Umbrella Sampling Methods

In nearly all cases, however, the Monte Carlo estimate of an ensemble average such as (18) is far from a routine matter. The nature of the difficulty can be described more precisely by rewriting such averages as one-dimensional integrals, e.g.. 

With an appropriate choice of weighting function, will be measurable over a very much wider range of than would be /o in an unbiased experiment. The required values of /o can then be extracted over this same wider range by using Eq. (24), provided an accurate estimate of is simultaneously 

As well as the free energy, the average energy itself is readily calculated from /o. To obtain other averages, such as pair correlation functions, obviously requires more elaborate filing of data during the Markov chain, but this can be done if necessary. 

It is worth mentioning that calculations such as these can be extremely economical, since often only small numbers of particles are needed in the sample. That is because the N-dependences of the Monte Carlo properties of the system and its (rather similar) reference system will be very much the same. The differences between the systems, measured by the Monte Carlo experiments, will therefore be rather insensitive to the size of the system. That is especially true for high densities This conjecture is borne out in practice accurate energies and free energies of the Lennard-Jones system can be obtained with only N = 32 particles.  

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Umbrella sampling method, see Free energy perturbation method... 

Major DT, Gao J (2007) An integrated path integral and free-energy perturbation-umbrella sampling method for computing kinetic isotope effects of chemical reactions in solution and in enzymes. J Chem Theory Comput 3 949—960... 

With the characterized mechanism, the next key question is the origin of its catalytic power. A prerequisite for this investigation is to reliably compute free energy barriers for both enzyme and solution reactions. By employing on-the-fly Born-Oppenheimer molecular dynamics simulations with the ab initio QM/MM approach and the umbrella sampling method, we have determined free energy profiles for the methyl-transfer reaction catalyzed by the histone lysine methyltransferase SET7/9... 

Fig. 4.1. Stratification consists of splitting the interval of interest into subintervals, thereby reducing the free energy barriers inside each window. The umbrella sampling method can bias the sampling and attempt to make it more uniform...

Tx and Tx are the kinetic energies of the atomic coordinates and X variables, respectively. The As are treated as volumeless particles with mass mx. Since the X variables are associated with the chemical reaction coordinates , the A-dynamics method can utilize the power of specific biasing potentials in the umbrella sampling method to overcome sampling problems that require conventional FEP calculations to be performed in multiple steps. 

Figure 10. Classical adiabatic free energy curve (solid line) forthe Fe /Fe electron transfer at the water/Pt(lll) interface calculated using the Anderson-Newns Hamiltonian and the molecular dynamics umbrella sampling method. Also shown by the dashed line is the parabolic fit of the data. (Reprinted from Ref. 14.)...

Using the umbrella sampling method, Rose and Benjamin determined the free-energy profile for Na and CUover the range from bulk water to... 

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

The weighted histogram analysis method was presented by Kumar et al. [90] as an extension of the umbrella sampling method to compute multi-dimensional PMFs. A detailed description of the method can be found in the references [95,... 

Next, consider the umbrella sampling method. Here, rather than perform one simulation, a series of canonical distributions are sampled,... 

In simulations, an alternative to perturbation theory is umbrella sampling. It is used to connect the two configurations of interest (e.g., a protein plus a bound ligand in solution versus a protein in solution and a free ligand in solution see Fig. 10(b)) by an appropriate configurational coordinate. Since the calculations correspond exactly to those used with a reaction coordinate in activated dynamics (Chapt. IV.E) we do not repeat the description of the umbrella sampling method.,70a... 

The free energies along the z-direction were calculated using the umbrella sampling method [32]. The distance along z was divided into a sequence of windows approximately 5 A wide. Two consecutive windows overlapped by at least 1.5 A. Approximately, six windows were needed to obtain a complete free energy profile. As mentioned above, each molecular dynamics trajectory yielded results for two windows. Thus three trajectories were needed per free energy profile. For each window, a molecular dynamics... 

A method to deal with such problems is the umbrella sampling method. In this method, a modified potential V ) is devised, = E( )-H where U( ) is a bridge... 

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