Multi-cell migration is important for tissue development and repair. An experimentally accessible example of multi-cell migration is provided by the classic scratch-wound assay. In this assay, a confluent monolayer is `injured' by forcibly removing a strip of cells, and the remaining monolayer `heals' through some combination of cell migration, spreading and proliferation. The scratch wound has been used for decades as a model of wound healing and an assay of cell migration, however the mechanisms that underlie the coherent expansion of cells in the surviving monolayer are still debated. Here we develop an agent-based computational model that predicts the most robust characteristics of healing in scratch wounds. The cells in our model are simple mechanical agents that respond to cell contact by redirecting migration and slowing division. We imbued model cells with crawling and growth dynamics and measured for individual L1 fibroblasts and found that simulated recovery occurs in a steady, sheet-like and division-independent fashion to mimic healing by L1s. The lack of cohesion and biochemical cell-cell communication in the model suggests that these factors are not strictly necessary for cells to migrate as a group. Instead, our analysis suggests that steady sheet migration can be explained by cell spreading in the monolayer.
Multi-cell migration is of great significance in biology. Coordinated cellular movements underlie tissue formation during embryogenesis (Martin and Parkhurst, 2004) and the repair of tissue after injury (Martin, 1997). Some metastatic cells migrate as a group (Hegerfeldt et al., 2002), and other cancers induce nearby endothelial cells to sprout new vasculature (Kim et al., 1993). Outside the animal kingdom, individual starving soil myxamoebae can gather together to form a collectively migrating multicellular slug (Bretschneider et al., 1999). Despite the widespread distribution and importance of multi-cell migration, in many contexts the mechanisms that coordinate the movement of cells as a group remain poorly understood. Particularly relevant as a model of sheet migration by epithelia, the classic scratch-wound assay (Todaro et al., 1965; Yarrow et al., 2005) is a valuable tool for studying multi-cell migration in vitro. In this assay, contact-inhibited cells are grown on a 2D surface until they form a confluent monolayer. A strip of cells is then removed by mechanically scraping the monolayer. The monolayer responds with cell spreading and migration into the denuded region until the wound is closed.
Although the scratch-wound assay has been used to help identify many molecules and signaling pathways important for cell migration, surprisingly few studies have attempted to reveal the impetus for the healing response. In one study, Sammak and colleagues demonstrated that a wave of elevated intracellular Ca2+ moves from leading cells rearward in scrape-wounded epithelial and endothelial sheets (Sammak et al., 1997). Because the Ca2+ waves die out in several seconds however, they cannot sustain a healing response that continues for many hours. To examine whether cellular injury was required to trigger the healing response, Block et al. used agarose strips to create gaps in epithelial monolayers (Block et al., 2004). Following strip removal, cellular sheets migrated at the same rates as sheets created in scraped monolayers. The findings of Block et al. support the idea that the release of spatial constraints can initiate the healing response, but they do not address the mechanism(s) that sustain collective migration in the scratch-wound assay.
In experiments on wounded epithelial monolayers, Farooqui and Fenteany eliminated several biochemical routes for cell-cell communication, including paracrine mechanisms and gap junction signaling, and found that wound healing rates were not significantly affected (Farooqui and Fenteany, 2005). The authors also discovered that epithelial cells, while maintaining apical contacts with the cells around them, appear to be actively and somewhat independently crawling during wound healing. With these data, Farooqui and Fenteany raised the possibility that crawling cells, even cells interior to the surviving monolayer, might be guided by a lower mechanical resistance in the direction of the wound. Some of the oldest observations from cell culture confirm that cells can sense physical contact and respond by slowing proliferation (Dulbecco and Stoker, 1970; Todaro et al., 1965; Turner and Sherratt, 2002) and redirecting migration (Abercrombie and Heaysman, 1954). It is also notable that epithelial cell cultures do not normally have free edges at steady state but instead expand to the physical limits of vessels in 2D cultures, and form balls, cysts and tubes in 3D cultures. Thus although it is reasonable that wound healing may be initiated and sustained by the sensation of free space, experimentally demonstrating this is difficult because eliminating all possible routes for biochemical communication is a seemingly impossible task.
