Structure of Membranes - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 13

To find the stress at which the membrane will burst suppose we cut the membrane
along some line of length . To prevent the two sides of the cut from separating,

a force, F , must be present which is proportional to

, then, the
proportionality constant γ is called the elastic tension in the wall. This tension is not
the surface tension, T , of a fluid, but it does play a similar role. The excess pressure
inside a bubble of radius R , over and above atmospheric pressure, is 2 T

. Writing F = γ

R .Itturns
out that a similar relation to this may be used to compute the excess pressure

/

P
inside a spherical pressure vessel, such as a membrane, if we replace the surface
tension T with the elastic tension γ . Thus we have:

2 R

P = (2.1)

The elastic stress, σ , inside the membrane is related to the elastic tension by

γ = D σ (2.2)

where D is the wall thickness. The reason for this is that the surface area of the cut
is D ,

which is the stress of the membrane wall. From the two equations above, the elastic
stress, σ , is given by

D and hence, the force per unit area on the surface of the cut is F / D = γ/

R P

2 D

σ =

(2.3)

so that the cell will burst when this stress exceeds the fracture stress of the material
from which the cell membrane is made.
A large amount of diffusion in biological organisms takes place through mem-
branes. These membranes are very thin, typically ranging from 65 10 − 10

× to

10 − 10 m across. Most membranes are selectively permeable; that is, they allow
only certain substances to cross them because there are pores through which sub-
stances diffuse. These pores are so small (from 7

100 ×

10 − 10 m) that only
small molecules can get through. Other factors contributing to the semi-permeable
nature of membranes have to do with the chemistry of the membrane, cohesive and
adhesive forces, charges on the ions involved, and the existence of carrier molecules.
Diffusion through membranes is a relatively slow process.
In order to provide a simple mathematical description of passive diffusion across a
membrane, we can apply Fick's Law to the transport of molecules across a membrane
of thickness

10 − 10 to 10 × ×

x . Assume also that the concentration of the solute on the left side is c L
and that on the right side is c R . The solute diffusion current, I , across the membrane,
according to Fick's Law, is given by [ 13 ]

D

I = k A c (2.4)

x

where c = c R − c L , k is the diffusion constant and A is the cross-sectional area of

the membrane.