On different notions of timescales in molecular
dynamics
Origin of Scaling Cascades in Protein Dynamics
Lara Neureither
June 8, 2017
IHP Trimester ”Stochastic Dynamics out of Equilibrium”
Lara Neureither
On different notions of timescales in molecular dynamics
Overview
1. Motivation
2. Definition of the timescales of interest: convergence rates and mean
first exit times
3. Review of results for overdamped Langevin equation
4. Approach for linear but irreversible systems: entropy production
Lara Neureither
On different notions of timescales in molecular dynamics
Origin of the scaling cascades in protein dynamics
Proteinstructure (Source: MaxPlanckForschung 4/2003)
Observation:
I
single point mutations → large non-local effects
Trying to understand the observations:
I study parameter sensitivies of timescales
I
study relations between timescales
Lara Neureither
On different notions of timescales in molecular dynamics
A model to simulate molecular dynamics
Overdamped Langevin equation
dXt = −∇V (Xt )dt +
p
2β −1 dBt
where
I
X ∈ R3N : positions of the atoms
I
V : R3N → R is the interaction potential
I
Bt is standard N−dimensional Brownian motion
I
β −1 ∈ R+ is the temperature.
Lara Neureither
On different notions of timescales in molecular dynamics
Illustration of the model
Overdamped Langevin equation
dXt = −∇V (Xt )dt +
p
2β −1 dBt
Associated probability density ρt and equilibrium distribution ρ∞ ∼ e −V β
Lara Neureither
On different notions of timescales in molecular dynamics
Quantities of interest
Convergence to equilibrium:
kρt − ρ∞ k ≤ c γ(t)
Lara Neureither
Mean first exit times (MFET):
E(τx (∂D)) = E(inf {t > 0 : Xt ∈
/ D})
On different notions of timescales in molecular dynamics
Decay towards equilibrium in L2µ−1
∞
dXt = −∇V (Xt )dt +
p
2β −1 dBt
Generator: L = β −1 ∇2 − ∇V · ∇
I
describes evolution of expectation values and probability densities
I
is self-adjoint wrt dµ∞ =
I
spec(−L) ⊂ {0} ∪ [λ, ∞)
I
this implies convergence towards equilibrium in L2µ−1 , i.e.
1 −βV (x)
dx,
Ze
i.e. hLf , g iµ∞ = hf , Lg iµ∞
∞
Z
2
kρt − ρ∞ kL2
−1
µ∞
=
2
2
−2λt
|ρt (x) − ρ∞ (x)| ρ−1
kρ0 − ρ∞ kL2
∞ (x)dx ≤ e
−1
µ∞
Problems:
I λ is not known
R
I ρ0 ∈ L2 −1 ⇔
ρ0 (x)2 e βV (x) dx < ∞
µ
∞
Lara Neureither
On different notions of timescales in molecular dynamics
Mean first exit times and eigenvalues of L
Theorem [Bovier et al. 2004]: Assumptions
I V has n minima x1 , . . . , xn
I ordering of minima according to energy barrier height possible
k−1
S
B β1 (xj ).
Define τxk (Sk−1 ) = inf {t ≥ 0 : Xt ∈ Sk−1 , X0 = xk } , Sk−1 =
j=1
Then L has n eigenvalues 0 = Λ1 > Λ2 > . . . > Λn and ∃ δ > 0 such that
1
1 + O(1 + e −βδ )
Λk = −
E(τxk (Sk−1 ))
p
= −c exp (−β∆(xk , {x1 , . . . , xk−1 })) 1 + O(1 + β −1 log β −1 ) , c > 0.
Lara Neureither
On different notions of timescales in molecular dynamics
What we would like to do
Convergence in terms of relative entropy instead of L2µ−1 :
∞
Z
Relative entropy
H(ρt |ρ∞ ) =
ρt (x) log
ρt (x)
ρ∞ (x)
dx
I
Relation to L1 norm via Csizsàr-Kullback/Pinsker inequality:
p
kρt − ρ∞ kL1 ≤ 2H(ρt |ρ∞ )
I
L1 is the natural norm for probability densities
I
relation to measurable quantities
I
applicable for any process that admits a pdf,
not only reversible processes
I
H is computable from simulation data
Lara Neureither
On different notions of timescales in molecular dynamics
Linear but possibly irreversible processes
dXt = AXt dt + σ
p
2β −1 dBt ,
X ∈ Rn , A ∈ Rn×n ,
σ ∈ Rn×m , Bt ∈ Rm , m ≤ n.
Conditions
(i) spec(A) ⊂ C− = {λ ∈ C : <(λ) < 0}
(ii) A and σ fulfill the Kalman
rank condition
rank( σ, Aσ, . . . , An−1 σ ) = n
=⇒ existence of unique positive invariant measure ρ∞ = N (0, Σ).
