The study of DNA topology relies heavily on both math and biology to unravel the shape and geometry of complex biological structures and to understand the enzymatic actions that affect DNA topology.

by Annie Bautista

Imagine

Untanglethis gossamer string remain untangled de-spite its tight packaging? And how does the tight compaction still allow accessibility for decoding and accurate replication? SF State Associate Professor Mariel Vazquez, a pio-neer in mathematical biology, studies DNA compaction in normal cells, as well as DNA rearrangements and other aberrations in cells with erroneous genetic behavior such as breast cancer cells. Her approach comes from pure mathematics and involves compu-tational tools. The outcome, however, could someday help others save lives.

compacting a 2-meter-long string into a tiny ball just 0.005 millimeters across—much narrower than the width of a human hair. The probability of the condensed string re-maining untangled is rather slim. Our own bodies, however, pull off this improbable feat every day. Each of our 100 trillion cells plays some active role in movement, diges-tion, sensation, and so on, and functions by means of genetic instructions encoded on a 2-meter-long “string” of DNA packed inside the microscopic nucleus. How does

UsingMathto

DNA

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In eukaryotic cells (such as human cells), DNA molecules must be “supercoiled,” or tightly folded, and also packaged in chromosomes. The packaging requires as-sistance, and proteins called histones supply it. The DNA of simple organisms such as bacteria is also supercoiled and subjected to various levels of compaction. Bacterial DNA is circular and is therefore prone to

becoming knotted. While the DNA molecule naturally exists in a

topologically stable conforma-tion, the long molecule must

unravel during DNA replication and

transcription. The cell is fully

equipped with enzymes to facilitate

the efficient duplication of the genetic code, as well

as the transcription of DNA into RNA and

the translation of RNA into pro-

teins. When the activity of these en-

zymes is altered or poorly regulated, knots and links

can occur within the ultra-long DNA molecule. This may result in

chromosomal abnormalities and cell death. To prevent DNA knots and links, which are

Noting this, researchers realized that deliberately blocking topoisomerase activity could potentially lead to cell death–and therefore allow for new types of antibiot-ics and anti-cancer drugs. Pharmaceutical researchers have designed certain drugs to inhibit the enzymes in bacterial cells: The broadspectrum antibiotic ciproxin, for example, which is widely used to fight bacterial infections, blocks type II topoi-somerases. Likewise, etoposides are used in combination with other drugs during che-motherapy for a variety of cancers, includ-ing lung cancer and lymphona. Etoposides cause unrepairable damage in tumor cells by blocking their topoisomerase activity. Drugs of this type can potentially cause an accumulation of supercoils, knots or links

obviously undesirable topological forms, cells use enzymes called topoisomerases.

Mariel Vazquez studies the way type II topoisomerases and enzymes with related activity called site-specific recombinases change the topology of DNA. To do so, she employs circular DNA molecules such as the circular chromosomes of the bacteria Escherichia coli (E. coli). Before a bacterial cell divides, its single circular chromosome is replicated, resulting in two interlinked DNA circles. When events such as knotting or linking occur in the circular strands, type II topoisomerases can return the DNA mol-ecules to their native, untangled state. In this way, they function to regulate the topology of DNA by removing unwanted knots and links in preparation for cell division.

In her studies, Vazquez focuses on the cell’s ability to faithfully replicate its DNA and to produce unlinked DNA molecules. Vazquez’s doctoral advisor from Florida State University, Dr. De Witt Sumners, likes to say that “Mother Nature’s solution to the entanglement problem is topoi-somerase.” The enzyme’s activity–passing one strand of DNA through another and alleviating torsional stress to eliminate knots and links–is especially important in newly replicated DNA. Because topoisomerases have such a pivotal role as safe-keepers of DNA topology, malfunctions can have catastrophic consequences.

she explains it, she solves the biological problem by “modeling the [underlying] process using pure math, and then translat-ing the output of the models back into their original context”–the cell.

Mathematicians have created tools from computational and analytical knot theory to provide a detailed picture of what DNA looks and acts like when packaged in confined environments, and in turn, how enzymes change the topology of DNA. “Imagine circular rubber bands as DNA,” says Vazquez, “ and then stuff them into a tiny box. Each band will produce some interesting configurations, but every time you pull [a band] out, it is still just a circle.” Now, she goes on, “imagine that little enzymes are placed inside the box. Put

program “Knotplot.” Vazquez has also au-thored “TangleSolve,” a Java applet for the topological analysis of site-specific recombi-nation. These programs allow researchers to take experimental data and reproduce and reinterpret it using rigorous mathematical analysis. Using these advanced programs, mathematicians are able to “investigate beautiful mathematical problems that arose from the original biological problems,” says Vazquez.

