Rules for coupled expression of regulator and effector genes in inducible circuits.

The induction of effector genes that encode enzymes is often controlled by the protein product of a regulator gene that is directly involved in the control of its own expression. This coupling of elementary gene circuits can lead to three patterns of regulator and effector gene expression. As effector gene expression increases, regulator gene expression can increase, remain the same, or decrease, and these are referred to as directly coupled, uncoupled, or inversely coupled patterns. To determine the relative merits of each pattern, we have constructed appropriate mathematical models for the alternative gene circuits and made well-controlled comparisons using a priori criteria to evaluate their functional effectiveness. We have considered both negatively and positively controlled systems that are induced by an intermediate of the regulated pathway. Different results are obtained in the two cases. Our results indicate that direct coupling is better than inverse coupling or uncoupling for negatively controlled systems, while inverse coupling is better than the other two patterns for positively controlled systems. These optimal forms of coupling promote a fast response to inducer. Our results also indicate that realization of the optimal forms of coupling is influenced by the subunit structure of regulator proteins and requires a low capacity for induction, i.e. the ratio of maximal to minimal level of effector gene expression is small. These results lead to testable predictions, which we have compared with experimental data from over 30 systems.

Completely uncoupled and perfectly coupled gene expression in repressible systems.

Two forms of extreme coupling have been documented for the regulation of gene expression in repressible systems governed by a regulator protein. The first form, complete uncoupling, is distinguished by a constant level of regulator protein. The second form, perfect coupling, is distinguished by a level of regulator protein that varies coordinately with the level of the regulated enzyme. To determine how these two forms of coupling influence the performance of a system, so that we might predict the conditions under which each evolves through natural selection, we have used a mathematical approach to compare systems with complete uncoupling and perfect coupling. Our comparisons, which are controlled so that alternative systems are free from irrelevant differences, are based on a priori criteria that are related to various aspects of a system's performance, such as temporal responsiveness. By examining the influence of physical constraints that are related to the subunit structure of regulatory proteins and that limit the cooperativity of regulatory interactions, we have extended an early theory of gene circuitry for repressible systems. We obtain new results and testable predictions that can be summarized as follows. For typical systems with a low gain, performance is better with perfect coupling than with complete uncoupling if the mode of regulation is negative and better with complete uncoupling than with perfect coupling if the mode of regulation is positive. For systems with a high gain, these preferred forms of coupling are prevented by the physical constraints on cooperativity, and other forms of coupling can be expected. Tests of our predictions are illustrated by using data available in the literature.

Subunit structure of regulator proteins influences the design of gene circuitry: analysis of perfectly coupled and completely uncoupled circuits.

Cells regulate expression their genome by means of a diverse repertoire of molecular mechanisms. However, little is known about their design principles or how these are influenced by underlying physical constraints. An early theory of gene regulation for inducible systems predicted that expression of the regulator and regulated proteins would be perfectly coupled (coordinate expression of regulator) when the regulator is a repressor and completely uncoupled (invariant expression of regulator) when the regulator is an activator. The experimental data then available tended to support these predictions, but there were notable exceptions. Here, we describe an extended theory, which takes into account the subunit structure of regulator proteins. The number of subunits determines the allowable range of values for the regulatory parameters, and, as a consequence, new rules for the prediction of gene circuitry emerge. The theory predicts perfectly coupled circuits with repressors, but only when the capacity for induction is "small"; it predicts completely uncoupled circuits with repressors when the capacity is "large". This theory also predicts completely uncoupled circuits with activators when the capacity for induction is small; it predicts perfectly coupled circuits with activators when the capacity is large. These new predictions are more fully in accord with available experimental evidence.

Demand theory of gene regulation. I. Quantitative development of the theory.

The study of gene regulation has shown that a variety of molecular mechanisms are capable of performing this essential function. The physiological implications of these various designs and the conditions that might favor their natural selection are far from clear in most instances. Perhaps the most fundamental alternative is that involving negative or positive modes of control. Induction of gene expression can be accomplished either by removing a restraining element, which permits expression from a high-level promoter, or by providing a stimulatory element, which facilitates expression from a low-level promoter. This particular design feature is one of the few that is well understood. According to the demand theory of gene regulation, the negative mode will be selected for the control of a gene whose function is in low demand in the organism's natural environment, whereas the positive mode will be selected for the control of a gene whose function is in high demand. These qualitative predictions are well supported by experimental evidence. Here we develop the quantitative implications of this demand theory. We define two key parameters: the cycle time C, which is the average time for a gene to complete an ON/OFF cycle, and demand D, which is the fraction of the cycle time that the gene is ON. Mathematical analysis involving mutation rates and growth rates in different environments yields equations that characterize the extent and rate of selection. Further analysis of these equations reveals two thresholds in the C vs. D plot that create a well-defined region within which selection of wild-type regulatory mechanisms is realizable. The theory also predicts minimum and maximum values for the demand D, a maximum value for the cycle time C, as well as an inherent asymmetry between the regions for selection of the positive and negative modes of control.

