Physicists at the University of Bristol have shown
that knots in proteins can be understood using 'virtual knots' - a
branch of knot theory previously considered as abstract and without
Earlier research into knotted proteins involved adding lines to close a
protein curve into a loop. As there is no obvious single way to do
this, researchers took averages over many different closure lines.
‘Knots in proteins can be understood using 'virtual knots' - a branch of knot theory previously considered as abstract and without application.’
Many of the processes essential to life involve proteins - long
molecules which 'fold' into three-dimensional shapes allowing them to
perform their biological role.
Consisting of strings of amino acids, a folded protein molecule
resembles a coiled, tangled piece of wire, which, as everyday experience
suggests, may be knotted.
The mathematical study of knots is called knot theory, a branch of
abstract mathematics which is related to other areas of maths such as
algebra. The knotted curves studied in knot theory have closed ends,
like a knot in a circle, but protein molecules do not.
Professor Mark Dennis, from the School of Physics, said: "Our
procedure, however, takes views of the protein curve from different
directions, that is, projections, which can be mathematically analysed
as virtual knots without adding extra lines. This captures the
essential ambiguity of where the ends of the protein curve are."
Viewing the protein curve in different directions results in
different projections, or 'shadows', of the curve. The virtual knotting
of each shadow can be identified mathematically from the sequence of
over and under crossings of the projection.
The various types of virtual or regular knot that occur in each
direction, not obvious without smoothing the projection, can be drawn on
a spherical map; the 'globe' of viewing directions is broken up into
'seas and islands' of different knot types. The three-dimensional
structure of the protein, essential to its function, can be better
understood from the different kinds of map that appear in this way.
When the protein knots are closed by extra lines, the seas and
islands are restricted to only a small number of 'classical' knot types - those of knotted circles. Since there are many more virtual knot
types than classical types (as they don't have to close), viewing the
knot 'virtually' offers a more subtle understanding about the protein
This work is part of the Scientific Properties of Complex Knots
(SPOCK) project, a collaboration between the University of Bristol and
The aim of the project is to create new computational tools and
mathematical techniques for the analysis, synthesis and exploitation of
knotted structures in a wide range of complex physical phenomena.