Transition Paths of
Conformational Change at
Minimal Cost
Robert Skeel, Ruijun Zhao
Department of Computer Science, Purdue University
July 2, 2010
Outline
Motivation
Objective
Summary
Some Numerical Results
Some Details
- the geometric property
- a key assumption
- implementation
Remarks
Motivation!
Catalytic domain of Src Tyrosine kinase
inactive!
active!
N-lobe!
N-lobe!
α-helix C!
α-helix C!
A-loop!
A-loop!
C-lobe!
C-lobe!
Biological questions
How does the mechanical signal of unbinding at an allosteric site
transmit through the molecule to induce conformational
change in the active site?
What are possible intermediate states involved in the transition that
can be used as targets for drug design?
What is the free energy difference between the two conformational
states?
Molecular dynamics!
Assume molecular system obeys Newtonian dynamics where M is
atomic masses and U(x) is potential energy.
Initial values of positions x and velocities are drawn from the
canonical ensemble with inverse temperature β.
Objective
Compute for minimal cost best possible
representative paths of conformational change
from conformation A to conformation B.
Representative path: center of an isolated cluster of
trajectories. Clustering might occur only with the use of a
reduced set of collective variables (colvars):
ζ1 = ξ1 (x), ζ2 = ξ2 (x), . . . , ζncv = ξncv (x)
e.g., phi and psi dihedrals
for alanine dipeptide.
Minimal cost: sampling is limited to the path in colvar space.
(Also, minimal programming effort through use of existing features
of simulators.)
Best possible: narrow tube (of constant cross-section area)
that maximizes flow rate of reactive trajectories—assuming
trajectories are confined to the tube.
Summary
Define
the free energy function F by
exp(−βF (ζ)) = ρξ (ζ)
where ρξ(ζ) is the probability density function of ζ = ξ(x),
the tensor D by
�
�
D(ζ) = mtot ξx M −1 ξxT ξ(x)=ζ
and the norm
|ω|ζ = (ω D(ζ)
T
−1
1/2
ω)
The formula
The maximum flux transition path (MFTP)
ζ = Z(s),
0 ≤ s ≤ 1,
minimizes
�1
1/2
exp(βF
(Z))(det
D(Z))
|Zs |Z
0
ds.
with Z(0) in conformation Aξ and Z(1) in conformation Bξ,
where Z = Z(s) and Zs = (d/ds)Z(s).
derived using transition path theory
many minima
For Comparison
Minimum resistance path (MRP) minimizes
�1
−1
2
exp(βF
(Z))|Z
|
|Z
|
s
s Z ds.
0
Path depends on how colvar space is parameterized (see The
geometric property).
Berkowitz, Morgan, McCammon and Northrup.
J. Chem. Phys., 1983.
Huo and Straub. The MaxFlux ..., J. Chem. Phys., 1997.
Minimum free energy path (MFEP) minimizes
�1
|D(Z)∇F (Z)|Z |Zs |Z ds.
0
Maragliano, Fischer, Vanden-Eijnden, and Ciccotti,
J. Chem. Phys., 2006.
Vanden-Eijnden and Heymann. J. Chem. Phys., 2008.
Minimum free energy path (MFEP) neglects finite temperature
effects–in the explicit degrees of freedom.
alanine dipeptide (OPLS-AA with GBSA)
MFEP & MRPs
density of paths from TPS
Jiménez and Crehuet, Theor. Chem. Account, 2007.
The MFTP is computationally most attractive:
The MRP has complicated dependence on Zs.
The MFEP has cusps and its minimization formulation may be
impractical.
minimization principle is an advantage
The M(F)EP with increasing resolution
Some Numerical Results
Three hole potential
The MFTP at different temperature (1/β in kcal/mol) and the minimum energy path (MEP).
The contour lines are separated by 0.25 kcal/mol.
Alanine dipeptide
The MFTP and the MFEP for alanine dipeptide in vacuum at 300 K.The colvar space is phi/psi
dihedrals. The contour lines are separated by 0.6 kcal/mol.
