Model for circadian oscillations in mammals involving interlocked negative and positive regulations of Per, Cry, Bmal1, and Rev-Erbα genes by their protein products. We focus on the case where BMAL1 exerts a direct negative feedback on the expression of its gene. The role of the Rev-Erbα gene product in the indirect regulation of Bmal1 expression by BMAL1 (indicated in gray) is considered in a second stage (the gray loop then replaces the direct negative feedback exerted by BMAL1). The kinetic equations governing the time evolution of the model are listed in Supporting Text, together with the definition and values of the parameters, given in the legend to Fig. 5 and in Table 1, respectively, which are published as supporting information on the PNAS web site, www.pnas.org.

Circadian oscillations in DD (A and B) and entrainment by LD cycles (C and D). (A) The mRNA of Bmal1 oscillates in antiphase with respect to the mRNAs of Per and Cry.(B) Corresponding oscillations of the PER, CRY, and BMAL1 proteins. (C) Oscillations of the mRNAs after entrainment by 12:12 LD cycles. The peak in Per mRNA occurs in the middle of the light phase. (D) Oscillations are delayed by 9 h and the peak in Per mRNA occurs in the dark phase when the value of parameter K_AC is decreased from 0.6 to 0.4 nM. Other parameter values correspond to the basal set of values listed in Table 1. In C and D, the maximum value of the rate of Per expression, v_sP, varies in a square-wave manner such that it remains at a constant low value of 1.5 nM/h during the 12-h-long dark phase (black rectangle) and is raised up to the high value of 1.8 nM/h during the 12-h-long light phase (white rectangle). The curves have been obtained by numerical integration of Eqs. 1–16 (see Supporting Text) of the model without REV-ERBα.

Relating the model to syndromes associated with disorders of the circadian oscillatory system. (A) Effect of the maximum rate of PER phosphorylation on the free running period in DD and on the phase of the oscillations in LD. The phase corresponds to the time (in h) at which the maximum in Per mRNA occurs after the onset of the light phase. Situations 1 and 2 show that different values of the control parameter can produce different phases after entrainment, even though they correspond to the same free running period in DD. Situation 1 corresponds to the entrainment shown in . The double-arrow lines show how to obtain the free running period and the phase in LD for a given value of the control parameter. Situations a–c indicate that decreasing (increasing) the rate of phosphorylation, V_phos, with respect to the basal situation b can produce a phase advance (delay) as well as a decrease (increase) in the free running period. The transition from b to a would correspond to the phase shift observed in FASPS (see text). The gray areas on the left and right refer to absence of entrainment. The data are obtained by integration of Eqs. 1–16 (see Supporting Text) of the model without REV-ERBα, for the basal parameter values listed in Table 1, with V_phos = V_1P = V_1PC = V_3PC. (B) Quasi-periodic behavior and phase jump outside the range of entrainment. The phase of the circadian oscillations does not lock to a constant value with respect to the 24-h LD cycle, as occurs in the case of entrainment (see ). Instead, the phase advances every day by <1 h. During 25 successive days the peak in Per mRNA falls within the light phase of the LD cycle, until it reaches the end of the preceding dark phase. Then, in only 4 successive days (horizontal arrows), it crosses the dark phase and reaches the end of the preceding light phase. Gray and white bars represent the dark and light phases of the LD cycles, respectively. The curve in B has been obtained as in , for the same parameter values except K_IB = 1.64 nM instead of 2.2 nM.

The possibility of oscillations due to multiple oscillatory mechanisms. (A) The oscillations in disappear in the absence of PER protein synthesis (k_sP = 0). The curves show the asymptotic, stable steady state reached after transients have subsided. (B) Sustained oscillations are restored in these conditions when the degree of cooperativity, m, of repression of Bmal1 by CLOCK-BMAL1 increases from 2 to 4. The fact that oscillations can occur in the absence of PER protein indicates the existence of another oscillatory mechanism that relies only on CLOCK-BMAL1 autoregulation. The curves are obtained by numerical integration of Eqs. 1–16 of the model without REV-ERBα, for the basal parameter values listed in Table 1, except k_sP = 0, m = 4, k_sB = 0.5 h^-1; for this choice of parameter values, the period is 19.83 h. The period varies from some 12 h to 40 h as the rate constant, k_sB, measuring BMAL1 synthesis decreases from 1.4 h^-1 to 0.1 h^-1. (C) Oscillations occurring solely due to the PER–CRY negative autoregulation can occur in the absence of negative feedback of BMAL1 on Bmal1 expression. The curves, showing the time evolution of Per, Cry, and Bmal1 mRNAs, were obtained as in , for the same parameter values except K_IB = 100 nM (the effect of the negative feedback of BMAL1 on Bmal1 transcription is negligible at such a high value of the inhibition constant), v_mB = 0.96 nM/h. The period of the oscillations is 23.91 h.