A Simulation of Multiple Cell Boundary Behaviors on an Asymmetrically Constrained Lattice

 A first graphical result of the COMSOLDIP finite element simulation has just come out. I'm so excited right now. The first picture below is a standard COMSOL generated picture of displacement contour. The upper right of the square shaped lattice is fixed - that is, the nodal points in the mesh will have fixed translational and rotational degrees of freedom - and the rest of the boundaries are all free to move. Because of this asymmetrical boundary condition, given the inward cell traction force around the focal adhesion boundary, the resulting strain distribution is asymmetric.

The figure below shows that there is a noticeable imbalance in the strain distribution on the lattice. The existence of higher strains on the upper left region is explained by its close proximity to the "free" boundary condition (this will allow the elements in this region to move more freely than those around the fixed boundary). Our assumption is that this asymmetry will be reflected on the boundary shape change of cell, thus the name, "mechanosensitive cell".


The above FEM result only shows one cell on the lattice. Nevertheless, my interest grows further to the point of planting multiple cells on a single lattice and a following expectation is to see some observable trend in cell shape changes. Of course, my FEM code is not mature enough to plant more than one cell simultaneously, but I went around to this issue by inserting an iteration loop such that every time at the end of a calculation the code will automatically change the cell location and repeat FEM analysis until it completes the simulation on all the locations I specify.



I decided to plant an array of 8 by 8 cells, all equidistant from each other, on the same lattice and ran the software for 8 hours. It usually takes four and a half hours but I accidentally aborted the code almost to the end of the run. Silly! I had to run again. Anyway, once I had all the 64 results, then it is pretty much a strict manual labor to gather and superimpose all the data. After taking about three hours of this brute force labor, the following picture is the result.


Just a side thing before I go on, I tried to pick a perfect color that goes together when I made the above figure. I eventually settled with the two colors (see figure), but I still don't like it. Any suggestion will be welcomed. The red circles marks the initial locations of each cell (they are just markers) and green loops mark deformed cell boundaries. The initial shape of the cells is a perfect circle. To me, they rather look like a contour plot of a differential system with many many poles (or sinks, whatever).

Digressing from something pretty much randomly is my specialty and since I already showed my magnificent specialty, let's go back to the figure again. Indeed there is some observable trend! It appears that, on the top boundary, the fixed constraint has prevented the cell from orienting in vertical directions. This makes sense since if the cells oriented in vertical directions, doing that would require relatively more energy, thus rather unnecessary and unnatural from the perspective of the cells. Unless the cells have some biological preferences, i.e.,resistances, the logical conclusion is that they would not align vertically. Since every cell has some degree of inward traction force around its membrane, the net result is the asymmetric change of cell boundary shapes. 

Based on my explanation, one can easily guess what will happen to the cells in the middle of the boundary where they are virtually far away from the asymmetrical boundary condition. In this region, cells do not feel the variation in lattice stiffness that the net result from the cell traction force will (and since traction force is uniform) be uniform around the focal adhesion boundary. The initial cell shape before the finite element simulation was assumed to be a perfect circle. Considering this in mind, the cells in the middle region should preserve their circular shapes which indeed is the case.

This figure shows a raw output graph from my FEM Displacement Integration Module.

2 comments:

faith said...

so nice your simulation and results, an I ask, how did you calculate the traction forces?
I read somewhere that cells in the soft regions are more round. It seems near to free boundary the region is soft and the cell is more cappibale to deform beside, in the middle aera the substrate is more stiff than free boundary ones. but your result are varce versa. am I rong? can you explane me how you estimate new shape for cell while it changes its shape? because I am working on cell proliferation and I have problem with my subroutine to define the cell hsape.

faith said...

so nice your simulation and results, an I ask, how did you calculate the traction forces?
I read somewhere that cells in the soft regions are more round. It seems near to free boundary the region is soft and the cell is more cappibale to deform beside, in the middle aera the substrate is more stiff than free boundary ones. but your result are varce versa. am I rong? can you explane me how you estimate new shape for cell while it changes its shape? because I am working on cell proliferation and I have problem with my subroutine to define the cell hsape.