Umbrella potential

To bias the sampling toward a region of interest that would not otherwise be significantly populated, a restraining potential UXq) is added to the potential energy of the system. U, is often referred to as an umbrella potential [37]. For concreteness, we assume the harmonic form... 

Eor many problems, the ideal umbrella potential would be one that completely flattens the free energy profile along q, i.e., UXq) = W(q). Such a potential cannot be determined in advance. However, iterative approaches exist that are known as adaptive... 

The foregoing approaches used an umbrella potential to restrain q. The pmf W(q) can also be obtained from simulations where q is constrained to a series of values spanning the region of interest [48,49]. However, the introduction of rigid constraints complicates the theory considerably. Space limitations allow only a brief discussion here for details, see Refs. 8 and 50-52. 

A powerful and general technique to enhance sampling is the use of umbrella potentials, discussed in Section IV. In the context of alchemical free energy simulations, for example, umbrella potentials have been used both to bias the system toward an experimentally determined conformation [26] and to promote conformational transitions by reducing dihedral and van der Waals energy terms involving atoms near a mutation site [67]. 

Umbrella sampling is aimed at overcoming the sampling problem by modifying the potential function so that different favorable states separated by energetic barriers are sufficiently sampled. An artificial umbrella potential is introduced ... 

The umbrella sampling technique28, 29 can be utilized to overcome these difficulties. An umbrella potential along the X coordinate can be expressed as... 

AB, = B, % = 1) - B, (A, =0) and Pum is the probability function of the hybrid potential with the umbrella potential. If (AB/ = ABj), the effect of the umbrella potential will be canceled completely. An iterative procedure is sometimes required to produce complete sampling of important configurations along the chemical coordinates. In such cases, WHAM,... 

In the iterative approach using a constant bias F, the free energy barrier is reduced each successive iteration and therefore produces complete sampling of important configurations along the coupling parameter (i.e., reaction coordinates). Furthermore, more complicated umbrella potentials (e.g. see Equation 21) may also be applied with the iterative procedure. 

The main difficulties of Umbrella Sampling are the choice of the biasing umbrella potentials and the verification of complete sampling for each window. 

Umbrella Sampling simulations were carried out over six windows for water (seven for DMF), corresponding to biasing umbrella potentials centered at different ri values of the reaction coordinate (rc)... 

With the equilibrated system, we began a series of canonical ensemble simulations by placing the umbrella potential center close to the intercalated... 

To ensure appropriate sampling for deriving the free energy surface W( ), simulations can be performed in which an umbrella potential is added to the Hamiltonian // that describes the system (see Figure 8). This is typically a harmonic potential of the form ... 

To compute free energy differences between configurations or potentials of mean force, umbrella potentials are used to sample phase space at increments of X (see Figure 8). Protocols such as multi-step thermodynamic perturbation (MSTP) or multi-configuration thermodynamic integration (MCTl), are used to compute free energies [86] ... 

Figure 8. Illustration of the use of the umbrella potential. The umbrella potential (dashed line) restrains the system to sample a defined region of the phase space (grey region). In the MCTI or MSTP protocol multiple simulations are performed with an umbrella potential that evolves with the coupling parameter X and allows a gradual sampling of the coordinate of interest.

Fig. 1.4. The potential of mean force (PMF) for the Tyr -isoalloxazine center-to-center distance, (a) Sampling histograms of the Tyr -isoalloxazine distance the solid line is the histogram with a restraint-free simulation. The dashed hues are histograms obtained with umbrella potentials, (b) The PMF generated with the data from (a) is shown as a solid line. Also shown is the experimental PMF open circles) from reference [6]...

The values of Eg (st), t = 1, - , n provides a first estimate of the free energy along the LFEP. This potential is used as a bias to perform a onedimensional umbrella sampling along the generalized reaction coordinate defined by the LFEP [5,7]. The forces on the atoms of position r due to the umbrella potential are... 

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