Research ArticleBIOPHYSICS

Frustration-induced phases in migrating cell clusters

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Science Advances  12 Sep 2018:
Vol. 4, no. 9, eaar8483
DOI: 10.1126/sciadv.aar8483
  • Fig. 1 Analyzing and modeling cell cluster phases.

    (A) Top: Experimental images of a cell cluster in each of the three phases, where the blue cells show positions at a certain time and red shows the positions of the same cells 15 s later. These positions are then used to calculate the cell velocities shown in yellow arrows. Bottom: Time series of the magnitudes of group polarization and angular momentum of the cell cluster. The colors along the bottom axis show the phase of the system with time (red, running; blue, rotating; green, random) for experimental data. (B) Schematic of the model. Green direction indicators show the directions of the neighbors of the gray cell, and the green indicator on the gray cell shows the alignment interaction (Embedded Image). The orange arrows show the Lennard-Jones interaction with each neighboring cell, and the red arrow is the total Lennard-Jones interaction (Embedded Image) on the gray cell. Finally, the blue spring denotes the surface tension interaction (Embedded Image). Note that it only exists between the gray cell and its second nearest neighbors that do not have cells interrupting the path between them. (C) Top: The proportion of time that the cluster spends in each phase [simulations (plain) and experiments (crosshatched)], along with a typical illustration of what each phase looks like in the simulations, with velocity vectors as black arrows. The cluster size for simulations is N = 37 cells, while experimental cluster sizes are distributed with a peak between 35 and 40 and a mean of about 50 (see fig. S7A). Bottom: Time series of the magnitudes of group polarization and angular momentum from simulations of a uniform cluster (dashed) and a cluster with behavioral heterogeneity (solid, corresponding to the point marked in Fig. 2B).

  • Fig. 2 Collective phase proportions of density-dependent propulsion model.

    (A) Proportion of time spent by the cluster (N = 37 cells) in each of the three phases plotted against propulsion p and noise Embedded Image for a cluster where all cells behave identically. (B) Phase diagram of the proportion of time spent in each of the three phases for a system with neighbor number–dependent propulsion where the rim cells (those with 3.67 neighbors) have a propulsion of prim = 8. pcore is the propulsion of core cells (those with 6 neighbors), and η is the magnitude of the noise. The black “x” shows the point where the time series and phase proportions shown in Fig. 1 (B, bottom, and C) are taken.

  • Fig. 3 Collective phase proportions of the coupled rim-core model.

    Proportion of time spent by the system in each of the three phases as a function of propulsion p and noise Embedded Image for a ring of 18 cells confined to a circle with propulsion prim = p. Dashed contour lines indicate regions (shaded) where the proportion of time spent in the corresponding phase exceeds 30% (blue) and 50% (red). Solid contour lines show the same contours but for the rim confined to a circle with prim = 8, coupled with a core of cells with pcore = p, and a full cluster size of N = 37. Note that the rotational phase only has nonzero values for the coupled system. The horizontal dashed line marks the noise value below which the rim alone would be ordered (greater than 30% running phase) with prim = 8, and the diagonal solid line marks the region above which a core with an average propulsion set by Eq. 7 (with pcore = p) is disordered (greater than 30% random phase).

  • Fig. 4 Defect dynamics and the transitions between phases.

    (A) Velocities of the rim cells of a 37-cell cluster that are confined to a circular shape projected onto the circle for a simulated cell cluster (left), as well as an experimental cell cluster with rim cell velocities projected onto a circle relative to the center of mass of the cluster (right). In the running phase (red panels), there are two defects of opposite signs in the velocity field, denoted by the orange and blue points. There are no defects in the rotating phase (blue panels). (B) Proportion of the number of defect pairs for each phase, with a peak at zero defect pairs for the rotating phase (blue) and one defect for the running phase (red). Simulation defect pairs are shown in solid bars, while the experimental defect pair counts are shown in the crosshatched bars. (C) Pair distribution function plotted against the separation between two defects when only one defect pair exists for parameters where the cluster primarily displays a running phase [note that g(r) is calculated over the whole simulation, independent of specific phases at any given point in time]. Inset: Pair distribution function for points in parameter space dominated by rotating (blue) and random (green) phases. Experimentally measured values of g(r) are shown as black points, with a linear fit shown by the dashed black line. A similar comparison of the results of the full-model simulation to the experiments shows no major difference (see section S11.)

  • Fig. 5 Cluster size and chemical gradient dependence.