Mariel Vazquez’s fusion of math and biology bridges many fields of research. In April 2002, the American Mathematical Society (AMS) Math Awareness month highlighted the importance of mathemat-ics in genomics research. As researchers complete the job of sequencing the entire

that seriously hinder the cell’s ability to manipulate its DNA genome and thus to divide normally and grow.

The study of DNA topology relies heavily on both math and biology to unravel the shape and geometry of complex biological structures and to understand the enzymatic actions that affect DNA topol-ogy. Vazquez applies topology, geometry and computational tools to study the shapes of circular DNA molecules in solution, as well as in confinement, and to understand the way enzymes such as type II topoisom-erases allow cells to function smoothly. As

this rubber band inside again, very tightly packed… The enzymes cut the DNA, transfer it through the break, reseal it, and do that many times. When you take the rubber band out it is very, very likely to be knotted.”

Jointly with her close collaborator, Javier Arsuaga, Vazquez has developed computer software of various kinds to understand the making and behavior of DNA knots and links. Together with their former postdoc, Rob Scharein, the two have contributed new knot theory software to the widely used computational knot theory

Mariel Vazquez, Associate Professor of Mathematics

Mathematicians have created tools from computational and analytical knot theory to provide a detailed picture of what DNA looks and acts like when packaged in confined environments, and in turn, how enzymes change the topology of DNA.

Page 34 lower left:

Computer representation of a DNA knot as a polygonal chain in the simple cubic lattice.

Page 35:

Random and spooling model of DNA packing in bacteriophage capsids. Colors indicate contiguous DNA regions.

Graphics courtesy of Dr. Vazquez.

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human genome, the AMS states in a press release, “…attention can shift from the job of finding genes…to understanding them.” Using mathematics to study the shape and geometry of DNA will “provide informa-tion about the biological function of DNA.” Vazquez’s software development and study of enzymes that change the topology of DNA is an important arm of that work.

Vazquez’s long-standing passion for math and biology laid the foundation for her educational and her professional career. During and after high school, she gravitated toward both subjects. She enrolled as an undergraduate at the National Autonomous University of Mexico (UNAM). At that time, however, they offered no interdisci-plinary programs in the two fields. Vazquez

focused on mathematics and fell in love with topology and geometry. However, her interest in biology remained. To graduate from UNAM in 1995, she completed a thesis called Applications of Knot Theory to the Study of DNA. She pursued her doctorate at Florida State University and in 2000, wrote a dissertation called Tangle

professor at SF State. In her laboratory, she has continued

her work on the tangle analysis of Xer recombination, which is the subject of the CAREER award that she obtained earlier this year from the National Science Foundation. Her group also undertakes the mathematical and computational analysis of

In her studies, Vazquez focuses on the cell’s ability to faithfully replicate its DNA and to produce unlinked DNA molecules.

In addition to her research, Vazquez teaches several courses and helps organize professional conferences. Every spring, she teaches a DNA topology course through the SF State math department. She has also organized many conferences and symposia in the last few years. These include the yearly Mod-ern Mathematics Workshop at SACNAS, SIAM mini-symposium on New Trends in Mathematical Biology at the joint meetings of the American Mathematical Society and the Mathematical Association of America (January 2010), the Fong Symposium at SF State in 2010, the San Francisco Interna-tional Meeting on DNA Topology (April 2009), and the annual Biol-ogy and Mathematics in the Bay Area (BaMBA) Day. BaMBA Days rotate between Bay Area Institu-tions. Stanford hosted BaMBA VI in November 2010. All of these events, says Vazquez, allow the public to learn the importance of applying fundamental mathematics to important biological problems.

Vazquez is convinced that an interdisciplinary approach helps solve such puzzles, and her research group at SF State reflects that con-viction. Her group and collaborators typically include chemists, bioengi-neers, computer scientists, biolo-gists, and mathematicians. “We try to work all together and it is very highly vertically integrated,” she says, meaning that “undergraduates work alongside graduate students, research technicians, research fel-lows, and faculty members.” This exposure helps biology majors become more comfortable with math and its utility in their research. “Many times, we see biolo-gists who start a research project and soon find that they need to learn statistics…how to program…more math. That’s when they realize that math is not so awful after all. Math can be fun,” Vazquez says, “and it is extremely useful nowadays if you’re a biol-ogy student. Acquiring strong mathematical skills is very important if you want to be competitive.” v

Analysis of Site-Specific Recombination: Gin and Xer systems. Next she worked as a Postdoctoral Fellow in Mathematics at UC Berkeley. While there, she split her time between mentoring undergraduate research projects in DNA topology and using math and computers to model the chromosomal rearrangements that result from radiation damage. In 2005, she became an assistant

how type II topoisomerases unknot DNA with funding from the National Institutes of Health (NIH). Jointly with Javier Ar-suaga, Assistant Professor of Mathematics at SF State, she has studied the topology of bacteriophage DNA with funding from the National Science Foundation. Arsuaga and Vazquez have funding from the National Center On Minority Health And Health Disparities to study CGH array data from breast cancer patients using techniques from computational topology. For the latter study, Vazquez and Arsuaga are searching for chromosomal changes associated with recurrent breast cancer in early stage pa-tients. They are also hoping to identify to-pological signatures in the DNA that allow an accurate prediction of who is most likely

to experience a cancer recurrence. Writ-ing in the journal Bioinformatics in 2002, mathematician Angela Torres states, “The fusion of mathematics and biology will result in a new era of molecular medicine, when the diagnosis, treatment and preven-tion of disease will be individual-specific and thus more successful.”

