Physics of the mitotic spindle

Newt cell

Khodjakov's figure

Microtubules-greenSpindle poles-magentaChromosomes-blueKinetochores-yellowKeratin-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 offat times that are tightlyregulated

General principles of design:

Robustness RedundancyRobustness, Redundancy

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: 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]

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

( , ) ( , ) ( , ) ( , )

( )

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

What moves and generates forces on MTs: molecular motors

/ ~ 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⎨ ⋅⎩

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= = =+ +

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

What motors do in the spindle:

Karsenti et al Nat Cell Biol 8 1204 2006

Problem #1: Mitotic spindle self-assemblyWhat is the optimal dynamic instability regime for the fastest search and capture?

Search

CaptureCapture

Mitchison & Kirschner, 1984-86

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

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 = =

‘Search and Capture’ can be biased

Odde Cur Biol 15 R328 2005

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

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 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

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

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

( )/ 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