the topology of dna-protein...

The Topology of DNA-Protein Interactions

Karin Valencia

Imperial College London

Applications of Topology Day, February 2011

Karin Valencia (Imperial College London) The Topology of DNA-Protein Interactions 1 / 19

The Biology of DNA: Primary, Secondary and Tertiary structures of DNA, Part I

Left: The primary structure of DNA. Middle: the secondary structure of DNA, the double helix. Right: A relaxed DNA ring.

Interplay between geometry and topology: Supercoiling

Tw + Wr︸︷︷︸

Geometric quantities; change under continuous deformations of the curve

= Lk︸︷︷︸

topological quantity; invariant under continuous deformations of the curve


The Biology of DNA: Tertiary Structure, Part II

DefinitionK ⊂ S3 is a knot if it is homeomorphic to S1 and K is the unknot if it is ambient isotopic to S1.

L ⊂ S3 is an n-component link if it is homeomorphic to ∪nS1 .

The central axis of DNA can be knotted or linked:

e.g.

replication (DNA copying)

recombination (DNA rearranging)


Biological methods for measuring DNA knotting and linking


Topology plays an important part in trying to understand enzymes that change

the topology of DNA

Many cellular processes, including recombination, transposition and replication of DNA, can transform supercoiled circular DNAinto a knot or a link. Understanding precisely which knots and links arise can help understand the details of the process.

Topological techniques for analysing enzyme mechanisms and product knots and catenanes:

The linking number, used to study the structure of negatively supercoiled circular DNA in solution.

Schuberts classification of 4-plats, used in to study interwinding in catenated and knotted DNA.

The Jones polynomial, a knot invariant that assigns a unique polynomial to each knot, is used in to work out arelationship between the polynomials associated with the substrate molecule and the product molecules obtained by sitespecific recombination.

We now look at how topology can help us understand the action of two important cellular enzymes on DNA: site-specific

recombinases and topoisomerases.


Site-specific recombination: Changing the topology of DNA

DefinitionSite-specific recombination is a cellular reaction responsible for the reshuffling of the genetic code. It is mediated by proteins thatrecognize two particular short (8-12 bp in length) DNA sequences.

It is important because:

It has a key role in a variety of biological processes. It can delete, invert and insert DNA segments. This corresponds to avariety of physiological processes, including crucial steps in viral infection.

They are rapidly becoming of pharmaceutical and agricultural interest.

Changing the topology of DNA is only a byproduct of the reaction. We use this as an advantage that helps us understand the

mechanisms of this reaction.


Site-specific recombination: Rough MechanismMinimally it needs:

one or two close circular DNA molecules, the substratefour proteins responsible for recognizing specific sites on the DNA and mediating the reaction

Mechanism1 Recognize and bind two copies of a specific DNA sequence2 Bring sites together to forming a synaptic complex3 Cut DNA at sites4 Exchange strands, possibly multiple times (processive recombination with serine recombinases)5 Reseal and release the DNA


Site-specific recombination: Two families


Model I: Predicting DNA knot and link products of site-specific recombinationgeneralization of a model by Dorothy Buck and Erica Flapan. (Joint work with Dorothy Buck.)

Input: Topology of the substrate and information about how the enzyme works.Goal: Given this information, use topological arguments to predict and characterize all possible products of the reaction.

Substrate: a twist knot

DefinitionIn the image bellow, a twisted double of an unknot K1 with v twists is called a twist knot with q twists K2 .

Twist knots are a family of knots that are ubiquitous in DNA.


Model I: Predicting DNA knot and link products

Theorem (Valencia)If recombination satisfies certain biologically reasonable assumptions, then products of site-specific recombination on a twist knotsubstrate must be in one of these families:

Denoted G1, G2, F (p, q, r, s, t, u), where the third family is a family of Montesinos knots and links ( ttu+1 , r

rs+1p

pq+1 )


Model I: Predicting DNA knot and link products

More precisely:


Model I: Predicting DNA knot and link productsAssumption I:

Biological Assumption 1The synaptic complex is a productive synapse, and there is a projection of the crossover sites which has at most one crossingbetween the sites and no crossings within a single site.

