Alchemical transformation

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. 

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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 ... 

The idea of a finite simulation model subsequently transferred into bulk solvent can be applied to a macromolecule, as shown in Figure 5a. The alchemical transformation is introduced with a molecular dynamics or Monte Carlo simulation for the macromolecule, which is solvated by a limited number of explicit water molecules and otherwise surrounded by vacuum. Then the finite model is transferred into a bulk solvent continuum... 

Hauck, Dennis William. Methods of alchemical transformation.. ... 

This guide to alchemical transformation presents dozens of practical methods to achieve physical, mental, and spiritual transformation. Included are workings with alchemical... 

The Book of Alchemy teaches its readers how to penetrate the obscure symbolic language of the alchemists. . . understand how alchemical transformation can initiate a profound change of consciousness, claimed by practitioners to bring eventual union with the Divine. . . practice traditional meditations and exercises. . . prepare herbal alchemical elixirs to benefit the body. . . and discover how the alchemists search for purity can become a twenty-first- century model for spiritual development"... 

Humburg, Burt. On the color changes in the "Great Work", or the alchemical transformation of matter, fhttp // www.alchemywebsite.com/humburg.html1. 

Osburn, Seth. Seven the seven stages of alchemical transformation.. CD Citrus Studios, 2002. 

Earth s spiritual energy with important revelations on how to access and benefit from that energy. The spiritual terrain of sacred sites gives us a galaxy on Earth to tap into and use for our own alchemical transformation" [Alchemy Journal]... 

Glass, J.M. Machiavelli s Prince alchemical transformation action archetype of regeneration. Polity 8, no. 4 (Summer 1976) 503-528. 

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...

In this section, we present a pseudocode for an FEP alchemical transformation based on the dual-topology paradigm. The steps followed in this algorithm, specifically (c)-(f), may be implemented independently of the core of the program that generates an ensemble of configurations at a given A state - either Monte Carlo or molecular dynamics. This is probably the simplest scheme, which may be improved in several ways, as will be discussed in Sect. 2.9. 

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. 

It is possible to treat the parameter A in the alchemical transformation as a dynamic variable using an extended ensemble [56]. For simplicity of implementation, it has been proposed to use two variables A0 and A i such that A +A = 1. The Hamiltonian function is then defined as [57, 58]... 

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. 

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...

Alchemical transformations have also been applied to the challenging case of G protein-coupled receptors (GPCRs), for which little structural information is available experimentally at the atomic level. Starting from a template of a seven-helix... 

Among the methods discussed in this book, FEP is the most commonly used to carry out alchemical transformations described in Sect. 2.8 of Chap. 2. Probability distribution and TI methods, in conjunction with MD, are favored if there is an order parameter in the system, defined as a dynamical variable. Among these methods, ABF, derived in Chap. 4, appears to be nearly optimal. Its accuracy, however, has not been tested critically for systems that relax slowly along the degrees of freedom perpendicular to the order parameter. Adaptive histogram approaches, primarily used in Monte Carlo simulations - e.g., multicanonical, WL and, in particular, the transition matrix method - yield superior results in applications to phase transitions,... 

Cellini reflects a belief common in the sixteenth century that metals in their natural state are watery, not solid. As Aristotle believed, pneuma is present in water, so all liquids have souls. Metals, in other words, are animated. When Cellini pours liquid bronze into the statue, it comes to life and simultaneously cures the sculptor of his own mortal illness. It represents the fulfillment of an impossible alchemical transformation. But how is a modernist to deal with such ideas ... 

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