The p53 pulse encoder counts by topological necessity: theory, evidence, and the molecular origin of the 5.5-hour clock: v1.0.0 - split-edition release

Abstract

After DNA damage, p53 does not signal how much damage there is by how much p53 there is: single cells fire a series of pulses whose amplitude is dose-invariant and whose number grows with dose. Why the encoder works this way has been open since the pulses were discovered. We show that the answer is topological. Pulse-generating circuits fall into two classes, pure negative feedback (NF) and excitable positive-feedback structures (EXC), and we prove that the classes are separable by three output signatures: onset continuity, hysteresis width, and the range sensitivity α = ∂lnA/∂lnD. NF onsets are Hopf-type, with A² linear in drive (verified at R² = 0.9926); EXC amplitudes are drive-independent exactly in the singular limit (6.5% variation over a 4.6-fold drive in the FitzHugh–Nagumo representative). The published p53 record decides the class: the same protein in the same cell line shows α = −0.001 ± 0.113 after double-strand breaks against α = +0.981 ± 0.114 after UV, 6.1 combined standard deviations apart, placing the DSB response deep in the EXC region. A Fisher-information census of the excitable encoder then shows that timing-type statistics carry nearly all parameter information while amplitude carries almost none (94.5% versus 5.5% at the representative calibration; the exact share is calibration-bounded, the ordering is not), that the noise amplitude is an exact null direction of all first-moment statistics (λ7/λ1 ≤ 4.7×10⁻¹⁹), and that dose reaches the nucleus near-losslessly as pulse count (I(D;N) = 3.07 of 3.17 bits). The 5.5 h period, 2.0 h interpulse interval and 1.25 h first-pulse delay decompose onto measured molecular rate constants, and an independent closed-form oscillator model lands on the same period (5.60 h) only under stressed-state parameters. A two-way dose-scan experiment with a frozen decision criterion is proposed as the decisive remaining test. Counting is not a design choice the cell happened to make; for this circuit class, it is the only channel left open.