Biomedical Engineering Reference
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Fig. 3.1 Steady state focal
adhesion distributions for a
circular cell adhered to a flat
gel substrate of Young's
moduli varying from 2 to
200 kPa, where R is the
radius of the cell
focal adhesion is taken as a region where ξ H o is high. The results are presented in
Fig. 3.1 , where axial symmetry permits the distributions to be plotted versus radial
position. We find that on a relatively soft gel of E
2 kPa, the cell does not develop
prominent focal adhesions. However, on the stiffer gels focal adhesions develop to
a much higher degree with their highest concentration at the periphery. The plot in
Fig. 3.1 demonstrates that cells form larger focal adhesions on stiffer substrates, in
qualitative agreement with many experimental studies, including that by Saez et al.
( 2005 ). To further elucidate this point we plot the cell total focal adhesion fraction
ξ T in steady state versus the stiffness of the substrate E in Fig. 3.2 (a). Here, the
cell total focal adhesion fraction ξ T is calculated by integrating the concentration
ξ H over the entire cell area and dividing the result by the total number of integrins
ξ o A on the cell membrane, where A is the area of the cell. As a result, ξ T is the
fraction of all integrins on the cell membrane that are in the high affinity state, and
thus in focal adhesions. As expected, ξ T increases for stiffer substrates, but reaches
an asymptote at approximately E
=
50 kPa. Any further rise in stiffness beyond this
level does not change the overall focal adhesion distribution, a trend also apparent
in Fig. 3.1 .
We compute the magnitude of the cell traction applied to the substrate as
ξ H (F 1 + F 2 ) 1 / 2 , and integrate this over the interface between the cell membrane
and the gel to obtain a total force magnitude F T applied by the cell to the substrate
(we note here to avoid confusion that the total force vector applied by the cell to the
gel is zero, but view the parameter F T as a useful gauge of the intensity of force gen-
eration associated with the contractile machinery of the cell). We find that the total
force F T increases as the substrate stiffness E is increased, as shown in Fig. 3.2 (b),
but reaches an asymptote beyond E =
=
50 kPa. In addition, we also establish a linear
correlation between focal adhesion assembly and force generation by plotting F T
versus ξ T (not shown).
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