PIEZO1 activity inhibits wound edge dynamics and leader cell formation

Efficient collective migration is driven by the formation of leader cells [33–35]. These highly specialized cells are distinct from their surrounding follower cells and play a key role in dictating collective dynamics [12]. In light of our previous finding that PIEZO1 activity inhibits wound closure [22], we sought to further characterize the effect of PIEZO1 on collective migration by investigating how PIEZO1 activity may affect leader cell formation. Since the emergence of leader cells drives collective migration and increased PIEZO1 activity results in delayed wound closure, we hypothesized that the number of leader cells would be affected by PIEZO1 activity levels. We generated scratch wounds in Piezo1-cKO, Piezo1-GoF, Yoda1-treated keratinocyte monolayers and their respective controls and utilized differential interference contrast (DIC) time-lapse imaging to examine the evolution of the wound margin (Fig 1C–1E). During reepithelialization, multicellular finger-like protrusions often form along the wound margin as cells work together to close the wound area [36]. At the front of these cellular outgrowths, leader cells can be identified by their specialized phenotypic morphology in which they display a larger size, increased polarity, and prominent lamellipodia (Fig 1F and S1 Fig) [11, 35]. Leader cells were manually identified in time-lapse images of wound closure, similar to methods other groups have used for leader cell quantification within migrating collectives [37]. Given the different genetic backgrounds between conditions (i.e., Piezo1-cKO, Piezo1-GoF, Yoda1-treated) and the differences observed in migration properties across the Control samples of these different backgrounds [22], keratinocytes are only compared to control conditions of the same genetic background for all analyses. As hypothesized, we found that in Piezo1-cKO monolayers, the monolayer edge shows an increase in the number of leader cells compared to those from Control_cKO keratinocyte monolayers (Fig 1C and 1G and S1A Fig). On the other hand in both Piezo1-GoF and Yoda1-treated monolayer conditions, where PIEZO1 activity is increased, the wound edge remains relatively flat throughout the imaging period due to a decrease in the formation of leader cells at the wound edge compared to respective control monolayers (Fig 1D, 1E and 1G, S1B and S1C Fig). Furthermore, we noticed in Piezo1-cKO monolayers, that leader cells appeared to recruit more follower cells as seen by an increase in the width of fingering protrusions relative to Control_cKO monolayers (Fig 1C and S1 Fig).

To quantify the effect that PIEZO1 activity has on wound edge dynamics and leader cell protrusions, we also measured the change in the length of the wound edge within a field of view over the course of the imaging period, similar to methods employed by other groups [38]. The presence of leader cells, which are located at the front of cellular outgrowths, increases the length of the wound edge. Therefore, a shorter wound edge length would indicate fewer leader cells while a longer wound edge would indicate an increase in leader cells along the wound margin. We found that Piezo1-cKO monolayers have a longer wound edge length relative to Control_cKO monolayers, which further supports our observation that the absence of PIEZO1 results in increased leader cells along the wound edge (Fig 1H, left). Conversely, in both Piezo1-GoF and Yoda1-treated monolayers we found that edge lengths are significantly shorter than the respective control monolayers (Fig 1H, middle, right). Thus, we find that PIEZO1 inhibits the formation of leader cells, resulting in a shorter and flatter wound edge, while the absence of the channel results in a longer and more featured wound edge due to an increase in leader cell protrusions.

Modeling PIEZO1’s influence on keratinocyte collective dynamics

Due to the many intricacies underlying the biological phenomena of collective cell migration, we adopted a theoretical approach as a framework for characterizing the biophysical relationship between PIEZO1 activity, leader cell initiation and wound closure. We designed a mathematical model of keratinocyte reepithelialization in order to study how PIEZO1 activity influences this process. We first separated reepithelialization into essential phenomenological components which could be incorporated into the design of the model as manipulable variables. As such we accounted for cell motility, and cellular retraction, a process central to the migration process and one which we previously found PIEZO1 activity promoted [22]. In our experimental data, we found that retraction varied in intensity such that in some instances it led to small regions of individual cells retracting while in other cases it led to the entire cell body pulling back away from the wound area [22]. Therefore we modeled retraction as a stochastic process at the leading edge associated with backward cell motion. We incorporated into our model design: (1) the average duration of retraction events at the monolayer edge, (2) the interval of time between sequential edge retractions, and (3) the strength of retraction. We also incorporated two hallmarks of collective cell migration: cell-cell adhesion and the coordination of keratinocyte migration direction, or coordinated directionality, both of which have been central to mathematical models proposed by other groups [32, 39–41]. Instead of modeling the mechanical forces involved in adhesion and retraction explicitly, we encoded the mechanistic effects such as cell motility, coordinated directionality, cell-cell adhesion, and retraction into model parameters. We then systematically manipulate these biological components of wound closure within our model and compare simulation results to experimental data garnered from scratch assays of PIEZO1 mutant keratinocytes (Table 1).

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Table 1. Coordinated directionality is the key model parameter which replicates PIEZO1 reepithelialization phenotypes.

Top: Summary table of monolayer experimental results on normalized wound closure and wound edge length. See also Fig 1. Bottom: Summary table of simulation results, depicting the effect of model parameters on normalized wound closure and wound edge length. For model parameters, single cell parameters (retraction strength, retraction duration, inter-retraction duration and cell motility) are separated from parameters which come from collective cell settings (cell-cell adhesion and coordinated directionality). A “+” indicates the wound feature is positively correlated with the model parameter, e.g., wound edge length increases with increased retraction strength, whereas “−” indicates a negative correlation, e.g., normalized wound closure is reduced with increasing retraction strength. Bolded italicized text denotes model parameters which correspond with experimental trends. See also Fig 2G and 2H and S3 Fig.

https://doi.org/10.1371/journal.pcbi.1011855.t001

Due to the inherent multivariate nature of our system, we utilized a partial differential equation (PDE) model to spatiotemporally describe PIEZO1’s effect on reepithelialization. The PDE governing collective cell migration, which describes behavior at the monolayer scale, is derived by upscaling a discrete model at the single cell level. Simulations of wound closure, obtained by solving the nonlinear PDE numerically, enables a deeper understanding of how each model parameter contributes at both the single cell and monolayer levels. Furthermore, integration of experimental data at the single cell and monolayer scales allows for calibration of the model. We present a dimensionless version of the model here. We rescale the cell density by its maximal value, which can be quantified by counting the maximum number of cells in squares of a grid in the monolayer region away from the wound edge, where we expect cell density to exhibit spatial and temporal uniformity. The characteristic length scale l is defined as the distance from the wound edge to the region where the cells reach the maximal density in the monolayer (typically ∼10 cell lengths). Hence, our computational domain is a small region around the wound edge. From the experimental data, we can extract a characteristic wound edge velocity v, which allows us to derive a characteristic time λ⁻¹ = l/v. See Section 6 in S1 Text for additional details.

The two-dimensional spatial discretization of a field of view containing a monolayer covered by a uniform grid of size h allows the labeling of indices (i, j) in space as x = x_i,j = (ih, jh), and cell density, ρ = ρ(x, t) = ρ(x, y, t), a function of space, x = (x, y)^T, and time, t, can be represented by ρ_i,j = ρ(x_i,j, t) at time t (Fig 2A). The dimensionless experimental field of view is a unit square domain: (see Section 6 in S1 Text for the details of nondimensionalization). By incorporating the essential biological components of reepithelialization (Table 1, bottom), we construct the following discrete master equation (Fig 2A, left; Eq 1), which demonstrates the change rate of cell density over time (Eq 1; left hand side) in response to the net flux of cells (Eq 1; right hand side): (1)

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Fig 2. Coordinated directionality is the key model parameter which replicates PIEZO1 reepithelialization phenotypes.

