Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

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Abstract

The physiological responses of cells to external and internal stimuli are governed by genes and proteins interacting in complex networks whose dynamical properties are impossible to understand by intuitive reasoning alone. Recent advances by theoretical biologists have demonstrated that molecular regulatory networks can be accurately modeled in mathematical terms. These models shed light on the design principles of biological control systems and make predictions that have been verified experimentally.

Introduction

Since the advent of recombinant DNA technology about 20 years ago, molecular biologists have been remarkably successful in dissecting the molecular mechanisms that underlie the adaptive behaviour of living cells. Stunning examples include the lysis–lysogeny switch of viruses [1], chemotaxis in bacteria [2], the DNA-division cycle of yeasts [3], segmentation patterns in fruit fly development [4] and signal transduction pathways in mammalian cells [5]. When the information in any of these cases is laid out in graphical form (http://discover.nci.nih.gov/kohnk/interaction_maps.html; http://www.csa.ru:82/Inst/gorb_dep/inbios/genet/s0ntwk.htm; http://www.biocarta.com/genes/index.asp), the molecular network looks strikingly similar to the wiring diagram of a modern electronic gadget. Instead of resistors, capacitors and transistors hooked together by wires, one sees genes, proteins and metabolites hooked together by chemical reactions and intermolecular interactions. The temptation is irresistible to ask whether physiological regulatory systems can be understood in mathematical terms, in the same way an electrical engineer would model a radio [6]. Preliminary attempts at this sort of modelling have been carried out in each of the cases mentioned above 7., 8., 9., 10., 11., 12.••, 13., 14., 15.••.

To understand how these models are built and why they work the way they do, one must develop a precise mathematical description of molecular circuitry and some intuition about the dynamical properties of regulatory networks. Complex molecular networks, like electrical circuits, seem to be constructed from simpler modules: sets of interacting genes and proteins that carry out specific tasks and can be hooked together by standard linkages [16].

Excellent reviews from other perspectives can be found elsewhere 17., 18.•, 19., 20., 21., 22., 23.•, 24.•, 25., and also book-length treatments 26., 27., 28., 29..

In this review, we show how simple signaling pathways can be embedded in networks using positive and negative feedback to generate more complex behaviours — toggle switches and oscillators — which are the basic building blocks of the exotic, dynamic behaviour shown by nonlinear control systems. Our purpose is to present a precise vocabulary for describing these phenomena and some memorable examples of each. We hope that this review will improve the reader’s intuition about molecular dynamics, foster more accurate discussions of the issues, and promote closer collaboration between experimental and computational biologists.

Section snippets

Linear and hyperbolic signal-response curves

Let’s start with two simple examples of protein dynamics: synthesis and degradation (Figure 1a), and phosphorylation and dephosphorylation (Figure 1b). Using the law of mass action, we can write rate equations for these two mechanisms, as follows:dRdt=k0+k1S−k2R,dRPdt=k1S(RT−RP)−k2RP.In case (a), S=signal strength (e.g. concentration of mRNA) and R=response magnitude (e.g. concentration of protein). In case (b), RP is the phosphorylated form of the response element (which we suppose to be the

Sigmoidal signal-response curves

Case (c) of Figure 1 is a modification of case (b), where the phosphorylation and dephosphorylation reactions are governed by Michaelis-Menten kinetics:dRPdt=k1S(RT−RP)Km1+RT−RPk2RPkm2+RP,In this case, the steady-state concentration of the phosphorylated form is a solution of the quadratic equation:k1S(RT−RP)(Km2+RP)=k2RP(Km1+RT−RP).The biophysically acceptable solution (0<RP<RT) of this equation is [30]:RP,ssRT=G(k1,S,k2,Km1RT,Km2RT),where the ‘Goldbeter-Koshland’ function, G, is defined as:

Perfectly adapted signal-response curves

By supplementing the simple linear response element (Figure 1a) with a second signaling pathway (through species X in Figure 1d), we can create a response mechanism that exhibits perfect adaptation to the signal. Perfect adaptation means that although the signaling pathway exhibits a transient response to changes in signal strength, its steady-state response Rss is independent of S. Such behaviour is typical of chemotactic systems, which respond to an abrupt change in attractants or repellents,

Positive feedback: irreversible switches

In Figure 1d the signal influences the response via two parallel pathways that push the response in opposite directions (an example of feed-forward control). Alternatively, some component of a response pathway may feed back on the signal. Feedback can be positive, negative or mixed.

There are two types of positive feedback. In Figure 1e, R activates protein E (by phosphorylation), and EP enhances the synthesis of R. In Figure 1f, R inhibits E, and E promotes the degradation of R; hence, R and E

Negative feedback: homeostasis and oscillations

In negative feedback, the response counteracts the effect of the stimulus. In Figure 1g, the response element, R, inhibits the enzyme catalysing its synthesis; hence, the rate of production of R is a sigmoidal decreasing function of R. The signal in this case is the demand for R — that is, the rate of consumption of R is given by k2SR. The steady state concentration of R is confined to a narrow window for a broad range of signal strengths, because the supply of R adjusts to its demand. This

Positive and negative feedback: oscillators

Oscillations often arise in systems containing both positive and negative feedback (Figure 2b,c). The positive-feedback loop creates a bistable system (a toggle switch) and the negative-feedback loop drives the system back and forth between the two stable steady states. Oscillators of this sort come in two varieties.

Complex networks: the cell cycle control system

The signal-response elements we have just described, buzzers, sniffers, toggles and blinkers, usually appear as components of more complex networks (see, for example, 7., 8., 9., 10., 11., 12.••). Being most familiar with the regulatory network of the eukaryotic cell cycle, we use that example to illustrate the issues involved in modelling realistic wiring diagrams.

A generic wiring diagram for the Cdk network regulating DNA synthesis and mitosis is presented in Figure 3a. The network, involving

Signaling in space

So far, we have considered only time-dependent signaling. But spatial signaling also plays important roles in cell physiology (e.g. cell aggregation, somite formation, cell division plane localization, etc.). Interestingly the same mechanism (autocatalysis plus negative feedback) that creates oscillations (broken symmetry in time) can also create spatial patterns (broken symmetry in space) 55., 56.. Two sorts of patterns may arise. If the inhibitor (or substrate) diffuses much more rapidly than

Conclusions

The life of every organism depends crucially on networks of interacting proteins that detect signals and generate appropriate responses. Examples include chemotaxis, heat shock response, sporulation, hormone secretion, and cell-cycle checkpoints. Although diagrams and informal hand-waving arguments are often used to rationalize how these control systems work, such cartoons lack the precision necessary for a quantitative and reliable understanding of complex regulatory networks. To reprogram

Update

Recent work includes an elegant theoretical and experimental study of NF-κB signaling [61••] and methods for deducing a molecular wiring diagram from a system’s transient response to small disturbances 62.•, 63.•.

References and recommended reading

Papers of particular interest, published within the annual period of review, have been highlighted as:

  • of special interest

  • ••

    of outstanding interest

Acknowledgements

We gratefully acknowledge financial support from the National Science Foundations of the USA (MCB-0078920) and Hungary (T 032015), from the Defense Advanced Research Project Agency (AFRL #F30602-02-0572), and from the James S McDonnell Foundation (21002050).

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