Download Hongyun Wang - Research Review Day

Exploring Protein Motors -- from what we can
measure to what we like to know
Hongyun Wang
Department of Applied Mathematics and Statistics
University of California, Santa Cruz
Baskin School of Engineering Research Review Day
October 12, 2007
The goal of mathematical studies …
Visscher, Schnitzer & Block (1999)
Chen & Berg (2000)
Yasuda, et al. (1998)
… is to decipher motor mechanism from measurements.
An example of macroscopic motor
(1) Intake
(2) Compression
(3) Expansion
(4) Exhaust
Torque of a single cylinder engine:
How to find the motor force
of a molecular motor?
Mathematical framework for
molecular motors
Mechanochemical models
Visscher, Schnitzer & Block (1999)
Mechanical motion:
dW t 
dx
  S x    L  2kBT 
dt
dt
Load
Force from
potential
Chemical reaction:
Yasuda, et al. (1998)
Chen & Berg (2000)
dS
 K x  S
dt
Brownian
force
force
SITE

SITE  ATP 
SITE  ADP

SITE  ADP P 
Mechanical motion and chemical reaction are coupled.
i
Characters of molecular motors
Molecular motors
•
Time scale of inertia << time scale of reaction cycle
•
Instantaneous velocity >> average velocity
•
Kinetic energy from average velocity << kBT
•
Use thermal fluctuations to get over bumps
Macroscopic motors
•
Time scale of inertia >> time scale of reaction cycle
•
Instantaneous velocity ≈ average velocity
•
Kinetic energy from average velocity >> kBT
•
Use stored kinetic energy (inertia) to get over bumps
Macroscopic motors use stored kinetic
energy to get over bumps
V
E

V
mV 2
E
V
 1% 
 1%
2
mV
V
This does not work in molecular motors!
Molecular motors use thermal
excitations to get over bumps
Thermal energy is huge!
The energy for accelerating a bottle of water to
100 miles/hour
can only heat up the bottle of water by
0.24 degree!
Mathematical equations
Langevin formulation
(stochastic evolution of an individual motor):
dx

dW t
S x    L
D
 2D
dt
kBT
dt
(mechanical motion)
dS
 K  x S
dt
(chemical reaction)
Fokker-Planck formulation
(deterministic evolution of probability density):
 S
 S 
  S  x    L
D 
S 

t
x 
k BT
x 
Diffusion 
Convection

N
 k  x 
j 1
Sj
Change of
occupancy
j
, S  1, 2,
,N
Efficiencies of a molecular motor
Thermodynamic efficiency of a motor working
against a conservative force
External
agent
Visscher et al (1999). Nature

Thermodynamic efficiency 
f v
TD 
,
G r
Energy output
Energy input
v  average velocity,
r  reaction rate,
 G   free energy drop of each cycle
Motor
system
Stokes efficiency of a motor working
against a viscous drag
Hunt et al (1994). Biophys. J.
Yasuda, et al (1998). Cell.
 Viscous drag is not a conservative force:
 Energy output = 0
The Stokes efficiency:
Stokes 
 v
2
 G r
Stokes efficiency  thermodynamic efficiency
(experimental observations)
Viscous stall load < Thermodynamic stall load
lim   v  f Stall
 
Hunt et al (1994). Biophys. J.
Visscher et al (1999). Nature
Which measurement is correct?
Stokes efficiency  thermodynamic efficiency
(theory)
Viscous stall load:
lim   v
 
  
0 exp  kBT s  ds

1
   sL  sL     sL  
 
 1
exp
ds

exp


0 0 
k BT

 k BT
1
 
s   ds
 
<
fStall
… implies that the motor force
is not uniform.
Motor potential profile
A mean-field potential
Fokker-Planck formulation
 S

  S  x   ƒ L
D 
S  S
t
x 
kBT
x
 N
   kS j  x   j , S  1, 2,, N
 j 1
At steady state, summing over S
   x   ƒ L
 
0D 
 
x  kBT
x 


N
x    S x 
Probability density of motor at x
N
1
  x  
 S  x S x 

x  S1
Motor force at x (averaged over
all chemical states).
S 1
The motor behaves as if it were driven by a single potential (x)
(x) is called the motor potential profile.
The potential profile is measurable
 x   x t 
x(t)x
… does not work well
A more robust formulation:
   x   ƒ L
 
0D 
 
x  kBT
x 




   x   f
v
 
L
  J 
D 

kBT
x 
L

  x
x
f x
J
1
 L  log    x    
ds
k BT
k BT
D 0  s
Therefore, we only need to reconstruct PDF.
Time series of motor positions
measured in single
molecule experiments
Reconstructing potential in a test problem
Extracted motor potential profile

A sequence of 5000 motor
positions generated in a
Langevin simulation

Summary
Everything depends on a proper mathematical model
and
analysis of the model!
Time scale of inertia
Newton’s 2nd law: m
Time scale of inertia:
m 
4 3
r ,
3
dv
dW t
  v  
S x    L  2kB T
dt
dt
t0 
m

  6 r
For a bead of 1 mm diameter:

t0 
m


2 2
r
9
t0  56  10 9 s  56 ns