In vitro surface coatings are widely used to mimic the role of extracellular ma- trix 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 coat- ings: 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 coat- ings. 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. Preprint submitted to Chemical Engineering Science 26 February 2020 Key words: Wound healing assay; Stopper assay; Surface coating; Fisher-Kolmogorov; Reaction-diﬀusion; Transport ∗Corresponding author. Email address: w1.jin@qut.edu.au (Wang Jin). 2 1 Introduction 1 In vitro surface coatings are used as a fundamental experimental approach to 2 mimic and study the eﬀect of the extracellular matrix (ECM) in the in vivo 3 environment. It is important to include eﬀects of the ECM in experimental 4 studies because the ECM interacts with cells both physically and chemically, 5 aﬀects cell behaviour, and plays an important role in biological processes such 6 as tissue development and cancer progression (Hay 2013; Horiguchi et al., 2012; 7 Screen et al., 2015; Vedula et al., 2013). Popular choices of coating reagents 8 include gelatin, poly-L-lysine (PLL), collagen, and ﬁbronectin (Liberio et al., 9 2014; McCarthy et al., 1983; McIntosh et al., 1988). The precise role of sur- 10 face coatings, however, remains unclear since experimental observations are 11 inconsistent. For example, gelatin is thought to increase both cell motility and 12 proliferation (McCarthy et al., 1983; McIntosh et al., 1988), while PLL can 13 have the opposite eﬀects (Liberio et al., 2014). In contrast, other studies sug- 14 gest that PLL may have no impact on cell motility or proliferation (Fischer 15 et al., 2007; Rangappa et al., 2000). 16 Among various types of in vitro experiments, wound-healing assays are widely 17 used to study cell migration and cell proliferation under diﬀerent conditions 18 including the use of diﬀerent surface coatings (Ascione et al., 2017; Liberio et 19 al., 2014; Tremel et al., 2009). Wound-healing assays are initiated by creating a 20 scratch in a monolayer of cells, followed by the observation of how the remain- 21 ing cells migrate and proliferate to close the gap over time. This type of assay 22 can be used to provide quantitative insight by measuring rates of the gap clo- 23 sure or measuring the increase in the numbers of cells in particular subregions 24 of the experiment (Johnston et al., 2014; Liberio et al., 2014; McCarthy et al., 25 1983; Treloar and Simpson, 2013). However, these measurements alone can- 26 not tease apart the intricate interplay between the eﬀects of cell migration and 27 cell proliferation that can sometimes lead to surprising results. For example, 28 Barrandon and Green (1987) measure the temporal change in the numbers of 29 cells in certain in vitro colonies and note a dramatic increase in cell numbers 30 3 when TGF- αis included. This increase in cell number is due to the interplay of 31 cell migration and cell proliferation, since TGF- αpromotes migration, which 32 then in turn provides space for the cells to proliferate (Barrandon and Green, 33 1987). 34 Mathematical models have been applied to mimic both in vitro and in vivo 35 experiments to test biological hypotheses and to predict experimental out- 36 comes (Jin et al., 2018a; Jove et al., 2019; Nardini et al., 2016; Sheardown and 37 Cheng, 1996; Tam et al., 2019; Villella et al., 2019). One of the most com- 38 monly used models to study collective cell migration is the Fisher-Kolmogorov 39 model (Fisher, 1937; Maini et al., 2004; Sherratt and Murray, 1990; Simpson 40 et al., 2006). The Fisher-Kolmogorov model describes the interplay between 41 cell migration and cell proliferation by assuming (i) cells migrate according to 42 undirected linear diﬀusion, and (ii) cells proliferate logistically to a carrying 43 capacity density. In previous studies, the Fisher-Kolmogorov model has been 44 calibrated to match experimental data from in vitro wound-healing assays, 45 and has been used to provide mechanistic insight into the role of the initial 46 seeding density, the wound geometry, and the shape of the migrating front 47 (Jin et al., 2016; Jin et al., 2018b; Sengers et al., 2007). 