Figure 1.
Modeling of cell contraction on 1D elastic substrate with mixed boundary conditions.
(a) An elastic bar is discretized with 3 nodes with concentrated forces applying on each node along with their respective displacements. Note that this general loading is used for deriving stiffness matrix which uniquely relates nodal forces to the nodal displacement subject to any boundary condition. (b) A cell applies contractile forces on nodes 1 and 2 (i.e with known, measured displacements) while node 3 is free. This set of inputs constitutes a Mixed Boundary Condition, in that a combination of nodal displacements and forces are given () and their respective unknowns (
) are computed by the model.
Figure 2.
Two cells applying contractile forces on 2D elastic substrate.
In our finite element scheme, all nodal displacement underneath the cells on the top surface of the gel are measured, while for the nodes outside the cells all tractions are assigned zero and thus their displacements are not necessary to measure. All nodal displacements at the bottom and side walls of the gel are assigned zero (not shown in the figure). These combinations of data inputs constitute the set of Mixed Boundary Conditions in our FEM simulation. The computed parameters of the model are nodal traction underneath the cells as well as displacements of the extracellular nodes (traction-free nodes on the surface).
Figure 3.
The traction for the nodes far from the cells are zero, however errors will be introduced if the cell boundary is poorly defined and there are nodes that fall outside the presumed cell boundary where cells may apply traction.
In cases where the cell boundaries cannot be identified due to imaging conditions, displacements should be prescribed for regions nearby the cells.
Figure 4.
Validation of the accuracy and uniqueness of the finite element solution to extract 3D traction force fields.
(a) A computational model with two regions representing 2 separated cells, each 20 µm in diameter and separated by half-cell distance 10 µm, was established. A self-equilibrated force field was applied within each region. The magnitude and directions of forces were indicated by arrows. (b) The resultant full displacement field was obtained by ANSYS. (c) The displacement fields underneath each cell were chosen and assigned to the same model. The boundary conditions of nodes outside the regions were set traction-free. (d) A new force field was obtained using the above mixed-boundary condition. The magnitude and directions of nodal forces were shown by arrows. (e-f) The node-by-node difference between initially applied forces and retrieved forces (in x and z direction, respectively) are shown. The difference is <10−2 nN (within 1%).
Figure 5.
Verification of the uniqueness of solution of the traction field computed from the experimental displacement field.
(a) Phase-contrast pictures of 2 spatially isolated MKF cells on 1 kPa PA gels, cultured after 1 day. Scale bar: 15 µm. (b) The displacement fields underneath each cell were chosen for computation. (c) A larger area enclosing both the cells and neighboring area was chosen where displacement field was prescribed. In both cases, the nodes outside the selected regions were set traction-free. (d-e) The traction field computed by above 2 cases were visualized and compared by 2D contour plots (d-e) and 3D bar representation (g-h). Also, the node-by-node difference of traction fields computed using 2 selected schemes was illustrated by both 2D contour plot (f) and 3D bar representation (i). Dashed lines in orange outline the cells boundaries.
Figure 6.
Comparison of mixed-boundary condition method and full-field displacement boundary condition method.
(a) Phase contrast picture of a single cell cluster to be studied. Scale bar: 50 µm. (b) The displacement field generated by cell cluster on the top surface of substrate. (c-e) The traction field calculated by mixed-boundary method, and full-field displacement boundary method (with iterative calculation 1 time and 2 times, respectively), were shown respectively. The difference of RMS of the traction between mixed-boundary method and full-field displacement boundary method with 1 time iteration was 1.6×10−1 kPa, less than 3.8% of the maximum computed traction at cell cluster and substrate interface. The difference of RMS of their nodal force was 0.2 nN, which was 0.25% of the maximum nodal force at cell cluster and substrate interface. Dashed lines in orange outline the cells boundaries. (f-g) Histograms of nodal traction and force obtained by the two methods demonstrated good agreement between each other. (h) Sum of net forces and absolute forces calculated by the above three conditions. The force equilibrium was best satisfied in mixed boundary condition method, which is 6.69% of total force. (i) Sum of surface nodal force distribution calculated by above three conditions. The RMS results of nodal force calculated by mixed BC method and 1-time iteration method agreed within 4.96%. (j) Sum of surface nodal traction distribution calculated by the above three conditions. The RMS results of nodal force calculated by mixed BC method and 1-time iteration method agreed within 9.27%.
