Theory and Modeling of
Reactive Events
Eric Vanden-Eijnden
Courant Institute
The evolution of many dynamical system can be viewed as a navigation on an
energy or free energy landscape.
The system tends to spend long period of time in the regions of low energy, and
it only rarely makes transition from one such region to another (metastability).
These transitions often are the most interesting part of the dynamics:
•
•
•
•
•
kinetic phase transitions;
conformation changes in macro-molecules;
chemical reactions;
regime changes in climate;
etc.
Understanding the long time dynamics of these systems is a challenge, both from
theoretical and computational viewpoints.
1.2
Simplest example of system displaying metastability:
Motion in a double-well potential, with two minima separated by a saddle point.
!
ẋ(t) = −∇V (x(t)) + 2β −1 η(t)
1.5
1
1
0.8
0.5
position
y
0.6
0.4
0.2
0
!0.5
0
!1
!0.2
!0.4
!1
!0.5
0
x
0.5
1
Examples of this type are easy to treat.
!1.5
0
2
4
6
time
8
10
4
!
x 10
Q̇ = P,
√
Ṗ = Q − Q3 + P + 2ε Ẇ (t)
However they are very simplistic ...
... actual energy landscape of many systems of interest is enormously complicated!
(and makes LD theory inapplicable directly).
Example: solvated alanine dipetide:
CH3
O
C"
C
C!
N
H
H
N
C
CH3
O
12 point particles ( = atoms ) + 252 water molecules (i.e. a dynamical system
with about 1e3 degrees of freedom).
%!!
$#!
$!!
!!!
#!
!
!#!
time series (in ns) of
dihedral angle psi
!$!!
!$#!
!%!!
!
!"#
$
&'()*+,-.
$"#
%
Calculations by Luca Maragliano.
Energy landscape is typically rugged, i.e.
There are many features of the potential on small scales (e.g. many critical points)
which are mostly irrelevant for the rare events. What matters are large scale
features (& LD theory does not apply directly).
Example: Rugged Mueller potential
!
dx(t) = −∇V (x(t), !)dt + 2β −1 dW (t)
V (x, !) = V0 (x) + !V1 (x/!)
More difficult if ε ≈ β-1
small but finite.
and ...
Entropic (i.e. volume) effects matter, presence of dead-ends, dynamical traps, etc
Example: a maze
A
B
Large deviation theory and extensions:
(Wentzell-Freidlin, Bovier et al., ...)
dx(t) = −∇V (x(t))dt +
!
2β −1 dW (t)
Assume that V(x) is a Morse function with growth condition at infinity.
Then dynamics is ergodic w.r.t. the invariant measure
dµ(x) = C −1 exp(−βV (x))dx
When β →∞ this measure becomes atomic on the minima of V(x)
Dynamics can be reduced to a continuous-time Markov chain (random walk on a
network) by mapping the trajectory x onto the index of the last local minimum it
visited
- elementary transitions are related to energy barriers and exponentially small in β;
- pathways of elementary transitions are predictable and follow minimum energy paths
solution of 0 = [∇V (γ)]⊥
Full theory can be obtained by analyzing the Witten complex associated with
the generator of the diffusion (Helffer, Kurchan, ...)
Complicated in practice (analysis must be done globally).
Transition Path Theory
Key concept: reactive trajectories, i.e. those trajectories by which the reaction occurs.
Conceptually, these reactive trajectories can be obtained by pruning a long ergodic
trajectory which oscillates between A and B.
A
B
Understanding the mechanism of the reaction
= characterizing the statistical mechanics properties of the reactive trajectories
Given a trajectory x(t), let R be the set
of times during which it is reactive (i.e.
red in the figure).
A
Two key objects:
Probability density of reactive trajectories defined as:
!
1
ρR (x) = lim
T →∞ T
0
T
δ(x − x(t))1R (t)dt
Probability current of reactive trajectories defined as:
1
JR (x) = lim
T →∞ T
!
