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The p53 protein regulates the transcription of many different genes in response to a wide variety of stress signals. Following DNA damage, p53 regulates key processes, including DNA repair, cell-cycle arrest, senescence and apoptosis, in order to suppress cancer. This Analysis article provides an overview of the current knowledge of p53-regulated genes in these pathways and others, and the mechanisms of their regulation. In addition, we present the most comprehensive list so far of human p53-regulated genes and their experimentally validated, functional binding sites that confer p53 regulation.

Smooth Stabilization Implies Coprime Factorization Absfract-This paper shows that coprime right factorizations exist for the input-to-state mapping of a continuous-time nonlinear system provided that the smooth feedback stabilization problem is solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result, a Lyapunovtheoretic definition is proposed for bounded input bounded output stability. The main technical fact proved relates the notion of stabilizability studied in the state-space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization.

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It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or ''remember'' transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex. Here, we show that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this open-loop, feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2͞Wee1 system and a more complicated five-variable mitogenactivated protein kinase cascade.O ne of the most important and formidable challenges facing biologists and mathematicians in the postgenomic era is to understand how the behaviors of living cells arise out of the properties of complex networks of signaling proteins. One interesting systems-level property that even relatively simple signaling networks have the potential to produce is bistability. A bistable system is one that toggles between two discrete, alternative stable steady states, in contrast to a monostable system, which slides along a continuum of steady states (1-9). Early biological examples of bistable systems included the lambda phage lysis-lysogeny switch and the hysteretic lac repressor system (10-12). More recent examples have included several mitogen-activated protein kinase (MAPK) cascades in animal cells (13-15), and cell cycle regulatory circuits in Xenopus and Saccharomyces cerevisiae (16)(17)(18). Bistable systems are thought to be involved in the generation of switch-like biochemical responses (13,14,19), the establishment of cell cycle oscillations and mutually exclusive cell cycle phases (17, 18), the production of self-sustaining biochemical ''memories'' of transient stimuli (20,21), and the rapid lateral propagation of receptor tyrosine kinase activation (22).Bistability arises in signaling systems that contain a positivefeedback loop (Fig. 1a) or a mutually inhibitory, doublenegative-feedback loop (which, in some regards, is equivalent to a positive-feedback loop) (Fig. 1b). Indeed, Thomas (23) conjectured that the existence of at least one positive-feedback loop is a necessary condition for the existence of multiple steady states; alternative proofs of this conjecture are given in refs. 24-27. However, the existence of positive loop...

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In the early embryonic cell cycle, Cdc2-cyclin B functions like an autonomous oscillator, whose robust biochemical rhythm continues even when DNA replication or mitosis is blocked. At the core of the oscillator is a negative feedback loop; cyclins accumulate and produce active mitotic Cdc2-cyclin B; Cdc2 activates the anaphase-promoting complex (APC); the APC then promotes cyclin degradation and resets Cdc2 to its inactive, interphase state. Cdc2 regulation also involves positive feedback, with active Cdc2-cyclin B stimulating its activator Cdc25 (refs 5-7) and inactivating its inhibitors Wee1 and Myt1 (refs 8-11). Under the correct circumstances, these positive feedback loops could function as a bistable trigger for mitosis, and oscillators with bistable triggers may be particularly relevant to biological applications such as cell cycle regulation. Therefore, we examined whether Cdc2 activation is bistable. We confirm that the response of Cdc2 to non-degradable cyclin B is temporally abrupt and switch-like, as would be expected if Cdc2 activation were bistable. We also show that Cdc2 activation exhibits hysteresis, a property of bistable systems with particular relevance to biochemical oscillators. These findings help establish the basic systems-level logic of the mitotic oscillator.

Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.