Physics of Brownian Motors: Swimming in Molasses and Walking in a Hurricane R. Dean Astumian...

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

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.



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)



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)


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)

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

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

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