(A) Schematic showing a simplified visual of the modeling approach and visualization of the simulation domain. In the semi-discrete master equation (left; Eq 1), transitional probabilities associated with cell influx are highlighted in blue, while cell efflux related transitional probabilities are in red. Corresponding arrows depict this process on the grid (middle), indicating that the net flux is equal to the change in cell density over time at grid point (i, j). D represents coordinated directionality, and R represents retraction. (B) Simulation snapshots taken at equidistant time intervals depicting the evolution of the wound edge until wound closure (the moment interfaces touched) under low (top), Control (middle) and increased (bottom) levels of retraction strength. Shaded areas represent cell monolayers, while unshaded areas denote the cell-free space. (C) Plots showing quantification of the normalized edge length of simulated wounds, a measurement indicative of the number of leader cells, under different levels of retraction strength as a function of time. Shorter lines indicate simulation ending earlier due to faster wound closure. (D, E) Same as for (B) and (C), but under different levels of coordinated directionality. (F) The proportion of wound closure cases under different retraction magnitudes. The proportion of open wound closure cases start to decline after increasing retraction strength to 40, and almost no closure cases occur as retraction strength approaches 60. See also S2 Fig. (G) Line graphs showing the mean of 100 simulation results depicting the effect of retraction strength on normalized wound closure (red; left axes) and normalized edge length (blue; right axes). Error bars depict the standard error of mean. (H) Similar to (G) but for coordinated directionality. In C, E, F-H, all numbers have no unit because the model is dimensionless. See also S3 Fig and Table 1.

https://doi.org/10.1371/journal.pcbi.1011855.g002

Here, the T’s are transitional probabilities per unit time associated with given directions of movement (i.e., , , and ) for cells migrating between adjacent grid points (e.g., from x_i,j to x_i+1,j for ). Each transitional probability accounts for cell motility, cell-cell adhesion, coordinated directionality, retraction events, and volume filling limitations.

In the discrete master equation (Eq 1), the transitional probability for cells traveling from x_i,j to x_i+1,j is defined as the following: (2) The term (1 − ρ_i+1,j) models the effects of volume filling, e.g., if the cell density at x_i+1,j has reached its maximal value, it restricts further cell movement into that point. The term (1 − αρ_i−1,j)(1 − αρ_i,j+1)(1 − αρ_i,j−1) models cell-cell adhesion from three directions that hinder the cell migration, where α ∈ [0, 1] is the adhesion coefficient, which is assumed to be the same in each direction [29]. In the last term , the vector models diffusive cell motion while models cell movement due to retraction. The dependence on h reflects diffusive () and advective () scaling of the equations, respectively. The diffusive component , scaled as , generates a diffusion flux that depends on the gradient of cell density. The advective component , scaled as , results in an advection velocity independent of cell density that mimics the influence of retraction events (see Section 4 in S1 Text for details). Further, d^↔ represents the magnitude of movement in the horizontal coordinate direction, while accounts for cell retraction. Note that the cell density term ρ_i,j in models the moving front that connects a region of zero cell density (wound) with a region of non-zero density (monolayer), e.g., [42]. The other transitional probabilities, , and are defined analogously. Hence, Eq 2 can be rewritten as (3)

A continuum limit can be obtained by taking h → 0 in the discrete master equation (Eq 1) to yield the partial differential equation (4) which is a diffusion-advection equation where the diffusion, D, models cellular locomotion and coordinated directionality, whereas the advection velocity, R, models retraction of the leading edge. The diffusion coefficient, or diffusivity, D, is a 2 × 2 positive definite matrix given by (5) where d > 0 models cell motility during collective migration. The diffusion decomposition w_II + w_AA combines the diffusion isotropy, where the identity matrix I = I₂ models the randomness of cellular migration, and diffusion anisotropy, where the matrix A models directed cellular migration. During wound closure, directional cues received from leader cells promote the migration of followers into the cell-free space to close the wound, thus promoting cells to have a higher probability of moving into the wound area and resulting in an anisotropic direction of diffusion.

The information regarding coordinated directionality is transmitted from the discrete level through the incorporation of distinct magnitudes of movement in the coordinate directions (d^↔ and d^↕ in Eqs 2 and 3) that influence the transitional probabilities in two directions. Considering that cells receive signals to migrate towards the wound gap, we assume a larger magnitude of movement in the vertical direction (d^↕ ≥ d^↔) based on our experimental configuration (Fig 2A, right). This assumption facilitates the following decomposition: (6) where the continuous coefficients of cell motility d, isotropic strength w_I, and anisotropic strength w_A (representing coordinated directionality) are derived from the discrete coefficients of the magnitudes of movement in the coordinate directions d^↔ and d^↕ through the following relation: (7) Here, the directionality assumption d^↕ ≥ d^↔ guarantees the weights w_I and w_A are non-negative and bounded by 1, and the convex weighting relation w_I + w_A = 1 naturally holds from the derivation of w_I and w_A. From these relations, we can observe how w_A measures coordinated directionality. Systems with stronger coordinated directionality, e.g., larger w_A that results from large relative differences between between d^↕ and d^↔, are more likely to migrate towards the direction of closure.

The scalar diffusion coefficient in Eq 5 is a polynomial of cell density ρ: (8) which is derived through a multi-scale modeling process from the scaled cell density ρ_i,j/h², cell-cell adhesion (e.g., (1 − αρ_i−1,j)(1 − αρ_i,j+1)(1 − αρ_i,j−1) in ) and volume filling (e.g., 1 − ρ_i+1,j in ). The adhesion coefficient, α, which lies in the range [0, 1], models the adhesion forces between adjacent cells, with a larger α corresponding to larger adhesion forces. Volume-filling limitations to cell movement are also modeled in to hinder cells from migrating into a cell-dense area. In order to maintain a positive diffusivity, the value of α is bounded by ∼ 0.66 from above (see the detailed derivation in Section 3 in S1 Text).

Analogous to the derivation of diffusion, retraction, R (Fig 2A, right), is derived from the component of the discrete transitional probability (Eq 3) by taking the limit h → 0: (9) where 1 − ρ and (1 − αρ)³ model the effects of volume filling and cell-cell adhesion respectively. The retraction magnitude and directions are modeled phenomenologically in Δr^↔ and Δr^↕ as being localized in space and time, motivated by our previous work [22]. In particular, we assume:

1. Retraction occurs locally along the wound edge. This means only a part of wound edge cells are involved in retraction events at each time, while the rest of the cells on the edge and cells within the monolayer away from the edge just migrate by diffusion.
2. Retraction occurs intermittently in time. This means no retraction event is endless, i.e., no regions retract indefinitely. Hence at a wound edge point, there is a finite interval of duration time for each retraction event, and there is also a finite interval of time between two consecutive retraction events.

Because the computational domain is a small region around the wound edge (∼10 cell lengths, see Section 6 in S1 Text), we assume there is one localized retraction region of fixed width that occurs at random times and locations on each side of the wound edge.

Following the localization assumptions (1) and (2), a choice for Δr^↔ and Δr^↕ is (10) where H is a Heaviside function (11) with threshold γ, which localizes the retraction to the wound edge (γ = 0.4 was adapted in the simulation). In particular, H(γ − ρ) = 0 turns off the retraction for ρ > γ, which is the high cell density region far away from the wound edge, while H(γ − ρ) = 1 turns on the retraction for ρ < γ, which is the low cell density region near the wound edge.

By labeling retraction events in chronological order with positive integers i = 1, 2, 3, …, indicator functions and localize the regions where the edge retracts in time and space, respectively. We take the retraction to be localized in a region Ω_i ≔ [c_i − ω_r/2, c_i + ω_r/2] × [0, 1] about a line segment x = c_i with width ω_r (ω_r = 0.2 was used in the simulation): (12) where we account for randomness by taking the uniform distribution . This allows the region Ω_i to randomly slide around [0, 1] to localize the retraction events.

