DNA TOPOLOGY De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL

Pedagogical School: Knots & Links: From Theory to Application

De Witt Sumners: Florida State University Lectures on DNA Topology: Schedule Introduction to DNA Topology Monday 09/05/11 10:40-12:40 The Tangle Model for DNA Site-Specific Recombination Thursday 12/05/11 10:40-12:40 Random Knotting and Macromolecular Structure Friday 13/05/11 8:30-10:30

What is mathematics good for? “No longer just the study of number and space, mathematical science has become the science of patterns, with theory built on relations among patterns and on applications derived from the fit between pattern and observation.” L.A. Steen, SCIENCE (1988)

Mathematics at the Beach –Farside--Larsen

Drinking and Deriving

What is Topology?

Isomers

Using Topology in Science

SYNTHETIC KNOT Dietrich-Buchecker & Sauvage, Ang. Chemie 28 (1989), 189

KNOT IN A PROTEIN J. Am. Chem. Soc. 118(1996), 8945

What is Knot Theory?

KNOT HISTORY

Knots and Catenanes

Crossover Number

Peter Guthrie Tait Photo: Roger

A Knot Zoo By Robert G. Scharein © 2005 Jennifer K. Mann

Torus Knots Rob Scharien A Knot Zoo—Knot Plot

Twist Knots Rob Scharien A Knot Zoo—Knot Plot

Prime and Composite Knots

CHIRALITY

CHIRAL AND ACHIRAL KNOTS

CROSSING SIGN CONVENTION

LINKING NUMBERS

UNKNOTTING NUMBER

STRAND PASSAGE METRIC Distance between two knots {K 1,K 2 } = minimum number of strand passages to convert from K 1 to K 2 Darcy &Sumners, Math. Proc. Camb Phil. Soc. 128 (2000), 497

STRAND PASSAGE METRIC TABLE Isabel Darcy

DNA Replication

Adenine-Thymine Base Pair

Cytosine-Guanine Base Pair

Chemical Orientation of Backbone Strands: 3’  5’

Mathematical Orientation of Backbone Strands Forget the chemical orientation 3’  5’: orient the axis and the strands in parallel to the axis DNA forms a ribbon Covalently closed circular DNA forms an orientable ribbon (because of the chemistry 3’  5’)

DNA Structure

DNA is Crowded in the Cell

Radial Loop Chromosome

Replication Obstruction

DNA ENTANGLEMENT: A DILEMMA FOR CELLS DNA must be long enough to encode organism complexity (50kb for e Coli; 3x10 9 base pairs for humans) DNA must be thin and flexible enough to fit inside the cell (or inside the cell nucleus) THE PROBLEM: DNA collisions can knot, link and drive DNA recombination, potentially destroying cellular control of DNA geometry and topology

HOW DOES THE CELL CONTROL DNA TOPOLOGY? The cell produces and uses a number of types of the enzyme TOPOISOMERASE to control cellular DNA topology and geometry

Strand Passage Topoisomerase

TOPOISOMERASE Hyscience August 9, 2007

TOPO I MECHANISM

2-Gate Model for Topoisomerase II

Crystal Structure of Topoisomerase

ROCA TOPOISOMERASE MODELS I.html

Site-specific Recombination Recombinase

LINEAR, RELAXED, SUPERCOILED DNA

TWIST

WRITHE & AVERAGE CROSSING NUMBER Writhe --average the sum of signed crossings over all projections (average number of crossings over all projections)

LK = TW + WR

TOPOISOMERSE AND LINKING

TOPO I vs TOPO II

DNA PLASMID REPLICATION

Enzyme Bound to DNA

Topological Enzymology Mathematics: Deduce enzyme binding and mechanism from observed products

Information We Seek

TOPOLOGICAL ENZYMOLOGY React circular DNA plasmids in vitro (in vivo) with purified enzyme Gel electrophoresis to separate products (DNA knots & links) Electron microscopy of RecA coated products Use topology and geometry to build predictive models

