Stochastic gradient descent on the epigenetic landscape: a unified framework for cellular plasticity, tumor heterogeneity, and the asymptotic irrelevance of fitness

Phenotypic plasticity, the ability of cells to switch between states, is central to development, differentiation, and therapy resistance. Although it is modeled at several scales, from compartmental ODEs to phenotype-structured PDEs and single-cell stochastic equations, a framework connecting these descriptions is missing. We present such a framework. Starting from a general n-compartment ODE model encompassing nonlinear growth and linear transitions between phenotypes, we show, with a new, elementary and generalizable proof of a recent theorem, that under uniform competition, the long-term population distribution is solely governed by transition rates. All phenotypes become selectively neutral at saturation, and the imprint of fitness differences during growth fades at an explicit rate. Restricting transitions to neighboring states transforms the model into a discretization of a phenotype-structured reaction-diffusion-advection PDE. In this continuum model, diffusion and advection are identified from switching rates, and fitness remains asymptotically irrelevant. Interpreting the advection velocity as the negative gradient of an effective epigenetic potential transforms the PDE into a Fokker--Planck equation and converts single-cell trajectories into stochastic gradient descent (SGD) in the Langevin sense on the phenotypic landscape. Non-local transitions, such as mutations, are incorporated via an integro-differential term, yielding a unified model with reaction, gradient flow, diffusion, and jumps. State-dependent noise reshapes the effective landscape without altering the underlying potential. This gives cancer a route to elevated plasticity that static, single-cell snapshots cannot distinguish from a changed landscape. This framework provides a physical interpretation of Waddington's landscape, where cells perform SGD, and cancer corresponds to a corrupted landscape.