Modeling Contact Inhibition of Locomotion of Colliding Cells Migrating on Micropatterned Substrates

<div><p>In cancer metastasis, embryonic development, and wound healing, cells can coordinate their motion, leading to collective motility. To characterize these cell-cell interactions, which include contact inhibition of locomotion (CIL), micropatterned substrates are often used to restrict cell migration to linear, quasi-one-dimensional paths. In these assays, collisions between polarized cells occur frequently with only a few possible outcomes, such as cells reversing direction, sticking to one another, or walking past one another. Using a computational phase field model of collective cell motility that includes the mechanics of cell shape and a minimal chemical model for CIL, we are able to reproduce all cases seen in two-cell collisions. A subtle balance between the internal cell polarization, CIL and cell-cell adhesion governs the collision outcome. We identify the parameters that control transitions between the different cases, including cell-cell adhesion, propulsion strength, and the rates of CIL. These parameters suggest hypotheses for why different cell types have different collision behavior and the effect of interventions that modulate collision outcomes. To reproduce the heterogeneity in cell-cell collision outcomes observed experimentally in neural crest cells, we must either carefully tune our parameters or assume that there is significant cell-to-cell variation in key parameters like cell-cell adhesion.</p></div

Reversal is robust, but chains require tuning.

<p>The percentage of collisions that result in chains is plotted; all other collisions create reversal, except in the marked region with <i>k</i>_<i>CR</i> ≤ 0.01<i>s</i>⁻¹ and <i>k</i>_<i>FR</i> ≤ 0.01<i>s</i>⁻¹ where mechanical interactions can dominate (discussed in the text). The parameters are <i>σ</i> = 2.25<i>σ</i>₀, <i>O</i>_<i>crit</i> = 0<i>μm</i>⁻² and <i>α</i> = 0.4<i>α</i>₀. We did 100 simulations for each point of the grid, which has a step size of 0.0025<i>s</i>⁻¹ for <i>k</i>_<i>CR</i> and 0.025<i>s</i>⁻¹ for <i>k</i>_<i>FR</i>. It should be noted that <i>k</i>_<i>FR</i> is an order of magnitude larger than <i>k</i>_<i>CR</i>.</p

Elements of our model.

<p>The cell shape is tracked by a phase field <i>ϕ</i>(<b>r</b>). The cell boundary (<i>ϕ</i> = 0.5 contour line) is plotted in black. On the left side the Rac concentration <i>ρ</i>(<b>r</b>) is shown, which defines the cell front. The inhibitor level <i>I</i>(<b>r</b>) is plotted on the right. To limit the internal fields to the inside of the cell, we plot <i>I</i>(<b>r</b>) × <i>ϕ</i>(<b>r</b>) (<i>ρ</i>(<b>r</b>) × <i>ϕ</i>(<b>r</b>), respectively). Throughout this work we use the shown color scales. To indicate the (static) micropattern, the contour line with <i>χ</i>(<b>r</b>) = 0.5 is displayed as a thick blue line.</p

Transition between sticking and reversal is sharp and depends on balance of adhesion and propulsion.

<p>We show the fraction of sticking events; all other events are reversals. The parameters are <i>k</i>_<i>CR</i> = 0.1<i>s</i>⁻¹, <i>k</i>_<i>FR</i> = 0<i>s</i>⁻¹ and <i>O</i>_<i>crit</i> = 0<i>μm</i>⁻². The simulations run for <i>T</i> = 2500<i>s</i>. For <i>α</i> the step size is 0.025<i>α</i>₀ and for <i>σ</i> it is 0.02<i>σ</i>₀ close to the transition and 0.25 further away. We did 100 simulations for points near the transition and otherwise 10.</p

Snapshots of different outcomes.

<p>In each panel <i>ρ</i>(<b>r</b>) is on the left, <i>I</i>(<b>r</b>) on the right and the edges of the adhesive micropattern are indicated in blue. <i>α</i> = 0.4<i>α</i>₀ for all cases. The outcomes are i) reversals, ii) sticking, iii) walk-past, and iv) chaining. Next to each outcome are the parameters of the snapshots and the rate of the outcome for the given parameters based on 100 simulations. We chose the parameters that yield the maximum rate for each outcome. <i>α</i>₀ = 1<i>pN</i>/<i>μm</i>³ and <i>σ</i>₀ = 1<i>pN</i>/<i>μm</i>. Times are measured relative to the time of first contact.</p

Schematic picture of parameters controlling the different outcomes.

<p>Schematic picture of parameters controlling the different outcomes.</p

Walk-past depends non-monotonically on adhesion, strongly on critical overlap.

<p>Percentage of collisions that lead to walk-past events is shown; 100 simulations are performed for each data point. The error bars are calculated with the binomial proportion confidence interval with the significance level 0.05. The parameters are <i>O</i>_<i>crit</i> = 0.15<i>μm</i>⁻² (upper panel), <i>σ</i> = 3.70<i>σ</i>₀ (lower), <i>α</i> = 0.4<i>α</i>₀, <i>k</i>_<i>FR</i> = 0.075<i>s</i>⁻¹ and <i>k</i>_<i>CR</i> = 0.004<i>s</i>⁻¹.</p

Basic experimental observations.

<p>Basic experimental observations.</p

Robust walk-past requires a balance of contact repolarization and front repolarization.

<p>Percentage of walk-past (left), reversal (middle), and chaining (right) events. It should be noted that <i>k</i>_<i>FR</i> is an order of magnitude larger than <i>k</i>_<i>CR</i>. We don’t include <i>k</i>_<i>CR</i> = 0<i>s</i>⁻¹ in this figure because these cases can have ambiguous outcomes. In the marked region in the lower left corner mechanical interactions can dominate and the collision outcomes do not always resemble any of the four experimental cases (details for both in text). The parameters are <i>σ</i> = 3.70<i>σ</i>₀, <i>O</i>_<i>crit</i> = 0.15<i>μm</i>⁻² and <i>α</i> = 0.4<i>α</i>₀. We performed 100 simulations for each point of the grid, which has a step size of 0.002<i>s</i>⁻¹ for <i>k</i>_<i>CR</i> and 0.025<i>s</i>⁻¹ for <i>k</i>_<i>FR</i>; the color maps have been interpolated.</p

Sketch of the impact of CR (left) and FR (right) repolarization mechanisms on a head-tail collision.

<p>Contour lines of <i>ϕ</i> are shown in black. The front of the cells with a high value of <i>ρ</i> is marked with red. The solid blue line shows edges of the micropattern <i>χ</i>. Cyan parts of the cell boundary mark inhibitor production near the cell edge from CR/FR. The inhibitor diffuses to other parts of the cell (cyan stripes) and inhibits <i>ρ</i> there. With CR the cells produce inhibitor when they are in contact with any part of another cell. This causes the left cell to repolarize. For FR, there is only inhibitor production when the cell is in contact with the front of another cell. Thus, no cell will repolarize in this head-tail collision.</p