One of the central puzzles regarding circadian rhythm is the nature of the cellular machinery responsible for it 1. Although numerous genes required for circadian rhythm have been identified in Drosophila 23 and other organisms, including cyanobacteria 4, the actual mechanism whereby their products give rise to stable 24-hour oscillations is not established in most cases. Two interesting features have recently been highlighted in reviews: first, that different organisms have different as well as (sometimes) homologous components in their circadian oscillators; and second, that even when components are homologous between organisms, they may function in different ways 156. Thus, there may be principles of organization and function that transcend the specific components involved.

Most circadian oscillators are thought to exist within single cells 178. Consistent with this, transcriptional-translational feedback circuits ("transcriptional-translational oscillators", or TTOs) are central to most models 14, although not to all 910. In a remarkable recent study, a circadian oscillator has been reconstituted that contains only three cyanobacteria-derived proteins in homogeneous solution 11, but this so far appears exceptional.

Ultradian oscillators, i.e. oscillators with periods much less than 24 hours, are ubiquitous in biology, and several authors have suggested that at least some circadian oscillators comprise coupled ultradian ones 1213. Examples of ultradian oscillations include 3-hour cycles of expression of the mammalian p53 protein 14, 2-hour periodicity in the expression of the Notch effector Hes1 in cultured cells 15, a 1.5–3 hour periodicity in the expression of NF-κB signaling molecule in mouse cells in culture 16, and a 40-minute cycle in general transcriptional activity in yeast 17. These systems are members of a broader collection of ultradian oscillators, examples of which include 18 oxygen consumption and other metabolic processes in Acanthamoeba castellanii, which have a period of 69 minutes, respiration in Dictyostelium, with a period of 60 minutes, and energy metabolism in yeast, which shows the same 40-minute period as much of its transcriptional activity 7.

The idea of generating slow rhythms from relatively fast biochemical processes goes back at least to 1960 19. The presence of 'beats' was noted in several experimental studies 2021, and has been suggested as a mechanism for producing circadian oscillations. It was also suggested that, at least in multicellular organisms, weak coupling of ultradian oscillators between cells can produce circadian oscillations 1213222324. The 'beats' mechanism has been largely ignored because of a number of critical arguments (cf. 24), but most of the criticisms predated the gene regulatory model of circadian oscillations. In this paper we invoke a phenomenon somewhat related to 'beats' as a way of using ultradian cycles to generate circadian ones within a single cell.

More recently, several models for TTO circadian oscillations have been developed that do not depend on ultradian oscillators as components. One of these 2526 comprises two genes, one producing a transcriptional activator and the other a repressor, each of which affects both itself and the other gene. In addition, the activator and repressor proteins combine into a dimer, which inactivates them both. Another model for a mammalian TTO, comprising interacting positive and negative regulatory loops, involves the products of Per, Cry, Bmal1, Clock and Rev-Erbα genes, and also produces circadian oscillations and entrainment to light-dark cycles 27. A similar model for the circadian oscillator in Drosophila involves a complex of the products of Per and Tim 28. These examples involve closely-interlinked TTO components. Interestingly, it was the circadian clock in Drosophila that prompted the modeling of circadian rhythms as coupled ultradian ones 12, and this proposal was based partly on data showing ultradian peaks in the power spectrum.

A model proposing that circadian oscillators have evolved from pre-existing ultradian ones involves five ultradian oscillators arranged in a loop 29. We describe here a different kind of coupled ultradian model, in which two independent ultradian TTOs drive a third oscillator by the combination of their protein products. In this model, the frequency of the output is related to the difference in frequencies between the two independent primary oscillators. Neither the early papers suggesting 'beats' as a mechanism 2021 nor the proposed mathematical models involving populations of ultradian oscillators 121324 include mechanistic or molecular details. In this paper, we demonstrate that realistic mechanisms and parameters taken from molecular biology can produce a circadian oscillator using ultradian component TTOs. The model also suggests a mechanism for its evolution.

