It has been over 25 years since publication of the famous review "Ultrametricity for physicists" (Rammal et al. Rev. Mod. Phys. 1986) with a new philosophy in physics of spin-glasses based on hierarchy and ultrametricity. I intend to introduce the recent breakthrough of this philosophy in protein physics. Proteins often refer as complex functional systems like molecular machines, but it is still not very clear how a polymeric molecule turns into a molecular machine.

The mini-course, in its core, focuses on multi-scale description of protein dynamics by p-aqic equation of ultrametric diffusion. I am going to start from two classical experiments related to protein dynamics and protein functioning, namely, the spectral diffusion in deeply frozen proteins and the CO rebinding to myoglobin. From this discussion we will get to know the puzzles coming up from the experiments and will specify where and how the problem of multi-scale description of protein dynamics appears. Next lecture will be devoted to ultrametric representation of protein energy landscape, p-adic description of protein dynamics and its physical meaning. I will focus substantially on p-adic equation of ultrametric diffusion and its solutions. Finally, on the third lecture, I will return back to the experiments and will show how physical observables are constructed under p-adic modeling. We will see that extremely complex picture of interrelated movements of thousands atoms in a protein molecule is surprisingly well described by the p-adic equation of ultrametric diffusion.

Lectures:

• An Introduction to Proteins and Protein Dynamics
  Proteins: How do they look and what they do? Two classical experiments: CO binding to myoglobin and spectral diffusion in frozen proteins. Puzzles and problems.
• p-Adic description of multi-scale dynamics on complex energy landscapes
  From tree-like presentation of high-dimensional rugged energy landscapes and basin-to-basin kinetics to ultrametric space of states, ultrametric diffusion, p-adic master equation, and the solutions.
• Applications
  Spectral diffusion in frozen proteins and first passage time distribution for ultrametric random walk. CO-rebinding to myoglobin and p-adic equations of the "reaction-diffusion" type. Concluding remarks about non-Archimedean biophysics: chromatin architecture, molecular machines, and origin of life.