$p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

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Vladik Avetisov

Ultrametric paradigm in biophysicss


In the early 1980s, at the beginning of the "spin-glass epoch", it was realized that the condensed systems with a large number of quenched "internal conflicts" (frustrations) can equilibrate via the tree-like splitting of the phase space into hierarchically nested domains [1]. When the temperature decreases, the most distant groups of spin-glass states are first factorized. Then, each of these groups splits into a number of subgroups, etc. The scales of hierarchically nested groups satisfy the strong triangle inequality, i.e., the equilibrated states obey ultrametric relations. The physical ground for the emergence of ultrametricity in spin-glass systems is quite general. In a complex system with an extremely large number of possible metastable states, the phase trajectories cover a tiny part of phase space without returning to already visited regions. For this reason, the factorization of spin-glass phases is similar to the random branching in a high-dimensional (infinitely dimensional, in the thermodynamic limit) space. The spin-glass phases are specified only by branching points because if two branches are diverged in high-dimensional spaces they never intersect again.
Shortly after, similar ideas were proposed to present a hug number of conformational states of a protein molecule [2]. Proteins are also highly frustrated condensed polymer systems characterized by extremely rugged energy landscapes of high dimensionality. Protein ultrametricity implies a tree-like representation of the protein energy landscape by means of ranging the local minima into hierarchically embedded basins of minima. Ultrametricity emerges here from the conjecture that the equilibration time within the basin is significantly smaller than the time necessary to escape the basin. As a result, the transitions between local minima obey the strong triangle inequality and protein dynamics is modeled via a jump-like random process that propagates in an ultrametric space.
It is interesting that approximation of the protein energy landscapes by self-similarly branching trees has been shown to be very fruitful for describing actual protein characteristics justified experimentally on an extremely wide temperature range from room temperatures up to the deeply frozen states [3]. With respect to ultrametricity and the self-similarity of the protein energy landscapes, it is important to keep in mind that proteins are functional systems. In actuality, proteins precisely operate at the atomic level using sparse-event statistics, which is exactly the bridge between non-local protein dynamics controlled by rugged high-dimensional energy landscape and low-dimensional directed movements of particular atomic fragments in a protein molecule.
In the context of high dimensionality, randomness, and sparse-event statistics, ultrametricity emerges in diverse areas (for review, see, for example, [4]). Concerning the latest manifestations of the ultrametic paradigm in biophysics, I intend to present new intriguing type of objects called crumpled (or fractal) polymeric globules. As it is suggested in [5], such objects closely relate to molecular machines.
1. G. Parisi, J. Phys. A 13, L115 (1980); J. Phys. A 13, 1101 (1980); M. MĀ“ezard, G. Parisi, and M. Virasoro, Spin glass theory and beyond (World Scientific, Singapore, 1987).
2. H. Frauenfelder, in Protein Structure: Molecular and Electronic Reactivity, Eds. by R. Austin et al. (New York: Springer, 1987), pp. 245; Nat. Struct. Biol. 2, 821 (1995)
3. V. A. Avetisov, A. Kh. Bikulov, and A. P. Zubarev, Proc. Steklov Inst. Math. 285, 3 (2014)
4. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I. V. Volovich, p-Adic Numbers, Ultrametric Analysis, and Applications. 1, 1 (2009).
5. V. A. Avetisov, V. A. Ivanov, D. A. Meshkov, S. K. Nechaev, Biophysical Journal, 107, p.2361(2014)