lec1 - university of british columbia department of ...keshet/lec1.pdf · title: microsoft...
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Physics of the mitotic spindle
Newt cell
Khodjakov's figure
Microtubules-greenSpindle poles-magentaChromosomes-blueKinetochores-yellowKeratin-red
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Kwon and Scholey Trends Cell Biol 14 194 2004
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Multiple parts of the spindle are involved in stochastic, yet robust and predictable ‘dance’
Force-balance model;Motors switch on and offat times that are tightlyregulated
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General principles of design:
Robustness RedundancyRobustness, Redundancy
Open system, consuming lots of energy
Multi-objective optimization: speed and accuracy
Inter-connectedness, impermanence
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What is the spindle made from: microtubules (MTs)
Dynamic instability: Dynamic instability:
switching between
phases of growth p g
and shrinkage
8
9
10
cron
)
8 nm
25 nmGTP
GDP
4
5
6
7M
T le
ngth
(mic
0 1000 2000 3000 4000 5000
2
3M
Time [s]
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Δx Probability that there is a growing MT at x
[ ]( ) ( ) ( ) ( ) ( )v tp x p x v t p x c t p x r t p xv t+
+ + + + + −+
ΔΔ = − Δ − − Δ ⋅ + Δ ⋅
Δ
Probability that there is a growing MT at x
c r [ ]( ) ( ) ( ) ( ) ( )v tp x p x v t p x c t p x r t p xv t−
− − − − + −−
ΔΔ = + Δ − + Δ ⋅ − Δ ⋅
Δ
x( ) ( ) ( ) ( ) ( )p x p x p x v tv c p x r p xt v t
+ + + ++ + −
Δ − − Δ= − − ⋅ + ⋅
Δ Δ( ) ( ) ( ) ( ) ( )
t v tp x p x v t p xv c p x r p x
t v t
+
− + − +− + −
−
Δ ΔΔ + Δ −
= + + ⋅ − ⋅Δ Δ
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( , ) ( , ) ( , ) ( , )
( )
p x t pv x t cp x t rp x tt x
p x t p
+ ++ + −
∂ ∂= − − +
∂ ∂∂ ∂
[ ]( ) exp /p x P x l=
( , ) ( , ) ( , ) ( , )p x t pv x t cp x t rp x tt x
− −− + −
∂ ∂= + −
∂ ∂
( , ) ( , ) 0,
0
p x t p x tt tdp
+ −
+
∂ ∂= =
∂ ∂⎧
[ ]( ) exp /p x P x l± ±= −
0v c P rPl+
+ −
⎧⎛ ⎞− + =⎜ ⎟⎪⎪⎝ ⎠⎨0,
0,
pv cp rpdx
dpv cp rpdx
++ + −
−− + −
⎧− − + =⎪⎪⎨⎪ + − =⎪⎩
0
lvcP r Pl−
+ −
⎪⎝ ⎠⎨
⎛ ⎞⎪ − + =⎜ ⎟⎪ ⎝ ⎠⎩dx⎪⎩
, v vl v c v rv c v r
− +− +
+
= >−0.9
1
P±− +
v vr c− +>
0.4
0.5
0.6
0.7
0.8
average growth per cycle is less than
l l− +>0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
xl
yaverage shortening per cycle
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What moves and generates forces on MTs: molecular motors
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/ ~ 4 / 8 ~ 0.5Bk T pN nm nm pNEnergyFδ ⋅⎧
⎨
Karsenti et al Nat Cell Biol 8 1204 2006
~ ~/ ~ 80 / 8 ~ 10ATP
gyFE pN nm nm pNStep δ⎨ ⋅⎩
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Mechanochemicalcycle
1
1k2k/0
2 2Bf k Tk k e δ−=
12
1dp k k2
11 1 2 2
21 1 2 2
dp k p k pdtdp k p k pdt
= − +
= −12
kpk k
=+
1 2 1p p+ =
1 1 2 2dt1 2k k+
01 2 1 2
2 2 /01 2 2 1
Bf k T
k k k kv k pk k k k e δ
δ δδ= = =+ +
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01 2
/0 ;Bf k T
k kvk k δ
δ=
+
Motor is characterized by the force-velocity relation
V,n/d
/02 1
4
1 2
4 ; 8 ;
100 10~ ; ~
Bf k T
B
k k ek T pN nm nm
k k
δ
δ+
≈ ⋅ ≈
f pN
1 2;sec sec
f,pN
G f S Bl kGroup of S. Block
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What motors do in the spindle:
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Karsenti et al Nat Cell Biol 8 1204 2006
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Problem #1: Mitotic spindle self-assemblyWhat is the optimal dynamic instability regime for the fastest search and capture?
