The linear potential energy Φ quantifies cell differentiation potency.

(a) Boxplot of the linear potential energies of cells sampled different time stages of the embryonic murine cerebral cortex development. (b) Trend in the addictive inverse of linear potential (circular points connected by black lines with y axis on left-hand side) and temporal score (triangle points connected by red lines with y axis on right-hand side) across cell types. (c) The linear potential energy Φ estimated by GraphFP. (d) The stationary probability distribution pss of the cell types.</p

GraphFP is robust to uncertainty presented in cell type labels.

GraphFP was applied to the murine cerebral cortex dataset based on the labelling of 4 cell types with a coarse resolution (a-c) and the labelling of 7 cell types with a fine resolution (d-f), separately. The estimated Φ (a), W (b) and charted probability flow (c) by GraphFP based on the labelling of 4 clusters (“A-Neurons”, “B-Young Neuron”, “C-APs/RPs”, “D-IPs”). Aggregated results of the estimated Φ (d), W (e) and charted probability flow (f) by GraphFP based on the labelling of 7 clusters, averaging the results from i) “3-APs/RPs” and “5-APs/RPs”, ii) “2-Young Neurons” and “6-Young Neurons” and iii) “4-IPs” and “7-IPs”, separately, resulting in the same dimensions as those based on the labelling of 4 cell types.</p

Details for the parameter estimation of GraphFP.

This document provides detailed description of the parameter estimation and pseudocode for the GrapFP algorithm. (PDF)</p

Runtimes of GraphFP on the mouse spinal cord injury dataset.

(DOCX)</p

Evaluation of GraphFP’s performance on quantifying the stochastic dynamics of cell-type frequencies with cell-cell interaction term (<i>W</i> ≠ 0) and without cell-cell interaction term (<i>W</i> = 0) on the murine cerebral cortex dataset.

Evaluation of GraphFP’s performance on quantifying the stochastic dynamics of cell-type frequencies with cell-cell interaction term (W ≠ 0) and without cell-cell interaction term (W = 0) on the murine cerebral cortex dataset.</p

Overview of GraphFP algorithm.

GraphFP takes the input of time series single-cell transcriptomic data incorporated with experimental temporal information. It identifies cell states/types, estimates the cell type frequencies at each time point, estimates the linear potential energy Φ and the cell-cell interaction matrix W based on the adjoint method. GraphFP outputs the stochastic dynamics of cell type frequencies p(t) on probability simplex in continuous time, the cell state transition potential energy, and the probability flows of cell state-transitions underlying the evolving cell population.</p

Application of GraphFP to the mouse spinal cord injury dataset.

Fig A. GraphFP reconstructs the cell state-transition energy landscape on the mouse spinal cord injury scRNA-seq dataset. Fig B. The linear potential energy Φ quantifies cell differentiation potency. Table A. Evaluation of GraphFP’s performance on quantifying the stochastic dynamics of cell type frequencies with cell-cell interaction term (W ≠ 0) and without cell-cell interaction term (W = 0) on the mouse spinal cord injury dataset. (PDF)</p

GraphFP accurately reconstructs the cell state-transition energy landscape of the murine cerebral cortex dataset.

(a) The gold standard trajectory of embryonic murine cerebral cortex development. (b) The t-SNE plot of cells from the murine cerebral cortex dataset, colored by their cell-type labels. (c) GraphFP estimated the linear potential energy Φ. (d) GraphFP estimated the cell-cell interaction matrix W. (e) Static linear potential energy landscape of cells on the t-SNE plot: cells are color-coded according to the linear potential energies Φs of their corresponding cell types. (f) The free energy (Eq (1)) of the system decreased over time. (g) The reconstructed potential energy landscape Ψ(t) of cell types (colored curves) over time. (h) The potential energies of the cell state pairs with the top 3 highest positive values of cell-cell interaction strengths wijs: “2-Young Neurons ← 1-Neurons” (left panel), “6-Young Neurons ← 3-APs/RPs” (middle panel), and “4-IPs ← 1-Neurons” (right panel). (i) The potential energies of the cell state pairs with the top 3 lowest negative values of cell-cell interaction strengths wijs: “6-Young Neurons ← 1-Neurons” (left panel), “4-IPs ← 3-APs/RPs” (middle panel), and “2-Young Neurons ← 4-IPs” (right panel).</p

GraphFP accurately quantifies the stochastic dynamics of the cell type frequencies by modelling cell-cell interactions.

GraphFP calculated the stochastic dynamics of the cell type frequencies p(t) with cell-cell interaction term (W ≠ 0; solid lines) and without cell-cell interaction term (W = 0; dashed lines). Triangle points are the estimated cell type frequencies at each time point where red represents the input data point to GraphFP, while blue represents the held-out data point to GraphFP. (a) Using all 4 time points as input. (b) Held-out E13.5. (c) Held-out E15.5. (d) Held-out E13.5 and E15.5.</p

GraphFP charts the probability flows of cell state-transitions.

The circle point represents cell type (point size is proportional to the cell-type frequency at each time point); the line between cell types represents probability flow from source cell type to target cell type (line width is proportional to the value of probability flow).</p