Modeling of Mechanosensing Mechanisms Reveals Distinct Cell Migration Modes to Emerge From Combinations of Substrate Stiffness and Adhesion Receptor-Ligand Affinity

Abstract

Mesenchymal cell migration is an integral process in development and healing. The process is regulated by both mechanical and biochemical properties. Mechanical properties of the environment are sensed through mechanosensing, which consists of molecular responses mediated by mechanical signals. We developed a computational model of a deformable 3D cell on a flat substrate using discrete element modeling. The cell is polarized in a single direction and thus moves along the long axis of the substrate. By modeling discrete focal adhesions and stress fibers, we implement two mechanosensing mechanisms: focal adhesion stabilization by force and stress fiber strengthening upon contraction stalling. Two substrate-associated properties, substrate (ligand) stiffness and adhesion receptor-ligand affinity (in the form of focal adhesion disassembly rate), were varied for different model setups in which the mechanosensing mechanisms are set as active or inactive. Cell displacement, focal adhesion number, and cellular traction were quantified and tracked in time. We found that varying substrate stiffness (a mechanical property) and adhesion receptor-ligand affinity (a biochemical property) simultaneously dictate the mode in which cells migrate; cells either move in a smooth manner reminiscent of keratocytes or in a cyclical manner reminiscent of epithelial cells. Mechanosensing mechanisms are responsible for the range of conditions in which a cell adopts a particular migration mode. Stress fiber strengthening, specifically, is responsible for cyclical migration due to build-up of enough force to elicit rupture of focal adhesions and retraction of the cellular rear. Together, both mechanisms explain bimodal dependence of cell migration on substrate stiffness observed in the literature.

Keywords: cell migration; computational modeling; discrete element method; mechanosensing; mesenchymal; traction force.

Figure 1

Schematic of cell model. (A) Tessellation of cell surface and substrate plane using triangles. Cell is a 3D object bound to a 2D substrate plane. (B) Cross-section of cell showing distinct cellular parts used in cell migration. (C) Forces involved in evolution of the mechanical system. The following relevant forces are indicated: cortical elastic spring (${\vec{F}}_{linear}$), cortical dissipation (${\vec{F}}_{dashpot}$), cortical bending (${\vec{F}}_{bend}$), local triangle and global cell area conservation (${\vec{F}}_A$), cell volume conservation (${\vec{F}}_{vol}$), membrane contact (${\vec{F}}_{MD}$), focal adhesion (${\vec{F}}_{FA}$), protrusion lamellipodium force (${\vec{F}}_{prot}$), internal counter force (${\vec{F}}_{cnt}$), and stress fiber (${\vec{F}}_{SF}$).

Figure 2

Implementation of mechanosensing mechanisms in the cell. (A) Focal adhesion (FA) disassembly rate (r_{off, FA}) decreases with force carried by FA (|${\vec{F}}_{FA}|$). Two values of parameter (ζ_FA) are displayed to demonstrate effect of focal adhesion maturation (FA_mat). (B) Step increase in force carried by ECM (|${\vec{F}}_{ECM}|$) bound to a FA and applied by a stress fiber (SF). Steps correspond to stalling in SF contraction. The stiffer the substrate, the faster a SF strengthens, taking more steps during its lifetime. Graphs shown are a result of a simulation of an isolated two-spring system representing the FA and ligand molecule (ideal case). Substrate stiffness values shown: soft (1.2 × 10⁻² N/m), med (2.4 × 10⁻² N/m), stiff (4.16 × 10⁻² N/m). (C) Dependence of factor φ on change in length of SFs (ΔL_fib). The factor is multiplied by SF force (${\vec{F}}_{SF}$), implementing a weakening with strain (i.e., SF shortening).

Figure 3

Representation of the position of the cell cortex (side and top view) at different time points (start, middle, and end of simulation) (A). Example representations of the cell displacement and evolution of shape during a 60-min interval for the two identified migration modes: (B) progressive retraction (mechanosensing setup: FA_mat[ON] || SF_str[ON], condition: k_ECM = 0.0416 N/m, ${\langle \lambda \rangle }_{FA}^0=0.3$ min) and (C) collective retraction (mechanosensing setup: FA_mat[ON] || SF_str[ON], condition: k_ECM = 0.5 N/m, ${\langle \lambda \rangle }_{FA}^0=33.3$ min). Shape shown for following intervals: t = 230 min (FA assembly), t = 250 min (collective rupture), t = 260 min (cell detachment), and t = 280 min (refractory period). Representations of the number of FAs at each interval (not cumulative), the cumulative number of ruptured FAs, and the traction values in time for corresponding simulations: (D) progressive retraction and (E) collective retraction (shading highlights the different events in a force build-up and rupture part of the migration cycle). The same time interval of 60-min was chosen for display to compare quantities in both retraction modes; this is the approximate duration of a single iteration of the migration cycle. The range in the y axes is also the same for each quantity in each mode for easier comparison.

Figure 4

Average displacement values (N = 5) for simulations for all four mechanosensing setups (FA_mat and SF_str being either ON or OFF); for each setup, 24 conditions were run, defined by parameter values for substrate stiffness (k_ECM, six values) and expected FA lifetime under no force (${\langle \lambda \rangle }_{FA}^0$, four values). Cells in conditions encased by blue lines displayed progressive retraction, while those encased by red lines displayed collective retraction. Values correspond to displacements over the entire duration of the simulations (24 h).

Figure 5

Summarized results for FA_mat[OFF] || SF_str[OFF]. (A) The top-left heat map presents the number of collective retraction events per condition; gray corresponds to conditions with no displacement. There was no collective rupture for this condition, and all cells displayed progressive retraction. (B) Other heat maps (top right) show the average actual lifetime of the FAs, the strengthening level of the SFs (n_{str, f}), the average number of FAs, and the cumulative number of ruptured FAs. (C) The scatter plots (bottom) show average traction over the last hour of simulation. Error bars correspond to standard error of the mean (N = 5).

Figure 6

Summarized results for FA_mat[OFF] || SF_str[ON]. (A) The top left heat map presents the number of collective retraction events per condition; gray corresponds to conditions with no displacement. Collective rupture for this condition required a stiff substrate and long-lived FAs. (B) Other heat maps (top right) show the average actual lifetime of the FAs, the strengthening level of the SFs (n_{str, f}), the average number of FAs, and the cumulative number of ruptured FAs. (C) The scatter plots (bottom) show average traction over the last hour of simulation. Error bars correspond to standard error of the mean (N = 5).

Figure 7

Summarized results for FA_mat[ON] || SF_str[OFF]. (A) The top-left heat map presents the number of collective retraction events per condition; gray corresponds to conditions with no displacement. There was no collective rupture for this condition, and all cells displayed progressive retraction. (B) Other heat maps (top right) show the average actual lifetime of the FAs, the strengthening level of the SFs (n_{str, f}), the average number of FAs, and the cumulative number of ruptured FAs. (C) The scatter plots (bottom) show average traction over the last hour of simulation. Error bars correspond to standard error of the mean (N = 5).

Figure 8

Summarized results for FA_mat[ON] || SF_str[ON]. (A) The top-left heat map presents the number of collective retraction events per condition; gray corresponds to conditions with no displacement. Collective rupture occurred over a larger range than any other setup and displayed a bimodal relation between cellular traction and substrate stiffness (k_ECM). (B) Other heat maps (top right) show the average actual lifetime of the FAs, the strengthening level of the SFs (n_{str, f}), the average number of FAs, and the cumulative number of ruptured FAs. (C) The scatter plots (bottom) show average traction over the last hour of simulation. Error bars correspond to standard error of the mean (N = 5).