48 In this work we apply a new modiﬁed stopper-based in vitro wound-healing 49 assay, called a stopper assay , to explore the role of two widely used surface 50 coatings: gelatin and PLL. Compared to other commercial experimental ap- 51 paratus (Liberio et al., 2014; Qi et al., 2019), one important feature of the 52 stopper assays is that we can study two diﬀerent coating conditions simulta- 53 neously in one experimental well. This allows us to make direct comparisons 54 of how diﬀerent coatings aﬀect gap closure in the same ﬁeld-of-view without 55 being concerned by potential diﬀerences caused by monitoring diﬀerent exper- 56 imental wells. Our experimental data describes the gap width and the increase 57 in the numbers of cells in particular subregions in the experiment. This data 58 indicates that the gap closes faster, and the cell population increases faster in 59 the gelatin- and PLL-coated regions. By calibrating the Fisher-Kolmogorov 60 model to match the experimental data, we ﬁnd that both gelatin and PLL 61 4 coatings signiﬁcantly increase the cell diﬀusivity without aﬀecting the rate of 62 cell proliferation. 63 2 Methods 64 2.1 Experimental methods 65 Stopper assays are performed in six–well experimental plates (Corning, USA), 66 in which each experimental well has a diameter of 35 mm. Schematics of the 67 stopper assays are illustrated in Figure 1. The lower semicircular surface of the 68 experimental well is covered with tape (8018, 3M) to form the uncoated control 69 area (Figure 1(a)). For the surface coating, sterile 0.1% (w/w) gelatin (Sigma, 70 USA) or 0.1 µg/mL poly-L-lysine (Sigma, USA) prepared in 1X phosphate- 71 buﬀered saline (PBS) is added to the well and incubated at 37 °C for 30 72 min. After the excess liquid is aspirated completely, the tape is removed, with 73 only the upper semicircular surface of the well coated (Figure 1(a)). In the 74 remainder of this work, we refer to the upper and lower semicircular surfaces 75 of the experimental well as the upper half and the lower half , respectively. 76 5 Upper Lower 0 h (f)Upper LowerUpper Lower Experimental well Stopper UpperStopper Medium Cells(g) (h)(b) (c) (a) (d) (e) (i) (j)48 h Fig. 1. Illustrations of in vitro stopper assays. (a) Experimental well in which the upper half is coated. (b) A stopper for creating the cell-free region. (c) Experimental well with a stopper placed in the middle. (d) Section of the stopper assay with a stopper in place. The thick green line indicates the upper half and the yellow dots indicate cells. (e) Section of the stopper assay after a cell monolayer is grown and the stopper is removed. (f) An illustration of how the data are retrieved from the stopper assay. The white space indicates the initially vacant area. The black rectangles correspond to regions where two experimental images are taken. (g)–(h) Experimental images taken at t= 0 h in the indicated upper and lower halves, shown in (f). (i)–(j) Experimental images taken at t= 48 h in the indicated upper and lower halves. The yellow line indicates edges of the gap. The scale bar corresponds to 500 µm. 6 We perform the experiments with the mouse embryonic ﬁbroblast cell line 77 NIH 3T3, purchased from the Bioresource Collection and Research Center 78 (BCRC), Taiwan. The cell culture medium consists of Dulbecco’s Modiﬁed 79 Eagle’s medium (DMEM, Gibco, USA) and 10% calf serum (CS, Invitrogen, 80 USA). Although the calf serum can adhere to the surface of the experimental 81 wells and potentially interact with surface coatings, many studies that use calf 82 serum do not consider this to be an explicit coating treatment (Gospodarowicz 83 et al., 1983; Liberio et al., 2014; Rangappa et al., 2000). Cells are incubated 84 in cell culture ﬂasks (Corning, USA) under 5% CO 2at 37 °C. 85 A customized I-shaped stopper (Figure 1(b)), made of