DefinitionA productive synapse is the recombinase complex B that meets the substrate J in precisely the two crossover sites.

Mathematical Assumption 1B ∩ J is precisely two arcs and there is a projection of B ∩ J with ≤ 1 crossings between the arcs sites and no crossings within asingle arc.


Model I: General idea of proof


Model I: Predicting DNA knot and link productsAn application of our model: We can distinguish between products of processive recombination and distributive recombination


Model II: Get information about the mechanism of site-specific recombinationThe Tangle Model, due to Claus Ernst and De Witt Sumners.

DefinitionA tangle is a pair (B3, t), B3 is a 3-ball with four distinguished boundary points and t is a pair of properly embedded arcs withendpoints the four points on ∂B3 .

the DNA-recombinase complex is regarded as the numerator closure of the sum of two tangles: N(O + P).

the enzyme action is viewed as tangle surgery: delete tangle P and replace it by tangle R.the knot types of the substrate and the product yield equations in the recombination variables O, T and R:

N(O + P) = substrate 01

N(O + R) = product b(p, q)

the aim of the tangle model is to find all possibilities of tangles for O, P and R that satisfy these equations.the key to solving the tangle equations: interplay between the double branched covers (dbc).


Model II: The tangle modelExample:

Products of Hin-mediated recombination on a mutant circular DNA substrate: 01 → 31 → 51 → a 7-noded knot and thecomposite knot 31]31.

Proposed tangle equations for processive Hin recombination on mutated sites (for P and R rational tangles.):

N(O + P) = b(1, 1) (1) N(O + R) = b(3, 1) (2)

N(O + R + R) = b(7, 3) (3) N(O + R + R + R) = K7-noded knot (4)

N(Q + P) = b(3, 1) (5) N(Q + R) = b(3, 1)](3, 1) (6)

Solutions:

Theorem (Mauricio)Given Equations (1) − (4), then either O = t 1

2and R = t−2 or O = t −1

2and R = t2 and under the hypotheses of the tangle

model, N(O + R + R + R) = b(11; 5) = 72.

Given that P = t0, R = t2 , the following are the complete tangle solutions for Q to equations (5), (6):

Rational tangles: There are no rational tangle solutions.

Prime tangles: There are no prime tangle solutions.

Locally knotted solutions: The only 4 possible solutions for Q illustrated below.


Regulating DNA topology: Knotting and linking and Topoisomerase IIType-2 topoisomerase (Topo II) is an enzyme whose primary function is to change DNA knot and link type.

This action appears as likely to increase as to decrease entanglements.

But they actually act on a manner that preferentially unknots and unlinks DNA!

G. Buck and L. Zechiedrich claim that type-2 topoisomerases have evolved to act at “hooked” juxtapositions of strands.

Mathematically, this is a question of “knot adjacency”.

DefinitionThe strand passage distance between any two links or knots K , L, denoted d(K , L) is the minimal number of crossing changes(+1 ↔ −1) needed to convert K to L, taken over all projections.

Julian Gibbons is currently looking at unknotting number 1 three-tangle pretzel knots using techniques including Heegaard-Floer

Homology.Karin Valencia (Imperial College London) The Topology of DNA-Protein Interactions 17 / 19

Conclusion

Topology has major applications in molecular cellbiology.

Thank you for listening!

I would like to thank Ulrike Tillmann for inviting me to give this talk.


References

G Buck, L Zechiedrich - DNA Disentangling by Type-2 Topoisomerases

D Buck, M Mauricio - Tangle solutions for composite knots, applications to Hinrecombination

D Buck, K Valencia - Predicting knot and catenane type of products of site-specificrecombination on twist knot substrates.

C Erns, DW Sumners - A calculus for rational tangles: applications to DNA recombination.

A Maxwell, A Bates - DNA Topology.

D Rolfsen - Knots and links

K Valencia, D Buck - Characterization of knots and links arising from site-specificrecombination on twist knots.

J Wang, Untangling the double helix.


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