Fluctuations of intracellular filaments Ines Weber, Ludger Santen, - - PowerPoint PPT Presentation

Ines Weber

Fluctuations of intracellular filaments

Ines Weber, Ludger Santen, Cécile Appert-Rolland and Grégory Schehr

Ines Weber

• Tubulin heterodimers (8 nm) polymerize

to form polarized protofilaments

• 13 protofilaments wrap in a helical way

into a hollow cylinder

• stiff filaments which are resistant to mechanical

stress like stretching and bending:

• bending rigidity
• elasticity
• MTs bend on a typical scale of millimeters under thermal fluctuations
• persistence length

Introduction

Microtubule structure

• B. Alberts et al.,

Molecular Biology of the Cell (Garland Science, 2002)

k ≈ 10−24Nm2 ǫ ≈ 1GPa Lp ≈ 1mm

Ines Weber

Highly bent microtubules in cells, bending on a much smaller length scale

• Non-equilibrium effect
• therwise fluctuations would have to

be on the scale of the persistence length Microtubule dynamics in vivo Active processes inside cells:

• Microtubule fluctuations due to activity of molecular motors

Introduction

Ines Weber

Introduction

• motor proteins can drag intracellular cargo along filament or link

two filaments Molecular motors

• unidirectional movement with discrete steps
• Kinesin: plus-end directed

Dynein: minus-end directed

• load-dependent:

reduced velocity for big forces, motor stops at stall force and detaches from the filaments at Fs ≈ 3pN Fd ≈ 6pN Walk along filaments by converting ATP into mechanical work.

http://multimedia.mcb.harvard.edu/

Ines Weber

• 2 types with oppositely directed

motion

• stochasticity & processivity
• finite force & path length
• flexible coupling to the filament

Microtubule

• semi-flexible filament modelled

as a worm-like chain ,

• no tensibility
• periodic boundary conditions

H = k 2 L ∂θ ∂s 2 ds Molecular motors

Modelling of microtubule fluctuations

load force Lspring = Lmax Lspring = 0   Lspring < Lmax 

Ines Weber

Estimation of filament‘s shape ti+1 ti−1 ti vi vi+1 xi ui(t) ti ti+1 Segment defines filament shape between sites and . E = k

N

• i=1

ti+1

ti

• ∂2

t ui(t)

2 dt Bending energy The energy writes with , . E = xtBx − 1 4Λt ˜ A−1Λ Λ = Λ(t, x) A, B = F(t)

⇔

At mechanical equilibrium: ui(t) = ait3

i + bit2 i + citi + di with

a, b, c, d = f(xi, xi+1, vi, vi+1) F ∼ ∂4

t ui(t)

Modelling of microtubule fluctuations

Ines Weber

rapid, step-like force fluctuations Force dependent rates p(F) = p0

• 1 − F

Fs

• wd(F) = wd0 exp

|F| Fd

• attachment rate

hopping rate detachment rate wa(F) = wa0

Modelling of microtubule fluctuations

Ines Weber

Dynamic of the filament

Influence of the filament rigidity

5×10

4

1×10

5

time [sec]

• 30
• 20
• 10

10 20 30 vertical distance [10nm] 5×10

4

1×10

5

time [sec]

• 30
• 20
• 10

10 20 30 vertical distance [10nm] 5×10

4

1×10

5

time [sec]

• 30
• 20
• 10

10 20 30 vertical distance [10nm]

flexible regime semi-flexible regime stiff regime k ≪ 1 k ≫ 1 k ≈ 1

Ines Weber

• 0,1
• 0,05

0,05 0,1 velocity [nm/s] 0,2 0,4 0,6 0,8 1 CDF

k = 0.001 k = 0.01 k = 0.1 k = 1 k = 10 k = 100 k = 1000

Results

Velocity distribution - influence of the rigidity

• gaussian fluctuations
• small rigidity causes high velocity
• motors cannot induce global drift

σMA =

• 1

Nmax − n

Nmax

• t=n

[x(t) − ˜ xn(t)]2

Non-stationary stochastic time series - trajectory of the filament standard deviation about the moving average x(t)

˜ xn(t) = 1 n

n−1

• k=0

x(t − k)

σMA

Ines Weber

Long-range correlations

exhibits a power law dependence , with the Hurst Parameter and the diffusion coefficient. H σMA = D nH D σMA negative correlation for , positive correlation for , uncorrelated Brownian process for . 0 < H < 0.5 H = 0.5 0.5 < H < 1 The exponent corresponds to a