0
T
ẋ(t)δ(x − x(t))1R (t)dt
B
rate (committor)
1.912·10
rate (volume integral) 1.924·10−2
rate (DNS)
2.079 · 10−2
var(DNS)
1.33·10−6
Table 2. Transition Rate(! = 0.6, 400 × 400Grid, N = 1000, τ = 10−5 )
A simple illustrative example in 2d:
2. Schulten-Potential. Now we consider the Smoluchowsky dynamics in a sime potential which exhibits metastability. This model system was first introducedTRANSITION
d studied in (?).
PATH THEORY : ILLUSTRATION VIA SIMPLE EXAMPLES
!3
x 10
12
14
2.5
2.5
10
12
2
2
8
1.5
1.5
10
1
6
1
4
0.5
8
0.5
0
6
0
2
!0.5
!0.5
4
0
!1
!1
2
!2
!1.5
!1.5
!1.5
!1
!0.5
0
0.5
1
1.5
!1.5
Figure 5. Contour plot of the potential
Left: triple-well potential;
!1
!0.5
0
0.5
1
1.5
0
Figure 6. Gibbs distribution
Right: Boltzmann equilibrium PDF.
0.9
2.5
0.8
2
1.5
1
Calculations by Philipp Metzner.
0.7
0.6
!1.5
!1
!0.5
0
0.5
1
1.5
8. Distribution
of AB-reactive probability
trajectories
PDF ofFigure
reactive
trajectories,
current and flux
TRANSITION PATH THEORY : ILLUSTRATION VIA SIMPLE EXAMPLES
7
!$
!5
x 10
!
2.5
2.5
"%
&#$
8
2
2
1.5
,
&
7
-!"%
+
"#$
1.5
6
1
*
"
1
)
5
0.5
%#$
0.5
0
$
4
%
0
!0.5
3
!0.5
(
!%#$
'
!1
!1
2
!"
&
1
!"#$
"
!1.5
!1.5
!1.5
!1.5
!1
!1
!0.5
!0.5
0
0
0.5
0.5
1
1
1.5
!
1.5
Figure 10. Probability current. For sake of visualization I com8
puted the current in a low resolution
Figure 9. A reactive AB-trajectory for ! = 0.6
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
!0.5
!0.5
!1
!1
!1.5
!1.5
!1
!0.5
0
0.5
1
1.5
!"
!%#$
%
%#$
"
"#$
PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE
2.5
!1.5
!"#$
!1.5
!1
!0.5
0
0.5
1
1.5
Another example:
!'
!
!
'
&
"$%
&
"'
&
"&
"$%
"
"#
"
#
#$%
#
!'
)
#$%
(
#
'
!#$%
&
!&
!#$%
!
!
!"
#
!"
"
#
%
!#$
!
"#$
"
!"#$
!!
*!"#
"
!
"
Probability current and flux in rugged Mueller potential
1.8
1.6
1.4
1.2
1
0.8
%
0.6
0.4
0.2
!"#
0
!0.2
!1.5
!1
!0.5
0
0.5
1
!
$"#
$
!!"#
!!
!$"#
$
$"#
!
The key object to quantify the statistical properties of the reactive trajectories is
(beside the equilibrium PDF) the committor function q(x) (aka capacitor, p-fold, ...)
whose value at point x is the probability to reach B first rather A starting from x:
6
PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE
0.9
q(x) = Px (τB < τA )
4
PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE
2.5
rate (committor)
1.912·10−2
rate (volume integral) 1.924·10−2
rate (DNS)
2.079 · 10−22
var(DNS)
1.33·10−6
Table 2. Transition Rate(! = 0.6, 400 × 400Grid, N = 1000, τ = 10−5 )
0.8
0.7
1.5
τA = inf{t : x(t) ∈ A},
3.2. Schulten-Potential. Now we consider the Smoluchowsky dynamics in a simple potential which exhibits metastability. This model system
1 was first introduced
and studied in (?).
