Analysis of the system design principles of signaling systems requires model systems where all components and regulatory interactions are known. Components of the Lac and Ntr systems were used to construct genetic circuits that display toggle switch or oscillatory behavior. Both devices contain an "activator module" consisting of a modified glnA promoter with lac operators, driving the expression of the activator, NRI. Since NRI activates the glnA promoter, this creates an autoactivated circuit repressible by LacI. The oscillator contains a "repressor module" consisting of the NRI-activated glnK promoter driving LacI expression. This circuitry produced synchronous damped oscillations in turbidostat cultures, with periods much longer than the cell cycle. For the toggle switch, LacI was provided constitutively; the level of active repressor was controlled by using a lacY mutant and varying the concentration of IPTG. This circuitry provided nearly discontinuous expression of activator.

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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 (k F ) and loss (k L ). 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 (10 9 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.

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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.

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.

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