Search
CaptureCapture
Mitchison & Kirschner, 1984-86
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p probability of a successful search
Mathematical analysis (Holy & Leibler, 1994)
p – probability of a successful search– average time of a successful search– average time of an unsuccessful search
τ - average search time
stut
(1 ) ( )
q – prob. to grow in the right direction, ~ 1/3p* - probability to grow to length d
2
(1 ) ( )1(1 ) ( 2 ) ...
s s u
s u s u
pt p p t tpp p t t t t
p
τ = + − + +−
− + + = +
* * /
/
1, , , ~ d lu
d l
tp p qp p ep
V l l
τ −<< ≈ =For Newt lung cell, distance between
i dl l d h d 10/
, , ,
min( ) 10 min at
d lg
ucat g g
V l lel tf V qV
de l d
τ
τ
= ≈ ≈ spindle pole and chromosome, d ~ 10 μm. Rescue frequency is very small, and average microtubule length, l ~ 10 μm. Growth rate, ~ 10 μm/min. TimegVmin( ) ~10 min at
g
l dqV
τ = = , μin prometaphase, τ ~ 10 min.
g
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Multiple chromosomes - greater time to capture:1) Capture of the last chromosome corresponds
t th l t h ti l ith f th
Monte-Carlo simulations: optimal unbiased ‘Search and Capture’ is not fast enough
to the longest search; time ~ logarithm of thenumber of chromosomes.
2) Geometric effect: a few-fold increase. ( ) ( ) ( )( )
( ),1 ,1
/ /
1 1
1 1
M
M
NNM NM
Np t p N t
P t P t P t
tτ τ
τ τ τ≤ = − > = − > =
( )/ /1 1 , Mu M up t p N t u
M
te eN p
τ τ τ− −− = − =
( ) ( )( ) ( )/, ,1 1 KK M u
NN p N tNM NK NMP t P t e ττ τ −≤ = ≤ = −
Numerical experiment:ln ( 92 : a few-fold increase)u
K KM
t N NpN
τ = =
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‘Search and Capture’ can be biased
Odde Cur Biol 15 R328 2005
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2
2
2
A AD pAT X∂ ∂
= −∂ ∂
: 0A BX L D DX X∂ ∂
= = =∂ ∂
2
2
B BD pAT X∂ ∂
= +∂ ∂
0 : A BX D D kX X∂ ∂
= − = =∂ ∂
1 /A A T t X D, , /A Aa T t X D pxp
= = =
2
2
a a a∂ ∂= −
∂ ∂
: 0ax lx∂
= =∂∂2t x∂ ∂ 0 : axx
λ∂= = −
∂
/ / 1, kl L D pA
λ= =kinase
RanGTP – a(x,t)
G ( )
pA Dp
1 2
0
x x
l l l
a C e C e
C e C e C e
−
−
= +
≈ =RanGDP – b(x,t) 1 2 1
1 2 2
0C e C e C eC C C λ
− ≈ =− + ≈ =
k X⎡ ⎤2 110 / 3mD Dμ
phosphatasex0
exp/
k XADp D p
⎡ ⎤≈ −⎢ ⎥
⎢ ⎥⎣ ⎦
~ 10 , ~ , / ~ 3sec sec
D p D p mμ μ
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Optimal biased ‘Search and Capture’ is fast enough:
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Unbiased, N=1000, 250, ,
Biased, N=1000
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angle=2*pi*(rand(N)-0.5); le=zeros(size(angle)); vel=ones(size(angle));% vectors for angles lengths velocities% vectors for angles, lengths, velocitiess=ones(size(angle)); cat=cata*ones(size(angle)); % vector for gr/sh state and cat
for k=1:10 % time loople=le+vel*dt; % length update
for j=1:N % MT loopx=le(j)*cos(angle(j)); y=le(j)*sin(angle(j)); % plus end coordinatesif ((x-0 5)^2+y^2)<0 25 cat(j)=cata; else cat(j)=10*cata; end % local cata freqif ((x-0.5) 2+y 2)<0.25 cat(j)=cata; else cat(j)=10 cata; end % local cata freq
if rand<cat(j)*s(j)*dt vel(j)= - vel(j); s(j)=0; end % catastropheif le(j)<0 le(j)=0; vel(j)=1; s(j)=1; angle(j)=2*pi*(rand-0.5); end % rescueif (abs(x-xk1)<0.05 & abs(y-yk1)<0.05)
vel(j)=0; s(j)=0; end % captureend
end
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The model inspired two recent studies:
Lenart et al, 2005:at large centrosome-chromosomedistances, “Search&Capture” is
t ffi i t d ti inot efficient, and actin-myosin“fishnet” mechanism works first
Caudron et al, 2005:Ran gradients exist andbias microtubule asters
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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
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Goshima et al JCB 171 229 2005
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Problem: merotelic attachments
( )/ lnT t pM K
attachment number numbertime of MTs of KTs
( )~ / lna uT t pM K
~ /ut L V 2~ /p A L ~A rl
unsuccessful probability K-fibersearch time of capture area
len e
lnr ct
number number time toof errors of errors correct
( )3 ln 1 t nL K
~ c ec
t nT lr
lL
V ( ) ln 1~ c ea c
t nL KT T lVMr l r
+ +
Cimini, Wollman 2005-07
r L