Solvated alanine dipeptide
Alanine decapeptide
alpha helix to pi helix transition
Energy along a selection of low energy paths
Chu, Trout, & B. Brooks, J. Chem. Phys., 2003.
Free energy along maximum flux transition path using alpha carbons
Some Details
The geometric property
Point in collective variable space = hyper-surface in configuration
space
The geometric property means that if we instead minimize the
integral using variables
'
'
" = # (x) = $ (#(x))
then the resulting path ζʼ = Zʼ(s) will satisfy Zʼ(s) = χ(Z(s)).
e.g., squaring colvars that measure distance yields same path.
This is achieved for the MFTP by using the metric tensor D-1 to define
distance from
! ζ to ζ+dζ as
|dζ|ζ
(and using this metric to measure cross-section area).
The MFTP is a geodesic with metric tensor
exp(2βF (ζ)) det(D(ζ))D(ζ)−1
The others?
Underlying assumption
It is assumed that paths ζ = ξ(x(t)) are those of the Brownian
dynamics
d
ζ
dτ
= −β D̄(ζ)∇F (ζ) + (∇ · D̄(ζ))T
√
1/2
+ 2D̄(ζ) η(τ )
where η(τ) is white noise, τ is fictitious time, and the
–
diffusion tensor D is a rescaling of D.
Role of committor
The cross sections of the narrow tube are taken to be
isocommittors because the surfaces of Aξ and Bξ are.
Current implementation
1. free energy approximated using stiff restraints
exp(−βF (ζ)) = ρξ (ζ) ≈ �δε (ξ(x) − ζ)�
where
δε (s) = (2πε2 )−1/2 exp(−s2 /(2ε2 ))
e.g., ε = 1 degree
Euler-Lagrange ``equationsʼʼ for the MFTP
D(Z)−1 Zs � − β∇ζ F+ (Z, Zs ) +
(D(Z)−1 Zs )s
|Zs |2Z
where
1
ncv −1
2
log(c0
det D(ζ)|ω|ζ )
F+ (ζ, ω) = F (ζ) −
2β
2. The simplified MFTP:
Neglect derivatives of D orthogonal to the path:
−1
D(Z)
Zs � − β∇ζ F (Z) +
(D(Z)−1 Zs )s
|Zs |2Z
3. piecewise linear discretization of path;
discretize Euler-Lagrange equation using upwinding
4. The semi-implicit simplified string method [5] is implemented.
E, Ren, and Vanden-Eijnden. J. Chem. Phys., 2007.
Vanden-Eijnden and Heymann. J. Chem. Phys., 2008.
35-50 iterations for alanine dipeptide.
MFEP requires 25% more--if it has a cusp.
5. sampling performed using 50+500 ps of Langevin or NoseHoover dynamics per image per iteration.
Remarks
Reiterating
A maximum flux transition path (MFTP) is the center of a
narrow tube (of constant cross-section area)
that yields the maximum reaction rate—if trajectories were
confined to the tube.
Claim: It is the best representation of trajectories obtainable
at minimal computing cost with modest programming effort.
An MFTP is a worthwhile improvement to an
MRP / MaxFlux path
* is more meaningful due to geometric property
* produces no unpleasant surprises
* requires no apologies
* and costs only a little extra.
Molecular simulators impede innovation
* they ought to be callable from a popular scripting
language, providing a variety of low-level functions,
supplemented by a set of canned scripts. e.g, we
have to continually reinvoke the program and would
have to use named pipes to avoid disk accesses.
* they ought to provide a broad array of basic functions:
force fields, constraints & restraints, colvars and their
derivatives, integrators & samplers, ..., e.g., we have
to use tricks to get derivatives of colvars
Computation is not the 3rd pillar
Rather, theory and computation are extremes on a
single spectrum of possible approaches. Notably,
modeling of rare events cannot be studied by direct
numerical simulation (DNS) alone.
Additionally, it is better not to dilute the role of
experiment.
Computation is a tool of the theoretician (and
experimentalist).
Acknowledgments
Juanfang “Janice” Shen, graduate student
Carol Post Lab: Carol, He Huang
Eric Vanden-Eijnden
Jhih-Wei Chu
NIH