Research ArticleSYSTEMS BIOLOGY

A statistical inference approach to reconstruct intercellular interactions in cell migration experiments

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Science Advances  11 Mar 2020:
Vol. 6, no. 11, eaay2103
DOI: 10.1126/sciadv.aay2103
  • Fig. 1 Motivation for ME models with bound constraints in cell-tracking experiments.

    (A) Two cells (whose boundaries are the red and green closed curves) and their centers of mass (red and green disks) are tracked through a grid of pixels (gray). Each center is assigned a nominal position, i.e., the center of the pixel where it is located (dashed red and green circles, respectively). (B) At a subsequent time, the green cell has moved to a neighboring pixel (curved black arrow). The nominal position of the cell has changed, and its displacement is the vector difference between the nominal position in (B) and in (A) (dashed black arrow), resulting in a well-defined direction of motion. Because the red cell has moved within the pixel, its nominal position is the same as in (A), its nominal displacement is null, and its direction of motion is not defined, thus leading to an uncertainty in all physical observables that involve directions of motion.

  • Fig. 2 Wound-healing experiment.

    (A) Fluorescence image showing the nuclei (red, H2B-mCherry) of HeLa cells migrating toward a wound located at the right edge of the image (scale bar, 100 μm). (B) Tracked cell trajectories: The instantaneous position of each cell is marked with a circle, and the respective time is specified by the color code. Only a few representative cells are shown for clarity. (C) Polar histograms for the angle θ and (D) mean-squared displacement R2(t)¯ versus time show that the motion is affected by a strong bias, which yields a mean-squared displacement growing quadratically in time: Experimental data are shown in dark color, and the best fit y = p1 + p2t + p3t2 with p1 = 33, p2 = 0.5, p3 = 0.01 is shown in bright color. The inset in (D) shows the number of tracks recorded at each instant of time to figure out the statistically significant time window. (E) Angle pairwise correlation θ̂(R) for the wound-healing experiment.

  • Fig. 3 Dendritic cell experiment.

    (A) Microscope image of cells showing chemokine-poor and chemokine-rich regions, i.e., zones 1 and 2, respectively, separated by green dashed lines (scale bar, 100 μm). (B) Tracked cell trajectories in zones 1 and 2: The instantaneous position of each cell is marked with a circle, and the respective time is specified by the color code. Only a few representative cells are shown for clarity. Cells in zone 1 (C and D) move isotropically and diffusively (best fit y = p1 + p2t with p1 = − 90, p2 = 31), while cells in zone 2 (E and F) feel a drift and move ballistically (y = p1 + p2t + p3t2 with p1 = − 306, p2 = 99, p3 = 0.6). The insets in (D) and (F) show the number of tracks recorded at each time. (G) Angle pairwise correlation θ̂(R) for the three experimental instances above.

  • Fig. 4 Analysis of the wound-healing experiment via ME method with bound constraints.

    (A) Full set of solutions of the KKT conditions in order of increasing entropy from left to right. Each solution is labeled with an integer shown on the abscissa, and the corresponding value of J (top), Hx (middle), and Hy (bottom) is represented with the color in the box. (B) Entropy per cell for each solution. (C) Modulus of the relative residual Δineq of the inequality KKT conditions shown for each solution, where residuals that are positive and negative are marked in blue and red, respectively. (D) Modulus of relative residual Δeq of the equality KKT conditions, shown for each solution. The numerical precision used in the calculation, ϵ, is marked in (C) and (D). Only the solutions with Δineq ≥ 0 [blue columns in (C)] are admissible, i.e., solutions 12 and 19. Among these two solutions, solution 12 is the only one with Δeq ∼ ϵ, see (D), and it thus constitutes the only admissible, ME solution (green rectangle). In (B), unphysical solutions with negative probability are ruled out and not shown.

  • Fig. 5 Statistical inference analysis of the dendritic cell experiment with high cell density with the ME method with bound constraints.

    (A to D) Analysis for data in zone 1, where we use the same notation and the same procedure to select the ME solution as in Fig. 4. (E to H) Analysis for zone 2. The numerical values of J and H for the MEb solutions are shown in table S2.

  • Fig. 6 Statistical inference analysis of the dendritic cell experiment with low cell density with the ME method with bound constraints.

    (A to D) Analysis for data in zone 1, where we use the same notation and the same procedure to select the ME solution as in Fig. 4. (E to H) Analysis for zone 2. The numerical values of J and H for the MEb solutions are shown in table S2.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/11/eaay2103/DC1

    Section S1. ME models

    Section S2. Tests of ME method with bound constraints

    Section S3. Visual comparison between experiments

    Section S4. Analysis of motional data

    Section S5. Estimate of positional uncertainty

    Fig. S1. Test of ME method with bound constraints on synthetic data for the XY model.

    Fig. S2. Test of ME method with bound constraints on synthetic data for the SP model.

    Fig. S3. Consistency test of ME method with bound constraints.

    Fig. S4. Visual comparison between tracks of the wound-healing and dendritic cell experiment.

    Fig. S5. Statistics of motional data for the wound-healing experiment.

    Fig. S6. Empirical distributions for cell velocities in the wound-healing experiment.

    Fig. S7. Statistics of motional data for zone 1 in the dendritic cell experiment.

    Fig. S8. Empirical distributions for cell velocities for zone 1 in the dendritic cell experiment.

    Fig. S9. Statistics of motional data for zone 2 in the dendritic cell experiment.

    Fig. S10. Empirical distributions for cell velocities for zone 2 in the dendritic cell experiment.

    Fig. S11. Estimate of the positional error.

    Table S1. Confidence intervals for the empirical averages in the wound-healing experiment.

    Table S2. Confidence intervals for the empirical averages in the dendritic cell experiment.

    References (3335)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. ME models
    • Section S2. Tests of ME method with bound constraints
    • Section S3. Visual comparison between experiments
    • Section S4. Analysis of motional data
    • Section S5. Estimate of positional uncertainty
    • Fig. S1. Test of ME method with bound constraints on synthetic data for the XY model.
    • Fig. S2. Test of ME method with bound constraints on synthetic data for the SP model.
    • Fig. S3. Consistency test of ME method with bound constraints.
    • Fig. S4. Visual comparison between tracks of the wound-healing and dendritic cell experiment.
    • Fig. S5. Statistics of motional data for the wound-healing experiment.
    • Fig. S6. Empirical distributions for cell velocities in the wound-healing experiment.
    • Fig. S7. Statistics of motional data for zone 1 in the dendritic cell experiment.
    • Fig. S8. Empirical distributions for cell velocities for zone 1 in the dendritic cell experiment.
    • Fig. S9. Statistics of motional data for zone 2 in the dendritic cell experiment.
    • Fig. S10. Empirical distributions for cell velocities for zone 2 in the dendritic cell experiment.
    • Fig. S11. Estimate of the positional error.
    • Table S1. Confidence intervals for the empirical averages in the wound-healing experiment.
    • Table S2. Confidence intervals for the empirical averages in the dendritic cell experiment.
    • References (3335)

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