1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Nov 13, 2013: DNA Topology II
Fall 2013 course offered through the
University of Iowa Division of Continuing Education
Isabel K. Darcy, Department of Mathematics
Applied Mathematical and Computational Sciences, University of Iowa

2. Biologically relevant topological structures of DNA
Biologically relevant topological structures of DNA. Depicted are schematics of the three topological forms of DNA that topoisomerases maintain and modulate.
Biologically relevant topological structures of DNA. Depicted are schematics of the three topological forms of DNA that topoisomerases maintain and modulate. For simplicity, each line represents a double-stranded DNA helix, as shown by the upper left inset. As indicated by the arrow sizes, most type-2 topoisomerases shift the DNA topology equilibrium toward relaxing, unknotting and decatenating. Bacterial DNA gyrase and archaeal reverse gyrase are unique enzymes that introduce supercoils into DNA. This figure is reproduced from (3).
Liu Z et al. Nucl. Acids Res. 2009;37:
© 2009 The Author(s)‏

3. Recall that linear DNA with sticky ends will circularize (if long enough) to form nicked circular DNA. A protein called ligase can seal these nicks creating closed circular DNA.

4. Note DNA in many in vitro (in the test tube) experiments is often about 4000 – 8000 bp, but one can also work with smaller mini-circular DNA. In the examples shown here, the DNA has 360 bp.
DNA prefers to have about 10 bp per turn (depends on salt conditions, often closer to 10.4 bp per turn, but for illustrative purposes we will assume salt conditions that give us an integer value of 10 bp per full 360 turn. Thus twist is approximately 360bp / (10 bp per turn) = 36. Linking number is also 36 and thus writhe = 0 since lk = tw + wr, 36 = The DNA is called relaxed when writhe is close to 0 and DNA has its preferred twist.
If a protein cuts the DNA and changed the linking number to 32, then the DNA supercoils (middle figure). In this example, twist is kept at 36 and thus write = -4, 32 = 36 – 4 In reality twist would also change, but not nearly as much as writhe. Think about a flat belt. It’s preferred twist is 0. If you add twist and close the belt, you get writhe, but also a little twist.
Note in the middle figure the DNA is underwound, lk = 32. This makes it easier for the DNA to unwind (right side) to allow replication or transcription

5. Pictures of DNA

6. DNA substrate = starting conformation of DNA before protein action
Usually unkotted, and
supercoiled
Even if DNA is linear, it is kept organized, attached to a scaffold, and thus one also sees supercoiling. The type of supercoilding shown is is called plectonemic
Meiotic double-strand breaks in yeast artificial chromosomes containing human DNA
Grzegorz Ira,  Ekaterina Svetlova, Jan Filipski
Nucl. Acids Res. (1998) 26 (10):

7. Structure of the nucleosome.
DNA is kept organized wrapped around proteins. Note the DNA crossings contributing to writhe. This kind of supercoiling is called solenoidal.
Structure of the nucleosome. In green, the histone tetramer (H3-H4)2, in blue and yellow, the two histone dimers (H2A-H2B). The figure has been obtained starting from the crystallographic structure [31], to which were added two short entering and exiting DNA segments (courtesy of Richard Lavery).
Barbi M et al. Interface Focus 2012;2:
©2012 by The Royal Society

8. One of the many possible models of chromatin fibre, in a more condensed (a) and a less condensed (b) configuration.
One of the many possible models of chromatin fibre, in a more condensed (a) and a less condensed (b) configuration. The regular spacing between nucleosomes allows for the formation of a compact structure, probably stabilized by stacking nucleosome–nucleosome interactions.
Barbi M et al. Interface Focus 2012;2:
©2012 by The Royal Society

9. Nicked circular DNA
computer simulation

10. supercoiled DNA
computer simulation

12. (J. Mann) http://www.sbs.utexas.edu/herrin/bio344/
Note links result after replication. A precatenante is shown on the right in a partially replicated molecule.
Postow L. et.al. PNAS;2001;98:

13. Topoisomerase II performing a crossing change on DNA:
Cellular roles of DNA topoisomerases: a molecular perspective, James C. Wang, Nature Reviews Molecular Cell Biology 3, (June 2002)

14. Proposed mechanisms for type I topoisomerases.
There are many different types of topoisomerases. Type 1 topos cut only one of the two backbones. They help keep DNA properly supercoiled.
Proposed mechanisms for type I topoisomerases. Shown is a segment of DNA in a (−) supercoiled molecule that is acted on by a type I topoisomerase (gray). (Upper) An enzyme-bridged mechanism for type IA topoisomerases. The free energy of (−) supercoiling promotes the melting of the DNA upon enzyme binding and the stabilization of a (−) crossing of single strands of DNA. After cleavage of one strand of DNA, the enzyme bridges the break by covalent attachment to one end and noncovalent binding to the other. Passage of the intact strand through the break inverts the sign of the crossing. The result is a ΔLk of 1 per round of catalysis. (Lower) A strand-rotation mechanism, as proposed for type IB topoisomerases. After binding to duplex DNA, the enzyme nicks one strand by addition across the phosphodiester bond and thereby becomes covalently attached to one end of the nick. The other end of the nick is not bound, resulting in rotation of the free end about the intact strand before rejoining of the DNA ends. The result is the relaxation of several supercoils and a nonunity ΔLk per round of catalysis.
Dekker N H et al. PNAS 2002;99:
©2002 by National Academy of Sciences

