2.3.1 Approximate Bayesian computation.

In practice, the likelihood for complex stochastic models is mathematically intractable and typically only accessible through simulation. Approximate Bayesian computation is a popular likelihood-free tool to estimate the left-hand side in Eq (6) [24, 26]. It approximates the likelihood, , using repeated simulation of the model, accepting simulated parameters θ only if the resulting model output, , is closer than some threshold, ϵ, to the data, : (7) where d is some distance function that quantifies the difference between the data, , and model output, . In many applications of ABC, the data are high-dimensional, meaning that, from a computational point of view, it is necessary to summarise the data in lower-dimensional summary statistics so that the distance in Eq (7) can be efficiently computed. In Section 2.3.3, we describe our choice of summary statistic for this problem.

We use a version of sequential Monte Carlo sampling (ABC-SMC) [24, 26] as our implementation of ABC. In brief, ABC-SMC propagates a sample from the prior distribution, π(θ), through a sequence of intermediate distributions to approximate the target distribution, . The intermediate distributions are a sequence of ABC approximations P_t for t ∈ {1, …, T}. We use a sequential importance sampling approach to ABC-SMC [24, 27, 28], where each Monte Carlo sample built at generation t is used to construct an importance distribution at generation t + 1. We define N_gen as the total number of parameters that are drawn from the importance distribution at each generation, and N_acc as the total number of parameters that are accepted. Given a Monte Carlo sample at generation t, the sample at generation t + 1 is obtained by using a perturbation kernel K(⋅|t) to propose N_gen perturbed parameters from the Monte Carlo sample at generation t, from which the N_acc proposed parameters with smallest distance to the real-world data are accepted into the Monte Carlo sample for generation t + 1. In this work, we use the Normal perturbation kernel proposed by Filippi et al. [9] and let N_gen = 2 ⋅ 10⁴ and N_acc = 500, meaning that each generation has an acceptance rate of 2.5%. The final sample, , forms the output of the ABC-SMC algorithm. All implementation details are outlined in Supplementary Information S1 Text Section S2.

2.3.2 Minibatch-ABC-SMC.

In some problems, one model simulation can be compared with arbitrarily many observations, using the distance function d. This is particularly the case when the model is deterministic and one realisation of the model suffices to capture the relevant behaviour of the model. When the model is stochastic, however, a single model realisation will likely not provide sufficient information on the distribution of model outputs to sensibly compare to observed data. An additional, and significant, problem occurs with time series data because the initial conditions across different experiments can be vastly different. This means that for each comparison with a different datapoint, a new model simulation with a specific initial condition is needed to ensure that the comparison between the summary statistics of the simulated and observed data is fair and informative. For high-throughput data, the implications of these additional computational costs are significant. As an illustration of this point, the data set in this work consists of many hundreds of observations with different initial wound location, initial cell densities and initial topology. Fig 2 illustrates the extent to which two wells aimed at studying the same genetic profile can differ. To apply Bayesian inference approaches to combine models to such data requires individual simulations, which mirror the initial conditions of each experiment separately.

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Fig 2. Two initial conditions from the assays conducted on wild-type cells.

Note the variation in both the location of the wound as well as the size of the initial wound, highlighting the need to carry out separate simulations, one for each initial condition.

https://doi.org/10.1371/journal.pcbi.1010191.g002

When observed data, , can only be compared to model simulations that share the same initial condition, the simulated data will equally consist of N_obs datapoints, such that , and shares its initial condition with . A scalar measure of the discrepancy between the observed and simulated data is then obtained by taking the mean of distances between the observed data points and their corresponding model simulations: (8)

When the size of observed data, N_obs, is large, i.e. N_obs ≫ 1, and the model needs to be simulated separately for each data point, ABC-SMC approaches fail to scale well with the number of observations: simulation time—which is normally a bottleneck—increases linearly with the number of observations.

