Thermodynamic cycle calculation

Finally, when performing thermodynamic cycle calculations, the same ensemble type should be used for all parts of the cycle. 

It is important to note that when one makes thermodynamic cycle calculations which involve the gaseous lanthanide atoms the saw-tooth -like variation of the heat of sublimation observed in fig. 6 will also be found in the calculated thermodynamic heat or energy quantity, for example the dissociation energies of the gaseous molecules R2, RAu, RO, RCj, etc. (Cater 1978). 

Born-Haber cycle A thermodynamic cycle derived by application of Hess s law. Commonly used to calculate lattice energies of ionic solids and average bond energies of covalent compounds. E.g. NaCl ... 

Fig. 11.9 Thermodynamic cycle used to calculate absolute free energies [Jorgensen et al. 1988].

Figure 1 Thermodynamic cycles for solvation and binding, (a) Solutes S and S in the gas phase (g) and solution (w) and bound to the receptor R in solution, (b) Binding of S to the receptors R and R. The oblique arrows on the left remove S to the gas phase, then transfer it to its binding site on R. This pathway allows the calculation of absolute binding free energies.

In the present case, each endpoint involves—in addition to the fully interacting solute—an intact side chain fragment without any interactions with its environment. This fragment is equivalent to a molecule in the gas phase (acetamide or acetate) and contributes an additional term to the overall free energy that is easily calculated from ideal gas statistical mechanics [18]. This contribution is similar but not identical at the two endpoints. However, the corresponding contributions are the same for the transfonnation in solution and in complex with the protein therefore, they cancel exactly when the upper and lower legs of the thermodynamic cycle are subtracted (Fig. 3a). 

Figure 6 Thermodynamic cycle for multi-substate free energy calculation. System A has n substates system B has m. The free energy difference between A and B is related to the substate free energy differences through Eq. (41). A numerical example is shown in the graph (from Ref. 39), where A and B are two isomers of a surface loop of staphylococcal nuclease, related by cis-trans isomerization of proline 117. The cis trans free energy calculation took into account 20 substates for each isomer only the six or seven most stable are included in the plot.

Subsequently, in Chapter 5, we shall show how the cooling quantities may be determined we give even more practical cycle calculations, with these cooling quantities (ip) being determined practically rather than specified ab initio. But for the discussions in this chapter, in which we assess how important cooling is in modifying the overall thermodynamics of gas turbine cycle analysis, it is assumed that tp is known. 

To further understand the thermodynamic philosophy of the improvements on the EGT cycle we recall the cycle calculations of Chapter 3 for ordinary dry gas turbine cycles—including the simple cycle, the recuperated cycle and the intercooled and reheated cycles. 

Figure 16.2 An exainple of a thermodynamic cycle for calculating differences in solvation energies...

For what is probably the earliest microscopic calculations of thermodynamic cycles in proteins see Ref. 12, that reported a PDLD study of the pKtt s of some groups in lysozyme. The use of FEP approaches for studies of proteins is more recent and early studies of catalysis and binding were reported in Refs. 11, 12, and 13 of Chapter 4. 

This calculation demonstrates that a nonpolar solvent can accelerate S 2 reactions. However, this is not what we are asking the relevant quantity is the overall activation energy for the reaction in a nonpolar enzyme which is surrounded by water. Thus, as is indicated in the thermodynamic cycle of Fig. 9.3, we should include the energy of moving the ionized R-O- from water to the nonpolar active site (AAg j1). Thus the actual apparent change in activation barrier is... 

The data for A P make it possible to calculate using Eq. (1) and ag+, calculated according tcthe thermodynamic cycle, Eq. (9). 

Energy effects associated with the dissolution of a given substance (which in the following is distinguished with the index " ) can be determined experimentally. They depend on the system s initial and final state, but not on the path taken by the process. Hence, for calculations, the device of thermodynamic cycles is often used, where the true path of the process is replaced by another path (which may even be a path that actually cannot be realized) for which the energy effects of the individual intermediate steps can be determined. 

SCHEME 7.25 Determination of the AE for tautomerization by thermodynamic cycle and calculation (inset). 

In principle, one can calculate pKa by making use of the thermodynamic cycle as shown in Figure 10-1 and a relationship between the free energy of deprotonation in aqueous phase and pKa,... 

Figure 10-1. Thermodynamic cycle for the absolute pKa calculation of small molecules. Subscripts gas and aq denote gas- and aqueous-phase, respectively...

Figure 10-2. Thermodynamic cycle for the pKa calculation of proteins. Subscripts mod and prot refer to the model compound and protein, respectively...

Since electrostatic effects dominate the thermodynamic cycle as shown in Figure 10-2, major development efforts have focused on the calculation of electrostatic energy for transferring the neutral and charged forms of the ionizable group from water with dielectric constant of about 80 to the protein with a low dielectric constant (see later discussions). This led to the development of continuum based models, where water and protein are described as uniform dielectric media, and enter into the linearized Poisson-Boltzmann (PB) electrostatic equation,... 

An alternative approach to calculating the free energy of solvation is to carry out simulations corresponding to the two vertical arrows in the thermodynamic cycle in Fig. 2.6. The transformation to nothing should not be taken literally -this means that the perturbed Hamiltonian contains not only terms responsible for solute-solvent interactions - viz. for the right vertical arrow - but also all the terms that involve intramolecular interactions in the solute. If they vanish, the solvent is reduced to a collection of noninteracting atoms. In this sense, it disappears or is annihilated from both the solution and the gas phase. For this reason, the corresponding computational scheme is called double annihilation. Calculations of... 

Fig. 4.15. Thermodynamic cycle for the binding of and. A) to a receptor molecule It,. Calculating AA3 and AA4 is often easier than AAi and AA2...

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