16
MAE 545: Lecture 6 (10/6) Growth dynamics of actin filaments and microtubules
MAE 545: Lecture 6 (10/6)
Growth dynamics of actin filaments and microtubules
2
Cytoskeleton in cells
Actin filament
Microtubule
(wikipedia)
Cytoskeleton matrix gives the cell shape and mechanical resistance to deformation.
3
7nm
Actin filaments
Persistence length `p ⇠ 10µm
Minus end(pointed end) Plus end
(barbed end)
Typical length L . 10µm
Actin treadmilling
actinmonomer
Dynamic fi laments423
Fig. 11.12 (a) If [ M ] c + = [ M ] c
− , both fi lament ends grow or shrink simultaneously. (b) If [ M ] c
+ ≠ [ M ] c − , there is
a region where one end grows while the other shrinks. The vertical line indicates the steady-state
concentration [ M ] ss where the fi lament length is constant.
the same time as the minus end shrinks. A special case occurs when the two
rates have the same magnitude (but opposite sign): the total fi lament length
remains the same although monomers are constantly moving through it.
Setting d n + /d t = −d n − /d t in Eqs. (11.5), this steady-state dynamics occurs at
a concentration [ M ] ss given by
[ M ] ss = ( k off + + k off
− ) / ( k on + + k on
− ). (11.7)
Here, we have assumed that there is a source of chemical energy to phos-
phorylate, as needed, the diphosphate nucleotide carried by the protein
monomeric unit; this means that the system reaches a steady state, but not
an equilibrium state.
The behavior of the fi lament in the steady-state condition is called tread-
milling, as illustrated in Fig. 11.13 . Inspection of Table 11.1 tells us that
treadmilling should not be observed for microtubules since the critical con-
centrations at the plus and minus ends of the fi lament are the same; that
is, [ M ] c + = [ M ] c
− and the situation in Fig. 11.12(a) applies. However, [ M ] c − is
noticeably larger than [ M ] c + for actin fi laments, and treadmilling should
occur. If we use the observed rate constants in Table 11.1 for ATP-actin
solutions, Eq. (11.7) predicts treadmilling is present at a steady-state actin
concentration of 0.17 µ M , with considerable uncertainty. A direct meas-
ure of the steady-state actin concentration under not dissimilar solution
conditions yields 0.16 µ M (Wegner, 1982 ). At treadmilling, the growth rate
from Eqs. (11.5) is
d n + /d t = −d n − /d t = ( k on + • k off
− – k on − • k off
+ ) / ( k on + + k on
− ), (11.8)
corresponding to d n + /d t = 0.6 monomers per second for [ M ] ss = 0.17 µ M .
4
Actin growth
k+on
k+o↵
k�o↵
k�on
dn�
dt= k�
on
[M ]� k�o↵
dn+
dt= k+
on
[M ]� k+o↵
growth of minus end growth of plus end
concentration of free actin monomers
no growth at no growth at
[M ]+c =k+o↵
k+on
[M ]�c =k�o↵
k�on
actin growsactin shrinks
Steady state regime
dn+
dt= �dn�
dt
front speeddn+
dt=
k+on
k�o↵
� k�on
k+o↵
k+on
+ k�on
⇡ 0.6s�1
[M ]ss
=k+o↵
+ k�o↵
k+on
+ k�on
⇡ 0.17µM
5
Distribution of actin filament lengths
k+on
k+o↵
k�o↵
k�on
total rate of actin monomer addition
kon
= k+on
+ k�on
ko↵
= k+o↵
+ k�o↵
total rate of actin monomer removal
Master equation@p(n, t)
@t= k
on
[M ]p(n� 1, t) + ko↵
p(n+ 1, t)� kon
[M ]p(n, t)� ko↵
p(n, t)
Continuum limit@p(n, t)
@t= �v
@p(n, t)
@n+D
@2p(n, t)
@n2
v = kon
[M ]� ko↵
D = (kon
[M ] + ko↵
)/2
drift velocitydiffusion constant
at large concentrations actin grows ( )
[M ] >ko↵
kon
= [M ]ss
v > 0
at low concentrations actin shrinks ( ) v < 0
[M ] <ko↵
kon
= [M ]ss
6
Distribution of actin filament lengths
k+on
k+o↵
k�o↵
k�on
total rate of actin monomer addition
kon
= k+on
+ k�on
ko↵
= k+o↵
+ k�o↵
total rate of actin monomer removal
What is steady state distribution of actin filament lengths at low concentration?
