Synergies of Multiple Zeitgebers Tune Entrainment

**Background:** Circadian rhythms are entrained by external cues called zeitgebers, most commonly light. In natural and clinical environments, multiple zeitgebers (light, food, drugs) act simultaneously on the clock. The mammalian core clock involves delayed negative feedback loops centered on Period (Per) and Rev-Erb genes. While most studies focus on single-zeitgeber entrainment, the synergistic effects of multiple coexisting zeitgebers are poorly understood. This study systematically investigates how two zeitgebers—light (acting on Per) and a Rev-Erb agonist drug—interact to entrain mathematical models of the mammalian circadian clock.

**Methods:** The authors used two intermediate-complexity ODE models of the mammalian circadian clock: (1) a modified Korenčič model (15 ODEs derived from a 5-gene DDE model, including Bmal1, Rev-Erbα, Per, Cry, Dbp) and (2) an Almeida model (8 ODEs). Both models incorporate transcriptional-translational feedback loops with E-boxes, R-boxes, and D-boxes. Zeitgebers were modeled as square waves modulating Per production (light) and Rev-Erb levels (drug). The free-running period of the modified Korenčič model was 24.95 h. Simulations ran for 80 zeitgeber cycles; periods, phases, and amplitudes were determined from the last 10 cycles using Lomb-Scargle periodograms. Key parameters varied included zeitgeber amplitude (r = 0.1–0.4), zeitgeber period (22–30 h), and phase difference between zeitgeber onsets (ΔΦ = 0–24 h).

**Key Results:** (1) Solitary light entrainment in the modified Korenčič model had an entrainment range of 1.7 h at r=0.2; solitary Rev-Erb agonist entrainment covered 3.6 h at equal amplitude. (2) The 12 h rule (phase of entrainment spanning ~180° across the entrainment range) held for both models and all zeitgeber inputs. (3) For coexisting zeitgebers with equal periods, the phase difference ΔΦ critically governed entrainment: ΔΦ=12 h produced synergistic effects with an entrainment range of 22.2–28.2 h, while ΔΦ=3 h produced a narrower range of 23.8–26.8 h. (4) At ΔΦ=12 h, entrained amplitudes were 1.5-fold higher than at ΔΦ=3 h. (5) Transient time to entrainment was shorter at synergistic phase differences: median 9 cycles at ΔΦ=15.5 h vs. 20 cycles at ΔΦ=4.0 h. (6) In the modified Korenčič model, the entrainment phase followed the drug cycle; in the Almeida model, phase control shifted between light and drug dominance depending on ΔΦ, with period doubling observed at some phase differences. (7) When zeitgeber periods differed (24 h light, 28.8 h drug), 5:6 frequency locking occurred with folded limit cycles and large day-to-day amplitude and phase modulations.

**Clinical Implications:** The study predicts that the phase relationship between Rev-Erb modulation and light input could be exploited to strengthen disrupted circadian rhythms in patients with low-amplitude rhythms (e.g., septic shock, cancer). Synergistic phase differences produce broader entrainment ranges, suggesting more robust clocks that can adapt to extreme chronotypes such as FASPS or extreme night owls. Faster transient recovery at optimal phase differences implies that jet lag or shift work disruption could be mitigated by appropriately timed drug-light schedules. The authors note that experimental investigation of circadian treatment schedules in complex clinical environments remains sparse, and mathematical modeling can help identify optimal timing and strength of zeitgebers.