polydimethylsiloxane 86 (PDMS, Dow Corning SylgardTM184 Silicone Elastomer, USA), is placed 87 into the experimental well so that it is perpendicular to the boundary of 88 the upper and lower halves (Figure 1(c)). Approximately 2.5 ×105cells are 89 seeded uniformly into the six–well experimental plates and incubated overnight 90 (Figure 1(d)). The stopper is then removed to create an initial gap (Figure 91 1(e)). After removing the stopper, cells are free to migrate and proliferate, 92 eventually leading to the closure of the initially vacant space in both the 93 upper and lower halves (Figure 1(g)-(i)). The gap width and the number of 94 cells in certain subregions are measured at ﬁve equally-spaced time points: 95 t= 0,12,24,36, and 48 h. Each experiment is performed three times so that we 96 can extract data from each replicate and average the data across the replicates. 97 In this work we use data describing individual experimental replicates as well 98 as the averaged data. To distinguish these two types of the data, we use 99 superscripts on certain variables to indicate the individual replicate number, 100 and we use the variables with a tilde to indicate the averaged data. 101 Before presenting our experimental and modelling results, we now describe 102 some of the terminology we use to describe the experimental protocol and ex- 103 perimental data. Some experiments do not involve any coating at all, and we 104 refer to these experiments as control experiments. Other experiments involve 105 applying diﬀerent coatings to the experimental well, of which the upper half is 106 coated with gelatin or PLL and the lower half is uncoated (Figure 1(f)). This 107 7 means that we have two diﬀerent types of control assays: one type corresponds 108 to the control experiments, and the other type corresponds to the uncoated 109 regions in the coating experiments. In Section 3.1 we plot the average data 110 for the control experiment as well as the average data in the upper and lower 111 halves of the wells for the experiments with gelatin and PLL coatings. When 112 we calibrate the solution of the Fisher-Kolmogorov model to match the ex- 113 perimental replicates in Section 3.3, we combine data from the two types of 114 control assays which we refer to as the regrouped control experiment . 115 2.2 Edge detection and cell counting methods 116 We use ImageJ to detect edges of vacant areas and measure the gap areas 117 from the experimental images at t= 0,12,24,36, and 48 h (Schindelin et al., 118 2015; Simpson et al., 2013). Details of the methods are described in Jin et al. 119 (2018b). Examples of experimental images, each of which has a length of 1960 120 µm and a width of 1288 µm, with detected gap edges superimposed, are shown 121 in Figure 2. In the remainder of this work we refer to the area contained in 122 the experimental image as the experimental ﬁeld-of-view , since these images 123 show relatively small subregions within the entire experimental well (Simpson 124 et al., 2018). Estimates of the gap width at each time point is calculated 125 by taking the gap area divided by the width of the experimental ﬁeld-of- 126 view. This calculation is performed at each time point for each experimental 127 replicate. Here we use W(r)(t) to denote the experimental measurement of 128 the gap width for replicate rat timet. We then average the data to give 129 /tildewiderW(t) = (1/R)/summationtextR r=1W(r)(t), whereRis the number of replicates. We report 130 the sample mean as well as the variability across the replicates by calculating 131 and reporting the sample standard deviation. 