Dynamic landscapes and statistical limits on growth during cell fate specification

The complexity of gene regulatory networks in multicellular organisms makes interpretable low-dimensional models highly desirable. An attractive geometric picture, attributed to Waddington, visualizes the differentiation of a cell into diverse functional types as gradient flow on a dynamic potential landscape. However, it is unclear under what constraints this metaphor is mathematically precise. Here, we consider the controlled growth of a single cell into a population with a target distribution over cell states. Expanding on a connection between stochastic control and optimal transport, we show that growth-maximizing regulatory strategies are described by time-dependent potential landscapes under certain generic growth-control trade-offs. Our analysis leads to a sharp bound on the time it takes for a population to grow to a target distribution with a certain size. We show how the framework can be used to compute regulatory strategies and growth curves in an illustrative model of growth and differentiation. The theory suggests a conceptual link between nonequilibrium thermodynamics, cellular decision-making during fate specification, and transport-based sampling methods from machine learning.