Dedifferentiation stabilizes stem cell lineages: From CTMC to diffusion theory and thresholds

TL;DR

This work develops a multiscale framework linking a density-dependent five-channel CTMC for stem and TD compartments to a diffusion approximation that preserves mechanistic noise structure. Through a LLN/Diffusion limit, it derives a chemical Langevin equation whose state-dependent covariance exactly matches the CTMC’s aggregated covariances, enabling rigorous analysis of demographic fluctuations. A no-dedifferentiation pathology reveals a diffusion-level dichotomy: in subcritical regimes, the stem population asymptotically extincts, while in supercritical regimes polynomial moments diverge, highlighting the rescue role of dedifferentiation. A deterministic PDE backbone yields equalization and ratio laws that interpret steady states and separate theorem-regime global convergence from threshold-like regime B bistability, which is corroborated by numerical experiments showing Ω^{-1/2} scaling and regime-specific phase portraits with biological implications for tissue homeostasis and therapy resistance.

Abstract

We study stem-terminally differentiated (TD) lineages in small niches where demographic noise from discrete division and death events is non-negligible. Starting from a mechanistic five-channel, density-dependent CTMC (symmetric self-renewal, symmetric differentiation, asymmetric division, dedifferentiation, TD death), we derive its mean-field limit and a functional CLT, obtaining a chemical Langevin diffusion whose explicit state-dependent covariance exactly matches the CTMC's aggregated channel-wise infinitesimal covariances. Within this diffusion approximation we remove the dedifferentiation flux and obtain a sharp dichotomy: in subcritical regimes the stem coordinate becomes extinct asymptotically almost surely, whereas in supercritical regimes polynomial moments diverge exponentially. This identifies, at the diffusion level, a structural failure mode of strictly hierarchical lineages under demographic fluctuations and clarifies how a cyclic return flux can rescue homeostasis. For interpretation we also derive an exact totals ODE backbone from a damage-structured transport model and obtain two steady-state constraints (ratio and equalization laws) linking compartment ratios to turnover and balancing dedifferentiation against fate bias. Numerical experiments corroborate the $Ω^{-1/2}$ fluctuation scaling, illustrate the pathology, and contrast theorem-regime global convergence with threshold (Allee-type) behaviour outside the theorem hypotheses.

Figures (6)

• Figure 1: Multiscale roadmap. Event-level lineage dynamics induce deterministic and stochastic limits whose structure explains both homeostatic constraints and diffusion-level pathologies.
• Figure 2: Diffusion approximation and system-size scaling. The black curve shows the deterministic mean-field trajectory. Colored curves show sample paths of the chemical Langevin diffusion \ref{['eq:cle_main']} for $\Omega=50$ (higher noise) and $\Omega=1000$ (lower noise). The reduction in fluctuation amplitude with increasing $\Omega$ is consistent with the $\mathcal{O}(\Omega^{-1/2})$ scaling predicted by Theorem \ref{['thm:fclt']}.
• Figure 3: Validation of diffusion approximation in a higher-density regime. The black curve shows the deterministic mean-field trajectory, while colored curves show chemical Langevin sample paths for $\Omega=50$ and $\Omega=1000$. The fluctuation magnitudes of trajectories decrease with $\Omega$, consistent with $\mathcal{O}(\Omega^{-1/2})$ fluctuations.
• Figure 4: No-dedifferentiation pathology in the diffusion approximation ($\lambda_R\equiv 0$). (a): subcritical regime (net stem drift uniformly negative) exhibiting asymptotic extinction of the stem coordinate. (b): supercritical regime (net stem drift uniformly positive) exhibiting rapid growth consistent with exponential moment divergence. These simulations illustrate Proposition \ref{['prop:no_dedifferentiation_main']} at the diffusion-approximation level and do not assert finite-time boundary hitting.
• Figure 5: Regime B phase portrait: saddle, separatrix, and Allee-type threshold (outside the theorem regime). Streamlines show the vector field of the totals ODE \ref{['eq:totals_ode_backbone']} under the Regime B parameter set \ref{['eq:regimeB_lambdaR']}. Solid and dashed curves denote the nullclines $\dot P=0$ and $\dot W=0$, respectively. The system admits two positive equilibria: a hyperbolic saddle at $(P_{\mathrm{sd}},W_{\mathrm{sd}})\approx(72.76,37.73)$ and a locally stable equilibrium at $(P_{\mathrm{st}},W_{\mathrm{st}})\approx(82.48,39.06)$. The stable manifold of the saddle (thick curve) acts as a separatrix dividing trajectories that recover to the positive equilibrium from those that collapse to extinction, producing an Allee-type threshold.

Theorems & Definitions (53)

• Lemma 3.1: Uniform LLN for Poisson processes
• Theorem 3.2: Mean-field limit
• proof : Proof sketch
• Theorem 3.3: FCLT / diffusion approximation
• Proposition 4.1: Existence and uniqueness up to exit
• proof : Proof sketch
• Proposition 4.2: Lyapunov non-explosion criterion
• proof : Proof sketch
• Remark 4.4
• Theorem 4.5: Polynomial moment bounds up to exit