(Non)Equilibrium computation
of free energy differences
using Langevin dynamics
Gabriel STOLTZ
CERMICS & MICMAC project team, Ecole des Ponts ParisTech
http://cermics.enpc.fr/∼stoltz/
Edinburgh, june 2010
– p. 1/33
Outline of the talk
A brief presentation of methods to compute free energy differences
Thermodynamic integration using Langevin dynamics
Nonequilibrium Langevin dynamics
Edinburgh, june 2010
– p. 2/33
Computing free energy differences
Edinburgh, june 2010
– p. 3/33
Microscopic description of a classical system
Positions q (configuration), momenta p = M q̇ (M diagonal mass matrix)
Microscopic description of a classical system (N particles):
(q, p) = (q1 , . . . , qN , p1 , . . . , pN ) ∈ E
For instance, E = T ∗ D = D × R3N with D = R3N or T3N
More complicated situations can be considered... (constraints defining
submanifolds of the phase space)
N
X
p2i
+ V (q1 , . . . , qN )
Hamiltonian H(q, p) =
2m
i
i=1
All the physics is contained in V
Canonical probability measure:
µ(dq dp) = Z
−1 −βH(q,p)
e
dq dp,
1
β=
kB T
Edinburgh, june 2010
– p. 4/33
Sampling the canonical measure
The aim is to compute an approximation of the high dimensional integral
Z
hAi =
A(q, p) µ(dq dp)
T ∗D
Restated as a one-dimensional integral using ergodic properties of an
irreducible dynamics for which the canonical measure is invariant:
1
lim
T →+∞ T
Z
0
T
A(qt , pt ) dt =
Z
A(q, p) µ(dq dp)
a.s.
T ∗D
Overdamped Langevin dynamics (momenta trivial to sample)
dqt = −∇V (qt ) dt +
r
2
dWt
β
Zero mass limit of the Langevin dynamics or the limit of the Langevin
dynamics when the friction goes to infinity (with suitable time rescaling)
Edinburgh, june 2010
– p. 5/33
Langevin dynamics
Stochastic perturbation of the Hamiltonian dynamics

 dq = M −1 p dt
t
t
 dpt = −∇V (qt ) dt−γ(qt )M −1 pt dt + σ(qt ) dWt
2
Fluctuation/dissipation relation σσ = γ
β
T
Invariance of the canonical measure when it is a stationary solution of the
Fokker-Planck equation ∂t ψ = L∗ ψ with
eβH
−βH
L = {·, H} +
∇p ·
divp γe
β
T
T
and {A1 , A2 } = (∇A1 )T J∇A2 = (∇q A1 ) ∇p A2 − (∇p A1 ) ∇q A2
Irreducibility amounts to controllability (Hörmander condition)
Numerical schemes obtained by a splitting strategy for instance (Verlet
scheme + partial randomization of momenta)
Edinburgh, june 2010
– p. 6/33
Metastability (1)
Numerical discretization of the overdamped Langevin dynamics:
q
n+1
n
n
= q − ∆t∇V (q ) +
s
2∆t n
G
β
1.5
1.5
1
1.0
x coordinate
y coordinate
where G n ∼ N (0, IddN ) i.i.d.
0.5
0
−0.5
−1
0.5
0.0
−0.5
−1.0
−1.5
−1.5
−1
−0.5
0
0.5
X coordinate
1
1.5
−1.5
0.0
2000
4000
6000
8000
10000
Time
Projected trajectory in the x variable for ∆t = 0.01, β = 8.
Edinburgh, june 2010
– p. 7/33
Metastability (2)
Although the trajectory average converges to the phase-space average,
the convergence may be slow...
Slowly evolving macroscopic function of the microscopic degrees of
freedom: reaction coordinate ξ(q) ∈ Rm with m ≪ N
Two origins : energetic or entropic barriers (in fact, free energy barriers)
y coordinate
x coordinate
8
4
0
−4
x coordinate
(a)
Entropic barrier.
−8
0.0
5000
10000
15000
20000
Time
(b)
Associated trajectory.
Edinburgh, june 2010
– p. 8/33
Metastability (3)
1.5
1.5
1
1.0
x coordinate
y coordinate
Assume the free energy F associated with the slow direction x has been
computed, and sample the modified potential V(x, y) = V (x, y) − F (x).
0.5
0
−0.5
0.0
−0.5
−1.0
−1
−1.5
−1.5
−1.5
0.5
−1
−0.5
0
0.5
x coordinate
1
1.5
0.0
2000
4000
6000
8000
10000
Time
Projected trajectory in the x variable for ∆t = 0.01, β = 8.
Many more transitions! The variable x is uniformly distributed.
Reweighting with weights e−βF (x) to compute canonical averages
Compute efficiently the free energy?
Edinburgh, june 2010
– p. 9/33
Computation of free energy differences
Alchemical transition: indexed by an external parameter λ (force field
parameter, magnetic field,...)

