Quantifying the role of diﬀerent surface coatings in experimental models of wound healing

. Abstract In vitro surface coatings are widely used to mimic the role of extracellular matrix in the in vivo environment. Diﬀerent eﬀects are reported for diﬀerent surface coatings, however, some of these results are inconsistent across the literature. To explore the role of diﬀerent surface coatings, we use a new modiﬁed stopper-based wound-healing assay, called a stopper assay , with two commonly used surface coatings: gelatin and poly-L-lysine (PLL). Our experimental data show the gap width decreases faster with the gelatin and PLL coatings. Similarly, the number of cells in certain subregions increases faster with these coatings. Unfortunately, neither of these observations provides deﬁnitive mechanistic insight into the role of the coatings. To provide such insight we calibrate the solution of the Fisher-Kolmogorov model to match the experimental data. Our parameter estimates indicate that both coatings signiﬁcantly increase cell motility without aﬀecting cell proliferation.

replicate. Here we use W (r) (t) to denote the experimental measurement of 128 the gap width for replicate r at time t. We then average the data to give   within Subregion 1 and Subregion 2 for replicate r at time t, respectively.

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We then average the data to give

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Since the Subregion 1 and Subregion 2 are located symmetrically about the 144 centre of the experimental field-of-view, we expect that the number of cells in 145 each subregion will be approximately equal. Therefore, we average the num-146 ber of cells in the two subregions for each experimental replicate to give We also average this quantity across the 148 experimental replicates to give N (t) = N 1 (t) + N 2 (t) /2. Again, we report 149 these averaged quantities and we approximate the uncertainty in these quan-150 tities by calculating and reporting the sample standard deviation. The Fisher-Kolmogorov model is a reaction-diffusion equation given by where C(x, t) [cells/µm 2 ] is the one-dimensional vertically averaged cell den-167 sity. To simplify Equation (1) we integrate both sides of the reaction-diffusion 168 equation with respect to y, and then divide both sides by Y to obtain We note that the second and third terms on the right side of Equation   It is worth noting that simplifying a two-dimensional reaction-diffusion model where C(x, t) is the cell density obtained by solving Equation (4) numerically. 195 We further average N 1 (t) and N 2 (t) to give    replicates, but first we will estimate the carrying capacity density and the 242 initial condition separately. we treat K as a constant.

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Unlike the carrying capacity density, we find that there are some differences 253 in the details of the initial conditions in the various experimental replicates.

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Therefore we specify a different initial condition for each replicate by counting 255 cells away from the initial gap. The initial condition for each replicate is given where W (r) (0) is the initial gap width in experimental replicate r, and C where j is an index indicating the time points, and W (r) max and N (r) max are the λ and ε that minimise E (r) , and we denote these estimates using an overbar,   Table 1 Estimates ofD,λ, and¯ from individual experimental data. All parameter estimates are given to two significant figures.

Numerical methods for solving the Fisher-Kolmogorov model
The one-dimensional Fisher-Kolmogorov equation is of the form the cell proliferation rate, and K [cells/µm 2 ] is the carrying capacity density.
To numerically solve Equation (S1), the spatial domain is discretised into M nodes using a central difference approximation with uniform spacing δx.
Here we denote C i as the cell density at a discretised node i, where i = 1, 2, 3, . . . , M .
The discretisation for the internal nodes at time t is as follows for i = 2, . . . , M − 1. We apply zero net flux boundary condition at both boundaries, i.e., C 2 = C 1 and C M = C M −1 . The initial condition is obtained by measuring cell densities from the experimental images at t = 0 h, with the average data and data for individual replicates given in Table S5 and   Table S16, respectively. The resulting system of nonlinear ordinary differential equations is integrated using a backward Euler method with constant time step We numerically solve the one-dimensional Fisher-Kolmogorov model (Equation (S1)) using a finite difference method, with C (r) (x, 0) and K measured from the experimental images. Using the numerical methods introduced in Section 1, we obtain the density profiles at t = 0, 12, 24, 36, and 48 h. We then compute W(t) and N (t), and with these estimates we calibrate D and λ for the regrouped control, gelatin, and PLL experiments. We systematically vary , from 0.01K − 0.25K, to identify the level set that minimises the leastsquares measure of the discrepancy between the data and the model solution, given in Equation (10) in the main manuscript.
To consider the variation in the parameter estimates, we calibrate the solution of Equation (S1) to match the W (r) (t) and N (r) (t) data from each individual replicate. We find that each case appears to have a well-defined minimum, from which we estimateD (r) ,λ (r) , and¯ (r) . We then average them across the replicates to giveD,λ, and¯ , which are listed in Table 1 in the main manuscript. In this supplementary material, we show histograms of the parameter estimates of the cell diffusivityD (r) and proliferation rateλ (r) for individual replicates in Figure S1. The estimated parameter values for individual replicates are listed in Table S1 -Table S3.  Table S1 Estimates ofD andλ for individual replicates in the regrouped control experiment.
All parameter estimates are given to two significant figures. The red column in each replicate indicates the level set which gives the minimum least-squares measure.

Regrouped control
Level set (% of K) 3 Experimental data describing the carrying capacity density, the initial cell density, and the initial gap width In this section, we discuss the measure of carrying capacity density, K, and initial cell density, C  Table S4.
To estimate C (r) 0 we count and average the number of cells in the same four identically-sized rectangular boxes for replicate r at t = 0 h. In Table S5 we show the estimates of C 0 for the control, gelatin, and PLL experiments, obtained by averaging C In addition, in Table S6 we show the data of the average initial gap width,    Table S5 Estimates of C 0 obtained by measuring and averaging cell numbers in the four identically-sized boxes in experimental images at t = 0 h.
All estimates are rounded to two decimal places.  4 Experimental data for individual replicates Table S7 -Table S9 list the data of W (r) (t) and N (r) (t) for the three replicates in the control experiment. Table S10 -Table S12 list the data of W (r) (t) and N (r) (t) for the three replicates in the gelatin experiment. Table S13 - Table S15 list the data of W (r) (t) and N (r) (t) for the three replicates in the PLL experiment. Table S16 shows the data of C (r) 0 and K (r) measured from individual replicates.                               Table S16 Experimental data of C (r) 0 and K for individual replicates in the three groups. All the data are given to two decimal places.