Pi, Pie, Knotted Structures, and Biophysics

Biophysicists have always found themselves closely connected to many allied disciplines and conducting research that blurs the lines between these disciplines. With March 14 officially marking the celebration of Pi, the Biophysical Society thought it was a good time to shed some light on the amazing discoveries mathematics research (and Pi!) has contributed to the field of biophysics.

The Society was fortunate to co-sponsor a meeting on the significance of knotted structures for functions of proteins and nucleic acids that included among the speakers many mathematicians.  We have asked three speakers from that meeting to share some thoughts on their work, their career trajectory, and of course, Pi, with us in honor of Pi day.

 Stu Whittington
University of Toronto

Stuart (Stu) Whittington’s interest in Biophysics was sparked when he realized that circular DNA molecules from bacteria could be knotted and that these knots could interfere with cellular processes such as replication. He says that he is “definitely not a biologist — indeed I never took a biology course in high school or in university.” His main research interests are in rigorous statistical mechanics, especially of models of polymers, and he likes models with a combinatorial (so that he can count instead of integrating) or topological flavor. Whittington first became interested in knotting in ring polymers when he heard De Witt Sumners talk about it in about 1986. Whittingon and Sumners have been working together on random knotting and linking ever since. They have proved some results about the inevitability of knotting in long flexible objects (like hose pipes and DNA molecules) and also about the inevitability of writhe. These ideas are easily understood by the general public (who hasn’t found random knots in an extension cord?) and are a useful way to engage the average person about the utility of mathematics in biophysics, as well as in chemistry and physics.

Chris Soteros
University of Saskatchewan

Chris Soteros focuses on lattice models of polymers and biopolymers.  She uses combinatorics, probability, and asymptotic analysis to study the models as well as exact enumeration and Monte Carlo computer simulation methods. In addition, analyzing the computer simulation data involves statistical methods. By using these mathematical approaches, Soteros and her colleagues can identify the minimal ingredients needed to incorporate into a model in order to observe trends similar to those obtained in polymer and biopolymer experiments.  Sometimes it is possible to prove results about the models.  This can lead to new insights into the experimental results or can significantly strengthen the evidence for what was previously conjectured.

Soteros, who identifies herself as a mathematician, got her start in biophysical research in 1988 when she was a postdoc working with Stuart Whittington.  At that time she had the opportunity to study a mathematical problem that was motivated by a DNA topology question. This work was in collaboration with De Witt Sumners and resulted in a paper:  C.E. Soteros, D.W. Sumners and S.G. Whittington, 1992. Entanglement Complexity of Graphs in Z3. Math. Proc. Camb. Phil. Soc., 111, 75-91.  Soteros has been studying related problems every since then.

One challenge Soteros finds in working at this interface is learning enough terminology from another discipline, such as molecular biology, to read journal articles in that discipline. It can also be a challenge to keep on top of the latest advances in more than one discipline.  On the other hand, these barriers are greatly reduced when one collaborates with others who have complementary expertise and who are open to bridging the communication gap.  Soteros feels fortunate to have been involved in many such collaborations.

Soteros is especially grateful for having many opportunities to learn from others at interdisciplinary conferences involving molecular biologists, physicists, chemists, mathematicians, and computer scientists. The September 2014 Biophysical Society’s thematic meeting, “Significance of Knotted Structures for Functions of Proteins and Nucleic Acids’’ Conference in Warsaw, Poland was a prime example.

From the perspective of other researchers, Soteros thinks her work is interesting when she and her colleagues prove or find strong evidence for hypotheses that were previously only conjectures, or when they propose a novel hypothesis or conjecture.  She offers an example, “We were able to prove for a model of polymers confined to a tube, that knotting is inevitable for very large polymers, regardless of whether they are stretched or compressed (M. Atapour, C. Soteros and S. Whittington, 2009. Stretched polygons in a lattice tube. J. Phys. A:  Math. Theor., vol. 42, 322002 (9pp)).

From the perspective of the public, Soteros’ work demonstrates that fairly simple mathematical models can be used to gain insights into DNA experiments. More broadly, improved understanding of enzyme action on DNA through collaborative efforts involving mathematicians, physicists and molecular biologists is expected to lead to improved cancer treatments.

