Biochemical Systems Theory Research Articles

Enzyme kinetics: from diastase to multi-enzyme systems

Preface Part I. Basic Steady State Enzyme Kinetics: 1. Derivation of a rate equation 2. A closer look at the basic assumptions 3. Enzyme inhibition 4. Reversible enzyme-catalyzed reactions Part II. Enzyme Reaction Sequence: 5. Multi-reactant enzymic reactions 6. Analysis of multi-reactant enzyme kinetics 7. Prediction of reaction sequence 8. Enzyme-catalyzed isotopic exchange 9. Kinetic isotope effect on steady state parameters 10. The effect of pH on enzyme kinetics Part III. Non-Hyperbolic Enzyme Kinetics: 11. The causes of non-hyperbolic enzyme kinetics 12. Analysis of non-hyperbolic enzyme kinetics 13. The effect of subunit interactions on enzyme kinetics Part IV. Control of Multi-Enzyme Systems: 14. Control of linear multi-enzyme systems 15. Control of branched multi-enzyme systems 16. Biochemical systems theory Part V. Solutions To Problems: Author index Subject index.

The tricarboxylic acid cycle in Dictyostelium discoideum. Two methods of analysis using the same data.

The results of using metabolic control theory versus biochemical systems theory to analyze a model of the tricarboxylic acid cycle in Dictyostelium are described. The two types of analysis give essentially the same results, except that smaller responses to perturbations are observed for the model analyzed by metabolic control theory. The basis for this difference is discussed.

Modeling approach to control of carbohydrate metabolism during citric acid accumulation by Aspergillus niger: I. Model definition and stability of the steady state

A model of the carbohydrate metabolism and the anaplerotic synthesis of oxalacetate in Aspergillus niger, under conditions of citric and accumulation, is presented. In this first article we set the stage for subsequent analysis within the framework of the biochemical system theory (BST): we formulate the model and develop the system representation in power law forms, showing that the steady state is locally stable. In the second article, the control structure of the system is described and a rationale for the optimization of the process is developed.

Analysis of systems influencing renal hemodynamics and sodium excretion. I. Biochemical systems theory.

In this article we present a new methodology--Biochemical Systems Theory and Analysis--as an alternative to traditional parametric statistical procedures for investigating differences between risk groups in a population. We review the systems theory and how it can be used to represent a model of processes influencing renal hemodynamics and sodium (Na+) excretion. We also discuss the potential for new measures of the biology of common diseases that can emerge from a synergism between systems theory and population-based statistical approaches.

The tricarboxylic acid cycle in Dictyostelium discoideum. Systemic effects of including protein turnover in the current model

The current model for the tricarboxylic acid cycle in Dictyostelium discoideum is based on extensive experimental studies of enzyme kinetics in vitro and of metabolite fluxes measured in vivo. In the previous papers (Shiraishi, F., and Savageau, M. A. (1992) J. Biol. Chem. 267, 22912-22918; 22919-22925; 22926-22933; 22934-22943) of this series we have carried out extensive analyses of the current model within the framework of biochemical systems theory with a view toward understanding the behavior of the integrated system. The model was found to be ill determined with respect to at least three of its features. In this paper we propose a minimal modification in the model that is consistent with previous experimental data but also includes recycling of amino acids for protein synthesis, one of the neglected features identified as important in the previous analysis. We again perform an analysis within the framework of biochemical systems theory to determine the systemic consequences of this change. The results show that the robustness of the modified model, as determined by the parameter sensitivities, is improved by 2 orders of magnitude over that of the previous model. Analysis of the dynamics shows that the turnover times for the pools of alanine, glutamate, and aspartate are reduced by 2 orders of magnitude and made more physiologically realistic. The distribution of flux is no longer rigidly fixed, and problems previously centered on the metabolism of pyruvate have been partially alleviated. Continued discrepancies lead us to question the degree to which kinetic data obtained with purified enzymes in vitro faithfully reflect the kinetic behavior of the integrated enzyme system in vivo. We must continue to re-examine the manner in which the kinetics of reactions in vivo are represented and to reassess the physical conditions that prevail in vitro and in vivo. Results in this paper direct our attention toward specific aspects of the system where these efforts should be focused. Thus, a minimal modification of the previous model has led to several improvements that make it more representative of the tricarboxylic acid cycle in D. discoideum, and the analysis in this paper leads to further predictions for improving the model.

Simulation and optimization of metabolic pathways

A program to simulate metabolic pathways on the basis of either kinetic data or data on the steady-state levels and fluxes using Biochemical Systems Theory is presented. Having established such a model it is possible to determine the rate-controlling enzymes and metabolites for the system. While this in itself yields extremely valuable information pinpointing the limiting fluxes through the system, a more usual objective is to maximize a production rate or flux or to minimize byproducts or toxic intermediates. For this reason the problem is posed as a constrained optimization problem. While we recognize that the optimum thus found may be unachievable in practical terms due to limitations in the extent to which cells can be manipulated, it is important in that it: (a) gives an upper or lower bound to the production rate or flux; and (b) indicates the ideal combination of variables which should be aimed at. The model system used to test the software is a pathway for the degradation of toluene encoded by the TOL plasmid from Pseudomonas putida transformed into Escherichia coli. Kinetic data has been obtained, partly from the literature and partly by approximation, and realistic metabolite and enzyme concentrations are chosen.

