A Phase-Gated AR(2) Framework for Colonic Crypt Renewal: A Temporal Analogue to Fibonacci Tissue Codes with Tuft Cell Readouts

This document introduces a Phase-Gated Autoregressive model of order 2 (PAR(2)) as a temporal framework for understanding colonic crypt renewal, complementing Bruce M. Boman's spatial Fibonacci tissue code. The PAR(2) model uses circadian rhythms, NOTCH signaling, and WNT/AXIN2 feedback as molecular mechanisms to describe renewal dynamics and cell-type proportions in colonic crypts. The framework explains how crypt-level variables exhibit Fibonacci-like patterns in time and predicts changes in crypt stability and tuft cell fractions under genetic perturbations (e.g., BMAL1 or AXIN2 knockout) and chemotherapy-induced injury. The study bridges spatial and temporal tissue renewal models, offering a unified approach to investigate colonic crypt dynamics.