132 8 0 h 24 h 48 h Control Gelatin PLL(a) (b) (c) )f( )e( )d( )i( )h( )g( Subregion 1 Subregion 2Subregion 1 Subregion 2Subregion 1 Subregion 2 Fig. 2. Experimental images at t= 0, 24, and 48 h from stopper assays. (a)–(c) Experimental images of the control experiment. (d)–(f) Experimental images of the gelatin–coated experiments. (g)–(i) Experimental images of the PLL–coated experiments. All images show the experimental ﬁeld-of-view in the upper half of the experimental well, indicated by the black arrow in Figure 1(f)–(g). The yellow line indicates the detected edge of the gap area produced by ImageJ. The rectangle highlighted in blue indicates the subregions in which cell numbers are counted over time. The scale bar corresponds to 500 µm. 9 We manually count the number of individual cells contained in two rectangular 133 subregions at t= 0,12,24,36, and 48 h. Each subregion, highlighted in blue 134 in Figure 2, measures 1288 µm×150µm. The two subregions are located 300 135 µm away from the left and right boundaries of the experimental ﬁeld-of-view, 136 respectively. For simplicity we refer to the left-most subregion as Subregion 137 1and the right-most subregion as Subregion 2 . These estimates are obtained 138 at each time point for each experimental replicate. Here we use N(r) 1(t) and 139 N(r) 2(t) to represent the experimental measurement of the number of cells 140 within Subregion 1 and Subregion 2 for replicate rat timet, respectively. 141 We then average the data to give/tildewiderN1(t) = (1/R)/summationtextR r=1N(r) 1(t), and/tildewiderN2(t) = 142 (1/R)/summationtextR r=1N(r) 2(t). 143 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 147 N(r)(t) =/parenleftBig N(r) 1(t) +N(r) 2(t)/parenrightBig /2. We also average this quantity across the 148 experimental replicates to give/tildewiderN(t) =/parenleftBig/tildewiderN1(t) +/tildewiderN2(t)/parenrightBig /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. 151 10 2.3 Mathematical model: the Fisher–Kolmogorov model 152 The Fisher-Kolmogorov model is a reaction–diﬀusion equation given by 153 ∂¯C(x,y,t ) ∂t=linear diﬀusion/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright D∇2¯C(x,y,t ) +logistic growth/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright λ¯C(x,y,t )/parenleftBigg 1−¯C(x,y,t ) K/parenrightBigg , (1) on 0≤x≤X, 0≤y≤Y, whereXandYare the horizontal length and verti- 154 cal height of the experimental ﬁeld-of-view, respectively (see Figure 3(a)–(b)), 155 andt≥0 is time. In this model ¯C(x,y,t )≥0 [cells/µm2] is the cell density, 156 D≥0 [µm2/h] is the cell diﬀusivity, λ≥0 [/h] is the cell proliferation rate, 157 andK > 0 [cells/µm2] is the carrying capacity density. Since all cells, both 158 inside and outside of the experimental ﬁeld-of-view, are uniformly seeded we 159 deﬁne our simulation domain to be a horizontal extension of the experimental 160 ﬁeld-of-view, by doubling its length to avoid boundary eﬀects. Therefore, we 161 solve Equation (1) on −X/2≤x≤3X/2, and 0≤y≤Y. Zero net ﬂux 162 boundary conditions are applied along the four boundaries (Jin et al. 2016). 163 Previously, Simpson (2009) showed that depth-averaging can be used to sim- 164 plify two-dimensional reaction-diﬀusion models into one-dimensional reaction- 165 diﬀusion models by averaging ¯C(x,y,t ) in the vertical direction, 166 C(x,t) =1 Y/integraldisplayY 0¯C(x,y,t ) dy, (2) whereC(x,t) [cells/µm2] 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 Yto obtain 169 ∂C ∂t=D∂2C ∂x2+D Y/parenleftBigg∂C ∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Y−∂C ∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle 0/parenrightBigg +λC/parenleftbigg 1−C K/parenrightbigg . (3) We note that the second and third terms on the right side of Equation (3) 170 vanish since ∂C/∂y = 0 along the boundaries where y= 0 andy=Y. 171 Therefore, Equation (3) reduces to 172 ∂C ∂t=D∂2C ∂x2+λC/parenleftbigg 1−C K/parenrightbigg . (4) 11 0X 0Y x (µm)y (µm) 0X x (µm)-X/2 3X/2 0Yy (µm) x1x2 x3x4(c)(b) (a)Fig. 3. Schematics showing the entire experimental well, the experimen- tal ﬁeld-of-view, and the simulation domain. (a) Entire experimental well. The white area shows the initial gap made by the stopper. The grey area shows the region where cells are uniformly seeded. (b) Experimental ﬁeld-of-view, where X= 1960µm,Y= 1288µm and the origin and coordinate system is shown in red. The yellow curve highlights the gap edges. (c) Simulation