Design principles of biochemical oscillators pdf

Delayed negative feedback is often used to model oscillatory responses in molecular cell biology. Because this regulatory motif has 3 or more components, it cannot be adequately represented by a two-variable state space, as in Figs. Please notice that we always plot the activator X on the vertical axis.

For the case of a delayed feedback loop, the trajectories do not obey the flow indicators Fig. If we are plotting X vs. Y Fig. Z Fig. In either case, the delay allows the trajectory to form a closed orbit around the steady state instead of spiraling into it.

It should be obvious that activator amplification may be just as effective. For the case of activator amplification Fig. For inhibitor amplification Fig. In either case, trajectories are now forced to wheel around the steady state onto a closed orbit curve c , a sustained oscillation. In this case— Fig. If Y is large, then X will be small, for sure, and if Y is small, then X may be large. But for intermediate values of Y , the steady state concentration of X can be either large or small, depending upon how the system got to the intermediate value of Y.

This bistability is reflected in the Z-shaped nullcline curve a on the left of Fig. For the case of inhibitor amplification Fig. In this case, Y can be a multi-valued function of X , and the Y-nullcline curve b becomes N-shaped.

In either case, the negative feedback between X and Y forces trajectories to rotate clockwise on our standard XY plane, and the positive feedback loop puts kinks in one of the nullclines to prevent trajectories from spiraling into the steady state. This class of oscillators appears commonly in the literature, from the earliest models of chemical oscillations 44 - 46 to recent models of mitosis-promoting factor MPF in frog egg extracts 47 , Recently, a synthetic oscillator based on an activator-amplified negative feedback loop has been built in bacteria by Jeff Hasty's group By rewiring an activator-amplified negative feedback loop D , we create new regulatory motifs C and E.

Hence, we describe these motifs as incoherently amplified negative feedback loops NFLs. The four cases are shown in Fig. In each case, the NFL may, of course, oscillate in its own right, but the additional positive feedback loop adds bistability and robustness to the mechanism The earliest models of glycolytic oscillations 52 - 54 belong to this class of oscillators Fig.

The Martiel-Goldbeter 55 model of cyclic AMP oscillations in slime mold cells is another example of the fourth motif in Fig. This is the fast activatory signal from Y to X. Furthermore, extracellular cAMP pushes the membrane receptor Z into an inactive state, which only slowly recovers activity after cAMP is destroyed by extracellular phosphodiesterase. This is the slow inhibitory signal from Y to X.

To model oscillations in p53 a transcription factor that coordinates intracellular responses to DNA damage , Ciliberto et al. Everything we have said so far might lead us to expect oscillations in this motif under the right choice of nonlinearities and rate constants. However, it is possible to show that motif G cannot generate oscillations for any choice of nonlinear rate equations or any parameter settings see supplementary information S3 box.

Motif H has two positive feedback loops, one with two components and one with three. We might expect this regulatory motif to exhibit bistability but not oscillations, and indeed this is the case see supplementary information S4 box. To this point we have exhausted all possible oscillatory motifs with 3 components and at most 4 links.

For example, motif I combines a Class 2 inhibitor-amplified negative feedback loop with a Class 1 three-component negative feedback loop. Either sub-motif may oscillate in its own right. Motif I has recently been used by Rust et al. Allowing for four components and 5 or 6 links opens new possibilities that we have not explored systematically.

Some oscillatory motifs are simple generalizations of topologies we have seen before. For example, motif J is just an activator-amplified negative feedback loop Class 2 ; the only difference from Fig. This motif is capable of oscillating By combining two or more oscillatory motifs in a common mechanism it is possible to create quite exotic behavior including chaos, as illustrated in Fig. As Fig. Viewing a trajectory in X-Y 1 -Y 2 space Fig. It would be beyond the scope of this review to go further into the subtleties of deterministic chaos, except to point out that the requirements for chaos seem to be quite undemanding.

Chaotic trajectories readily arise in systems of coupled oscillators. Since multiple, coupled oscillators are likely to be common companions in the complex reaction networks underlying cell physiology, it is surprising that deterministic chaos has not been identified more often in experimental data 5. Perhaps the chaotic trajectories of individual cells are averaged out when large populations of cells are monitored. When the behavior of single cells is monitored with fluorescent proteins, the possibility of deterministic chaos is likely to be swamped by the white noise of molecular fluctuations in small volumes a single cell.