The retractions are assumed to occur at particular times τ_i with durations . Accordingly, we take (13) where s_i is the speed (or strength) of the retraction, and the next retraction occurs at where is the inter-retraction duration. To account for randomness, we assume: (14) where denotes the normal distribution with mean μ and standard deviation σ, and all random variables are independent and identically distributed (iid). Thus, the mean strength of the retraction forces is μ_s and a single retraction event is sustained for a random duration with mean μ_r. Any subsequent retraction will only start after a random idle duration with mean μ_nr. The corresponding variances are , , and , respectively. To ensure that our model incorporates only physically meaningful events, any negative duration or strength values that arise during the simulation are promptly discarded.

In summary, Δr^↔ and Δr^↕ are designed to model retractions such that cell movement would be governed by a diffusion-advection equation that guides the migrating cells in the retraction region near the wound edge: (15) where D is the diffusivity (Eq 5) and where the retraction strength, s, regulates the magnitude of advection velocity. On the other hand, cells far from the wound edge (e.g., interior of the monolayer) migrate following a simple diffusion equation (16) with the same diffusivity (Eq 5). In fact, our model passes retraction information from the discrete to the continuous level. For example, if then there is no retraction at x_i,j, e.g., monolayer region far away behind the wound edge. In this case, the governing equation is a pure diffusion equation without advection (Eq 16) since Δr^↔ = r^← − r^→ = 0, which appears in the continuum limit. In the retraction region, increases and decreases, so Δr^↔ = r^←−r^→ ≠ 0 and the governing equation turns into a diffusion-advection equation (Eq 15).

Note that both Heaviside function H(γ − ρ) and characteristic functions and are discontinuous. To preserve differentiability, we smooth H using a hyperbolic tangent function (Eq. 34 in Section 5 in S1 Text) and smooth and using generalized bell-shaped functions (Eq. 38 in Section 5 in S1 Text) and (Eq. 35 in Section 5 in S1 Text). In addition to the definition of Δr^↔ and Δr^↕ (Eq 10) given above, there are alternative choices that can be adapted to interpolate the advection velocity between the retraction and non-retraction regions. However, the qualitative results of the model are not sensitive to the choice of Δr^↔ and Δr^↕ under assumptions (1) and (2) and the model reduces to Eq 15 near the wound edge and to Eq 16 far from the wound edge.

Since we only model a subset of the observation domain in the experiment, e.g., the region close to the wound edges as opposed to the whole experimental domain, we impose the following conditions at the boundaries of the computational domain: (17) Horizontally on the top and bottom of the domain, time-dependent Dirichlet boundary conditions at x = (x, 0) and x = (x, 1) assign cell densities to the boundary points by functions g₀ and g₁, which mimic the effect of cells that flow into the observation domain area from the monolayer roughly perpendicular to the wound edge. The functions g₀ and g₁ are random functions of space and time (See Section 1 in S1 Text for the definitions of g₀ and g₁). Whilst vertically on the left and right sides of the domain, no-flux (Neumann) boundary conditions are used to approximate a net balance of cell influx and efflux into the observation domain roughly parallel to the wound edge, as suggested by the experiments.

The initial condition is generated by solving the PDE without retraction events for a short time period, which produces a banded heterogeneous monolayer with a cell-free region in the middle mimicking the initial wound (see Section 2 in S1 Text for details).

Summarizing, the model depends on the following biological parameters: (1) the mean retraction duration, μ_r, (2) the mean inter-retraction duration, μ_nr, (3) the mean retraction strength, μ_s, (4) cell motility, d, in the absence of retraction (pure diffusion context), (5) cell-cell adhesion, α, and (6) the strength of coordinated directionality, w_A. The governing equation (Eq 4) is a nonlinear stochastic PDE, where stochasticity arises from the random coefficients. We solve the equations numerically using a finite difference method to obtain the cell density ρ(x, t) on the simulation domain over time until wound closure. Multiple simulations are performed under each condition to quantify the variability for the subsequent data analysis (See Methods Section Numerical scheme for details), from which we investigate how each model parameter influences collective migration during reepithelialization.

Coordinated directionality is the key model parameter which replicates PIEZO1 reepithelialization phenotypes

Simulations of wound closure provide insight into how individual model parameters affect the wound closure process (Fig 2B–2E). Thus, through a parameter study, we can explore the effects of model parameters on two experimentally-measured phenotypes affected by PIEZO1 activity during keratinocyte reepithelialization: (1) the rate of normalized wound closure and (2) normalized wound edge length, a measurement to characterize the degree of cellular protrusions and retractions during wound healing, which is correlated with leader cell presence (Fig 1G and 1H; [26]). These metrics were chosen because they can be directly measured and compared to experimental data. During simulations we found that wounds would fail to close if parameters exceed a reasonable range (Fig 2F and S2 Fig). For instance, when retraction strength is set too high, cells are unable to overcome retractions of the wound edge which causes wounds to remain open indefinitely (Fig 2F). This model prediction is consistent with experimental results where Yoda1 treatment sometimes resulted in an increase in wound area during wound closure assays (Fig 1B; [22]).

By plotting the average rate of wound closure and edge length across multiple simulations we can see how the setting of individual model parameters compares to experimental trends we observe (Fig 2G and 2H and S3 Fig). We find that increasing the retraction strength parameter hinders wound closure, a result which is in line with the mechanism proposed in Holt et al., 2021 [22] (Fig 2G). However, our parameter study also shows that increased retraction strength results in a longer wound edge length, suggesting an increase in leader cell-like protrusions along the simulated wound margin. This contradicts our experimental observations in which a shorter wound edge length with fewer leader cells accompanies delayed wound closure (Fig 2G; Table 1). Similarly, we find that lower retraction strength elicited faster wound closure with shorter edge lengths due to fewer leader cell-like protrusions which also contradicts our experimental results (Fig 2G). Together, these results indicate that there is more to PIEZO1’s role in cell migration than retraction alone.

To identify possible contributors of wound closure regulation influenced by PIEZO1 activity, we performed an extensive parameter study in which we made adjustments to the model parameters of: cell-cell adhesion, retraction duration, inter-retraction duration, cell motility and coordinated directionality. We found that manipulation of all parameters aside from coordinated directionality fail to replicate the observed experimental results, i.e., faster wound closure accompanying a longer edge length, or conversely, delayed closure occurring with a shorter edge length (Table 1; S3 Fig). By increasing the coordinated directionality parameter within our model, wounds close faster with longer edge lengths due to the presence of more leader cell-like protrusions, replicating experimental observations in Piezo1-cKO monolayers (Fig 2H). On the other hand, under low coordinated directionality parameter conditions cells migrate more aimlessly, with formation of fewer leader-cell like protrusions along the wound edge and with inhibited closure, similar to observations from Piezo1-GoF and Yoda1-treated wounds (Fig 2H). Taken together, our parameter study predicts that while other model parameters, including retraction strength, affect keratinocyte migration, coordinated directionality plays a key role in modeling PIEZO1 inhibition of keratinocyte reepithelialization.

