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VeloTree: Inferring single-cell trajectories from RNA velocity fields with varifold distances

Elodie Maignant, Tim Conrad, Christoph von Tycowicz

Abstract

Trajectory inference is a critical problem in single-cell transcriptomics, which aims to reconstruct the dynamic process underlying a population of cells from sequencing data. Of particular interest is the reconstruction of differentiation trees. One way of doing this is by estimating the path distance between nodes -- labeled by cells -- based on cell similarities observed in the sequencing data. Recent sequencing techniques make it possible to measure two types of data: gene expression levels, and RNA velocity, a vector that quantifies variation in gene expression. The sequencing data then consist in a discrete vector field in dimension the number of genes of interest. In this article, we present a novel method for inferring differentiation trees from RNA velocity fields using a distance-based approach. In particular, we introduce a cell dissimilarity measure defined as the squared varifold distance between the integral curves of the RNA velocity field, which we show is a robust estimate of the path distance on the target differentiation tree. Upstream of the dissimilarity measure calculation, we also implement comprehensive routines for the preprocessing and integration of the RNA velocity field. Finally, we illustrate the ability of our method to recover differentiation trees with high accuracy on several simulated and real datasets, and compare these results with the state of the art.

VeloTree: Inferring single-cell trajectories from RNA velocity fields with varifold distances

Abstract

Trajectory inference is a critical problem in single-cell transcriptomics, which aims to reconstruct the dynamic process underlying a population of cells from sequencing data. Of particular interest is the reconstruction of differentiation trees. One way of doing this is by estimating the path distance between nodes -- labeled by cells -- based on cell similarities observed in the sequencing data. Recent sequencing techniques make it possible to measure two types of data: gene expression levels, and RNA velocity, a vector that quantifies variation in gene expression. The sequencing data then consist in a discrete vector field in dimension the number of genes of interest. In this article, we present a novel method for inferring differentiation trees from RNA velocity fields using a distance-based approach. In particular, we introduce a cell dissimilarity measure defined as the squared varifold distance between the integral curves of the RNA velocity field, which we show is a robust estimate of the path distance on the target differentiation tree. Upstream of the dissimilarity measure calculation, we also implement comprehensive routines for the preprocessing and integration of the RNA velocity field. Finally, we illustrate the ability of our method to recover differentiation trees with high accuracy on several simulated and real datasets, and compare these results with the state of the art.

Paper Structure

This paper contains 25 sections, 3 theorems, 29 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Trajectory inference, a tree inference problem
  3. RNA velocity
  4. Our method
  5. Organization of the article
  6. Background
  7. Varifold distance between curves
  8. Distance-based tree inference
  9. Methods
  10. Algorithm
  11. Dimensionality reduction
  12. Preprocessing of the RNA velocity field
  13. Smoothing
  14. Tangent projection
  15. Integrating the RNA velocity field
  16. ...and 10 more sections

Key Result

Proposition 1

Let $i, j$ be two nodes of $T$. Let $i=i_1,\dots,i_N=j$ denote the shortest path from $i$ to $j$. Then we have the equivalence where $\sigma_x \asymp \sigma_t$ means here that there exist $k_1, k_2 > 0$ such that $k_1 \sigma_x \leq \sigma_t \leq k_2 \sigma_x$. $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: RNA velocity field of a bifurcating trajectory simulated with the dyngen library robrecht_dyngen_2021. The 2D visualization is generated via principal component analysis (PCA), superimposed with the streamplot function of matplotlib for a better rendering of the RNA velocity field. The colors correspond to cell states as labeled by dyngen.
  • Figure 2: Smoothed RNA velocity field of the bifurcating trajectory pictured in Fig. \ref{['fig:velocity']}.
  • Figure 3: RNA velocity field of the bifurcating trajectory after projection.
  • Figure 4: Two branching integral curves of the velocity field of the bifurcating trajectory.
  • Figure 5: A rooted tree embedded in the plane.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of lemma \ref{['lemma:1']}
  • proof : Proof of lemma \ref{['lemma:2']}