Biology Reference
In-Depth Information
supports neutral stability, the outcome will be dictated by the complementary sub-
set. With this in mind, predictions about the stability of symmetry with microtubule
length distributions can be obtained by mere inspection of the stability and instability
domains already mapped.
The instability domain in the (
q
max
,
S
/
R
,
L
/
R
) space is bounded and embedded in
the domain of stability (Fig.
12
). Furthermore, it lies above the domain of neutral
stability (
S
/
R
< 2−2
L
/
R
). At large
q
max
, there is a considerable separation between the
domains of instability and neutral stability; at small
q
max
, they almost touch.
The support of a microtubule length distribution is represented in this parameter
space by a vertical segment. If such a segment lies entirely within or outside the
instability domain, the central symmetry will be, correspondingly, unstable or sta-
ble. For example, any distributions whose support falls entirely to the right of the
L
/
R
= 1 isoline in Fig.
12b
will predict stable central positioning of the spindle.
Further, unless the distribution is unusually sharply concentrated at the interme-
diate values, the instability domain can be considered as touching the neutral stabil-
ity domain for smaller
q
max
. Therefore, distributions whose support falls below the
upper bounding surface of the instability domain (Fig.
12b
) predict instability of
symmetry, if
q
max
is not large.
In the likely case of descending exponential distributions (Gliksman et al.
1992
),
the contribution of the few exceptionally long microtubules may prove negligible.
In that case, the prediction can be further simplified: For larger
q
max
and
S
/
R
, stabil-
ity is predicted, and for smaller
q
max
and
S
/
R
, instability is predicted.
Among the theoretically possible structures and equilibria, several can be con-
sidered paradigmatic, based on qualitative examination of images of spindles in the
experimental literature. Firstly, there is the case of a long spindle with short astral
microtubules that radiate from the poles in a wide angle. The equilibrium conforma-
tion is plotted in Fig.
13a
. According to the preceding analysis, in this regime (large
S
/
R
, small
L
/
R
, large
q
max
), the symmetric equilibrium is the only equilibrium, and it
is stable. Awaiting measurements motivated by the theory, it can be said that this
regime appears common among the variety of equally dividing cells. The HeLa
cultured cells are one example (Théry et al.
2005
).
Another characteristic example is the structure with long astral microtubules that
radiate in a wide angle. This regime exhibits the bistability between the symmetric
and asymmetric equilibria. The alternative conformations are illustrated in
Fig.
13b, c
. This example is characterized by a comparatively small
S
/
R
and large
L
/
R
. In this respect it is reminiscent (in the asymmetric conformation) of the first
division in the invertebrate models of development that include the mussel
Unio
and
the roundworm
Caenorhabditis
(Lillie
1901
; Hyman and White
1987
; Symes and
Weisblat
1992
; Grill et al.
2001
). For comparison, Fig.
13d
displays the only stable
equilibrium that exists in the regime with
L
≈
R
and a spindle of medium length,
which is asymmetric. In terms of the spindle proportions and position, this last
example is reminiscent of the mouse oocyte (Schuh and Ellenberg
2008
).
The example of bistability in particular raises the question of the absolute mag-
nitude of the collective spindle forces. The natural unit of force,
N
EI/
R
2
, equals
22 pN when
N
= 100, EI = 26 pN μm
2
, and
R
= 10 μm. The barrier for switching from
