Elucidating the role of matrix stiffness
On classical two-dimensional substrates, single cells attach and migrate by arranging their cytoskeletal components throughout a progressive cycle of attachment, protrusion, and disruption to coordinate forces at their leading and trailing edges to achieve cellular movement.This process is highly sensitive to the cell’s immediate external environment comprised of physical cues that produce variations in migration, division, differentiation, and apoptosis (3–15).To discretize this equation by the finite element method, one chooses a set of basis functions defined on Ω which also vanish on the boundary.One then approximates , so all its eigenvalues are real.However, the extracellular matrix (ECM) is an interwoven fibrous mesh and a cell’s interaction with it can be categorized in two ways, interaction characterized by stretching over and making contact with the whole mesh, representative of bulk behavior, or interaction with the fibrils or bundles of fibers that make up the bulk structure.The ECM network consists of individual fibrils (30–70 nm in diameter) that can bundle into fibers 200 nm to 1 m in diameter (26–28).and independent of geometry), but little is known on how cells respond to subtle changes in local geometry and structural stiffness (N/m).Over a wide range of increasing structural stiffness (2 to 100 m N/m), cells exhibited decreases in migration speed and average nucleus shape index of ~57% (from 58 to 25 m).
In general, to each scalar elliptic operator L of order 2k, there is associated a bilinear form B on the Sobolev space H In order to implement the finite element method on a computer, one must first choose a set of basis functions and then compute the integrals defining the stiffness matrix.
Moreover, it is a strictly positive-definite matrix, so that the system AU = F always has a unique solution.
(For other problems, these nice properties will be lost.) Note that the stiffness matrix will be different depending on the computational grid used for the domain and what type of finite element is used.
A wide variety of reductionist in vitro systems, including culture systems of flat wells coated with adhesion proteins, two- (2D) and three-dimensional (3D) gels, and fiber networks of different diameters, have been developed in an attempt to capture the in vivo physiological state (10,11,16–19).
From the mechanistic viewpoint, these culture systems capture the native stiffness (ability to resist deformation) of the tissue-cell model commonly known as Young’s modulus (N/m), which is geometry-independent.
Geometry is embedded in structural stiffness, which relates the size, organization, and shape of a material to its ability to resist deformation and is represented in units of N/m.