domain. The inner rect- angular region (solid lines) shows the actual experimental ﬁeld-of-view. The outer rectangular region (dashed lines) shows the extended simulation domain. The blue rectangular subregions indicate the areas where cell numbers are counted. It is worth noting that simplifying a two–dimensional reaction–diﬀusion model 173 into a vertically averaged one–dimensional model can introduce an error. How- 174 ever, this error vanishes when the initial condition is uniform in the vertical 175 direction (Simpson, 2009). Since the stopper assays do not involve any cell 176 density gradients in the vertical direction, there is no error associated with 177 averaging the two–dimensional reaction–diﬀusion equation into the simpler 178 one–dimensional reaction–diﬀusion equation. 179 12 2.3.1 Estimating gap width using the Fisher-Kolmogorov model 180 One feature of the Fisher-Kolmogorov model is that its solutions do not have 181 compact support (McCue et al., 2019). A level set /epsilon1∈[0,K], needs to be 182 nominated to represent edges of the gap. The choice of /epsilon1also aﬀects the com- 183 putation of gap width, W(t). To estimateW(t) from the numerical solution 184 of Equation (4), we calculate the xcoordinates, xlandxr, at left and right 185 side of the gap, respectively, which satisﬁes C(xl,t) =C(xr,t) =/epsilon1. With this 186 data the gap width is given by W(t) =xr−xl, providedxr> x l. Otherwise 187 W(t) = 0. When we calibrate the mathematical model to a particular exper- 188 imental replicate we use a superscript to denote the particular replicate. For 189 example,W(r)(t) denotes the gap width predicted by the model for experi- 190 mental replicate r. 191 2.3.2 Estimating the number of cells using the Fisher-Kolmogorov model 192 The number of cells within Subregion 1 and Subregion 2 can be calculated by 193 integrating the cell density, C(x,t), over the two subregions, giving 194 N1(t) =Y/integraldisplayx2 x1C(x,t) dx, andN2(t) =Y/integraldisplayx4 x3C(x,t) dx, (5) whereC(x,t) is the cell density obtained by solving Equation (4) numerically. 195 We further average N1(t) andN2(t) to give 196 N(t) =N1(t) +N2(t) 2. (6) Again, when we calibrate the mathematical model to a particular experimental 197 replicate we use a superscript to denote the particular replicate. For example, 198 N(r)(t) denotes the number of cells predicted by the model for experimental 199 replicater. 200 13 3 Results and discussion 201 3.1 Experimental estimates of/tildewiderW(t)and/tildewiderN(t) 202 Using the methods described in Section 2.2, we obtain estimates of/tildewiderW(t) 203 and/tildewiderN(t) for the gelatin, PLL, and control experiments (Figure 4(a)–(b)). 204 This average data indicates that/tildewiderW(t) decreases fastest for the gelatin–coated 205 experiments and slowest for the control experiments. We also see similar trends 206 in terms of/tildewiderN(t) data. Starting with/tildewiderN(0) = 0 in all cases, the experiments 207 with gelatin coating lead to the fastest increase in/tildewiderN(t) while the control 208 experiments lead to the slowest increase in/tildewiderN(t). 209 t [h]010002000W(t) [μm]Control Gelatin (upper) Gelatin (lower) PLL (upper) PLL (lower) 0150300N(t) [cells] 04 8 24 t [h]04 8 24(a) (b) ~ ~ ** * Fig. 4. Time evolution of /tildewiderW(t)and/tildewideN(t) (a) Time evolution of /tildewiderW(t) for the control, gelatin, and PLL experiments. (b) Time evolution of /tildewideN(t) for the control, gelatin, and PLL experiments. Data points correspond to the sample mean and the error bars indicate the sample standard deviation. The asterisk indicates the data with which the null hypothesis is rejected at the 5% level. To statistically identify the signiﬁcance of diﬀerence in both the/tildewiderW(t) and/tildewiderN(t) 210 data between the control and coating experiments, we conduct a two-sample 211 t-test for/tildewiderW(48) and/tildewiderN(48), respectively, with the null hypothesis that the 212 data in the control and coating experiments comes from independent random 213 samples from normal distributions with equal means and equal but unknown 214 variances (De Winter, 2013). The null hypothesis is rejected at the 5 % level 215 (p<0.05) for the experiments with gelatin coating in the/tildewiderW(48) data, and for 216 14 the experiments with gelatin and PLL coatings in the/tildewiderN(48) data. We obtain 217 much larger pvalues when considering results for the lower half of the wells 218 in the gelatin- and PLL-coated experiments, where the surface is uncoated. 