Supporting Material. A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition

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1 Supporting Material A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition S. Aland 1,, H. Hatzikirou 1,2, J. Lowengrub,, A. Voigt 1,2,4 1 Department of Mathematics, TU Dresden, Germany; 2 Center for Advancing Electronics Dresden (CfAED), TU Dresden, Germany; Department of Mathematics and Center for Complex Biological Systems, UC Irvine, USA; 4 Center for Systems Biology Dresden (CSBD), Dresden, Germany Corresponding authors: sebastian.aland@tu-dresden.de, lowengrb@math.uci.edu One-mode approximation A detailed analysis of the standard PFC model in Eq. (2) was performed in (4) for the case of q = 1. We also refer the reader to introductions on PFC modeling in (5, 11). For q = const the analysis follows along the same lines. We can represent the periodic solution by a one-mode approximation with the lowest-order harmonic expansion ρ = A(cos(2πx/a) cos(2πy/ a) cos(2πy/ a)/2) + ρ 0 where ρ 0 is the average epithelial cell density, A measures the amplitude of the epithelial cell density field and a is the distance between the cells. Recall that the density ρ actually represents the deviation from a reference value ρ (see Sec. MATERIALS AND METHODS in the main text). Using this expression in the PFC energy in Eq. (1) and minimizing with respect to A and a leads to A ± = 4 5 ( ρ 0 ± 1 15r 6ρ 2 0 ), a = 4π q, where solutions that minimize the energy are A = A + for ρ 0 > 0 and A = A for ρ 0 < 0. From the obtained values for a and A we see that the distance between cells scales with the square root of q, whereas the amplitude of the epithelial cell density field is not affected by q. The maxima of the above one-mode approximation are easily calculated as ρ peak = ρ 0 1.5A. We will use this value as ρ max. We can obtain the elastic properties of the hexagonal phase from the one-mode approximation by considering the energy cost of a deformation of the equilibrium state following (4). The resulting elastic constants are C 11 / = C 12 = C 44 = A 2 /16, independent of q. From these coefficients, we obtain the Poisson ratio ν = 1/ and the Young s modulus E = A 2 /2. These calculations are only valid within the one-mode approximation, assuming a perfectly hexagonal packing of the epithelial cells. As shown in (4) these approximations are in good agreement with the elastic properties obtained from the full PFC model, at least if r is small. In the simulations presented in this paper, we are within this regime, but far away from an equilibrium hexagonal state, which should be kept in mind by parametrizing the model according to experimental measurements of E. Mechanical properties of epithelial cell colonies have previously been considered in (8) within a onedimensional approach by using a homogeneous and isotropic material, parametrized by ν and E. However, in (8) the cells are coupled to a substrate and therefore the results can not be compared with our simulations in the current setting. The fixed Poisson ratio is within the range of typical soft materials (6). Using r = 0.9 and ρ 0 = 0.54 we obtain E = 0.4. Below, we demonstrate the effects of changing the Young s modulus E (see Fig. SI.). 1

