Writhing across the protein universe

Abstract

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.