We describe a model that uses transcriptional-translational oscillators of relatively fast (ultradian) frequencies to drive a forced oscillator with a period of approximately 24 hours, i.e. a circadian oscillator. The ultradian oscillators differ in their frequencies, and their products are coupled to force the output oscillator. It is only necessary that the primary oscillators are periodic – sinusoidal oscillators with the same period as the nonlinear transcriptional-translational systems described will drive the forced oscillator in the same way, with a similar fine structure.

The two primary oscillators may differ qualitatively, to avoid having either one alone able to drive the forced oscillator. For example, ultradian cycling of the cellular redox state might alter the effectiveness of a transcription activator with its own independent ultradian rhythm. Indeed, an effect of redox state on a transcription activator of circadian gene expression is known 32. Because the primary oscillators in our model work in a product fashion, rather than, say, being additive, it is not necessary that their individual products have similar concentration ranges to drive the fifth gene with a circadian period.

It is difficult to relate the parameters in this model to actual values in cells undergoing circadian rhythm, much less to components of circadian oscillators themselves, many of which remain unknown. However, the parameters (Figure 2 legend) are based on plausible values. The most critical values are the degradation rates of mRNA and, to a lesser extent, protein. We have used 8 minutes for the half-life of mRNAs of the primary oscillators, which is similar to several eukaryotic and prokaryotic mRNAs: c-fos mRNA has been reported to have a half-life of 6.6 minutes in NIH 3T3 cells 41 and 9 minutes in human fibroblasts 42, and the average for E. coli mRNA has been reported to be 6.8 minutes 43. The stabilities of individual mRNAs in a cell can differ by orders of magnitude, but the short half-life used in our model is not unrealistic.

The parameter for protein turnover in the model corresponds to a half-life of about 10 minutes. Although the half-life of the average protein in eukaryotic cells is many hours, much faster turnover is found for some proteins, including reported 12 and 18-minute half-lives for rat liver ornithine decarboxylase and δ -aminolevulinate synthetase, respectively 44. The corresponding value for Tim, a component of the Drosophila circadian system, is 20 minutes 38. The half-life of p53 is 16 minutes in a keratinocyte cell line 45, and that of N-myc is 30 minutes 46. Although prokaryotic proteins typically have half-lives in the order of hours, there are exceptions. For example, 48 proteins of Caulobacter turned over much more quickly than the cell cycle time of 120 minutes 47, and the lambda repressor protein in E. coli has a half-life of about 60 minutes 48. More generally in E. coli, the majority of proteins turn over slowly, but some are much shorter-lived 49. In the represillator model of Elowitz and Leibler, the critical proteins were taken to have a half-life of about 10 minutes 36. In any case, our proposed mechanism is not ultimately dependent on the shorter half-lives we have chosen but on the ratio of the periods of the primary oscillators.

The light-dependent mechanism of phase-resetting in the model is based on the properties of the Drosophila Cryptochrome protein, which induces light-activated degradation of Tim protein that is part of that organism's circadian oscillator 38. In our model, and using the parameters of Figure 2, the half-life of proteins P₂ and P₄ are reduced from 10 minutes in the dark to 5.1 in light through the coupling factor k₂₉. A more realistic version would probably have the effect of light-driven degradation restricted to only one of the primary oscillators, but we have not pursued this variation.

The output of the model (gene 5 in Figure 1) could provide the kind of circadian timing that would be analogous to the "master regulators" that control the timing of cell cycle events in Caulobacter 33. The evolution of such a circadian system might begin with the development of ultradian TTOs, which themselves have important regulatory value, like that of the NF-κB system 1635. The creation of a forced oscillator that responds to the products of two such ultradian oscillators depends on their individual frequencies, the strength of their interactions, and the binding strengths between their products and the transcription control site of the forced oscillator. Thus, the development of a circadian oscillator could occur independently of the functions of the primary oscillators, allowing for the development of a new, beneficial trait (circadian rhythm) without significantly affecting the primary systems. A different model for evolution of circadian systems based on the development of synchronized metabolic pathways has been proposed by Roenneberg and Merrow 29.