Demand theory of gene regulation. II. Quantitative application to the lactose and maltose operons of Escherichia coli.

Induction of gene expression can be accomplished either by removing a restraining element (negative mode of control) or by providing a stimulatory element (positive mode of control). According to the demand theory of gene regulation, which was first presented in qualitative form in the 1970s, the negative mode will be selected for the control of a gene whose function is in low demand in the organism's natural environment, whereas the positive mode will be selected for the control of a gene whose function is in high demand. This theory has now been further developed in a quantitative form that reveals the importance of two key parameters: cycle time C, which is the average time for a gene to complete an ON/OFF cycle, and demand D, which is the fraction of the cycle time that the gene is ON. Here we estimate nominal values for the relevant mutation rates and growth rates and apply the quantitative demand theory to the lactose and maltose operons of Escherichia coli. The results define regions of the C vs. D plot within which selection for the wild-type regulatory mechanisms is realizable, and these in turn provide the first estimates for the minimum and maximum values of demand that are required for selection of the positive and negative modes of gene control found in these systems. The ratio of mutation rate to selection coefficient is the most relevant determinant of the realizable region for selection, and the most influential parameter is the selection coefficient that reflects the reduction in growth rate when there is superfluous expression of a gene. The quantitative theory predicts the rate and extent of selection for each mode of control. It also predicts three critical values for the cycle time. The predicted maximum value for the cycle time C is consistent with the lifetime of the host. The predicted minimum value for C is consistent with the time for transit through the intestinal tract without colonization. Finally, the theory predicts an optimum value of C that is in agreement with the observed frequency for E. coli colonizing the human intestinal tract.

Development of fractal kinetic theory for enzyme-catalysed reactions and implications for the design of biochemical pathways.

Recent evidence has shown that elementary bimolecular reactions under dimensionally-restricted conditions, such as those that might occur within cells when reactions are confined to two-dimensional membranes and one-dimensional channels, do not follow traditional mass-action kinetics, but fractal kinetics. The power-law formalism, which provides the context for examining the kinetics under these conditions, is used here to examine the implications of fractal kinetics in a simple pathway of reversible reactions. Starting with elementary chemical kinetics, we proceed to characterise the equilibrium behaviour of a simple bimolecular reaction, derive a generalised set of conditions for microscopic reversibility, and develop the fractal kinetic rate law for a reversible Michaelis-Menten mechanism. Having established this fractal kinetic framework, we go on to analyse the steady-state behaviour and temporal response of a pathway characterised by both the fundamental and quasi-steady-state equations. These results are contrasted with those for the fundamental and quasi-steady-state equations based on traditional mass-action kinetics. Finally, we compare the accuracy of three local representations based on both fractal and mass-action kinetics. The results with fractal kinetics show that the equilibrium ratio is a function of the amount of material in a closed system, and that the principle of microscopic reversibility has a more general manifestation that imposes new constraints on the set of fractal kinetic orders. Fractal kinetics in a biochemical pathway allow an increase in flux to occur with less accumulation of pathway intermediates and a faster temporal response than is the case with traditional kinetics. These conclusions are obtained regardless of the level of representation considered. Thus, fractal kinetics provide a novel means to achieve important features of pathway design.

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.

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)].

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.

Mathematics of organizationally complex systems.

Our approach to the development of an appropriate formalism for organizationally complex systems has been to search for a general formalism that would retain the essential nonlinear features (at least in approximate form) and yet would be amenable to mathematical analysis. The power-law formalism, described in detail elsewhere, leads naturally to a system of nonlinear differential equations, which is called an "S-system" because it captures the saturable and synergistic properties intrinsic to biological and other organizationally complex systems. Some of the advantages of this formalism and its implications for complex systems are discussed. Although the power-law formalism was originally developed as an "approximation", there are now several examples of "exact" representation by S-systems. In fact, a wide range of nonlinear equations can be recast in the form of S-systems. Such recasting and the use of algorithms optimized for S-systems greatly improves the efficiency of solution over that obtainable with conventional algorithms.

A theory of alternative designs for biochemical control systems.