2 The evolution equation for the density field In this section we we motivate the modifications of Eq. (2) that lead to Eq. (). To ensure the number of cells is conserved between mitotic or apoptotic events, we need local mass conservation for each cell. Hence, mass must be added when a local maximum shrinks and mass must be removed when a local maximum grows. The simplest way to do this is to add a source term to the standard PFC model. Hence, we propose the following evolution equation t ρ =η δe N δρ + α (ρ max ρ i ) max(ρ, 0)χ i, (SI.1) i=1 where α is a relaxation constant. Equation (SI.1) is designed so that for each cell, the local maximum of ρ stays close to the value ρ max, which approximates the cell density peak at equilibrium and can be a priori calculated from an one-mode approximation, see SI. Note that the evolution equation (SI.1) only prevents disappearance of cells since the source term (last term on the RHS) is restricted to the cell region by χ i. Analogously, a similar source term is added in the region without cells to ensure that no cells nucleate there. The resulting evolution equation reads: t ρ = η δe N δρ + α (ρ max ρ i ) max(ρ, 0)χ i + β(ρ min ρ)χ 0, (SI.2) i=1 where β is a relaxation constant and ρ min is a reference value of the density for the region without cells that can also be estimated from the one-mode approximation, see SI. In the literature on PFC models in condensed matter physics other approaches to ensure a conservation of particles/cells have also been discussed (, 10). Our results show that the dynamics of the modified PFC model is very similar to that of the original PFC mode. In particular, the source and sink correction terms on the RHS of Eq. (??) at each time step in the numerical scheme are very small because the the diffusional process of disappearance and nucleation of particles/cells is slow compared with other processes. Further, these corrections can be interpreted as penalty terms that seek to maintain the peak values of the density field at ρ max and the bulk value at ρ min. Phase diagram By comparing the free energies (for constant q) of a hexagonal cell ordering in equilibrium and a constant density, a phase diagram can be constructed. The phase diagram is shown in Fig. SI.1) in terms of the parameters ρ 0 and r. There are three phases: (i) constant density phases where there are no cells; (ii) coexistence phases where regions of hexagonally-arranged cells can coexist in equilibrium with regions where there are no cells; and (iii) hexagonal phases where there are only hexagonally-arranged cells in equilibrium. Results are independent of q. Numerical algorithm In the following we briefly mention some details of the numerical implementation of the model. The algorithm for one time step is summarized in Fig.??. The finite-difference scheme proposed in (1) is employed to solve the PFC equation. For the standard PFC equation this scheme is unconditionally energy-stable. The discretized equations are solved by an efficient nonlinear multigrid method proposed in (7). For the standard PFC equation this algorithm is first-order accurate in time and second-order accurate in space. We obtain similar results for the modified PFC equation used here. According to the phase diagram of the standard PFC equation we choose r = 0.9 together with ρ 0 = This choice ensures coexistence between a hexagonal epithelial cell packing and a region without cells in equilibrium, with a sharp transition between both phases, see SI for details. The parameters imply that ρ max = ρ peak = 0.8, the Poisson s ratio is ν = 1/ and the Young s modulus is E = 0.4 (see SI). Since the results are not sensitive to the value of ρ min, we take ρ min = 0.7. Other numerical parameters are time step t = 50s and grid size x = 0.59µm and we use N = 2048 grid points in each direction. The spatial domain is thus a square with area 1.46mm 2. 2

3 r Phase diagram constant coexistence E=0.411 E=0.75 E=0.20 hexagonal ρ 0 Figure SI.1: The phase diagram for the phase field crystal model. The region labelled coexistence marks the ranges of the parameters r and ρ 0 where regions with hexagonal cell ordering in equilibrium can coexist with regions with constant density (e.g., no cells). Note that ρ 0 actually represents the deviation from a reference value ρ (recall Sec.??). The blue circles indicate the three distinct parameter combinations together with the corresponding Young s moduli. To extract the peak positions from the density field we exploit that peak positions are known from the previous time step. For every i, all grid points in a neighborhood of the old peak position x i are traversed (here we use the 50 closest grid points). Then x i is set to the position where ρ assumes the largest value. For the generation of Voronoi cells, we use a weighted Voronoi tesselation where the distance to all peak positions is calculated for every grid point and divided by the radius of the corresponding cell. The closest peak position x i is used to label the grid point belonging to Voronoi cell i. Parameter variation We next demonstrate how the results of our simulations depend on the model parameters, including the mobility, the choice of cleavage plane, and the Young s modulus. Effect of mobility and cleavage plane. Fig. SI.2 shows the epithelial cell colony at final time t = 5.77d for various parameter combinations. The cleavage plane seems to have little impact on the colony morphologies (top) or the total number of cells (Fig. SI.2A). The only noticeable influence is on the colony boundary, which seems more rough for the worst angle of the cleavage plane (recall Fig.??). The colony boundary is extracted from the epithelial cell density field ρ, or the discrete representation χ i, using a segmentation algorithm based on a Mumford-Shah energy (9) with the initial noisy representation of the colony defined by 1 χ 0. This approach was shown to be robust for crystalline materials in (1, 2) and works well also in the present situation. The most striking observation from Fig. SI.2 is that the cell colony grows larger with increasing mobility since a larger mobility makes it easier for the cells to move. Fig. SI.2B shows that increasing the mobility delays the transition from exponential to quadratic growth making the transition occur at larger cell populations. Accordingly, the epithelial cell density (Fig. SI.2C) and the median of the epithelial cell area (Fig. SI.2D) remain constant for a longer time. Hence larger mobilities lead to a longer free-growth regime. A larger mobility makes it easier for the bulk epithelial cells to push outer cells aside to gain enough space to grow. Thus, bulk epithelial cells are less compressed and contact inhibition sets in later than for lower mobilities. The longer lasting exponential growth is not only reflected by the total colony area but also by the increased total number of epithelial cells for larger η (Fig. SI.2A). As shown in Fig.?? in the main text, the results collapse onto a single curve when plotted against a shifted time t and the colony size is rescaled. Accordingly, t shift = 16h, -24h and A ref = 0.41,.79 for η = 5, 20, respectively. Effect of elastic parameters To demonstrate the influence of E on our results we also consider r = 0.6, 0.8, 0.95 and keep ρ peak = 0.8 fixed, which results in a change of ρ 0 depending on ρ peak as indicated above. The resulting Young s moduli are E = 0.20, 0.75, 0.411, respectively. The three