D = (kon
[M ] + ko↵
)/2
drift velocitydiffusion constant
@p⇤(n, t)
@t= �v
@p⇤(n, t)
@n+D
@2p⇤(n, t)
@n2= 0
v = kon
[M ]� ko↵
< 0
p⇤(n) =|v|D
e�|v|n/D =1
ne�n/n
n =D
|v| =(k
o↵
+ kon
[M ])
2(ko↵
� kon
[M ])average actin
filament length
7
Actin filament growing against the barrierwork done against the barrier for insertion of
new monomer
W = Fa
effective monomer free energy potential without barrier
away fromfilament
attachedto the tip
�
k+on
⇠ 4⇡D3
a
k+o↵
/ e��/kBT
effective monomer free energy potential with barrier
�
W
away fromfilament
attachedto the tip
k+on
(F ) ⇠ k+on
e�Fa/kBT
k+o↵
(F ) ⇠ k+o↵
F
k+on
(F )
k+o↵
(F )
2a
��W
8
Actin filament growing against the barrierwork done against the barrier for insertion of
new monomer
W = Fa
effective monomer free energy potential with barrier
�
W
away fromfilament
attachedto the tip
Growth speed of the tip
k+on
(F ) ⇠ k+on
e�Fa/kBT
k+o↵
(F ) ⇠ k+o↵
Maximal force that can be balanced by growing filament
v+(Fmax
) = 0 Fmax
=kBT
aln
✓k+on
[M ]
k+o↵
◆
F
k+on
(F )
k+o↵
(F )
2a
v+(F ) =dn+(F )
dt= k+
on
[M ]e�Fa/kBT � k+o↵
k+on
⇠ 10µM�1s�1
k+o↵
⇠ 1s�1
[M ] ⇠ 10µMa ⇡ 2.5nmFmax
⇠ 8pN
(stall force)
9
Microtubules
25nm
Persistence length Typical length L . 50µm
`p ⇠ 1mm
10
Microtubule dynamic instability
Wikipedia
catastrophe occurs when protective cap disappears
11
Simple model of microtubule growth
What is the average growth speed and average diffusion constant
for such dynamic system?
First let’s ignore all molecular details and assume that microtubules switch at fixed rates between
growing and shrinking phases
(for simplicity ignore diffusion during individual growing or shrinking event)
vg
vs
x
rres rcat
@pg(x, t)
@t
= �vg@pg(x, t)
@x
� rcatpg(x, t) + rresps(x, t)
@ps(x, t)
@t
= +vs@ps(x, t)
@x
+ rcatpg(x, t)� rresps(x, t)
M. Dogterom and S. Leibler,PRL 70, 1347-1350 (1993)
Typical values in a tubilin solution of concentration :10µM
vg ⇡ 2µm/min
vs ⇡ 20µm/min
rcat ⇡ 0.24min�1
rres ⇡ 3min�1
12
Fokker-Plank equation in Fourier spectrum
@p(x, t)
@t
= �v
@p(x, t)
@x
+D
@
2p(x, t)
@x
2
Master equation
Fourier spectrum
i!p̃(k,!) = +ivkp̃(k,!)�Dk2p̃(k,!)
i⇥! � vk � iDk2
⇤p̃(k,!) = 0
!(k) = vk + iDk2Only those Fourier modes are
nonzero that satisfy dispersion relationp̃(k,!)
p(x, t) =
Zdkd! e
i!t�ikx
p̃(k,!)
Initial condition
p(x, 0) =
Zdk e
�ikx
p̃(k,!(k))
Time evolution
At large times only small k components are relevant!
p(x, t) =
Zdk e
�ik(x�vt)e
�Dk
2t
p̃(k,!(k))
13
Average growth speed and average diffusion constant can be determined
from dispersion relation for such system:
Master equation
vg
vs
x
rres rcat
@pg(x, t)
@t
= �vg@pg(x, t)
@x
� rcatpg(x, t) + rresps(x, t)
!(k) = vk + iDk2 + · · ·
Simple model of microtubule growth
@ps(x, t)
@t
= +vs@ps(x, t)
@x
+ rcatpg(x, t)� rresps(x, t)
14
Master equationvg
vs
x
rres rcat
@pg(x, t)
@t
= �vg@pg(x, t)
@x
� rcatpg(x, t) + rresps(x, t)
Simple model of microtubule growth
Fourier spectrump
g,s
(x, t) =
Zdkd!e
i!t�ikx
p̃
g,s
(k,!)
@ps(x, t)
@t
= +vs@ps(x, t)
@x
+ rcatpg(x, t)� rresps(x, t)
Only those Fourier modes are nonzero, that correspond to the matrix with zero determinant!
(! � vgk � ircat)(! + vsk � irres) + rresrcat = 0
i
✓! � vgk � ircat irres
ircat ! + vsk � irres
◆✓p̃g(k,!)p̃s(k,!)
◆= 0
15
vg
vs
x
rres rcat
Simple model of microtubule growthDispersion relation
± i
2
q(rcat + rres)2 + 2ik(rcat + rres)(vg + vs)� k2(vg + vs)2
!(k) =1
2
✓k(vg � vs) + ircat + irres
◆
!(k) = vk + iDk2 + · · ·
We are interested in effective behavior at large timescales, which correspond to small .
Taylor expand for small k to find: !
average growth speed average diffusion constant
v =rres
(rres + rcat)vg �
rcat(rres + rcat)
vs D =rresrcat(vg + vs)2
(rres + rcat)3
probability of growing
probability of shrinking
16
vg
vs
x
rres rcat
Simple model of microtubule growth
average growth speed
v =rres
(rres + rcat)vg �
rcat(rres + rcat)
vs
x
t
v > 0
v < 0
unbounded growth
bounded growth
bounded growth
p
⇤g
(x) =v
s
(vg
+ v
s
)
1
L
e
�x/L
p
⇤s
(x) =v
g
(vg
+ v
s
)
1
L
e
�x/L
L =vgvs
(vsrcat � vgrres)/ 1
|v|
average filament length
v < 0
