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

• The framework bridges models of microscopic cellular wear and macroscopic population games, which are rarely treated together.Source: Author summary · p. 2
• Homeostasis is represented as a Nash equilibrium in which stem and mature cells have no competitive growth advantage.Source: Author summary · p. 2
• Application to mouse intestine regeneration predicts how tissues heal after severe injury.Source: Author summary · p. 2
• Aggregate cell-population snapshots can be used to calculate otherwise hidden tissue healing rates.Source: Author summary · p. 2

• Rapidly renewing tissues must maintain macroscopic stability while cells accumulate damage and experience stress.Source: §1 · pp. 2–3
• Structured PDEs capture continuous damage and division but are infinite-dimensional and difficult to convert into simple tissue-level laws.Source: §1.1 · p. 3
• Game-theoretic payoff functions are often phenomenological, leaving unclear whether tissue stability is biological or merely mathematical.Source: §1.1 · p. 3

Contributions

• Under uniform TD mortality, the PDE total-mass balances close exactly into a planar ODE, making the induced game an emergent property of lineage dynamics.Source: §1.2 · p. 3
• Homeostasis is mathematically equivalent to a Nash equilibrium when stem and TD phenotypes equalize their per-capita net growth rates.Source: §1.2 · p. 4
• The intestinal-crypt case study models dedifferentiation-driven regeneration and estimates dedifferentiation flux from static aggregate flow-cytometry observables.Source: §1.2 · p. 4

Falsifiable readouts

• The framework is falsifiable because measured totals must satisfy closed balance laws, while perturbations can test the predicted extinction–growth threshold.Source: §1.3 · p. 5
• At steady state, the Ratio and Equalization Laws connect stem-to-TD abundance, mortality, feedback, and dedifferentiation rates to aggregate measurements.Source: §1.3 · p. 5

• Stem and TD densities depend on time and a scalar damage coordinate, with total masses obtained by integrating over damage.Source: §2.1 · p. 5
• Damage increases through transport, while division produces stem, TD, or mixed daughters with deterministic inherited damage fractions.Source: §2.1 · pp. 5–6
• TD cells dedifferentiate into the stem compartment, and their mortality may depend on damage through δ(x).Source: §2.1 · p. 6

Feedback regulation

• Hill-type feedback maps regulate self-renewal and differentiation probabilities as tissue burden changes.Source: §2.2 · p. 7

Mortality assumptions

• The model distinguishes general bounded mortality, used for well-posedness and balance laws, from uniform mortality, used for exact closure.Source: §2.2 · p. 7
• Uniform mortality is a structural identifiability condition rather than a claim that real tissues have damage-independent death.Source: §2.2 · p. 7

Well-posedness

• Characteristic trajectories and integrating factors support a mild formulation, while Lipschitz feedback, positivity, and mass bounds yield a unique global solution.Source: §2.3 · p. 8

Balance laws

• Theorem 3.1 establishes unique global nonnegative PDE solutions and balance laws for the total masses.The balance laws hold for almost every t > 0 under the stated assumptions.Source: §3.1 · p. 9

Failure of closure

• The balance laws are not closed in general because mortality depends on the full differentiated-cell distribution W(t, ·), not only its total mass.Additional weighted moments are then required, and finite-dimensional closures are typically approximate.Source: §3.2 · p. 10, §3.3 · p. 10

Exact reduction

• Uniform mortality δ(x) ≡δ converts the mortality flux into δ ¯W(t), yielding an exact planar ODE for total stem and differentiated-cell masses.The nonnegative quadrant is forward invariant and solutions are global.Source: §3.2 · p. 10

Interpretation and testing

• Uniform mortality is a structural identifiability condition for exact closure, not a literal claim that differentiated-cell death is damage-independent in vivo.Exact closure requires the mortality flux to be expressible as a function of ¯W(t) alone.Source: §3.3 · pp. 10–11
• Aggregate trajectories that violate the closed balance laws beyond discretization error empirically falsify the uniform-mortality closure for that regime.The figure contrasts total-mass-dependent mortality with distribution-shape-dependent mortality.Source: §3.3 · p. 11, Figure 3 · p. 11

Replicator mapping

• The exact two-compartment closure induces replicator dynamics with stem and differentiated phenotypes as strategies and per-capita net growth rates as state-dependent payoffs.The game-theoretic structure is derived from PDE lineage bookkeeping rather than postulated.Source: §4 · p. 12, §4.1 · p. 12, §4.2 · p. 13
• The sign of FP −FW determines selection on stemness: FP > FW increases stem frequency s, whereas otherwise s decreases.This follows from the replicator equation ˙s = s(1 −s)(FP −FW ).Source: §4.1 · p. 12, §4.2 · p. 13

