Biology Reference
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FIGURE 15.6 The inherent instability of a long cylinder made of a surface under tension, as demonstrated with
children's bubble-blowing apparatus.
1/R 2 ), and therefore the mean curvature ) will be the same every-
where. Two well-known forms that satisfy the condition of constant mean curvature are
the plane (zero curvature) and the hollow sphere ('cyst'), with R 1 ¼
everywhere, so (1/R 1 þ
R 2 . Other shapes are
also possible and can be explored by dipping an arbitrarily distorted wire loop into a solution
of detergent; the resulting shapes will be of minimum surface area and, although they may
include complex curves of convex (R positive) and concave (R negative) types, the mean
curvature will be constant everywhere.
Modelling epithelia as tense films does, however, reveal a serious problem with the
stability of one of the most common of all epithelial configurations, the hollow tube. At first
sight, there is nothing wrong with a tense film taking up a cylindrical configuration; it will,
after all, have a constant curvature throughout its length. Unlike a plane or sphere, though,
a cylinder is inherently unstable to small disturbances. This fact may be demonstrated by
drawing detergent solution between two wire rings ( Figure 15.6 ). Once the ratio of length
to diameter increases beyond about
, the middle section of the cylinder constricts so that
its radius diminishes. Constancy of curvature is maintained because the increasing positive
p
1/R 2 ). 41
) The mean curvature of a surface is defined as 0.5 (1/R 1
þ
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