Z
−βH1 (q,p)
e
dq dp 

 ;
ZT ∗ D
∆F = −β −1 ln 


−βH0 (q,p)
e
dq dp
T ∗D
Typically, Hλ = (1 − λ)H0 + λH1 . Other parametrizations possible (see
Gelman and Meng, Stat. Sci., 1998)
(given) reaction coordinate ξ : R3N → Rm (angle, length,. . . ):

Z
−βH(q,p)
e
δξ(q)−z1 (dq) dp

 Σ(z)×R3N
−1
.

Z
∆F = −β ln 

−βH(q,p)
e
δξ(q)−z0 (dq) dp
Σ(z)×R3N
n
with Σ(z) = q ∈ D
o
ξ(q) = z
Edinburgh, june 2010
– p. 10/33
Cartoon comparison of the methods (reaction coordinate case)
(a)
(c)
Histogram method
Nonequilibrium switching dynamics
(b)
Thermodynamic integration
(d)
Adaptive dynamics
Edinburgh, june 2010
– p. 11/33
Some elements on the scientific landscape
We focus on the reaction coordinate case
Histogram methods: WHAM (Kumar et al.), MBAR (Chodera/Shirts)
Thermodynamic integration in the Hamiltonian case (Carter et al., den
Otter/Briels, Sprik/Ciccotti) and HMC (Hartmann/Schütte) or for
overdamped Langevin dynamics (Ciccotti/Lelièvre/Vanden-Eijnden)
Nonequilibrium methods: overdamped case (Lelièvre/Rousset/Stoltz) or
steered versions (potentials Vλ (q) = V (q) + K(ξ(q) − λ)2 )
Adpative methods: adaptive biasing force (Darve/Pohorille, Chipot/Hénin),
nonequilibrium metadynamics (Bussi/Laio/Parrinello), Wang-Landau,
self-healing umbrella sampling (Marsili et al., Dickson et al.), etc
Aims of this work:
Thermodynamic integration with Langevin dynamics
Nonequilibrium Langevin dynamics
Edinburgh, june 2010
– p. 12/33
Thermodynamic integration with
Langevin dynamics
Edinburgh, june 2010
– p. 13/33
Constrained Langevin dynamics (1)
Consider the following Langevin process:

−1

dq
=
M
pt dt,
t




dpt = −∇V (qt ) dt − γ(qt )M −1 pt dt + σ(qt ) dWt + ∇ξ(qt ) dλt ,





ξ(qt ) = z
2
Standard fluctuation/dissipation relation σσ = γ
β
T
dξ(qt )
Hidden velocity constraint:
= vξ (qt , pt ) = ∇ξ(qt )T M −1 pt = 0
dt
The corresponding phase-space is Σξ,vξ (z, 0) where
n
o
6N Σξ,vξ (z, vz ) = (q, p) ∈ R ξ(q) = z, vξ (q, p) = vz
An explicit expression of the Lagrange multiplier can be found by
computing the second derivative in time of the constraint
Edinburgh, june 2010
– p. 14/33
Constrained Langevin dynamics (2)
Reformulation of the constrained Langevin dynamics as


