Potential of mean constraint force

The steepest descent (4.48) requires calculation of dA/d . Laio and Parrinello chose to estimate this term by performing short constrained molecular dynamics simulations and computing the potential of mean constraint force (see Carter et al. [45]). 

Here Vij denotes the distance between atoms i and j and g(i) the type of the amino acid i. The Leonard-Jones parameters Vij,Rij for potential depths and equilibrium distance) depend on the type of the atom pair and were adjusted to satisfy constraints derived from as a set of 138 proteins of the PDB database [18, 17, 19]. The non-trivial electrostatic interactions in proteins are represented via group-specific dielectric constants ig(i),g(j) depending on the amino-acid to which atom i belongs). The partial charges qi and the dielectric constants were derived in a potential-of-mean-force approach [20]. Interactions with the solvent were first fit in a minimal solvent accessible surface model [21] parameterized by free energies per unit area (7j to reproduce the enthalpies of solvation of the Gly-X-Gly family of peptides [22]. Ai corresponds to the area of atom i that is in contact with a ficticious solvent. Hydrogen bonds are described via dipole-dipole interactions included in the electrostatic terms... 

Elber, R. (1990). Calculation of the Potential of Mean Force Using Molecular Dynamics with Linear Constraints An Application to a Conformational Transition in a Solvated Peptide. I. Chem. Phvs. 93(6) 4312-4321. 

Other methods arrive at the potential of mean force by calculating the derivative of the free energy with respect to the constrained generalized coordinate in a series of simulation runs. A mean force, (F) = —can be integrated numerically to yield the corresponding PMF. These simulations involve constraints, and an appropriate correction term must therefore be accounted for in the calculation of the PMF. For a system with no constraints and consisting of N particles with Cartesian coordinates r = (ri, T2,. .., rjv), the mean force can be written as ... 

As noted above, the density-of-states method described earlier can be extended to yield accurate estimates of the potential of mean force. The weight factors that dictate the walk in the space can be computed on the fly during a simulation in a self-consistent manner. The simulation can be performed without any constraints, which means that the resulting weights can be used directly as in (21) to give the potential of mean force. One can also accumulate the forces acting on the particles that define the reaction coordinate and then use (24) to get the PMF. The computed PMF is therefore available as a continuous function of... 

R. Elber, Calculation of the potential of mean force using molecular dynamics with linear constraints An application to a conformational transition in a solvated dipeptide. J. Chem. Phys. 93, 4312 (1990). 

Nevertheless, the ability to constrain dihedral angles can be useful in simulations in which one needs to assess the behavior of a molecule as a function of conformation. An important application of this type of constraint is the evaluation of potentials of mean force (PMF), which provide the work required to move the system along a predefined dihedral coordinate. The thermal probability is directly related to this free energy profile. PMF provide a means of estimating relative stabilities of conformers as well as the kinetics of conformational changes. 

The relationship in Eq. [28] can be used for a single solute pair if simulations are performed using the umbrella sampling technique.In this method, the solute-solute distance is constrained to lie within a small window, allowing for the efficient calculation of g r) over that restricted distance range. The effect of the constraint is subtracted from the calculated u> r) after the fact. Results from several overlapping windows may be combined to yield the full potential of mean force. 

According to MP2/6-31G(d,p) calculations, the He-B distance in HeBBHe is rather short at 1.270 A, and may be compared with the standard value for a B-H bond (1.21 A). Inspection of the diagonalized force-constant matrix showed one degenerate negative eigenvalue for the linear HeBBHe molecule at all levels of theory. Geometry optimization without linear constraints resulted in dissociation. This means that HeBBHe is not a minimum on the respective potential energy hypersurface [11]. 

To determine the free-energy profile (rather then the potential energy profile as above), one performs a (typically short) constant temperature (canonical ensemble) FPMD simulation of the system (while constraining the reaction coordinate to the prescribed value) and records the mean force of constraint along the chosen reaction coordinate. This process is then repeated for a number of points along the reaction coordinate connecting the reactant and... 

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