Topology, DNA, and Quantum Computing

1. Topology, DNA, and Quantum Computing Hsin-hao Su 2007.2.7

2. What? Topology? • I heard about geometry, numbers, trigonometry, statistics, and probability (Las Vegas, hehehe!). But, what did you say? Topology?

3. Topology • Topology is also called “rubber band geometry” or “clay geometry”. It is the study of geometric figures that are shrunk, stretched, twisted or somehow distorted. One of the main parts of topology is that of classifying surfaces. Picture courtesy to www.lz95.org/sl/finearts/talbert/clayjugs.htm

4. Topologist • It has been said that a topologist doesn’t know the difference between a donut and a mug! • A donut and coffee don’t just taste good. They are the same thing! =

5. Smooth Deformation • If an object can be smoothly deformed (think of morphing) to another one, topologically, they are the same. Picture courtesy to http://www-xray.ast.cam.ac.uk/~jgraham/hypo/h12/topology.html

6. Rules of “Morphing”

7. Homotopy Equivalent • Two spaces are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Picture courtesy to mathworld.wolfram.com/BoySurface.html

8. Which One Is Allowed?

9. Holes!!! • Without tearing, cutting and gluing, we cannot create a hole or eliminate a hole. • Topologists use the number of holes to distinguish geometric objects. Pictures courtesy to http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/SurAndHome.htm

10. Genus • The number of holes (or handles) is called “genus” in topology. • Or, how many cuts do you need to eliminate all the holes? Pictures courtesy to http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/SurAndHome.htm

11. Euler Characteristic Number • Another useful invariant in topology is “Euler Characteristic Number”. • The classical Euler Characteristic Number is defined by , where V is the number of vertices, E is the number of edges, and F is the number of faces of a polyhedron.

12. Euler Characteristic of a Cube • For a cube, we have 8 vertices, 12 edges, and 6 faces. Thus, we have that the Euler Characteristic number equals

13. Euler Formula • In general, for a polyhedron, the Euler Characteristic Number is always 2. This is called the Euler Formula. • Tetrahedron • Octahedron

14. Euler Characteristic of a Sphere • According to our previous discussion, a sphere is the same thing as a cube. We should expect the same Euler Characteristic, 2.

15. Euler Characteristic of a “Donut” • To find the Euler Characteristic of a torus (donut), We start from how to make one. • According to my secret undercover in the Dunkin Donuts, we roll out a dough and then form a ring to make a donut. Picture courtesy to http://motivate.maths.org/conferences/conf11/c11_project4.shtml

16. Euler Characteristic of a Torus • By using a rectangular paper to form a torus, we can count to get 1 vertex, 2 edges, and 1 face. Therefore, the Euler Characteristic is Picture courtesy to http://www.search.com/reference/Torus

17. Möbius Band • Möbius Band is a surface which has only one side and one edge. • If you are walking on a Möbius Band, you cannot tell that you are in the “inside” or “outside”.

18. How to Make a Möbius Band • Simply starts with a strip. Twist it and glue edges together. • If you put your pencil on the strip (don’t lift if off) and then draw a line, you will see that one line covers the whole strip! • If cut it through this line, you get a larger Möbius Band.

19. Euler Characteristic of a Möbius Band • From previous discussion, a Möbius Band has only 1 edge and 1 face. • Look at the paper. • We can see that there are two vertices. But, we create two more edges from these two vertices. Therefore, the Euler Characteristic of a Möbius Band is

20. Relation with Genus • Can we use Euler Characteristic to distinguish geometric objects? • Yes! Euler Characteristic is an invariant of topological objects. Also, we have a relation between Euler Characteristic and the genus. Picture courtesy to www.indiana.edu/~minimal/maze/handles.html

21. Euler Characteristic and Genus

22. It’s Fun … But, Why Topology? • Topology plays a central role in mathematics since it is the tool to study continuity. • It is looked like a mathematician’s personal habit. Any real-world application?

23. Applications • Biologists use it to understand how enzymes cut and recombine DNA. • Quantum Computer Scientists use it to construct a fault-tolerant bit. Picture courtesy to www.ericharshbarger.org/lego/mini_dna.html Picture courtesy to www.q-pharm.com/home/contents/backup/quantum_comp

24. DNA • The basic structure of duplex DNA consists of two molecular strands that are twisted together in a right-handed helix, while the two strands are joined together by bonds. Picture courtesy to whyfiles.org/075genome

25. DNA • DNA can exist in nature in linear form (i.e. as a long line segment) or in closed circular form (i.e. as a simple closed curve). Pictures courtesy to www.ucalgary.ca/~zleonenk

26. Supercoiling • DNA that is circular can be either supercoiled or relaxed.The supercoiled form is much more compact. Picture courtesy to www.cbs.dtu.dk/staff/dave/roanoke Picture courtesy to biology200.gsu.edu/houghton/4595%20

27. Picture courtesy to http://www.maths.uq.edu.au/~infinity/Infinity7/supercoiling.html

28. DNA Replication • Supercoiling allows for easy manipulation and so easy access to the information coded in the DNA. When a cell is copying a DNA strand, it will uncoil a strand, copy it and then recoil it. Picture courtesy to http://www.maich.gr/natural/staff/sotirios/topo.html

29. DNA Replication • DNA replication begins with a partial unwinding of the double helix at a part known as the replication fork. • An enzyme known as DNA helicase does this. Picture courtesy to http://www.virtualsciencefair.org/2004/mcgo4s0/public_html/t2/dna.html

30. Knots • Let us play a rubber band again. • A knot is a closed curve in three-dimensional space or paths that you can trace round and round with your finger. It is as though the two free ends of tangled rope have been spliced together. • Two knots are considered the same if one can be moved smoothly through space, without any cutting, so that it is identical to the second.