dqt = M −1 pt dt,

dpt = −∇V (qt ) dt + ∇ξ(qt )f M (qt , pt ) dt − γP (qt )M −1 pt dt + σP (qt ) dWt ,
rgd
Constraining force (projection of the conservative force + centrifugal term)
−1
M
T
−1
−1
−1
frgd
(q, p) = G−1
(q)∇ξ(q)
M
∇V
(q)
−
G
(q)Hess
(ξ)(M
p,
M
p)
q
M
M
where GM (q) = ∇ξ(q)T M −1 ∇ξ(q)
Projected fluctuation/dissipation matrices
T
(σP , γP ) := (PM σ, PM γ PM
)
T
−1
where PM (q) = Id − ∇ξ(q)G−1
(q)∇ξ(q)
M
M
Edinburgh, june 2010
– p. 15/33
Properties of the projected Langevin dynamics
Generator of the dynamics: L = Lham + Lthm with
1
Lham = { · , H}Ξ ,
Lthm = eβH divp e−βH γP ∇p ·
β
Invariant measure
−1 −βH(q,p)
µΣξ,vξ (z,0) (dq dp) = Zz,0
e
σΣξ,vξ (z,0) (dq dp),
where σΣξ,vξ (z,vz ) (dq dp) is the phase space Liouville measure of
Σξ,vξ (z, vz ) induced by the symplectic matrix J
Reversibility: Law(qt , pt ; 0 ≤ t ≤ T ) = Law(qT −t , −pT −t ; 0 ≤ t ≤ T )
Detailed balance condition (momentum flip S(q, p) = (q, −p))
Z
Z
ϕ1 LΞ (ϕ2 ) dµΣξ,vξ (z,0) =
(ϕ2 ◦ S) LΞ (ϕ1 ◦ S) dµΣξ,vξ (z,0)
Σξ,vξ (z,0)
Σξ,vξ (z,0)
Ergodicity (convergence of the longtime trajectorial averages)
The normalizing constant of this canonical distribution defines a rigid free
energy (more on this later)
Edinburgh, june 2010
– p. 16/33
Numerical discretization
Splitting into Hamiltonian part + constrained Ornstein-Uhlenbeck process
Midpoint scheme for the momenta (reversible for the canonical measure
with constraints)
r
∆t
∆t
γ M −1 (pn + pn+1/4 ) +
σ G n + ∇ξ(q n ) λn+1/4 ,
pn+1/4 = pn −
4
2
with the constraint ∇ξ(q n )T M −1 pn+1/4 = 0
RATTLE scheme (symplectic)