τB = inf{t : x(t) ∈ B}
0.5
0.6
0.5
12
2.5
10
2
0
0.4
8
1.5
1
!0.5
0.5
0.3
6
4
0
!1
0.2
2
!0.5
0
0.1
!1.5
!1
!2
!1.5
!1.5
!1
!0.5
0
0.5
1
1.5
!1.5
!1
!0.5
0
0.5
1
1.5
Figure 5. Contour plot of the potential
Figure 8. Reaction coordinate
Thm (E, V.-E.): a.s. as T→∞:
1
T
1
T
!
T
0
6
2.5
δ(x − x(t))1R (t)dt → Z −1 e−βV (x) q(x)(1 − q(x))
2
0
!
!4
x 10
T
5
1.5
δ(x − x(t))1R (t) ◦ dx(t) → Z
4
1
−1 −βV (x)
e
0.5
∇q(x)
3
0
!0.5
2
The key object to quantify the statistical properties of the reactive trajectories is
(beside the equilibrium PDF) the committor function q(x) (aka capacitor, p-fold, ...)
whose value at point x is the probability to reach B first rather A starting from x:
6
PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE
0.9
q(x) = Px (τB < τA )
4
PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE
2.5
rate (committor)
1.912·10−2
rate (volume integral) 1.924·10−2
rate (DNS)
2.079 · 10−22
var(DNS)
1.33·10−6
Table 2. Transition Rate(! = 0.6, 400 × 400Grid, N = 1000, τ = 10−5 )
0.8
0.7
1.5
τA = inf{t : x(t) ∈ A},
3.2. Schulten-Potential. Now we consider the Smoluchowsky dynamics in a simple potential which exhibits metastability. This model system
1 was first introduced
and studied in (?).
τB = inf{t : x(t) ∈ B}
0.5
0.6
0.5
12
2.5
10
2
0
0.4
8
1.5
1
!0.5
0.5
q(x) is THE reaction coordinate!
0.3
6
4
0
!1
0.2
2
!0.5
0
0.1
!1.5
!1
!2
!1.5
!1.5
!1
!0.5
0
0.5
1
1.5
!1.5
!1
!0.5
0
0.5
1
1.5
Figure 5. Contour plot of the potential
Figure 8. Reaction coordinate
Thm (E, V.-E.): a.s. as T→∞:
1
T
1
T
!
T
0
6
2.5
δ(x − x(t))1R (t)dt → Z −1 e−βV (x) q(x)(1 − q(x))
2
0
!
!4
x 10
T
5
1.5
δ(x − x(t))1R (t) ◦ dx(t) → Z
4
1
−1 −βV (x)
e
0.5
∇q(x)
3
0
!0.5
2
The maze example:
Committor
Committor
1
1
A
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
B
0
B
0
Effective current
Effective current
A
A
B
B
B
The main issue becomes the computation and analysis of the committor function q(x).
The main issue becomes the computation and analysis of the committor function q(x).
This can be done by sampling using the original (physical) dynamics, or any artificial
dynamics with the same committor (indeed, just as different dynamics can have the
same equilibrium distribution, they can also have the same q(x)).
The main issue becomes the computation and analysis of the committor function q(x).
This can be done by sampling using the original (physical) dynamics, or any artificial
dynamics with the same committor (indeed, just as different dynamics can have the
same equilibrium distribution, they can also have the same q(x)).
Or, it can be done by direct manipulations on the equation for q(x) under specific
assumption (e.g. small temperature, localized tubes concentrating the flux of
reactive trajectories, etc.)
The main issue becomes the computation and analysis of the committor function q(x).
This can be done by sampling using the original (physical) dynamics, or any artificial
dynamics with the same committor (indeed, just as different dynamics can have the
same equilibrium distribution, they can also have the same q(x)).
