Computational model of 3D cell migration based on the molecular clutch mechanism

The external environment is a regulator of cell activity. Its stiffness and microstructure can either facilitate or prevent 3D cell migration in both physiology and disease. 3D cell migration results from force feedbacks between the cell and the extracellular matrix (ECM). Adhesions regulate these force feedbacks by working as molecular clutches that dynamically bind and unbind the ECM. Because of the interdependency between ECM properties, adhesion dynamics, and cell contractility, how exactly 3D cell migration occurs in different environments is not fully understood. In order to elucidate the effect of ECM on 3D cell migration through force-sensitive molecular clutches, we developed a computational model based on a lattice point approach. Results from the model show that increases in ECM pore size reduce cell migration speed. In contrast, matrix porosity increases it, given a sufficient number of ligands for cell adhesions and limited crowding of the matrix from cell replication. Importantly, these effects are maintained across a range of ECM stiffnesses’, demonstrating that mechanical factors are not responsible for how matrix microstructure regulates cell motility.

129 with "1" indicating seeded cells, "2" indicating cells generated from generation 1 and so 130 on. 140 Environment drag, F drag , from the ECM viscous environment is: 142 where c = 6 is a cell shape parameter, assuming a spherical cell of radius, , in 143 an infinitely viscous medium; = 55 Pa-s is the effective viscosity of the medium; is 144 cell velocity and implies a velocity-dependent opposing force, associated with the 145 viscoelastic character of the surrounding ECM (31).

146
Assuming force balance between and , the total distance a cell can 147 move at each timestep is computed by multiplying the velocity, , by the time step. The 148 model then evaluates a probability for migration, based on the ratio between the 149 distance between neighboring pores in the matrix and the distance the cell can move, 150 as: Based on the migration 151 probability, the model evaluates each cell movement. If a cell successfully migrates to a 152 neighboring pore, the origin pore and destination pore are updated to account for 153 crowding during the simulation.

154
Following migration, cells have a chance to proliferate. In the model, cell 155 proliferation follows a constant pre-set probability or replication rate and, in most cases, 156 preferentially occurs following migration. However, in cases in which crowding prevents 157 migration, cells may proliferate without migrating. Newly created daughter cells can also 158 migrate, following the identical migration steps as the parent cell. However, unlike the 159 parent cell, daughter cells cannot proliferate after migration.

162
Integrin adhesions between each cell and the surrounding matrix surface are 163 treated as implicit molecular clutches that dynamically bind and unbind matrix ligands, 164 depending on the contact area between each cell and its pore. Each bond between 165 integrin and matrix ligands has a defined stiffness, which depends upon matrix rigidity,  In order to evaluate the effect of matrix microstructure on 3D cell migration, we first 208 ran simulations in which matrix rigidity was fixed and pore size was systematically varied.
209 We tested pore sizes from 60 µm to 120 µm, which models the pore sizes in soft tissue 214 µm/hr to a maximum of ~ 20 µm/hr (Fig. 4A).

215
We next tested the combined effects of pore size and porosity. Cells moved the 216 fastest at small pore sizes and high porosities, equating to a greater number of small 217 pores (Fig. 4B). With Y = 6.3 kPa and high porosity (80-95%), increasing pore size caused 218 a rapid decrease in cell migration speed from ~ 60 µm/hr to 10 µm/hr (Fig. 4B). With Y = 219 6.3 kPa and small pore size (60 µm), decreasing porosity from 95% to 50% decreased 220 migration speed from ~ 60 µm/hr to ~ 30 µm/hr (Fig. 4B). This suggests that cell crowding 221 limits migration. At low porosity, the same number of cells concentrate in fewer pores, so 222 movement into neighboring pores is limited since more of the neighboring pores are full.
223 With larger pore sizes (100 µm -120 µm), migration occurred at a slow rate of ~10 µm/hr 224 and was not further reduced by decreased porosity (Fig. 4B). Thus, when pores are large 225 and cells can make few contacts with their environment, reducing the space for this limited 226 contact does not reduce the migration speed.

227
We tested the conclusion that a larger ratio between surface area available for cell-

247
We next sought to understand how different pore sizes affect migration speed as 248 a function of lattice stiffness. In order to examine whether the biphasic trend of migration 249 speed relative to stiffness was independent of pore size, we varied pore size from 60 to 250 100 µm, a range that presented differences in cell motion at all porosities (Fig. 4B).
251 Migration speed decreased with increasing pore size, consistent with results in Fig. 4.
252 While the biphasic relation between migration speed and matrix stiffness was maintained 253 at all tested pore sizes, the magnitude of change with the different stiffnesses decreased 254 with increasing pore size (Fig. 5B). This finding is consistent with changes in adhesion 255 number and traction force with pore size, which also present a biphasic relation with 256 stiffness (Fig. S1A-B).

