Period as a function of global transcription rates in common circadian oscillator models. The parameter ρ is a multiplicative factor that controls all transcription rates, ρ=1 corresponds to the published nominal parameter values. Two simple low-dimensional models (
A,
B) and two more detailed models (
C,
D) are shown. (A) The
three species Gonze-Goldbeter delayed negative feedback model with a messenger (m), a cytoplasmic (C) and nuclear repressor (N) shows onset of oscillation at ρ=0.43 (Hopf bifurcation). Oscillations are kept when the transcription rate is raised by at least 20-fold. Period length initially rapidly increases with transcription, and then levels off at
T∼34 h. The variation of the period with increased transcription ρ is positive around ρ=1. The period lengthens with increasing ρ around ρ=1. (B) A
minimal two species relaxation (hysteresis-based) oscillator given by the equations for an activator (A) and a repressor (R):

with parameters
s=0.064 [
A]/h, d
A=0.32/h,
f=5.8 [
A]/h,
e=1.6 [
R]
−1/h,
k=0.64 [
A]
−1/h and d
R=0.15/h. Here, transcription and translation processes are taken together. The model has an infinite period bifurcation near ρ=0.27 and a Hopf bifurcation at ρ=2.2. The period shortens with increasing ρ around ρ=1. (C) The
16-dimensional mammalian model by Leloup and Goldbeter. With its standard parameters, this model has a very narrow range of oscillation around ρ=1 within ∼10% of variation of the transcription rates in either direction (Hopf bif urcations at ρ=0.94 and ρ=1.14). Furthermore, the model shows a cyclic fold in a narrow window around ρ=1.1 with coexistence of two stable (plain) and one unstable limit cycle (dashed line). The period lengthens with increasing ρ around ρ=1. (D)
The 74-dimensional mammalian Forger–Peskin model. Here, the model has both
Per1 and
Per2 genes, and it is thus also possible to simulate the
Per1 knockout phenotypes (dashed line). In contrast to the experimental data, the
Per1 mutant oscillators display slightly longer periods. Transcription can be reduced almost to zero while keeping oscillations. The period shortens with increasing ρ around ρ=1. Oscillation amplitudes for the same four models are shown in Supplementary Figure 4.