219 This result conﬁrms our assumption that the two diﬀerent types of control 220 experiments we consider lead to indistinguishable results. 221 In summary, our experimental data shows that diﬀerent coatings lead to diﬀer- 222 ent estimates of/tildewiderW(t) and/tildewiderN(t). For example, both coatings lead to/tildewiderN(t) that 223 increases faster than/tildewiderN(t) for the control experiments. One naive way to inter- 224 pret this diﬀerence is that the coatings stimulate cell proliferation. However, as 225 we will show in Section 3.3, when we carefully interpret the experimental data 226 using the Fisher–Kolmogorov model it suggests that the coatings stimulate 227 cell migration but not cell proliferation. 228 To match the number of cells data with the solution of the Fisher-Kolmogorov 229 model, we integrate both sides of Equation (4) with respect to xacross the 230 two subregions to give 231 dN1 dt=ﬂux into the left boundary of Subregion 1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright −D∂C ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x1+ﬂux out of the right boundary of Subregion 1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright D∂C ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x2+population change in Subregion 1 due to cell proliferation/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright λ/integraldisplayx2 x1C/parenleftbigg 1−C K/parenrightbigg dx , (7) dN2 dt=ﬂux into the left boundary of Subregion 2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright −D∂C ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x3+ﬂux out of the right boundary of Subregion 2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright D∂C ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x4+population change in Subregion 2 due to cell proliferation/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright λ/integraldisplayx4 x3C/parenleftbigg 1−C K/parenrightbigg dx . (8) Equations (7)–(8) show that the rate of change of cell numbers in particu- 232 lar subregions is driven by a combination of the net ﬂux of cells across the 233 boundaries of the subregion and the eﬀect of proliferation within the sub- 234 region. Therefore, the increase in N(t) is driven by both the migration term 235 and the proliferation term in Equation (4). Similarly the decrease in the width 236 of the gap,W(t), is also driven by the combined migration and proliferation 237 terms in Equation (4). Without carefully interpreting the experimental data 238 15 using a mathematical model it could be very diﬃcult to separate the eﬀects 239 of cell migration from cell proliferation in these kinds of experiments. We will 240 now calibrate the solution of Equation (4) to match the data for individual 241 replicates, but ﬁrst we will estimate the carrying capacity density and the 242 initial condition separately. 243 3.2 Specifying the carrying capacity density and the initial condition 244 We directly measure the carrying capacity density, K, from the experimental 245 images (Supplementary Material). Direct counting of maximum cell densi- 246 ties at the ﬁnal time point of the experiments gives K= 2.43±0.36×10−3247 cells/µm2. We ﬁnd that our estimates of Kare very similar for the control, 248 PLL- and gelatin-coated experiments so we have pooled all these estimates 249 together (Supplementary Material). Since we ﬁnd that Kdoes not vary sig- 250 niﬁcantly between diﬀerent replicates or between diﬀerent coating conditions, 251 we treatKas a constant. 