Or, it can be done by direct manipulations on the equation for q(x) under specific
assumption (e.g. small temperature, localized tubes concentrating the flux of
reactive trajectories, etc.)
The second approach is the one taken in the string method (E, Ren, V.-E.) based on:
Variational formulation: The committor function is the minimizer of:
!
e−βV (x) |∇q(x)|2 dx,
Ω
among all q(x) such that q(x) = 0 in A and q(x) = 1 in B.
Lower bound argument to find the flowline of max-flux:
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
!
!
(x) |∇q(x)|2 dx ≥
(x) |∇q(x)|2 dx
e−βV
e−βV
!
!
B
!Ω
!! δ≤
e−βV (x)|∇q(x)|2dx
e−βV (x)|∇q(x)|2dx
−βV
(x) |∇q(x)|
2 dx2
Ω (x) |∇q(x)|2 dx ≥
B (x)
e−βV
e−βV
|γ % · ∇q(x)|
dx
! δ
Ω
Bδ
!Bδ≤
e−βV (x)|γ $ · ∇q(x)|2dx
Bδ (x) |γ % · ∇q(x)|2 dx
≥
e−βV
!
Optimizing
the bound:2
−βV (x)
inf
q
!Ω
e
!
Bδ
−βV (x) |γ!% · ∇q(x)|2 dx
!
|∇q(x)|
dx
≥
sup
inf
e
−βV (x) |∇q(x)|2 dx
q 2dx
γ
B
inf
e−βV (x)|∇q(x)|
≤
sup
inf
e
δ
!
q Ω
q# B
"
#−1
"!
γ
−1
!
δ
−βV
(x)
2
−βV
(x)
%
2
" |γ · ∇q(x)|#−1
inf
e
|∇q(x)| dx δ&1
≥ sup inf
e
dx βV (γ)
q Ω
inf
e
ds
≈ γsupq BδeβV (γ)ds! βV=
(γ)
γ
γ " γ ≈ sup # e
"ds !γ
#−1
−1
!
γ
γ
δ&1
eβV (γ)ds
= inf eβV (γ)ds
≈ sup
γ γ
γ
γ
Lower bound argument to find the flowline of max-flux:
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
!
!
(x) |∇q(x)|2 dx ≥
(x) |∇q(x)|2 dx
e−βV
e−βV
!
!
B
!Ω
!! δ≤
e−βV (x)|∇q(x)|2dx
e−βV (x)|∇q(x)|2dx
−βV
(x) |∇q(x)|
2 dx2
Ω (x) |∇q(x)|2 dx ≥
B (x)
e−βV
e−βV
|γ % · ∇q(x)|
dx
! δ
Ω
Bδ
!Bδ≤
e−βV (x)|γ $ · ∇q(x)|2dx
Bδ (x) |γ % · ∇q(x)|2 dx
≥
e−βV
!
Optimizing
the bound:2
−βV (x)
inf
q
!Ω
e
!
Bδ
−βV (x) |γ!% · ∇q(x)|2 dx
!
|∇q(x)|
dx
≥
sup
inf
e
−βV (x) |∇q(x)|2 dx
q 2dx
γ
B
inf
e−βV (x)|∇q(x)|
≤
sup
inf
e
δ
!
q Ω
q# B
"
#−1
"!
γ
−1
!
δ
−βV
(x)
2
−βV
(x)
%
2
" |γ · ∇q(x)|#−1
inf
e
|∇q(x)| dx δ&1
≥ sup inf
e
dx βV (γ)
q Ω
inf
e
ds
≈ γsupq BδeβV (γ)ds! βV=
(γ)
γ
γ " γ ≈ sup # e
"ds !γ
#−1
−1
!
γ
γ
δ&1
eβV (γ)ds
= inf eβV (γ)ds
≈ sup
γ γ
γ
γ
Finding the flowline of max-flux reduces to the problem of computing a geodesic.
