Alchemical free energy simulations

Another variant that may mrn out to be the method of choice performs the alchemical free energy simulation with a spherical model surrounded by continuum solvent, neglecting portions of the macromolecule that lie outside the spherical region. The reaction field due to the outer continuum is easily included, because the model is spherical. Additional steps are used to change the dielectric constant of that portion of the macromolecule that lies in the outer region from its usual low value to the bulk solvent value (before the alchemical simulation) and back to its usual low value (after the alchemical simulation) the free energy for these steps can be obtained from continuum electrostatics [58]. 

Finally, an alchemical free energy simulation is needed to obtain the free energy difference between any one substate of system A and any one substate of system B, e.g., Ai- In practice, one chooses two substates that resemble each other as much as possible. In the alchemical simulation, it is necessary to restrain appropriate parts of the system to remain in the chosen substate. Thus, for the present hybrid Asp/Asn molecule, the Asp side chain should be confined to the Asp substate I and the Asn side chain confined to its substate I. Flat-bottomed dihedral restraints can achieve this very conveniently [38], in such a way that the most populated configurations (near the energy minimum) are hardly perturbed by the restraints. Note that if the substates AI and BI differ substantially, the transfomnation will be difficult to perform with a single-topology approach. 

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]. 

QM/MM Alchemical Free Energy Simulations Challenges and Recent Developments... 

Keywords alchemical free energy simulation combined quantum mechanical/ molecular mechanical potential generalized ensemble simulation conformational sampling long-range electrostatic interaction... 

Li, H.Z., Fajer, M., Yang, W. Simulated scaling method for localized enhanced sampling and simultaneous "alchemical" free energy simulations A general method for molecular mechanical, quantum mechanical, and quantum mechanical/molecular mechanical simulations. J. Chem. Phys. 2007,126, 024106. 

Seehger D, de Groot B (2010) Protein thermostability calculations using alchemical free energy simulations. Biophys J 98 2309-2316... 

Aleksandrov, A., Thompson, D., Simonson, T. (2010). Alchemical free energy simulations for biological complexes Powerful but temperamental. Journal of Molecular Recogtiition, 23,117. 

Here, 7 runs over all simulations and k, I run over all bins. These equations can be solved iteratively, assuming an initial set oi fj (e.g., fj =1), then calculating p°i from Eq. (34) and updating Ihe fj by Eq. (35), and so on, until thep°i no longer vary, i.e., the two equations are self-consistent. Erom the p°i = P(qt, sp and Eq. (27), one then obtains the free energy of each bin center (q, sp. Error estimates are also obtained [46]. The method can be applied to a one-dimensional reaction coordinate or generalized to more than two dimensions and to cases in which simulations are run at several different temperatures [46]. It also applies when the reaction coordinates are alchemical coupling coordinates (see below and Ref. 47). 

Fig. 2.12. Enthalpy, entropy, and free energy differences for the ethane —> ethane zero-sum alchemical transformation in water. The molecular dynamics simulations are similar to those described in Fig. (2.7). 120 windows (thin lines) and 32 windows (thick lines) of uneven widths were utilized to switch between the alternate topologies, with, respectively, 20 and lOOps of equilibration and 100 and 500 ps of data collection, making a total of 14.4 and 19.2 ns. The enthalpy (dashed lines) and entropy (dotted lines) difference amount to, respectively, —0.1 and +1.1 kcalmol-1, and —0.5 and +4.1 calmol-1 K For comparison purposes, the free energy difference is equal to, respectively, +0.02 and —0.07kcalmol I, significantly closer to the target value. Inset Convergence of the different thermodynamic quantities...

At the end of the chapter, techniques for alchemical transformations were presented. We showed that, in order to avoid rapid changes in free energy and improve the efficiency of the calculation, the parametrization of the Hamiltonian is critical and soft-core potentials should be used [see (4.50)]. A popular approach is the technique of A dynamics which leads to an improved sampling. In this approach A is a variable in the Hamiltonian system [see (4.51)]. Umbrella sampling, metadynamics or ABF can be used to reduce the cost of A dynamics simulations. 

Calculations of relative free energies of binding often involve the alteration of bond lengths in the course of an alchemical simulation. When the bond lengths are subject to constraints, a correction is needed for variation of the Jacobian factor in the expression for the free energy. Although a number of expressions for the correction formula have been described in the literature, the correct expressions are those presented by Boresch and Karplus.21... 

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