GEL ELECTROPHORESIS

Rec A Coating Enhances EM

RecA Coated DNA

DNA Trefoil Knot

DNA (2,13) TORUS KNOT

T4 TWIST KNOTS

GIN KNOTS Kanaar et al. CELL 62(1990), 553

TOPOISOMERASE CAN MAKE OR BREAK DNA KNOTS AND LINKS Stoichiometry (ratio of topoisomerase to DNA substrate) determines outcome: Many copies of topoisomerase to each copy of DNA plasmid--knots and links are created Few copies of topoisomerase to each copy of DNA plasmid (cell physiology conditions)--knots and links are destroyed (below thermodynamic equilibrium values!)

Topoisomerase I Experiment Dean et al. J. Biol. Chem. 260 (1985), 4795

Topoisomerase Knots

Right and Left Hand Trefoils

Torus and Square Knots

Gel Mobility of DNA Knots

Conclusions

GEL VELOCITY IDENTIFIES KNOTS

Toposides--Chemotherapy Replication Fork Topoisomerase

LANCE ARMSTRONG Testicular cancer (metastatic) Oct Topotecan Hydrochloride (topoisomerase inhibitor) Tour de France winner

© 2008 Jennifer K. Mann

Effect of DNA knotting on gene function © 2008 Jennifer K. Mann

Hin Substrate 26 bp Recognition & Cleavage Sites Wild-type (Inversion, no knot) TTCTTGAAAACCAAGGTTTTTGATAA AAGAACTTTTGGTTCCAAAAACTATT Mutant (knot) TTCTTGAAAACCATGGTTTTTGATAA AAGAACTTTTGGTACCAAAAACTATT (2 bp overhangs) © 2008 Jennifer K. Mann

Processive Hin Recombination © 2008 E. Lynn Zechiedrich

Hin Recombination © 2008 E. Lynn Zechiedrich & Jennifer K. Mann

Processive Hin Recombination 1(360 o )  Trefoil or 3-Noded Twist 2(360 o )  5-Noded Twist 3(360 o )  7-Noded Twist In theory: n(360 o )  2+(2n-1) = (2n+1) Noded Twist Distributive Hin Recombination 1(360 o ) + 1(360 o )  6-Noded Composite 1(360 o ) + 2(360 o )  8-Noded Composite 1(360 o ) + 1(360 o ) + 1(360 o )  9-Noded Composite # # # 3 1 # 3 1 © 2008 Jennifer K. Mann

DNA KNOTS ARE BAD FOR THE CELL Plasmid containing ampicillin-resistance gene is knotted by in vivo Hin recombination (producing DNA trefoil knots) Knotting promotes replicon loss by blocking DNA replication Knotting blocks gene transcription Knotting causes mutation at rate 3-4 orders of magnitude higher than unrecombined substrate Knotted DNA is potentially toxic and may drive genetic evolution R.W. Deibler, J.K. Mann, D.W. Sumners, L. Zechiedrich. Hin-Mediated DNA Knotting and Recombining Promote Replicon Dysfunction and Mutation, BMC Molecular Biology 8 (2007), 44

DNA KNOTTING IS REPLICATION OBSTRUCTION

DNA KNOTTING IS LETHAL!

Potential Models for how knots affect DNA metabolism © 2008 Jennifer K. Mann

1. Knots inhibit gene function by blocking replication and transcription Conclusions: 2. Knots induce DNA rearrangements -In an essential gene, knots can be lethal -Do knots promote evolution? -Do knots account for the genomic instability associated with chemotherapy? © 2008 Jennifer K. Mann

TOPOISOMERASE RESOLVES KNOTS IN HEALTHY CELLS In wild-type eColi no endogenous knotted DNA plasmids are found In eColi with a point mutation on a TOPO II gene, there is a compensatory mutation on another TOPO I gene, the cell lives, and endogenous DNA knots appear (about 3- 5%, all trefoils) Shishido et al JMB (1987)

Thank You National Science Foundation Burroughs Wellcome Fund