Fig 1.
Experimental migration trajectories toward higher FGF concentration (indicated by the black arrow) for single cells (left) and clusters of
~10 Drosophila retinal progenitor cells in microchannels prepared with nearly steady FGF gradient (upward in plot orientation). A chemotactic index (CI) greater than 0.5 indicates positive directional migration along concentration gradients [17,19]. Small clusters of cells tend to migrate toward the higher FGF concertation (Average CI = 0.61 ± 0.15) while individual cells appear to migrate more randomly (Average CI = 0.22 ± 0.20). Each plot shows 3 trials (raw data provided in Pena 2019 supplemental information [17]); distances traveled are normalized to the number of cells in the cluster. The curves are cubic splines to aid the eye.
Fig 2.
Schematic of 1D Agent Based Model (ABM) for 5 cell cluster.
Concentration gradient influences intercellular cohesion to produce a net force on cluster. Similar results are obtained using either (a) Kelvin-Voigt or (b) Standard Linear Solid models of intercellular interactions.
Fig 3.
Simulations of 5-cell clusters with intercellular cohesion gradient.
(a) The outermost cells, N1 & N5, experience an outward extension of the same distance at the same time, and each colored line shows subsequent positions. The periodic extension causes the cluster to elongate until it reaches a natural equilibrium state. The black dashed line shows the center of mass, which remains stationary. (b) The same simulation in which the outermost cells move both up and down in phase, and (c) the same in which the outermost cells move in and out in phase. As expected, in all cases the center of mass remains stationary except for small transient oscillations. Key parameters are as follows: gradient strength (ratio of strongest to weakest cohesion strengths) = 10, cluster size = 5, protrusion amplitude (length of protrusions) = 1, protrusion period (frequency of protrusions) = 15. Protrusion amplitude is a fixed value in this simulation, whereas all other simulations have a variable amplitude using a Gaussian whose standard deviation is the specified protrusion amplitude.
Fig 4.
Analysis of motion from Fig 3A due to single outward step of outermost cells, starting form an “Initial” state at rest.
During “Extension” phase, the upper and lower cells are instantaneously extended outward (here by 2 cell radii), leaving the central cell body (gray) and the center of mass (dashed) stationary. During the subsequent “Relaxation” phase, the lower cell, feeling very weak cohesive tension with its neighbor, moves and affects the central cell body little, while the upper cell, subject to strong tension, rapidly equilibrates to an average position with the neighboring central cell body. After each cycle of extension and relaxation, the bottom cell is displaced 2 radii, and the two upper cells are displaced 1 radius, leaving the center of mass unmoved.
Fig 5.
Stochastic extension of outermost cells – (a) For stochastic extension distances (red arrows) that are short compared with the distance to the nearest cell, inward and outward motions are equally probable.
Dashed lines indicate mean extension distances, , where σ is the standard deviation of a Gaussian governing extension probabilities, shown to the right. (b) Extensions cannot reach beyond a neighboring cell a distance ∆ away, so as extension distances grow (or distances between neighbors shrink), extension probabilities will be truncated, as shown in blue. Consequently, the mean inward distance traveled by the outermost cell will become shorter than the mean outward distance. (c) In the presence of a cohesion gradient, the stronger cohesion end of a cluster will exhibit closer neighboring cells, and so will lead asymmetric migration of the cluster. The weaker cohesion end will exhibit more distant neighbors, and so will generate more symmetric, smaller displacement, migration.
Fig 6.
Simulations of 5-cell cluster with Gaussian fluctuations.
(a) Outermost cells, N1 and N5, experience displacements according to Gaussian fluctuations. The colored lines show the subsequent positions of each cell within the cluster. Note the upper 4 cells migrate consistently upward, toward the higher internal cohesion gradient, while the lower cell wanders almost independently. (b) Comparison between centers of mass subject to regular extensions (gray, from Fig 3), Gaussian fluctuations without a cohesion gradient (blue), and Gaussian fluctuations with a gradient (red). Solid lines indicate the average cluster position at successive times, and the dotted line are the least squares fit to the data. Only the case with the gradient migrates persistently, as shown by the positive slope (red dotted line). Key parameters are as follows: gradient strength = 10, cluster size = 5, protrusion amplitude = 1, protrusion period = 5. (c) Mean Squared Displacement analysis of N = 100 simulations of cell clusters without a gradient (blue) and with a gradient (red). Final positions range from -1 to 7 radii away from the initial position for clusters with a gradient vs ±3 radii away for clusters without a gradient. Cyan dashed lines indicate linear regression: and quadratic regression:
. Key parameters are as follows: gradient strength = 10, cluster size = 5, protrusion amplitude = 1, protrusion period = 15.
Fig 7.
Cluster speeds calculated from 100 simulations per parameter generated from stochasticity displacements of end agents only. Main plots show Standard Linear Solid (SLS), insets show Kelvin-Voigt (KV) model results. Violin plots show the distribution of all data point, and standard errors are shown for each parameter value. (a) Speed (unit distance relative to the radius of agents per computational timestep) vs cohesive gradient strength (the ratio of the strongest to weakest attraction in a linear gradient). Cluster speed grows from zero for gradient strength of 1 (equal leading and trailing cell cohesion), to an asymptote at large ratio. Cluster size = 5, protrusion amplitude = 0.5, protrusion period = 15. (b) Speed vs number of cells, N, in cluster produces a 1/N dependence as predicted by Eq. (6). Note N=2 has no gradient and migrates randomly. Gradient strength = 10, protrusion amplitude = 0.5, protrusion period = 15. (c) Speed vs protrusion amplitude (proportional to σ in Eq. (6)). Gradient strength = 10, cluster size = 5, protrusion period = 15. (d) Speed vs protrusion period of protrusion. Gradient strength = 10, cluster size = 5, protrusion amplitude = 0.5.
Fig 8.
Effect of internal migration on cluster motion.
100 simulation means of cluster migration speed for three cases using Standard Linear Solid (SLS) model. Green: persistent motion of cluster is seen when only outermost cells in cluster wander stochastically. All other parameters from Fig 7C. Red: identical simulation in which all cells wander randomly, demonstrating that wandering of internal cells interferes with cluster migration. Blue: hybrid simulation in which outermost cells wander as in the green case, but all other cells migrate with σ reduced by 75%. Evidently, persistent cluster migration depends on suppression of wandering of interior cells.