Here we investigate the possibility of mechanical coordination in migrating sheets by comparing an agent-based computational model to scratch-wound experiments. In agent-based models, simulated cells modify their behavior (migration, death, division, etc.) according to internal and environmental variables and pre-programmed rules that may be deterministic or stochastic (Walker et al., 2004). The behavior of the system emerges from a great number of individual interactions between cells and their environment, and comparisons to the macroscopic behavior of experimental systems test the appropriateness of the underlying rules. To test the hypothesis of mechanical coordination, we created simulated cells that were simple mechanical agents; our cells can sense when they are crowded and respond with lower division rates and migration toward less crowded areas. We developed the model in conjunction with experiments on L1 fibroblasts for which we could directly measure many of the model parameters. We choose L1 fibroblasts because we found that they exhibit reproducible, contact-inhibited growth curves, that they reproducibly heal scratch wounds as a collective sheet, and that they do not create adherens junctions in confluent cultures (not shown). By neglecting cohesive junctions typical of epithelial sheets, we can address the question of collective migration in a model with few unknown processes and free parameters.
Surveying the vast literature on the scratch-wound assay for both epithelial and non-epithelial cells, two generalizations can be made: (1) after a short (2-4 hours) refractory period following wounding, rectangular wounds close at a steady rate (Nishio et al., 2005; Pullar et al., 2006; Richards et al., 2004; Shanley et al., 2004; Takeda et al., 2004); and (2) cell division increases in response to wounding (Todaro et al., 1965), but does not affect closure rates (Block et al., 2004; Farooqui and Fenteany, 2005; Sponsel et al., 1994). Grown in simulated culture to confluence and then `wounded' by deleting a strip of cells, populations of our agents reproduce these characteristics of collective cell migration, as well as other qualitative behaviors we observe for L1 cells. Quantitative agreement with L1 closure rates occurs if simulated cells avoid overlap through stochastic rather than deterministic rules. Our work suggests that the most robust features of sheet migration are explainable by the ability of cells to sense – and migrate away from – physical contacts.
Constructing model cells
We sought to construct an agent-based model in which simulated cells faithfully mimicked the measurable dynamics of individual L1 fibroblasts. Two relevant behaviors of L1 fibroblasts could be readily incorporated into the model: their motility in sparse cultures and their density-dependent division rates. Each of these can be described mathematically with key parameters assessed experimentally. A third behavior, the migration response to cell-cell contacts, is more enigmatic. Beyond the original descriptions of contact-inhibited motility by Abercrombie (Abercrombie and Heaysman, 1954), little has been done to describe the impact of cellular collisions on migration. Lacking details, we construct a minimalist description of cell-cell collisions and assume that our ability to match experiments (wound healing) will depend strongly on the reasonableness of our description.
Motility in sparse cultures
To characterize cell motility we conducted time-lapse video microscopy of low-density cultures at 37°C, and applied custom cell-tracking algorithms to acquire trajectories of hundreds of crawling cells. Considering single-cell motility as a semi-persistent random walk (Othmer et al., 1988), we calculated the mean squared displacement along each trajectory to determine the characteristic speed and persistence time (time between direction changes) for each cell. Distributions of both speed and persistence are both positively skewed (Fig. 1A,B), so we assigned simulated cells the median values of the distributions rather than the means.
For sparse motility, simulated cells can be represented as points on a two-dimensional surface whose coordinates change with each iteration of the simulation. To mimic the speed of L1 cells, each cell was advanced a distance sΔt per iteration, where s is the median L1 speed and Δt is the time interval that an iteration represents. To mimic the semi-persistent nature of L1 migration, each cell continued moving in one of 12 principal directions unless selected to choose a new direction randomly. The probability of selecting a new direction at each time step given by P(Δt)=1–e–(Δt/τ) where Δt is the time step and τ is the persistence time. With a time step of τ/100 or smaller, we found that the analysis of simulated trajectories reproduced the median speeds and persistences of L1 trajectories (Fig. 1C,D). The higher variability of the experimental cells is expected because the distribution includes both stochastic variations and phenotypic diversity. By contrast, all simulated cells are assigned the same underlying motility parameters and so variability can only arise from stochastic differences in trajectories.
