Fig 1.
Illustration of our computational model of isotropic cell spreading.
(A) 3D rendering shows the geometry and defines coordinates for an axisymmetric spreading cell on a flat surface. The enlarged inset (right) illustrates the stress balance at the free cell boundary. (B) Pre-contact adhesion stress effectively pulls the membrane onto the flat surface. The enlarged insets conceptually depict how the adhesion stress is computed for different ligand densities ρl. (C) Cortical tension and membrane curvature give rise to an effective inward normal stress. (D) The protrusion stress acts normal to the cell membrane and is concentrated at the region of the membrane closest to the substrate. (E) The cross-sectional snapshot of a simulation illustrates the mesh composed of quadrilateral elements used in the calculations, with tighter element packing closer to the flat surface. The computational domain only includes half of this mesh due to axial symmetry. Boundary regions are labeled Γ, and the vertical dashed line is the symmetry axis (r = 0) (F) This snapshot of a simulation illustrates the fluid velocity (vector field) and relative pressure (heat map) computed for a given cell shape with known boundary stresses.
Fig 2.
Summary of experimental results.
(A) The contact region of a spreading neutrophil was imaged using reflection interference contrast microscopy (RICM). Contact regions in this example are traced in yellow. The scale bar in the first image denotes 10 μm. (B) Contact area is plotted as a function of time for the spreading cell shown in (A). The spreading speed is defined as the maximum slope extracted from a sigmoidal fit, and the maximum contact area is given by averaging the contact area at the plateau. (C) Mean contact area was computed at discrete time values for aligned curves of cells spreading on different densities of IgG. Mean quantified IgG densities are reported in the figure legend as IgG molecules per μm2. Error bars represent standard error of the mean. (D) Average spreading speed was quantified for the different IgG densities. There is no significant difference between mean slope values. (E) Maximum contact area increases slightly as a function of IgG density. Statistically significant differences in maximum contact area values were verified using ANOVA followed by a Tukey post hoc test. Error bars in both (D) and (E) indicate standard deviation. The IgG density was quantified for each condition as described in [24].
Fig 3.
Predictions of the Brownian zipper model.
(A) A model cell spreading on the highest tested ligand density (100%) quickly approaches a spherical cap morphology. Time stamps are included for each snapshot. (B) Curves of contact area vs. time show that the spreading rate decreases monotonically. The contact area ultimately approaches steady-state values predicted by the Young-Dupre equation (dashed lines). Predicted equilibrium shapes are included on the right. (C) Log-log plot of contact area vs. time is nearly linear over the initial spreading phase. The slope of this line corresponds to the exponent of a power law describing contact area growth as a function of time. The dashed line shows a slope of 0.5 for reference.
Fig 4.
Relationship between ligand density and maximum contact area, as predicted by the Young-Dupre equation.
The adhesion energy γ, and therefore the ligand density ρIgG, can be expressed as a function of the contact area of the final spherical cap formed by a viscous droplet. The tested adhesion strengths are marked as red dots. Actual experimental values are shown for comparison. Error bars indicate standard deviation.
Fig 5.
Predictions of the protrusive zipper model in the absence of adhesion stress.
(A) For purely protrusion-driven spreading, lamellar pseudopods form as the protrusion stress increases over time. Time stamps of the the shown sample shapes are included. (B) Contact area increases sigmoidally over time for the protrusive zipper model, in good agreement with our measurements of the time course of the mean contact area on the highest density of IgG. Filled circles indicate where the shapes shown in panel A were computed. (C) Protrusion stress (governed by Eq (11)) and cortical tension (governed by Eq (1)) increase during cell spreading. (D) Varying the parameter s0 determines the thickness of the leading lamella, resulting in a thin pseudopod for s0 = 0.4 μm (top) and coordinated global cell deformation for s0 = 2.0 μm (bottom). The sample shapes are shown at 120 s. All other protrusive zipper simulations in this paper were carried out using s0 = 0.8 μm.
Fig 6.
Predictions of the protrusive zipper model with varying adhesion stress.
(A) Contact area increases faster and reaches higher values for higher ligand densities. The protrusion-only curve is shown for reference as a dashed line. (B) Magnified views of the leading edge of the model cell highlight subtle, adhesion-dependent changes of the pseudopod morphology. The shapes of the leading cell edge are shown for ρl = 0% and ρl = 100% at times ranging from 50 s to 130 s. On higher ligand density, both the extent of spreading and the dynamic contact angle increase. Other than that, the cell shapes exhibit little dependence on adhesion strength within the full protrusive zipper model.
Fig 7.
Predictions of the protrusive zipper model with discrete adhesion sites.
(A) Two time series (times shown at the left) of simulation snapshots illustrate how the progression of the model cell’s leading edge is affected by the spacing of discrete adhesion sites. Unoccupied binding sites are depicted by empty circles, whereas filled circles indicate that the cell membrane is locally attached. (B) Contact-area-versus-time curves share similar slopes over three orders of magnitude of ligand density, but plateau at different maximum contact areas.
Fig 8.
Direct comparison of experimental data with the predictions of the protrusive zipper model with discrete adhesion.
(A) Time courses of the cell-substrate contact area predicted by the protrusive zipper model with discrete adhesion are overlaid on experimental results for two different ligand densities. The model curves are not the result of nonlinear fits but simply the predictions based on reasonable choices of parameter values. (B) The predicted ligand-density-dependent increase in maximum contact area agrees well with experimental results obtained in frustrated phagocytosis experiments. Modeling only ligands as discrete entities results in an apparent overestimation of the maximum contact area at the highest IgG density (solid blue curve), indicating that the number of Fcγ receptors rather than the number of IgG ligands becomes the limiting factor of the maximum adhesion strength in this regime. A model version that accounts for the discrete nature of both ligands as well as receptors (Appendix F in S1 Appendices) improves the agreement with the data at the highest IgG density (dashed red curve). Error bars indicate standard deviation.