Page 36 top:A torus knot diagram.

Page 36 bottom:Disentangling DNA by recombination is illustrated by two black arrows (sites) targeted by a blue ball (enzyme) along a polygonal chain (DNA).

Page 37:Visual representation of a cloud of points in 3-dimensions. The cloud of points arises from genomic data of breast cancer patients. Loops indicate contiguous genomic regions.

Graphics courtesy of Dr. Vazquez.

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The College of Science & Engineering is proud to an-nounce that Mariel Vazquez had been awarded a highly prestigious National Science Foundation CA-REER grant totaling $600,000. These awards are extremely com-petitive, with only about two dozen awarded in mathematics each year in the entire country, including just a few per year in biomathematics. The College of Science & Engi-neering has been remarkably suc-cessful with these unusual awards, with 11 of our faculty receiving

these awards over the last eight years. The National Sci-ence Foundation website includes this description of these awards: The Faculty Early Career Develop-ment (CAREER)

Program is a Foundation-wide ac-tivity that offers the National Sci-ence Foundation’s most prestigious awards in support of junior faculty who exemplify the role of teacher-scholars through outstanding re-search, excellent education and the integration of education and research within the context of the mission of their organizations.

Topological mechanism of DNA unlinking by the XerCD-FtsK system Principle Investigator: Mariel Vazquez 

topology is the study of knotting, linking and supercoiling of circular DNA molecules. The bacterial chromosome is circular and replication invariably results in the formation of interlinked daughter chro-mosomes. Error-free unlinking is required to ensure proper segregation at cell division and stable plasmid inheritance. Type II topoisomerases unlink replication links. In Escherichia coli, in the absence of topo IV (a type II topoisomerase credited with chromosome unlinking), the site-specific recombination system XerCD mediates sister chromosome unlinking. This reaction is activated at the division septum by a power-ful translocase FtsK, which coordinates the last stages of chromosome segregation. The mechanism by which the XerCD-FtsK complex simplifies the topology of DNA remains unclear. The main objective of the proposed studies is to characterize the topological mechanism of DNA unlinking by the XerCD-FtsK system using knot theory, low-dimensional topology, and computer simulations. There is evidence that after being activated by FtsK, the enzymes XerCD unlink DNA in a stepwise manner. The tangle method will be used to find possible topological pathways of DNA unknotting and unlinking by site-specific recombination on small substrates. A computer model of DNA recombination will be developed, adapted to the Xer-FtsK system, and combined with the analytical results to analyze experimental data obtained from the Sherratt lab. The research is highly interdisciplinary and involves close collaboration with groups in Japan, Canada and the UK. Such collaborations will facilitate state-of-the-art student cross-training. Basic information about DNA topology will be disseminated to the general public, including elementary school children and visitors to the California Academy of Sciences.

  DNA replication is the basis for biological inheritance. In bacteria, reproduction starts with replication of the chromosome into two identical daughter molecules, followed by segregation of the newly replicated chromosomes and division of the parent cell into two daughter cells. In circular chromosomes, problems of entanglement during DNA linking complicate the process of chromosome segregation. In Escherichia coli, DNA unlinking is typically mediated by the enzyme topoIV, which is an important drug target for quino-lone antimicrobial agents. Understanding DNA unlinking by Xer recombination, in addition to providing a more complete picture of the chromosome segregation process, is highly relevant for drug design. Mathe-matical and computational tools are very useful for studying the action of enzymes that change the topol-ogy of DNA. In this project such tools will be used to characterize all unlinking pathways and to reveal the mechanism of unlinking by Xer. The educational goal is to develop new and effective ways to disseminate knowledge related to DNA topology and its biological significance, as well as to increase public awareness of the critical role of mathematics in understanding biological processes. The proposed plans include the creation of Math Circles for elementary school children in San Francisco and the development of a series of educational materials for public consumption in collaboration with the California Academy of Sciences. This will culminate in the production of an exhibit on DNA topology for the general public in the Califor-nia Academy of Sciences Museum.

DNAPublic Abstract

Above left: Computer representation of a negatively supercoiled DNA molecule with two recombination sites.

Above right: Right-handed torus link with two anti-parallel sites. Links of this form arise from DNA replication.

Page 39: Stepwise pathway of DNA unlinking by XerCD-FtsK.

Graphics courtesy of Dr. Vazquez.

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