Projecting proteins and random walks

: knotting in open curves via virtual knots

  • Keith Alexander

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


In this thesis we develop a new method of knot recognition for open curves
based on taking many projections and identifying them as virtual knots, an
extended class of knotted objects which exist ‘in-between’ classical knot types.
We call this method virtual closure. We explore how virtual closure differs
from a method we call sphere closure which involves joining the ends of the
curve to many far away points, finding that virtual closure is more sensitive to
knotting and provides more complex and detailed conformational information.
An important distinction we find is between curves which present a single
dominant knot type across closures, which we call strongly knotted, and more
ambiguous curves which are knotted but with many different knots depending
on the closure chosen, which we call weakly knotted.
We perform a knotting survey of all proteins in the Protein Data Bank using
virtual closure. Compared to previous sphere closure surveys, we find 25%
more knotted proteins. Of all the knotted proteins, 40% are found to be weakly
knotted under virtual closure, many more than under sphere closure, hinting
that the knotting in proteins is more ambiguous than was previously thought.
We then investigate the knotting of random walks, finding that weak knot-
ting is very rare in unconfined walks, but increasingly common in isotropically
confined walks both on and off-lattice. We determine that weak knotting is
essentially length independent, instead depending only on the degree of con-
finement - the ratio of average radius of gyration of unconfined walks to con-
fined walks of the same length. The greater the degree of confinement, the more
likely that knotting is weak. We reduce the number of confined dimensions,
moving from walks in the sphere, to the tube and then to slits, finding that
overall knotting and weak knotting become less common.
Date of Award6 Nov 2018
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorSimon Hanna (Supervisor) & Mark Dennis (Supervisor)

Cite this

Projecting proteins and random walks: knotting in open curves via virtual knots
Alexander, K. (Author). 6 Nov 2018

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)