Theorem[Arnold, Erb 2014]
∗
H(ρt |ρ∞ ) ≤c · H(ρ0 |ρ∞ )e −2λA t ,
λ∗A = min {|<(λ)k : λ ∈ spec(A)} , c ≥ 1.
Lara Neureither
On different notions of timescales in molecular dynamics
Mean first exit times and eigenvalues of the covariance
dXt = AXt dt + σ
p
2β −1 dBt
Interested in: τx (∂D) = inf {t > 0 : Xt ∈
/ D} , D = {x : |x| < 1} .
Theorem [Zabczyk ’85]
Assume that conditions (i) and (ii) are fulfilled s. th. ρ∞ = N (0, Σ)
exists. Let λ∗Σ = max {λ : λ ∈ spec(Σ)} > 0, E = {v : Σv = λ∗Σ v }.
Then for large β, i.e. small temperature
lim β −1 log E(τx (∂D)) =
β→∞
1
−→ exit time
2λ∗Σ
and for any η > 0 lim P(dist(Xτx (∂D) , E ) ≤ η) = 1 −→ exit path
β→∞
We can show that
λ∗A ≥ (2λ∗Σ )−1 λ+
σ,
T
λ+
σ = min{λ > 0 : λ ∈ spec(σσ )}.
Lara Neureither
On different notions of timescales in molecular dynamics
Analysis of relaxation behaviour
Splitting up:
Z
H(ρt |ρ∞ ) =
=
log
ρt (x)
ρ∞ (x)
ρt (x) dx
1
−1
T −1
Tr (Σt Σ−1
∞ ) − n −Tr (log(Σt Σ∞ )) + µt Σ∞ µt .
{z
} | {z }
2|
{z
}|
=b(t)
=a(t)
|
{z
Covariance
=c(t)
}
|
{z
Mean
}
T
At
Same structure in all terms: z T e A t Σ−1
∞ e z.
1
For a(t) and b(t) : z = (Σ0 − Σ∞ ) 2 ,
for c(t) : z = x0 .
Lara Neureither
On different notions of timescales in molecular dynamics
Some examples for different relaxation behaviour
high temperature
low temperature, x0 = EVec(A)
1
1.4
relative entropy
1.2
0.6
0.4
H(0)*e -2 t
a(t)
b(t)
c(t)
0.6
H/H 0
H/H 0
0.8
relative entropy
0.8
H(0)*e -2 t
a(t)
b(t)
c(t)
1
0.4
0.2
0.2
0
0
-0.2
-0.4
-0.2
0
1
2
3
4
5
0
1
2
low temperature, x0 6= EVec(A)
4
5
low temperature, A = AT
1
2
relative entropy
c*H(0)*e -2 t
a(t)
b(t)
c(t)
H(0)*e -2 t
a(t)
b(t)
c(t)
0.6
H/H 0
1
relative entropy
0.8
H(0)*e -2 t
1.5
H/H 0
3
time
time
0.4
0.2
0.5
0
0
-0.2
0
1
2
3
4
5
time
0
1
2
3
4
5
time
Lara Neureither
On different notions of timescales in molecular dynamics
Understanding plateaus in the entropy decay
Necessary and sufficient condition for the existence of a plateau:
→ degeneracy of the noise, i.e. det σσ T = 0.
For c(t) this translates to:
ċ(t) = 0 ⇔ ż · ∇p = 0 ⇔ z moves along contour lines of the potential p
−1
Here zi (t) = e −λi t (Sx0 )i , p(z) = z T S −T Σ−1
z with S such that
∞S
−1
SAS = diag (λ1 , . . . , λn ).
det(σσ T ) = 0
1
0
a
b
c
c at discrete times
0.6
40
35
-0.5
30
-1
25
S2
H/H 0
0.8
0.4
-1.5
0.2
-2
20
15
0
z at discrete times
10
-2.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time
1
1.5
2
2.5
3
3.5
4
S1
Lara Neureither
On different notions of timescales in molecular dynamics
From degenerate to isotropic noise
−1
c(t) = z T S −T Σ−1
z, zi = e −λi t zi (0), λ1 = 1, λ2 = 10.
∞S
det(σσ T ) = 0
1
0
a
b
c
c at discrete times
0.6
40
35
-0.5
30
-1
25
S2
H/H 0
0.8
0.4
-1.5
0.2
-2
0
-2.5
20
15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z at discrete times
1
1.5
2
time
2.5
3
10
3.5
4
S1
det(σσ T ) = 1
1
H/H 0
0.6
0.4
S2
a
b
c
c at discrete times
0.8
0
35
-0.5
30
25
-1
20
-1.5
15
0.2
-2
0
-2.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time
z at discrete times
1
1.5
2
2.5
10
3
3.5
4
S1
Lara Neureither
On different notions of timescales in molecular dynamics
Identification of slow and fast?
I
Can we identify slow and fast dof?
I
Can we get estimates on the marginals?
I
Can we get hierarchichal order of timescales?
Lara Neureither
On different notions of timescales in molecular dynamics