Interaction of motor proteins
with obstacles
Helicase unwinding of DNA
M. D. Betterton
Department of Applied Mathematics
University of Colorado at Boulder
joint work with Frank Jülicher
MPIPKS, Dresden
http://www.mpipks-dresden.mpg.de/mpi-doc/julichergruppe/
Not all motors move on an
infinite periodic track
The Polymerization Ratchet
Growth of a polymer near a wall
wall
F
polymer
given enough
space, next
monomer can bind
Peskin, Odell, and Oster, Biophys J 65, 316 (1993)
The Polymerization Ratchet
Dogterom and Yurke, Science 278, 856 (1997)
Mogilner and Oster, Eur Biophys J 28, 235 (1999)
Carlsson, Phys Rev E 62, 7082 (2000)
Kolomeisky and Fisher, Biophys J 80, 149 (2001)
and important applications to cell
motility, …
Kinesin-related ATPase
Localizes to microtubule ends
Hunter et al. Mol. Cell 11 445, (2003)
http://www.mpi-cbg.de/research/groups/howard/projects.html
MCAK
Accelerates MT depolymerization 100x
Appears to processively depolymerize
MCAK off rate 0.054 s-1
Tubulin dimer off rate 1 s-1
Hunter et al. Mol. Cell 11 445, (2003)
http://www.mpi-cbg.de/research/groups/howard/projects.html
MCAK
http://fajerpc.magnet.fsu.edu/Education/2010/Lectures/26_DNA_Transcription.htm
Nucleic-acid motors
RNA Polymerase
http://ntri.tamuk.edu/cell/ribosomes.html
NA-based motors
Ribosome
http://www.stanford.edu/group/blocklab/Exo2.gif
NA-based motors
Exonuclease
NA-based motors
Helicase
Interacting Hopping Model
Hopping rates:
k
Position: n
k 

m
1D lattice
Two fluctuating degrees of freedom
Hypothesize interaction potential
k
Position: n
k 

m
0 m
U ( m  n)  
 m
n
n
Interaction changes rates
Detailed balance
 n ,mn ,m 1 
 e
 n ,m 1n ,m 
k
k

n , m  n 1, m

n 1, m  n , m

[U ( m  n 1) U ( m  n )]
kT
k
 e
k
[U ( m  n 1) U ( m  n )]
kT
Simplest interaction
Exclusion interaction
Steric inhibition
n=m forbidden
k

n=m-1
Steric Inhibition
k

n=m-1
[U (0) U (1)]
kT
 mm1 
 e
 m 1m 
  0 when U (0)  
Hard-wall Potential
1 
 e
0 

1

0

[U (0) U (1)]
kT
k
k
 e
k
k
[U (0) U (1)]
kT
 1  0
U (0)     
k1  0
Questions
How does changing the interaction
potential change the motion of the
complex?
Is there an optimal potential for fastest
motion?
Helicase opens dsNA
Motor protein – fueled by ATP hydrolysis
Can open duplexes of DNA-DNA, DNA-RNA, or
RNA-RNA
+ATP
+ATP
(Assumes strands
don’t re-anneal)
Cellular Role of Helicases
All cellular processes involving nucleic acids
Replication
Transcription
Translation
RNA processing
DNA repair
Important for Genome Stability
helicase binds
3’
5’
single strand
double strand
helicase translocates
helicase moves junction
displaces strand
Bird et al. Nucl Acids Res 26, 2686 (1998)
Dillingham et al. Biochemistry 39, 205 (2000)
Dillingham et al. Biochemistry 41, 643 (2002)
Mechanism
Passive
Doesn’t interact directly
with duplex
Waits for fluctuation to
advance
Inhibits closing
Active
Interacts with duplex
Destabilizes duplex
Increases opening rate
Hard wall
Lohman and Bjornson, Ann Rev Biochem 65, 169 (1996)
Singleton and Wigley, J. Bacteriol 184, 1819 (2002)
Mutation Studies support idea
of an active mechanism
Mutate PcrA residues which touch
duplex
Unwinding rates decrease 10–30x
Soultanas et al. EMBO Journal 19, 3799 (2000)
Helicase motion
k3’
k+
5’
Helicase motion
k3’
k+
5’
If out of equilibrium can have k+>k-
PcrA: k+-k- =80 bases/s
Dillingham et al. Biochemistry 41, 643 (2002)
DNA ss-ds junction motion
 
