge lies in the reverse engineering of gene networks, and goes from determining the structure of static networks, to reconstructing the dynamics of interactions from time series data. Here, we focus our attention on the latter area, and in particular, on parameterizing a dynamic network of oriented interactions between genes. By basing the parameterizing approach on a known power-law relationship model between connected genes (S-system), we are able to account for non-linearity in the network, without compromising the ability to analyze network characteristics. In this article, we introduce the S-System Parameter Estimation Method (SPEM). SPEM, a freely available R software package (http://www.picb.ac.cn/ClinicalGenomicNTW/temp3.html), takes gene expression data in time series and returns the network of interactions as a set of differential equations. The methods, which are presented and tested here, are shown to provide accurate results not only on synthetic data, but more importantly on real and therefore noisy by nature, biological data. In summary, SPEM shows high sensitivity and positive predicted values, as well as free availability and expansibility (because based on open source software). We expect these characteristics to make it a useful and broadly applicable software in the challenging reconstruction of dynamic gene networks.     

Systems metabolic engineering: Genome-scale models and beyond

The advent of high throughput genome-scale bioinformatics has led to an exponential increase in available cellular system data. Systems metabolic engineering attempts to use data-driven approaches – based on the data collected with high throughput technologies – to identify gene targets and optimize phenotypical properties on a systems level. Current systems metabolic engineering tools are limited for predicting and defining complex phenotypes such as chemical tolerances and other global, multigenic traits. The most pragmatic systems-based tool for metabolic engineering to arise is the in silico genome-scale metabolic reconstruction. This tool has seen wide adoption for modeling cell growth and predicting beneficial gene knockouts, and we examine here how this approach can be expanded for novel organisms. This review will highlight advances of the systems metabolic engineering approach with a focus on de novo development and use of genome-scale metabolic reconstructions for metabolic engineering applications. We will then discuss the challenges and prospects for this emerging field to enable model-based metabolic engineering. Specifically, we argue that current state-of-the-art systems metabolic engineering techniques represent a viable first step for improving product yield that still must be followed by combinatorial techniques or random strain mutagenesis to achieve optimal cellular systems.