15. Topoisomerases are involved in
Replication
Transcription
Unknotting, unlinking, supercoiling.
Targets of many anti-cancer drugs.

16. Type II topoisomerases are proteins which cut one segment of DNA allowing a second DNA segment to pass through before resealing the break.
They can create DNA knots, but they are far more likely to unknot the DNA. They keep the DNA nice so the DNA will be well-organized and well-packed. When packing for a trip, it takes more room and is harder to work with a knotted extension cord.

17. a knot table

18. Knot table reproduced by type 1 topoisomerase acting on nicked ds DNA.
Duplex DNA knots produced by Escherichia coli topoisomerase I. Dean FB, Stasiak A, Koller T, Cozzarelli NR., J Biol Chem Apr 25;260(8):

19. Type 2 topo acting on supercoiled DNA
Type 2 topo acting on supercoiled DNA. note how the action of topo traps DNA supercoils. Normally topo unknots instead of creating knots, but at high concentrations topo can create knots.

20. http://www. sciencemag. org/content/277/5326/690. full. pdf, Science
The action of 4 different topos are shown acting on nicked DNA. Note that 3 of them (human
Experimentally, Type-II topoisomerases (with ATP hydrolysis) lower knotted and catenated DNA populations to steady-state levels well below that at topological equilibrium.
7-kb pAB4 DNA: knots: 90 times lower
catenanes: 16 times lower
10-kb P4 DNA: knots: 50 times lower

21. What products would you predict if topoisomerase acted exactly once on the following 5 crossing knot?

22. .

23. Lynn Zechiedrich and
Jennifer Mann,
unpublished result. .

24. control needed for publication

25. How can topoisomerase acting locally determine best action for global result, unknotting the DNA??

26. Model of type II topoisomerase action.
Model of type II topoisomerase action. The enzyme (green) bends a G segment of DNA (red) into a hairpin. The entrance gate for the T segment of DNA (yellow) is inside the hairpin. Thus, the T segment can pass through the G segment only from inside to outside the hairpin.
Vologodskii A V et al. PNAS 2001;98:
©2001 by National Academy of Sciences

27. J Mol Biol Jul 23;340(5):933-9.
DNA disentangling by type-2 topoisomerases.
Buck GR, Zechiedrich EL.

28. The hooked juxtaposition hypothesis.
The hooked juxtaposition hypothesis. The hypothesis put forth by Buck and Zechiedrich stipulates that type-2 topoisomerases unknot and decatenate by selective segment passages at hooked juxtapositions but not at free juxtapositions. Schematized here, as an example, is the hypothesized decatenating mechanism by type-2 topoisomerases of two daughter chromosomes, one red and one blue. Shown is the perspective of the small topoisomerase and the global linkage is hard to ascertain globally. Type-2 topoisomerase (schematically represented by a green circle) catalyzes segment passage specifically at a hooked juxtaposition (top row), which is more likely to occur when the two chromosomes are linked globally. The type-2 enzyme will not act at a free juxtaposition (bottom row), which is more likely to occur when two chromosomes are not linked globally (71). Although depicted here for decatenation, this model is the same for knot-generated juxtapositions as well.
Liu Z et al. Nucl. Acids Res. 2009;37:
© 2009 The Author(s)‏

29. The juxtaposition-centric computational approach.
The juxtaposition-centric computational approach. (a) The hooked, half-hooked and free juxtapositions in the simple cubic lattice (Z3) model. The schematics in (b and c) illustrate how conformational enumeration and sampling are conducted in the juxtaposition-centric approach. The geometry of a preformed juxtaposition (tube-like drawings) remains unchanged during a simulation, while the conformational possibilities of the rest of the chain(s) (dashed curves) are either enumerated exhaustively for short chains or sampled statistically using Monte Carlo techniques for longer chains. The connectivity of the dashed curves to the preformed juxtapositions in (b) are for the studies looking at two separate chains, which consider the decatenating potentials, whereas those in (c) are for one-chain studies for the corresponding unknotting potentials. In addition to the three juxtapositions shown here, the juxtaposition-centric approach has been applied to several thousand lattice juxtapositions (84,85).
Liu Z et al. Nucl. Acids Res. 2009;37:
© 2009 The Author(s)‏

31. Recombination: