Problem #3: spindle positioning
Oegema and Hyman Cell division 2006 WormBook
How many motors pull spindle to the cortex?
Grill et al 2003 Science 301 518
Grill and Hyman Dev Cell 8 461 2005
Spindle oscillates as it moves in the posterior direction
Pecreaux et al Curr Biol 2006 16 2111
Kozlowski et al Cell 2007 129 499
‘Tug of war’ between opposing force generators
The key idea behind the model is that the rate of
detachment of the force generators from MTs
is load dependent
Grill et al Phys Rev Lett 2005 94 108104
Pecreaux et al Curr Biol 2006 16 2111
ζ y& = − Ky + ( f + − f − )
y
p& + = − koff p+ + kon (1 − p+ )
p& − = −koff p− + kon (1 − p− )
p− , f −
⎛
v⎞
f = f 0 ⎜1 − ⎟
⎝ v0 ⎠
f ± = f 0 m ϕ y&
kon = const ; koff = k off(0) exp [ f / f c ] ≈ k off(0) (1 − α y& )
ζ y& = − Ky + α ( p+ − p− ) − αβ y& ( p+ + p− )
p& + ≈ k − ( a − by& ) p+
p& − ≈ k − ( a − byy& ) p−
λ ~ (α s1 − s2 ) ±
s1 , s2 , s3 > 0
(α s1 − s2 )
2
− s32
y = 0, p+ = p− = k / a
⎡ y ⎤ ⎡ y ⎤ ⎡ y0 ⎤
⎢ p ⎥ = ⎢ p ⎥ + ⎢ p ⎥ eλt
⎢ +⎥ ⎢ +⎥ ⎢ 1⎥
⎢⎣ p− ⎥⎦ ⎢⎣ p− ⎥⎦ ⎢⎣ p2 ⎥⎦
exp ( γ t ) × sin ( wt )
Interplay between spindle pole movements and microtubule dynamics
Microtubules Contact the Cortex Briefly
before Undergoing Catastrophe
Dynamic Microtubules Associate with Each
Other to Form Persistent Astral Fibers
Microtubule Fibers Contact the Cortex at Fixed
Positions
more MT contacts are made on the side of the
cortex that is approached by the aster than on
the receding side.
Kozlowski et al Cell 129 499 2007
f
v
y
0.6
2
y
1.5
f ( y& )
1
0.4
0.2
0.5
y&
0
0
y& = − Ky + f ( y& )
-0.2
-0.5
-0.4
-1
-0.6
-1.5
-2
-20
-15
-10
-5
0
5
10
15
20
-0.8
-2
1
⎡ f ( y& ) − y& ⎤⎦
K⎣
-1.5
-1
-0.5
0
0.5
1
1.5
2
Another example of oscillations:
Directional instability phenomenon
Joglekar and Hunt Biophys J 2002 83 42
Gardner and Odde Cur Opin Cell Biol 2006 18:639
Problem #4: spindle length
f
v
f
v
Two Opposing Motors
v
Can two opposing motors compete?
-
FK
k Klp
v
FN
n Ncd
+
⎛
v ⎞
nf n ⎜1 − ⎟ = kζ k v
⎝ Vn ⎠
Vn x
n
,x =
v=
x + ε n (1 − x )
n+k
εn =
ζ kVn
fn
≈ 4.5
⎛
v ⎞
kf k ⎜1 − ⎟ = nζ n v
⎝ Vk ⎠
V (1 − x )
n
,x =
v= k
n+k
(1 − x ) + ε k x
εk =
ζ nVk
fk
≈ 0.25
ζ k ~ ζ n ~ kt ~ 0.05 pN×sec/nm
t ~ 0.01/sec, k ~ 1 pN/nm
S2
ς
S3
dS3
ς
= k5 − k 6 S 3
dt
ς
Nédélec J Cell Biol 2002 158 1005
dS 2
= k3 − k 4 S 2
dt
dS4
= k7 ( L − S 4 ) − k8 S 4
dt
L
S1
dS1
ς
= ⎡⎣ k1 ( L − S1 ) − k2 S1 ⎤⎦
dt
Important here is that due to transport properties of the motors there is a selection
of the motors of the same p
polarity
y at p
parallel overlapping
pp g MT p
pairs.
Nédélec J Cell Biol 2002 158 1005
Nédélec J Cell Biol 2002 158 1005
Ambrose et al Mol Biol Cell 2005 16 1584
Force balance models
Motors
45
4,5
MT dynamics
1,6
,
“spindle matrix”
3
cortex forces
2
Extrinsic mechanisms
What determines
stable length of
bi-polar
p
spindle?
p
depletion of molecules
1
Concentration gradient of
morphogens models
Intrinsic
mechanisms
Balance of dynein (outward) and ncd (inward) forces explains pole
separation and transient steady state in interphase - prophase
S
Fdyn
Fncd
Sharp
p et al., Mol. Biol. Cell. 2000 11:241
Cytrynbaum et al., Biophys. J. 2003 84:757
Geometry questions:
where are dynein, ncd,
actin?
Are MT asters asymmetric
y
and how are they made
asymmetric?
Mechanical q
questions:
how strong are the forces?
dS 2
= ( Fdyn
y − k ncd S )
dt ς
S ≈ S0 (1 − e − t / T )
?