252 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. 254 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 256 by 257 C(r)(x,0) =  C(r) 0, x≤/parenleftBig X−W(r)(0)/parenrightBig 2, 0/parenleftBig X−W(r)(0)/parenrightBig /2<x</parenleftBig X+W(r)(0)/parenrightBig /2, C(r) 0, x≥/parenleftBig X+W(r)(0)/parenrightBig /2,(9) whereW(r)(0) is the initial gap width in experimental replicate r, andC(r) 0258 is the cell density away from the initial gap in experimental replicate r. We 259 extract these quantities from each experimental replicate and report them in 260 the Supplementary Material document. 261 16 3.3 Calibrating the cell diﬀusivity and the cell proliferation rate 262 In this section we calibrate the solution of Equation (4) to match both the 263 W(r)(t) andN(r)(t) data from each individual experimental replicate. We 264 solve Equation (4) using a ﬁnite diﬀerence method, with our measurements 265 ofC(r)(x,0) andK(Supplementary Material). In Figure 5(a)–(c) we plot 266 the solution of Equation (4), C(x,t) att= 0,12,24,36, and 48 h, for repli- 267 cate 2 of the regrouped control, and experiments with gelatin and PLL coat- 268 ings, respectively. Using these solutions we compute W(t) andN(t), and with 269 these estimates we calibrate D,λand/epsilon1using MATLAB’s lsqcurveﬁt algo- 270 rithm (MathWorks, 2020) to provide the best match to the experimental data 271 (Supplementary Material). We use a least–squares measure of the discrepancy 272 between the data and the model solution, given by 273 E(r)(D,λ,/epsilon1 ) =5/summationdisplay j=1/bracketleftBiggW(tj)−W(r)(tj) W(r) max/bracketrightBigg2 +/bracketleftBiggN(tj)−N(r)(tj) N(r) max/bracketrightBigg2 , (10) wherejis an index indicating the time points, and W(r) maxandN(r) maxare the 274 largest gap width and number of cells in experimental replicate r, respectively. 275 For the experiments with gelatin and PLL coatings, we have r= 1,2, and 276 3, while for the regrouped control experiment, r= 1,2,..., 9. Equation (10) 277 measures the diﬀerence between the model predictions and experimental data, 278 both scaled by the maximum value in the experimental observations. This 279 approach of scaling normalises the two diﬀerent types of the data so that both 280 of them are in the same order of magnitude. Calibrating the solution of the 281 Fisher-Kolmogorov model to match the data allows us to estimate values of D, 282 λandεthat minimise E(r), and we denote these estimates using an overbar, 283 ¯D(r),¯λ(r)and ¯/epsilon1(r)(Supplementary Material). Since these estimates do not vary 284 too much across the diﬀerent replicates, we average them across the replicates 285 to give ¯D,¯λ, ¯/epsilon1, together with estimates of variability across the replicates in 286 Table 1. 287 17 Gap width [μm] Number of cells [cells]Gap width [μm] Number of cells [cells] C(x,t) [cells/μm ] 2C(x,t) [cells/μm ] -980 0 1960 2940x [μm]02.4C(x,t) [cells/μm ]×10-32 0 4802000 1000 24Gap width [μm] t [h] 0300Number of cells [cells]0 48 24 t [h]-980 0 1960 2940x [μm]02.4 0 4802000 1000 24 t [h] 0300 0 48 24 t [h]200 100200 100-980 0 1960 2940x [μm]02.4×10-3 0 4802000 1000 24 t [h] 0300 0 48 24 t [h]200 100Experimental replicatesRegrouped control Gelatin PLL (a) (b) (c) (d) (e) (f) (g) (h) (i) 2×10-3 Model prediction Fig. 5. Comparing calibrated solutions of the Fisher-Kolmogrov model and the experimental data. (a)–(c) Numerical solutions of Equation (4) at t= 0,12,24,36, and 48 h, using ¯Dand¯λ, withC(2)(x,0) andW(2)(0) for experimental replicate 2. The arrow indicates the direction of increasing t. The orange dashed vertical lines indicate the experimental ﬁeld-of-view. (d)–(f) Time evolution ofW(2)(t) from the calibrated solution of Equation (4) superimposed on the data, W(r)(t),r= 1,...,R . (g)–(i) Time evolution ofN(t) from the calibrated solution of Equation (4) superimposed on the data, N(r)(t),r= 1,...,R . In (d), (g), R= 9. In (e), (f), (h), (i),R= 3. 