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Discrete Delay Model for the Mammalian Circadian Clock

A three-variable discrete delay model is proposed for the circadian rhythm of the mammals with BMAL1, PER-CRY complex and REV-ERBαprotein concentrations as the dynamical variables. The delay model is phenomenological in nature rather than the precise description of all the underlying complex processes. The goal of this paper is to study the effects of delay in the circadian rhythms of mammals that appears in both the positive and negative feedback loops of the model. The delay model exhibits 24-hour limit cycle oscillations, entrainment to light-dark (LD) cycles and phase response curves. The model is also found to exhibit quasiperiodic and chaotic oscillations under LD cycles when delay is varied. These are linked to non-24-hour sleep-wake syndrome and cancer incidence. The mutations in Bmal1–/– , PerBrdm1, Rev-Erbα–/–are explained in terms of delay, whereas the double mutations PerBrdm1/Cry2–/–and Cry1–/–/ Cry2–/–are explained in terms of the strength of delayed positive and negative regulations. The delay model in essence captures the core mechanism of the mammalian circadian rhythms with a smaller number of variables and parameters.