To measure density-dependent division rates, we followed L1 cells in time-lapse microscopy (37°C) as they grew from sparse to confluent cultures for more than 5 days. We used custom, automated image-processing routines to extract cell densities every 30 minutes. Consistent with contact-inhibited growth, L1 cell density increased with time in a sigmoidal fashion (Fig. 2A). A transformed version of the growth curve was directly useful for our model. First, we calculated the per capita growth rate by dividing the local slope on the growth curve by the corresponding cell density. We then plotted the per capita growth rate as a function of the mean intercellular spacing. The resulting plot (Fig. 2B) has two separable phases, a phase of approximately constant per capita division rate at large cell spacings and a sharp decline in division rate at smaller cell spacings. Viewed in this way, the growth data suggest a range of dimensions over which cells display sensitivity to crowding. Fitting a line to each phase and extrapolating to their intersection gives an upper limit of ∼65 μm to the range and a minimum spacing where division stops of ∼47 μm. We assume that these numbers represent cell diameters as cells transition from spread morphologies at subconfluence to rounded morphologies at full confluence, and so we advance our description of cells in the model from points to circles containing compressible and incompressible regimes (Fig. 2C).
By allowing cells to overlap in their compressible domains, the intercellular spacing naturally decreases in simulations as cells become crowded. In this way we can directly apply the per capita division rates implied by the line fits in Fig. 2B to determine the probability per unit time that a particular cell divides (see Materials and Methods). Cells chosen to divide are replaced with two daughter cells each with half the diameter of the mother cell. Cells grow until their incompressible diameters reach 47 μm (see Materials and Methods). Simulated cultures grown with these rules produce per capita growth plots that mimic those of real cells provided the compressible zone extends between 2 and 9 μm beyond the incompressible core (Fig. 2D). Respecting the uncertainty in this value, we examine the effect of compressible zones ranging between 2 and 9 μm in the wound-healing simulations described below.
Motility in crowded cultures
While individual cell migration is well described as a semi-persistent random walk (Othmer et al., 1988), we found no comparable description of cell migration in crowded cultures where collisions between cells occur. Lacking guidance from the literature, we simply programmed simulated cells to move in a manner that minimizes overlap of their compressible dimension. Should contact occur between two sparse cells, both cells choose randomly from a range of unobstructed directions for their subsequent migration. This simple rule gives cells a behavior reminiscent of Abercrombie's description of contact-inhibited motility (Abercrombie and Heaysman, 1954) and results in simulated cultures that mimic real L1 cultures by growing sparsely rather than in colonies (supplementary material Movie 1). Simulated cells in crowded cultures may overlap neighbors in all directions and migrate to minimize, but cannot eliminate, overlap. Recognizing that the dimensions for cell sensitivity to division and migration could be different, we define `division' and `sensing' radii to represent these two zones of sensitivity respectively, and consider the effects of varying each radius for wound healing. In the text and figures that follow we use the notation x/y to identify simulated cells constructed with a sensing radius x and division radius y. Real cells round up as they approach confluence and repolarize to migrate away from obstacles. With a compressible annulus at their border, simulated cells have a somewhat fuzzy shape. `Overlapped' regions can be thought of as portions that deform because of contact with a neighbor, and as with real cells, this deformation is transduced into changes in cell behavior.
Having defined the rules that govern individual cell behaviors, we now examine the dynamics of a population of simulated cells arranged as a wounded monolayer. First we note that L1 cells, the cells that the model agents are designed to mimic, heal rectangular wounds at a steady rate that is independent of the amount of division within the monolayer or the size of the wound (Fig. 3A). Trial-to-trial variations in division rates occur naturally in our scratch-wound assays. We suspect that the variation arises because monolayers appearing similarly confluent can have small differences in intercellular spacing that map to dramatic differences in per capita division rate (Fig. 2B). To compare simulated and L1 wound healing, we imbued simulated cells with the properties of L1 cells (Table 1), grew them to near maximal density, and then deleted cells to match the geometry of denuded areas in L1 scratch-wound experiments. One figure (Fig. 3B,C), one table (supplementary material Table S1) and one supplemental movie (supplementary material Movie 2) examine the influence of cell parameters on simulated wound healing.
