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 .

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