For example, could be the conditional probability of finding the atoms of a valine and a serine at a given distance from each other, giving rise to the free energy difference . The total free energy difference of a protein, , is then claimed to be the sum of all the pairwise free energies: where the sum runs over all amino acid pairs (with ) and is their corresponding distance. In many studies does not depend on the amino acid sequence. Conceptual issues Intuitively, it is clear that a low value for indicates that the set of distances in a structure is more likely in proteins than in the reference state. However, the physical meaning of these statistical PMFs has been widely disputed, since their introduction. The main issues are: The wrong interpretation of this "potential" as a true, physically valid potential of mean force; The nature of the so-called reference state and its optimal formulation; The validity of generalizations beyond pairwise distances. Controversial analogy In response to the issue regarding the physical validity, the first justification of statistical PMFs was attempted by Sippl. It was based on an analogy with the statistical physics of liquids. For liquids, the potential of mean force is related to the radial distribution function , which is given by: where and are the respective probabilities of finding two particles at a distance from each other in the liquid and in the reference state. For liquids, the reference state is clearly defined; it corresponds to the ideal gas, consisting of non-interacting particles. The two-particle potential of mean force is related to by: According to the reversible work theorem, the two-particle potential of mean force is the reversible work required to bring two particles in the liquid from infinite separation to a distance from each other. Sippl justified the use of statistical PMFs—a few years after he introduced them for use in protein structure prediction—by appealing to the analogy with the reversible work theorem for liquids. For liquids, can be experimentally measured using small angle X-ray scattering; for proteins, is obtained from the set of known protein structures, as explained in the previous section. However, as Ben-Naim wrote in a publication on the subject: [...] the quantities, referred to as "statistical potentials," "structure based potentials," or "pair potentials of mean force", as derived from the protein data bank (PDB), are neither "potentials" nor "potentials of mean