Fig 1.
Cell-substrate interaction model is illustrated on the right.
The biological details of the mechanisms involved in cell spreading are reduced to four major model components, labeled (a)-(d) in the right panel titled “Reduced mathematical model”. Both the cell and the substrate are two dimensional. The rheology of the cell is determined by local concentrations of filaments, organelles, and additional macromolecules, and upon applying an external force via microindentation, the cytoplasmic forces undergo relaxation [26]. We assume the average material properties of the cell to be hypoelastic (a), and we assume that the deformable substrate is linearly elastic (d). Actin-based cell movement [24, 27] is described by an active rate of deformation tensor (b). The interaction between cell and substrate occurs via the FAs, which are modeled as collections of discrete linear springs (c). A description of the dynamic FA complex is simplified and written as stress dependent evolution of a single scalar field representing the volume fraction of the entire FA complex [4].
Fig 2.
In [15], specific parameter values are not provided and a dimensionless system of equations is solved. In the graph we use parameter values C1 = 1, μb = 0, ϵb = 1, β = 1, and ΔG = 0. Note that in the function graphed we add , whereas in Eq (8) the term
is subtracted. This is because compressive stresses, which are assumed in [15] to activate FA formation, are negative, and fa is denoted to be a positive force parameter.
Table 1.
Parameter values used in simulations.
Fig 3.
Comparison of simulations to cell spreading in a confined environment.
Our simulations are shown and corresponding experimental results can be found in Fig 3 of Wang et al. [39] (see center and rightmost panel in middle row.) We illustrate numerically computed substrate displacements and FA tractions. FA traction magnitudes are shown as contours and traction vectors are also illustrated. We note that numerical results compare well to experiments both qualitatively and quantitatively. Cell and substrate Young’s modulus: Ec = 0.5 kPa, Es = 1 kPa. FA springs do not rupture.
Fig 4.
Comparison of simulations to contraction of a cardiomyocyte.
Our simulations are shown in the left panel, and experiments from Hersch et al. [40] are on the right (with permission under CC-BY 4.0 license). Experimental result figures were modified from original by including scale bar and removing letter referring to original figure from top left corner. Scale bar = 10 μm. The substrate displacements from simulations compare well to those obtained experimentally both qualitatively and quantitatively. FA tractions, whose magnitudes are illustrated as contours, compare well qualitatively. Cell and substrate Young’s modulus: Ec = 20 kPa, Es = 15 kPa. FA springs rupture at a stretch of 0.4 μm.
Fig 5.
Initial FAs are placed at a radial distance of R = 4 μm (left) or R = 7 μm (right). From this initial configuration FA location evolves in a stress dependent manner as described by Eq (9).
Fig 6.
Effects of critical FA-cell area ratio and substrate stiffness on cell response.
The horizontal axis represents variations in critical FA-cell area ratio, while line colors/markers correspond to different substrate stiffnesses; the legend for all figures is illustrated in the top left graph. The first column of graphs corresponds to data for cells with central, weak FAs (R = 4), the second column corresponds to peripheral, strong FAs (R = 7, no FA break), and the third column corresponds to peripheral, weak FAs (R = 7).
Fig 7.
Dependence of FA volume fraction ϕ and bulk stress on spread cell radius.
We consider four different substrate Young’s modulus, FA-cell area ratio combinations: black- Es = 2.5 kPa, 10%, blue- Es = 2.5 kPa, 30%, red- Es = 100 kPa, 10%, magenta- Es = 100 kPa, 30%. A legend for all graphs is in the top left panel, and the initial FA configurations for data illustrated in a given row are shown in the left-most column.
Fig 8.
Spatiotemporal evolution of ϕ (top row), cell stress (middle row), and substrate displacement (bottom row) at time = 30, 60, 90 min.
FA-cell area critical ratio: 30%; Es = 2.5 kPa; No FA breakage. Black outlines indicate the location of the FAs. Vectors in cell stress contour graphs (middle row) show direction of forces arising from FA attachments. Note that as the cell spreads these forces are oriented toward the cell center.
Fig 9.
Comparison of computed total compressive forces and experimentally measured attachment forces.
(a) Radial compressive forces are determined from the computed radial stress data illustrated in Fig 7. Radial compressive forces are obtained by integrating the average bulk stresses over the cell area and averaging over the angle 2π. Different colors correspond to different combination of substrate Young’s modulus and critical FA-cell area ratio. (b) Attachment forces computed from adhesion strength data in Fig 5 in [10]. Attachment forces were computed by multiplying experimentally measured attachment strengths by total attachment area. Attachment strengths were determined by counting remaining numbers of adherent cells at the end of a spinning platform experiment. Three attachment configurations were considered: A 6 μm circular patch with a total adhesive area of 28 μm2, a 10 μm ring with a total adhesive area of 28 μm2, and a 10 μm circle with total adhesive area of 78 μm2. Each spinning platform experiment was performed in triplicate. Attachment forces computed from average attachment strengths and corresponding standard deviations are illustrated. More details can be found in [10]. Experiments confirm that as total attachment area increases and as attachments form closer to the periphery, attachment forces increase.