Control of metabolism: where is the theory?

Evolution of Henrik Kacser's thought: Early publications on the organization of the whole system

Although the papers of Kacser and Burns (1973) and Heinrich and Rapoport (1974a,b) are commonly taken as the birth of metabolic control analysis, many of the ideas in them are foreshadowed in earlier papers, from 1956 onwards, when Kacser first argued for taking a systemic view of genetics and biochemistry.

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.

Dynamic metabolic control theory. A methodology for investigating metabolic regulation using transient metabolic data

The purposes of the dynamic metabolic control theory are to provide a theoretical basis for estimating the control coefficients using the transient metabolic responses and to gain insights into the metabolic regulation in the transient states. The numerical application of this theory is relatively straightforward: it involves a standard linear regression and a matrix multiplication. Although the equations are exact only for linear kinetics, they yield relatively good estimates of the control coefficients for nonlinear systems.

Enzyme-enzyme interactions and control analysis. 1. The case of non-additivity: monomer-oligomer associations

Two usual assumptions of the treatment of metabolism are: (a) the rates of isolated enzyme reactions are additive, i.e., that rate is proportional to enzyme concentration; (b) in a system, the rates of individual enzyme reactions are not influenced by interactions with other enzymes, i.e. that they are acting independently, except by being coupled through shared metabolites. On this basis, control analysis has established theorems and experimental methods for studying the distribution of control. These assumptions are not universally true and it is shown that the theorems can be modified to take account of such deviations. This is achieved by defining additional elasticity coefficients, designated by the symbol pi, which quantify the effects of homologous and heterologous enzyme interactions. Here we show that for the case of non-proportionality of rate with enzyme concentration, (pi ii not equal to 1), the summation theorems are given by (Formula: see text). The example of monomer-oligomer equilibria is used to illustrate non-additive behaviour and experimental methods for their study are suggested.

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. 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.

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)].

Metabolic control therapy 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.

The matrix method of metabolic control analysis: its validity for complex pathway structures

The sensitivities of the variables of a metabolic system (such as fluxes and concentrations) to variations in enzyme concentration are expressed in metabolic control analysis as control coefficients. The matrix method is a system of writing matrix equations that generate expressions for the control coefficients in terms of the characteristics of the components (principally the enzymes). Previously, the matrix method has been considered in terms of simple pathway structures; here we justify its applicability to complex pathways, such as those with multiple branches. It is shown that this requires modification of the branch point relationship to take account of changes of flux along the limbs of the branch and of stoichiometric factors. The method of deriving the flux control coefficients with respect to different fluxes in the system is extended to cope with these circumstances.