Biology Reference
In-Depth Information
to include the mechanical properties of the boundary that is more complex than the
bilayer membrane shell.
The cell boundary mechanics was captured in an abstracted two-dimensional
form in models that aimed to predict self-organization of multiple adjacent cell
boundaries into nonrandom cellular arrangements characteristic of many tissues
(Käfer et al.
2007
; Farhadifar et al.
2007
). The deformation mechanics of the single
animal cells can be described as governed by line tension (Evans and Yeung
1989
).
This force was found insufficient in the cited computational models for tissue orga-
nization, and realistic predictions were obtained by adding phenomenological elas-
tic terms to the effective energy function alongside the line-tension terms. These
postulatory terms are not needed in a theory that includes a mechanistic model of
the elastic cytoskeleton of the enclosed cell body. Also, to complete the physical
description of the boundary dynamics per se, one must include the quasi-elastic
effects of oncotic pressure which is associated with the volume enclosed by the
boundary.
A theoretical model with the requisite characteristics was constructed by
Arkhipov and Maly (
2006a
) and analyzed further by Baratt et al. (
2008
). The model
minimizes an energy function of a free-floating cell
E
f
comprised of the microtu-
bule bending energy
E
b
, surface energy of the boundary
E
s
, and oncotic volume
energy
E
v
:
EEEE
f
=++
b
s
v
E
b
is obtained by integration of squared curvature along each microtubule and
summation over the microtubules in the cell, with the microtubule bending rigidity
as a factor.
E
s
is calculated as a product of the measured cortical tension of unat-
tached cells and the area of the boundary enclosing the microtubules.
E
v
is calcu-
lated using the measured oncotic pressure. The extracellular pressure is assumed to
remain constant as the cell volume changes. The work against it that is associated
with the changing volume enclosed by the boundary is trivial to calculate. The intra-
cellular oncotic pressure changes with the changing concentration of macromole-
cules, to which the cell boundary is impermeable. It therefore changes inversely
proportionally to the volume. The term of the volume energy function that is the
work of the intracellular oncotic pressure when the cell volume
V
deviates from the
oncotic equilibrium volume
V
eq
is then Π
V
eq
ln(
V
/
V
eq
), where Π is the equilibrium
oncotic pressure. The calculations of Arkhipov and Maly and Baratt et al. assumed
that the extracellular pressure equals the equilibrium oncotic pressure, as is proba-
bly the case in mammalian tissues and in blood or lymph. In the future the model
could be adapted to specific extracellular pressures to capture, for example, the
conditions of experiments in cell culture more precisely.
The model is spatially discretized and the conformation corresponding to a mini-
mum of
E
f
is found numerically. Multiple runs of the minimization algorithm starting
from a random set of conformations converge, depending on the physical parameters,
around one conformation or several very similar conformations. With the free
