Free energy calculations: An efficient adaptive biasing potential - - PowerPoint PPT Presentation

Free energy calculations: An efficient adaptive biasing potential method

• P. Fleurat-Lessard - B. Dickson
• F. Legoll - T. Leli`

evre - G. Stoltz ENS Lyon - ENPC

mDoS – p. 1

Abstract

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Free Energy

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Free Energy (kcal/ mol)

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C

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put ed Fr ee Ener gy

Φ (degree) Ψ ( d egr ee)

1 2 3 4 5 6 0.5 1 1.5 2 2.5 3 3.5 4

Time (ns)

M ean Error ( kcal/ m o l)

ABF

Deconvolution

mollification

We have recently introduced an efficient sampling and free energy calculation technique within the adaptive biasing potential (ABP) framework. The adaptive bias potential is computed from the population along the reaction coordinate: e−βAα(ξ∗) ∝

• exp
• − |ξ(x) − ξ∗|2

α2

• e−βV (x) dx

This approximation introduces two parameters: strength of mollification and the zero of energy of the bias potential. We present here two extensions to deal with complex systems. The first extension consists in using a local and adaptive Gaussian height. In particular, adapting the height with the bias evolution rate prevents getting trapped in narrow but deep wells.

mDoS – p. 2

Introduction

Rare events Bias to enhance sampling Reaction coordinate: ξ Adaptive Bias: Adaptive Biasing Potential (or DoS):

A(ξ) is constructed

Adaptive Biasing Force:

• nly ∂A(ξ)

∂ξ

is constructed

mDoS – p. 3

Our approach: mDoS

We bias DoS:

e−βA(ξ∗) = Z−1

• X

δ(ξ(x) − ξ∗) e−βV (x) dx

Main idea:

δ(ξ) → δα(ξ) =

1 α√π exp

• − ξ2

α2

• mollified Density of States (mDoS):

e−βAα(ξ∗) = Z−1

• X

δα(ξ(x) − ξ∗) e−βV (x) dx

mDoS – p. 4

mDoS advantages

Non-locality Derivatives easy to compute:

∂Aα(ξ∗) ∂ξ∗ = −kBT

• X

∂ξ∗δα(ξ(x) − ξ∗) e−βV (x) dx

• X

δα(ξ(x) − ξ∗) e−βV (x) dx

with ∂ξ∗

j δα(ξj(x) − ξ∗) =

2 α2(ξj(x) − ξ∗ j )δα(ξ(x) − ξ∗)

No derivatives of ξ ! Icing on the cake: e−βAα(ξ) = e−βA(ξ) ∗ δα(ξ)

⇒ Error can be greatly reduced by deconvolution.

mDoS – p. 5

The bias Vb(ξ)

Vb(ξ, t) = −Aα(ξ, t) does not converge

Normalizing Aα(ξ, t) → ∆Aα(ξ, t) with:

∆Aα(ξ, t) = Aα(ξ, t) − min

ξ∗ [Aα(ξ∗, t)]

Convergence boost:

eβVb(ξ,t) = e−β∆Aα(ξ,t) eβc,

eβc can be seen as the Gaussians height

mDoS – p. 6

Application: alanine dipeptide

AMBER ff94 T=300K: Langevin Implicit solvent: Generalized-Born model RC: (ξ1, ξ2) = (Ψ, Φ)

Aref(ξ1, ξ2): 120ns of unbiased trajectory.

mDoS – p. 7

Influence of α

1 2 3 4 5 6 7 0.5 1 1.5 2 2.5 3 3.5 4 |A-Aref| (kcal/mol) t (ns) α=0.8 α=2 α=5 α=10 α=20

Non local and good sampling

mDoS – p. 8

Influence of c

1 2 3 4 5 6 7 8 9 10 11 0.5 1 1.5 2 2.5 3 3.5 4 |A-Aref| (kcal/mol) t (ns) c=0 c=5kT c=15kT c=30kT

mDoS – p. 9

Other approaches: Metadynamics

Well-tempered metadynamics

Barducci, Bussi and Parrinello PRL 100, 020603 (2008) V meta

b

(ξ, τ) =

• t′≤τ

h(ξ, t′) exp

• −(ξ − ξt′)2

2w2

• with h(ξ, t) = ω exp[−V meta

b

(ξ, t)/kB∆T]τG.

but Gaussians add up to A(ξ)

mDoS – p. 10

Other approaches: ABF

Adaptive Biasing Force

Darve, Rodriguez-Gomez, Pohorille J. Chem. Phys. 2008, 128, 144120.

ABF(ξ∗) = − d dt

• Mξ

dξ dt

• ξ∗

, More standard: Fj(ξ∗) = N

• i=1

∇V · G−1

ji ∇ξi − β−1∇ · (G−1 ji ∇ξi)

• ξ∗

,

Parameter free

lim

t→∞ ABF = ∂A(ξ)

∂ξ

mDoS – p. 11

Comparison

0.2 0.4 0.6 0.8 1 1.2 1.4 1 2 3 4 |A-Aref| (kcal/mol) t (ns) α=5 ABF Metadyn.

mDoS – p. 12

Comparison: deconvolution

0.2 0.4 0.6 0.8 1 1.2 1.4 1 2 3 4 |A-Aref| (kcal/mol) t (ns) α=5 α=5 deconv α=10 deconv Metadyn.

mDoS – p. 13

Conclusion and Perspectives

mDos is a new method: Very efficient Easy to implement Perspectives: Multi-replica with selection Chemical reactions (high barriers) Going to nD, n>2 Put it in PLUMED !

Dickson et al. J. Phys. Chem. B 2010, 114, 5823-5830.

mDoS – p. 14