Deca-Alanine Stretching
Free Energy Calculation from
Steered Molecular Dynamics Simulations
Using Jarzynski’s Eqaulity
Sanghyun Park
May, 2002
Jarzynski’s Equality
T
 = i
T
 = (t)
 = f
heat Q
work W
 = end-to-end distance, position of substrate along a channel, etc.
2nd law of thermodynamics:
Jarzynski (1997):
W  F = F(f) - F(i)
 exp (-W)  = exp (-F)
difficult to estimate
Derivation of Jarzynski’s Equality
Process described by
a time-dependent Hamiltonian H(x,t)
Trajectory xt

External work W = 0 dt tH (xt,t)
If the process
(1) is Markovian: tf(x,t) = L(x,t) f(x,t)
(2) satisfies the balance condition: L(x,t) exp[-H(x,t)] = 0
Then, exp { - [ F() - F(0) ] } =  exp (-W) 
Isothermal MD schemes (Nose-Hoover, Langevin, …)
satisfies the conditions (1) and (2).
Cumulant Expansion of Jarzynski’s Equality
F = - (1/) log e-W
statistical error & truncation error
= W - (/2) ( W2 - W2 )
+ (2/6) ( W3 - 3W2W + 2W3 ) + •••
p(W)
shift  2 / kBT
shift/width   / kBT
W  p(W)
W2  p(W)
W3  p(W)
e-W  p(W)
width 
big in strong nonequilibrium
W
Helix-Coil Transition of Deca-Alanine in Vacuum
Main purpose:
Systematic study of the methodology of free energy calculation
- Which averaging scheme works best
with small number (~10) of trajectories ?
Why decaalanine in vacuum?
- small, but not too small: 104 atoms
- short relaxation time  reversible pulling  exact free energy
end-to-end distance (Å)
Typical Trajectories
v = 0.1 Å/ns
v = 10 Å/ns
v = 100 Å/ns
time
time
time
30
20
10
Force (pN)
600
0
-600
end-to-end distance (Å)
F = W
TS = E - F
E, F, S (kcal/mol)
reversible !
E
F
TS
# of hydrogen bonds
work W (kcal/mol)
Reversible Pulling (v = 0.1 Å/ns)
end-to-end distance (Å)
PMF and Protein Conformations
PMF (kcal/mol)
Irreversible Pulling ( v = 10 Å/ns )
10 blocks of 10
trajectories
PMF (kcal/mol)
winner
end-to-end distance (Å)
end-to-end distance (Å)
PMF (kcal/mol)
Irreversible Pulling ( v = 100 Å/ns )
10 blocks of 10
trajectories
PMF (kcal/mol)
winner
end-to-end distance (Å)
end-to-end distance (Å)
Umbrella Sampling w/ WHAM
Weighted Histogram Analysis Method
SMD
trajectory
Biasing potential:
Choice of t:
Weighted Histogram Analysis Method
P0i(x), i = 1,2,…,M: overlapping
local distributions
P0(x): reconstructed overall
distribution
Underlying potential:
U0(x) = -kBT ln P0(x)
To reconstruct P0(x) from P0i(x) (i=1,2,…,M)
Ni = number of data points in distribution i,
Biasing potential: Us(x,t) = k(x-vt)2
NIH Resource for Macromolecular Modeling and Bioinformatics
Theoretical Biophysics Group, Beckman Institute, UIUC
SMD-Jarzynski
Umbrella Sampling
(equal amount of simulation time)
simple analysis coupled nonlinear equations (WHAM)
uniform sampling of the reaction coordinate nonuniform sampling of the reaction coordinate
II. Finding Reaction Paths
Reaction Path - Background
Typical Applications of Reaction Path
 Chemical reaction
 Protein folding
 Conformational changes of protein
Reaction Path - Background
Reaction Path
steepest-descent path
for simple systems
R
complex systems
P
?
Reactions are stochastic: Each reaction event takes a different path
and a different amount of time.
 Identify a representative path, revealing the reaction mechanism.
Reaction Path - Accomplishments
Reaction Path Based on the Mean-First Passage Time
Reaction coordinate r(x):
the location of x in the progress of reaction
r(x) = MFPT (x) from x to the product
MFPT (x)  average over all reaction events
reaction path // -
Reaction Path - Accomplishments
Brownian Motion on a Potential Surface
three-hole potential U(x)
0
-1
-2
R
-3
P
-4
Smoluchowski equation: tp = D(e-U (eU p))
 -D(e-U ) = e-U
Reaction Path - Accomplishments
Brownian Motion on a Potential Surface
=1
=8
Reaction Path - Accomplishments
Brownian Motion on a Potential Surface
=8



=1