Irreversibility in unbranched pathways: preferred positions based on regulatory considerations.

It has been observed experimentally that most unbranched biosynthetic pathways have irreversible reactions near their beginning, many times at the first step. If there were no functional reasons for this fact, then one would expect irreversible reactions to be equally distributed among all positions in such pathways. Since this is not the case, we have attempted to identify functional consequences of having an irreversible reaction early in the pathway. We systematically varied the position of the irreversible reaction in model pathways and compared the resulting systemic behavior according to several criteria for functional effectiveness, using the method of mathematically controlled comparisons. This technique minimizes extraneous differences in systemic behavior and identifies those that are fundamental. Our results show that a pathway with an irreversible reaction located at the first step, and with all other reactions reversible, is on average better than an otherwise equivalent pathway with all reactions reversible, which in turn is on average better than an otherwise equivalent pathway with an irreversible reaction located at any step other than the first. Pathways with an irreversible first reaction and low concentrations of intermediates (one of the primary criteria for functional effectiveness) exhibit the following profile when compared to fully reversible pathways: changes in the concentration of intermediates in response to changes in the level of initial substrate are equally low, the robustness of the intermediate concentrations and of the flux is similar, the margins of stability are similar, flux is more responsive to changes in demand for end product, intermediate concentrations are less responsive to changes in demand for end product, and transient times are shorter. These results provide a functional rationale for the positioning of irreversible reactions at the beginning of unbranched biosynthetic pathways.

Extending the method of mathematically controlled comparison to include numerical comparisons.

MOTIVATION: The method of mathematically controlled comparison has been used for some time to determine which of two alternative regulatory designs is better according to specific quantitative criteria for functional effectiveness. In some cases, the results obtained using this technique are general and independent of parameter values and the answers are clear-cut. In others, the result might be general, but the demonstration is difficult and numerical results with specific parameter values can help to clarify the situation. In either case, numerical results with specific parameter values can also provide an answer to the question of how much larger the values might be. In contrast, a more ambiguous result is obtained when either of the alternatives can have the larger value for a given systemic property, depending on the specific values of the parameters. In any case, introduction of specific values for the parameters reduces the generality of the results. Therefore, we have been motivated to develop and apply statistical methods that would permit the use of numerical values for the parameters and yet retain some of the generality that makes mathematically controlled comparison so attractive. RESULTS: We illustrate this new numerical method in a step-by-step application using a very simple didactic example. We also validate the results by comparison with the corresponding results obtained using the previously developed analytical method. The analytical approach is briefly present for reference purposes, since some of the same key concepts are needed to understand the numerical method and the results are needed for comparison. The numerical method confirms the qualitative differences between the systemic behavior of alternative designs obtained from the analytical method. In addition, the numerical method allows for quantification of the differences and it provides results that are general in a statistical sense. For example, the older analytical method showed that overall feedback inhibition in an unbranched pathway makes the system more robust whereas it decreases the stability margin of the steady state. The numerical method shows that the magnitudes of these differences are not comparable. The differences in stability margins (1-2% on average) are small when compared to the differences in robustness (50-100% on average). Furthermore, the numerical method shows that the system with overall feedback responds more quickly to change than the otherwise equivalent system without overall feedback. These results suggest reasons why overall feedback inhibition is such a prevalent regulatory pattern in unbranched biosynthetic pathways.

Effect of overall feedback inhibition in unbranched biosynthetic pathways.

We have determined the effects of control by overall feedback inhibition on the systemic behavior of unbranched metabolic pathways with an arbitrary pattern of other feedback inhibitions by using a recently developed numerical generalization of Mathematically Controlled Comparisons, a method for comparing the function of alternative molecular designs. This method allows the rigorous determination of the changes in systemic properties that can be exclusively attributed to overall feedback inhibition. Analytical results show that the unbranched pathway can achieve the same steady-state flux, concentrations, and logarithmic gains with respect to changes in substrate, with or without overall feedback inhibition. The analytical approach also shows that control by overall feedback inhibition amplifies the regulation of flux by the demand for end product while attenuating the sensitivity of the concentrations to the same demand. This approach does not provide a clear answer regarding the effect of overall feedback inhibition on the robustness, stability, and transient time of the pathway. However, the generalized numerical method we have used does clarify the answers to these questions. On average, an unbranched pathway with control by overall feedback inhibition is less sensitive to perturbations in the values of the parameters that define the system. The difference in robustness can range from a few percent to fifty percent or more, depending on the length of the pathway and on the metabolite one considers. On average, overall feedback inhibition decreases the stability margins by a minimal amount (typically less than 5%). Finally, and again on average, stable systems with overall feedback inhibition respond faster to fluctuations in the metabolite concentrations. Taken together, these results show that control by overall feedback inhibition confers several functional advantages upon unbranched pathways. These advantages provide a rationale for the prevalence of this control mechanism in unbranched metabolic pathways in vivo.

Comparing systemic properties of ensembles of biological networks by graphical and statistical methods.

MOTIVATION: When dealing with questions that concern a general class of models for biological networks, large numbers of distinct models within the class can be grouped into an ensemble that gives a statistical view of the properties for the general class. Comparing properties of different ensembles through the use of point measures (e.g. medians, standard deviations, correlation coefficients) can mask inhomogeneities in the correlations between properties. We are therefore motivated to develop strategies that allow these inhomogeneities to be more easily detected. RESULTS: Methods are described for constructing ensembles of models within the context of a Mathematically Controlled Comparison. A Density of Ratios Plot for a given systemic property is then defined as follows: the y axis represents the value of the systemic property in a reference model divided by the value in the alternative model, and the x axis represents the value of the systemic property in the reference model. Techniques involving moving quantiles are introduced to generate secondary plots in which correlations and inhomogeneities in correlations are more easily detected. Several examples that illustrate the advantages of these techniques are presented and discussed.

Systemic properties of ensembles of metabolic networks: application of graphical and statistical methods to simple unbranched pathways.

MOTIVATION: Mathematical models are the only realistic method for representing the integrated dynamic behavior of complex biochemical networks. However, it is difficult to obtain a consistent set of values for the parameters that characterize such a model. Even when a set of parameter values exists, the accuracy of the individual values is questionable. Therefore, we were motivated to explore statistical techniques for analyzing the properties of a given model when knowledge of the actual parameter values is lacking. RESULTS: The graphical and statistical methods presented in the previous paper are applied here to simple unbranched biosynthetic pathways subject to control by feedback inhibition. We represent these pathways within a canonical nonlinear formalism that provides a regular structure that is convenient for randomly sampling the parameter space. After constructing a large ensemble of randomly generated sets of parameter values, the structural and behavioral properties of the model with these parameter sets are examined statistically and classified. The results of our analysis demonstrate that certain properties of these systems are strongly correlated, thereby revealing aspects of organization that are highly probable independent of selection. Finally, we show how specification of a given behavior affects the distribution of acceptable parameter values.

Rules for the evolution of gene circuitry.

Cells possess the genes required for growth and function in a variety of contexts. In any given context there is a corresponding pattern of gene expression in which some genes are OFF and others ON. The ability of cells to switch genes ON and OFF in a coordinate fashion to produce the required patterns of expression is the fundamental basis for complex processes like normal development and pathogenesis. The molecular study of gene regulation has revealed a plethora of mechanisms and circuitry that have evolved to perform what appears to be the same switching function. To some this implies the absence of rules. However, simple rules capable of relating molecular design to the natural environment have begun to emerge through the analysis of elementary gene circuits. Two of these rules are reviewed in this paper. These simple rules have the ability to unify understanding across several different levels of biological organization--molecular, physiological, developmental, ecological.

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.

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.

Application of biochemical systems theory to metabolism in human red blood cells. Signal propagation and accuracy of representation.