31. Picture courtesy to http://www.tiem.utk.edu/~gross/bioed/webmodules/DNAknot.html The first knot is merely a loop of string that has been twisted, an "unknot". It could easily be unknotted by pulling on the string to form a single loop. The 2nd knot, however, is clearly a knot. The only way to get rid of the knot would be to cut through it and retie the string. The 3rd knot is even more complicated.

32. Picture courtesy to http://www.c3.lanl.gov/mega-math/workbk/knot/knbkgd.html

33. Picture courtesy to http://www.c3.lanl.gov/mega-math/workbk/knot/knbkgd.html

34. Crossing Points • Each crossing point is assigned a + or - sign, depending on the orientation of the crossing point. • If the strand passing over a crossing point can be turned counter-clockwise less than 180 to match the direction of the strand underneath, then the sign is positive (+); if the strand on top must be rotated clockwise, it is negative (-). Picture courtesy to mathworld.wolfram.com/Writhe.html/

35. Writhe • The writhe of a knot is the sum of all signs of its crossing points.

36. Unknotting Number • The only way to untie a mathematical knot is to cut through the knot so that the strand that was lying on top is now underneath. • This is equivalent to changing the sign of a crossing point. • The number of times on must allow one strand of a knot to pass through another (in order to unknot it), is called the unknotting number. Picture courtesy to www.tricksecrets.com/images/333.jpg

37. Picture courtesy to http://www.tiem.utk.edu/~gross/bioed/webmodules/DNAknot.html

38. Electron Microscopes • Scientists use electron microscopes to take pictures of DNA. Underlying and overlying segments are distinguished by using a protein coating. The flattened DNA is then visualized as a knot. • The unknotting number and ideal crossing number can then be estimated. Picture courtesy to http://www.tiem.utk.edu/~gross/bioed/webmodules/DNAknot.html

39. Difficulties • It is possible to experimentally determine the outcome of an enzyme action. • But, there is no known method to actually observe the action of an enzyme. Picture courtesy to www.readingwithtlc.com/Pages/whorisk.htm

40. The Action of Enzymes on DNA • When an enzyme acts on a DNA, one possible result is that one (or a pair of) strand(s) of DNA will pass through the other strand. • The DNA become more or less supercoiled.

42. Topoisomerase • An enzyme called topoisomerase is used to unpack and pack in DNA by changing the number of twists in DNA. • Brown and Cozzarelli (1979) use topology to determine how the topoisomerase works to supercoil DNA.

44. Oh! Topology! • Topology gives cell biologists a quantitative and invariant way to measure properties of DNA. • Knot theory has helped to understand the mechanisms by which enzymes unpack DNA. • Measuring changes in crossing number helps to understand the termination of DNA replication and the role of enzymes in recombination.

45. Coffee or Tea Time • “A mathematician is a device for turning coffee into theorems“ -- Alfréd Rényi (Hungarian mathematician, 1921-1970) (It is often attributed to Paul Erdös)

46. Quantum Computing • Quantum computing is a new field in computer science which relies on quantum physics by taking advantage of certain quantum physics properties of atoms that allow them to work together as quantum bits, or qubits. • By interacting with each other, qubits can perform certain calculations exponentially faster than conventional computers. Picture courtesy to http://www.theteacherplace.com/gallery.php

47. Qubits • Quantum computer operates on information represented as qubits, or quantum bits. • A qubit, so-called superposition state, can be any proportion of 0 and 1. • We can think of the possible qubit states as points on a sphere. Picture courtesy to www.uni-ulm.de/qiv/forschung/qubit.html

48. Superposition • 3 bits have 8 combinations of values. Only one of those values can be stored in a digital 3-bit set • But in a 3-qubit set we can store all of them thanks to the superposition property of the qubits! Picture courtesy to www.uni-ulm.de/qiv/forschung/qubit.html

49. Anyon • Ordinarily, every particle in quantum theory is neatly classified as either a boson--a particle happy to fraternize with any number of identical particles in a single quantum state--or a fermion, which insists on sole occupancy of its state. • Almost 30 years ago researchers proposed a third category, "anyons," where a limited number of particles could inhabit a single state. • Frank Wilczek used the term anyons in 1982 to describe such particles, since they can have "any" phase when particles are interchanged. Hall of Quantum Effects Four voltage "gates" on this semiconductor surface created a central disk with "quasiparticles" having one-fifth of an electron's charge (red) surrounded by a ring of one-third charge quasiparticles (blue). Measurements revealed that the quasiparticles are neither fermions nor bosons. Courtesy to http://focus.aps.org/story/v16/st14

50. Decoherence • When a qubit interacts with the environment, it will decohere and fall into one of the states. This makes the extraction of the results calculated by an operation to a set of qubits really difficult. • Furthermore, the problem increases in large qubit systems and causes the potential computing ability of quantum computers to drastically fall.