Free energy alchemical

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). 

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

A very simple version of this approach was used in early applications. An alchemical charging calculation was done using a distance-based cutoff for electrostatic interactions, either with a finite or a periodic model. Then a cut-off correction equal to the Born free energy, Eq. (38), was added, with the spherical radius taken to be = R. This is a convenient but ill-defined approximation, because the system with a cutoff is not equivalent to a spherical charge of radius R. A more rigorous cutoff correction was derived recently that is applicable to sufficiently homogeneous systems [54] but appears to be impractical for macromolecules in solution. 

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]. 

In the earlier sections, we have developed the theoretical framework for the FEP approach. In this section, we outline some specific methodologies built upon this framework to calculate the free energy differences associated with the transformation of a chemical species into a different one. This computational process is often called alchemical transformation because, in a sense, this is a realization of the inaccessible dream of the proverbial alchemist - to transmute matter. Yet, unlike lead, which was supposed to turn into gold in the alchemist s furnace, the potential energy function is sufficiently malleable in the hands of the computational chemist that it can be gently altered to transform one chemical system into another, slightly modified one. 

Fig. 2.9. The thermodynamic cycle used for the determination of protein-ligand relative binding free energies. Instead of carrying the horizontal transformations one can mutate the ligand in the free state - i.e., the left, vertical alchemical transformation , and in the bound state -i.e., the right, vertical alchemical transformation. This yields the difference in the binding free energies. AA I, jI,I j T. binding A mutation mutation...

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...

From (2.70), it follows that the free energy cannot be divided simply into two terms, associated with the interactions of type a and type b. There are also coupling terms, which would vanish only if fluctuations in AUa and AUb were uncorrelated. One might expect that such a decoupling could be accomplished by carrying out the transformations that involve interactions of type a and type 6 separately. In Sect. 2,8.4, we have already discussed such a case for electrostatic and van der Waals interactions in the context of single-topology alchemical transformations. Even then, however, correlations between these two types of interactions are not... 

In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

So far we have discussed various techniques for computing the PMF. The other type of free energy calculation commonly performed is alchemical transformation where two different systems are compared. Such calculations have many applications such as Lennard-Jones fluid with and without dipoles for each particles, comparison of ethanol (CH3CH2OH) and ethane thiol (CH3CH2SH), replacing one amino acid by another in a protein, changing the formula for a compound in drug discovery, etc. 

It is often the case that alchemical transformations are used to compare the binding affinity of two ligands and Jz 2 to a receptor molecule R. For example Jz i and -S 2 might be two putative inhibitors of an enzyme R. If AA (respectively, AA2) is the free energy of binding (respectively, Jz 2) to R, we can define the relative binding affinity by AAA = AA2 — AA4. 

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. 

Lu, N.D., Woolf, T.B., Overlap perturbation methods for computing alchemical free energy changes variants, generalizations and evaluations, Mol. Phys. 2004, 102, 173-181... 

Fig. 12.1. Thermodynamic cycle for ligand binding. Solutes L and L in solution (below) and bound to the receptor P (above). Vertical legs correspond to the binding reactions. Horizontal legs correspond to the alchemical transformation of L into L . The binding free energy difference can be obtained from either route AAA = AA4 — A A3 = AAi — A An...

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... 

Blondel. A. 2004. Ensemble Variance in Free Energy Calculations by Thermodynamic Integration Theoiy. Optimal Alchemical Path, and Practical Solution. J. Compul. Chem., 25, 985. 

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... 

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