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Inferring stochastic dynamics with growth from cross-sectional data

Stephen Zhang, Suryanarayana Maddu, Xiaojie Qiu, Victor Chardès

TL;DR

This work tackles inferring stochastic dynamical systems with growth from cross-sectional population snapshots. It introduces unbalanced probability flow inference (UPFI), which combines a Lagrangian probability-flow representation of the Fokker–Planck equation with denoising score matching and an unbalanced Sinkhorn loss to jointly recover drift and growth from growth-affected distributions. The method demonstrates strong performance on high-dimensional bistable systems, simulated gene regulatory networks, and lineage-tracing single-cell RNA-seq data, outperforming several baselines and providing interpretable growth and regulatory insights. The work also analyzes non-identifiability between drift and growth in linear-quadratic settings and recommends engineering regularization and autonomy priors to achieve reliable inference in practice.

Abstract

Time-resolved single-cell omics data offers high-throughput, genome-wide measurements of cellular states, which are instrumental to reverse-engineer the processes underpinning cell fate. Such technologies are inherently destructive, allowing only cross-sectional measurements of the underlying stochastic dynamical system. Furthermore, cells may divide or die in addition to changing their molecular state. Collectively these present a major challenge to inferring realistic biophysical models. We present a novel approach, \emph{unbalanced} probability flow inference, that addresses this challenge for biological processes modelled as stochastic dynamics with growth. By leveraging a Lagrangian formulation of the Fokker-Planck equation, our method accurately disentangles drift from intrinsic noise and growth. We showcase the applicability of our approach through evaluation on a range of simulated and real single-cell RNA-seq datasets. Comparing to several existing methods, we find our method achieves higher accuracy while enjoying a simple two-step training scheme.

Inferring stochastic dynamics with growth from cross-sectional data

TL;DR

This work tackles inferring stochastic dynamical systems with growth from cross-sectional population snapshots. It introduces unbalanced probability flow inference (UPFI), which combines a Lagrangian probability-flow representation of the Fokker–Planck equation with denoising score matching and an unbalanced Sinkhorn loss to jointly recover drift and growth from growth-affected distributions. The method demonstrates strong performance on high-dimensional bistable systems, simulated gene regulatory networks, and lineage-tracing single-cell RNA-seq data, outperforming several baselines and providing interpretable growth and regulatory insights. The work also analyzes non-identifiability between drift and growth in linear-quadratic settings and recommends engineering regularization and autonomy priors to achieve reliable inference in practice.

Abstract

Time-resolved single-cell omics data offers high-throughput, genome-wide measurements of cellular states, which are instrumental to reverse-engineer the processes underpinning cell fate. Such technologies are inherently destructive, allowing only cross-sectional measurements of the underlying stochastic dynamical system. Furthermore, cells may divide or die in addition to changing their molecular state. Collectively these present a major challenge to inferring realistic biophysical models. We present a novel approach, \emph{unbalanced} probability flow inference, that addresses this challenge for biological processes modelled as stochastic dynamics with growth. By leveraging a Lagrangian formulation of the Fokker-Planck equation, our method accurately disentangles drift from intrinsic noise and growth. We showcase the applicability of our approach through evaluation on a range of simulated and real single-cell RNA-seq datasets. Comparing to several existing methods, we find our method achieves higher accuracy while enjoying a simple two-step training scheme.
Paper Structure (32 sections, 8 theorems, 90 equations, 5 figures, 6 tables, 1 algorithm)

This paper contains 32 sections, 8 theorems, 90 equations, 5 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stochastic population dynamics with growth
  3. Problem statement
  4. Methodology
  5. Unbalanced Probability Flow Inference (UPFI)
  6. The problem of identifying simultaneously drift and fitness
  7. Loss function and uniqueness of the minimum
  8. Results
  9. High-dimensional bifurcating system
  10. Simulated gene regulatory networks
  11. Lineage-tracing single-cell RNA-seq dataset
  12. Discussion
  13. General results for Ornstein-Uhlenbeck processes
  14. OU process with affine fitness
  15. OU process with quadratic fitness
  16. ...and 17 more sections

Key Result

Proposition 2.0

Consider an OU process with quadratic fitness, whose density satisfies where $\bm{A}_t \in \mathbb{R}^{d \times d}$, $\bm{e}_t, \bm{c}_t \in \mathbb{R}^{d}$, and $b_t \in \mathbb{R}$ are generic, $\bm{D}_t \in \mathbb{R}^{d \times d}$ is symmetric positive definite, and $\bm{\Gamma}_t \in \mathbb{R}^{d \times d}$ is symmetric negative semi-definite. If $\rho_0 = m_0 \m

Figures (5)

  • Figure 1: Overview. (i) Stochastic population dynamics with growth, governed by a Fokker-Planck equation with source. (ii) Score matching trains a neural network to model the contribution of noise. (iii) Neural-ODE based learning of the Fokker-Planck characteristics. (iv) Learned characteristics and corresponding SDE trajectories.
  • Figure 2: Non-identifiability of growth and drift in the Gaussian case. 8-dimensional OU process with quadratic growth. Left: autonomous drift and growth produces the same marginals as a non-autonomous system.
  • Figure 3: 10-dimensional bistable system. (a) Population snapshots over time, shown in the first coordinate $x_0$. (b) True and inferred force fields shown in $(x_0, x_1)$, coloured by fate probabilities. (c) True and inferred growth rates shown in $(x_0, x_1)$. (d) Sampled trajectories from true and learned dynamics with growth suppressed (pure drift-diffusion process). The fraction of trajectories terminating in the upper (resp. lower) regions are indicated. Without growth both branches are equiprobable.
  • Figure 4: Simulated regulatory networks. (a) (i) True and inferred vector fields for 7-dimensional bifurcating system, coloured by fate probabilities. (ii) Learned causal graphs using neural graphical model within UPFI (PFI) frameworks with true interactions shown in red. (iii) Precision-recall curve quantification of prediction accuracy. (b)(i-iii) Same as (a) for 11-dimensional HSC system.
  • Figure 5: Monocyte-neutrophil development. (a) Temporal snapshots of monocyte-neutrophil fate determination, shown using SPRING coordinates and celltype annotations from the original publication. (b) Learned vector fields: RNA velocity vector field learned from spliced-unspliced data, all others from temporal snapshots. Cosine distance (relative to RNA velocity field) for Mon, Neu cells shown. (c) Learned growth rates. (d) Fate probabilities empirically estimated from lineage tracing data and predicted from learned dynamics. Pearson correlation (relative to lineage tracing data) shown.

Theorems & Definitions (13)

  • Proposition 2.0: OU process with quadratic fitness
  • Corollary 2.0
  • Theorem 2.1: Loss function for OU processes with quadratic fitness
  • Proposition A.1
  • proof
  • Proposition A.1: OU process with quadratic fitness
  • proof
  • Corollary A.1
  • proof
  • Theorem B.1: Loss function for OU processes with quadratic fitness
  • ...and 3 more