Fig 1.
Conceptual illustration of inference problem.
(a) Vector field (grey), sample trajectory (green) and observed cell state (red) drawn from ground truth process. (b) Sampled snapshot with labelled source (red) and sink (blue) states. (c) Inferred state transition probabilities. (d) Fate probabilities calculated from Markov chain.
Fig 2.
Illustration of the splitting scheme for decomposing Eq (2) into growth (Eq (4)) and transport (Eq (5)).
Composing the effects of growth and transport must maintain the steady-state profile ρeq. The coupling induced by transport is recovered by matching ρG(⋅, Δt) and ρT(⋅, Δt).
Fig 3.
(a) Illustration of the potential Ψ in the first two dimensions of the space . (b) Examples of simulated particle trajectories
following the drift-diffusion process. (c) Snapshot particles
shown in the first two dimensions of
, with the value of R indicated. Source and sink regions correspond to R > 0 and R < 0 respectively. (d) Evolution of the dynamics recovered by StationaryOT.
Fig 4.
Accuracy of inferred dynamics for potential-driven system.
(a) Colours representing estimated fate probabilities towards each of the wells {z0, z1, z2} are displayed on the snapshot coordinates. (b-d) Correlation with ground truth fate probabilities. (e) Comparison of estimated MFPT (in terms of Markov chain steps) to ground-truth MFPT (in continuous time units). (f) Comparison of recovered velocities to ground truth velocity.
Fig 5.
Effect of parameter choices on inference for potential-driven system.
(a) Correlation for varying regularisation parameter ε for entropic and quadratic regularisations. For entropic regularisation, the theoretically optimal value of ε is indicated in red. (b) Summarised correlations for systematic perturbation of flux rates towards each of the wells {z0, z1, z2}. Note that the simplex represents the perturbation applied to the true flux rates rather than the flux itself, so the centre (1/3, 1/3, 1/3) of each simplex corresponds to using the true flux rates.
Fig 6.
Inferred dynamics in the space of paths for potential-driven system.
(a) Collections of 100 sample paths from the ground truth process Eq (1) as well as StationaryOT outputs for both entropic and quadratic OT with optimal and sub-optimal ε. The vertical axis corresponds to a projection 〈x, u〉 of the 10-dimensional state space onto a convenient 1-dimensional subspace defined by u = (cos(π/12), sin(π/12), 0, …, 0). (b) W2 error on paths for StationaryOT reconstructions, shown for 5 repeated samplings of 250 paths.
Fig 7.
(a) Illustration of potential-driven (Ψ) and non-conservative (f) components of the overall drift v. (b) Examples of simulated particle trajectories following the drift-diffusion process. (c) Snapshot samples shown in the first two dimensions of
, with source (R > 0) and sink (R < 0) regions indicated. (d) Comparison of fate probabilities towards the sinks in the first quadrant. (e). Correlation of estimated fate probabilities to ground truth fates with (λ = 0.25) and without incorporation of velocity data (λ = 0). (f) Summary of fate probability correlation as a function of λ ∈ [0, 1].
Fig 8.
Inferred dynamics in the space of paths for non-conservative system.
Collections of 100 sample paths drawn from the ground truth process Eq (1), as well as StationaryOT output with and without velocity estimates for both entropic and quadratic OT. We indicate the initial condition π0 as dots.
Fig 9.
Effect of parameter choices on inference for non-conservative system.
(a) Correlation of estimated fate probabilities to ground truth as a function of noise η. (b) Sensitivity of entropic and quadratic regularisations to the choice of ε.
Fig 10.
The Arabidopsis root tip system.
(a) While individual cells divide (green), elongate (blue), and are displaced from the bottom 0.5 cm (red) as the root grows, cell populations remain in equilibrium. (b) The structure of the Arabidopsis thaliana root tip by developmental zone (left) and lineage (right) (Illustrations modified from the Plant Illustration repository [44]).
Fig 11.
Arabidopsis atlas cell annotations and StationaryOT output.
Developmental zone (left) and lineage annotations (centre) shown on a UMAP embedding. Putative fate probabilities from StationaryOT with entropic regularisation are visualised on the right, where each cell is coloured by putative fate and its saturation based on the magnitude of that probability. For over 80% of cells the putative fate matched the annotation, with the magnitude of the probability increasing later in development.
Fig 12.
Comparison of StationaryOT performance to other methods.
Proportion of cells where the maximum probability matches the annotation by developmental zone and lineage.
Fig 13.
Comparison of fate probabilities found by StationaryOT to other methods.
Fate probabilities for StationaryOT with entropic and quadratic regularisations compared to the annotation, as well as PBA and CellRank output. The colour indicates the maximum fate probability (putative fate) of each cell and the colour saturation shows the magnitude of the fate probability.
Fig 14.
Output of StationaryOT on full atlas dataset.
Atlas annotation on the full dataset (1.1 × 105 cells) shown in UMAP coordinates compared to fate probabilities computed on the full dataset respectively using the subsampling approach (using entropic regularisation for each subproblem) and memory-efficient GPU implementations of StationaryOT with entropic and quadratic regularisations.