Alchemical transformation calculations

Figure 16.2 A thermodynamic cycle. If the path being used to calculate free energy moves horizontally employing methods like MM-PBSA, MM-GBSA or LIE, these calculations are referred to as endpoint calculations . Until recently, endpoint calculations were considered better suited for diverse ligands. If the path used to calculate free energy is vertical, methods like FEP, TI or NE are employed. These calculations are referred to as alchemical transformation calculations because ligand 1 is transformed through a series of steps into ligand 2.

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. 

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. 

Fig. 4.16. The A dynamics method for alchemical transformations was developed by Guo and Brooks [57] for rapid screening of binding affinities. In this approach the parameter A is a dynamic variable. Techniques like ABF or metadynamics [34] can be used to accelerate this type of calculation. A dynamics was used by Guo [57] to study the binding of benzamidine to trypsin. One simulation is sufficient to gather data on several benzamidine derivatives. Substitutions were made at the para position C5 (H, NH2, CH3 and Cl). The hydrogen atoms are not shown for clarity...

In this chapter, we have discussed various methods to calculate the PMF and alchemical transformation. 

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. 

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