Nash/homeostasis equivalence

• At homeostasis, payoff equalization gives stem and differentiated phenotypes identical per-capita net growth, so neither has an aggregated selective advantage.This is the model’s induced Nash-equilibrium interpretation.Source: §4.2 · pp. 12–13, §4.3 · p. 13
• An interior equilibrium occurs if and only if FP = FW = 0, while FP = FW alone identifies a replicator rest point for the stem frequency.The additional zero common growth-rate condition distinguishes a fixed point from frequency stationarity.Source: §4.2 · p. 13

Closed-form laws and observables

• The Ratio and Equalization Laws express equilibrium through algebraic identities among measurable compartment sizes and effective division, death, and dedifferentiation rates.Steady-state sizes can be measured by cell counting or flow cytometry, while rates can be estimated using proliferation, death, and lineage-tracing assays.Source: §4.3 · pp. 13–15
• Once δ and λP(¯W∗) are estimated, the steady-state ratio and dedifferentiation rate required to offset differentiation losses become directly testable predictions.The framework therefore connects aggregate measurements to effective lineage plasticity rates.Source: §4.3 · pp. 13, 15

Dimensionality reduction

• Uniform TD death closes the PDE balance laws exactly to a planar ODE system on total stem and differentiated-cell populations.The closed quadrant is forward invariant and solutions are global.Source: §5 · p. 15

Extinction–growth threshold

• The threshold ∆pcrit = −ˆλR/δ separates locally stable extinction from a growth regime in which the origin is a saddle with a one-dimensional unstable manifold.At equality, the determinant vanishes and one eigenvalue is zero.Source: §5.1 · p. 16

Uniqueness

• The Ratio law defines a ratio curve, and substituting it into the Equalization law produces a scalar root condition for interior equilibria.Under the stated slope condition, the residual is strictly decreasing and has at most one root.Source: §5.2 · pp. 16–17

Global stability

• Negative divergence excludes nontrivial periodic orbits, and every bounded interior trajectory converges to the unique interior equilibrium when it exists.The global conclusion is stronger than a local eigenvalue calculation.Source: §5.3 · p. 18, Global structure (not merely local… · p. 18

Scaling invariance

• Uniform amplification of Hill feedback gains preserves the equilibrium stem-to-TD ratio while rescaling total abundances by 1/A.The scaling symmetry separates tissue composition from the level of homeostasis.Source: §5.4 · pp. 18–19

Verification framework

• The verification framework separates numerical equation-solving accuracy from certified checks of theorem-predicted qualitative behavior.Deterministic, parameter-locked scripts emit the reported quantities and map analytical results to computational tests.Source: §6 · p. 20

Numerical schemes

• Conservative finite-volume remapping preserves the mass identity required for balance-law residuals and PDE-versus-ODE closure checks.The method uses exact overlap fractions rather than pointwise interpolation, while negative-value corrections are monitored through induced mass defects.Source: §6.1 · pp. 20–22

Verification tests

• At T = 5, balance-law residuals are O(10^-3), decrease under grid refinement, and remain independent of xmax ∈{10, 15, 20}.The residual magnitudes confirm the balance laws to the accuracy of the spatial discretization.Source: §6.4 · p. 24

Closure verification

• With xmax = 20, PDE-versus-ODE total-mass discrepancies decrease at the expected first-order rate under grid refinement, with zero positivity-fix mass defect.The remaining discrepancy is attributed to the O(∆x) upwind scheme rather than closure failure.Source: §6.6 · p. 26

Analytical predictions

• Eight trajectories converge monotonically to the unique interior equilibrium, while the replicator field has a stable Nash frequency and the divergence is strictly negative throughout the sampled domain.The gain-scaling test additionally confirms equilibrium mass rescaling by 1/A and invariant composition, with ratio errors ≤3 × 10^-15.Source: §6.7 · p. 27, §6.8 · pp. 27–28, §6.10 · p. 29

Parameter scans

• The extinction-growth scan resolves the threshold ∆pcrit = −0.6, and increasing δ shifts the equilibrium toward smaller total mass.The threshold separates the extinction regime from the positive equilibrium branch in the admissible parameter window.Source: §6.9 · pp. 28–29

Calibration

• The calibration targets 14 stem cells and 278 differentiated cells, corresponding to a total crypt cellularity of approximately 292 cells.The 14-cell stem target is consistent with approximately 14 ± 2 Lgr5hi stem cells reported for murine small-intestinal crypts.Source: §7.1 · p. 31, §7.3 · pp. 32–33

Model setup

• The crypt is modeled by identifying P with Lgr5+ CBC stem cells and W with all remaining crypt-resident differentiated cells, including transit-amplifying progenitors.This is an aggregate coarse-graining rather than a one-to-one transcriptional cell-state model.Source: §7.2 · p. 32

Calibration

• 4.9643 day^-1 is the calibrated aggregate effective production rate from the Ratio law, not the literal single-cell Lgr5+ CBC cycling rate.The reduction incorporates the transit-amplifying cascade into the effective differentiated compartment.Source: §7.3 · p. 32