Drawing inspiration from minibatch methods in SGD in machine learning [12–14], we propose a minibatch approach to computing the distance in Eq (8): rather than using all N_obs observations to compute the distance whenever a parameter θ is proposed in ABC-SMC, we suggest computing the distance between observed and simulated data based on randomly chosen batches from the data. This is done by choosing a batch size, N_bs, such that N_bs ≪ N_obs, that controls the number of datapoints considered in each comparison between model simulations and data. Formally, this amounts to drawing N_bs random numbers i_j independently with replacement, from {1, …, N_obs}, obtaining the corresponding data point and simulating a model output, , that shares the same initial condition. Then, the discrepancy between model and data can be approximated by (9)

This approach immediately reduces simulation time by a factor N_obs/N_bs. The minibatch approach is simple to implement within the importance sampling part of the ABC-SMC framework described in Section 2.3.1. At each generation, t, an importance distribution is given (for t = 1 this is the prior distribution). Then, N_gen parameters are sampled according to this distribution. For each parameter a minibatch of size N_bs is sampled from the data and for each observation within that minibatch a corresponding model simulation is generated given the parameter and observed initial condition. Mathematically, if is the initial condition corresponding to observed data point , the model simulates a corresponding output . The discrepancy between simulated and observed data is computed for each parameter using Eq (9). The algorithm is summarized in Algorithm 1.

Algorithm 1: Minibatch-importance sampling

Input: Data, ; number of parameters to sample, N_gen; number of parameters to accept, N_acc; batch size, N_bs; prior distribution, π(θ); importance distribution, ; model, f(•|θ); distance function, d.

Output: N_acc weighted samples .

1 for i = 0, …, N_gen do

2  Increment i ← i + 1;

3  Generate parameter vector θ_i according to given importance distribution ;

4  for k = 1, …, N_bs do

5   Draw data index ξ uniformly at random from {1, …, N_obs};

6   Simulate ;

7  end

8  Calculate ϵ_i using Eq (9).

9 end

10 From , select the N_acc parameters with lowest value of ϵ_i;

11 Set w_i = π(θ_i)/q(θ_i) for all accepted parameters θ_i.

At this point, we remark that an alternative to using minibatch ABC-SMC might be to a priori select a subset of the data (e.g., using stratified sampling) and use ABC-SMC on this subset. While reducing the amount of data to be processed would certainly reduce the computational burden of ABC-SMC, this approach also vastly reduces the amount of available information. Moreover, subsampling a priori from the data might introduce bias into the problem, as it is up to the practitioner to define a sensible subset of the data. In Supplementary Information S1 Text Section S4.5, we carefully examine the quality of the posteriors generated by subsampling from the data in comparison to those generated using minibatch ABC-SMC. We find that a priori selected subsets give rise to more variability in the estimated posterior mean, and higher posterior variance across all parameters, minibatch ABC-SMC with minibatches of the same size. We conclude that our approach is to be preferred over simply reducing the amount of available information by subsampling the data.

Our method contains the minibatch size, N_bs, and the number of generations, T, as tunable hyperparameters. In machine learning applications, it has been noted that inference with smaller batch sizes when optimizing an objective function leads to less overfitting to the data, as the algorithm is better able to observe the full range of parameter values that yield results that resemble the full distribution of the data, rather than the parameters that only produce summary statistics that are close to the average observed in the data set [13, 14]. At the same time, using a very small batch size might not provide a good estimate of the model distribution, meaning that careful consideration of the batch size is necessary. This will be the focus of Section 3.1. For the number of generations, we choose T = 4. This hyperparameter choice comes from performing test runs of minibatch ABC-SMC with various batch sizes and numbers of generations and computing the Kullback-Leibler (KL) divergence [29] between the successive posteriors as a measure of change of the distribution from one generation to another, as is done by Filippi et al. [9]. As a general stopping criterion, we suggest to choose the smallest T so that the KL divergences for the subsequent generations converge to a fixed value. In this specific problem, given the model and the data, this amounts to T = 4, indicating that the distributions do not change significantly after this generation. In other problems, a different number of generations will typically be needed. For details on how the KL divergence between successive generations was calculated, we refer the reader to Supplementary Information S1 Text Section S2.3.

