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Applications of Knot Theory in Bioinformatics - A Quick Reader


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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.

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Applications of Knot Theory in Bioinformatics - A Quick Reader

  1. 1. Knot Knot Knot ! Applications of Knot Theory in Bioinformatics
  2. 2. Maths Meets Molecules !
  3. 3. -Don Knuth “There are millions and millions of unsolved problems. Biology is so digital and incredibly complicated, but incredibly useful.”
  4. 4. -Victor Hugo “Where the telescope ends, the microscope begins. Which one of the two has the grander view? ”
  5. 5. -Joel Cohen “Mathematics if Biology’s Next microscope, only better; Biology is Mathematics’ Next Physics, Only Better.”
  6. 6. 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.
  7. 7. What is a knot ? What is a link ?? Knots are closed curves in 3D Links are collections of non-intersecting knots
  8. 8. Knots are mathematically defined as embeddings of a circle in three dimensional Euclidean space
  9. 9. Two knots are equivalent if one can be transferred to the other via a deformation of R upon itself known as ambient isotropy.
  10. 10. 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.
  11. 11. Knot Polynomials Laurent Polynomials Alexander Polynomials Jones Polynomials
  12. 12. Hence Knot Theory deals with closed structures
  13. 13. However, the vast majority of biological or synthetic polymers are open chains.
  14. 14. 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.
  15. 15. Nature tend to avoid knots. Knotted protein backbone are rare and the physical mechanism governing their formation is largely unknown.
  16. 16. 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.
  17. 17. 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.
  18. 18. Trefoli Knot Fold
  19. 19. 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.
  20. 20. 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.
  21. 21. 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
  22. 22. Links and Knots A link is the union of a finite number of disjoint knots in a three dimensional space.
  23. 23. Types of Links Trivial Link Hopf link Whitehead link Borromean link
  24. 24. Most important rationale for including links is that certain operations are invariant to the class of links but not to the class of knots.
  25. 25. Links Metrics Linking Number Twisting Number Writhing Number
  26. 26. 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.
  27. 27. 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.
  28. 28. 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
  29. 29. 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.
  30. 30. 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.
  31. 31. 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.
  32. 32. 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.
  33. 33. 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.
  34. 34. 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.
  35. 35. Knots and Proteins
  36. 36. Knots are rare in proteins despite their length. When knotted proteins do occur, they have a significant effect on the protein stability or folding.
  37. 37. 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.
  38. 38. Protein Folding
  39. 39. Knotted proteins have received lots of attention due to their interesting topological novelty as well as its puzzling folding mechanisms.
  40. 40. Understanding the difference between knotted and unknotted protein structures may offer insights into how proteins fold.
  41. 41. Knots and Enzymes
  42. 42. 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
  43. 43. 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
  44. 44. 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.
  45. 45. 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.
  46. 46. RKnots A flexible R package providing tools for the detection and characterisation of topological knots in biological polymers.
  47. 47. RNA != Knots Unlike other biopolymers, RNA the long strand that is the cousin of DNA tend not to form Knots.