Metadynamics

The expression for A + depends on whether crystallization is diffusion-limited or collision-limited, in full analogy with the nucleation case. In the diffusion-limited case, Vg reads  

MTD is based on the assumption that the relevant processes can be described by a small set of coarse-grained variables, called collective variables (CVs), describing the essential degrees of freedom of the system. The CVs f ( r ) themselves are 

The description of the free energy surface (FES) in the coarse-grained d-dimensional space of the CVs is obtained by integrating out the irrelevant degrees of freedom of the system (and neglecting the constant contribution due to the kinetic energy)  

The hills are usually chosen to be repulsive Gaussian funcAons. In this case, the bias potential Vb can be written as (for d — 1)  

Assuming that the size of the hills is small compared to the size of the free energy basins of the system, after a sufAciently long time the high barriers of the FES will 

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In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

Several other techniques are available to compute the PMF. In this section we discuss the steered force molecular dynamics (SMD) [35-37] and the metadynamics of Laio and Parrinello [30-34]. 

Metadynamics defines coarse-grained variables which are assumed to be slow coordinates of the system. Those coordinates are similar to the order parameters considered earlier in this chapter. The coarse variables are evolved independently following a steepest-descent equation. In the case of a single variable, Laio and Parrinello [34] use... 

Fig. 4.16. The A dynamics method for alchemical transformations was developed by Guo and Brooks [57] for rapid screening of binding affinities. In this approach the parameter A is a dynamic variable. Techniques like ABF or metadynamics [34] can be used to accelerate this type of calculation. A dynamics was used by Guo [57] to study the binding of benzamidine to trypsin. One simulation is sufficient to gather data on several benzamidine derivatives. Substitutions were made at the para position C5 (H, NH2, CH3 and Cl). The hydrogen atoms are not shown for clarity...

Many other techniques have been developed for the PMF such as SMD, which applies an external force to drive the system along a certain path or metadynamics which biases the system away from free energy minima in an attempt to sample more efficiently transition states and rare events. 

At the end of the chapter, techniques for alchemical transformations were presented. We showed that, in order to avoid rapid changes in free energy and improve the efficiency of the calculation, the parametrization of the Hamiltonian is critical and soft-core potentials should be used [see (4.50)]. A popular approach is the technique of A dynamics which leads to an improved sampling. In this approach A is a variable in the Hamiltonian system [see (4.51)]. Umbrella sampling, metadynamics or ABF can be used to reduce the cost of A dynamics simulations. 

Micheletti, C. Laio, A. Parrinello, M., Reconstructing the density of states by history-dependent metadynamics, Phys. Rev. Lett. 2004, 92, 170601-170604... 

Stirling, A. Iannuzzi, M. Laio, A. Parrinello, M., Azulene-to-naphthalene rearrangement the Car-Parrinello metadynamics method explores various reaction mechanisms, Eur. J. Chem. Phys. Phys. Chem. 2004, 5, 1558-1568... 

Another apparent difference between the various free energy methods lies in the treatment of order parameters. In the original formulation of a number of methods, order parameters were dynamical variables - i.e., variables that can be expressed in terms of the Cartesian coordinates of the particles - whereas in others, they were parameters in the Hamiltonian. This implies a different treatment of the order parameter in the equations of motion. If one, however, applies the formalism of metadynamics, or extended dynamics, in which any parameter can be treated as a dynamical variable, most conceptual differences between these two cases vanish. 

V. Spiwok, B. Kralova, and I. Tvaroska, Modelling of P-D-glucopyranose ring distortion in different force fields a metadynamics study, Carbohydr. Res., 345 (2010) 530-537. 

Gervasio FL, Laio A, Parrinello M (2005) Flexible docking in solution using metadynamics. J Am Chem Soc 127(8) 2600-2607... 

Laio A, Parrinello M (2006) Computing free energies and accelerating rare events with metadynamics. In Ferrario M, Ciccotti G, Binder K (eds) Computer simulations in condensed matter from materials to chemical biology, vol 1, Springer. Berlin, Heidelberg, New York, pp 315-347... 

R. Martonak, A. Laio, M. Bernasconi, C. Ceriani, P. Raiteri, F. Zipoli, and M. Parrinello (2005) Simulation of structural phase transitions by metadynamics. Zeitschrift fiir Kristallographie 220, p. 489... 

Computing Free Energies and Accelerating Rare Events with Metadynamics... 

A. Laio and M. Parrinello Computing Free Energies and Accelerating Rare Events with Metadynamics, Lect. Notes Phys. 703, 315—347 (2006)... 

Merging Results from Different Metadynamics Calculations.340... 

Minimal Free Energy Path with Metadynamics.341... 

The capability of metadynamics to provide quantitative informations on the equilibrium properties of a system is based on the observation that the... 

In this chapter we provide a general introduction to metadynamics without entering into the details of the choice of the collective variables, which is, as we already underlined, a system-specific issue. In Sect. 1 we describe the... 

The parameters w and 5s determine the accuracy and efEciency of the free energy reconstruction, as we will discuss in Sects. 1.1 and 3. After the collective coordinates are updated using (3), an ensemble with values is prepared, and the new forces are evaluated in another MD or MC run. In this manner the collective variables will perform a metadynamics in the CV space, visiting configurations very different from the ones they would have explored under the action of the thermodynamic force alone. In the following we refer to the point that explores the CV space as a walker. 

The example of Fig. 1 also provides a demonstration of the two different manners in which metadynamics can be exploited ... 

If the aim is to estimate the free energy in a predefined region in the CV space, metadynamics should be stopped when the walker has explored this whole region and its motion becomes diffusive. This can be detected by... 

If the metadynamics is terminated at position s, the reconstructed free energy will present a bump in a region around s whose spread depends on the correlation time of metadynamics. This effect can be lessened if the contribution of the Gaussians placed at the end of the dynamics are weighted less. Therefore, after the metadynamics with constant w is terminated the free energy is reconstructed from Fr (s) =... 

These prescriptions can be formalized in the following modified metadynamics equations, which replace (3) and (4) ... 

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