Junction motion
 


G
Junction at Junction at
base m
base m+1
Closing lowers energy

e

G

kT
1
e 
7
2
Effects neglected
Helicase binding/unbinding
DNA flexibility
Different biochemical states of helicase
DNA sequence variability
Effects of randomness on unzipping
Lubensky and Nelson Phys Rev E 2002
Effects of randomness on motor protein motion
Kafri, Lubensky and Nelson cond-mat 2003
Computing Unwinding Rate
P (n, m, t )
probability of finding
helicase at n, junction at m, time t
dP(n, m, t )


 ( mm 1   mm1  knn1  knn1 ) P(n, m, t )
dt

 kn 1n P(n  1, m, t )   m 1m P (n, m  1, t )
 kn1n P(n  1, m, t )   m 1m P(n, m  1, t )
Simulation of Full Equations
Junction
Closing rate = 0.1/time step
Opening rate 0.1/7
Helicase
Forward hop rate = Closing rate/100
Backward hop rate = Forward rate/40
Start with uniform junction position
Run for 25 closing times
Junction position m
Thanks to Alex Barnett
Helicase position n
Simulation of Full Equations
NA closes quickly compared to
helicase hop
Speed up movie 500x
Junction position m
Thanks to Alex Barnett
Helicase position n
Separate dynamics
j  m  n difference
l  m  n midpoint
Rates depend on j only
U ( m  n)  U ( j )
 j,j,k ,k

j

j
P j (t )   P( j, l , t )
l
Difference-variable equation
dP j
dt
 (k  k   j   j )P j

j

j
 j 1P j 1   j 1P j 1 DNA

j 1 j 1
k P

j 1 j 1
k P
Helicase
Difference-variable equation
dP j
dt
 (k  k   j   j )P j

j

j
( j 1  k ) P j 1 Increase j

j 1
(  j 1  k

j 1
)P j 1 Decrease j
Difference-variable dynamics equilibrate
quickly compared to midpoint motion
Boundary conditions:
zero-current solution
 j k
P j 1  
  j 1  k


j

j 1

 P j

Steady-State Unwinding Rate
Find current the average current in l
Unwinding velocity
1


v   ( j  k j   j  k j )P j
2 j
Forward
rate at j
Backward Prob
rate at j
at j
Hard-Wall Opening
v HW
 k   k
v HW 
  k
k 
 0 when
  0.14
k 
Effective chemical potential of opening must be larger
than the free energy change of DNA closing
Numbers: upper bound
Assume k  0
k

v HW
 1

 k  11 bp/s

Bp at junction is open 1/7 of the time
When helicase tries to move forward, it
succeeds with probability 1/7
Varying step size
Hard-wall
unwinding
velocity drops
rapidly with
increasing step
size
Questions
How does changing the interaction
potential change the unwinding rate?
Is there an optimal potential for fastest
opening?
The Step
Energetic Cost Uo for
dsDNA and helicase to
overlap by one base
0  U
 e
1 
Relatively faster opening
slower forward hop

0

1

o
k
k Uo
 e
k
k
Determining Rates
A degree of freedom remains
1   e
0   e
 fU o
 ( f 1)U o
0  f 1
f 0
f 1
1  
 0   eU
Increase
opening
o
1   e U
0  
o
Decrease
closing
One-Step Unwinding Rate
v1
c  (1  c)e  fU o

1
U o
v HW c  (1  c)e
  k 
c

  k 
One-step active opening
faster than passive
Multi-step Staircase
Each step height Uo
For larger number of steps
n, v increases more
rapidly
n
vn

v HW
c n  (1  c)e  ( f 1)U o  c n  j e  jU o
j 1
n
c  (1  c) c
n
j 1
n  j  jU o
e
Multi-step Staircase
For large n,
maximum at
U max   ln c  2
Optimal potential
cancels base pairing
energy
Opening neutral
Helicase crystal
structures suggeset n
of 5-10
Thanks to Seth Fraden
Interaction potential
Worse than Hard Opening:
Negative Step Height
-1 0 1 2
Difference variable m-n
Well depth of 2kT decreases velocity
to 0.2 of hard-wall velocity
Force-velocity curves
Soft opening,
one step
Force changes
base-pairing
energy
G  Go  Fl
Result strongly
depends on
step height
Summary
Simple model for motor protein-obstacle
interactions
Comparison of active and passive helicase
unwinding
Predict changes if vary
k+/kBase-pair free energy
Speed decrease from active to passive
Model: factor of 7 (hard to soft)
Model: factor of 35 (soft to well)
Experiment: factor of 10-30