∆t
n
n
n+1/2
n+1/4
n+1/2

∇V
(q
)
+
∇ξ(q
)
λ
,
p
=
p
−



2
q n+1 = q n + ∆t M −1 pn+1/2 ,



 pn+3/4 = pn+1/2 − ∆t ∇V (q n+1 ) + ∇ξ(q n+1 ) λn+3/4 ,
2
with ξ(q n+1 ) = z and ∇ξ(q n+1 )T M −1 pn+3/4 = 0
∆t
Overdamped limit obtained when
γ = M ∝ Id
4
Metropolization of the RATTLE part to eliminate the time-step error
Edinburgh, june 2010
– p. 17/33
Thermodynamic integration (1)
The free energy can be estimated from constrained samplings as
Z
1
e−βH(q,p) δξ(q)−z (dq) dp
F (z) = − ln
β
Σ(z)×R3N
Z
1
M
(z) − ln
= Frgd
(det GM )−1/2 dµΣξ,vξ (z,0) + C
β
Σξ,v (z,0)
ξ
with the rigid free energy (constraints on both q and p)
Z
1
M
e−βH(q,p) dµΣξ,vξ (z,0)
Frgd
(z) = − ln
β
Σξ,v (z,0)
ξ
Extension to the case of molecular constraints
Thermodynamic integration through the computation of the mean force
Z
M
M
(q, p) µΣξ,vξ (z,0) (dq dp)
frgd
(z) =
∇z Frgd
Σξ,vξ (z,0)
Edinburgh, june 2010
– p. 18/33
Thermodynamic integration (2)
Longtime (a.s.) convergence
1
lim
T →+∞ T
Z
0
T
M
dλt = ∇z Frgd
(z)
No second order derivatives of ξ needed!
Variance reduction: keep only the Hamiltonian part of λt
Numerical discretization: approximate the mean force using only the
Lagrange multipliers from the RATTLE part:
N−1
N−1
X
X
1
1
M
n n
M
frgd (q , p ) ≃
(λn+1/2 + λn+3/4 )
∇z Frgd (z) ≃
N n=0
N∆t n=0
Consistency result
λn+1/2 + λn+3/4
∆t M n n+1/4
M
=
frgd (q , p
) + frgd
(q n+1 , pn+3/4 ) + O(∆t3 )
2
Edinburgh, june 2010
– p. 19/33
Application: Solvatation effects on conformational changes (1)
2 2
(r − r0 − w)
Two particules (q1 ,q2 ) interacting through VS (r) = h 1 −
w2
Solvent: particules
through the purely repulsive potential
interacting
σ 12 σ 6
VWCA (r) = 4ǫ
−
+ ǫ if r ≤ r0 , 0 if r > r0
r
r
|q1 − q2 | − r0
Reaction coordinate ξ(q) =
, compact state ξ −1 (0),
2w
stretched state ξ −1 (1)
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Edinburgh, june 2010
– p. 20/33
Application: Solvatation effects on conformational changes (2)
0.0
20
−0.5
Free energy
Mean force
10
0
−10
−1.5
−2.0
−20
−30
−0.2
−1.0
−2.5
0.0
0.2
0.4
0.6
Reaction coordinate
0.8
1.0
1.2
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Reaction coordinate
Left: Estimated mean force. Right: Corresponding potential of mean force.
Parameters: β = 1, N = 100 particles, solvent density ρ = 0.436, WCA interactions σ = 1 and
ε = 1, dimer w = 2 and h = 2. Mean force estimated at the values zi = zmin + i∆z, with
zmin = −0.2, zmax = 1.2 and ∆z = 0.014, by ergodic averages obtained with the projected
dynamics with Metropolis correction (time T = 2 × 104 , step size ∆t = 0.02, scalar friction γ = 1).
Edinburgh, june 2010
– p. 21/33
Nonequilibrium Langevin dynamics
Edinburgh, june 2010
– p. 22/33
Presentation of the dynamics (1)
Idea: start at equilibrium and perform a switching from the initial to the
final state in a finite time T
Schedule z(t) for t ∈ [0, T ] and nonequilibrium dynamics

−1
dq
=
M
pt dt,

t




dpt = −∇V (qt ) dt − γP (qt )M −1 pt dt + σP (qt ) dWt + ∇ξ(qt ) dλt ,




 ξ(q ) = z(t),
(Cq (t))
t
with equilibrium initial conditions (q0 , p0 ) ∼ µΣξ,vξ (z(0),ż(0)) (dq dp)
T
Projected fluctuation/dissipation relation (σP , γP ) := (PM σ, PM γ PM
) so
that the noise act only in the direction orthogonal to ∇ξ
Hidden constraint on the reaction coordinate velocity vξ (q, p) = ż(t)
M
A computation shows dλt = frgd
(qt , pt ) dt + G−1
M (qt )z̈(t) dt
Not the same dynamics as in Latorre/Hartmann/Schütte
Edinburgh, june 2010
– p. 23/33
Presentation of the dynamics (2)
Backward dynamics: start at equilibrium for the final value of the
schedule, switching with a time-reversed schedule t′ 7→ z(T − t′ )
Initial conditions (q0b , pb0 ) ∼ µΣξ,vξ (z(T ),ż(T )) (dq dp) and evolution

b
−1 b
′
dq
p
′ = −M
′ dt ,

t
t




dpbt′ = ∇V (qtb′ ) dt′ − γP (qtb′ )M −1 pbt′ dt′ + σP (qtb′ ) dWtb′ + ∇ξ(qtb′ ) dλbt′ ,