4 best angle random angle worst angle Figure SI.2: (Top row:) Epithelial cell colony morphologies at final time t = 5.77d for different cleavage planes (best,random and worst angle (from left to right) and mobilities (η = 5, 10, 20 plotted together, (A) The total number of cells at final time for different cleavage planes and mobilities. (B) Total area of the spreading colony. The solid black line corresponds exponential growth with average cell cycle time 0.75d, (C) Average cell density for different mobilities, (D) Median of cell area distribution. Since the cleavage plane was found to have little influence, results in (B), (C) and (D) are shown for the best angle mechanism only. distinct parameter sets are marked in the phase diagram in Fig. SI.. All parameter sets are within the coexistence regime of the hexagonal phase and a constant phase. Simulation results in Fig. SI. shows that increasing E delays the transition from exponential to quadratic growth by increasing the ability of cells to push each other. This extends the free-growth regime in a manner analogous to that observed when the mobility is increased. As shown in Figs. SI.B and D, the results still collapse onto a single curve when plotted against a shifted time t and with rescaled colony size. Accordingly, tshif t = 0.6d, 0.2d and Aref = 0.45, 0.77 for E = 0.2, 0.75, respectively. Movie We provide a movie showing the evolution of the colony in the reference configuration (η = 10, best angle). Time is scaled with a factor of 21.8h/s. References 1. Rainer Backofen and Axel Voigt. A phase field crystal study of heterogeneous nucleation - application of the string method. European Physical Journal - Special Topics, 22(): , Benjamin Berkels, Andreas Ra tz, Martin Rumpf, and Axel Voigt. Extracting grain boundaries and macroscopic deformations from images on atomic scale. Journal of Scientific Computing, 5(1):1 2, Pak Yuen Chan, Nigel Goldenfeld, and Jon Dantzig. Molecular dynamics on diffusive time scales from the phase-field-crystal equation. Physical Review E, 79():05701,

5 Figure SI.: Colony area (A) and cell density (C) for different values of the Young s modulus of the epithelial cell cluster. Results collapse on one another when plotted in rescaled variables (see text) in (B) and (D) and show good agreement with experiments from (12) (see text). 4. K.R. Elder and M. Grant. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E, 70:051605, Heike Emmerich, Hartmut Löwen, Raphael Wittkowski, Thomas Gruhn, Gyula I. Tth, Gyrgy Tegze, and Lszl Grnsy. Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Advances in Physics, 61(6):665 74, N. Guz, M. Dokukin, V. Kalaparthi, and I. Sokolov. If Cell Mechanics Can Be Described by Elastic Modulus: Study of Different Models and Probes Used in Indentation Experiments. Biophysical Journal, 107:564, Z Hu, SM Wise, C Wang, and John S Lowengrub. Stable and efficient finite-difference nonlinearmultigrid schemes for the phase field crystal equation. Journal of Computational Physics, 228(15):52 59, A.F. Mertz, S. Banerjee, Y. Che, G.K. German, Y. Xu, C. Hyland, M.C. Marchetti, V. Horsley, and E.R. Dufresne. Scaling of traction forces with the size of cohesive cell colonies. Phys. Rev. Lett., 108:198101, David Mumford and Jayant Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(5): , Simon Praetorius and Axel Voigt. A phase field crystal approach for particles in a flowing solvent. Macromolecular Theory and Simulations, 20(7): , Nikolas Provatas and Ken Elder. Phase-Field Methods in Materials Science and Engineering. Wiley, Alberto Puliafito, Lars Hufnagel, Pierre Neveu, Sebastian Streichan, Alex Sigal, D. Kuchnir Fygenson, and Boris I. Shraiman. Collective and single cell behavior in epithelial contact inhibition. Proceedings of the National Academy of Sciences, 109():79 744, SM Wise, C Wang, and John S Lowengrub. An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM Journal on Numerical Analysis, 47(): ,