Simulation and optimization of metabolic pathways

A program to simulate metabolic pathways on the basis of either kinetic data or data on the steady state levels and fluxes using Biochemical Systems Theory is presented. Having established such a model it is possible to determine the rate-controlling enzymes and metabolites for the system. While this in itself yields extremely valuable information pinpointing the limiting fluxes through the system, a more usual objective is to maximise a production rate or flux or to minimise byproducts or toxic intermediates. For this reason the problem is posed as a constrained optimization. While we recognise that the optimum thus found may be unachievable in practical terms due to limitations in experimental techniques, it is important in that it (a) gives an upper or lower bound to the production rate or flux, and (b) indicates the ideal combination of variables which should be aimed at. The model system used to test the software is a pathway for the degradation of toluene encoded by the TOL plasmid from Pseudomonas putida transformed into Escherichia coli. Kinetic data has been obtained, partly from the literature and partly by approximation, and realistic metabolite and enzyme concentrations are chosen.

Interpretation of Nutrient-Response Relationships in Rats

Nutrient-response curves are nonlinear and, as is true of other curves that define the rates of biological reactions, these curves can be described by rational polynomials. These nutrient-response curves presumably reflect the metabolic fate of the nutrient. Data reported for the accumulation of body nitrogen in rats fed three different proteins are interpreted in terms of the biochemical systems theory and metabolic control theories. Analysis of the rational polynomials that define the nutrient-response curves permits partitioning the nutrient between a number of grossly defined metabolic pathways. The power law formalism of the biochemical systems theory is discussed with respect to the well-established relationship between body weight and basal energy requirement.

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.

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.

Constraints among molecular and systemic properties: Implications for physiological genetics

Physiological genetics attempts to relate the molecular genetic properties of an organism--the genotype--to its integrated or physiological behavior--the phenotype. There has been relatively little progress in this field when compared to the neighboring fields of molecular and population genetics. This is due in part to the large number of highly non-linear interactions that characterize such systems. Biochemical Systems Theory is one approach that shows promise in dealing with the large number of non-linear interactions in a systematically structured manner. A variant of this approach has stressed the use of specific mathematical constraints, called summation and connectivity relationships, among molecular and systemic properties. In particular, the summation relationship has been used to argue that the predominance of recessive mutations is the inevitable consequence of the kinetic structure of enzyme networks and need not be attributed to natural selection. In order to put in broader perspective the implications of such constraints for physiological genetics, we have presented in this paper the outlines of the larger theory and the set of generalized steady state constraints that follow from first principles within this theory. The results show that the summation relationship suffers from a number of fundamental limitations that make it invalid for analyzing realistic biological systems. It also is shown that the more general constraint relationships, while valid, provide nothing new that cannot be obtained directly from the explicit solutions that are available within the larger theory. Thus, one can conclude that approaches based directly on the underlying equations of the system are superior to those based upon constraint relationships as a foundation for the development of physiological genetics.

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.

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. 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 system 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 Crabtee 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 that 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.

Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. I. unbranched pathways

It is shown that metabolic control theory (MCT), is its present form, is a particular case of general sensitivity theory, which studies the effects of parameter variations on the behavior of dynamic systems. It has been shown that metabolic control theory is obtained from this more general theory for the particular case of steady-state and linear relationships between velocities and enzyme concentrations. In such conditions the relationships between elasticities and flux control coefficients are easily obtained. These relationships are in the form of a matrix product constructed in a priori form. Relationships between combined response coefficients and concentration control coefficients are presented. The use of implicit methodology from general sensitivity theory provides a generalization of MCT, which is applied to unbranched pathways. For this particular case, provided the matrices have been properly constructed, the matrix of global properties (flux and concentration control coefficients) can be obtained by inversion of the matrix of local properties (elasticities). The theorems of MCT (concentration summation, flux summation, flux connectivity, and concentration connectivity) applicable for unbranched pathways are directly obtained by inspection of the matrix product. With these results, the present theoretical basis of MCT is extended with a more structured framework that allows a wider range of application. The results make clearer the relatedness of MCT to the more general approach provided by biochemical systems theory (BST).

Metabolic control theory and biochemical systems theory: Different objectives, different assumptions, different results

The claim by Savageau et al. (1987 a, b, Math. Biosci. 86, 127-145, 147-167) that the theory of metabolic control associated with Kacser & Burns (1973, Symp. Soc. Exp. Biol. 27, 65-104) and with Heinrich & Rapoport (1974, Eur. J. Biochem. 42, 89-102) is no more than a special case of the biochemical systems theory of Savageau and colleagues is examined. It is shown to be based on a misconception of the objectives and assumptions of metabolic control theory. In particular, the control and elasticity coefficients that play a central role in metabolic control theory are not constants and cannot be treated as constants. Consequently they cannot in general be equated with the kinetic orders that appear in biochemical systems theory, though they do correspond at the point where the two theories are tangential to one another.

Biochemical systems theory and metabolic control theory: 1. fundamental similarities and differences

Biochemical Systems Theory (BST) was developed in the late 1960s to explicate the integrated behavior of intact biochemical systems—specific dynamic behavior as well as general principles of design—in relation to the properties of their underlying molecular elements. This approach was used successfully in a number of biochemical and other biological applications throughout the 1970s and 1980s. A related approach, Metabolic Control Theory (MCT), was proposed in the mid 1970s. Its developments generally have followed without reference the analogous developments in BST, and its proponents have treated the two approaches as if they were unrelated. Detailed comparison of the fundamental structures of BST and MCT shows that, although there are some superficial differences, both in fact are based upon the same underlying formalism. Molecular descriptions in MCT comprise a special case of those in BST. Systemic descriptions differ with respect to the level of aggregation assumed. The aggregation at the level of net increase or net decrease of each system constituent found in BST is shown to produce the more revealing and useful theory, and results presented elsewhere [41] suggest that this level of aggregation also provides a more accurate description of the system. At this fundamental level, MCT represents a special case of BST, for the content and range of validity of BST are more inclusive than those of MCT.

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.

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.