PIEZO1 activity is predicted to regulate wound closure by hindering coordinated directionality

Through numerical simulation, our modeling parameter study reveals how altering individual model parameters one at a time while keeping the remaining parameters at their base values (S4 Fig) affects wound closure. However, experimental results reveal that PIEZO1 activity may alter more than one model parameter, which may generate compensating effects that reduce the contribution of coordinated directionality in the experimental setting. Therefore, we sought to further constrain the mathematical model by incorporating model parameters derived from experimental data. To this end, we utilized and expanded upon analyses performed on single migrating keratinocytes in our previous study [22], to compile an experimental dataset characterizing PIEZO1’s effect on: cell motility, retraction duration, inter-retraction duration and cell retraction strength (Fig 3A and S5 Fig; Table 2). Cell motility parameters were calculated by extracting cell speed information from single cell tracking experiments which were previously performed using single Piezo1-cKO and Piezo1-GoF keratinocytes [22] (Fig 3A, left and S5 Fig; Table 1). We expanded upon this work by also tracking individually migrating Yoda1-treated and DMSO-treated keratinocytes to incorporate the effect of Yoda1 on cell motility into our model predictions. Similar to our observations in Piezo1-GoF keratinocytes, Yoda1 treatment had no effect on the motility of single migrating keratinocytes compared to DMSO-treated control cells (S6 Fig). To find the average duration of retractions and intervals between successive retractions for all experimental conditions (Piezo1-cKO, Piezo1-GoF, Yoda1-treatment and the respective controls), we utilized two analysis methods performed in our previous study [22]: (1) kymographs (Fig 3A, right), which graphically depict the retraction and inter-retraction durations of the leading edge of migrating keratinocytes, and (2) a cell protrusion quantification software, ADAPT [43], which quantifies the strength of retraction events at the leading edge. Thus, from these measurements (Fig 3A and S5 Fig; Table 2), we can calibrate our model parameters based on experimental measurements, enabling us to make experimentally relevant predictions regarding PIEZO1’s influence on wound closure behavior.

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Fig 3. PIEZO1 activity is predicted to regulate wound closure by hindering coordinated directionality.

(A) Schematic depicting experimentally measured features used to generate the single cell migration dataset. Left, representative still image of migrating keratinocyte with overlaid cell trajectory. Trajectory is derived from tracking cell motility during time-lapse experiments. Color denotes passage of time such that yellow is the starting position and purple denotes track end position. Cell boundary is in white. Scale bar = 100 μm. Kymographs (right) taken at the leading edge of migrating cells (e.g., similar to black box in the left image) are used to obtain information regarding inter-retraction duration and retraction duration. The cell protrusion quantification software, ADAPT [43] was used to gain information regarding retraction strength. Scale bar = 10 μm, Time bar = 5 min. (B) Cumming plots showing simulation results using the calibrated model (^CM) to predict how PIEZO1 affects normalized wound closure (left plots) and wound edge length (right plots) in simulated Control_GoF monolayers (dark gray), Piezo1-GoF monolayers without altered coordinated directionality parameters (white), and Piezo1-GoF monolayers with coordinated directionality decreased (green). See Methods Section Model parameter adjustment for the details. (C) Similar to B but using simulation results from DMSO-treated monolayers (black), Yoda1-treated monolayers without altered coordinated directionality parameters (white), and Yoda1-treated monolayers with coordinated directionality decreased (red). (D) Similar to B but using simulation results from Control_cKO monolayers (light gray), Piezo1-cKO monolayers without altered coordinated directionality parameters (white), and Piezo1-cKO monolayers with coordinated directionality increased (purple). In B-D, n = 100 simulation results for each condition, and ^CM denotes “Calibrated Model”. To account for differences between control cases, data are normalized by rescaling to the mean of the corresponding control. Larger normalized wound closure indicates faster wound closure, while a smaller normalized wound closure indicates slower wound closure. Similarly, a larger normalized edge length indicates a more featured wound while a smaller normalized edge length indicates a flatter or less featured wound. Black check marks at the top of each plot condition indicate that simulation results match experimental trends while a red cross indicates simulation fails to match the experiment trends. See also Table 3. For comparison with experimental data see Fig 1B, 1G and 1H.

https://doi.org/10.1371/journal.pcbi.1011855.g003

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Table 2. PIEZO1 activity affects single cell migration.

Summary table presenting experimental results obtained from quantitative analysis of single cell migration experiments (e.g., kymograph, cell protrusion analyses, single cell tracking assays). A “+” indicates an increase, “−” indicates a decrease, and “∼” indicates no statistically significant change between Control and Test condition. All data aside from DMSO-treated and Yoda1-treated cell motility (S6 Fig) was initially published in [22]. Actual data values for each condition can be found listed in S5 Fig.

https://doi.org/10.1371/journal.pcbi.1011855.t002

To calibrate our model, we created a respective simulation control for each experimental condition (Piezo1-cKO, Piezo1-GoF and Yoda1-treated) by fixing the values of model parameters to a basecase, where the frequency of retraction was set to be the same as the corresponding experimental control. For a given experimental condition, the model parameters related to retraction (retraction duration, inter-retraction duration, retraction strength) and cell motility were adjusted from the control condition by the same proportions as their experimentally-measured changes relative to the control condition (Fig 3A and S5 Fig; Table 2). In particular, the mean retraction and inter-retraction durations μ_r and μ_nr, the cell motility d and the mean retraction strength μ_s are changed proportionally in the model (see Methods Section Model parameter adjustment for details). With cell-cell adhesion and coordinated directionality unchanged compared to Control_GoF, we find that while we can replicate simulated monolayers of Piezo1-GoF keratinocytes having slower wound closure compared to simulated Control_GoF monolayers, we fail to observe the expected decreasing change in leader cell-like protrusions as indicated by a smaller simulated monolayer edge length (Fig 3B). However, by lowering the collective migration parameter of coordinated directionality, we recapitulate the experimental phenotype of both a shorter edge length and slower wound closure in simulated Piezo1-GoF monolayers (Fig 3B). On the other hand, we see that model simulations calibrated by the single cell migration dataset for both Piezo1-cKO and Yoda1-treated keratinocytes reproduce the expected experimental trends; however, by incorporating changes to coordinated directionality we observe a stronger effect (Fig 3C and 3D). Notably, we observe that adjustment of cell-cell adhesion parameters, another model parameter integral to collective migration, fails to replicate all experimental results, reinforcing that coordinated directionality plays a primary role in PIEZO1’s effect on reepithelialization (Table 3; S7 Fig). The more retraction regions generated, the slower the wound healing process and the longer the wound edge length. Matching the experimental results when PIEZO1 signaling is upregulated still required a decrease in coordinated directionality. Taken together, these studies demonstrate that only by including alterations to coordinated directionality are we able to mimic all experimental phenotypes.

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Table 3. Coordinated directionality recovers monolayer closure behavior from single cell data.

Summary table depicting simulation results using the calibrated model (^CM) to predict how PIEZO1 affects normalized wound closure and normalized edge length with altered adhesion and coordinated directionality parameters. A “+” indicates a parameter set has a predicted increase upon an experimental measure while a “−” indicates a predicted decrease. Double signs (+ + /−−) represent a stronger observed effect on the simulated measure than single signs (+ /−). Red font and cross mark indicate that model predictions calibrated by the “Single Cell Migration” dataset do not match experimental trends (Table 1), while a check mark ✔indicates that model predictions are consistent with experimental results. See also Fig 3, S7 and S8 Figs.

https://doi.org/10.1371/journal.pcbi.1011855.t003

To test whether the effect of higher PIEZO1 activity hindering coordinated directionality is sensitive to the details of the mathematical model of cell-cell adhesion, we also considered a phenomenological continuum modeling framework (see Section 7 in S1 Text for details) in which the diffusion coefficient was assumed to be a decreasing function of cell-cell adhesion instead of upscaling from a discrete model. This alternative approach follows that of Amereh et al., 2021 [44] and does not rely on upscaling from a discrete model. We repeated all the simulations using this new model of adhesion and found that the results are qualitatively consistent with the upscaled adhesion model we originally considered (S14 and S15 Figs), supporting the idea that the hindering effect of cell-cell adhesion can be modeled in various ways without altering the qualitative results.

