# A hybrid Ornstein–Uhlenbeck–Branching framework unifies microbial and pediatric tumor evolution

## TL;DR

A new model combines two evolutionary theories to better understand how tumors and microbes evolve, showing how stability and randomness shape their development.

## Contribution

A hybrid OU–Branching framework is introduced to unify microbial and tumor evolution under stabilizing and stochastic forces.

## Key findings

- The model successfully captured mutation dynamics and lineage patterns in E. coli evolution experiments.
- priA lineages showed high plasticity, while recG lineages exhibited constrained diversification and increased turnover.
- Simulated therapies revealed oscillatory traits and population rebounds, suggesting potential for evolution-aware treatment strategies.

## Abstract

Pediatric tumors can relapse despite low mutation burdens, suggesting hybrid evolutionary dynamics shaped by stochastic variability and stabilizing forces. We develop a hybrid Ornstein–Uhlenbeck (OU)–Branching framework that couples mean-reverting stochastic trait dynamics with demographic birth–death processes to model lineage diversification under effective stabilizing constraints.

Using Escherichia coli long-term evolution experiment (LTEE) lineages (WT, priA, recG), we parameterized the equilibrium mean (μ), mean-reversion strength (θ), and diffusion scale (σ) on the log10 mutation-frequency axis via replicate-grouped likelihood inference. We performed forward simulations for predictive envelopes, uncertainty quantification, phase-plane dynamics, and OU–Branching lineage networks. We also ran illustrative in silico therapy simulations under fixed OU parameters with exposure-modulated birth/death rates.

The fitted model recapitulated lineage-specific mutation dynamics and branching architectures. priA exhibited elevated stochastic dispersion and drift-prone behavior consistent with a high-plasticity regime, whereas recG showed constrained diversification and increased lineage turnover consistent with a collapse-prone regime. Illustrative therapy simulations generated oscillatory trait trajectories, suppression–rebound population dynamics, clonal pruning, and extinction-versus-persistence regimes.

Although Y is directly observed as log10 mutation frequency in LTEE, in tumors Y can represent a longitudinally measurable phenotypic state (e.g., drug-tolerance scores from single-cell data, MRD/VAF-derived burden proxies, or pathway activity states). The balance between stabilizing strength (θ) and stochastic variability (σ) provides a quantitative axis governing plasticity and persistence, motivating future calibration to clinical longitudinal data for evolution-aware, patient-specific modeling.