Michael A. Savageau

Recasting nonlinear differential equations as S-systems: a canonical nonlinear form

An enormous variety of nonlinear differential equations and functions have been recast exactly in the canonical form called an S-system. This is a system of nonlinear ordinary differential equations, each with the same structure: the change in a variable is equal to a difference of products of power-law functions. We review the development of S-systems, prove that the minimum for the range of equations that can be recast as S-systems consists of all equations composed of elementary functions and nested elementary functions of elementary functions, give a detailed example of the recasting process, and discuss the theoretical and practical implications. Among the latter is the ability to solve numerically nonlinear ordinary differential equations in their S-system form significantly faster than in their original form through utilization of a specially designed algorithm.

Design of gene circuits: lessons from bacteria

Researchers are now building synthetic circuits for controlling gene expression and considering practical applications for engineered gene circuits. What can we learn from nature about design principles for gene circuits? A large body of experimental data is now available to test some important theoretical predictions about how gene circuits could be organized, but the data also raise some intriguing new questions.

Design principles for elementary gene circuits: Elements, methods, and examples

The control of gene expression involves complex circuits that exhibit enormous variation in design. For years the most convenient explanation for these variations was historical accident. According to this view, evolution is a haphazard process in which many different designs are generated by chance; there are many ways to accomplish the same thing, and so no further meaning can be attached to such different but equivalent designs. In recent years a more satisfying explanation based on design principles has been found for at least certain aspects of gene circuitry. By design principle we mean a rule that characterizes some biological feature exhibited by a class of systems such that discovery of the rule allows one not only to understand known instances but also to predict new instances within the class. The central importance of gene regulation in modern molecular biology provides strong motivation to search for more of these underlying design principles. The search is in its infancy and there are undoubtedly many design principles that remain to be discovered. The focus of this three-part review will be the class of elementary gene circuits in bacteria. The first part reviews several elements of design that enter into the characterization of elementary gene circuits in prokaryotic organisms. Each of these elements exhibits a variety of realizations whose meaning is generally unclear. The second part reviews mathematical methods used to represent, analyze, and compare alternative designs. Emphasis is placed on particular methods that have been used successfully to identify design principles for elementary gene circuits. The third part reviews four design principles that make specific predictions regarding (1) two alternative modes of gene control, (2) three patterns of coupling gene expression in elementary circuits, (3) two types of switches in inducible gene circuits, and (4) the realizability of alternative gene circuits and their response to phased environmental cues. In each case, the predictions are supported by experimental evidence. These results are important for understanding the function, design, and evolution of elementary gene circuits. (c) 2001 American Institute of Physics.

Concepts relating the behavior of biochemical systems to their underlying molecular properties.

The concepts of amplification, parameter sensitivity, and feedback effectiveness in biochemical systems are presented and explored using a method of mathematical analysis previously developed for biochemical systems. These are properties of the intact system that can in no way be determined solely by experiments on the individual parts or reactions that comprise the system. These properties are important because they are functions only of the parameter values of the system, and thus may be used to characterize the system for comparative purposes. Furthermore, since these properties are independent of the concentration variables, they may be used to predict the behavior of the system over a wide range of conditions that would alter these concentrations. Empirical findings are also presented in support of this conclusion. Finally, and of most importance, these properties enable us to relate knowledge at one level of organization in biochemical systems to that at another. This is because the values of these properties can be predicted, using this theory, from knowledge of the component parts of the system, and they may also be obtained from direct measurements in the intact system. These two results must agree if the system is properly understood.

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

Abstract 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.