    (A) Proportion of time spent in the rotating phase by a system with neighbor number–dependent propulsion as a function of pcore and noise Embedded Image for prim = 8. The black horizontal dashed line marks the noise value below which the rim alone would be ordered (corresponding to the dashed transition line in Fig. 3 with prim = 8). The diagonal lines show the noise value above which a uniform system with average propulsion p would be disordered (red, N = 19; blue, N = 37; green, N = 61). The shaded regions are where the clusters spend at least 30% time in the rotational phase, with the same color scheme (blue, N = 37; green, N = 61; note that there is no red shaded region). Inset: Dependence of the proportion of rotating phase on system size. The shaded red region shows the range of dependence for a spread of parameter values marked with black crosses in the main figure. The experimental measurements are shown as the blue points. (B) Experimental image of a cell cluster where cells are colored red or green for visualization. The red cell labeled by the white arrow moves from the rim of a cluster into the core between the top and bottom images over a 2-min time period. (C) Fluidity of the cluster measured as the rate of exchange between the core and rim cells of the cluster, for several system sizes, for both simulations (plain bars) and experimental data (crosshatched bars). Inset: Contours for the fluidity of the cluster over the pcore-η parameter space. (D) Simulated proportion of each phase (see legend) plotted with increasing chemical gradient (gcr in Eq. 5, where r is the cell diameter), along with experimental data in the inset. The concentration gradient of chemokine in the experiments is measured in (ng/ml)/mm and shown on the x axis for the inset.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/9/eaar8483/DC1

    Section S1. Model details

    Section S2. Lattice-induced rotations

    Section S3. Phase characterization

    Section S4. Density-dependent propulsion

    Section S5. Rim-core cluster model

    Section S6. Size dependence

    Section S7. Rim cell exposed edge

    Section S8. Varying rim propulsion

    Section S9. Solid body–like rotations

    Section S10. Fluidity versus gradient

    Section S11. Comparison of defects in experiment to full model

    Fig. S1. Schematic illustration of the chemical gradient force on the gray cell.

    Fig. S2. Collective phase proportions of a cluster with monodisperse cell sizes.

    Fig. S3. The propulsion of cells plotted against distance from the cluster center of mass.

    Fig. S4. Effects of density-propulsion relationship on phases.

    Fig. S5. Rim-core model phase proportions, with the rim cells confined to a circle.

    Fig. S6. Rim-core model phase proportions, with the rim cells unconfined.

    Fig. S7. Cluster size dependence of all phases.

    Fig. S8. Schematic for rim cell definition.

    Fig. S9. Collective phase proportions with varying rim propulsion.

    Fig. S10. Rotational slip of outer rim around the inner core.

    Fig. S11. Cluster fluidity as a function of chemical gradient.

    Fig. S12. Defect dynamics and the transitions between phases for the full model.

    Movie S1. Lattice-induced rotations for a crystalline cell cluster, which only occurs when the cells are of identical sizes and noise is sufficiently low.

    Movie S2. A system with the same parameters as movie S1 but with polydisperse cell sizes with a spread of 10% of the average cell size.

    Movie S3. Experimental cell cluster transitioning between the three phases of motion: running, rotating, and random.

    Movie S4. Defect dynamics as a cluster transitions from the rotating phase to the running phase and back again.

    Reference (40)

  • Supplementary Materials

    The PDF file includes:

    • Section S1. Model details
    • Section S2. Lattice-induced rotations
    • Section S3. Phase characterization
    • Section S4. Density-dependent propulsion
    • Section S5. Rim-core cluster model
    • Section S6. Size dependence
    • Section S7. Rim cell exposed edge
    • Section S8. Varying rim propulsion
    • Section S9. Solid body–like rotations
    • Section S10. Fluidity versus gradient
    • Section S11. Comparison of defects in experiment to full model
    • Fig. S1. Schematic illustration of the chemical gradient force on the gray cell.
    • Fig. S2. Collective phase proportions of a cluster with monodisperse cell sizes.
    • Fig. S3. The propulsion of cells plotted against distance from the cluster center of mass.
    • Fig. S4. Effects of density-propulsion relationship on phases.
    • Fig. S5. Rim-core model phase proportions, with the rim cells confined to a circle.
    • Fig. S6. Rim-core model phase proportions, with the rim cells unconfined.
    • Fig. S7. Cluster size dependence of all phases.
    • Fig. S8. Schematic for rim cell definition.
    • Fig. S9. Collective phase proportions with varying rim propulsion.
    • Fig. S10. Rotational slip of outer rim around the inner core.
    • Fig. S11. Cluster fluidity as a function of chemical gradient.
    • Fig. S12. Defect dynamics and the transitions between phases for the full model.
    • Legends for movies S1 to S4
    • Reference (40)

    Download PDF

    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mov format). Lattice-induced rotations for a crystalline cell cluster, which only occurs when the cells are of identical sizes and noise is sufficiently low.
    • Movie S2 (.mov format). A system with the same parameters as movie S1 but with polydisperse cell sizes with a spread of 10% of the average cell size.
    • Movie S3 (.mov format). Experimental cell cluster transitioning between the three phases of motion: running, rotating, and random.
    • Movie S4 (.mp4 format). Defect dynamics as a cluster transitions from the rotating phase to the running phase and back again.

    Files in this Data Supplement:

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