Figure 7.
(a-c) The convergence test was performed to determine the maximum fine mesh size needed to obtain the accurate solution.
The mesh sets with different Δx and Δy (Δx = Δy = 3.23 µm, 4.84 µm, and 6.45 µm, respectively) were tested respectively. The traction distribution map and traction magnitude histograms from three mesh-size displayed uniform feature patterns. Dashed lines in orange outline the cells boundaries. (d) All three cases showed sum ratio of net forces within 7%, satisfying the force equilibrium requirement. (e) The root mean square (RMS) difference of traction between 3.23 and 4.84 µm meshes was about 64.06 Pa (1.28% of maximum computed traction), and the difference between 4.84 and 6.45 µm mesh sizes was about 192.7 Pa (3.86% of maximum traction). The comparison indicates that when mesh size is reduced to 4.84 µm or below, the traction output starts to show minimum variation.
Figure 8.
Traction and force maps of single human colon cancer cell (HCT-8) cluster.
The cluster behaves as a single contractile unit. (a) - (c): Phase-contrast image, traction and nodal force map of a well-spread pre-MLP HCT-8 cancer cell cluster. The cells were cultured on 2 kPa hydrogel substrates. The distance between the nodes is about 5 µm. Scale bar: 60 µm. Colors of contour represent the magnitude of traction stress. Vectors indicate the direction of traction force at each node and arrow lengths represent the magnitude of node force. Dashed lines in orange outline the cluster boundary. (d) A free body diagram visualizes the mechanics of this long-distance force transmission. The cell cluster exerted contractile force on the substrate through the adhesion sites of the outer cells. The inner cells transmitted the force possibly through cell-cell junctions and cortical actin.
Figure 9.
Traction maps of two neighboring human colon cancer cell (HCT-8) clusters.
Their interior traction domains are dynamic. (a) - (c): Phase-contrast image, traction and nodal force maps of two independent cancer cell clusters cultured on 2 kPa flexible hydrogel. Cells were on culture day 5. Each cluster generated high traction well within the periphery, leaving the periphery almost traction-free. (d) - (f): Phase-contrast image, traction stress and nodal force maps of the merged pre-MLP HCT-8 cancer cell cluster after 24 hours (6th culture day). Following merging, many more cells in both the clusters participated in generating traction, and the net force increased by about 20 folds, although the direction of the net force did not change. Scale bar: 40 µm. Dashed lines in orange outline the cluster boundaries.
Figure 10.
Evidence of cell-cell compression in monkey kidney fibroblast (MKF) cluster.
(a) Several MKF cell clusters on 1 kPa PA gel. (b)-(c) The displacement and traction field produced by the clusters on the top surface of the substrate. The traction by the small clusters is negligible compared to those generated by the larger ones. Dashed lines in orange outline the cluster boundaries. (d) Nodal forces computed for the largest cluster. The finite element grid size is about 5 µm. There are regions in the cluster, shown by dashed lines, where repulsive forces appear on the substrate, i.e., cells “push” against each other. (e) To explain the cell-cell compression, a free body diagram is shown to reveal the intercellular force and cell-substrate traction force of 2 neighboring cells on the substrate. As the substrate is soft, the cells have less likelihood of spreading or wetting the substrate, but can adhere to the substrate due to the fibronectin functionalization. As cell proliferation and growth occur within the cluster, the cells push against their neighbors, generating an outward force on the substrate. Scale bar: 50 µm.