Writhing across the protein universe

The function of a protein is primarily determined by its specific 3D structure, which is itself informed by the sequence of amino acids that make up the protein. Thus, being able to predict the final 3D shape of a protein from its sequence is of vital importance to researchers. This thesis discusses methods for studying protein structure on various scales, motivated in large part by the development of Carbonara; a software for rapid refinement of protein structure based on experimental solution scattering data. This software fills the gap where machine learning methods such as AlphaFold are unable to produce predictions which accurately capture a proteins structure and dynamics in near native conditions in solution. The local geometry of a protein is well understood to be tightly constrained by its chemistry, we provide constraints on the super secondary and tertiary scale. To achieve this, we present a novel method for smoothing the protein’s backbone curve which produces a minimal representation of the underlying entanglement of its secondary structure elements. By studying the distribution of writhe for these smoothed backbone curves we find clear limits on their entanglement. We show that a large scale helical geometry is responsible for proteins which have maximal entanglement relative to this bound. We show that helical geometries are also dominant as a super secondary motif within proteins, linked to their structural and thermal stability. We show that there is a clear lower bound on the expected amount of absolute entanglement of the backbone as a function of its secondary structure. This insight was key to the development of Carbonara, with this lower bound acting as a penalty to produce biologically plausible predictions. This is a vital step in Carbonara’s pipeline, allowing the coarse grained model to be safely passed into all-atomistic molecular dynamics simulations. We present the framework for a complementary model to Carbonara which uses gradient descent to optimise the backbone curve model.