Free energy thermodynamic cycle

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.

Consider the enzyme-catalyzed and noncatalyzed transformation of the ground state substrate to its transition state structure. We can view this in terms of a thermodynamic cycle, as depicted in Figure 2.4. In the absence of enzyme, the substrate is transformed to its transition state with rate constant /cM..M and equilibrium dissociation constant Ks. Alternatively, the substrate can combine with enzyme to form the ES complex with dissociation constant Ks. The ES complex is then transformed into ESt with rate constant kt , and dissociation constant The thermodynamic cycle is completed by the branch in which the free transition state molecule, 5 binds to the enzyme to form ESX, with dissociation constant KTX. Because the overall free energy associated with transition from S to ES" is independent of the path used to reach the final state, it can be shown that KTX/KS is equal to k, Jkail (Wolfenden,... 

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,... 

We have seen several examples of a technique for separation of gas mixtures which, in contrast with most commercial processes, requires no physical transfer of solvent, handling of solids, or cycling of temperature or pressure. The energy requirements can also be far lower The thermodynamic minimum work of separation is, under isothermal conditions, the free energy difference between the process stream and byproduct, or permeate, stream. When this difference is due only to the partial pressure difference of component 1, it becomes ... 

Fig. 2.6. The thermodynamic cycle for estimating the hydration free energy, zl/Ihydration, of a small solute (the right side of the figure). One route is the direct evaluation of A/lhydrauon along the upper vertical arrow. The solute, originally placed in vacuum (a) is moved to the bulk water (b). Another route consists of annihilating, or creating, the solute both in vacuo and in the aqueous medium and corresponds to the vertical lines in the thermodynamic cycle. As suggested by the cycle, these two routes are formally equivalent, as A/lhydrnt,ion = A A0 — A A1...

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. 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. 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...

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