Biophysics Master Course, Fall 2002

Some of the physics cells have to deal with:

Random walks, diffusion and Brownian motion

Background reading:

— Frederick Reif: Statistical and Thermal Physics, Chpt. 1 (random walks), Chpt. 15 (fluctuations, Brownian motion)

— Howard Berg: Random Walks in Biology, Chpts. 1, 2 (diffusion), Appendix A (distributions)

— Jonathon Howard, Mechanics of Motor Proteins and the Cytoskeleton, Chpt. 4 (diffusion), Chpt. 16 (motor models)

— Richard Feynman, Feynman Lectures I, Chpt 41 (Brownian motion),Chpt. 46 (thermal ratchet)

— Landau, Lifschitz, Volume V, Statistical Physics, Chpt. 12 (fluctuations, pretty advanced)

— Frederick Gittes, Christoph Schmidt, Signals and noise in micromechanical measurements. In Laser Tweezers in Cell Biology. Methods in Cell Biology, 55: 129-156, Academic Press, San Diego, CA, 1998 (power spectral analysis).

— Gittes, F., Schmidt, C.F. (1998), Thermal Noise Limitations on Micromechanical Experiments, Eur. Biophys. J., 27: 75-81 (spectral analysis, other noise)

Optical trapping of 0.2 µm silica beads in water

Bacterial motility, E. coli from Howard Berg lab, courtesy Linda Turner

Intracellular transport in Reticulomyxa,video: M. Schliwa, M. Koonce

N

N

HC

C

H

O

H

Adenine(Base)

NH2

N

N

CHC

C

O

O O

O

i

Inorganicphosphate, P H O

O O O

2

B

R ibose(Sugar)

OHOH

OCH

Adenosine Triphosphate, ATP,Adenosine Diphosphate, ADP

(Nucleotide)

Change inf reeenergy: ΔG ≈ 13 kcal/mol ≈ 22 k T

-

P-

Triphosphate

P

O

OP

--

2D random walk, 18050 steps

Intracellular Transport on Cytoskeletal Tracks

1 m

Cell Body

Synapse

Axon

Vesicles with motors

Active transport: v ≈ 1µm/s, T ≈ 10 days

Diffusion: T = x2/6D ≈ 26,000 years

Microtubules

The Main Motor Protein Families

(asymmetric) track: actin filaments, microtubules

Cargo:Vesicles,Organelles

Motors:myosins,kinesins,dyneins

Fuel:ATP

The Feynman Thermal Ratchet

Pforward~exp(-/kT1)Pbackward~exp(-/kT2)

works only if T1>T2 !!

motor protein conformational change: µsdecay of temperature gradient over 10 nm: ns

wrong model

rel ≈ Cl2/(42)

Brownian Ratchet (A.F. Huxley ‘57)

CargoThermal motion

Track

Net transport

perpetuum mobile? Not if ATP is used to switch the off-rate.

Motor

Three-bead assay with ncd

Myosin: averaged power strokes(Veigel et al. Nature ‘99, 398, 530)

Myosin Power Stroke

Mechano-chemical cycle:

M*ATP

M*ADP*Pi

M

attach

working stroke

detach

recovery stroke

ADP+Pi

ADP

Pi

actinmyosin

Conformational Change of Single ncd Molecule

-5

0

5

10

0 400 800 1200 1600

time [ms]

2 µM ATP

release

DeCastro, Fondecave, Clarke, Schmidt, Stewart, Nature Cell Biology (2000), 2:724

ADP ADPADP*Pi

~ 7 nm

-50

0

50

100

150

200

250

300

-1

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60

time [s]

160

176

192

208

224

240

6 6.5 7 7.5 8 8.5 9

time [s]

Stepping and Stalling of a Single Kinesin Molecule

~ 6 pN stall force

~ 8 nm steps

Svoboda, Schmidt, Schnapp, Block, Nature (1993), 365: 721

<x2(t)> - <x(t) > 2

Randomness parameter

r:= lim t -> ∞ d <x(t)>

1 -> 2 -> 3 -> 4 -> 5 -> 6k k k k k

r = 0

r = 1

1 -> 2k

t=const.

“clockwork”

t exponentially distributed

Poisson process

Randomness parameter for single kinesin

(Visscher,Schnitzer, Block (‘99),Nature 400, 184)

Thermal Motion of a Trapped/Tethered Particle

var( x ) = x

2

− x

2

=

kB

T

S ( f ) =

kB

T

2

γ ( fc

2

+ f

2

)

fc

=

2 γ

, S0

=

4 γ kB

T

2

trappe d bea d attached tomotor: (var x ) =

kB

T

trap

+ motor

-200-150-100-50

050

100150200

0 0.05 0.1 0.15 0.2 0.25 0.3

Displacement [nm]

Time [s]

0.01

0.1

1

10

100

1 10 100 10 3

Power spectral density, S(f) [nm

2

/Hz]

fc

Frequency, f [Hz]

S(f) ≈ S0

slope = -2

Time series:

Spectrum:

Efficiency, Invertability andProcessivity of Molecular Motors

F. Jülicher, Institut Curie, Paris

http://www.curie.fr/~julicher

A. ParmeggianiL. Peliti (Naples)A. Ajdari (Paris)J. Prost (Paris)

v = dx J ii

∑0

l

∫

Mechano-chemical coupling

∂tPi +∂xJ i =− ωijj≠i∑ Pi + ωji

j≠i∑ Pj

J i =−μ(kBT∂xPi +Pi∂xWi −fextPi )

M

M-ADPM-ADP-PM-ATP

mean velocity

1

2

3

4

x

Example: weakly bound state

ATP

ADP-P ADP

strongly bound

weakly bound

Example: identical shifted states

la

U

UkBT

=20

al

=0.1

ξ =7.4⋅10−4kg/s

l =16nmω12 =ωe(Δμ−ΔW) /kT

ω21=ω

Δμ =μATP −μADP −μP

μi ≈μi0 +kTln(Ci /Ci

0)

ω12 ω21

chemical free energy of hydrolysis:

Dissipation rates

motionwithin a state:

˙ Q i = dxJ i0

l

∫ ∂xHi ≥0

chemical transitions:

˙ Q α

= dx(ω120

l

∫ P1 −ω21P2)(H1 −H2 +Δμ)≥0

Hi =Wi − fextx

+kBT ln(Pi)

total internal dissipation of the motor:

˙ Q = ˙ Q 1 + ˙ Q 2 +Qα ≥0

Efficiency of energy transduction

η =−fextvrΔμ

fext

v

Δμ rforce

chemical energy

velocity

chemical rate

A. Parmeggiani, F. Jülicher, A. Ajdari and J. Prost, PRE 60, 2127 (1999)

Energy conservation:

˙ Q =rΔμ +fextv

chemical work

mechanical work

total internal dissipation