Most generally we find that simulated wound healing proceeds in two distinct phases: an early phase of steady edge advancement characterized by coherent cell expansion, followed by a phase with slowing edge advancement characterized by cell scatter. The two phases are most obvious in simulations with cell division turned off (supplementary material Movie 3; Fig. 4A,B). In these simulations it is clear that the duration of the early expansion phase depends on the value of the sensing radius, although the velocity of expansion is largely independent of this value (see also supplementary material Table S1). Turning on division extends the duration of the linear expansion phase – often eliminating the second phase entirely – but does not affect the velocity of linear expansion (see supplementary material Movie 2 and Fig. 3B). Similarly, increasing the intrinsic division rate tenfold has no effect on the linear expansion (Fig. 3B), nor does growing simulated cultures to a lower degree of confluence so that the division increases ∼twofold during healing (see supplementary material Table S2). Thus, like L1 cells, simulated monolayers begin healing with a steady rate of expansion that is independent of division. Increasing the intrinsic cell speed both shortens the duration of steady expansion (Fig. 3B) and increases its rate (Fig. 3B,C). Changes in persistence time have no impact on healing rates (Fig. 3B) because collisions that cause cells to alter their directions occur on much shorter time scales.
Although division does not influence the initial rate of healing, it does sustain linear expansion by replenishing cell overlap depleted by migration. With division present, the diffusive phase of wound healing does not occur unless the motility-sensing radius is smaller than the division radius (supplementary material Table S1 and Fig. 4B,C). In these cases, once cells escape the zone over which they influence each other's movements, division remains inhibited and cannot sustain the steady expansion of the monolayer. Cells near the boundary escape the slowing expansion of the bulk monolayer. Since L1 cells heal linearly until wound closure (Fig. 3A), we can eliminate cases where sensing radii are smaller than division radii as possible descriptors of L1 cells. Simulated cells without a sensing radius become rigid, diffusing objects that do not heal significantly on the time scale of L1 wound healing (Fig. 3B). Thus the rectification of individual cell movement via a contact-sensitive border is an absolute requirement for our simulations to mimic the linearity and time scale of L1 wound healing.
Our examinations so far explain the steady, division-independent, healing rates seen in L1 experiments as an emergent property of single-cell dynamics. Despite this qualitative agreement, there is a quantitative disagreement in the rate of wound closure between the simulation and the experiment: model cells imbued with the speed, persistence and sensing/division radii that maintain linearity until closure, heal wounds at a rate ∼35% faster than L1 cells (supplementary material Table S1; Fig. 3C). In additional wound-healing experiments we blocked L1 division with the mitotic inhibitor mitomycin C (MMC), but we also found that MMC slowed cell speeds and increased persistence times for sparse L1 cells. Employing the MMC median speed and persistence time in wound-healing simulations with division turned off, we again found that simulations were ∼33% faster than the corresponding MMC experiments (Fig. 3C).
One possible explanation for the disparity between experimental and simulated healing rates is that intercellular cohesions exist despite the lack of adherens junctions, and that these interactions slow sheet expansion. We tested the effects of cohesion by running simulations that required that cells maintain contact with at least one neighbor. This limited form of cohesion slowed healing by only ∼10% (Fig. 3C): a result that is consistent with the fact that standard simulations – like healing by real L1 cells – display very little cell scatter. Because instantaneous cell speed is the most important determinant of healing rates in simulations, a second possibility is that cells within recovering L1 monolayers move more slowly than the sparse L1 cells we measured for input model speeds. Simulations indicate that a ∼25% slower intrinsic cell speed could account for the 33% disparity between simulation and experimental healing rates (Fig. 3C). Unfortunately we could not quantify the speeds of individual cells within experimental monolayers because our cell-identification algorithms were not robust enough to isolate and track cells in highly crowded cultures.
We also considered a third possibility: that cells are less than perfectly efficient in their movements to minimize crowding. Given the stochastic nature of cell migration, the baseline assumption that cells instantly respond to cell contact to avoid collisions is probably too idealized. To test this idea, we ran simulations in which cells occasionally moved in non-ideal directions. We found that simulations quantitatively match both MMC and normal L1 wound-healing experiments if cells move to minimize overlap with 80% of the steps they take (Fig. 3C); the remaining 20% of the time, individual cells move in a manner that produces less than the maximal expansion of the monolayer and thus the average healing rates slows. Under these conditions, movies of simulated wound healing show strong visual similarity to experimental wound healing (Fig. 5; supplementary material Movie 3). In addition to quantitative agreement in healing rates, these simulations match experiments by exhibiting a steady advancement of the wound edge, a small degree of cell separation shortly before wound closure, and cell division throughout the monolayer. More detailed models and high-resolution observations of cell interactions will be needed to rigorously test our ideas on the efficiency of contact avoidance and non-specific cohesions. Still, with only a small quantitative disparity and excellent qualitative agreement between simulations and experiments, the mechanisms exhibited by our model deserve consideration as potential mechanisms for real sheet migration.
