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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)
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
--
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
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
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: 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