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WIREs Comput Mol Sci
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Umbrella sampling

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Abstract The calculation of free‐energy differences is one of the main challenges in computational biology and biochemistry. Umbrella sampling, biased molecular dynamics (MD), is one of the methods that provide free energy along a reaction coordinate. Here, the method is derived in a historic overview and is compared with related methods like thermodynamic integration, slow growth, steered MD, or the Jarzynski‐based fast‐growth technique. In umbrella sampling, bias potentials along a (one‐ or more‐dimensional) reaction coordinate drive a system from one thermodynamic state to another (e.g., reactant and product). The intermediate steps are covered by a series of windows, at each of which an MD simulation is performed. The bias potentials can have any functional form. Often, harmonic potentials are used for their simplicity. From the sampled distribution of the system along the reaction coordinate, the change in free energy in each window can be calculated. The windows are then combined by methods like the weighted histogram analysis method or umbrella integration. If the bias potential is adapted to result in an even distribution between the end states, then this whole range can be spanned by one window (adaptive‐bias umbrella sampling). In this case, the free‐energy change is directly obtained from the bias. The sampling in each window can be improved by replica exchange methods; either by exchange between successive windows or by running additional simulations at higher temperatures. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 932–942 DOI: 10.1002/wcms.66 This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods

Separation of the reaction coordinate (dashed line) between two states (here represented by two minima on the potential energy surface) into distinct windows. The system is mainly sampled perpendicular to the reaction coordinate in each window.

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Weights of weighted histogram analysis method (WHAM) and umbrella integration of three windows in a real simulation of the enzyme para‐hydroxybenzoate hydroxylase (PHBH).49 A maximum in the free energy is found between the second and third windows.

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Global free energy (thick solid curve) and the contributions Ai of some of the windows (thin dashed curves). For clarity, only every third window is shown. At the bottom, the biased distributions Pib as obtained from the simulation are shown (thin solid curves). Relatively few bins (100) have been used to generate this figure.

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