The scratch-wound assay has been used for nearly half a century as an in vitro model of wound healing and as a tool to discover factors important to cell migration (Soderholm and Heald, 2005; Todaro et al., 1965). Remarkably, the mechanism(s) that coordinate the collective migration of the surviving cells remain poorly understood. Many possibilities can be imagined including biochemical communication through paracrine signaling or gap junctions, selective activation of cells at the wound edge, and cohesion between migrating cells. The significance of our model is that it does not include these complexities and yet accounts for the robust characteristics of sheet migration. Thus it is reasonable to assume that in cells that heal wounds in a steady, division-independent fashion, the primary impetus for sheet migration is a simple release of mechanical constraints. This hypothesis has been raised and supported in several investigations (Block et al., 2004; Farooqui and Fenteany, 2005; Kornyei et al., 2000; Todaro et al., 1965), but definitive experiments that isolate such a mechanical impetus are elusive because it is impossible to eliminate all imaginable routes for biochemical communication.
To test the idea that mechanical influences could sustain the migration of a cellular sheet, we developed a simulation in which cells could be represented as simple mechanical agents. Our model cells lack all forms of biochemical or junctional communication and divide and crawl in contact-dependent fashions. Constraining the model with measurements on L1 fibroblasts, we reproduce the qualitative features of L1 sheet migration with no free parameters other than a small uncertainty in the dimensions over which cells sense each other. Although it might be expected that physical constraints imposed by cell crowding should bias cell spreading and migration into the denuded region in simulations, it is not obvious that spreading should be steady and division independent. Our parametric studies indicate that this phase of healing is due to the release of the initial compaction of cells grown to confluence. The rate of steady expansion is limited by the rate of cell migration, and the duration of this phase is limited by the amounts of initial compaction and cell division. Although healing can occur by cellular diffusion (i.e. scatter), this mechanism is much slower. Thus the arrangement of cells into a pre-packed monolayer is advantageous for quickly restoring continuity after wounding.
We conducted our studies on fibroblasts to eliminate the need to represent cohesive junctions between model cells; nonetheless, the model describes a mechanism for sheet migration that appears relevant to epithelial sheets. Indeed the phenomenon of steady sheet advancement that is independent of cell proliferation is documented mostly in studies on epithelial cells both in vitro (Block et al., 2004; Farooqui and Fenteany, 2005; Richards et al., 2004; Takeda et al., 2004) and in vivo (Krawczyk, 1971; Pullar et al., 2006; Takeda et al., 2004). A recent review of corneal epithelial cells describes the healing process as the flattening and migration of cells as an intact sheet until the wound is covered (Zieske, 2001). Because cells can remain in contact with one another as they flatten, cohesive tight and adherens junctions in epithelia might not hinder healing that occurs by such a mechanism. Finally we note that although in recent years the term `collective migration' has been defined as a mode of migration in which cells maintain intercellular junctions (Friedl, 2004; Friedl et al., 2004), our model illustrates that a combination of cell spreading and migration can lead to wound healing that resembles collective cell migration without the need to assume intercellular cohesion.
Our work also highlights the significance of the familiar but still unexplained phenomena of contact inhibition. The classic observation that non-transformed mammalian cells arrest their growth in confluent cultures must have a mechanism broad enough to apply to many types of non-transformed mammalian cells (Castor, 1969; Folkman and Moscona, 1978; Todaro and Green, 1963). Thus while there is evidence in some cells that particular cell-cell adhesion molecules contribute to contact-inhibition (Grazia Lampugnani et al., 2003), our model embodies the more general idea that individual cell shape controls the cell cycle (Castor, 1969; Chen et al., 1997; Folkman and Moscona, 1978).
The mechanisms explaining contact inhibition of motility are also enigmatic, but likely more significant for wound healing. Our model suggests a phenomenological description that must be tested in future work. For example, the model predicts that the distance over which cells can influence one another's movements (`the sensing radius') governs the transition between linear and diffusive regimes of healing. Non-traditional `wounds' such as patches or strips might limit the capacity of a monolayer to expand and be useful in testing this prediction.
