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FIGURE 12.5 The generalized Born energy between two equal charges. When the two charges coincide, the GB energy matches with the sum of the Born energies of both charges. When the distance between the two charges is large, the GB energy behaves like Coulomb. The functional for the GB energy is designed so that it goes smoothly (continuously) between these two extremes.

set of decoys. Goldstein and collaborators (Goldstein et al. 1992) developed an analytical method based on spin glass theory for determining the parameters that maximize the stability of the native structure relative to an average alternative structure, which they refer to as the foldability of the protein sequence. Note that the concept of foldability is related to the concept of specificity: A protein sequence will fold into a given target structure if it is stable for that structure and specific to that structure, that is, incompatible with any competing conformations. To account for both effects, we need to ensure that the energy of the sequence in the target conformation remains lower than in other competing conformations, which requires that all competing folds are known. Alternatively, Goldstein and coworkers introduced an "average" competing fold, and defined the difference in energy for a given sequence between its target or native structure and this competing fold as its foldability (Goldstein et al. 1992). This approach has been further extended to include other definitions of foldability, depending on how the competing folds are defined (Sasai 1995; Mirny and Shakhnovich 1996; Thomas and Dill 1996). Alternatively, putative energy terms can be derived from amino-acid pairing frequencies observed in known protein structures. This approach was initially proposed by Tanaka and Scheraga (Tanaka and Scheraga 1976) and subsequently extended by Miyazawa and Jernigan to account for solvent effects (Miyazawa and Jernigan 1985). Sippl and coworkers (Hendlich et al. 1990; Sippl 1990) noted that the conformation of small protein fragments is often defined by their flanking residues. To explain this observation, they introduced a residue-based energy function that depends on the separation of residues in both 3D space and along the protein sequence. Using this energy function, they modeled why certain protein fragments with identical sequences adopt different conformations in different proteins. These potentials are referred to as either log-odd potentials, if only statistical information is considered, or potentials of mean force, if a physical model based on statistical mechanics is used.

The Potential of Mean Force

For a state variable X of a physical system in equilibrium, the probability that X takes the value x is given by the Boltzmann law:

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