Robustness of circadian rhythms with respect to molecular noise

Abstract

We use a core molecular model capable of generating circadian rhythms to assess the robustness of circadian oscillations with respect to molecular noise. The model is based on the negative feedback exerted by a regulatory protein on the expression of its gene. Such a negative regulatory mechanism underlies circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. The model incorporates gene transcription into mRNA, translation of mRNA into protein, reversible phosphorylation leading to degradation of the regulatory protein, transport of the latter into the nucleus, and repression of gene expression by the nuclear form of the protein. To assess the effect of molecular noise, we perform stochastic simulations after decomposing the deterministic model into elementary reaction steps. The oscillations predicted by the stochastic simulations agree with those obtained with the deterministic version of the model. We show that robust circadian oscillations can occur already with a limited number of mRNA and protein molecules, in the range of tens and hundreds, respectively. Entrainment by light/dark cycles and cooperativity in repression enhance the robustness of circadian oscillations with respect to molecular noise.

Figure 1

Core model for circadian rhythms. The model represents a prototype for the molecular mechanism of circadian oscillations based on negative autoregulation of gene expression. The model incorporates gene transcription, transport of mRNA (M_P) into the cytosol where it is translated into the clock protein (P₀) and degraded. The clock protein can be reversibly phosphorylated from the form P₀ into the forms P₁ and P₂, successively. The latter form is degraded or transported into the nucleus (P_N), where it exerts a negative feedback of cooperative nature on the expression of its gene. The model accounts for circadian oscillations of per mRNA and PER protein in Drosophila (5, 6) but does not aim at providing a detailed picture of the mechanism of circadian rhythmicity in this organism, where additional gene products are at work (–3). Similar results are obtained in a more extended model incorporating the formation of complexes between various clock proteins (–11). The model can apply also to circadian oscillations of frq mRNA and FRQ protein in Neurospora (8).

Figure 2

Circadian oscillations predicted by the negative-feedback model schematized in Fig. 1. (A) Oscillations obtained in the absence of noise. The curves are generated by numerical integration of the five kinetic equations governing the time evolution of the deterministic model (see refs. and and Appendix). Sustained oscillations of mRNA (M_P) and nuclear (P_N) and total clock (P_t) protein presented (Left) correspond to the evolution toward a limit cycle shown as a projection onto the (M_P, P_N) plane (Right). (B) Oscillations generated by the model in the presence of noise for Ω = 500 and n = 4. The results shown correspond to those obtained in A for the deterministic model. The data, expressed in numbers of molecules of mRNA and of nuclear and total clock protein, are obtained by stochastic simulations of the detailed reaction system corresponding to the deterministic model schematized in Fig. 1. Here the number of mRNA molecules oscillates between a few and 1,000, whereas nuclear and total clock protein oscillate in the ranges of 200–4,000 and 800–8,000, respectively. Robust circadian oscillations occur in these conditions despite the presence of molecular noise, with a mean period of 24.4 h and a standard deviation of 1.3 h. The decomposition of the deterministic model into elementary steps, the method of stochastic simulation, and parameter values are listed in Appendix.

Figure 3

Effect of number of molecules on the robustness of circadian oscillations. Shown in rows A–C are the oscillations in the numbers of molecules of mRNA and nuclear clock protein (Left), the projection of the corresponding limit cycle, and the histogram of periods of 1,200 successive cycles for Ω varying from 100 (A) to 50 (B) and 10 (C). The curves are obtained by stochastic simulations as described for Fig. 2B for n = 4. For period histograms, the period was determined as the time interval separating two successive upward crossings of the mean level of mRNA or clock protein. In A and B, the decrease in the numbers of mRNA and protein molecules still permits robust circadian oscillations [see histograms in which the mean value (μ) and standard deviation (σ) of the period are indicated in h], whereas at still lower numbers of molecules (C) noise begins to obliterate rhythmic behavior. For the oscillations in A, an average of one molecule of mRNA is produced every 2.5 min, whereas one molecule of clock protein is synthesized per mRNA molecule every 30 min; the average number of mRNA molecules is of the order of 60.

Figure 4

Robustness of circadian oscillations measured by half-life of autocorrelations. (A) Time evolution of the autocorrelation function, with indication of half-life time, for the oscillations obtained for Ω = 100 and n = 4 in Fig. 3A. (B) Phase of maximum in mRNA in the presence of molecular noise. The histogram is obtained under free-running conditions for 1,200 successive periods for the case considered in A. (C) Half-life of autocorrelations increases in a linear manner with the parameter Ω that provides a measure of the number of molecules present in the system. The oscillations corresponding to Ω = 500, 100, 50, and 10 are shown in Figs. 2B and 3A–C, respectively. (D) Influence of the degree of cooperativity of repression on the robustness of circadian oscillations. The half-life of autocorrelations is determined as a function of n for the case Ω = 100. In A–C, as well as in other figures, the degree of cooperativity n is equal to 4. In D, the standard deviation σ of the period distribution (shown for n = 4 in Fig. 3A) goes from 7.7 h (mean period μ = 26.9 h) for n = 1 to 2.6 h (μ = 21.2 h), 1.9 h (μ = 20.3 h), and 2.7 h (μ = 24.8 h) for n = 2, 3, and 4, respectively.

Figure 5

Effect of molecular noise on circadian oscillations under conditions of periodic forcing by an LD cycle. The data are obtained for Ω = 100 and n = 4 and should be compared with the results shown in Figs. 3A and 4 A and B in the case of continuous darkness. (A) Circadian oscillations in the numbers of mRNA and nuclear clock protein molecules. (B) Histogram of periods with mean value (μ) and standard deviation (σ) indicated in h. (C) Time course of the autocorrelation function. (D) Histogram of the time corresponding to the maximum number of mRNA molecules over a period. In A, periodic forcing is achieved by doubling during each light phase the value ascribed during the dark phase to the parameter (k_d3) measuring the probability of the protein degradation step (see Table 1, which is published as supporting information on the PNAS web site). Histograms and autocorrelations are determined for some 1,200 successive cycles.