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Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis

Zhenyi Zhang, Yuhao Sun, Qiangwei Peng, Tiejun Li, Peijie Zhou

TL;DR

This work surveys how dynamical systems concepts—ranging from Markov chains to SDEs and PDEs, plus generative OT and Schrödinger-bridge frameworks—can be applied to scRNA-seq and spatial transcriptomics across time. It synthesizes methods for both snapshot and temporally resolved data, detailing pseudotime, RNA velocity, and OT-based trajectories, and extends these ideas to spatial contexts with FGW-OT and spatiotemporal OT. Key contributions include a unified view of discrete and continuous dynamics, discussion of vector-field and energy-landscape interpretations, and outlines of extensions to cell–cell interactions and developmental landscapes. The practical impact lies in guiding the construction of interpretable, multi-scale models of cellular development, differentiation, and tissue dynamics, with potential applications in developmental biology and disease modeling.

Abstract

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schrödinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective.

Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis

TL;DR

This work surveys how dynamical systems concepts—ranging from Markov chains to SDEs and PDEs, plus generative OT and Schrödinger-bridge frameworks—can be applied to scRNA-seq and spatial transcriptomics across time. It synthesizes methods for both snapshot and temporally resolved data, detailing pseudotime, RNA velocity, and OT-based trajectories, and extends these ideas to spatial contexts with FGW-OT and spatiotemporal OT. Key contributions include a unified view of discrete and continuous dynamics, discussion of vector-field and energy-landscape interpretations, and outlines of extensions to cell–cell interactions and developmental landscapes. The practical impact lies in guiding the construction of interpretable, multi-scale models of cellular development, differentiation, and tissue dynamics, with potential applications in developmental biology and disease modeling.

Abstract

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schrödinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective.

Paper Structure

This paper contains 42 sections, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview of the Data and Models
  3. Spatiotemporal scRNA-Seq Data
  4. Snapshot scRNA-Seq Data
  5. Temporally and Spatially Resolved scRNA-Seq
  6. Models for Cell-State Transitions
  7. Discrete Dynamics: Markov Chain Model
  8. Continuous Dynamics: From Trajectories to Population Dynamics
  9. Dynamic Modeling of Single-Cell Transcriptomics
  10. Snapshot Single-Cell RNA-Seq
  11. Pseudotime Methods
  12. Discrete Dynamics Modeling
  13. Continuous Dynamics Modeling
  14. Steady-State Assumption: Parameter Estimation in Velocyto la2018rna
  15. Dynamic Inference: Parameter Estimation in scVelo bergen2020generalizing
  16. ...and 27 more sections

Figures (4)

  • Figure S1: Overview of the data and models. (a) Discrete and Continuous Model: The discrete model constructs Markov chains between cells with dynamics encoded in a transition matrix, while the continuous model describes single-cell motion via stochastic differential equations (SDEs) and cell population dynamics through a corresponding partial differential equation (PDE). (b) Datasets: Snapshot data is an $n \times g$ matrix $\boldsymbol{X}$ ($n$: cell count, $g$: gene count); temporal data provides gene expression matrices $\boldsymbol{X}_i$ at time points $i \in \{0,\ldots,T-1\}$; spatial data additionally records coordinates for each cell in $\boldsymbol{X}_i$.
  • Figure S2: Dynamic modeling of snapshot single-cell transcriptomics.
  • Figure S3: Dynamic modeling of temporally-resolved single-cell transcriptomics.
  • Figure S4: Dynamic modeling of spatial transcriptomics.