Proper use of the adaptive biasing force (ABF) algorithm in free-energy calculations needs certain prerequisites to be met, namely, that the Jacobian for the metric transformation and its first derivative be available and the coarse variables be independent and fully decoupled from any holonomic constraint or geometric restraint, thereby limiting singularly the field of application of the approach. The extended ABF (eABF) algorithm circumvents these intrinsic limitations by applying the time-dependent bias onto a fictitious particle coupled to the coarse variable of interest by means of a stiff spring. However, with the current implementation of eABF in the popular molecular dynamics engine NAMD, a trajectory-based post-treatment is necessary to derive the underlying free-energy change. Usually, such a posthoc analysis leads to a decrease in the reliability of the free-energy estimates due to the inevitable loss of information, as well as to a drop in efficiency, which stems from substantial read-write accesses to file systems. We have developed a user-friendly, on-the-fly code for performing eABF simulations within NAMD. In the present contribution, this code is probed in eight illustrative examples. The performance of the algorithm is compared with traditional ABF, on the one hand, and the original eABF implementation combined with a posthoc analysis, on the other hand. Our results indicate that the on-the-fly eABF algorithm (i) supplies the correct free-energy landscape in those critical cases where the coarse variables at play are coupled to either each other or to geometric restraints or holonomic constraints, (ii) greatly improves the reliability of the free-energy change, compared to the outcome of a posthoc analysis, and (iii) represents a negligible additional computational effort compared to regular ABF. Moreover, in the proposed implementation, guidelines for choosing two parameters of the eABF algorithm, namely the stiffness of the spring and the mass of the fictitious particles, are proposed. The present on-the-fly eABF implementation can be viewed as the second generation of the ABF algorithm, expected to be widely utilized in the theoretical investigation of recognition and association phenomena relevant to physics, chemistry, and biology.

Conspectus The observation of complex structural transitions in biological and abiological molecular objects within time scales amenable to molecular dynamics (MD) simulations is often hampered by significant free energy barriers associated with entangled movements. Importance-sampling algorithms, a powerful class of numerical schemes for the investigation of rare events, have been widely used to extend simulations beyond the time scale common to MD. However, probing processes spanning milliseconds through microsecond molecular simulations still constitutes in practice a daunting challenge because of the difficulty of taming the ruggedness of multidimensional free energy surfaces by means of naive transition coordinates. To address this limitation, in recent years we have elaborated importance-sampling methods relying on an adaptive biasing force (ABF). In this Account, we review recent developments of algorithms aimed at mapping rugged free energy landscapes that correspond to complex processes of physical, chemical, and biological relevance. Through these developments, we have broadened the spectrum of applications of the popular ABF algorithm while improving its computational efficiency, notably for multidimensional free energy calculations. One major algorithmic advance, coined meta-eABF, merges the key features of metadynamics and an extended Lagrangian variant of ABF (eABF) by simultaneously shaving the barriers and flooding the valleys of the free energy landscape, and it possesses a convergence rate up to 5-fold greater than those of other importance-sampling algorithms. Through faster convergence and enhanced ergodic properties, meta-eABF represents a significant step forward in the simulation of millisecond-time-scale events. Here we introduce extensions of the algorithm, notably its well-tempered and replica-exchange variants, which further boost the sampling efficiency while gaining in numerical stability, thus allowing quantum-mechanical/molecular-mechanical free energy calculations to be performed at a lower cost. As a paradigm to bridge microsecond simulations to millisecond events by means of free energy calculations, we have applied the ABF family of algorithms to decompose complex movements in molecular objects of biological and abiological nature. We show here how water lubricates the shuttling of an amide-based rotaxane by altering the mechanism that underlies the concerted translation and isomerization of the macrocycle. Introducing novel collective variables in a computational workflow for the rigorous determination of standard binding free energies, we predict with utmost accuracy the thermodynamics of protein–ligand reversible association. Because of their simplicity, versatility, and robust mathematical foundations, the algorithms of the ABF family represent an appealing option for the theoretical investigation of a broad range of problems relevant to physics, chemistry, and biology.

A robust importance-sampling algorithm for mapping free-energy surfaces over geometrical variables, coined meta-eABF, is introduced. This algorithm shaves the free-energy barriers and floods valleys by incorporating a history-dependent potential term in the extended adaptive biasing force (eABF) framework. Numerical applications on both toy models and nontrivial examples indicate that meta-eABF explores the free-energy surface significantly faster than either eABF or metadynamics (MtD) alone, without the need to stratify the reaction pathway. In some favorable cases, meta-eABF can be as much as five times faster than other importance-sampling algorithms. Many of the shortcomings inherent to eABF and MtD, like kinetic trapping in regions of configurational space already adequately sampled, the requirement of prior knowledge of the free-energy landscape to set up the simulation, are readily eliminated in meta-eABF. Meta-eABF, therefore, represents an appealing solution for a broad range of applications, especially when both eABF and MtD fail to achieve the desired result.

To improve sampling of the configurational entropy change upon protein-ligand binding, we have introduced a new set of coarse variables describing the relative orientation and position of the ligand via a global macromolecular orientational procedure, onto which geometrical restraints are applied. Evaluating the potential of mean force for the different coarse variables, the experimental standard binding free energy for three decapeptides associated with the SH3 domain of the Abl kinase is reproduced quantitatively.

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