physics of brownian motors: swimming in molasses and walking in a hurricane r. dean astumian...
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Physics of Brownian Motors: Swimming in Molasses and Walking in a Hurricane
R. Dean Astumian
Department of Physics and Astronomy
University of Maine
Principles of directed motion of molecular motors and pumps
Molecular motors operate in an environment where viscosity dominatesInertia, and hence where velocity if proportional to force. AnalogiesWith electric circuits and chemical networks are much more appropriate Than models based on cars, turbines, judo throws, and the like.
Directed motion and pumping arises by a cycling through parameterspace
The role of chemical free energy is to assure that the orderof breaking and making bonds when “substrate” binds isnot the microscopic reverse of the order of breaking and making bonds when product is released
Strong coupling and effective motion in the face of thermal noiseIs attained by reduction of dimensionality.
Time scales are determined by depth of energy wells and height of Barriers. Moving between wells occurs by thermal activation over
Barriers.
where
for
ATP
ATP
ATP
ADP + Pi
ADP + Pi
ADP + Pi
Na+
Na+
linear motor
rotary motor
ion pump
Bivalves, bacteria, and biomotors
Scallop Theorem - swimming in molasses
Reynolds number
10-2 m
10-2 m/s
Purcell, AJP 1977
Inertial force/viscous force
r1
r2 A
A = constant
A
Adiabatic
Non-adiabatic
ChairBoat
Closed
Closed
Open
Open
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Nano-patio furniture
Energetics - Stability vs. Lability
First formed last broken; thermodynamic control
First formed first broken; kinetic control
Trajectory
r1
r2
r1
r2
1 2
2
11
2Thermodynamic kinetic
Parametric modulation
€
vcm = dxcm /dt = FdΔx /dt = FdΔxeq∫ + FdδΔx∫adiabatic non-adiabatic
Dephosphorylated
12 2 1
Phosphorylated
€
dΔx /dt = −2k(γ1−1 +γ 2
−1)(Δx− Δxeq )
€
F =(γ1
−1 − γ 2−1)
2(γ1−1 +γ 2
−1)€
dxcm /dt = −k(γ1−1 − γ 2
−1)(Δx− Δxeq )
Energetics - Stability vs. Lability
G0ATP = 7.3 kcal/mole; GATP (phys) = 12. kcal/mole
Ghyd = 1-5 kcal/mole
RT = .6 kcal/mole (300 K)
Stability: First formed last broken; thermodynamic control
Lability: First formed first broken; kinetic control
Engineering - many weak bonds vs. a few strong bonds
12 2 1
12 2 1
F
xeq
Modularity -the binding and spring cassettes are independent
€
dF /dt = −τ1−1(F− F )
dΔxeq /dt = −τ 2−1(Δxeq − Δxeq )
Reduce dimensionality and then use thermal noise to assist motionIn one direction but not another by a brownian ratchet.
What is a Brownian ratchet?
Ultimately energy comes from chemical reaction, but the momentumIs borrowed from the thermal bath
x2x1
X2 Fx1
Fx2
X1
(n,n-1) (n,n-2)
ab
k+k-
ATP
ADP + Pi
m n
ssds
Three Key Features of Brownian Motor Mechanisms:
• spatial or temporal asymmetry
• external energy source
• thermal noise
S
S
P
kBA
kAB
kAB
kBA
S
S
P
Brownian Motor
white black
R
L
N
2(T)
7, 11 (8) 7, 11 (8)11 (2) 11 (2)
2,3,12 (4) 2,3,12 (4)2,4,12 (5) 2,4,12 (5)
other (24)other (29) other (29)
white black
R
L
N
1(H)
other (24)
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BW Wlose win2/36
2/36 2/36
5/36
5/36
5/36
8/36
8/36
8/36
4/36
4/36
4/36
game 1(H):
game 2(T):
BW Wlose win
lose
lose
win
win
lose
win
1(H)
2(T)
1(H)
ln(36/2)
ln(36/4)
ln(36/8)
ln(36/5)
ln(8/5)
ln(2/4)ln(5/8)
ln(4/2)
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lose
win
combined gameln(4.5/36)
(5/4.5ln )
(4.5/5ln )
(5/36ln )
>>b
Influencing intramolecular motion with an alternating electric field
Bermudez et al., Nature 406: 608 (2000)
Unidirectional rotation in a mechanically interlocked molecular rotor
Leigh et al., Nature 424: 174 (2003)
Unidirectional rotation in a mechanically interlocked molecular rotor
Leigh et al., Nature 424: 174 (2003)
+ -
k φ
k
k
/k φ
+
-
-
-
-
-
-
-
+
+
+
+
+
+
+
-
-
-
-
-
-
-
+
+
+
+
+
+
+
Conclusions:
A very general framework for biological energy transduction takes advantage of the linearity of many processes includinglow Reynolds number motion.
Velocities depend on time only through the parameters so thedistance transported per cycle of modulation can be expressedas a loop integral.
The direction of motion is controlled by the trajectory of the transitions between the “equilibrium” states, and NOT by theirstructure.
Electrical analogies are more appropriate than automotive ones
Acknowledgements
• Marcin Kostur, University of Silesia and University of Maine
• Martin Bier, Univ. of Chicago, East Carolina University
• Imre Derenyi, Curie Institute, Paris
• Baldwin Robertson, NIST• Tian Tsong, University of
Minnesota
• NIH, NSF MRSEC,MMF
Na+-K+ transport ATPase
3
2
Peter Lauger, “Ion motive ATPases”
Serpersu and Tsong, (1984) JBC 259, 7155; Liu, Astumian, and Tsong, (1990) JBC 265, 7260
dQ/dt = I1(t) + I2(t)
F = I1(t)/[I1(t) + I2(t)]
Inet = <F dQ/dt>
Causality vs. Linkage
S
P
I I
Inet = kesc u sin( cos( ~ ~
1 + ~
Robertson and Astumian, Biophys. Journ. (1990), 58:969
exp() = exp(V(t) M/d)
Astumian, Chock, Tsong, Chen, and Westerhoff, Can Free EnergyBe Transduced from Electrical Noise?, PNAS 84: 434 (1987)
Na+-K+ transport ATPase
3
2
Hysteresis of BPTI - MD simulation
Protein Response to External Electric Fields: Relaxation, Hysteresis, and Echo, Dong Xu,† James Christopher Phillips, and Klaus Schulten*J. Phys. Chem. 1996, 100, 12108-12121
I1 = k1(1-Q) - k1 Q = 0
I2 = k2(1-Q) - k2 Q = 0
k1 k2
k1 k2
= 1
At equilibrium
A corollary
But u and drop out!
Iad = F dQeq/dt = F dQeq
Qeq exp(
F = exp(u
Electrical or Photocontrol ofthe Rotary Motion of a Metallacarborane
M. Frederick Hawthorne et al. SCIENCE VOL 303 19 MARCH 2004, 1649-1651
Rotation angle
Rotation angle
m v – F = - v + (2 k T 1/2 (t)
Baseballs, beads, and biopolymers
r = 10-1 m r = 10-5 m r = 10-8 m
Newtonian Aristotelian Brownian
hurricanemolassesNewton