From Microscopic Damage to Macroscopic Games: A Dimensionality Reduction of Stem Cell Homeostasis

TL;DR

This paper develops a mathematical framework that reduces complex tissue damage models to simple equations, revealing how tissues maintain balance and predict regenerative behavior, validated through a case study of intestinal crypts.

Contribution

It introduces an exact dimensionality reduction of a PDE model into a low-dimensional system, linking microscopic damage dynamics to macroscopic tissue homeostasis as a Nash equilibrium.

Findings

The model recovers experimental crypt regeneration behavior.

It predicts lineage plasticity rates from static cell counts.

The framework provides testable predictions for tissue regeneration.

Abstract

Tissues must maintain macroscopic homeostasis despite the continuous microscopic accumulation of cellular damage. Theoretical models of this process often suffer from a disconnect between microscopic biophysics and macroscopic phenomenological games. Here, we bridge this gap by deriving an exact dimensionality reduction of a physiologically structured partial differential equation (PDE) into a low-dimensional dynamical system. Under the condition of uniform mortality, we mathematically demonstrate that tissue homeostasis operates as an induced Nash equilibrium, where the per-capita net growth rates of stem and differentiated phenotypes perfectly equalize. This reduction yields closed-form algebraic rules, the Ratio and Equalization Laws, that map continuous microscopic state dynamics to measurable macroscopic observables. To demonstrate the biological utility of this framework, we present a concrete, falsifiable case study of the murine intestinal crypt. By modeling crypt regeneration following irradiation-induced stem cell depletion, our framework successfully recovers the experimentally observed reliance on progenitor dedifferentiation. Furthermore, the model generates explicit, testable predictions, enabling the in vivo estimation of hard-to-measure lineage plasticity rates directly from aggregate static cell counts. This work provides a rigorous, predictive mathematical foundation for understanding how fast-renewing tissues filter microscopic noise to sustain macroscopic regenerative capacity.