On different notions of timescales in molecular dynamics

Transcription

1 On different notions of timescales in molecular dynamics Origin of Scaling Cascades in Protein Dynamics June 8, 217 IHP Trimester Stochastic Dynamics out of Equilibrium

2 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

3 Origin of the scaling cascades in protein dynamics Proteinstructure (Source: MaxPlanckForschung 4/23) Observation: single point mutations large non-local effects Trying to understand the observations: study parameter sensitivies of timescales study relations between timescales

4 A model to simulate molecular dynamics Overdamped Langevin equation dx t = V (X t )dt + 2β 1 db t where X R 3N : positions of the atoms V : R 3N R is the interaction potential B t is standard N dimensional Brownian motion β 1 R + is the temperature.

5 Illustration of the model Overdamped Langevin equation dx t = V (X t )dt + 2β 1 db t Associated probability density ρ t and equilibrium distribution ρ e V β

6 Quantities of interest Convergence to equilibrium: ρ t ρ c γ(t) Mean first exit times (MFET): E(τ x ( D)) = E(inf {t > : X t / D})

7 Decay towards equilibrium in L 2 µ 1 Generator: L = β 1 2 V dx t = V (X t )dt + 2β 1 db t describes evolution of expectation values and probability densities is self-adjoint wrt dµ = 1 Z e βv (x) dx, i.e. Lf, g µ = f, Lg µ spec( L) {} [λ, ) this implies convergence towards equilibrium in L 2, i.e. µ 1 ρ t ρ 2 L 2 µ 1 Problems: = λ is not known ρ L 2 µ 1 ρ t (x) ρ (x) 2 ρ 1 (x)dx e 2λt ρ ρ 2 L 2 µ 1 ρ (x) 2 e βv (x) dx <

8 Mean first exit times and eigenvalues of L Theorem [Bovier et al. 24]: Assumptions V has n minima x 1,..., x n ordering of minima according to energy barrier height possible Define τ xk (S k 1 ) = inf {t : X t S k 1, X = x k }, S k 1 = k 1 B 1 (x j). β Then L has n eigenvalues = Λ 1 > Λ 2 >... > Λ n and δ > such that 1 ( Λ k = 1 + O(1 + e βδ ) ) E(τ xk (S k 1 )) = c exp ( β (x k, {x 1,..., x k 1 })) (1 + O(1 + β 1 ) log β 1 ), c >. j=1

9 What we would like to do Convergence in terms of relative entropy instead of L 2 : µ 1 Relative entropy H(ρ t ρ ) = ( ρt (x) ρ t (x) log ρ (x) ) dx Relation to L 1 norm via Csizsàr-Kullback/Pinsker inequality: ρ t ρ L 1 2H(ρ t ρ ) L 1 is the natural norm for probability densities relation to measurable quantities applicable for any process that admits a pdf, not only reversible processes H is computable from simulation data

10 Linear but possibly irreversible processes Conditions dx t = AX t dt + σ 2β 1 db t, X R n, A R n n, (i) spec(a) C = {λ C : R(λ) < } (ii) A and σ fulfill the Kalman rank condition rank( [ σ, Aσ,..., A n 1 σ ] ) = n σ R n m, B t R m, m n. = existence of unique positive invariant measure ρ = N (, Σ). Theorem[Arnold, Erb 214] H(ρ t ρ ) c H(ρ ρ )e 2λ A t, λ A = min { R(λ) : λ spec(a)}, c 1.

11 Mean first exit times and eigenvalues of the covariance dx t = AX t dt + σ 2β 1 db t Interested in: τ x ( D) = inf {t > : X t / D}, D = {x : x < 1}. Theorem [Zabczyk 85] Assume that conditions (i) and (ii) are fulfilled s. th. ρ = N (, Σ) exists. Let λ Σ = max {λ : λ spec(σ)} >, E = {v : Σv = λ Σ v}. Then for large β, i.e. small temperature lim β β 1 log E(τ x ( D)) = 1 2λ exit time Σ and for any η > lim P(dist(X τ x ( D), E) η) = 1 exit path β We can show that λ A (2λ Σ) 1 λ + σ, λ + σ = min{λ > : λ spec(σσ T )}.

12 Analysis of relaxation behaviour Splitting up: H(ρ t ρ ) = ( ρt (x) log ρ (x) =a(t) ) ρ t (x) dx = 1 [ Tr(Σt Σ 1 ) n Tr(log(Σ t Σ 1 )) + µ T t Σ 1 ] µ t. 2 }{{}}{{} =b(t) } {{ } Covariance Same structure in all terms: z T e AT t Σ 1 e At z. For a(t) and b(t) : z = (Σ Σ ) 1 2, for c(t) : z = x. } {{ } } =c(t) {{ } Mean

13 Some examples for different relaxation behaviour high temperature low temperature, x = EVec(A) H/H relative entropy H()*e -2 t a(t) b(t) c(t) H/H relative entropy H()*e -2 t a(t) b(t) c(t) time time low temperature, x EVec(A) low temperature, A = A T H/H relative entropy H()*e -2 t c*h()*e -2 t a(t) b(t) c(t) H/H relative entropy H()*e -2 t a(t) b(t) c(t) time time

14 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 =. For c(t) this translates to: ċ(t) = ż p = z moves along contour lines of the potential p Here z i (t) = e λ i t (Sx ) i, p(z) = z T S T Σ 1 S 1 z with S such that SAS 1 = diag(λ 1,..., λ n ). det(σσ T ) = H/H a b c c at discrete times S time z at discrete times S 1 15

15 From degenerate to isotropic noise c(t) = z T S T Σ 1 S 1 z, z i = e λ i t z i (), λ 1 = 1, λ 2 = 1. det(σσ T ) = H/H a b c c at discrete times S time z at discrete times S 1 15 det(σσ T ) = 1 H/H a b c c at discrete times time S z at discrete times S

16 Identification of slow and fast? Can we identify slow and fast dof? Can we get estimates on the marginals? Can we get hierarchichal order of timescales?