Integration of Molecular Signaling into Multiscale Modeling of Cancer - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 381

degradation, and at the same time, E-cadherin is sequestered into the cytoplasm by
endocytosis. Upregulation of soluble b-catenin is related to cell migration and the
epithelial-mesenchymal transition (EMT) [ 49 ], a process where a well ordered and
polarized layer of cells changes into an unstructured configuration to facilitate
collective cell migration. Sufficiently large concentrations of soluble b-catenin
then move from the cytoplasm into the nucleus, where it interacts with tran-
scription factors which modify cell behavior, e.g., by promoting cell proliferation.
It has been observed that invasive cells show a higher nuclear accumulation of
soluble b-catenin. In particular, as proposed in [ 50 ], b-catenin may translocate into
the nucleus above a certain concentration threshold, leading to downregulation of
E-cadherin-mediated adhesion.

2.2.2 Microenvironment

A lattice-free environment is constructed to explore linking the intracellular
dynamics of E-cadherin and b-catenin interaction pathways, physical forces on
the cells, and the extracellular microenvironment. Each cell is considered as an
individual entity in which intracellular dynamics are governed by the a set of
mass conservation chemical equations, using the same form of Eq. ( 1 ). A sim-
plified b-catenin pathway which captures the key features of the cell adhesion
process is implemented. Since cells in isolation tend to aggregate, it is assumed
that an invasive cell can change into a noninvasive state again if it comes into
contact with other cells. Cell movement is modeled by a stochastic equation of
motion:

cv i ¼ X
jnni

F ij þ X
jnni

F ij þ f i ð t Þ ;

ð 2 Þ

where v i is the velocity of the cell i at time t; F ij and F ij are the repulsive force and
adhesive force of cell j on cell i, respectively, and the sums are over the nearest
neighbors in contact with cell i; f i ð t Þ represents a noise term for cell i at time t; and
c is the cell-substrate friction constant. Interested readers are encouraged to refer to
the original article [ 48 ] for more details on each term. Briefly, this governing
equation accounts for the influence of forces and a random contribution to the
locomotion which results from the local exploration of space. The adhesion forces
between cells are controlled by the density of E-cadherin in the cell membrane
within the cell-cell contact zone.

2.2.3 Cell Migration Determination upon Expression Levels of b-Catenin

As explained above, upregulation of soluble b-catenin is assumed to interact with
transcription factors in the nucleus, and intend to induce cell migration [ 51 ]. In
fact, both attachment and detachment of cells lead to an exchange of E-cadherin