Homeostatic equilibrium

• The calibrated equilibrium is (P̄*, W̄*) = (14.0000, 278.0000), with equalization and ratio residuals below 10^-17.The reduced system reproduces the prescribed aggregate homeostatic state to floating-point precision.Source: §7.3 · p. 33, §7.4 · p. 33
• The equilibrium stem fraction is 4.79%, where stem and differentiated phenotypes have equal per-capita net growth rates.Thus, feedback-regulated lineage dynamics maintain the stem phenotype without intrinsic competitive dominance.Source: §7.4 · p. 33

Regeneration

• After irradiation-like stem depletion, recovery is initially driven by differentiated-to-stem return flux, matching experimentally documented progenitor dedifferentiation.The model predicts a transient positive stem-versus-differentiated payoff difference that restores the homeostatic composition.Source: §7.5 · pp. 33–35

Predictions and limitations

• The case study generates experimentally testable predictions linking epithelial loss, stem-to-differentiated ratios, and lineage plasticity rates.The framework is presented as a proof-of-principle rather than a formal statistical fit to one harmonized cohort.Source: §7.6 · p. 35, §7.7 · p. 37
• The deliberate lumping of transit-amplifying cells prevents direct comparison between the effective production rate and the CBC single-cell cycling rate.A three-compartment stem/TA/TD model would map more literally to crypt biology but would abandon the exact two-dimensional closure.Source: §7.7 · pp. 36–37

Measurable laws

• The Ratio law links the steady-state stem-to-differentiated ratio to differentiated-cell clearance and the effective stem-cycle rate.The clearance rate can be measured with labeled-cohort survival or apoptosis assays, while the effective cycle rate can be estimated from EdU/BrdU incorporation.Source: Experimental interpretation of the… · p. 37
• The Equalization law provides a balance statement for the differentiated-to-stem return flux required to maintain steady tissue composition.Lineage tracing or fate mapping can estimate the return rate under steady state or controlled micro-injury.Source: Experimental interpretation of the… · p. 37

Experimental implications

• Increasing effective differentiated-to-stem return should shift the steady ratio along the algebraic laws.This provides a perturbation logic for testing plasticity experimentally.Source: Experimental interpretation of the… · p. 37

Scope of the reduction

• The exact evolutionary-game derivation relies on damage-independent mortality, δ(x) ≡ δ.This assumption corresponds biologically to removal driven primarily by extrinsic stochastic factors such as mechanical shedding or random differentiation-linked loss.Source: On the assumption of uniform mortality · pp. 37–38
• When mortality depends on damage, exact moment closure fails, although homeostatic stability may not necessarily be destroyed.The reduced system is therefore a transparent theoretical benchmark, while future moment-closure approximations could extend the framework.Source: On the assumption of uniform mortality · p. 38

Regularity assumptions

• The feedback maps are bounded and Lipschitz on [0, ∞) under Assumption 2.1.This regularity condition supports the stated analytical framework.Source: §A.1 · p. 38

Lipschitz control

• The complementary feedback map p_3 is Lipschitz with constant L_p3 := L_p1 + L_p2.This follows from p_3(·) = 1 − p_1(·) − p_2(·).Source: §A.1 · pp. 38–39

Global bounds

• The model bounds p_i between 0 and p̂_i, p_3 between 0 and 1, and the rates λ_P and λ_R by their hatted maxima.These are the global bounds labeled (A1).Source: §A.1 · p. 38

Mass preservation

• The reduction relies on nonlocal birth operators preserving L1-mass in each deterministic inheritance branch.This is the key structural ingredient for deriving total-mass balance laws.Source: §A.2 · p. 39

Change of variables

• Lemma A.1 supplies the change-of-variables identity for α ∈ (0, 1) and integrable f.The substitution y = x/α with dx = α dy underlies the kernel calculations.Source: §A.2 · p. 39

Mass formulas

• Lemma A.2 obtains the birth-operator mass formulas by applying Lemma A.1 term by term and using p_1 + p_2 + p_3 = 1.The partition-of-probability identity closes the aggregate mass accounting.Source: §A.2 · p. 39

Boundary flux cancellation

• Under homogeneous inflow u(t, 0) = 0, integrating the transport equation over x ≥0 produces no boundary injection at x = 0.The result applies to weak solutions with v > 0 and integrable source term g.Source: §A.3 · pp. 39–40

Conclusion

• The resulting identity holds for almost every t in (0, T) in the distributional sense used to derive balance laws.Source: §A.3 · p. 39

Proof strategy

• A cutoff test function ψR equal to one on [0, R] and supported in [0, R + 1] localizes the weak formulation before taking R →∞.The cutoff derivative contribution vanishes by dominated convergence because u(t, ·) ∈L1.Source: §A.3 · p. 40