Physics of the mitotic spindle
Khodjakov's figure
Newt cell
Microtubules-green
Spindle poles-magenta
Chromosomes-blue
Kinetochores-yellow
Keratin-red
Kwon and Scholey Trends Cell Biol 14 194 2004
Multiple parts of the spindle are involved in
stochastic, yet robust and predictable ‘dance’
Force-balance model;
Motors switch on and off
at times that are tightly
regulated
General principles of design:
Robustness Redundancy
Robustness,
Open system, consuming lots of energy
Multi-objective optimization: speed and accuracy
Inter-connectedness, impermanence
What is the spindle made from: microtubules (MTs)
Dynamic instability:
switching between
phases of growth
p
g
and shrinkage
8 nm
10
25 nm
GDP
MT length (miccron)
M
GTP
9
8
7
6
5
4
3
2
0
1000
2000
3000
Time [s]
4000
5000
Δx
c r
Probability that there is a growing MT at x
Δp+ ( x) =
v+ Δt
[ p+ ( x − v+ Δt ) − p+ ( x)] − cΔt ⋅ p+ ( x) + r Δt ⋅ p− ( x)
v+ Δt
Δp− ( x) =
v− Δt
[ p− ( x + v− Δt ) − p− ( x)] + cΔt ⋅ p+ ( x) − r Δt ⋅ p− ( x)
v− Δt
x
Δp+ ( x)
p ( x) − p+ ( x − v+ Δt )
= −v+ +
− c ⋅ p+ ( x) + r ⋅ p− ( x)
Δt
v+ Δt
Δp− ( x)
p+ ( x + v− Δt ) − p+ ( x)
= +v−
+ c ⋅ p+ ( x) − r ⋅ p− ( x)
Δt
v− Δt
∂p+ ( x, t )
∂p+
= −v+
( x, t ) − cp+ ( x, t ) + rp− ( x, t )
∂t
∂x
∂p− ( x, t )
∂p−
= v−
( x, t ) + cp+ ( x, t ) − rp− ( x, t )
∂t
∂x
p± ( x) = P± exp [ − x / l ]
∂p+ ( x, t ) ∂p− ( x, t )
=
= 0,
∂t
∂t
dpp+
⎧
−
v
0
⎪⎪ + dx − cp+ + rp− = 0,
⎨
⎪v dp− + cp − rp = 0,
+
−
⎪⎩ − dx
⎧⎛ v+
⎞
−
c
⎟ P+ + rP− = 0
⎪⎜ l
⎪⎝
⎠
⎨
⎪cP − ⎛ v− + r ⎞ P = 0
⎟ −
⎪⎩ + ⎜⎝ l
⎠
v− v+
l=
, v− c > v+ r
v− c − v+ r
1
P±
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
l
0
0.5
x
1
1.5
2
2.5
3
3.5
4
4.5
5
average growth per
cycle
y is less than
average shortening
per cycle
v− v+
>
r
c
l− > l+
What moves and generates forces on MTs: molecular motors
F~
Karsenti et al Nat Cell Biol 8 1204 2006
gy ⎧ k BT / δ ~ 4 pN ⋅ nm / 8nm ~ 0.5 pN
Energy
~⎨
Step
⎩ E ATP / δ ~ 80 pN ⋅ nm / 8nm ~ 10 pN
Mechanochemical
cycle
1
k2
k2 = k20 e − f δ / kBT
k1
dp1
= − k1 p1 + k2 p2
dt
dp2
= k1 p1 − k2 p2
dt
p1 + p2 = 1
2
k1
p2 =
k1 + k2
v = δ k2 p2 =
δ k1k2
k1 + k2
=
δ k1k20
k20 + k1e f δ / kBT
Motor is characterized by the force-velocity relation
v=
δ k1k20
k + k1e
0
2
f δ / k BT
;
k BT ≈ 4 pN ⋅ nm; δ ≈ 8nm;
V,n/d
100
104
k1 ~
; k2 ~
sec
sec
f pN
f,pN
G
Group
off S.
S Block
Bl k
What motors do in the spindle:
Karsenti et al Nat Cell Biol 8 1204 2006
Problem #1: Mitotic spindle self-assembly
What is the optimal dynamic instability regime for the fastest search and capture?
Search
Capture
Capture
Mitchison & Kirschner, 1984-86
Mathematical analysis (Holy & Leibler, 1994)
p – probability of a successful search
ts– average time of a successful search
tu– average time of an unsuccessful search
τ - average search time
q – prob. to grow in the right direction, ~ 1/3
p* - probability to grow to length d
τ = pts + (1 − p) p(ts + tu ) +
(1 − p ) 2 p (ts + 2tu ) + ... = ts +
p << 1, τ ≈
l=
Vg
f cat
tu
, p = qp* , p* ~ e − d / l
p
, tu ≈
min(τ ) =
1− p
tu
p
d /l
l
l
le
, τ≈
,
Vg
qVg
de
~10
10 min at l = d
qVg
For Newt lung cell, distance between
spindle
i dl pole
l and
d chromosome,
h
d ~ 10 μm.
