CELLS AND MEMBRANES - Nanoindentation with Biological Applications - page 327

outer porous glycoprotein shell of mouse oocytes and measure the force-
displacement curves of the deformed egg. In combination with a simple
membrane model (stretching deformation only), the quantitative elastic
modulus and piercing of the egg wall before and after fertilization is
reported the first time in the literature.

3. Theories
3.1. Mechanical models for single cells

3.1.1. Balloon model

A typical cell is taken to be an incompressible liquid enclosed by a thin
spherical elastic membrane, similar to a balloon. The bending inertia of
the cell membrane is usually ignored so that membrane stretching serves
as the only deformation mode. Several theoretical works are based on the
same assumption, though various forms are reported in the literature. For
example, Yao et al . 17 developed a model to account for the turgor
pressure within a bacterium under AFM-indentation ( Fig. 10-5 ). In their
model, a uniform tensile deformation of the cell membrane is assumed,
and therefore, with no surprise, the calculation shows a linear force-
displacement relation,

2 π pR

F

k = =

(10-2)

w

R + c R + 2 c

log +

t

c R + c

where F is the force applied via an AFM tip with radius c , p is turgor
pressure, and R is the radius of the cell. The total indentation depth at the
pole, w t , is given by

F

w t = w 0 + 2 w c ≈ w 0 + 2 (10-3)

2 π pR

where w 0 is the depth of the indentation dimple and w c is the height of
the collapsed cap at one pole.
Sen et al. 18 developed a similar theoretical model which also ignores
bending effects. Adhesion between the cell membrane and substrate is,