Ines Weber

Long-range correlations

0,001 0,01 0,1 1 10 100 1000

filament rigidity k

0,4 0,42 0,44 0,46 0,48 0,5 0,52 0,54 0,56 0,58 Hurst Parameter H

L = 50 L = 100 L = 200

Hurst Parameter

• quasi Brownian motion
• maximal cooperation of

motors at k ≈ 30

0,001 0,01 0,1 1 10 100 1000

filament rigidity k

0,001 0,01 0,1 Diffusion coefficient D

L = 50 L = 100 L = 200

Diffusion coefficient

• Transition from semi-flexible

to stiff regime at k ≈ 30

• semi-flexible filament + uncoordinated motors
• sequential update
• velocity depends on the stiffness and on the force ratio
• quasi Gaussian fluctuations,
• diffusion coefficient indicates at least two fluctuation regimes

H ≈ 0.5

Ines Weber

First Summary

Microtubule fluctuations ➡ Motors are not able to induce global drift and persistent movement. Coordinated motor motion needed ?

• semi-flexible filament + coordinated motors
• Tug-of-war model inspired by Müller, Klumpp, and Lipowsky

➡ parallel update

Ines Weber

Tug-of-war model

Cargo transport is provided by a tug-of-war between two motor species

Tug-of-war as a cooperative mechanism for bidirectional cargo transport by molecular motors. Müller, Klumpp, Lipowsky, PNAS,2008

deterministic cargo motion load-dependent velocity binding to the cargo load-dependent unbinding wa p =    pF

• 1 − F

Fs

• ,

for F ≤ Fs pB

• 1 − F

Fs

• ,

for F ≥ Fs wd = wd0 exp |F| Fd

Ines Weber

Tug-of-war model

• no motion

equal number of bound + and - motors

• small fluctuations around initial position
• fast + and - motion with

interspersed pauses

both motor types are bound

• cargo switches between fast upwards/

downwards drift with fluctuating pauses

Tug-of-war as a cooperative mechanism for bidirectional cargo transport by molecular motors. Müller, Klumpp, Lipowsky, PNAS,2008

Cargo transport is provided by a tug-of-war between two motor species

Ines Weber

• 0,005

0,005 velcocity [nm/s] 0,2 0,4 0,6 0,8 1 CDF

L = 50* L = 100* L = 200* L = 50** L = 100** L = 200**

k = 1000, f = 5

*parallel, ** sequential

filaments become more active

• stiff filaments and large force ratio
• parallel update
• ATP-ADP cycle defines intrinsic time scale
• collective motor motion and attachment, hierarchic detachment
• no deterministic filament motion

f = Fs Fd Model

Tug-of-war model for filaments

Ines Weber

Tug-of-war model for filaments

Collective motor dynamics seems to be needed to induce filament drift

250 500 time [sec]

• 0,2

0,2 0,4 0,6 0,8 distance [10nm]

L = 50, k = 1000

sequentiell update

250 500 time [sec]

• 0,2

0,2 0,4 0,6 0,8 distance [10nm]

L = 50, k = 1000

parallel update

sequential update parallel update

Ines Weber

Tug-of-war model for filaments

single motor motion

• many uncoordinated motors
• large filament deformation, no global motion
• motor energy is absorbed in filament buckling

collective motor motion

• competitive motors
• filament stays straight and shows drift
• similar to fast + and - motion in

tug-of-war model

250 500 time [sec]

• 0,2

0,2 0,4 0,6 0,8 distance [10nm]

L = 50, k = 1000

sequentiell update

sequential update

250 500 time [sec]

• 0,2

0,2 0,4 0,6 0,8 distance [10nm]

L = 50, k = 1000

parallel update

parallel update

Results

Ines Weber

Summary

Coordinated motor motion

• semi-flexible filament + coordinated motors
• Tug-of-war model
• stiff filaments and large force ratio needed

➡ parallel update

• semi-flexible filament + uncoordinated motors
• velocity depends on the stiffness and on the force ratio
• quasi Gaussian fluctuations,
• diffusion coefficient indicates at least two fluctuation regimes

H ≈ 0.5 Uncoordinated motor motion ➡ Motors are not able to induce global drift and persistent movement. ➡ sequential update ➡ Coordinated motor motion is needed for persistent movement.

Ines Weber

Thanks to ...

• Annette Kraegeloh and Katharina Narr

Leibniz Institute for New Materials

• Fluorescent Microscopy
• Alexandra Kiemer and Birgit Wahl

Universität des Saarlandes

• Fluorescent Microscopy
• Joachim Weickert

Universität des Saarlandes

• Image Enhancement and Deconvolution