ξ(qtb′ ) = z(T − t′ )
The generator of the forward dynamics is (Gram matrix Γ = {Ξ, Ξ},
ζ = (z, ż))
Lft = Lham
+ Lthm
+ {·, Ξ} Γ−1 ζ̇(t)
Ξ
Ξ
while the generator of the backward dynamics is Lbt′ = R LfT −t′ R (where
R : φ 7→ φ ◦ S is the momentum flip operator with S(q, p) = (q, −p))
Edinburgh, june 2010
– p. 24/33
Generalized free energy and work
M
Rigid free energy Frgd
(z, vz ) = −
1
ln
β
Z
Σξ,vξ (z,vz )
e−βH(q,p) dµΣξ,vξ (z,vz )
ξ,v
Actual free energy recovered from the difference F (z) − Frgdξ (z, vz ),
which equals, up to an unimportant additive constant:
Z
β T −1
1
(det GM (q))−1/2 exp
vz GM (q)vz µΣξ,vξ (z,vz ) (dq dp)
− ln
β
2
Σξ,v (z,vz )
ξ
Work performed during the switching: several expressions
Z T
ż(t)T dλt
Force times displacement: W0,T {qt , pt }0≤t≤T =
0
Z
T
Energy variations: W0,T {qt , pt }0≤t≤T =
w(t, qt , pt ) dt where
0
d
T −1
w(t, q, p) = ζ̇(t) Γ {Ξ, H} (q, p) =
(q, p) with Φ
H ◦ Φt,t+h dh
h=0
the flow of the switched Hamiltonian dynamics
Edinburgh, june 2010
– p. 25/33
Jarzynski-Crooks relation
For any bounded path functional ϕ[0,T ] ,
−βW0,T ({qt ,pt }t∈[0,T ] )
E ϕ[0,T ] ({qt , pt }0≤t≤T ) e
Zz(T ),ż(T )
=
Zz(0),ż(0)
{q b′ , pb′ } ′
E ϕr
[0,T ]
t
t
0≤t ≤T
where ( · )r denotes the composition with the operation of time reversal of
paths:
ϕr[0,T ] {qtb′ , pbt′ }0≤t′ ≤T = ϕ[0,T ] {qTb −t , pbT −t }0≤t≤T
This leads in particular to the following free energy estimator
−β [W0,T ({qt ,pt }t∈[0,T ] )+C(T,qT )]
1 E e
F (z(T )) − F (z(0)) = − ln
−βC(0,q
)
0
β
E e
1
1
ln det GM (q) − ż(t)T G−1
with the corrector C(t, q) =
M (q)ż(t)
2β
2
Standard methods can then be used (bridge estimators, etc)
Edinburgh, june 2010
– p. 26/33
Some elements on the proof
The proof relies on the following nonequilibrium detailed balance condition
Z
−βH
b
f
dσΣξ,vξ (z(t),ż(t))
ϕ1 Lt (ϕ2 ) − ϕ2 LT −t (ϕ1 ) e
Σξ,vξ (z(t),ż(t))
=
Z
Σξ,vξ (z(t),ż(t))
+
d
dt
Z
βw(t, ·)ϕ1 ϕ2 e−βH dσΣξ,vξ (z(t),ż(t))
Σξ,vξ (z(t),ż(t))
ϕ1 ϕ2 e−βH dσΣξ,vξ (z(t),ż(t))
!
Left-hand side: standard (equilibrium) detailed balance contribution
Right-hand side: evolution of the measure and work correction
Edinburgh, june 2010
– p. 27/33
Numerical schemes: splitting strategy
Fluctuation/dissipation part (no Lagrange multiplier needed)
r
∆t
∆t
γP (q n )M −1 (pn+1/4 + pn ) +
σP (q n )G n
pn+1/4 = pn −
4
2
Hamiltonian part for the forward evolution

∆t
n+1/2
n+1/4
n
n n+1/2


p
=
p
−
∇V
(q
)
+
∇ξ(q
)λ
,


2




n+1
n
−1 n+1/2

q
=
q
+
∆t
M
p
,





ξ(q n+1 ) = z(tn+1 ),
(Cq )




