Free-energy Cycles

It is instructive to draw up a free-energy cycle for the cell reaction (3) so as to illustrate the dominant energy terms in the single electrode reaction (1) ... 

Changes in society have always been of creative interest for artists. The Russian sculptor and bionics specialist Vadim Kosmatschof creates visionary works that deal with one of the most topical subjects of our time ecological balance. For many years now, the artist has devoted himself to the visualization of an emission-free energy cycle, raising public awareness of ecological contexts with his kinetic sculptures in the constructivist tradition. 

Figure 11.9 Free energy cycle for computation of p/f, values (where n is an integer). This cycle is...

Figure 12.3 The vertical sides of this free-energy cycle correspond to free energies of aqueous solvation, while the horizontal sides correspond to chemical mutations that are not physically realistic but are accessible by FEP. The difference between the two vertical quantities must be equal to the difference between the two horizontal quantities. While the former difference is easier to measure, the latter is easier to compute...

A free-energy cycle finding particularly widespread use is one for evaluating differences in interactions between enzymes (or other molecular hosts) and alternative molecules in tlieir active sites. By mutating one substrate into another, both in the presence of the enzyme and isolated in solution, differences in free energies of binding may be determined (Figure 12.4). An example is provided in Section 12.6 as a case study. 

Figure 12.4 Differential binding free-energy cycle. The difference in binding free energies for two different substrates, S and S, is equal to the difference in mutation free energies for changing S into S in solution, and E S into E S in solution. The leftmost vertical free-energy change is zero, since the free enzyme is a constant independent of substrate...

Figure 12.6 Free-energy cycle associated with the binding of biotin and fluorobiotin analogs to avidin. What issues arise in choosing a force field for explicit simulation of these systems What methods are better suited to computing the vertical legs of the cycle and what methods the horizontal ones ...

The free energy cycle used in this approach contains several intermediates (see Figure 2). State 1 is the solvated QM representation of atom A, state 2 is a solvated... 

Scheme 1 Free-energy cycle diagram for mutation of two different molecules (A and B) in two different solvent mediums.

Figure 5 Free energy cycle for the conversion of [d(G3T4G3) ]2 from the Na to the ifr form in aqueous solution. The free energy difference between the di-Na quadruplex and the di-K quadruplex of—1.7 kcal moF was determined under conditions of equilibrium cation exchange by NMR spectroscopy. The relatively large difference between the free energy of dehydration for two Na ions that for two Kf ions, with respect to the free energy difference between the Na and Ffr forms of [d(G3T4G3)]2, revealed that Na ions are actually more favorably coordinated by G-quartets than ions. However, is selectively bound over Na because of its less unfavorable free energy of dehydration (Reproduced from ref. 36 copyright American Chemical Society)...

Table 2 contrasts the free energy cycle for association of a proton with NH3 (a very strong gas phase electrostatic dominated interaction) with that of two methane molecules (a very weak gas phase association). As one can see, the aqueous solution free energies are very different from those in the gas phase. 

FIGURE 9.2 The free energy cycle [75,76] used to calculate the QM/MM free energy difference between systems A and B, AGq / A B). The free energy difference between A and B is... 

Fig. 8.1 Free-energy cycle for the redox reaction (M M+ + e ), where M(g) denotes molecule M in gas phase, M(s) denotes the solvated molecule, and IP denotes ionization potential...

Fig. 9.2 Ionization potential (IP) and thermodynamic (TD) free energy cycle for the oxidation of solvent S...

In two papers Shao et al. predicted the E ox of sulfones and functionalized sulfones [60, 61]. Influenced by [28], the solvent E ox was taken as the difference of two free energy cycles - one for solvent oxidation and one for Li" reduction - and in part invoking tabulated data as input for the reference cycle [60]. However, new to the approach was the calculation of AGsoiv(Li ) at the experimentally determined solvent dielectric constants, which introduced a solvent dependence in the floating, rather than fixed, reference potential. The reference potentials depended on the specific computational approach, but were approximately 2.5-3.0 eV. The reference potential difference between the low (e=8) and high (e=95) dielectric solvents was 0.5 eV [60]. Thus, both the absolute numbers and the variation with solvent were large compared to the fixed LE/Li reference of 1.4-1.5 eV used by other authors [16, 57]. 

In certain contexts, the calculation of free energy differences is difficult to access computationally. Examples would include problems in which a large amount of solvent would need to be displaced in a chemical association, one in which a large conformational change occurs, or one in which a complex chemical intermediate is present. In these cases, one of which is described below in the context of acid-base chemistry, the method can often be readily combined with a hypothetical free energy cycle[46], with individual legs that are each readily evaluated computationally. 

Figure 23.8 A free energy cycle for calculating the standard Gibbs free energy of reaction. The dashed line shows the indirect (two-step) route.

Draw a Gibbs free energy cycle to calculate the standard Gibbs free energy change of decomposition of sodium hydrogencarbonate. 

The Gibbs free energy cycle is shown in Figure 23.9. 

Figure 23.9 The free energy cycle for the decomposition of sodium hydrogencarbonate. The dashed line shows the two-step route.

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