Comparative analysis of prototype two-component systems with either bifunctional or monofunctional sensors: differences in molecular structure and physiological function

Signal transduction by a traditional two‐component system involves a sensor protein that recognizes a physiological signal, autophosphorylates and transfers its phosphate, and a response regulator protein that receives the phosphate, alters its affinity toward specific target proteins or DNA sequences and causes change in metabolic activity or gene expression. In some cases the sensor protein, when unphosphorylated, has a positive effect upon the rate of dephosphorylation of the regulator protein (bifunctional sensor), whereas in other cases it has no such effect (monofunctional sensor). In this work we identify structural and functional differences between these two designs. In the first part of the paper we use sequence data for two‐component systems from several organisms and homology modelling techniques to determine structural features for response regulators and for sensors. Our results indicate that each type of reference sensor (bifunctional and monofunctional) has a distinctive structural feature, which we use to make predictions regarding the functionality of other sensors. In the second part of the paper we use mathematical models to analyse and compare the physiological function of systems that differ in the type of sensor and are otherwise equivalent. Our results show that a bifunctional sensor is better than a monofunctional sensor both at amplifying changes in the phosphorylation level of the regulator caused by signals from the sensor and at attenuating changes caused by signals from small phosphodonors. Cross‐talk to or from other two‐component systems is better suppressed if the transmitting sensor is monofunctional, which is the more appropriate design when such cross‐talk represents pathological noise. Cross‐talk to or from other two‐component systems is better amplified if the transmitting sensor is bifunctional, which is the more appropriate design when such cross‐talk represents a physiological signal. These results provide a functional rationale for the selection of each design that is consistent with available experimental evidence for several two‐component systems.

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.

Duplication Frequency in a Population of Salmonella enterica Rapidly Approaches Steady State With or Without Recombination

Tandem duplications are among the most common mutation events. The high loss rate of duplication suggested that the frequency of duplications in a bacterial population (1/1000) might reflect a steady state dictated by relative rates of formation (kF) and loss (kL). This possibility was tested for three genetic loci. Without homologous recombination (RecA), duplication loss rate dropped essentially to zero, but formation rate decreased only slightly and a steady state was still reached rapidly. Under all conditions, steady state was reached faster than predicted by formation and loss rates alone. A major factor in determining steady state proved to be the fitness cost, which can exceed 40% for some genomic regions. Depending on the region tested, duplications reached 40–98% of the steady-state frequency within 30 generations—approximately the growth required for a single cell to produce a saturated overnight culture or form a large colony on solid medium (109 cells). Long-term bacterial populations are stably polymorphic for duplications of every region of their genome. These polymorphisms contribute to rapid genetic adaptation by providing frequent preexisting mutations that are beneficial whenever imposed selection favors increases in some gene activity. While the reported results were obtained with the bacterium Salmonella enterica, the genetic implications seem likely to be of broader biological relevance.

Biochemical systems theory and metabolic control theory: 2. the role of summation and connectivity relationships

Abstract 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.

Molecular mechanisms of multiple toxin-antitoxin systems are coordinated to govern the persister phenotype.

Significance Persisters are drug-tolerant bacteria that account for the majority of bacterial infections. They are not mutants, but rather slowly growing cells in a heterogeneous population. Evidence links them to the toxin–antitoxin systems present in nearly all bacteria. To explore the connection, we have created a system-level model of toxin–antitoxin systems that includes molecular mechanisms, stochastic fluctuations, variable growth rate, and population dynamics. The results quantitatively describe how a noisy environment can give rise to a bet-hedging subpopulation of persisters that always exists, not just in reaction to stress. Furthermore, multiple toxin–antitoxin systems can cooperate to increase the persister frequency. Toxin–antitoxin systems are ubiquitous and have been implicated in persistence, the multidrug tolerance of bacteria, biofilms, and, by extension, most chronic infections. However, their purpose, apparent redundancy, and coordination remain topics of debate. Our model relates molecular mechanisms to population dynamics for a large class of toxin–antitoxin systems and suggests answers to several of the open questions. The generic architecture of toxin–antitoxin systems provides the potential for bistability, and even when the systems do not exhibit bistability alone, they can be coupled to create a strongly bistable, hysteretic switch between normal and toxic states. Stochastic fluctuations can spontaneously switch the system to the toxic state, creating a heterogeneous population of growing and nongrowing cells, or persisters, that exist under normal conditions, rather than as an induced response. Multiple toxin–antitoxin systems can be cooperatively marshaled for greater effect, with the dilution determined by growth rate serving as the coordinating signal. The model predicts and elucidates experimental results that show a characteristic correlation between persister frequency and the number of toxin–antitoxin systems.