Whether any existing circadian oscillators depend on ultradian ones as suggested here or in earlier work 121329 is unproven, but evidence consistent with this model can be seen in power spectral analyses of some circadian systems, including the activity profile of Drosophila 12 and the secretion of ghrelin in rats 50, both of which show higher frequency components in addition to the main circadian frequency.

Amongst the arguments that have been brought forward against 'beats' as a mechanism is that coupled oscillators of similar frequencies will undergo mutual entrainment and that the 'beats' will be lost 24. In our model, oscillators are coupled indirectly and weakly, through the formation of a protein heterodimer. In the case of weak coupling, Pavlidis 24 has argued that the relative phases of the primary oscillators would be random and too much variability of behavior would result. In the model presented here, the primary oscillators do not undergo mutual entrainment, and the output is not dependent on the initial phase relationship between them.

It has also been argued that models based on beats are not robust because small changes in the periods of the primary oscillators lead to large changes in the circadian period 2451. In the absence of directly pertinent data, it is difficult to determine whether this is a significant problem. However, the enzymes that carry out biochemical reactions have well defined rate constants, which do not normally change, and thus a shift in frequency would not be expected in such a model. A more fundamental concern is that real reactions are stochastic, and especially under cellular conditions with small numbers of some molecules (for example, the genes involved), this might lead to instability in oscillators of this type. We have therefore also cast the model into stochastic terms, and the results indicate that the system is robust to stochastic fluctuations (work in progress). Finally, a TTO model can provide temperature compensation, since the increase in reaction rates typical of biological processes may be opposed by a decrease in the rate of formation of DNA-binding protein dimers, as has been documented for the leucine zipper transcriptional oscillator GCN4 30.

The effect of light on the primary oscillators would be selected on the basis of the benefit of making the levels of certain gene products lower or higher in daylight than at night, and could be achieved by a light-sensitive protease such as the Cryptochrome of Drosophila 38 before the evolution of the circadian oscillator. Over time, the development of a circadian rhythm might impart larger benefits to the organism. In cyanobacteria, for example, matching of the free-running period to the light-dark cycle time provides a selective advantage 52, which is presumably the basis for its evolution. In Arabidopsis, matching between the circadian period and the light-dark cycle results in plants that fix carbon at a higher rate and grow and survive better than those that lack such a match 53.

Cellular oscillators based on metabolic pathways have also been described. Almost 40 years ago, Chance and colleagues described oscillations in glycolytic pathways both in yeast and yeast extracts. In intact cells the oscillations had a high damping factor, but with a judicious choice of long-lasting carbohydrate substrate, enzyme extracts could maintain oscillations for very long times. Furthermore, the basic short period oscillations (in the order of 10 minutes) were sometimes superimposed on slower periodicities that were two or even more times the fundamental frequency 54. These authors suggested that similar oscillations might be basic regulators of biological clocks. In general, however, oscillators that depend on extracellular substrates are not attractive for this purpose, since the oscillations will fluctuate or even extinguish depending on the levels of those substrates 55. Mechanisms that are entirely intracellular in terms of substrates and products, such as the one described here, are more likely to provide stable primary oscillators. The only necessary communication with the outside world is through a light-sensitive mechanism to reset the phase of the driven oscillator.

Independent transcriptional-translational oscillators with relatively short (ultradian) periods can be coupled to generate a circadian oscillator using conventional mechanisms of molecular genetics and reasonable values of parameters describing these mechanisms. The resulting circadian oscillator can be entrained by 24-hour light-dark cycles. The model suggests that evolution of such a circadian oscillator would occur under selective pressure without significantly perturbing the underlying components.

Differential equations were solved numerically using the XPPAUT software described by Ermentraut http://www.math.pitt.edu/~bard/xpp/xpp.html.

The author(s) declare that they have no competing interests.

VP proposed the original problem of generating circadian oscillations with relatively short-lived molecular processes and wrote the bulk of the paper; RI and RE proposed the coupled oscillator approach, and developed the ordinary differential equation model and the analysis of its behavior. All three authors worked to bring the model to fruition through discussions and analysis of simulations.