We note that the random sampling of minibatches ensures that with very high probability every datapoint in the dataset will be used in each generation. Since for each of the N_gen parameters in each generation there are N_bs datapoints sampled uniformly at random, the probability that one generation of minibatch ABC-SMC fails to contain a given datapoint, say i_⋆ ∈ {1, …, N_obs}, can be computed. Denoting this event , a simple computation shows that this probability can be bounded by (10) since (1 + x/n)ⁿ < e^x for any integer n. For N_gen = 2 ⋅ 10⁴, (11) meaning that the event that in one generation not all datapoints are used is essentially negligble.

All simulations were carried out using a development branch of the parallel open source ABC-SMC software Pakman [30]. Each ABC-SMC experiment was performed on a high performance compute (HPC) cluster using five 48-core Cascade Lake (Intel Xeon Platinum 8268 CPU @ 2.90GHz) nodes.

2.3.3 Summary statistics.

To compute the distance d between observed and simulated data, we employ the summary statistics proposed by Browning et al. [17], with the aim to capture three key features: population size increase, spatial structure and density profile [17]. To capture population size increase, we use the population size, N(t), at t = 24h. To capture spatial structure, we use the pair correlation function, P(r, t), which describes the density of pairs of agents separated by a distance r at time t, relative to the expected density of pairs if the population were uniformly distributed [17]. For our discrete data, the pair correlation function describes the relative number of pairs separated by distances in the range (j − 1)Δr < r < jΔr. Following [17], we compute (12)

In Eq (12), L and H are length and height, respectively, of the field of view. Following the recommendation in [17], we choose Δr = 5μm, and calculate the pair correlation function up to a distance of 100μm. To capture the density profile, we compute the density across cross-sections of the wound by dividing the field of view into 64 different bins in the direction parallel to the scratch and computing the cell density in each bin. Using 64 bins results in a bin width of approximately 24μm, or, one cell diameter. Assigning a centroid for each cell, and using this centroid to define the location of cell i as y_i, this results in a one-dimensional density ρ(j, t) at time t, where (13)

These three distance functions allow us to define the distance between an observed data point and a simulated data point as (14)

3.1.1 Variability of the posterior distributions.

In Supplementary Information S1 Text Section S4, we report the posterior means of the resulting posterior distributions, and their variance, at each batch size, as well as representative plots of the posterior distributions. At any value of the batch size, minibatch ABC-SMC correctly identifies the features of each parameter regime. For instance, for the dataset for parameter regime I, the estimated intrinsic motility rate, m, is consistently estimated as higher than that of the dataset for parameter regime II. Conversely, the net intrinsic proliferation rate, p − d, is correctly estimated as higher in parameter regime II than in parameter regime I. For the density-dependent interaction term, γ_m, the minibatch ABC-SMC implementation discerned at any batch size that the effect of crowding on motility is positive in regime I, and negative in regime II. Increasing the batch size improved the quality of the posterior distributions, although the improvements diminish rapidly as the batch size, N_bs, is increased. There is a critical value—in Fig E in S1 Text this is typically around N_bs = 10—after which increasing the batch size, N_bs, does not futher reduce the variability of the estimated posterior means. Similarly, after this critical value, the capacity of minibatch ABC-SMC to identify the true posterior mean does not improve, even when the batch size is increased to the full data size. Similarly, the standard deviation of the posterior distributions diminishes rapidly when the batch size is increased initially, but then stays relatively constant across larger batch sizes. In both parameter regimes, this is the case for N_bs = 10. Taken together, analysis of the posterior means and posterior standard deviations suggests that a minibatch implementation of ABC-SMC provides a robust approximation of the posterior distribution at a fraction of the computational cost. In the synthetic data set, the computational speedup of using a batch size of 10 samples compared to the full data set amounts to a factor 117/10 = 11.7.

