I got into protein modelling not long before AlphaFold2 first released. At that time some of the prevailing methods for protein structure prediction came from highly interpretable energy functionals that arose from a particularly beautiful intersection of statistical mechanics and biology. These “Potts” models are going to be the centre of a larger discussion in this blog on state-based discretisations of proteins, how they’ve shaped modern deep learning methods and whether there is still more to learn from them.
In the age of black box deep learning, does the Potts model still have a place?
The Potts/Ising Model
The Ising model is a well established popular theoretical physics model of ferromagnetism. Simply put, given a lattice of atoms each capable of adopting 1 of 2 spins (up and down) ferromagnetism arises when their spins align and their associated magnetic moments point in the same direction. The Ising model tries to parameterise the local and non-local relationships between atoms and their spin states such that we can learn the Hamiltonian of the system and its different configurations under the magnetic field. The Hamiltonian takes the following form for a system of N atoms
$$
E = -\sum_{i}^Nh_ix_i – \sum_{i<j}^N J_{ij}x_i x_j,
$$
where J is the “coupling energy” between any two atoms x_i and x_j, and h represents the magnetic field, or more appropriately for our purposes it can be framed as a single-site field dictating how an individual atom independently acts within the model. You might recognise the form this binary spin model takes as it arises naturally across the sciences including in Hopfield networks and graphical models.