Biology Reference
In-Depth Information
Fig. 12
Symmetry instability
domain. In the three-
dimensional space of
structural parameters
(
q
max
,
S
/
R
,
L
/
R
),
L
/
R
is
understood as the vertical
dimension
in this graph. Isolines
corresponding to the
indicated values of
L
/
R
are
shown. (
a
) Surface bounding
the domain from below.
(
b
) Surface bounding the
domain from above.
Reproduced from Maly
(
2012
) under the Creative
Commons Attribution
License
special case of
L
=
R
. Specifically, for both long and short microtubules, the (
q
max
,
S
/
R
) cross-sections of the symmetry instability domain are restricted to
q
max
< π/2.
For short microtubules, longer spindles exhibit unstable symmetry. Note, how-
ever, that the stability of spindles that are particularly short (
S
< 2(
R
−
L
)) is only
neutral in this case, because in their symmetric position, the astral microtubules do
not contact the boundary. For long microtubules, instability of symmetry is exhib-
ited by shorter spindles. In this case, it is of note that the longer spindles exhibit
bistability between the symmetric and asymmetric equilibria, as was illustrated in
Fig.
10b
(dashed curve).
For
L
≈
R
, the range of instability of symmetry extends through
q
max
= π. This
behavior, which has been illustrated in the special case of
L
=
R
, is retained for
L
taking values between approximately 0.9 and 1.05
R
. Although this range is narrow
in absolute terms, it seems to be particularly relevant, because inspection of images
in the literature suggests that
L
is often similar to
R
in the morphogenetically impor-
tant instances of large cells in early embryos (Lillie
1901
; Hyman and White
1987
;
Symes and Weisblat
1992
; Grill et al.
2001
).
It appears plausible that if each microtubule length from a set supports stability
of symmetry, then any distribution of lengths over that set will also support stability
of symmetry. Conversely it may be posited that if each length by itself supports
instability, any distribution will also support instability. Obviously, if any subset
