BIO5571
Cell shape. Page 1
2011
III. CELL SHAPE
III.A. Cell Size
The first morphological parameter considered is the size of a spherical cell. As you would expect, cell
membrane properties scale with the surface area of the cell, since the circuit elements of the surface are in
parallel to each other. To compare membrane properties between cells of different sizes, the following
quantities are used:
Specific membrane resistance
R
M
ohm·cm
2
Specific membrane conductance
G
M
S/cm
2
Specific membrane capacitance
C
M
F/cm
2
For a spherical cell of radius a cm,
r
M
= R
M
/(surface area) = R
M
/ (4
π
a
2
)
c
M
= C
M
×
4
π
a
2
Most cells have C
M
≈ 10
-6
F/cm
2
.
Values for R
M
vary from ~10
3
ohm·cm
2
to ~5x10
4
ohm ·cm
2
.
Cells with the lower values for R
M
are often called "leaky" cells, with the higher values "tight."
III.B. Non-spherical cells - general points
The basic point is fairly straightforward. For a spherical cell, the cell membrane resistance is larger than the
resistance between any points in the cytoplasm or any points in the extracellular fluid immediately outside the
cell membrane. Therefore, all of the membrane resistance and capacitance can be treated as being in parallel,
and the entire cell can be viewed as being at the some potential. This is not true for many cells: the resistance
between points in the cytoplasm can get quite large, and the cell is not necessarily equipotential. Therefore,
we must be concerned about voltages and currents between points inside the cell, as well as across the cell
membrane. This might appear to be a trivial problem, but it has many ramifications when it is considered
carefully.
The problem of current flow in cells of various shapes soon gets mathematically complicated. A book which
deals with it in detail, but needs mathematical training to follow, is Electric Current Flow in Excitable Cells
(Jack, Noble and Tsien, Clarendon Press, Oxford, 1975).
The general approach is to divide a cell of complicated shape up into a large number of small pieces, and to
use calculus to solve equations describing the current and voltage for each piece. In principle, this is
analogous to assembling a large number of spherical cells with the appropriate circuit elements and hooking
them together in a particular network. For example, after a spherical cell the next simplest cell is an infinitely
long, uniform cable - a squid axon, an unbranched dendrite, or a muscle fiber, for examples. Such a structure
could be represented by a strong of identical beads hooked together with internal longitudinal resistors r
i
and
external resistors r
x
.
In the mathematical analysis, each bead is made infinitesimally small and the resulting system of differential
equations solved. However, an intuitive appreciation of the effects of cell geometry can be gained by
applying principles discussed for spherical cells to the network of beads.