Nonetheless, experimentalists and theoreticians should be open to the possibility of deterministic chaos in their data and models of complex reaction networks with multiple sources of oscillation. By modelling specific examples of oscillatory processes, we have drawn a number of general conclusions about the design principles of biochemical oscillators.

In biochemical reaction kinetics there are many sources of nonlinearity that are conducive to oscillations Fig. With these ideas in mind we classify biochemical oscillators according to the topology of the positive and negative feedback loops in the reaction mechanism.

For systems with three components and three-or-four links no self-activation links , we identify three classes of oscillators: delayed negative feedback loops, amplified negative feedback loops, and incoherently amplified negative feedback loops. Our classification scheme is by no means complete, and oscillator motifs may be more complicated than any of our classes. Also, we have neglected mixed-mode effects, e. Mixed-mode interactions may easily generate complex oscillations and chaos.

If a reaction mechanism contains one of the regulatory motifs we have identified, then it may exhibit oscillatory behavior, provided the rate constants are properly tuned. Because interaction motifs and reaction rate constants are under genetic control, it is possible for biochemical oscillators to evolve.

Indeed, it is likely that biochemical oscillations have arisen repeatedly from basic, homeostatic, negative feedback loops by serendipitous genetic changes that destabilized the steady state and generated sustained oscillations. Maladaptive oscillations would have been quickly weeded out by selection, but weakly deleterious or adventitious oscillations may have been co-opted by evolving populations for beneficial physiological purposes.

Almost surely, mechanisms of circadian rhythms evolved many times independently by this scenario. On the other hand, bistability and oscillations that govern cell cycle events are much more highly constrained, and the underlying mechanism which seems to be universal across all eukaryotic cells must have been derived from a single common ancestor.

In a genetic disease, like sickle cell anemia, a mutant gene encodes a defective protein that cannot do its essential job in some important aspect of physiology. In a dynamical disease, a mutant gene encodes a modified protein that still does its job but at a different rate, thereby causing a homeostatic control mechanism to break out into pathological oscillations, or an oscillatory control system to spiral into a stable steady state.

Some cyclic blood disorders may be dynamical diseases of the first kind, and some sleep disorders may be dynamical diseases of the second kind. Finally, we hope that this review will help biochemists and molecular biologists to understand better the mechanisms underlying cellular oscillations and to recognize the importance of quantitative modelling in studying these oscillations. Supplementary Material :.

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Abstract Cellular rhythms are generated by complex interactions among genes, proteins and metabolites. They are used to control every aspect of cell physiology, from signalling, motility and development to growth, division and death. We consider specific examples of oscillatory processes and discuss four general requirements for biochemical oscillations: negative feedback, time delay, sufficient 'nonlinearity' of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions.

Positive feedback is one mechanism to delay the negative-feedback signal. Biological oscillators can be classified according to the topology of the positive- and negative-feedback loops in the underlying regulatory mechanism. Free full text. Nat Rev Mol Cell Biol. Author manuscript; available in PMC Dec PMID: John J. Author information Copyright and License information Disclaimer. Correspondence to B. Copyright notice.

See other articles in PMC that cite the published article. Go to:. Supplementary Material. Table 1 Survey of Biochemical Oscillators. Open in a separate window. Figure 1. Time-delayed negative feedback oscillator a Protein level is determined by opposing processes of synthesis and degradation. Time delay by a series of intermediates Our simple model of negative feedback on gene expression, Eq.

Figure 2. Multi-component, negative feedback oscillator a Negative feedback between mRNA and protein, as described by kinetic equations 3. Time delay by positive feedback Time delay is a sort of memory: protein synthesis rate at the present time depends on protein concentration over some time in the past. Figure 3. Hysteresis-driven, negative feedback oscillator a mRNA and protein in a negative feedback loop, as in Figure 2 , and the protein inhibits its own degradation.

Figure 4. Sources of nonlinearity a Oligomer binding. Figure 5. Class 1: Delayed Negative Feedback Loops By delayed negative feedback we refer to 3 or more components connected in a single loop by positive and negative links, with an odd number of inhibitory links. Other possibilities? More complex topologies and oscillatory behaviours To this point we have exhausted all possible oscillatory motifs with 3 components and at most 4 links.

Figure 6. Chaotic oscillators a Activator-amplification with two negative feedback loops in parallel. Supplementary Material Click here to view. Supplementary Material : 1. Computer Programs 2. Negative Feedback with Explicit Time Delay 3.

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