We also tested the sensitivity of our conclusion regarding the role of PIEZO1 on coordinated directionality with respect to the details of the mathematical model of cell motilities and retraction processes. We re-calibrated the model using cell motilities from monolayer experiments (S9 Fig) rather than single cell experiments (S5 Fig). Consistent with the single cell data, cell motility within the monolayer increased in Piezo1-cKO compared to Control_cKO. In Piezo1-GoF and Yoda1-treated conditions, the cell motilities decreased in the monolayer (compared to respective controls), which is different from the behavior of single cells where the motilities were the same. Nevertheless, the re-calibrated model still predicts that PIEZO1 activity decreases coordinated directionality (see Section 8 in S1 Text for details).

We also varied the magnitudes of the retraction processes (retraction duration, inter-retraction duration and retraction strength). Because our previous experimental work [22] indicated qualitative consistency between single-cell and monolayer retraction processes but did not measure these features quantitatively in monolayers, we tested three different magnitudes of these processes in the model (S18 Fig), rather than using magnitudes drawn directly from the single cell experiments as previously done. The results still predict that PIEZO1 hinders coordinated directionality (S18 Fig, also see Section 8 in S1 Text for details).

Thus, our model predicts that PIEZO1 activity affects coordinated directionality within monolayers such that increased PIEZO1 activity inhibits the cells ability to move cohesively during collective migration, ultimately delaying wound closure. On the other hand, in monolayers which lack PIEZO1 expression, cells are predicted to have stronger directionality signals and recruit more follower cells to close the wound faster.

PIEZO1 activity inhibits persistence of direction during keratinocyte collective migration

To test our model’s prediction we first utilized a cell tracking assay to examine the motility of individual cells during collective migration. To track the movement of individual cells within monolayers we utilized the live-cell DNA stain SiR-Hoechst to label individual nuclei within monolayers [45]. After imaging collective cell migration over the course of several hours, we tracked the movement of individual nuclei and analyzed the resulting cell trajectories (Fig 4A–4C). The mean squared displacement (MSD) is a common metric for analyzing cell displacement as a function of time. Replicating our single cell migration observations [22], we observe that individual tracked nuclei within Piezo1-cKO monolayers have MSDs that are greater than that of Control_cKO cells, demonstrating a larger area explored (Fig 4D). Measurement of the instantaneous cellular speed reveals that, similar to our previous observations, Piezo1-cKO cells migrate faster relative to littermate Control_cKO cells (S9(A) Fig). On the other hand, cells from both Piezo1-GoF and Yoda1-treated monolayers have MSDs that are significantly smaller (Fig 4E and 4F). This effect is distinct from our observation in single migrating cells, where we observed that Piezo1-GoF keratinocytes migrate farther than Control_GoF cells [22], and that Yoda1-treatment has no difference relative to DMSO-treated control cells (S6(A) Fig). Moreover, in both Piezo1-GoF and Yoda1-treated monolayers we observe that increased PIEZO1 activity inhibits migration speed (S9(B) and S9(C) Fig). This observation also differs from our single cell migration observations in which both Piezo1-GoF and Yoda1-treated keratinocytes have no difference in migration speed compared to respective control cells. Our observed differences for PIEZO1’s effect on speed and MSD between single cell and collective migration results may be attributed to additional mechanical information from cell-cell interactions during collective migration affecting activation of PIEZO1.

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Fig 4. PIEZO1 activity inhibits persistence of direction during keratinocyte collective migration.

(A-C) Representative field of view depicting individual cell trajectories derived from tracking (A) Control_cKO (left) and Piezo1-cKO (right) keratinocytes, (B) Control_GoF (left) and Piezo1-GoF (right) keratinocytes, and (C) DMSO-treated (left) and 4 μM Yoda1-treated (right) keratinocytes during collective migration experiments. Trajectory color depicts individual cell trajectories. Scale bar = 100 μm. (D-F) Average mean squared displacement (MSD) plotted as a function of time for: (D) Control_cKO (gray) and Piezo1-cKO (purple) keratinocytes, (E) Control_GoF (gray) and Piezo1-GoF (green) keratinocytes, and (F) DMSO-treated (gray) and 4 μM Yoda1-treated (red) keratinocytes. All error bars plotted as SEM, in some instances error bars are smaller than symbols. (G-I) Average velocity autocorrelation measurement of: (G) Control_cKO (gray) and Piezo1-cKO (purple) keratinocytes, (H) Control_GoF (gray) and Piezo1-GoF (green) keratinocytes, and (I) DMSO-treated (gray) and 4 μM Yoda1-treated keratinocytes, plotted as a function of time (* denotes p value<0.0001 as calculated via Kolmogorov–Smirnov test). For Control_cKO (n = 66 unique fields of view) and Piezo1-cKO (n = 85 unique fields of view) data plotted in A, D, G, images taken from three independent experiments. For Control_GoF (n = 56 unique fields of view) and Piezo1-GoF (n = 51 unique fields of view) data plotted in B, E, H, images taken from four independent experiments. For DMSO-treated (n = 32 unique fields of view) and 4 μM Yoda1-treated (n = 31 unique fields of view) keratinocyte data plotted in C, F, I, images taken from three independent experiments. Plotted n denotes the number of individual cell trajectories. See also S9 Fig.

https://doi.org/10.1371/journal.pcbi.1011855.g004

Since coordinated directionality can, in part, be inferred by how straight the trajectories of cells in a collectively migrating group are, we measured the directional persistence of individual cell trajectories. While coordinated directionality refers to how cohesively cells migrate in a similar direction, directional persistence refers to the directed migration of individual cells or, more simply, how straight individual cell trajectories are. Notably, these two elements are often seen to co-occur in monolayers which show increased wound closure efficiency [46]. The directional persistence of a cell can be quantified by measuring the velocity autocorrelation of cell trajectories [47]. The randomness in direction of a cell’s trajectory is indicated by how rapidly its velocity autocorrelation function decays: autocorrelation curves which decay slower indicate cells that have straighter migration trajectories. Measurement of the velocity autocorrelation shows that Piezo1-cKO keratinocytes migrating in cell monolayers move straighter than Control_cKO cells (Fig 4G), similar to our previous findings in single migrating cells. In both Piezo1-GoF and Yoda1-treated keratinocytes, cells move less straight than their respective controls (Fig 4H and 4I). This finding also differs from findings in single cell migration results wherein Yoda1-treatment does not change directional persistence (S6(C) Fig) while the Piezo1-GoF mutation induces straighter trajectories during single cell migration [22].

Taken together, our results show that PIEZO1 activity inversely correlates with both cell speed and the persistence of migration direction during keratinocyte collective migration. Our observation that the directional persistence of individual keratinocytes within a monolayer is inhibited by PIEZO1 activity during collective cell migration provides initial support for our model’s prediction that coordinated directionality is affected by PIEZO1 activity.

Increased PIEZO1 activity inhibits the coordination of cellular motion

The coordinated movement of keratinocytes during wound reepithelialization depends on the large-scale interactions of multiple cells as they work together to close wounds. While tracking individual cells in a monolayer provides useful information regarding the locomotion of individual cells, it does not fully describe the dynamics of collectively migrating cells. To further validate our model’s prediction that PIEZO1 activity inhibits coordinated directionality and to characterize the effect of PIEZO1 on large scale cellular interactions during wound closure we utilized particle image velocimetry (PIV). PIV is an optical method of flow visualization which allows us to dynamically map the velocity fields of migrating keratinocytes within a monolayer during wound closure [20, 48, 49]. By isolating the individual velocity vectors comprising a monolayer’s vector field and mapping the frequency of vector directions for samples from different conditions (e.g., Piezo1-cKO, Piezo1-GoF, and Yoda1-treatment), we can visualize how PIEZO1 affects the coordinated directionality and overall coordination of motion between cells during wound closure (Fig 5A–5C). Probability density distributions of velocity directions from Fig 5A–5C illustrate that Piezo1-cKO cells flow towards the wound margin (denoted by 0 degrees) to a greater extent than littermate Control_cKO cells (Fig 5D). Conversely, Piezo1-GoF and Yoda1-treated cells flow towards the wound margin to a lesser extent than their corresponding Controls (Fig 5E and 5F).