257
Since pore size reduced migration speed and reduced the effect of the biphasic 258 relationship between matrix stiffness and migration speed, we hypothesized that ligand 259 density controls migration speed via the molecular clutch mechanism. Ligand density 265 (meaning that all integrins become ligated) to 40% (less than half integrins become 266 ligated) corresponds to a proportional decrease in cell migration speed (Fig. 5C). Using

cells with larger diameters shifts the migration speed towards higher values at all ligand
268 densities, because of the larger surface area between cell and pore (Fig. S1C).
269 Accordingly, systematic variations of cell diameter or ligand density show a biphasic 270 relation of cell speed versus Y, where the peak is proportional to the relative cell surface 271 (Fig. S2A) and the number of adhesions, respectively (Fig. S2B). Collectively, these 272 results support the existence of a biphasic relation between cell migration speed and 273 matrix stiffness in different conditions of environment microstructure and ligand density.
274 The relation between cell migration speed and matrix stiffness has magnitude 275 proportional to the number of adhesions and cell traction force. The peak migration speed 276 corresponds to the stiffness at which the maximum number of adhesions is allowed, which 277 is also modulated by pore size (Fig. S1). This further indicates that the amount of surface 278 area available for the cells to bind, together with ligand density, determines the effective 279 number of possible adhesions mediating cell motion in 3D.

282
We next tested the effect of cell proliferation on 3D cell migration. Using a 283 physiological proliferation rate of 10 -7 s -1 and increasing pore size from 60 to 120 µm, our 284 model shows a reduction in average cell speed from 65 µm/hr to 10 µm/hr (Fig. 6A). This 285 results from the larger ratio between surface area and volume of smaller pores, allowing 286 cells to form more adhesions for the same number of cells in the pore. Additionally, this 287 is consistent with previous experiments on mouse fibroblasts in collagen-288 glycosaminoglycan scaffolds (5). More generally, using pore sizes smaller than 100 µm, 289 systematic variations in proliferation rate from 1 to 2.5 10 -7 s -1 reduced average migration 290 speed from ~ 65 µm/hr to ~ 55 µm/hr (Fig. 6A). At all proliferation rates, using variations 291 in stiffness from 2 to 8 kPa, a peak in migration speed was consistently observed for 4 292 kPa, (Fig. 6B), similar to runs with fixed cell density (no proliferations) (Fig. 5A). However, 293 both the maximum migration speed and the spread of this peak as a function of the 294 proliferation rate decreased with increasing proliferation rate ( Fig. 6B and Fig. S3A-B).
295 Smaller pores enhanced cell migration at all proliferation rates (Fig. S3C). Increasing the 296 proliferation rate increases the overall cell density, thereby preventing cells from migrating 297 to neighboring pores due to crowding. As a result, we expect the overall migration rate to 298 decrease.

299
In order to better understand why cell proliferation decreased cell migration speed, 300 we analyzed the distribution of velocities for different cell generations. The distribution of 301 cell migration speed shifted towards lower values by increasing cell generation (Fig. 6C), 302 indicating that crowding effects limit cell migration. Quantification of the fraction of 303 immotile cells at different proliferation rates showed that cells that cease moving, 304 decrease with increasing stiffness, up to 4 kPa, then increase (Fig. S3E). This 305 corresponds to the first increase in cell speed, followed by decreases as a function of 306 stiffness (Fig. S3D). When cells proliferate, the matrix becomes crowded, and cells do not 307 find space to move to the neighboring lattice points. This inhibited their displacement, 308 which decreased overall migration speed. Collectively, these data indicate that increased 309 cell proliferation decreases migration but does not affect how this speed varies neither as 310 a function of the matrix microstructure nor its mechanical properties. In fact, the decrease 311 of cell speed with increasing pore size and its biphasic relation with matrix stiffness is 312 maintained at all tested proliferation rates (Fig. 6A-B). 348 By contrast, the goal of our model is to understand fundamentally how cells move in 3D.
349 Hence the algorithm treats migration first and differentiation is an emergent property.    The 3D environment is generated as an object from a class known as Scaffold.
457 This class stores and keeps track of the properties of the 3D environment. To generate 458 the lattice, the total 3D volume, including empty and non-empty spaces, , is 459 calculated by raising the input side dimension of the environment to the third power: The total volume that is "porous" (empty) is calculated by multiplying the input percentage 462 of porosity by . Therefore, given a specific pore size, the porosity determines the 463 total empty (porous) volume in the lattice, as: The volume of each pore, , is calculated from the pore diameter, , with the 466 assumption that the pore is spherical: The 3D matrix is initially seeded with cells randomly throughout all the pores in 484 the matrix.
485 Data Generation