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Metadynamics is a powerful technique for enhancing sampling in molecular dynamics simulations and reconstructing the free‐energy surface as a function of few selected degrees of freedom, often referred to as collective variables (CVs). In metadynamics, sampling is accelerated by a history‐dependent bias potential, which is adaptively constructed in the space of the CVs. Since its first appearance, significant improvements have been made to the original algorithm, leading to an efficient, flexible, and accurate method that has found many successful applications in several domains of science. Here, we discuss first the theory underlying metadynamics and its recent developments. In particular, we focus on the crucial issue of choosing an appropriate set of CVs and on the possible strategies to alleviate this difficulty. Later in the second part, we present a few recent representative applications, which we have classified into three main classes: solid‐state physics, chemical reactions, and biomolecules. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 826‐843 DOI: 10.1002/wcms.31

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  • Molecular and Statistical Mechanics > Free Energy Methods
Figure 1.

Example of metadynamics simulation in a one‐dimensional model potential. The time t is measured by counting the number of Gaussians deposited. (Top) Time evolution of the collective variables during the simulation. (Bottom) Schematic representation of the progressive filling of the underlying potential (thick line) by means of the Gaussians deposited along the trajectory. The sum of the underlying potential and of the metadynamics bias is shown at different times (thin lines).

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Figure 1.

(Bottom) Free‐energy surface of the dissociation process of SC‐558 from cyclooxygenase‐2 as a function of the distance and dihedral collective variables. Isoenergy lines are drawn every 2 kcal/mol. (Top) The four main free‐energy basins A–D found during the metadynamics simulation. Basin A is the crystallographic pose, basin B an alternative pose, basin C corresponds to the gate site, and basin D is an external pose. The ligand and the main interacting residues are displayed as licorice, whereas the protein is represented as green cartoon with the α‐helices forming the gate colored in orange. The interacting waters are shown as spheres, whereas hydrogens are not displayed for clarity.

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Figure 2.

(a–c). Projection onto the Ramachandran plot of the configurations sampled during a well‐tempered metadynamics simulation of alanine dipeptide in vacuum (white dots) for different choices of ΔT [600 K (a), 1800 K (b), and 4200 K (c)]. The underlying color map shows the reference free‐energy landscape. (d) Estimate of the free‐energy difference between the two metastable minima C7ax (1.22, −1.22) and C7eq (−1.45, 1.29) as a function of the simulation time. Angles are measured in radians.

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Figure 3.

Example of the effect of neglecting a slow degree of freedom in the collective variables (CVs) set. (Top) Model two‐dimensional potential with relevant barriers both in s1 and s2. (Bottom) Representation of a metadynamics simulation using only s1 as CV. The sum of the underlying one‐dimensional free‐energy F(s1) (thick line) and of the metadynamics bias is shown at different times (thin lines). Neglecting s2 in the CVs set causes strong hysteresis in the reconstructed free energy.

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Figure 4.

(a and b). Representation of the path collective variables. Collection of points at constant s (a) and z (b) are shown for the case of a reference path designed in the two‐dimensional model potential of Figure 3.(c) Free energy as a function of s and z. The two minima A and B are projected onto (0,0) and (1,0).

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Figure 5.

A combination of Figures 1b and 4 of Sun et al.60 showing the structural evolution of solid CO2 obtained with ab initio metadynamics simulations. (a) Structural phase transformation of CO2 from Phase II (P42/mnm) at 60 GPa and 600 K, resulting in a layered structure (P‐4m2). (b) Snapshots of the transformation from Phase III (Cmca) to an α‐cristobalite‐like structure (P41212) at 80 GPa and 300 K.

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Figure 6.

Free‐energy surface for the calculation of the solid–liquid interface free energy γsl for a Lennard–Jones potential. The two collective variables (CVs) discriminate between a solid (s ∼ 0.85) or liquid (s ∼ 0) state of two halves of the supercell. γsl can be obtained from the difference between the plateau (in which an interface is present) and the two minima (which correspond to homogeneous configurations).

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Figure 7.

The molecular mechanism of the catalase reaction in Helicobacter pylori catalase obtained from quantum mechanics/molecular mechanics CPMD metadynamics simulations. R is the Heme+•FeIV = O···H2O2 reactant complex (also named Compound I···H2O2), R′ is the HemeFeIV = O···HOO intermediate formed after low‐barrier hydrogen abstraction and P is the HemeFeIII···H2O ···O2 product state. Conversion from R to R′ was spontaneous, whereas R′→P was activated by metadynamics. The two‐dimensional free‐surface energy displays two possible pathways, depending on the mechanism of the second hydrogen atom transfer.

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Figure 8.

(a) Reaction mechanisms for the hydrolysis of diglycine in hot‐pressurized water (HPW). (b) Corresponding schematic free‐energy profiles at ambient (black line and filled triangles) and at extreme conditions (blue line and open squares). (c) Reconstructed free‐energy surface for this reaction at HPW is presented as a volumetric data for selected free energy (contour) values (in kJ/mol); 4′cis is having a cis, and 4 and 4′ are having a trans peptide bond.

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Figure 9.

(Left, upper) The image shows an actin trimer. The bottom (blue) subunit is the subunit in which the folding of the DNase‐I binding (DB) loop was studied. The neighboring actin subunits (green and silver) make significant contacts with the DB loop, and were found to be important in the overall stability of the helix. (Left, lower) The close‐up view shows the neighborhood surrounding the DB loop. The unfolded (blue transparent) and folded (red helix) states are shown. (Right, lower) Free‐energy profiles for folding the DB loop in monomeric actin as a function of the bound nucleotide. (Right, upper) The transition states are shown in ribbon representation along with representative unfolded and folded structures.

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