Volume 2, Issue 1
On the Mathematics of RNA Velocity I: Theoretical Analysis

Tiejun Li, Jifan Shi, Yichong Wu & Peijie Zhou

CSIAM Trans. Appl. Math., 2 (2021), pp. 1-55.

Published online: 2021-02

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  • Abstract

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 RNA velocity analysis from the mathematical point of view, which will help to understand the rationale and drawbacks of different proposals. The current paper is devoted to this purpose. We present a thorough mathematical study on the RNA velocity model from dynamics to downstream data analysis. We derived the analytical solution of the RNA velocity model from both deterministic and stochastic point of view. We presented 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 and ending-cells identification, and the development routes finding. The overall analysis aims at providing a mathematical basis for more advanced design and development of RNA velocity type methods in the future.

  • Keywords

RNA velocity, stochastic model, continuum limit, kNN density estimate.

  • AMS Subject Headings

60J28, 62P10, 92B15

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-1, author = {Tiejun and Li and and 14458 and and Tiejun Li and Jifan and Shi and and 14459 and and Jifan Shi and Yichong and Wu and and 14460 and and Yichong Wu and Peijie and Zhou and and 14461 and and Peijie Zhou}, title = {On the Mathematics of RNA Velocity I: Theoretical Analysis}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {1}, pages = {1--55}, abstract = {

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 RNA velocity analysis from the mathematical point of view, which will help to understand the rationale and drawbacks of different proposals. The current paper is devoted to this purpose. We present a thorough mathematical study on the RNA velocity model from dynamics to downstream data analysis. We derived the analytical solution of the RNA velocity model from both deterministic and stochastic point of view. We presented 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 and ending-cells identification, and the development routes finding. The overall analysis aims at providing a mathematical basis for more advanced design and development of RNA velocity type methods in the future.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2020-0001}, url = {http://global-sci.org/intro/article_detail/csiam-am/18653.html} }
TY - JOUR T1 - On the Mathematics of RNA Velocity I: Theoretical Analysis AU - Li , Tiejun AU - Shi , Jifan AU - Wu , Yichong AU - Zhou , Peijie JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 1 EP - 55 PY - 2021 DA - 2021/02 SN - 2 DO - http://doi.org/10.4208/csiam-am.SO-2020-0001 UR - https://global-sci.org/intro/article_detail/csiam-am/18653.html KW - RNA velocity, stochastic model, continuum limit, kNN density estimate. AB -

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 RNA velocity analysis from the mathematical point of view, which will help to understand the rationale and drawbacks of different proposals. The current paper is devoted to this purpose. We present a thorough mathematical study on the RNA velocity model from dynamics to downstream data analysis. We derived the analytical solution of the RNA velocity model from both deterministic and stochastic point of view. We presented 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 and ending-cells identification, and the development routes finding. The overall analysis aims at providing a mathematical basis for more advanced design and development of RNA velocity type methods in the future.

Tiejun Li, Jifan Shi, Yichong Wu & Peijie Zhou. (2021). On the Mathematics of RNA Velocity I: Theoretical Analysis. CSIAM Transactions on Applied Mathematics. 2 (1). 1-55. doi:10.4208/csiam-am.SO-2020-0001
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