Human erythrocytes are among the simplest of cells. Many of their enzymes have been characterized kinetically using steady-state methods in vitro, and several investigators have assembled this kinetic information into mathematical models of the integrated system. However, despite its relative simplicity, the integrated behavior of erythrocyte metabolism is still complex and not well understood. Errors will inevitably be encountered in any such model because of this complexity; thus, the construction of an integrative model must be considered an iterative process of assessment and refinement. In a previous study, we selected a recent model of erythrocyte metabolism as our starting point and took it through three stages of model assessment and refinement using systematic strategies provided by biochemical systems theory. At each stage deficiencies were diagnosed, putative remedies were identified, and modifications consistent with existing experimental evidence were incorporated into the working model. In this paper we address two issues: the propagation of biochemical signals within the metabolic network, and the accuracy of kinetic representation. The analysis of signal propagation reveals the importance of glutathione peroxidase, transaldolase, and the concentration of total glutathione in determining systemic behavior. It also reveals a highly amplified diversion of flux between the pathways of pentose phosphate and nucleotide metabolism. In determining the range of accurate representation based on alternative kinetic formalisms we discovered large discrepancies. These were identified with the behavior of the model represented within the Michaelis-Menten formalism. This model fails to exhibit a nominal steady state when the activity of glutathione peroxidase is decreased by as little as 9%. Our current understanding, as embodied in this working model, is in need of further refinement, and the results presented in this paper suggest areas of the model where such effort might profitably be concentrated.

Model assessment and refinement using strategies from biochemical systems theory: application to metabolism in human red blood cells.

Models of biochemical systems are typically formulated with kinetic data obtained from isolated enzymes studied in vitro, and one has always to question whether or not all the relevant metabolites, processes and regulatory interactions have been identified and whether the parameter values obtained in vitro reflect the actual intracellular environment. In this paper we extend and further test strategies for model assessment and refinement that take advantage of the power-law formalism, which provides the systematic structure underlying biochemical systems theory. Our purpose is three fold. First, we introduce an algorithm for systematically scanning a model for putative errors, which, if corrected, would reconcile its behavior with the experimental system. Second, we further test the working hypothesis that systems in nature are selected to be robust and, hence, that the profile of parameter sensitivities can be used to identify poorly defined regions of a model. Third, we illustrate the use of these strategies within the context of a relatively large and realistic biochemical system--the metabolic pathways of the human red blood cell. Our results show that the reference model we have used is neither locally stable nor robust. The algorithm identifies a number of putative regulatory interactions that, when added to the model, are capable of stabilizing the nominal steady state. We include one of these, the feedback inhibition of hexokinase by fructose-6-phosphate, in a first refinement of the model because there is experimental support for it in the literature. Careful re-examination of the most sensitive section in this model, the pathways of nucleotide metabolism, reveals two mechanisms that were omitted from the reference model: membrane transport of adenosine and inosine, and regulation of phosphoribosyl pyrophosphate synthetase by adenosine diphosphate, 2,3 diphosphoglycerate and 5-phosphoribosyl-1-pyrophosphate. It was also found that the concentration of inorganic phosphate had been inappropriately assumed to be a constant. Modifications to correct these deficiencies produced a second refinement of the model whose parameter sensitivities are reduced on average by 10-fold. Although these refinements are modest and there is substantial room for further improvement, this application identified several biochemically relevant features of the model that had been overlooked. It also points to nucleotide metabolism as the area most in need of further experimental study.

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.

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.

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.

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.

Influence of fractal kinetics on molecular recognition.

Molecular recognition is a central issue for nearly every biological mechanism. The analysis of molecular recognition to date has been conducted within the framework of classical chemical kinetics, in which the kinetic orders of a reaction have positive integer values. However, recent theoretical and experimental advances have shown that the assumptions inherent in this classical framework are invalid under a variety of conditions in which the reaction environment may be considered nonideal. A good example is provided by reactions that are spatially constrained and diffusion limited. Bimolecular reactions confined within two-dimensional membranes, one-dimensional channels or fractal surfaces in general exhibit kinetic orders that are noninteger. An appropriate framework for the study of these nonideal phenomena is provided by the Power-Law formalism, which includes as special cases the Mass-Action formalism of chemical kinetics and the Michaelis-Menten formalism of enzyme kinetics. The Power-Law formalism is an appropriate representation not only for fractal kinetics per se, but also for other nonideal kinetic phenomena, provided the range of variation in concentration is not too large. After defining some elementary concepts of molecular recognition, and showing how these are manifested in classical kinetic terms, this paper contrasts the implications of classical and fractal kinetics in a few simple cases. The principal distinction lies in the ability of fractal kinetics to nonlinearly transform, rather than proportionally transmit, the input S/N ratio. As a consequence, fractal kinetics create a threshold for the input signal below which no recognition occurs and above which amplified recognition takes place.(ABSTRACT TRUNCATED AT 250 WORDS)

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.