3.1.2 Reliability of minibatch ABC-SMC using experimental data.

Finally, to compare performance on real-life data from the scratch assay, we perform inference on the Mock type data set in S1 Text Section S4, where we vary the batch size from 1 to 117 and find no meaningful differences between the different batch sizes in terms of their ability to identify the posterior means consistently. With growing batch size, just as with synthetic data, the variance of the posterior distribution decreases and shows a similar behaviour to the synthetic control data around a batch size of N_bs = 10. We conclude that choosing a minibatch size of N_bs = 10 allows for minibatch ABC-SMC to reliably identify the posterior mean.

While some genetic perturbations have a large number of observations and the choice of N_bs = 10 offers a good balance between computational complexity, robust inference of the posterior mean and low variance of the posterior distribution, some perturbations have very few observations in the high-throughput screen used in this work. As such we address the suitability of ABC-SMC in identifying mechanistically relevant differences between the different gene knockdowns by selecting 40 datapoints from each of the Mock and CDH5 knockdown experiments at random and without replacement, and dividing them into 20 pairs of two observations for each of the datasets. We perform ABC-SMC on each of the resulting miniature data sets and record their posterior distributions. We find that while the spread of the posteriors is indeed larger for each of the experimental conditions than in the case when more datasets are included, the posterior means can still identify meaningful differences between the two different genetic perturbations. This suggests that as few as just two observations can be used to confidently estimate model parameters.

Finally, we assess how well our IBM predicts experimentally observed wound healing behaviour when parametrised according to the posteriors generated using minibatch ABC-SMC. We sample 1000 parameters from the posterior distribution obtained using minibatch ABC-SMC with N_bs = 10 on the full Mock dataset and select one Mock initial condition uniformly at random. For this condition, we perform forward model simulations, one with each sampled parameter, and record the summary statistics defined in Section 2.3.3, which are the density profile, pair correlation function, and number of alive cells at t = 24h. From these forward simulations, we compute the mean and standard deviation for the density profile and pair correlation function. In Fig 3, we display the experimental data for density profile and pair correlation together with a range of one standard interval from the posterior mean. We also show a histogram of simulated cell counts at t = 24h, compared to the experimentally observed cell count at t = 24h. The posterior predictive ranges for the density profile and pair correlation are excellent fits to the data, and the simulated cell count histogram is centered around the experimentally observed cell count.

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Fig 3. Posterior predictive check for the model.

Left: experimentally observed density profile for a single datapoint sampled at random from the Mock dataset (red points) displayed against the posterior range of one standard deviation away from the posterior mean (blue shaded region). The gray line is the smoothed density profile generated using a Gaussian kernel. Middle: experimental pair correlation function (red line) displayed against the posterior range away of one standard deviation away from the posterior mean (blue shaded region). Right: experimentally observed cell count at t = 24h (red vertical line) compared to the histogram of model simulations of cell count (blue). All summary statistics have an excellent fit to the experimental data.

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3.3.1 Clustering analysis reveals functional subgroups.

