Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method

Lesage, Adrien; Lelièvre, Tony; Stoltz, Gabriel; Hénin, Jérôme

doi:10.1021/acs.jpcb.6b10055

Cited by 133 publications

(204 citation statements)

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“…where Z A ∈ (0, ∞) thanks to (38), is the unique invariant distribution of the biased SPDE (39), see for instance [16]. (12) in the case where the diffusion process is governed by a SPDE:…”

Section: The Spde Casementioning

confidence: 99%

“…The observations made in Section 1.1, concerning the system (2) and the consistency result, Theorem 1.1, can be generalized as follows.First, contrary to (1), the state space of the dynamics may not be compact. It may also be of infinite dimension.The most important generalization concerns the type of diffusion processes which are considered: our general framework also encompasses the following examples (this list is not exhaustive): (hypoelliptic) Langevin dynamics with position and momenta variables, extended dynamics -where an auxiliary variable is associated with the mapping ξ, see [38]) -and Stochastic Partial Differential Equations (SPDEs) -which are infinite dimensional diffusion processes. It may also be possible to study diffusions on smooth manifolds, however to simplify the presentation this situation is not treated.Abstract notation and analysis allow us to treat simulataneously these examples in a general framework.…”

mentioning

confidence: 99%

“…The most important generalization concerns the type of diffusion processes which are considered: our general framework also encompasses the following examples (this list is not exhaustive): (hypoelliptic) Langevin dynamics with position and momenta variables, extended dynamics -where an auxiliary variable is associated with the mapping ξ, see [38]) -and Stochastic Partial Differential Equations (SPDEs) -which are infinite dimensional diffusion processes. It may also be possible to study diffusions on smooth manifolds, however to simplify the presentation this situation is not treated.…”

mentioning

confidence: 99%

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Convergence analysis of adaptive biasing potential methods for diffusion processes

Benaı̈m

Bréhier

2019

Communications in Mathematical Sciences

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This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages.We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results , we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context.One of the main limitations of standard MCMC approaches comes from the fact that the ergodic dynamics are metastable whenever µ is multimodal. In the example above, this happens when V has several local minima. A direct simulation is not able to efficiently and accurately sample the rare transitions between the metastable states, hence the need for advanced Monte-Carlo methods.Many strategies have been proposed, analyzed and applied, to overcome this issue. The associated variance reduction approaches may be divided into two main families. On the one hand, Importance Key words and phrases. adaptive biasing, self-interacting diffusions, free energy computation. 1To simplify notation, we omit the dependence of A ⋆ with respect to the parameter β.The unknows in (2) are the stochastic processes t → X t ∈ T d , t → µ t ∈ P(T d ) (the set of Borel probability distributions on T d , endowed with the usual topology of weak convergence of probability distributions), and t → A t ∈ C ∞ (T m ) (the set of infinitely differentiable functions on T m ). Initial conditions X t=0 = x 0 and µ t=0 = µ 0 are prescribed.An important observation is that the third equation in (2) introduces a coupling between the evolutions of the diffusion process X t and of the probability distribution µ t . Thus the system defines a type of self-interacting diffusion process. However, a comparison with [4] and subsequent articles [5], [6] and [7], reveals a different form of coupling. One of the aims of this article is to study the new arguments which are required for the study of the system (2).The most important quantity in (2) is the random, time-dependent, probability distribution µ t . Observe that its construction requires two successive operations to be performed. First, a weighted occupation measure µ t = µ 0 + t 0 exp −βA r • ξ(X r ) δ Xr dr is computed. Second, this measure is 3 normalized to define a probability distribution, µ t = µt x∈T d µt(dx) . The weights exp −βA r • ξ(X r ) in the definition of µ t are chosen so as to obtain the following consistency result for the es...

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Section: The Spde Casementioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Convergence analysis of adaptive biasing potential methods for diffusion processes

Benaı̈m

Bréhier

2019

Communications in Mathematical Sciences

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“…Because of its popularity and versatility, the expanded ensemble method has been the subject of numerous studies [15][16][17][18][19][20][21] over recent years. To evaluate the thermodynamic expectations, several estimators have been proposed, namely, the (standard) binning estimator, 6 a standard reweighting estimator 21 , a self-consistent reweighting estimator 17 called simulated tempering weighted histogram analysis method (STWHAM), a histogram reweighting estimator 22 called corrected z-averaged restraints (CZAR) and an adiabatic reweighting (AR) estimator. 20,21 Self-consistent reweighting estimators are known under various names such as the Bennett acceptance ratio (BAR) method 23 , the wheighted histogram analysis method 24,25 (WHAM), the reverse logistic regression 26 , bridge sampling, 27 the multi-state BAR method 28 , binless WHAM 29 and the global likelihood method.…”

Section: -4mentioning

confidence: 99%

“…We therefore consider that the adaptive sampling process has been successfully completed: the biasing potential is frozen and does not depend on the simulation time anymore. We then show how to condition the binning, 6 standard reweighting, 21 self-consistent reweighting 17 and histogram reweighting 22 estimators of thermodynamic expectations, and demonstrate that the conditioning procedure systematically leads to the formulation of the adiabatic reweighting estimator. By comparing the asymptotic variances of the involved estimators, we eventually deduce that the conditioning procedure ensures systematic variance reduction, entailing that the adiabatic reweighting estimator is optimal among the large class of considered estimators.…”

Section: -4mentioning

confidence: 99%

Estimating thermodynamic expectations and free energies in expanded ensemble simulations: Systematic variance reduction through conditioning

Athènes

Terrier

2017

The Journal of Chemical Physics

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Markov chain Monte Carlo methods are primarily used for sampling from a given probability distribution and estimating multi-dimensional integrals based on the information contained in the generated samples. Whenever it is possible, more accurate estimates are obtained by combining Monte Carlo integration and integration by numerical quadrature along particular coordinates. We show that this variance reduction technique, referred to as conditioning in probability theory, can be advantageously implemented in expanded ensemble simulations. These simulations aim at estimating thermodynamic expectations as a function of an external parameter that is sampled like an additional coordinate. Conditioning therein entails integrating along the external coordinate by numerical quadrature. We prove variance reduction with respect to alternative standard estimators and demonstrate the practical efficiency of the technique by estimating free energies and characterizing a structural phase transition between two solid phases.

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Triggering Tautomerization of Curcumin by Confinement into Liposomes

et al. 2019

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Although solvent polarity, pH and temperature have been reported to affect the keto-enol equilibrium of curcumin, the synergistic effect of concentration and confinement is demonstrated for the first time herein. To this end, curcumin was encapsulated into vesicles of soy lecithin of about 100 nm in diameter. The keto form is favoured in the dilute regime (up to 2 μM). Above this concentration, tautomerization is shifted towards the enol form, as shown by fluorescence and compression isotherm studies. Moreover, molecular modeling and free-energy profiles, together with simulation of the optical properties of the curcumin isomers, allowed us to dissect the interaction of the keto and enol forms with a model membrane and confirm the concentration-triggered tautomerization of curcumin under confinement.

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Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method

Cited by 133 publications

References 21 publications

Convergence analysis of adaptive biasing potential methods for diffusion processes

Convergence analysis of adaptive biasing potential methods for diffusion processes

Estimating thermodynamic expectations and free energies in expanded ensemble simulations: Systematic variance reduction through conditioning

Triggering Tautomerization of Curcumin by Confinement into Liposomes

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