Proteins are fundamental macro-molecules that are involved in a variety of vital functions in living organisms. They primarily consist of a linear amino-acid sequence, which allows the molecule to fold into a 3D structure and perform its function thanks to its chemical and physical properties. In this work, we are interested in understanding the sequence statistics and its evolution over different timescales. An interplay of mutations and selection shapes the amino-acid variety over the course of history. Understanding the stochastic dynamics of protein evolution is essential to the comprehension of the diversification of life and the emergence of new protein functions.