My Session on the applications of Knot Theory and the Concepts of Tangles and Lasso in the world of bioinformatics and proteomics for the Bioinformatica Indica 2018 organised by Kerala University Department of Computational Biology and Bioinformatics.

1. Knot Knot Knot !
Applications of Knot Theory in Bioinformatics

3. Maths Meets Molecules !

4. -Don Knuth
“There are millions and millions of unsolved
problems. Biology is so digital and incredibly
complicated, but incredibly useful.”

5. -Victor Hugo
“Where the telescope ends, the microscope
begins. Which one of the two has the grander
view? ”

6. -Joel Cohen
“Mathematics if Biology’s Next microscope,
only better; Biology is Mathematics’ Next
Physics, Only Better.”

7. Introduction
Geneticists have discovered that
DNA can form knots and links that
can be described mathematically.
By understanding knot theory more
completely, scientists are becoming
more able to comprehend the
massive complexity involved in the
life and reproduction of the cell.
In recent times, developments in
polymer invariants for links and
knots have been used to describe
the structure of DNA and to
characterise the action of
recombinases.

8. What is a knot ?
What is a link ??
Knots are closed curves in 3D
Links are collections of non-intersecting knots

10. Knots are mathematically defined as
embeddings of a circle in three
dimensional Euclidean space

12. Two knots are equivalent if one can be
transferred to the other via a deformation
of R upon itself known as ambient
isotropy.

14. The discovery of Jones Polynomial by Vaughn Jones
in 1984 revealed deep connections between knot
theory and mathematical methods in statistical
mechanics and quantum field theory.

15. Knot Polynomials
Laurent Polynomials
Alexander Polynomials
Jones Polynomials

16. Hence Knot Theory deals
with closed structures

17. However, the vast majority of
biological or synthetic polymers
are open chains.

18. In this context, the definition of knot is
relaxed and transferred to open curves. A
chain is knotted if it does not disentangle
after being pulled from both the ends.

19. Nature tend to avoid knots. Knotted
protein backbone are rare and the physical
mechanism governing their formation is
largely unknown.

20. In the last 30 years, knot theory has also become a tool in
applied mathematics.
Chemists and biologists use knot theory to understand the
chirality of molecules and actions of enzymes on DNA. A
closely related theory of tangles has been used in studying
the action of certain enzymes on DNA.
Long strands of DNA floating in cells nucleus can easily
become tangled, just as a long extension cord does when
left in a heap. Knotted DNA makes it harder for a cell to
read genes. Recent research findings tells us that Brain
branches have knots and cuts.

21. Knot Theory
Fundamentals
The simplest knot of all is
the unknotted circle, called
an unknot or trivial knot and
denoted C.
The next simplest knot is
called a trefoil knot
In a projection of a knot into
a plane we call the places
where the knot crosses
itself in the graphs the
crossings.

23. Trefoli
Knot Fold

24. Knot Terminologies
The crossing number of a
knot K, denoted c(K) is the
smallest number of
crossings that occur in
any projection of the knot.
If a knot is nontrivial it has
more than one crossing in
a projection.
A figure eight knot has
four crossings.

25. Knot Terminologies
An orientation on a knot is
defined by choosing a direction
to travel around the knot.
Certain knots posses projects
in which crossings alternate
between under and overpasses
as one travels around the knot
in a fixed direction
We call this type of knot an
alternating knot. The trefoil
knot and figure eight knot are
alternating.

27. Knots and Reidmeister
He was the first person to prove
rigorously that knots exist that are
distinct from unknot
He showed that all knot formations
can be reduced to a sequence of
three types of moves
Twist move : Put in or take out a
twist in a knot
Poke move : Add or remove two
crossings
Slide move: Slide from one side
of crossing to the other side

28. Links and Knots
A link is the union of a finite number of disjoint
knots in a three dimensional space.

29. Types of Links
Trivial Link
Hopf link
Whitehead link
Borromean link

31. Most important rationale for including links
is that certain operations are invariant to
the class of links but not to the class of
knots.

