Modeling single cell trajectory using forward-backward stochastic differential equations

Recent advances in single-cell sequencing technology have provided opportunities for mathematical modeling of dynamic developmental processes at the single-cell level, such as inferring developmental trajectories. Optimal transport has emerged as a promising theoretical framework for this task by computing pairings between cells from different time points. However, optimal transport methods have limitations in capturing nonlinear trajectories, as they are static and can only infer linear paths between endpoints. In contrast, stochastic differential equations (SDEs) offer a dynamic and flexible approach that can model non-linear trajectories, including the shape of the path. Nevertheless, existing SDE methods often rely on numerical approximations that can lead to inaccurate inferences, deviating from true trajectories. To address this challenge, we propose a novel approach combining forward-backward stochastic differential equations (FBSDE) with a refined approximation procedure. Our FBSDE model integrates the forward and backward movements of two SDEs in time, aiming to capture the underlying dynamics of single-cell developmental trajectories. Through comprehensive benchmarking on multiple scRNA-seq datasets, we demonstrate the superior performance of FBSDE compared to other methods, highlighting its efficacy in accurately inferring developmental trajectories.Author summary: In this study, we address the challenge of modeling the trajectories of single-cell gene expression over time. A “trajectory” in this context refers to the dynamic evolution of gene expression within a single cell, and this challenge is exacerbated by the fact that the gene expression of sequenced cells cannot be observed in subsequent time points. To overcome this limitation, we employ innovative stochastic differential equations (SDEs), a technique recently utilized in trajectory modeling studies, to more accurately capture the dynamic evolution of single-cell gene expression. Rigorously tested on multiple single-cell RNA sequencing datasets, our enhanced SDE system demonstrates promising performance in capturing these intricate gene expression dynamics, thereby advancing our capacity to glean critical insights across various biological contexts.