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Fig 5. Increased PIEZO1 activity inhibits coordinated cellular motion.

(A-C) Representative mean Particle Image Velocimetry (PIV) flow fields derived from time-lapse images of labeled nuclei from collectively migrating monolayers of: (A) Control_cKO (Top) and Piezo1-cKO (Bottom) keratinocytes, (B) Control_GoF (Top) and Piezo1-GoF (Bottom) keratinocytes, and (C) DMSO-treated (top) and 4 μM Yoda1-treated keratinocytes (Bottom) during time-lapse scratch assay experiments. An individual flow field comprises either the upper or lower monolayer sheet of a scratch assay. Flow fields are oriented such that for the Y-direction, 0 μm is positioned within the cell free region. (D-F) Distribution plots showing the probability density of velocity vector direction for: (D) Control_cKO (gray; κ = 0.51) and Piezo1-cKO (purple; κ = 0.71) monolayers, (E) Control_GoF (gray; κ = 1.01) and Piezo1-GoF (green; κ = 0.71) monolayers, and (F) DMSO-treated (gray; κ = 0.61) and Yoda1-treated (red; κ = 0.30) monolayers. The curves depicted in the figure represent the fitting of von Mises distributions, where a smaller reported κ corresponds to less directed migration while a larger κ indicates an increase in directed migration. For D-F, p value calculated by Chi-squared test. (G) Cummings plot showing the mean angular deviation, or the variability in velocity direction isolated from PIV flow fields in: Control_cKO vs. Piezo1-cKO monolayers (left; p value calculated via two-sample t-test; Cohen’s d = -0.7), Control_GoF vs. Piezo1-GoF monolayers (middle; p value calculated via two-sample t-test; Cohen’s d = 0.43) or DMSO-treated vs. 4 μM Yoda1-treated monolayers (right; p value calculated via two-sample t-test; Cohen’s d = 1.14). Data are normalized such that 1 indicates highly random velocity directions and 0 indicates highly uniform velocity directions. (H-J) Spatial autocorrelation, C(Δr), of the radial velocity component, which is a measure of the spatial coordination of neighboring cells in monolayers, plotted as a function of increasing length scales of: (H) Control_cKO (gray) and Piezo1-cKO (purple) keratinocytes, (I) Control_GoF (gray) and Piezo1-GoF (green) keratinocytes, and (J) DMSO-treated (gray) and Yoda1-treated (red) keratinocytes. For H, I, J * denotes a statistically significant difference, and ns denotes not statistically significant as determined by one way ANOVA test. Specific p values for plotted points can be found in S11 Fig. See also S10 Fig. (K) Local spatial coordination, C(Δr = 150 μm), of keratinocytes where the correlation value is set at 150 μm to measure the coordination of motion with neighboring cells in: Control_cKO vs. Piezo1-cKO monolayers (left; p value calculated via two-sample t-test; Cohen’s d = 0.62), Control_GoF vs. Piezo1-GoF monolayers (middle; p value calculated via two-sample t-test; Cohen’s d = -0.7) or DMSO-treated vs. 4 μM Yoda1-treated monolayers (right; p value calculated via Mann-Whitney test; Cohen’s d = -1.4). n in B, C, E, F, H, I, J and K denotes the number of monolayer sheets imaged. For Control_cKO and Piezo1-cKO data plotted in A, D, G (left), H, and K (left), images are taken from three independent experiments. For Control_GoF and Piezo1-GoF data plotted in B, E, H (middle), I, and K (middle), images are taken from four independent experiments. For DMSO-treated and 4 μM Yoda1-treated keratinocyte data plotted in C, F, H (right), J, and K (right), images are taken from two independent experiments.

https://doi.org/10.1371/journal.pcbi.1011855.g005

To fit our vector direction datasets, we employed the von Mises distribution by minimizing the mean squared error with the von Mises probability density function (Eq 21 in the Methods Section). The resulting fitted curves (Fig 5D–5F) provide the best approximation of the data by adjusting the distribution parameters, including the mean (μ) and the concentration (κ, indicating the strength of directed migration in our experimental context). A smaller κ value corresponds to a flatter bell curve and a distribution closer to uniform, indicating less directed migration. Conversely, a larger κ value results in a sharper bump in the probability density function, indicating an increase in directed migration. We find that Piezo1-cKO cells show a higher κ (κ = 0.71) than Control_cKO (κ = 0.51) indicating that Piezo1-cKO cells move with increased coordination relative to Control_cKO cells (Fig 5D). On the other hand, we find that both Piezo1-GoF and Yoda1-treated monolayers show a loss in directed migration as illustrated by the broader distribution of isolated vector directions and a lower calculated κ value for experimental conditions (κ = 0.71 for Piezo1-GoF, κ = 0.30 for Yoda1-treated) relative to the respective control populations (κ = 1.01 for Control_GoF, κ = 0.61 for DMSO-treated; Fig 5E and 5F).

PIEZO1’s effect on coordinated directionality can be further parameterized by measuring the angular deviation, or the variability in velocity direction for all vectors within a PIV vector field. Thus, the range of the angular deviation indicates how coordinated the direction of cellular motion is within an entire monolayer field of view such that a higher angular deviation indicates less coordination. We observe that Piezo1-cKO monolayers have a lower average angular deviation value relative to Control_cKO monolayers, indicating a smaller spread in velocity direction (Fig 5G, left). This is opposed to Piezo1-GoF and Yoda1-treated monolayers which both show a higher angular deviation than the respective controls, further signifying that PIEZO1 activity promotes less directional migration (Fig 5G, middle, right). We note that any difference in the angular deviation between control conditions can likely be attributed to different genetic backgrounds between control conditions.

Recognizing that the synchronized movement of groups of cells during collective migration relies upon the coordination of migration direction across individual cells, we next looked at how PIEZO1 activity affects the distance over which cells align, or correlate, their motion within a monolayer. To do this, we determine how alike the velocity of nearby cells is by calculating the average spatial autocorrelation of velocity vectors, (C(Δr)), which measures the degree of correlation between velocity vectors of cells at increasing length scales within a monolayer (Fig 5H–5K). If keratinocytes within a monolayer are migrating together with high directional uniformity we expect a higher autocorrelation value, while a lower autocorrelation value indicates that individual keratinocytes are moving more independently of one another. Therefore, the decay rate of the average spatial autocorrelation curve indicates how coordinated a given cell’s direction of motion is to that of another cell located at iteratively increasing distances away (Fig 5H–5J). Measurement of the spatial autocorrelation in Piezo1-cKO and Control_cKO monolayers illustrate that Piezo1-cKO cells show an increase in coordination with cells at greater distances relative to Control_cKO cells, as indicated by a slower decay of the average Piezo1-cKO autocorrelation curve (Fig 5H). The length constant, or distance at which the spatial autocorrelation reaches a value of 0.37, was estimated by fitting an exponential curve to our experimental dataset. Calculations of the length constant for Piezo1-cKO cells show an increase in coordination by 21.47 μm farther than Control_cKO (Fig 5H and S10 Fig). To further quantify the coordination between nearby cells we measure the spatial autocorrelation values at 150 μm, the distance of a few cell-lengths away. Measurement of local autocorrelation values in Piezo1-cKO keratinocytes cells show an increased level of coordination of locomotion with neighboring cells compared to cells in Control_cKO monolayers (Fig 5H and 5K). In contrast, both Piezo1-GoF and Yoda1-treated monolayers exhibit less coordinated movement with neighboring cells when compared to control cells (Fig 5I–5K). Length constants in Yoda1-treated and Piezo1-GoF cells show a 58.560 μm and 85.54 μm decrease, respectively, in their coordination of motion relative to the respective control monolayers (Fig 5I and 5J and S10 Fig). Therefore, we find that PIEZO1 activity disrupts the distance over which cells coordinate their motion during wound closure which inhibits the efficiency of collective migration.