32. Links Metrics
Linking Number
Twisting Number
Writhing Number

33. Tangle
A tangle in a knot or a link projection is a region in the projection
plane surrounded by a circle such that the knot or link crosses the
circle exactly four times.

34. Tangle Terminology
Simplest tangles are are the
infinity tangle and the 0-tangle
The family of tangles that can
be converted to the trivial
tangle by moving the end
points of the strings is the
family of rational tangles.
An algebraic tangle is any
tangle obtained by additions
and multiplications of rational
tangles.

35. Tangles and Mutations
Mutation can turn a knot into
another, however it cannot
turn a nontrivial knot into a
trivial knot.
The mutants and tangles will
be used to understand
knotting in DNA.
Tangles has been applied to
study protein - DNA binding

36. It is still unclear whether
knots are selected in
evolution for their utility
In one case, that of ubiquitin hydrolase, the existence of a five crossing knot is
speculated to serve as a protection against degradation by the proteasome as
ubiquitin hydrolase tries to rescue other proteins from degradation.

37. Knots and DNA
DNA packing can be visualised as two very long strands that have intertwined millions of
times, tied into knots and subjected to successive coiling. For replication or transcription to
take place, DNA must first unpack itself so that it can interact with enzymes. It will be easier
if DNA is neatly arranged rather than tangled up in knots.

40. DNA Structure -
Fundamentals
DNA is double stranded molecule composed of
two polarised strands which run in antiparallel
directions and wind around a central common
axis.
One is entwined about the other such that an
overall helical shape results in a plectonemic
helix. This structure is to be contrasted with a
paranemic helix in which a pair of coils lie side by
side without interwinding.

41. Forms of DNA - 1
Supercoiled (knotted) DNA
Double stranded ( linear ) DNA can
have tertiary or higher order
structure
Superhelixicity is referred to as
DNA’s tertiary structure, which is
essentially knotted.
Only topologically closed domains
can undergo supercoiling.
Eukaryotic DNAs in association
with nuclear proteins acquire
superhelical conformation in
chromosomes.

42. Forms of DNA - 2
Negative Supercoiling
Supercoils formed by a deficit in link,
result from under winding, unwinding
or subtractive twisting
Negative supercoiling facilitates DNA
strand separation during replication,
recombination, and transcription.
Positive Supercoiling
Formed by an increase in link result
from tighter winding or overwinding of
the DNA helix. Strand separation is
difficult in this case.

44. Forms of DNA - 3
Relaxed DNA
Circular DNA without
superhelical twist is
known as a relaxed
molecule.
DNA in its relaxed
ideal state usually
assumes the B
configuration.

47. Knots and Proteins

48. Knots are rare in proteins despite their
length. When knotted proteins do occur,
they have a significant effect on the
protein stability or folding.

49. As knots are rare in real proteins, knot finding
programs may be useful in protein structure
prediction methods to filter predicted models, where
knotted structure occurs more frequently.

50. Protein
Folding

51. Knotted proteins have received lots of
attention due to their interesting
topological novelty as well as its puzzling
folding mechanisms.

53. Understanding the difference between
knotted and unknotted protein structures
may offer insights into how proteins fold.

54. Knots and Enzymes

57. Shape Similarity
Proteins provide a rich domain
in which to test theories of
shape similarity
Proteins can match at different
scales and in different
arrangements
Sometimes the detection of
common local structure is
sufficient to infer global
alignment of two proteins; at
times it provides false
information

58. Knot theory and PPI
Knot theory has many
applications in molecular
biology
Proteins such as
recombinanases and
topoisomerases can knot
and link circular DNA
molecules

59. Knots and Polymers
The topological study of
knotted biopolymers is
an active interdisciplinary
field of research.
In polymers, knots
influence both material
properties and polymer
chain dynamics.

64. PyKnot
A plugin that work seamlessly within PyMOL molecular viewer and
gives quick results including the knots invariants, crossing numbers,
and simplified knot projections and backbones.

65. RKnots
A flexible R package providing tools for the detection and
characterisation of topological knots in biological polymers.

66. RNA != Knots
Unlike other biopolymers, RNA the long strand that
is the cousin of DNA tend not to form Knots.