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

Jiguang Yu, Louis Shuo Wang, Shihan Ban

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.

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

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.
Paper Structure (56 sections, 14 theorems, 77 equations, 16 figures, 5 tables, 1 algorithm)

This paper contains 56 sections, 14 theorems, 77 equations, 16 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The challenge of scale separation: Physics vs. Games
  3. Contributions
  4. Most direct experimental settings and falsifiable readouts
  5. Damage-structured PDE model
  6. State variables and processes
  7. Feedback maps and modeling assumptions
  8. Well-posedness setting and weak/mild solutions
  9. Balance laws and exact reduction
  10. Balance laws for totals (general delta(x))
  11. Why the inflow boundary removes flux terms
  12. Exact two-dimensional closure under uniform death
  13. What breaks when delta(x) is non-constant
  14. Replicator mapping and game-theoretic interpretation
  15. Frequency dynamics
  16. ...and 41 more sections

Key Result

Theorem 3.1

Assume Assumption ass:basic. Then the PDE system eq:PDE_model--eq:IC admits a unique global nonnegative solution $(P,W)\in C([0,\infty);L^1([0,\infty))\times L^1([0,\infty)))$. Moreover, the total masses satisfy, for a.e. $t>0$, with initial conditions $\bar{P}(0)=\|P_0\|_{1}$ and $\bar{W}(0)=\|W_0\|_{1}$.

Figures (16)

  • Figure 1: From microscopic heterogeneity to macroscopic game theory.(a) Microscopic scale: Stem ($P$, blue) and TD ($W$, orange) cells are heterogeneous populations distributed over a damage coordinate $x$. (b) Exact reduction: The infinite-dimensional PDE dynamics (top surface) collapse exactly onto a 2D subsystem for totals (bottom phase plane) under uniform mortality. (c) Macroscopic law: Homeostasis emerges as a Nash equilibrium where per-capita growth rates (payoffs) equalize. Green arrows indicate the replicator flow.
  • Figure 2: Mechanisms of damage dynamics and regulation.(a) Conveyor-belt dynamics: Damage accumulates via transport (gray arrows). Division (blue arc) resets damage state ($\alpha x < x$), while dedifferentiation (red) couples the compartments. (b) Feedback logic: Hill-type regulation of self-renewal ($p_1$) and differentiation ($p_2$) probabilities stabilizes the tissue burden $\bar{W}$ at the homeostatic load $\bar{W}^*$.
  • Figure 3: Why uniform mortality enables exact dimensional reduction.(a) Uniform death ($\delta(x) \equiv \delta$): Mortality flux depends only on total mass (area), ensuring exact closure regardless of distribution shape. (b) Variable death ($\delta(x) \nearrow$): Mortality flux depends on the distribution's shape (e.g., age structure), breaking the ODE closure. This figure also serves as the mechanistic counterfactual: any observed deviation of total-mass trajectories from \ref{['eq:reduced_balance']} (beyond numerical error) indicates a non-constant effective mortality and hence failure of exact closure.
  • Figure 4: The geometry of homeostasis.(a) Intersection of laws: The unique Nash equilibrium is determined by the intersection of the Ratio Law (purple) and Equalization Law (teal) curves. (b) Replicator stability: The Nash point is a stable interior rest point of the induced replicator dynamics in the parameter regime shown. Green/red zones indicate regions where stem/TD phenotypes have a fitness advantage ($F_P > F_W$ or $F_W > F_P$), driving the frequency $s$ toward equilibrium.
  • Figure 5: Global qualitative structure and scaling symmetry.(a) Extinction-growth threshold (Theorem 5.1): A stability threshold at the origin separates the extinction regime (red zone, stable origin) from the growth regime (green zone, stable interior equilibrium). The critical threshold occurs exactly at $\Delta p_{\mathrm{crit}}$. (b) Gain-scaling symmetry (Proposition 5.6): Under uniform amplification of feedback gains ($A \to 2A$), the equilibrium point shifts along a ray of constant stem-to-TD ratio (dashed line), demonstrating that tissue composition is invariant to global sensitivity scaling while total mass scales as $1/A$.
  • ...and 11 more figures

Theorems & Definitions (30)

  • Theorem 3.1: Balance laws for totals
  • proof : Proof sketch (derivation of \ref{['eq:balance_laws_general']})
  • Corollary 3.2: Exact two-compartment closure for $\delta(x)\equiv\delta$
  • proof : Proof sketch
  • Remark 3.3: Uniform death as an identifiability/closure condition (not a biological claim)
  • Theorem 4.1: Interior equilibrium $\Longleftrightarrow$ payoff equalization
  • proof
  • Theorem 4.2: Homeostatic laws under uniform death
  • proof
  • Theorem 5.1: Structural extinction--growth threshold
  • ...and 20 more