Entanglement in biology; how nature controls the topology of proteins and DNA (13w5133)

Arriving in Banff, Alberta Sunday, November 17 and departing Friday November 22, 2013


Kenneth Millett (University of California, Santa Barbara)

(University of Saint Thomas)

(University of Saskatchewan)

(University of Lausanne)

(University of Warsaw)


This workshop will focus on the mathematics associated with a specific
array of cutting edge problems arising from molecular biology studies
of proteins, DNA, and other biopolymers that show promise for
immediate progress. Our proposed timing, Spring 2013, will serve as a
launching pad for the research of some carefully chosen graduate
students, post-docs, and new professors who will be invited to participate in the

Proteins with knots and slipknots in crystalized proteins have been
discovered over the last 10 years by several research groups and they
are gradually being recognized as significant structural
motifs. During this time, advances in understanding their (i)
complexity, (ii) topology, (iii) stability, and (iv) free energy
landscape (folding and unfolding pathways), have been made by
researchers working at the interface of mathematics, biology,
and physics. This is just the beginning of a new field. The most
complicated knotted protein yet discovered has six crossings and a
simple trefoil knot has been found in 11 different protein families.
However an effective nomenclature for knotted/slipknotted proteins has not
yet been agreed upon by the research community. Thus, defining
knotting in open chains and, in particular, in proteins, is an
important problem, one that is likely to be more applicable in natural systems than the
knotted loops studied in traditional knot theory.

The first experimental work of Jackson on the knotted proteins YibK, a
member of the family of the methyltransferases, showed that it is
extremely difficult to unknot knotted proteins. However folding is
very fast and efficient when the knot is already located in the
backbone. The group of Yeates engineered a protein containing a knot
that bas an almost-identical unknotted twin. They found that the
designed knotted protein folds 20 times slower than its unknotted
twin. Their results showed the direct influence of topological
constraints on the free energy landscape of knotted proteins. Although
there has been tremendous progress in understanding knotted protein
behavior, the folding pathway is still not understood experimentally.

These questions are ripe for significant progress through the applicaton of
mathematics and physics. For example, numerical simulations
demonstrated that a knotted protein can fold in realistic time scales
at a rate 20 times slower than an unknotted twin. The free energy
landscape also is composed of slipknots.

Additionally, theoretical mechanical manipulations have showed that
knots in protein chains behave in different ways than knots in DNA
or homopolymers. The tightening of knots in proteins was confirmed
recently by the experimental group of M. Rief however the tangling
is beyond experimental resolution.

These results open up many avenues of computational and
experimental research. Moreover, these novel numerical results suggest
a wealth of questions and conjectures that may be fruitfully addressed
by theoretical arguments from physics and mathematics.

This is not the first time that biology has received help from mathematics
and physics. Huge progress in the understanding of DNA behavior,
including the effects of (i) storage (in viral capsids, eukaryotic
nuclei, or bacterial cells), (ii) entanglement (knots and links),
(iii) replication, (iv) transcription into RNA, and (v) repair and
recombination (including site specific and general), and (vi)
relationship between the local geometry of chain juxtaposition and
global topology in any polymer, have been made as the result of close
collaborations between mathematics, biology, and physics.

The function of tangling and knotting in proteins is expected to
be related to the protein's physical properties, and this relationship
needs to be explained mathematically. This will open new subjects and increase
opportunities for multi-discipinary research between mathematicians and those working
at the interface of the biological sciences. Bringing a cadre of researchers
working at the frontier of polymer science to the Banff International
Research Station for Mathematical Innovation and Discovery provides
the opportunity to bridge these fields.

For the proposed 2013 workshop we plan to include some biologists and
experimentalists studying proteins with non-trivial topology as well
as mathematicians. We expect that this will be the first opportunity
for many of the invitees from these different disciplines to meet each
other. Thus the proposed workshop will not only lead to the
advancement of existing collaborations at the interface between
biology, mathematics, and physics but also will encourage the
development of new ones.

Solving the mystery of protein knotting is the main goal of the conference.
Additionally we plan to complement the focus of the conference
with DNA knotting problems, as currently
there are many biologists, physicists and mathemticians
working on knotted DNA, whose experience will be invaluable also in addressing
questions concerning knotted proteins.

The specific topics to be included are:

1. Defining knots and slipknots in proteins/open chains.
Proteins create open chains, which, from a strictly topological
perspective, are not knotted. Formally defining knotting in open
chains and, in particular efficiently detecting knotted regions
within a protein, is a complex task. This definition is further
complicated by the existence of slipknotted proteins, i.e. unknotted
proteins that contain a knotted subchain. The main goal in defining
knots in proteins is to capture the influence of these regions on protein function.
Knots and slipknots appear to arise in proteins less frequently than in
other systems, such as DNA or homopolymers. Explaining why they are so
rare is an important question.

2. The folding mechanism of knotted and slipknotted proteins

It was shown that protein models based on random contacts produce
knotted proteins with greater frequency than is seen in protein
structure databases. Knotted proteins are likely avoided during
evolution. Why some have remained is an evolutionary
curiosity. Topological constraints in such cases can lead to large
energetic or entropic barriers, which have not been characterized.
Recent experiments of T. Yeates provides an example of twin proteins
whose energy landscapes are virtually identical but whose kinetics are
vastly different. This finding certainly will inspire researchers to tyr to
understand both the kinetic and thermodynamic properties of knotted
proteins. A reduced description of the protein, in terms of a
continuum freely-jointed, wormlike chain model is a first step in
exploring how topological constraints affect the efficency of knotting
as the length and persistance length of the polymers increases. The
coarse-grained models have been extremely useful for understanding the
behavior of proteins with trivial topology and the next step is to
explore the efficiency of the protein knotting mechanism and the
stability using these models.

3. Stability and spatial structure in open chains

Recent results of Millett, Rawdon, Stasiak, based on polymers modeled
as freely fluctuating polygons, show how topology affects the average
sizes and shapes of polymers and how these effects evolve as the
length of the polymers increases. Recent experimental work of Yeates
and numerical simulations of Sulkowska (based on N-acetylornithine
transcarbamylase, protein without knot and human ormithine
transcarbamylase, protein with knot), showed larger mechanical
stability within proteins containing knots when compared to twin
unknotted proteins. This indicates that the existence of knotted
proteins could be related to their biological stability. The goal
will be to understand how the geometry of a configuration is related
to knotting and knotting stability, and whether there is any
geometrical spatial structure of chains that leads to entanglement.

4. Spatial structure and topology in open chains, interpretation of
single molecule experiments.

Works of Rief and Itaki have shown that knotted proteins can be tied
or untied experimentally by mechanical manipulation with AFM. These
experiments open many possibilities to understanding entanglement in
proteins. The spatial theoretical description is an essential step in
understanding the entanglement in proteins. In particular, parameters
such as the size and shape will help in interpreting experimental