Restraining potentials

Fig. 9.24 A restraining potential that does not penalise struetures in which the distance lies between the leaver and upper distances di and and uses harmonie functions outside this range (left). The harmonic potentials may also he replaeed by linear restraints further from this region (right).

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... 

In many applications, a single biasing potential is not sufficient to cover the whole range of and simultaneously produce good sampling. Thus a set of restraining potentials, C/ ( ), are used to shift the local minima in the desired direction. In this windowing approach, the potential of mean force, ), in each window takes the form... 

In order to overcome the problems that can occur in sampling configurations of the unbound ligands when using a multiple topology model, we consider two types of restraining potentials. For simplicity, both restraining potentials disappear at the bound states... 

The summation is taken over the bound state (X2=l) of the A-dynamics trajectory, including the restraining potential. Unfortunately, with this biasing potential the effect of the restraint (/ , ) becomes too large to yield reasonable convergence as the number of the unbound ligands increases. 

Another type of restraining potential for ligand i (/ , ) is defined as a function of x, and A,. In this case, the restraining potential does not depend directly on the environment atom coordinates... 

Because / " = 0 when A] = 1, and none of the restraining potentials depend on the environment atoms, the partition function for the system when Af = 1 can be expressed as follows... 

Using this relationship, AAA can be written as two terms. The first term involves the probability that a ligand is in the dominant state (A2=i) during the A-dynamics simulation and in the presence of the restraining potential. The second term corresponds to the partition function of the restraining potential (the umbrella correction)... 

The second term in Equation 41 is constant and may be estimated using free energy simulation or semi-empirical methods. When the ligands are similar and the entropy terms associated with the restraining potential are expected to cancel, the second term can be approximated by an internal energy difference... 

This potential was developed to ensure that the molecules inside the sphere never escape and maintain a fully solvated system during molecular dynamics. Here, es, Rs, ew and Rw are the van der Waals constants for the solvent and the wall and rj is the distance between the molecule i and the center of the water sphere, Ro is the radius of the sphere. The quantities A, B and Rb are determined by imposing the condition that W and dW/dr, vanish at r, = Ro. The restraining potential W is set to zero for r, < R0. The van der Waals parameters Es, ew, Rs and Rw can also be specifically defined for different solvents. The constants Awaii and Cwan are computed using a well depth of es = ew = 0.1 kcal and the radius of Rs = Rw = 1.25 A. For the other set of simulations, especially for the hydride ion transfer, we applied periodic boundary conditions by using a spherical boundary shell of 10.0 A of TIP3P40 water to cover the edges of the protein. 

The introduction of the external potential Vex, in Equation 4 is designed to mimic the effect of the surrounding (implicit) bulk solvent on the system by restricting the movement of any explicit water molecules.49 Thus, Vex[ is interpreted as arising from the force exerted on the explicit atoms by the implicit surrounding bulk solvent. This restraining potential has the simple harmonic form,49... 

We found that smaller shells of dynamically-restrained water molecules yielded active sites that are in broad structural terms similar to those obtained using much larger solvent shells of unrestrained molecules, without the artifacts and distortions that were apparent when restraining potentials were not used.48 This trend suggests that shells of restrained water would be an efficient model for bulk solvent in both active-site MD simulations and unconstrained-protein dynamics.49 Moreover, we found that the value of... 

UCW = capped water, TW = tethered water (see text), k = force constant for restraining potential (kcal/mol/A2). b Radius (A) of solvation sphere. 1 Numbers of dynamical water molecules within solvation sphere. d Mean and standard error for the forward (i.e. 8-methyl-N5-deazapterin —> 8-methylpterin) and reverse mutation of the electrostatic force field Cutoff for protein-ligand and solvent-ligand interaction all other interactions are subject to a 9 A cutoff. 

These authors also reported theoretical calculations of this frequency-dependent rotational relaxation. The theory of Auer et al. [98] using the quadratic electric field map, originally developed for HOD/D2O, was extended to the H0D/H20 system [52]. As before [38], the orientation TCF was calculated for those molecules within specified narrow-frequency windows (those selected in the experiment) at t = 0. TCFs for selected frequency windows, up to 500 fs, are shown in Fig. 8. One sees that in all cases there is a very rapid decay, in well under 50 fs, followed by a pronounced oscillation. The period of this oscillation appears to be between about 50 and 80 fs, which corresponds most likely to underdamped librational motion [154]. Indeed, the period is clearly longer on the blue side, consistent with the idea of a weaker H bond and hence weaker restraining potential. At 100 fs the values of the TCFs show the same trend as in experiment, although the theoretical TCF loses... 

To address this problem in [36] we introduce, in the spirit of the extended Lagrangian approach [49,50,52], an auxiliary variable s coupled to the system by an harmonic restraining potential of the form k s — s x)). We also assign... 

Wang, J., Deng, Y., and Roux, B. (2006) Absolute binding free energy calculations using molecular dynamics simulations with restraining potentials. Biophys. J. 91, 2798-2814. 

This potential is shown schematically in Figure 9.24. dj and d are the lower and upper distances that are considered to be consistent with the experimental data, (dj +d )/2 is thus the assigned target distance obtained from a measurement of the NOESY intensity and the error associated with that measurement is (d — di)/2. A distance between d and du incurs no penalty. Outside this region the restraint is applied using two harmonic potentials. These restraining potentials may have different force constants and so be of different steepness. In some functional forms, the harmonic potential is eventually replaced by a linear function, as illustrated in Figure 9.24. 

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