Materials and Methods
Mouse 3T3-L1 fibroblasts (ATCC, CL-173) were maintained in culture in DMEM supplemented with 10% FBS at 37°C in a 10% CO2 environment. When being prepared for experiments, cells were plated at low density (∼5000 cells/cm2) onto glass coverslips and allowed to adhere overnight (for subconfluent cell trajectory and density experiments) or to grow to confluence (3-5 days; for wound studies).
Time-lapse microscopy was used to acquire images of L1 fibroblast motility, growth in culture, and wound healing. During experiments, cells were maintained at 37°C in a microscope stage-top incubator to enable time-lapse microscopy. The media used during experiments was phenol-free non-CO2-buffered L-15 media supplemented with 10% FBS, and prewarmed to 37°C. After the cell coverslip was transferred, a thin layer of mineral oil was laid over the media to prevent evaporation. Cells can be cultured and observed for several days under these conditions. Acquired images were 640×480 pixels at 0.226 pixels/μm for nearly all data acquired.
Cell trajectory analysis
Cell trajectories were assembled using custom automated cell identification and tracking routines written in MATLAB. Briefly, images were thresholded to identify the dark centers of cells, the centroids of these clusters of dark pixels were identified in each frame, and the centroids were connected from frame to frame to construct trajectories of cells over time. Coverslip drift was corrected for by manually tracking stationary points and subtracting stationary point displacements from cell trajectories. We retained trajectories giving a very good fit to the persistent walk formula (r2>0.995). The same fits and criteria were used to process the trajectories of simulated cells. The speed and persistence time (the average time between direction changes) can be obtained from trajectories by averaging the squared displacement along the trajectory as a function of time interval (Othmer et al., 1988).
Cell density measurements
L1 fibroblasts exhibit contact inhibition of growth, in which cell crowding leads to decreased proliferation. To accurately represent this behavior in simulated cells, we needed to measure how the division rate of a cell depends on the cell density. To measure cell densities over time, low density cell cultures were prepared as described above and images were acquired every 30 minutes over several days. Using custom MATLAB routines, the images were background corrected and thresholded, with a collection of at least six dark pixels constituting a cell. A count of all identified cells divided by the observation area gives the cell density in that frame. This automated method was validated against hand counts (see supplementary material Fig. S1 for details and demonstration).
Per capita division rate vs intercell spacing
Plots of per capita division rate vs average intercellular spacing are created by transforming cell density vs time plots according to: where D(t) is the cell density at time t. The derivative of the density curve at time t is found by finding the slope of a line fitted through a window [t–10 hours, t+10 hours]. The quantity sin(π/3) is a geometrical factor which compensates for the fact that it is possible to pack more circles of a given diameter in a given area than squares with the same side length. The spacing therefore represents the average hexagonal spacing rather than the average orthogonal spacing.
Pharmacological inhibition of cell division
For both wound healing and subconfluent migration experiments, MMC solutions were prepared fresh and used within 36 hours. Cells were pretreated with 5 μM MMC in the culture medium 24 hours before observation. During observation, L-15 medium also contained 5 μM MMC. After the 24-hour pretreatment, no cell divisions were observed. Cell migratory behavior began to degrade about 60 hours after pretreatment, and by 72 hours, cell crawling essentially ceased, accompanied by widespread cell death. Accordingly, MMC data was only obtained between 24 and 48 hours after pretreatment.
Wound width measurement
To quantify wound edge advancement over time and wound healing rate, measurements of the wound width over time were necessary. For healing experiments, confluent monolayers of L1 fibroblasts on glass coverslips were wounded by scraping with a pipette tip or cell scraper, then transferred immediately to the microscope stage-top incubator. Images were acquired every five minutes until wound closure. The wound edge was extracted from images in an automated fashion using custom MATLAB routines. Details of the processing and a demonstration can be found in supplementary material Fig. S2. The quality of wound edge identification can be evaluated in supplementary material Movie 3.
Wound edge advancement rate and linearity calculations
Wound edge advancement rates were calculated by finding the slope of a best-fit line to all or part of the wound edge advancement vs time curves (hereafter `wound curves'). For untreated L1 wounds and simulations with division, the entire wound curve was used for the fit. For experimental MMC-treated wound curves, there was a transient rapid initial phase of closure, then a slower roughly linear phase, and then a final slowing phase as cells ceased migration and died due to MMC toxicity. The region from 5 to 24 hours post-wounding was chosen for fitting to exclude the initial transient phase and to be sure of completely excluding the final phase. Most simulated wounds without division exhibit breaking (wound curve nonlinearity), and since this is an artifact of the size of simulation, the post-breaking data was not used for rate fitting. The times at which breaking occurred were estimated by visual inspection of the wound curves.
