★★★ Advanced

🧬 Molecular Clock — PER/CRY/BMAL1 Oscillator

Simulate the transcription-translation feedback loop driving the mammalian circadian clock. PER/CRY proteins inhibit their own transcription via BMAL1:CLOCK, generating self-sustained ~24 h oscillations modelled by Goodwin ODEs.

Period: — h mRNA (M): — PER/CRY cytoplasm: — PER/CRY nucleus: — Phase: —

Hill coeff. n 12

mRNA degradation k_d 0.50

Protein degradation v_d 0.95

Transport k_1 (cyt→nuc) 0.50

Transport k_2 (nuc→cyt) 0.10

Speed 4×

How it works

The Goodwin oscillator models three coupled variables: mRNA (M), cytoplasmic PER/CRY protein (P_c), and nuclear PER/CRY (P_n). Nuclear P_n inhibits its own gene transcription via a Hill function with exponent n. High n (≥ 8) gives the sharp feedback needed for sustained oscillations. The period is set by how fast mRNA and protein are degraded — slower degradation → longer period. Tau mutation (CK1ε) speeds up protein degradation, shortening the clock to ~20 h. Below n = 8 the oscillation damps out — the clock becomes arrhythmic.

The Physics

Goodwin oscillator: dM/dt = v_s/(1+(P/K)^n) − k_d·M; dP_c/dt = k_s·M − v_d·P_c/(K_m+P_c) − k_1·P_c + k_2·P_n; dP_n/dt = k_1·P_c − k_2·P_n. Sustained oscillations require Hill coefficient n ≥ 8 (real clock uses ~12 sequential phosphorylation steps as effective n). Period determined by mRNA/protein degradation rates.