Rescue frequency is very small, and
average microtubule length, l ~ 10 μm.
Growth rate,, Vg~ 10 μ
μm/min. Time
in prometaphase, τ ~ 10 min.
Monte-Carlo simulations: optimal unbiased ‘Search and Capture’ is not fast enough
Multiple chromosomes - greater time to capture:
1) Capture of the last chromosome corresponds
t the
to
th longest
l
t search;
h time
ti
~ llogarithm
ith off th
the
number of chromosomes.
2) Geometric effect: a few-fold increase.
PNM ,1
( t ≤ τ ) = 1 − PNM ,1 ( t > τ ) = 1 − ( P ( t > τ ) )
(
1 − e − pτ / tu
)
NM
= 1 − e − pτ N M / tu ,
PNM , NK ( t ≤ τ ) = ( PNM ,1 ( t ≤ τ ) )
τ =
Numerical experiment:
τ =
NK
(
tu
NM p
= 1 − e − pτ N M / tu
NM
)
=
NK
tu
ln N K ( N K = 92 : a few-fold increase)
pN M
‘Search and Capture’
can be biased
Odde Cur Biol 15 R328 2005
∂A
∂B
=D
=0
∂X
∂X
∂A
∂B
X = 0 : −D
=D
=k
∂X
∂X
1
A = Aa
A , T = t , X = D / px
p
∂A
∂2 A
=D
− pA
2
∂T
∂X
∂B
∂2B
=D
+ pA
∂T
∂X 2
X = L:D
∂a
=0
∂a ∂ a
∂
x
= 2 −a
∂a
∂t ∂x
x = 0:
= −λ
∂x
k
l = L / D / p 1, λ =
A Dp
x=l:
2
kinase
RanGTP – a(x,t)
RanGDP
G
– b(x,t)
( )
a = C1e x + C2 e − x
C1el − C2 e − l ≈ C1el = 0
−C1 + C2 ≈ C2 = λ
D ~ 10
μ m2
sec
,p~
phosphatase
0
x
1
, D / p ~ 3μ m
sec
A≈
⎡
k
X ⎤
exp ⎢ −
⎥
Dp
D / p ⎥⎦
⎢⎣
Optimal biased ‘Search and Capture’ is fast enough:
Unbiased,, N=1000,, 250
Biased, N=1000
angle=2*pi*(rand(N)-0.5); le=zeros(size(angle)); vel=ones(size(angle));
% vectors for angles,
angles lengths
lengths, velocities
s=ones(size(angle)); cat=cata*ones(size(angle)); % vector for gr/sh state and cat
for k=1:10 % time loop
le=le+vel*dt; % length update
for j=1:N % MT loop
x=le(j)*cos(angle(j)); y=le(j)*sin(angle(j)); % plus end coordinates
if ((x-0
((x-0.5)
5)^2+y^2)<0
2+y 2)<0.25
25 cat(j)=cata; else cat(j)=10*cata;
cat(j)=10 cata; end % local cata freq
if rand<cat(j)*s(j)*dt vel(j)= - vel(j); s(j)=0; end % catastrophe
if le(j)<0 le(j)=0; vel(j)=1; s(j)=1; angle(j)=2*pi*(rand-0.5); end % rescue
if (abs(x-xk1)<0.05 & abs(y-yk1)<0.05)
vel(j)=0; s(j)=0; end % capture
end
end
The model inspired two recent studies:
Lenart et al, 2005:
at large centrosome-chromosome
distances, “Search&Capture” is
nott efficient,
ffi i t and
d actin-myosin
ti
i
“fishnet” mechanism works first
Caudron et al, 2005:
Ran gradients exist and
bias microtubule asters
The way to accelerate assembly: grow MTs from both the centrosome and chromosome
Nedelec et al Cur Opin Cell Biol 2003 15 118
O’Connell and Khodjakov Journal of Cell Science 120, 1717, 2007
Goshima et al JCB 171 229 2005
Problem: merotelic attachments
attachment number
time
of MTs
number
of KTs
Ta ~ ( tu / pM ) ln K
p ~ A / L2
tu ~ L / V
unsuccessful probability
search time of capture
l
r
ne
ne
number
of errors
number
of errors
A ~ rl
K-fiber
area
tc
time to
correct
tc ne
l
r
L3 ln K 1 tc ne
+
l
(Ta + Tc ) ~
VMr l
r
Tc ~
V
l
r
L
Cimini, Wollman 2005-07