The set of posterior means for the different gene knockdowns is a set of points in five-dimensional parameter space, and it can be studied to reveal patterns of cell behaviours across the different gene knockdown experiments. One popular method of understanding the spatial structure of data, when it cannot be readily visualised due to the dimension of the data space, is K-means clustering. The idea behind K-means clustering is to divide a set of points in Euclidean space into K clusters such that the sum of the pairwise distances within each cluster is minimised. With other words, it is an unsupervised method to assign labels to the datapoints such that points with the same label are close to each other given some defined distance function. Mathematically, given K clusters, the classification problem is to subdivide the data into K distinct clusters, , where the clusters are found by solving the optimization problem (15) where |S_i| denotes the number of points in cluster i, and ‖•‖ denotes the Euclidian distance. The number of clusters, K, is a hyperparameter that needs to be carefully chosen. Increasing the number of clusters will always decrease the total sum of pairwise distances, however it also leads to a loss of interpretability of the cluster labels and potentially overfits the data. A common practice in data science is to use the elbow method [34], which was first employed by Thorndike [35]. The elbow method amounts to choosing K as the elbow of the curve of the minimum of the objective function in Eq (15) as K is varied. The interpretation of picking the elbow of the curve, in clustering, corresponds to choosing K such that adding futher clusters does not provide a significantly better fit to the data. We perform K-means clustering for K = 1, …, 10, and find that the marginal improvement of the sum of pairwise distances rapidly decreases after K = 3 (see Supplementary Information S1 Text Section S6). Therefore, we perform clustering with three clusters. To further analyse robustness, we assess the difference between clusters when we set K = 2, 4, 5 to find that the addition of extra clusters yields further sub-clustering within the clusters found at K = 2. From this we conclude that K = 3 is an good choice to balance mechanistic insight with more detailed spatial information.

The pairwise marginal distributions of the posterior means for the different gene knockdowns is shown in Fig 6, where the colour key is defined by the cluster labels. The separation between the clusters is best seen in the pairwise martinal posterior distribution of the intrinsic motility parameter, m, and the density-dependent motility parameter, γ_m. The marginal posterior distributions of the two parameters show that the cluster labels correspond to changes in the sign of the density-dependent motility parameter, γ_m, and the magnitude of the intrinsic motility parameter, m. Interpreting this separation mechanistically allows us to identify three functionally distinct clusters in the data. We identify Cluster I as knockdowns with low intrinsic motility, and the motility is enhanced by high local cell density (negative values for γ_m). In Cluster II, knockdowns have low intrinsic motility, and the motility is inhibited by high local cell density (positive values for γ_m). In Cluster III, knockdowns have high intrinsic motility, and the motility is enhanced by high local cell density (positive values for γ_m).

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Fig 6. Pairwise distributions of the posterior means for each of the knockdown experiments in the high-throughput screen.

Diagonal plots show the distribution of the posterior means of each of the parameters, while off-diagonal plots show the pairwise distributions of the different knockdown posterior means. The colour key denotes the different clusters found using K-means clustering with K = 3. Star denotes Mock, diamond CDC42 knockdown and plus CDH5 knockdown.

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Furthermore, the pairwise plot of the intrinsic net proliferation parameter, p − d, and the density-dependent motility parameter, γ_p, shows that for knockdowns with a value of p − d close to zero, there is a wide range of possible values for the density-dependent proliferation parameter, γ_p, whereas the variability of γ_p diminishes for p − d further away from zero. As such, the mechanistic interpretation of this finding is that high intrinsic proliferation rates are associated with lower density-dependent inhibition of proliferation. Fig 6 also suggests that there is a coupling between a higher increased intrinsic motility, m, and the bias parameter, γ_b: knockdowns in Cluster III also show the highest values of γ_b. This coupling is much less pronounced when the intrinsic net proliferation rate is compared to the density-dependent crowding term, γ_b. In summary, analysis of the posterior means suggests that high intrinsic motility, but not high intrinsic net proliferation, are associated with a stronger sensing of local cell density gradients through the bias parameter, γ_b. Finally, the pairwise plots show that most of the functional differences between the knockdowns can be described in terms of the intrinsic motility rate, m, the net proliferation rate, p − d, and the density-dependent motility rate, γ_m. In the remainder of this section, we explore the impact of these parameters on the wound healing outcomes by analysing the observed cell count and wound area changes.