Lower bound argument to find the flowline of max-flux:
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
Given a curve γ connecting A and B, let Bδ = {x : d(x, γ) ≤ δ}.
!
!
(x) |∇q(x)|2 dx ≥
(x) |∇q(x)|2 dx
e−βV
e−βV
!
!
B
!Ω
!! δ≤
e−βV (x)|∇q(x)|2dx
e−βV (x)|∇q(x)|2dx
−βV
(x) |∇q(x)|
2 dx2
Ω (x) |∇q(x)|2 dx ≥
B (x)
e−βV
e−βV
|γ % · ∇q(x)|
dx
! δ
Ω
Bδ
!Bδ≤
e−βV (x)|γ $ · ∇q(x)|2dx
Bδ (x) |γ % · ∇q(x)|2 dx
≥
e−βV
!
Optimizing
the bound:2
−βV (x)
inf
q
!Ω
e
!
Bδ
−βV (x) |γ!% · ∇q(x)|2 dx
!
|∇q(x)|
dx
≥
sup
inf
e
−βV (x) |∇q(x)|2 dx
q 2dx
γ
B
inf
e−βV (x)|∇q(x)|
≤
sup
inf
e
δ
!
q Ω
q# B
"
#−1
"!
γ
−1
!
δ
−βV
(x)
2
−βV
(x)
%
2
" |γ · ∇q(x)|#−1
inf
e
|∇q(x)| dx δ&1
≥ sup inf
e
dx βV (γ)
q Ω
inf
e
ds
≈ γsupq BδeβV (γ)ds! βV=
(γ)
γ
γ " γ ≈ sup # e
"ds !γ
#−1
−1
!
γ
γ
δ&1
eβV (γ)ds
= inf eβV (γ)ds
≈ sup
γ γ
γ
γ
Finding the flowline of max-flux reduces to the problem of computing a geodesic.
(NB: line of max-flux is the MEP from LD theory in small noise limit)
String method in a nutshell:
Parametrize the curve e.g. by normalized arc-length;
Evolve it using a time-splitting method:
- one step of steepest descent along gradient of objective function (or CG,
BFGS, etc.);
- one step of interpolation-reparametrization to control the parametrization of
the curve
String method in a nutshell:
Parametrize the curve e.g. by normalized arc-length;
Evolve it using a time-splitting method:
- one step of steepest descent along gradient of objective function (or CG,
BFGS, etc.);
- one step of interpolation-reparametrization to control the parametrization of
the curve
Application: hydrophobic collapse of a polymeric chain
in collaboration with Tommy Miller (Caltech) and David Chandler (UC Berkekeley)
Chain made of 12 monomers of size 7.2 A solvated in a periodic box of size 99.5 A x
99.5 A x 116.1 A containing 34,000 rigid water molecules modeled by SPC/E.
Collective variables = monomer positions + local density field
- in total over 129,000 collective variables
Ref.
A
0.2
0.4
0.7
1.1
1.4
MFEP identified by the string method
Free energy
Dominated by work done by the solvent degrees of freedom.
Dynamical trajectories initiated from the transition state region
-150 ps
-90 ps
-60 ps
-30 ps
0 ps
30 ps
60 ps
90 ps
150 ps
Summarizing:
Reactive events can be understood from a probabilistic (i.e. statistical mechanistic)
viewpoint. In the context of reactive events, this means focusing on the statistical
mechanics description of the reactive trajectories;
Concepts for probability theory permit to define precisely the concept of reaction
coordinate to describe the transition from a reactant state A to a product state B in
terms of the committor function;
Open the door to accelerated computing strategies (i.e. with biased/artificial
dynamics) to analyze rare reactive events like e.g. the string method.
Acknowledgments:
Weinan E, Weiqing Ren, Bob Kohn,
Maddalena Venturoli, Luca Maragliano, Giovanni Ciccotti, Ron Elber,
David Chandler, Tommy Miller, ...
Funded by NSF, ONR and DARPA.