Single cell profiling methods are powerful tools for dissecting the molecular states of cells, but the destructive nature of these methods has made it difficult to measure single cell expression over time. When cell dynamics are asynchronous, they can form a continuous manifold in gene expression space whose structure is thought to encode the trajectory of a typical cell. This insight has spurred a proliferation of methods for single cell trajectory discovery that have successfully ordered cell states and identified differentiation branch-points. However, all attempts to infer dynamics from static snapshots of cell state face a common limitation: for any measured distribution of cells in high dimensional state space, there are multiple dynamics that could give rise to it, and by extension, multiple possibilities for underlying mechanisms of gene regulation. Here, we enumerate from first principles the aspects of gene expression dynamics that cannot be inferred from a static snapshot alone, but nonetheless have a profound influence on temporal ordering and fate probabilities of cells. On the basis of these unknowns, we identify assumptions necessary to constrain a unique solution for the dynamics and translate these constraints into a practical algorithmic approach, called Population Balance Analysis (PBA). At its core, PBA invokes a new method based on spectral graph theory for solving a certain class of high dimensional differential equation. We show the strengths and limitations of PBA using simulations and validate its accuracy on single cell profiles of hematopoietic progenitor cells. Altogether, these results provide a rigorous basis for dynamic interpretation of a gene expression continuum, and the pitfalls facing any method of dynamic inference. In doing so they clarify experimental designs to minimize these shortfalls.
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