Here, we have uncovered the inhibitory impact of PIEZO1 activity on coordinated cell directionality within a monolayer. This discovery naturally raises additional questions about the consequences of heterogeneous PIEZO1 activity, for instance within monolayers comprising cells with high and low PIEZO1 activity. To explore this, we extended the original model to investigate the impact of heterogeneous PIEZO1 activity in monolayers (we describe the model in Section 9 in S1 Text for details). This new model considers the migration of two cell types, each governed by its own set of equations with distinct model parameters, while interacting through cell-cell adhesion and volume-filling effects. Through simulations involving Control_cKO and Piezo1-cKO, as well as Control_GoF and Piezo1-GoF, we observed a correlation between the distribution of edge cells in the monolayers and the level of PIEZO1 activity. In particular, cells with reduced PIEZO1 activity (e.g., Piezo1-cKO) are over-represented at the wound edge (S19(B) Fig), which is consistent with the faster wound closure observed in homogeneous Piezo1-cKO monolayers. In contrast, cells with enhanced PIEZO1 activity (e.g., Piezo1-GoF) are underrepresented at the leading edge of the monolayer (S19(D) Fig), which is also consistent with the slower wound closure observed in homogeneous Piezo1-GoF monolayers. Testing these model predictions in experiments is deferred to future work.

Taken together, our experimental findings support our model predictions that PIEZO1 inhibits coordinated directionality during collective migration. Moreover, we identify that PIEZO1 activity negatively contributes to leader cell formation and the distance by which keratinocytes can coordinate their migration during 2D epithelial sheet migration.

Ethics statement

All studies were approved by the Institutional Animal Care and Use Committee of University of California at Irvine and The Scripps Research Institute and performed within their guidelines.

Animals

Keratinocyte samples from Piezo1-cKO and Piezo1-GoF mice were a gift from Dr. Ardem Patapoutian’s lab, the Scripps Research Institute. Piezo1-tdTomato reporter mice (Piezo1-tdTomato; JAX stock 029214), Piezo1-cKO and Piezo1-GoF mice were generated in previous studies [22, 81].

Keratinocyte isolation and culture

Primary keratinocytes were isolated from the upper dorsal skin of P0-P5 mice as previously described [22]. Briefly, dissected tissue was allowed to dissociate for 15–18 hours. After dissociation, the epidermis was separated and incubated in Accutase (CellnTec CnT-Accutase-100) for 30 minutes at room temperature. Subsequently, the epidermis was transferred to a dish of CnT-Pr media (CellnTec), supplemented with 10% FBS and 1% penicillin/streptomycin where the epidermis was minced and then agitated using a stir plate for 30 min. After agitation, cells were strained through a 70 μm cell strainer (Falcon). Strained cells were spun down and resuspended in CnT-Pr media (CellnTec) supplemented with ISO-50 (1:1000) (CellnTec) and Gentamicin (50 μg/ml) (Thermo Fisher).

Isolated keratinocytes were seeded directly onto the glass region of #1.5 glass-bottom dishes (Mat-Tek Corporation) coated with 10 μg/ml fibronectin (Fisher Scientific, CB-40008A). For single cell migration experiments, isolated cells were sparsely seeded onto the glass region at 1.5 × 10⁴ cells/dish while for monolayer scratch assay experiments, isolated cells were densely seeded onto the glass region at a density of 1.5x10⁵ cells/dish. One day after seeding, CnT-Pr supplemented culture media (see above) was switched to Cnt-Pr-D media (CellnTec) to promote keratinocyte differentiation. Keratinocytes were imaged 3 days after primary isolation, allowing at least 2 days for keratinocyte differentiation in Cnt-Pr-D media (CellnTec).

Microscopy

For in vitro image acquisition, an Olympus IX83-ZDC inverted microscope equipped with a SOLA light engine (Lumencor) was utilized. For time-lapse imaging experiments, a full enclosure stage-top incubator system (Tokai Hit) enabled cells to be imaged at 37°C with 5% CO₂ to maintain optimal cell health. μManager, an open-source microscopy controller software, was used for microscope hardware control and image acquisition [82, 83]. For all experimental data, images were taken using a UPlanSApo 10× dry objective with a numerical aperture of 0.40 and acquired using a Hamamatsu Flash 4.0 v2+ scientific CMOS camera.

Immunofluorescence staining

For immunostaining of healing monolayers in S12 Fig, scratch wounds were generated in confluent monolayers of isolated keratinocytes and then treated with either 4 μM Yoda1, or the equivalent concentration of the solvent DMSO, before allowing the monolayers to collectively migrate. 24 hours after initial wounding the monolayers, monolayers were fixed and then immunostained for total levels of Rac1 (S12 Fig, left) and RhoA (S12 Fig, right). Immunostaining was performed as previously described [84] using the following antibodies: Mouse anti-Rac1 (Millipore Cat#05–389-25UG, 1:200), Rabbit anti-RhoA (Proteintech Cat#10749–1-AP, 1:100), Donkey anti-Mouse 647 (Abcam Cat#AB150107, 1:500), Goat anti-Rabbit 488 (Life Sciences Cat#A32731, 1:500). Nuclei were stained by Hoechst (Invitrogen Cat#H1399) at 1μg/mL for 5 minutes.

Single cell migration assay

As previously described [22], time lapse sequences of DIC images were taken at 5 minute intervals. In brief, sparsely seeded keratinocytes were allowed to migrate for 16.67 hr at 37°C with 5% CO₂ in fibronectin-coated glass-bottom dishes. Cell centroids were tracked using Cell Tracker (https://celltracker.website/index.html, Piccinini2016-dx) and resulting trajectories were analyzed using the cell trajectory analysis software, DiPer [47].

Wound closure assay

Primary keratinocytes were cultured for 3 days until they formed a confluent monolayer. Prior to imaging experiments, cell nuclei were labeled by addition of SiR-Hoechst [45] (1 μM; Cytoskeleton Inc.) to Cnt-Pr-D+1.2 mM Ca²⁺ bath media for 1 hour prior to imaging. As previously described, monolayer scratches were generated using a 10 μl pipette tip and resulting cell debris was removed by performing three successive washes of culturing media [22, 85]. Time-lapse imaging series of wound closure were acquired by taking sequential DIC and fluorescence images at multiple positions. 1 μM SiR-Hoechst remained in the Cnt-Pr-D+1.2 mM Ca²⁺ bath media throughout the imaging period. For Yoda1 experiments, 4 μM Yoda1 or, as a control, the equivalent concentration of DMSO was supplemented to bath media prior to imaging. Leader cells display broad lamellipodia and are located at the front of protrusions along the leading edge of healing monolayers. During identification, leader cells were identified by manually reviewing time lapse image series and counting the number of cells located at the front of fingering protrusions at the leading edge which display increased polarization and large, prominent lamellipodia. Example leader cells identified during manual review are denoted by white arrows within S1 Fig. The number of leader cell formations is reported at the time point when either the wound interfaces touch or the imaging period finishes.