Fig 7 illustrates how the clusters can help tease apart the different mechanisms at play in wound healing for each of the gene knockouts. For each of the clusters, we select the knockouts with greatest dissimilarity with the other clusters. These are EPHA7, USP18 and ITPR1 for Clusters I, II, and III, respectively. While the EPHA7 and USP18 knockdowns have similar degrees of wound closure, the posterior means for the EPHA7 knockdown suggest that incomplete wound closure is mainly due to net cell death during the experiment, which is consistent with previous studies [36]. The high value for γ_m for USP18 knockdown suggests that cells are unable to move into the wound area due to defects in cell adhesion. This indicates that USP18 might be required for cell adhesion turnover through deubiquitination. Moreover, ITPR1 is known to regulate apoptosis and our results confirm its dual role on cell motility and proliferation [37].

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Fig 7. Identifying functional subgroups among perturbations.

A: Three-dimensional scatter plot of the posterior means for the intrinsic motility parameter, m, intrinsic net proliferation parameter, p − d, and density-dependent motility parameter, γ_m, for each of the samples in the high-throughput screen. Star denotes Mock, diamond denotes CDC42 knockdown and plus denotes CDH5 knockdown. B, top to bottom: wound mask for one final condition of the USP18, EPHA7, and ITPR1 knockdowns, respectively.

https://doi.org/10.1371/journal.pcbi.1010191.g007

In summary, Bayesian inference with a large screen of gene knockdowns can be used to characterise the functional role of the genes under investigation given a sufficiently detailed mathematical model. Given such a characterisation (in our work, this is the division into functional clusters), experimentally testable hypotheses can be generated to further refine and explore our understanding of the genes under consideration.

3.3.2 Density-dependent patterns of proliferation reproduce experimental patterns of cell count changes.

Changes in cell counts are a simple summary statistic to extract from scratch assay data [17, 20, 21] and provide an indication of the average net proliferation rate in a large population of cells, such as the cell monolayer used for scratch assays. However, computing an average net proliferation rate does not reflect more complex mechanisms, such as contact inhibition, that may influence the change in cell numbers throughout the experiment. In this section, we turn to the question of how density-dependent effects are observed in the high-throughput measurements in cell count changes for each of the knockdowns. Let c₀, c₂₄ denote the cell counts at t = 0h, 24h for a given knockdown. The fold change in cell count, , is given by . By the definition of the IBM, cell count changes occur as a function of the proliferation parameters p, d, and γ_p. In fact, Eq (4) relates the local proliferation rate to the intrinsic and density-dependent proliferation rates and hence relates the expected number of cells to the values of the different model parameters. In the case where density-dependent effects are ignored, i.e. γ_p = 0, one would expect that the expected value of the cell fold change, , is given by , since p − d is the net proliferation rate measured in h⁻¹. Fig 8 shows that this curve fails to describe the observed fold change in cell numbers across the screen, indicating that density-dependent effects affect fold changes in observed cell counts. In this work, density-dependent effects affect the proliferation rate through the sum in Eq (4). This sum depends on the cell locations at each simulation step, meaning that it cannot be determined from the available data. However, by assuming local cell density is constant across the assay, and by approximating local cell density as the average of the local cell densities at t = 0h and t = 24h, this difficulty can be overcome. Letting F be the area of the field of view, and w₀, w₂₄ the wound area at t = 0h, t = 24h, respectively, one can approximate the sum in Eq (4) by an integral such that (16)

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Fig 8. Observed cell fold change as a function of intrinsic net proliferation rate, p − d.

When density-dependent effects are ignored (blue) line, the estimate of cell fold change as a function of net intrinsic proliferation overestimates cell fold change. When density-dependent effects are considered, model predictions accurately capture cell fold change.

https://doi.org/10.1371/journal.pcbi.1010191.g008

Under this approximation, a coarse-grained estimate for the fold change in cell numbers over the 24-hour experiment is given by (17)

For each sample in the high-throughput screen, we compute the value of through Eq (17) and fit an exponential curve through the obtained fold estimates. In Fig 8 we show that the density-corrected cell fold estimate predicts the observed cell fold change as a function of the intrinsic motility parameter well. We conclude that density-dependent effects contribute to the observed cell fold change by modulating with the intrinsic net proliferation rate and that the IBM can shed light on the nature of density-dependent effects for each of the knockdowns.