Any theory of alternative designs must include (a) an appropriate language for describing alternatives, (b) methods of analysis for relating system behavior to changes in design elements, and (c) methods for critically comparing the behavior of alternatives. Over the past two decades we have been reasonably successful in developing a theory of alternative designs that meets these basic requirements. The language for describing alternatives is provided by the S-system of differential equations discussed elsewhere in this volume. The methods of analysis consist of adaptations of conventional methods from control theory and newly developed algorithms specific for S-systems. The methods for making critical comparisons perhaps are less familiar and will be treated in some detail. Several applications of this theory to alternative designs for biochemical control are reviewed.

Proteins of Escherichia coli come in sizes that are multiples of 14 kDa: domain concepts and evolutionary implications.

Initial attempts to correlate the distribution of gene density (number of gene loci per unit length on the linkage map) with the distribution of lengths of coding sequences have led to the observation that 46% of approximately 1000 sampled proteins in Escherichia coli have molecular masses of n X 14,000 +/- 2500 daltons (n = 1, 2, ...). This clustering around multiples of 14,000 contrasts with the 36% one would expect in these ranges if the sizes were uniformly distributed. The entire distribution is well fit by a sum of normal or lognormal distributions located at multiples of 14,000, which suggests that the percentage of E. coli proteins governed by the underlying sizing mechanism is much greater than 50%. Clustering of protein molecular sizes around multiples of a unit size also is suggested by the distribution of well-characterized HeLa cell proteins. The distribution of gene lengths for E. coli suggests regular clustering, which implies that the clustering of protein molecular masses is not an artifact of the molecular mass measurement by gel electrophoresis. These observations suggest the existence of a fundamental structural unit. The rather uniform size of this structural unit (without any apparent sequence homology) suggests that a general principle such as geometrical or physical optimization at the DNA or protein level is responsible. This suggestion is discussed in relation to experimental evidence for the domain structure of proteins and to existing hypotheses that attempt to account for these domains. Microevolution would appear to be accommodated by incremental changes within this fundamental unit, whereas macroevolution would appear to involve "quantum" changes to the next stable size of protein.

Significance of autogenously regulated and constitutive synthesis of regulatory proteins in repressible biosynthetic systems.

The functional implications of the different modes of regulation have been examined systematically. The results lead to certain predictions. The regulatory protein in repressor-controlled systems is constitutively synthesised. In activator-controlled systems synthesis of the regulatory protein is autogenously regulated. There is favourable agreement between these predictions and published experimental evidence.

Optimal design of feedback control by inhibition: dynamic considerations.

The local stability of unbranched biosynthetic pathways is examined by mathematical analysis and computer simulation using a novel nonlinear formalism that appears to accurately describe biochemical systems. Four factors affecting the stability are examined: strength of feedback inhibition, equalization of the values among the corresponding kinetic parameters for the reactions of the pathway, pathway length, and alternative patterns of feedback interactions. The strength of inhibition and the pattern of feedback interactions are important determinants of steady-state behavior. The simple pattern of end-product inhibition in unbranched pathways may have evolved because it optimizes the steady-state behavior and is temporally most responsive to change. Stability in these simple systems is achieved by shortening pathway length either physically or, in the case of necessarily long pathways, kinetically by a wide devergence in the values of the corresponding kinetic parameters for the reactions of the pathway. These conclusions are discussed in the light of available experimental evidence.

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.

Reconstructionist molecular biology.

In the editorial inaugurating this journal, Levine (1989) pointed to a new reductionism in biology, which--unlike the old reductionism that led to specialization and isolation of areas concerned with different aspects of a complex biological problem--is providing a renewed sense of unity. This development is the result of widespread use of common experimental methodology and the emergence of signal transmission and differential gene expression as themes that are central to many areas of modern biology. I describe here a set of complementary developments in molecular biology that focus attention on the problems of complexity and organization. Simple examples are given that illustrate the difficulty of relating systemic behavior to the properties of the underlying molecular determinants, and the outlines of a general approach to this problem are presented. These developments, together with those highlighted by Levine, are leading us to a new, more integrative intellectual paradigm whose fruits will be the elucidation of fundamental issues concerning network function, design, and evolution that cannot be addressed by the current paradigm.

Michaelis-Menten mechanism reconsidered: implications of fractal kinetics.

The Michaelis-Menten formalism assumes that the elementary steps of an enzymatic mechanism follow traditional mass-action kinetics. Recent evidence has shown that elementary bimolecular reactions under dimensionally-restricted conditions, such as those that might occur in vivo when reactions are confined to two-dimensional membranes and one-dimensional channels, do not follow traditional mass-action kinetics, but fractal kinetics. A Michaelis-Menten-like reaction operating under conditions of dimensional restriction is shown to exhibit new types of synergism and noninteger kinetic orders. These properties are likely to have an important influence on the behavior of intact biochemical systems, which is largely dependent upon the kinetic orders of the constituent biochemical reactions.