Contents - Max-Planck-Institut für Physik komplexer Systeme

lost
KC
MT
05:19 05:23 05:35 05:41
SPB
1µm
Figure 2: Astral microtubules pivot around the spindle pole body.
Time-lapse images of a cell in mitosis, in which kinetochores were labeled
in red (Ndc80-tdTomato) and microtubules in green (α-tubulin-
GFP). The cell was kept at 4 ◦ C to depolymerize microtubules. Subsequently,
the temperature was increased to 25 ◦ C, in order to allow for
microtubule re-polymerization. The time after the temperature increase
is shown in minutes and seconds. The schemes below the images
show the position of the kinetochore (red), microtubules (green),
and spindle pole bodies (grey). Kinetochore capture occurs at the
last frame. Note that the microtubule that captured the kinetochore
changed its angle with respect to the spindle.
To reveal the mechanism of KC capture, we first quantified
the pivoting of astral MTs as a time series of the
angle between the astral MT and the horizontal axis of
the image. To distinguish whether this movement is directed
or random, we calculated the mean squared angular
displacement, and found that it scales with time
to the power of ∼0.8. The exponent smaller than one
suggests that the angular movement is random, and in
particular subdiffusive. In addition to the movement of
astral MTs, we observed movement of lost KCs, which
was also subdiffusive [3]. Taken together, our results
indicate that the movement of the lost KC, as well as of
the astral MTs, plays an important role for KC capture.
In order to test whether the process of KC capture
could be driven by the observed random movement of
astral MTs and the KC, we introduce a simple threedimensional
model consisting of an SPB, a MT, and a
KC. We use a spherical coordinate system, by placing
the origin at the SPB, which is assumed to be stationary.
The orientation of the coordinate system is fixed
with respect to the cell. One end of the MT is at the origin,
while the orientation of the MT is described by the
inclination angle θ and the azimuth angle ϕ. The KC
is placed at an arbitrary position (θk, ϕk), and its size
is described by the parameter δ. Therefore, KC capture
occurs when the MT and the KC overlap in the angular
space, i.e., |θ − θk| < δ/2 and |ϕ − ϕk| < δ/(2sinθk).
This description implies that the distance between the
KC and the SPB is smaller than the length of the MT,
as well as that the KC can bind along the length of the
MT. The angular diffusion of the MT is described by the
following stochastic differential equations
dθ
= Dcosθ
dt sinθ + √ 2Dξθ
dϕ
dt =
√
2D
sinθ ξϕ
(1)
(2)
Here, D is the angular diffusion coefficient of the MT of
a length L, and ξθ, ξϕ is a Gaussian white noise obeying
〈ξi(t)ξj(t ′ )〉 = δijδ(t − t ′ ), i,j = θ,ϕ. In the model, we
use the reflecting boundary conditions 0 ≤ θ ≤ π and
0 ≤ ϕ ≤ π. We keep the KC fixed, while its diffusion is
described by including it in the diffusion coefficient D
of the MT.
We numerically solved Eqs. (1) and (2) for a set of random
initial positions of the MT and calculated the time
it takes a MT to reach the KC for the first time. The
number of lost KCs decreases exponentially with time,
and for a typical values of free parameters δ = 0.2 rad
and D = 0.005 rad 2 /s, half of the lost KCs are captured
in 5 minutes (Fig. 3). The agreement between the capture
time obtained by the model and by the experiment
shows that the kinetics of KC capture can be explained
by random movement of astral MTs and of the KC.
We found that thermal fluctuations drive angular
movement of MTs, as well as the movement of the KC.
We propose this movement of MTs as a search strategy
by which MTs explore space in order to find a KC. This
mechanism allows for KC capture with only a few astral
MTs.
% of cells with lost KC
70
60
50
40
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11
Time (min)
Figure 3: Results from the model: The kinetics of kinetochore capture
can be explained by random movement of astral microtubules and of
the kinetochore. The fraction of cells with lost kinetochores is shown
as a function of time elapsed since the temperature was increased
from 4 ◦ C to 25 ◦ C. Circles represent the experimental data and the
line shows the result from the model.
[1] T. Mitchison, and M. Kirschner, “Dynamic instability of microtubule
growth,” Nature, vol. 312, pp. 237–242, 1984.
[2] Y. Gachet, C. Reyes, T. Courtheoux, S. Goldstone, G. Gay, C. Serrurier,
and S. Tournier, “Sister kinetochore recapture in fission
yeast occurs by two distinct mechanisms, both requiring Dam1
and Klp2,” Mol Biol Cell, vol. 19, pp. 1646–1662, 2008.
[3] I.M. Tolić-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede,
and K. Berg-Sorensen, “Anomalous diffusion in living yeast
cells,” Phys Rev Lett , vol. 93, pp. 078102, 2004.
2.28. Kinetochores are Captured by Microtubules Performing Random Angular Movement 97