Figure 2. Correlating information flow, collective and local heterogeneity in single cell information transfer and multicellular adaptation.

(A) Schematics of cell-cell communication. Generic estimation of the asymmetric mutual influence between a pair of cells from their fluctuating time series. The influence of cell i on cell j is defined as the extent to which the past signal of cell i improves the prediction of cell j’s signal beyond the past signal of j alone and is determined using the pairwise asymmetric Granger Causality statistical test.

(B) Visualization of the spatial single cell heterogeneity of the degree rank (in-degree + out-degree). The color scale is linear.

(C) Heterogeneity in degree rank distribution. The Kernel Density Estimation (KDE) of the degree rank distributions of 10 null models that considered random shuffling of GC edges while preserving the probability for an edge (green) versus the experimentally observed degree rank distribution (red). The raw distribution (input to KDE) is shown in Fig. S3A, with Rinku index ~29 versus ~25 for the observed and null model correspondingly.

(D) Example of collective (top) and local (bottom) heterogeneity for three different network structures. Networks are ordered from left-to-right according to their heterogeneity levels measured with (Jacob et al., 2017) (collective) and (Estrada, 2010) (local). Graph (I) node degree ranks are (2,2,2,2,2,2): local heterogeneity = 0, collective heterogeneity = 0. Graph (II) node degree ranks are (1, 4, 2, 2, 3): local heterogeneity ≈ 0.38, collective heterogeneity ≈ 1. Graph (III) node degree ranks are (1,1,1,1, 4): local heterogeneity = 1, collective heterogeneity ≈ 0.59. See Methods for full details.

(E) Pairwise associations between two heterogeneity measures (local heterogeneity, collective heterogeneity), adaptation rate and GC edge probability. Edges color represents the level of association, as quantified by the magnitude of correlation coefficients. Color scale is linear. Note that some edges reflect positive correlations (e.g., collective heterogeneity - GC edge probability) while others reflect negative correlations (e.g., local heterogeneity – adaptation rate). N = 23 biological replicates, across shear stress levels, that passed the stationarity criterion were considered to calculate correlations. See full data (with signed correlations) in Fig. S4.