Wound edge length analysis

Monolayer sheets were segmented from images taken during wound closure assays using a custom deep-learning based U-net architecture written in Python [86] (https://github.com/Pathak-Lab/PIEZO1-Collective-Migration). The length of the segmented wound edge was calculated by taking the cumulative euclidean distance between all detected pixel positions along the segmented monolayer leading edge. Due to any possible differences in edge length which might arise when manually making scratches in monolayers, each field of view’s edge length was normalized by dividing the edge length at T_final, the time point when either the wound interfaces touch or the imaging period finishes by T₀, the starting edge length at the starting time point for a field of view. This normalized edge length was used as a measure of the prevalence of leader cells along the wound edge for a given condition.

Image analysis

Using the open-source image analysis software Fiji [87] the signal-to-noise ratio of SiR-Hoechst images was increased using Contrast Limited Adaptive Histogram Equalization (CLAHE) (https://imagej.net/plugins/clahe) prior to further analysis. For some images which had poor labeling of SiR-Hoechst, the denoising algorithm Noise2Void was also used to further increase the signal-to-noise ratio of nuclei images [88] (S13 Fig).

Individual cell tracking

We combined the deep learning-based object detection method StarDist with the cell tracking software TrackMate to perform automated tracking of cells within monolayers [89–91]. Cell trajectories harvested using TrackMate were then exported for further analysis. Due to the technical limitations surrounding Microsoft Excel’s ability to handle large datasets, we developed Cell_Pyper (https://github.com/Pathak-Lab/PIEZO1-Collective-Migration) a Pythonic analysis pipeline based on the open-source algorithm DiPer [47] to analyze the Mean Squared Displacement (MSD), Speed and Velocity autocorrelation of harvested cell trajectories.

For efficient computation of a trajectory’s MSD, MSDs are computed according to Eq 18 (Eq. 4.11 in [92]) where r(k) ≡ r(kΔt) is a cell trajectory consisting of N_t timepoints and the MSD is calculated for timestep m. (18)

As described by Gorelik & Gautreau (Eq. 6 and 7 in [47], Velocity Autocorrelation analysis is calculated according to Eqs 19 and 20 for a trajectory consisting of N timepoints with a time-step of Δt = 5 min. A normalization factor (Norm; Eq 19) is initially calculated for velocity vector v_i with starting coordinates (x_i, y_i) which is used to calculate the average velocity autocorrelation coefficient v_ac with step size n. (19) (20)

Particle image velocimetry analysis

Particle Image Velocimetry (PIV) analysis was performed using the Python implementation of OpenPIV [93] (https://github.com/Pathak-Lab/PIEZO1-Collective-Migration). We use multiple passes of interrogation window sizes, initially using first-pass calculations with a 64 pixel x 64 pixel (55.2 μm x 55.2 μm) window followed by two iterations of 32 x 32 (27.6 μm x 27.6 μm) pixel windows and two iterations of 16 x 16 pixel (13.8 μm x 13.8 μm) windows. Each interrogation window was computed with a 50% overlap. A signal-to-noise filter (Threshold = 1.3) was used on detected velocity vectors to remove any vector outliers. Outputs produced by OpenPIV analysis were then used to generate PIV flow fields as shown in Fig 5A–5C. Working from the flow fields, individual PIV vectors were isolated and PIV vector direction was calculated and normalized to 0° to account for differences in angles of scratches made in monolayers (https://github.com/Pathak-Lab/PIEZO1-Collective-Migration). Vector direction distributions are illustrated as the probability density distribution across experimental replicates in Fig 5D–5F. The von Mises distribution was employed to fit vector direction datasets by minimizing the mean squared error between the vector direction data and the von Mises probability density function. The resulting fitted curves represent the best approximation of the data by adjusting the parameters μ (mean) and κ (concentration or strength) of the von Mises distribution. The parameter μ represents the location where the distribution is clustered, while parameter κ indicates the level of directionality in our experimental context. The probability density function of the von Mises distribution for the vector direction angle x is expressed as: (21) where I₀(κ) represents the modified Bessel function of the first kind with order 0 (22) which is selected to ensure the distribution integrates to unity: (23)

The variance of PIV vector directions within a field of view was calculated as the mean angular deviation, z, where z is defined in Eq 24 (Eq. 2 in [48]). Outputs of this equation are bounded such that zero indicates no variability in vector direction within a flow field and one indicates high variability in vector direction. (24)

The spatial autocorrelation function, C, is computed according to Eq 25 (Eq. 4 in [48]) using the radial velocity component of a given PIV vector, v, within a vector flow field at varying length scales, r. (25)

For measurement of the local autocorrelation in vector direction, the spatial autocorrelation at Δr = 150 μm was used to capture correlation of motion at multiple cell lengths. Length constants were calculated by using OriginLab to fit an exponential function whose exponent is a 2^nd order polynomial (Eq 26) to the spatial autocorrelation dataset and calculating the distance at which C(Δr) ≈ 0.37. (26)

Numerical scheme

In order to solve the governing equation (Eq 4), we firstly carry out a forward time discretization (with size Δt) on the left hand side ∂ρ/∂t by (ρ(x_i,j, t + Δt) − ρ(x_i,j, t))/Δt. In terms of space discretization (right hand side), the transitional probability is proved to be separable (Eq 3) in the discrete model, which allows us to work on the diffusion part and advection part separately: for the diffusion part, a natural discretization is directly given by the discrete model (e.g., centered finite differences); for the advection part, we apply a 2nd order weighted essentially non-oscillatory (WENO) method [94, 95] to discretize the equation. Hence, an explicit finite difference scheme was used to update the cell density at the nth time step iteratively on the simulation domain [0, 1] × [0, 1] until wound closure.

Model parameter adjustment

In Fig 3B–3D and S7 Fig, the experimentally derived “Single Cell Migration” dataset guides the changes in model parameters of retraction strength, retraction duration, inter-retraction duration and cell motility when PIEZO1 activity is altered. However, cell-cell adhesion and coordinated directionality were not measured directly in the experiments and instead are inferred by trying to match model and experimental results.

While both cell-cell adhesion and coordinated directionality are designed to range from 0 to 1 in our model, the feasible adhesion coefficient actually needs to be bounded above by 0.66 in order for the diffusivity to be positive definite (see Section 3 in S1 Text for detailed derivation). Since the dependence of wound closure rate and wound edge length with respect to individual model parameters was already numerically shown to be a monotonic function of these parameters (Fig 2G and 2H and S3 Fig), it is sufficient to directly use the extrema of the model parameters: 1 for increased coordinated directionality, 0.66 for an increased adhesion, and 0 in the case that coordinated directionality and/or adhesion is decreased. For example, when matching experiments and simulations requires an increased coordinated directionality, we take w_A = 1. Because of the dependency of the outcomes (wound closure rate and edge lengths), if increasing a model parameter to its maxima fails to match the experimental trends, it would be impossible to match with smaller values.

The base values of cell-cell adhesion and coordinated directionality are taken to be 0.2 and 0.4, respectively. In Fig 3B–3D, the adhesion coefficient is fixed at the value 0.2, while in S7 Fig, coordinated directionality is fixed at the value 0.4. The base values of the model parameters can be found in S4 Fig.

Statistical analysis

P values, statistical tests, and sample sizes are declared in the corresponding figures. All datasets were tested for normality using the Shapiro-Wilk test prior to statistical analysis. The two-sample t-test was used where data were modeled by a normal distribution and a nonparametric test was used in the case of non-normal distributions. Cumming estimation plots were generated and Cohen’s d value was calculated using the DABEST python [96] (https://github.com/ACCLAB/DABEST-python). The Cohen’s d effect size is presented as a bootstrap 95% confidence interval (95% CI) on a separate axes. p values for Fig 5G–5I are declared in S11 Fig.