∆t

n+3/4
n+1/2
n+1
n+1 n+3/4

p
=
p
−
∇V
(q
)
+
∇ξ(q
)λ
,


2





∇ξ(q n+1 )T M −1 pn+3/4 = z(tn+2 ) − z(tn+1 ) ,
(Cp )
∆t
which defines a symplectic map
z(tn+1 ) − z(tn )
z(tn+2 ) − z(tn+1 )
Φn : Σξ,vξ z(tn ),
→ Σξ,vξ z(tn+1 ),
∆t
∆t
Edinburgh, june 2010
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Discrete Jarzynski-Crooks equality: Alchemical case
Idea in the alchemical case
Discrete schedule {λ(0), . . . , λ(tNT )} with NT ∆t = T
0
0
Initial conditions (q , p ) ∼ µ0 (dq dp) =
Z0−1
exp −βHλ(0) (q, p) dq dp
Successive updates of the parameter and the configuration:
Change the parameter from λ(n∆t) to λ((n + 1)∆t)
Update work: W n+1/2 = W n + Hλ((n+1)∆t) (q n , pn ) − Hλ(n∆t) (q n , pn )
Update the configuration using the Verlet map Φn+1 associated with
the Hamiltonian Hλ((n+1)∆t)
W n+1 = W n+1/2 + Hλ((n+1)∆t) (q n+1 , pn+1 ) − Hλ((n+1)∆t) (q n , pn )
Total work: W n = Hλ(n∆t) (q n , pn ) − Hλ(0) (q 0 , p0 )
Z
n
−βW n
=
It can be checked that E e
for any value of ∆t such that the
Z0
Verlet scheme is stable
Extension to path functionals
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Discrete Jarzynski-Crooks equality: The reaction coordinate case
Discrete schedule {z(0), . . . , z(tNT )}
Initial conditions (q 0 , p0 ) ∼ µΣ
(q b,0 , pb,0 ) ∼ µ
ξ,vξ
“
z(t )−z(t )
z(t0 ), 1 ∆t 0
” (dq dp)
and
„
« (dq dp)
z(tN +1 )−z(tN )
T
T
Σξ,vξ z(tNT ),
∆t
Initial work W 0 = 0, and work update
W n+1 = W n + H(q n+1 , pn+3/4 ) − H(q n , pn+1/4 )
Time discretization error free estimator of the free energy difference:
Z
z(NT ),
Zz(t
z(tN +1 )−z(tN )
T
T
∆t
0 ),
z(t1 )−z(t0 )
∆t
n n
−βW NT
E ϕ[0,NT ] ({q , p }0≤n≤NT ) e
=
′
′
E ϕr[0,NT ] ({q b,n , pb,n }0≤n′ ≤NT )
∆t
∆t
Overdamped limit
γ=M =
Id: no bias due to the finite time-step in
4
2
the estimator (compare to Lelièvre/Rousset/Stoltz, 2007)
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Application: Solvatation effects on conformational changes
4
Free energy
3
2
1
0
−1
0.0
0.2
0.4
0.6
0.8
1.0
Reaction coordinate
Estimated free energy profiles for T = 1 with K = 105 realizations (top curve),
T = 10 with K = 104 and T = 100 with K = 103 (smoothest curve).
Same parameters as before, except ∆t = 0.01. Schedule z(t) = zmin + (zmax − zmin ) Tt with
zmin = −0.1 and zmax = 1.1.
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References
Edinburgh, june 2010
– p. 32/33
References
The main references for this work:
T. L ELIÈVRE , M. R OUSSET
AND
G. S TOLTZ,
Langevin dynamics with
constraints and computation of free energy differences, arXiv preprint
1006.4914 (2010)
T. L ELIÈVRE , M. R OUSSET
AND
G. S TOLTZ Free energy computations: A
Mathematical Perspective,
Imperial College Press.
Other recent works on adaptive computation of free energy differences:
Free energy methods for efficient
exploration of mixture posterior densities, HAL preprint 00460914 (2010)
N. C HOPIN , T. L ELIÈVRE
AND
G. S TOLTZ,
Free
energy calculations: An efficient adaptive biasing potential method,
J. Phys. Chem. B 114(17), 5823-5830 (2010)
B. D ICKSON , F. L EGOLL , T. L ELIÈVRE , G. S TOLTZ
AND
P. F LEURAT-L ESSARD,
Workshop “Simulation of hybrid dynamical systems and applications to
molecular dynamics” (IHP, Paris, September 27-30th)
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