Free energy hypersurface

Polypeptide chains exist in an equilibrium between different conformations as a function of environment (solvent, other solutes, pH) and thermodynamic (temperature, pressure) conditions. If a polypeptide adopts a structurally ordered, stable conformation, one speaks of an equilibrium between a folded state, represented by the structured, densely populated conformer, and an unfolded state, represented by diverse, sparsely populated conformers. Although this equilibrium exists for polypeptide chains of any size, its thermodynamics and kinetics are typically different for oligopeptides and proteins. This can be broadly explained with reference to the different dimensionalities of the free-energy hypersurfaces of these two types of molecules. 

Figure 8.1 Simplified representation of slice of free-energy hypersurface for C10H22 is shown. Free energy vs. configuration (conformation) for two molecules, ensembles of which represent two isomeric compounds n-decane (A) and isodecane (B) are given.

It is now instructive to ask why the achiral calamitic SmC a (or SmC) is not antiferroelectric. Cladis and Brand propose a possible ferroelectric state of such a phase in which the tails on both sides of the core tilt in the same direction, with the cores along the layer normal. Empirically this type of conformational ferroelectric minimum on the free-energy hypersurface does not exist in known calamitic LCs. Another type of ferroelectric structure deriving from the SmCA is indicated in Figure 8.13. Suppose the calamitic molecules in the phase were able to bend in the middle to a collective free-energy minimum structure with C2v symmetry. In this ferroelectric state the polar axis is in the plane of the page. 

Figure 3.1. Free energy hypersurface G T, P,...) in the parameter hyperspace.

Since 0 is a function of q, Eq. (50) by itself does not exclude the possibility of the existence of more than one equilibrium value q. However, should more than one q exist, the Gibbs free energy hypersurface would necessarily include a spinodal region, and this is impossible for systems that are homogeneous over the whole composition space. Hence, in such systems the equilibrium composition is guaranteed to be unique. 

In the ideal case of s independent degrees of freedom, the free energy hypersurface, W, along the coordinates is the sum of each individual free energy surface, W, along the coordinates 0 (ZID biasing). Consequently the expression for the biasing potential will be ... 

The same concepts may be recast in a different form. We may define a solvation free energy hypersurface ... 

After the parenthesis on general aspects of reactions in solution, in this Section we shall go back to methods and evaluate the free energy hypersurface G(R) defined in eq.(7). Our analysis will consider each component of (j(R), eq.(7), separately, using the formalism of the PCM method developed in Pisa as reference. The combination of two or more terms in the same calculation will be examined at the appropriate places. 

Figure 1 The first-order distribution function of the Landau free energy hypersurface, a cluster cluster of size 14.5 A, showing the transformation...

Figure 4.8 Classical Marcus theory section across the reaction coordinate X through the free energy hypersurface of the reaction complex R and product complex P for an ET reaction, showing the activation barrier AG, the reorganisation energy A and the free energy of reaction AG°.

Finally, the choice of the spacer is crucial because minor structural variations may dramatically affect the issue of the thermodynamic assembly process with lanthanides, a situation encountered when the potential free energy hypersurface is rather flat. The bis-tridentate ligand L14, which is designed to produce triple-stranded helicates, indeed gives the... 

P, Cieplak, D. A, Pearlman, and P. A. Kollman,/. Chem, Phys, 101,627 (1994). Walking on the Free Energy Hypersurface of the 18-Crown-6 Ion System Using Free Energy Derivatives. 

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