18 Table 1 Estimates of ¯D,¯λ, and ¯/epsilon1from individual experimental data. All parameter estimates are given to two signiﬁcant ﬁgures. ¯D(µm2/h) ¯λ(/h) ¯/epsilon1(% ofK) Control 600±240 0.057±0.0061 17±2.7 Gelatin 1000±140 0.061±0.019 12±2.9 PLL 1000±390 0.059±0.012 17±2.9 Our estimates of ¯D,¯λand ¯/epsilon1are within previously reported ranges for ﬁ- 288 broblast cells (Jin et al., 2018b; Simpson et al., 2013; Tremel et al., 2009). 289 Comparing estimates of ¯Dand ¯λindicates that both the gelatin and PLL 290 coatings stimulate cell migration by about 70% compared to the regrouped 291 control experiment. However, the proliferation rate ¯λdoes not vary over the 292 three groups. Therefore, we ﬁnd that gelatin and PLL coatings lead to in- 293 creased cell motility, whereas neither coatings have any signiﬁcant impact on 294 cell proliferation. 295 Furthermore, our estimates of ¯Dfor the gelatin and PLL coatings are very 296 similar, suggesting that the impact of gelatin and PLL coatings on cell motility 297 is similar. Results in Figure 5(d)–(i) show W(2)(t) andN(2)(t) obtained using 298 the calibrated parameter values, superimposed with the W(r)(t) andN(r)(t) 299 data for the individual experimental replicates. These results conﬁrm that 300 the calibrated solution of the Fisher–Kolmogorov model is consistent with the 301 experimental data. 302 4 Conclusions 303 In vitro experimental approaches often study the eﬀect of surface coatings sim- 304 ply by comparing images of experiments with coatings to control experiments 305 without coatings. While such comparisons provide information about the net 306 eﬀect of the coating, the simple observations do not provide any insight into 307 19 how the coating aﬀects the cell-level mechanisms. For example, observing that 308 a particular coating increases the rate of gap closure provides no insight into 309 whether the increase in the closure rate is driven by an increase in cell motility, 310 an increase in cell proliferation, or a combined increase in cell motility and 311 cell proliferation. 312 In this work we take a diﬀerent, more quantitative approach to assess the 313 impact of diﬀerent coatings on in vitro wound–healing experiments. We are 314 motivated to take this approach because we aim to understand how diﬀerent 315 coatings aﬀect diﬀerent mechanisms. We consider three types of experiments 316 including control experiments without any coating as well as experiments with 317 gelatin and PLL coatings. In each experiment we extract two types of data: 318 (i) the gap width as a function of time; and (ii) the number of cells contained 319 within particular subregions as a function of time. By carefully calibrating 320 the solution of the Fisher–Kolmogorov model to match both types of data we 321 obtain estimates of the cell diﬀusivity, D, and the cell proliferation rate, λ. 322 Comparing estimates of Dandλbetween the control experiments and the 323 experiments with gelatin and PLL coatings indicates that these two coatings 324 increase cell migration by approximately 70% whereas the coatings have neg- 325 ligible impact on cell proliferation. 326 Overall, our results suggest that care ought to be taken when interpreting the 327 experimental data. For example, simply counting the number of cells within 328 particular subregions of the experiment shows that the number of cells in- 329 creases more dramatically in the experiments with PLL and gelatin coatings. 330 Our modelling suggests that this increase in cell number is driven by the coat- 331 ings stimulating cell migration without inﬂuencing the rate of cell proliferation. 332 This conclusion might not be obvious without interpreting our experimental 333 data with a mechanistic mathematical model. 334 Acknowledgments . We are supported by the Australian Research Council (DP170100474) 335 and the Taiwan Ministry of Science and Technology (MOST 106-2313-B-002- 336 031-MY3, MOST 107-2221-E-030-001-MY2). Wang Jin is supported by a QUT 337 20 Vice Chancellor’s Research Fellowship. We thank two reviewers for their help- 338 ful comments. 339