Are proteins ultrametric?

The question posed in the title of this article has been raised by Hans Frauenfelder over 20 years ago [1]. When studying the ligand-rebinding kinetics of myoglobin, Frauenfelder discovered that below the room temperature, at which the protein dynamics limit the rebinding rate, the reaction kinetics obeys a power law or a stretched exponent [2]. In order to explain this fact, he supposed that the protein energy landscape (EL) has a great number of local minima corresponding to conformational substates (CSs) with nearly the same energies. With respect to the transition rates between CSs, the local minima were assumed to be clustered into hierarchically embedded basins of minima: the smaller basins separated by smaller activation barriers are pooled into larger basins that are separated by higher barriers. In other words, Frauenfelder suggested that the protein CSs and the protein EL clearly do exhibit some taxonomic order. Since hierarchical taxonomy can be described using non-Archimedean (ultrametric) distances, i.e. distances satisfying the strong triangle inequality, an ultrametric space is introduced to describe the protein CSs. Hence, the protein dynamics are associated with an ultrametric random process. The term " protein ultrametricity " should be understood in the same sense. Frauenfelder's hypothesis, which has attracted a great deal of interest (see, for example [3,4]), is regarded as one of the most profound ideas put forward to explain the nature of protein attributes that has been proposed in the last decades. To date, however, no theoretical validation of protein ultrametricity has been found that would be accepted by the whole scientific community. In earlier theoretical works [5,6], some models were proposed to describe a random walk over an ultrametric space (ultrametric diffusion), but these models were confronted with difficulties in applications to the ligand-rebinding kinetics of myoglobin. More consistent ultrametric approaches were developed in [7,8], where p-adic numbers and p-adic pseudo-differential equations were used to describe ultrametric diffusion. It was shown in [7,8] that the specific features of the ligand