We introduce a powerful method for exploring the properties of the multidimensional free energy surfaces (FESs) of complex many-body systems by means of coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates. A characteristic feature of these dynamics is the presence of a history-dependent potential term that, in time, fills the minima in the FES, allowing the efficient exploration and accurate determination of the FES as a function of the collective coordinates. We demonstrate the usefulness of this approach in the case of the dissociation of a NaCl molecule in water and in the study of the conformational changes of a dialanine in solution.M olecular dynamics (MD) and the Monte Carlo simulation method have had a very deep influence on the most diverse fields, from materials science to biology and from astrophysics to pharmacology. Yet, despite their success, these simulation methods suffer from limitations that reduce the scope of their applications. A severe constraint is the limited time scale that present-day computer technology and sampling algorithms explore. In particular, there are many circumstances where the free energy surface (FES) has several local minima separated by large barriers. Examples of these situations include conformational changes in solution, protein folding, first-order phase transitions, and chemical reactions. In such circumstances a simulation started in one minimum will be able to move spontaneously to the next minimum only under very favorable circumstances. A host of methods have been suggested to lift this restriction and explore the FES (1-13) or to characterize the transition state (14, 15). Here we propose a solution to this problem by combining the ideas of coarse-grained dynamics (16, 17) on the FES (10, 12) with those of adaptive bias potential methods (2, 11), obtaining a procedure that allows the system to escape from local minima in the FES and, at the same time, permits a quantitative determination of the FES as a byproduct of the integrated process. MethodologyWe shall assume here that there exists a finite number of relevant collective coordinates s i , i ϭ 1,n where n is a small number, and we consider the dependence of the free energy Ᏺ(s) on these parameters. Practical examples of appropriate choices of these variables will be given below. The exploration of the FES is guided by the forces F i t ϭ ϪѨᏲ͞Ѩs i t . To estimate these forces efficiently, we introduce an ensemble of P replicas of the system, each obeying the constraint that the collective coordinates have a preassigned value s i ϭ s i t , and each evolved independently at the same temperature T. Since the P replicas are statistically independent, the estimate of thermodynamic observables (e.g., the forces on the constraints) is improved with respect to an evaluation on a single replica, and it can be parallelized in a straightforward manner. The constraints are imposed on each replica via the standard methods of constrained molecular dynamics (18) by adding to the Lagra...

Cluster analysis is aimed at classifying elements into categories on the basis of their similarity. Its applications range from astronomy to bioinformatics, bibliometrics, and pattern recognition. We propose an approach based on the idea that cluster centers are characterized by a higher density than their neighbors and by a relatively large distance from points with higher densities. This idea forms the basis of a clustering procedure in which the number of clusters arises intuitively, outliers are automatically spotted and excluded from the analysis, and clusters are recognized regardless of their shape and of the dimensionality of the space in which they are embedded. We demonstrate the power of the algorithm on several test cases.

Metadynamics is a powerful algorithm that can be used both for reconstructing the free energy and for accelerating rare events in systems described by complex Hamiltonians, at the classical or at the quantum level. In the algorithm the normal evolution of the system is biased by a history-dependent potential constructed as a sum of Gaussians centered along the trajectory followed by a suitably chosen set of collective variables. The sum of Gaussians is exploited for reconstructing iteratively an estimator of the free energy and forcing the system to escape from local minima. This review is intended to provide a comprehensive description of the algorithm, with a focus on the practical aspects that need to be addressed when one attempts to apply metadynamics to a new system: (i) the choice of the appropriate set of collective variables; (ii) the optimal choice of the metadynamics parameters and (iii) how to control the error and ensure convergence of the algorithm.

The possibility of observing chemical reactions in ab initio molecular dynamics runs is severely hindered by the short simulation time accessible. We propose a new method for accelerating the reaction process, based on the ideas of the extended Lagrangian and coarse-grained non-Markovian metadynamics. We demonstrate that by this method it is possible to simulate reactions involving complex atomic rearrangements and very large energy barriers in runs of a few picoseconds.

We present a fully Hamiltonian and computationally efficient scheme to include the electrostatic effects due to the classical environment in a Car-Parrinello mixed quantum Mechanics/molecular mechanics ͑QM/MM͒ method. The polarization due to the MM atoms close to the quantum system is described by a Coulombic potential modified at short range. We show that the functional form of this potential has to be chosen carefully in order to obtain the correct interaction properties and to prevent an unphysical escape of the electronic density to the MM atoms ͑the so-called spill-out effect͒. The interaction between the QM system and the more distant MM atoms is modeled by a Hamiltonian term explicitly coupling the multipole moments of the quantum charge distribution with the classical point charges. Our approach remedies some of the well known deficiencies of current electrostatic coupling schemes in QM/MM methods, allowing molecular dynamics simulations of mixed systems within a fully consistent and energy conserving approach.

By suitably extending a recent approach [Bussi, G.; et al. J. Am. Chem. Soc. 2006, 128, 13435] we introduce a powerful methodology that allows the parallel reconstruction of the free energy of a system in a virtually unlimited number of variables. Multiple metadynamics simulations of the same system at the same temperature are performed, biasing each replica with a time-dependent potential constructed in a different set of collective variables. Exchanges between the bias potentials in the different variables are periodically allowed according to a replica exchange scheme. Due to the efficaciously multidimensional nature of the bias the method allows exploring complex free energy landscapes with high efficiency. The usefulness of the method is demonstrated by performing an atomistic simulation in explicit solvent of the folding of a Triptophane cage miniprotein. It is shown that the folding free energy landscape can be fully characterized starting from an extended conformation with use of only 40 ns of simulation on 8 replicas.

Graphical abstract

Metadynamics is a powerful technique that has been successfully exploited to explore the multidimensional free energy surface of complex polyatomic systems and predict transition mechanisms in very different fields, ranging from chemistry and solid-state physics to biophysics. We here derive an explicit expression for the accuracy of the methodology and provide a way to choose the parameters of the method in order to optimize its performance.