On the Mathematics of RNA Velocity I: Theoretical Analysis (CSIAM Transactions on Applied Mathematics, 2021)
The RNA velocity provides a new avenue to study the stemness and lineage of cells in the development in scRNA-seq data analysis. Some promising extensions of it are proposed and the community is experiencing a fast developing period.However, in this stage, it is of prime importance to revisit the whole process of RNAvelocity analysis from the mathematical point of view, which will help to understandthe rationale and drawbacks of different proposals. The current paper is devoted tothis purpose. We present a thorough mathematical study on the RNA velocity modelfrom dynamics to downstream data analysis. We derived the analytical solution ofthe RNA velocity model from both deterministic and stochastic point of view. Wepresented the parameter inference framework based on the maximum likelihood estimate. We also derived the continuum limit of different downstream analysis methods,which provides insights on the construction of transition probability matrix, root andending-cells identification, and the development routes finding. The overall analysisaims at providing a mathematical basis for more advanced design and developmentof RNA velocity type methods in the future.
Keywords: RNA velocity; stochastic model; continuum limit; kNN density estimate.