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 Free Energy C om put ed Fr ee Ener gy m o l) 6 14 14 150 150 ABF 150 150 Free Energy (kcal/ 5 12 12 kcal/ 75 75 75 75 4 ee) 10 10 Error ( mollification egr 3 8 8 d 0 0 Ψ ( 0 0 2 M ean 6 6 1 -75 -75 -75 -75 4 4 mol) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 2 2 -150 -150 -150 -150 Time (ns) 0 0 -150 -150 -75 -75 0 0 75 75 150 150 -150 -150 -75 -75 0 0 75 75 150 150 Φ (degree) Deconvolution 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: − | ξ ( x ) − ξ ∗ | 2 � � � e − βA α ( ξ ∗ ) ∝ e − βV ( x ) dx exp α 2 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: only ∂A ( ξ ) is constructed ∂ξ mDoS – p. 3
Our approach: mDoS We bias DoS: � e − βA ( ξ ∗ ) = Z − 1 δ ( ξ ( x ) − ξ ∗ ) e − βV ( x ) dx X Main idea: � � − ξ 2 1 δ ( ξ ) → δ α ( ξ ) = α √ π exp α 2 mollified Density of States (mDoS): � e − βA α ( ξ ∗ ) = Z − 1 δ α ( ξ ( x ) − ξ ∗ ) e − βV ( x ) dx X mDoS – p. 4
mDoS advantages Non-locality Derivatives easy to compute: � ∂ ξ ∗ δ α ( ξ ( x ) − ξ ∗ ) e − βV ( x ) dx ∂A α ( ξ ∗ ) X = − k B T � ∂ξ ∗ δ α ( ξ ( x ) − ξ ∗ ) e − βV ( x ) dx X 2 j δ α ( ξ j ( x ) − ξ ∗ ) = α 2 ( ξ j ( x ) − ξ ∗ j ) δ α ( ξ ( x ) − ξ ∗ ) with ∂ ξ ∗ No derivatives of ξ ! Icing on the cake: e − βA α ( ξ ) = e − βA ( ξ ) ∗ δ α ( ξ ) ⇒ Error can be greatly reduced by deconvolution. mDoS – p. 5
The bias V b ( ξ ) V b ( ξ, t ) = − A α ( ξ, t ) does not converge Normalizing A α ( ξ, t ) → ∆ A α ( ξ, t ) with: ξ ∗ [ A α ( ξ ∗ , t )] ∆ A α ( ξ, t ) = A α ( ξ, t ) − min Convergence boost: e βV b ( ξ,t ) = e − β ∆ A α ( ξ,t ) e βc , e βc can be seen as the Gaussians height mDoS – p. 6
Application: alanine dipeptide AMBER ff94 T=300K: Langevin Implicit solvent: Generalized-Born model RC: ( ξ 1 , ξ 2 ) = (Ψ , Φ) A ref ( ξ 1 , ξ 2 ) : 120ns of unbiased trajectory. mDoS – p. 7
Influence of α 7 α =0.8 Non local α =2 6 α =5 and good sampling α =10 5 α =20 |A-A ref | (kcal/mol) 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 t (ns) mDoS – p. 8
Influence of c 11 c=0 10 c=5kT c=15kT 9 c=30kT 8 |A-A ref | (kcal/mol) 7 6 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 t (ns) mDoS – p. 9
Other approaches: Metadynamics Well-tempered metadynamics Barducci, Bussi and Parrinello PRL 100 , 020603 (2008) − ( ξ − ξ t ′ ) 2 � � � V meta h ( ξ, t ′ ) exp ( ξ, τ ) = b 2 w 2 t ′ ≤ τ with h ( ξ, t ) = ω exp[ − V meta ( ξ, t ) /k B ∆ T ] τ G . b 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. � d � �� dξ ABF ( ξ ∗ ) = − M ξ , dt dt ξ ∗ More standard: � N � � F j ( ξ ∗ ) = ∇ V · G − 1 ji ∇ ξ i − β − 1 ∇ · ( G − 1 ji ∇ ξ i ) , i =1 ξ ∗ Parameter free t →∞ ABF = ∂A ( ξ ) lim ∂ξ mDoS – p. 11
Comparison α =5 1.4 ABF Metadyn. 1.2 |A-A ref | (kcal/mol) 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 t (ns) mDoS – p. 12
Comparison: deconvolution α =5 1.4 α =5 deconv α =10 deconv 1.2 Metadyn. |A-A ref | (kcal/mol) 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 t (ns) 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
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