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Joint trajectory and network inference via reference fitting

Stephen Y Zhang

TL;DR

This work proposes an approach for leveraging both dynamical and perturbational single cell data to jointly learn cellular trajectories and power network inference and can infer directed and signed networks from time-stamped single cell snapshots.

Abstract

Network inference, the task of reconstructing interactions in a complex system from experimental observables, is a central yet extremely challenging problem in systems biology. While much progress has been made in the last two decades, network inference remains an open problem. For systems observed at steady state, limited insights are available since temporal information is unavailable and thus causal information is lost. Two common avenues for gaining causal insights into system behaviour are to leverage temporal dynamics in the form of trajectories, and to apply interventions such as knock-out perturbations. We propose an approach for leveraging both dynamical and perturbational single cell data to jointly learn cellular trajectories and power network inference. Our approach is motivated by min-entropy estimation for stochastic dynamics and can infer directed and signed networks from time-stamped single cell snapshots.

Joint trajectory and network inference via reference fitting

TL;DR

This work proposes an approach for leveraging both dynamical and perturbational single cell data to jointly learn cellular trajectories and power network inference and can infer directed and signed networks from time-stamped single cell snapshots.

Abstract

Network inference, the task of reconstructing interactions in a complex system from experimental observables, is a central yet extremely challenging problem in systems biology. While much progress has been made in the last two decades, network inference remains an open problem. For systems observed at steady state, limited insights are available since temporal information is unavailable and thus causal information is lost. Two common avenues for gaining causal insights into system behaviour are to leverage temporal dynamics in the form of trajectories, and to apply interventions such as knock-out perturbations. We propose an approach for leveraging both dynamical and perturbational single cell data to jointly learn cellular trajectories and power network inference. Our approach is motivated by min-entropy estimation for stochastic dynamics and can infer directed and signed networks from time-stamped single cell snapshots.
Paper Structure (21 sections, 1 theorem, 22 equations, 6 figures)

This paper contains 21 sections, 1 theorem, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Dynamical inference
  4. Reference fitting
  5. Modelling perturbations
  6. Identifiability and necessity of regularisation
  7. Approximation of the transition kernel
  8. Interpretation as a ground cost learning problem
  9. Interpretation as an inverse optimal transport problem
  10. Results
  11. Reference fitting using transient dynamics
  12. Simulated data with knockouts
  13. CRISPR perturbation time-series
  14. Discussion
  15. Supplementary Material
  16. ...and 6 more sections

Key Result

Proposition 1

Consider the problem eq:rf_objective_appendix where $K_A^\sigma$ is taken to be the approximate reference kernel, i.e. If $\mathcal{R}$ is bounded below and $\mathcal{R}(A) \to \infty$ whenever $\| A \|_F \to \infty$, then eq:rf_objective_appendix has at least one minimiser.

Figures (6)

  • Figure 1: (a) Entropic optimal transport and reference fitting with an Ornstein-Uhlenbeck (OU) family. (b) Given a series of observed population snapshots and starting with a pure Brownian reference, iterative fitting of the reference process allows to progressively improve an estimate of the underlying dynamics. (c) Ground truth and recovered interactions for a 8-dimensional non-equilibrium OU process. (d) Temporal dynamics inferred by reference fitting and standard entropic OT, as well as the true vector field shown in the two leading principal components. In the right two panes, the family of marginals starting from a fixed point (green triangle) are shown.
  • Figure 2: (a) Trifurcating synthetic network, nodes coloured by centrality. Activating (inhibitory) edges are shown in red (blue). (b) Wild-type samples and knockouts, coloured by simulation time. (c) Inferred networks from reference fitting without (AUPR = 0.54) and with knockouts (AUPR = 0.84), compared to the ground truth. (d) Network inference accuracy (averages over 10 datasets) as measured by AUPRC using reference fitting with and without knockouts, compared to alternative methods.
  • Figure 3: (a) Haematopoietic stem cell (HSC) network from pratapa2020benchmarking. Activating (inhibitory) edges are shown in red (blue). (b) Inferred networks from reference fitting without (AUPR = 0.29) and with knockouts (AUPR = 0.73), compared to the ground truth. (c) Force-layout dimensionality reduction of wild-type trajectory coloured by time, showing four stable states. (d) Network inference accuracy (averaged over 10 datasets) as measured by AUPRC using reference fitting with and without knockouts, compared to alternative methods.
  • Figure 4: (a) Single cells from wild type and knockout conditions across timepoints, coloured by knockout condition. (b) Wild-type and Pou5f1 (Oct4) knockout cells coloured by timepoint, showing temporal dynamics of knockout propagation. (c) Inferred network on subset of 103 TFs, shown as a signed adjacency matrix. (d) Top 25 TFs in network inferred using reference fitting with knockouts, by out-edge eigenvector centrality. TFs for which knockout data were used shown in red. (e) Inferred network (thresholded top 2.5% of edges) coloured by out-edge eigenvector centrality. (f) Precision-recall curves for subnetwork, using ChIP-seq database as reference.
  • Figure 5: AUPRC scores for reference fitting in (a) trifurcating and (b) HSC systems, with different numbers of knockouts and different regularisation strengths $\lambda$
  • ...and 1 more figures

Theorems & Definitions (5)

  • Proposition 1: Existence of minimisers
  • proof
  • Remark 1: Alternating scheme
  • Remark 2
  • Remark 3: Convergence of alternating scheme