Linearity exponents were calculated over the same region of wound curves as the associated edge advancement rates. The exponent is a measure of the degree of a single term polynomial passing through the origin and fit to the data. It assumes that the data is of the form: and finds the exponent m by fitting a line to the data on a loglog plot: The exponent is the slope of this line. An exponent of 1 indicates the original data was linear, whereas a value of 0.5 is expected for diffusion. Since the fit assumes the data passes through the origin, partial wound curves were rigidly translated so they began at the origin.
To test the hypothesis that cells exhibiting constitutive motility combined with simple mechanical sensing can perform collective migration, we constructed an agent-based model in which single cells move, grow, and divide in response to the creation of space around them. Simulated cell characteristics were constrained, wherever possible, by measurements on L1 fibroblasts. These measurements and the development of the model are described in the Results section. Parameters and their default values are described in Table 1. The final algorithm can be summarized as follows.
In each simulation step: (1) it is determined which cells are dividing, and those cells are divided (using cell division routine below); (2) all cells decide which direction they want to go in (using the direction determination routine below); (3) if the result of a cell going the direction it wants to go would lead to cells overlapping their incompressible regions, then that cell is stopped (i.e. does not move in this simulation step); (4) all other cells update their positions by moving one step in their desired direction; (5) repeat.
Cell division routine
When a cell divides, it is replaced by two cells that have half the diameter of the original cell. This generally results in cells that are smaller than the incompressible diameter. These cells are considered `immature'. Whenever there is space around an immature cell, its incompressible region grows to fill the space or until it reaches the standard incompressible diameter, whichever comes first. When the incompressible region reaches the standard incompressible diameter, it is considered `mature'. Only mature cells have a chance of dividing. The division rate of a mature cell is proportional to the amount of space it has beyond its incompressible size, up to the division radius (which corresponds to its maximal division rate). If cell A has closest neighbor cell B, then the division rate for cell A (dA) is given by: where DAB is the distance between the centers of cells A and B. For example, if the centers of cells A and B are 50 μm apart and both are mature (i.e. IA=IB=Im=47 μm), then the amount of space between their incompressible cores is 3 μm. The two cells share this space equally, and so cell A has 1.5 μm of space beyond its incompressible core. If the division radius for this simulation is 9 μm and the maximum division rate is 0.04 divisions/hour, then the division rate for cell A is (1.5/9)*0.04=0.01 divisions/hour. If the space between incompressible cores is greater than double the division radius, then the division rate is just the maximal division rate. The probability of a cell with division rate d dividing in the current simulation step (Pcd) is Pcd=1–e–Δt/d where Δt is the simulation step duration [Δt=L/s; typically (0.093 μm)/(0.76 μm/minute)=0.12 minutes].
Direction determination routine
All cells initially want to go the same direction they went in the previous step. They pick a new random direction if a random number is drawn for them which is less than the probability of a direction change (Pdc), Pdc=1–e–Δt/p, where p is the persistence time (Pdc is typically ∼0.01).
If a cell can sense obstacles (i.e. some other object extends into this cell's sensing radius), then its desired direction is further influenced by this contact. Each cell considers the expected effect of moving in all directions, and chooses the direction that results in the furthest closest neighbor (i.e. the direction which makes the new closest neighbor the greatest distance away). Usually the cell chooses this direction with 100% efficiency, but in some simulations we explore the effect of lowering this to 80% efficiency. This means that when a cell is being guided by contacts, 20% of the time it goes in a random direction rather than in the direction that minimizes overlap.
Automated cell density and wound edge extraction measurements
Techniques for automated cell density measurements and wound edge extraction are demonstrated in supplementary material Figs S1 and S2, respectively. Routines were written as custom MATLAB packages and are freely available from http://mcgrathlab.urmc.rochester.edu/resources/.
The authors are grateful for prior funding from the Whitaker Foundation in support of this work. A special thanks to Kristen Nicholson for heroic efforts obtaining manual cell counts and to Jeffrey Zuber for providing the polish we needed for a final algorithm.
Supplementary material available online at http://jcs.biologists.org/cgi/content/full/120/5/876/DC1
- Accepted December 21, 2006.
- © The Company of Biologists Limited 2007