3.3.3 Finding mechanistic contributions to wound closure.

While the fold change in cell counts is associated with the proliferation parameters, the fold change in wound size depends delicately and nonlinearly on all model parameters. We define the wound area fold change, , as w₂₄/w₀. Note that means that the wounds at t = 0h, 24h are equally large, and means the wound is fully closed. In this section we wish to understand the different roles of the various mechanistic terms in the model and explain the range of observed wound closure rates. In Fig 9, the observed wound area fold change is plotted against the observed cell fold change. While wound closure is not completely independent of increases in cell numbers, the data suggest that the dependence of wound closure on cell number increases is not as strong as might be expected if wound closure is thought to be primarily driven by proliferation: values of wound area fold change of approximately 0.75 are observed across knockdowns with 50% cell death all the way to 25% increases in cell counts, i.e. for net positive and net negative cell number changes, there can be similar amounts of wound closure. To investigate which mechanistic features of the model are key in wound closure, Fig 10 shows the observed wound area fold change, plotted against the different model parameters. Surprisingly, high intrinsic motility rates do not necessarily correlate with enhanced wound closure: the datapoints with labels corresponding to high intrinsic motility and density-enhanced motility all have much higher intrinsic motility posterior means than Mock, but their wound closure rates are not, in general, higher than Mock. At any given value of the intrinsic motility, m, a wide range of different wound closure extents are observed. In contrast, a stronger coupling is observed between the extent of wound closure and the density-dependent terms: when motility is strongly enhanced by local crowding, i.e. large, negative values of γ_m, very low values of wound area fold change are observed, whereas the wound area fold change is generally high when γ_m is high. Likewise, when γ_p is increased, wound area fold changes increase as well. These findings suggest that, rather than the intrinsic rates of movement and proliferation, it is the density-dependent interactions between cells that determine the degree to which wound healing is accomplished.

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Fig 9. Observed wound area fold change as a function of cell fold change.

Across the range of cell fold changes, starkly different rates of wound closure are observed.

https://doi.org/10.1371/journal.pcbi.1010191.g009

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Fig 10. Observed wound area fold change plotted against the different model parameters.

The colour key denotes the different clusters found using K-means clustering with K = 3. Star denotes Mock, diamond denotes CDC42 knockdown and plus denotes CDH5 knockdown.

https://doi.org/10.1371/journal.pcbi.1010191.g010

While Fig 10 provides information as to how the different model parameters influence wound closure, it is to be expected that there is an interplay between the intrinsic rates and their density-dependent counterparts. To investigate how different model parameters combine to give rise to wound closure, we investigate the dependence of wound closure on pairs of parameters. In Fig 11, wound area fold change is plotted as a function of the parameter pairs (m, p − d), (p − d, γ_p), (m, γ_m).

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Fig 11. Observed wound area fold change plotted as a function of combinations of posterior means.

Each three-dimensional plot shows the multivariate dependence of wound area fold change as a function of various parameters. The colour key denotes the different clusters found using K-means clustering with K = 3. Star denotes Mock, diamond denotes CDC42 knockdown and plus denotes CDH5 knockdown.

https://doi.org/10.1371/journal.pcbi.1010191.g011

Fig 11 suggests that high intrinsic motility rates can counteract low intrinsic proliferation rates to close the wound rapidly, but not vice versa. Moreover, the density-dependent mechanistic parameters have a profound impact upon wound area fold change. Between intrinsic motility, m, and density-dependent motility, γ_m, density-dependent interactions clearly play a crucial role. At the same time, across the range of values for the intrinsic net proliferation rate, p − d, the extent of density-dependent proliferation, γ_p, influences wound healing by increasing the extent of wound closure at similar levels of p − d.