Specific Adhesion of Soft Elastic Materials - Structural Interfaces and Attachments in Biology - page 148

8.3 Modeling Strategy
Like most engineering materials, cells deform when subjected to external forces,
and focal adhesions (FAs) can be dissociated when strong mechanical forces are
applied. However, does cell adhesion behave like a continuous or discrete multi-
scale-system? Or, should cell adhesion be modeled mechanically as a continuum or
a discrete multi-scale system? It may seem contradictive, but the answer largely
depends on the relevant biological tissues and scales involved. The underlying
assumption for treating a material as a continuum is that the smallest dimension to
be considered is much larger than the space over which structures and properties
may vary significantly. In biological cells, adhesion is mediated by the formation
and rupture of specific molecular bonds between ligands and receptors. As the
typical FA size is only about ten times larger than the bond spacing, the random
association/dissociation processes of these discrete binder molecules may signifi-
cantly change the distribution of adhesion stress and adhesion strength. This effect
reflects the coupling between the elastic deformation of the cell-ECM system and
the variance of bond number, adding a new phenomenon of direct relevance to
either the continuum or pure stochastic treatment of adhesion.
We recall the continuum modeling of adhesion between elastic media, which has
been an active research topic in contact mechanics for a few decades. For the
adhesive contacts of elastic spheres, the JKR [ 24 ] and DMT [ 25 ] models are very
useful in modeling the adhesive contact at two opposite extremes, whether surface
forces are short-ranged compared to resulting elastic deformations [ 26 ]. The
Maugis-Dugdale model [ 27 ], which is based on a cohesive description of the
surface forces, describes the transition between the JKR and DMT theories.
To maintain a stable adhesion state, one usually needs to know the strength of a
particular adhesion between two elastic media. In the case of a pulling load applied
to the adhered elastic bodies (Fig. 8.1 ), a stress concentration is expected to occur at
the edge of the contacting region. An increase of the load then enlarges the intensity
of stress concentration and eventually drives the edge cracks to propagate and break
the joint. In this case, the carrying capacity of the joint is not used most efficiently
because only a small fraction of material is highly stressed at any instant of loading,
and failure occurs by incremental crack propagation. The maximum strength should
correspond to an optimal adhesion state that at pull-off, the interfacial stress is
uniformly distributed over the contact region with a magnitude equal to the
theoretical adhesion strength. Gao and Yao [ 28 ] suggest that a robust, shape-
insensitive optimal adhesion becomes possible only when the adhesion size is
small enough. Below a critical structural size, the material fails no longer by
propagation of a pre-existing crack, but by uniform rupture at the limiting strength
of the binding molecules.
At the other extreme, from a statistical mechanics point of view, a single
molecular bond has only a finite lifetime. The time scale associated with individual
association/dissociation events takes minutes under small stretching forces, e.g.,
below 5 pN for biotin-streptavidin bonds [ 29 , 30 ]. Recent single molecule