As for Pi, Soteros does use it in her work.  She says, “Some simple examples come immediately to mind. We studied how knot reduction in a model of topoisomerase action on DNA depends on the opening angle at the strand-passage site (M. L. Szafron and C. E. Soteros, 2011. The effect of juxtaposition angle on knot reduction in a lattice polygon model of strand passage. Fast Track Communication, J. of Phys. A: Math. Theor., Vol. 44 (322001), (11 pp)).   In the calculations, the angles were calculated in radians where 2Pi radians corresponds to 360 degrees.  Also, in the statistical analysis of our computer simulation data, we use the central limit theorem; this uses the standard Normal (or Gaussian) distribution that has density function function 1/√2π e^(-〖x〗^2/2).  Also, we use polygons on the simple-cubic lattice to model polymer and biopolymer configurations.  The bond angles on this lattice are all either Pi/2 or Pi.”

Soteros looks forward to celebrating Pi day with some PIE!

Note:  Soteros wanted BPS blog readers  to be aware of two upcoming meetings focused on the type of research she has described in this post: May 18-29, 2015 Graduate Summer School in Applied Combinatorics  and June 1-4, 2015 The Canadian Discrete and Algorithmic Mathematics Conference (CanaDAM)


A planar projection of a 3-dimensional 5100-edge simple-cubic lattice polygon sampled from computer simulations for the paper Cheston M., McGregor K., Soteros C., and Szafron M., 2014. New evidence on the asymptotics of knotted lattice polygons via local strand-passage models. J. Stat. Mech.: Theor. Exp., 2014(2): P02014. It is a polygon with knot-type 5_1 where the “knotted part’’ is in the bright green part of the polygon at the right of the image. This is a randomly chosen 5_1 lattice polygon that illustrates what we expect to occur most often for large polygons, namely that the “knotted part” is relatively localized within the polygon. The image was created by M. Szafron using Rob Scharein’s KnotPlot software and the colour of an edge indicates its depth in the direction perpendicular to the plane of the image.

Eric Rawdon
University of St. Thomas

Eric Rawdon studies the knotting and tangling that occurs in physical systems, e.g. with DNA, proteins, or subatomic glueballs. He says he has always been drawn to computers, and gravitates towards problems that have some computational aspect. Most recently, he and his colleagues have been studying knotting in proteins, trying to understand which proteins are knotted, how they are knotted, and why they are knotted.

From Rawdon’s perspective as a trained mathematician, he thinks the things that biologists are able to do in the lab are amazing.  He notes, “I am too clumsy or impatient to deal with such messy experiments.  So I try to understand certain knotting behavior in physical systems by stripping away less relevant details.  For example, proteins are chains of amino acids.  If you analyze proteins at the atomic level, it is a mess (or at least that is how it looks to me), there are atoms and bonds everywhere. So we simplify the situation and model the protein as a chain of line segments. This is a coarse model and gives you a sort of long distance view of how the system is behaving.”

Like Soteros, Rawdon got his start in biophysical research early in his career.  His PhD advisor, Jon Simon (retired mathematician, University of Iowa), seemed to be drawn to mathematical problems in the sciences.  At conferences, Simon introduced Rawdon to many different people working in many different fields.  Rawdon never expected to do “applied” mathematics, but the biologists, chemists, and physicists brought such interesting questions to the table that he couldn’t help himself.  Early in his career he started working with Ken Millett (mathematician, University of California Santa Barbara). Like Simon, Millett had done some hard math but was open to mathematical problems in the sciences. Millett and Rawdon then team up with Andrzej Stasiak (biologist, University of Lausanne, Switzerland).

When it comes to identifying himself, Rawdon isn’t really sure what to call himself.  “I was trained as a mathematician, but I tend to publish in biology, chemistry, and physics journals.  So honestly, I’m not sure what to call myself.  But deep down inside, I think I am more mathematician than anything.”

Rawdon has continued to attend the interdisciplinary conferences he was introduced to as a graduate student.  He says the meetings he goes to typically have researchers from a wide-variety of fields, and the researchers tend to be open to interdisciplinary work. He met Joanna Sulkowska (physicist, University of Warsaw and one of the organizers of the BPS thematic meeting on Knotted Structures), at one such meeting, and she is the one who interested him in knotted proteins.

When working with scientists trained in other disciplines, Rawdon finds the biggest barrier to be language He offers the following example:  “My collaborator Andrzej Stasiak and I occasionally have disagreements over email, only to discover later that we were not in disagreement at all.  We simply both had interpreted the others’ words in terms of the language of our own fields.  If I say “protein topology” to a biologist and a mathematician, they are likely to interpret the phrase very differently.”

When asked if he has had any surprise research findings, Rawdon recounts the following collaboration from 2012:

“My collaborators and I were searching for knots in proteins. For each protein, we would generate a picture that encoded the knotting.  We created a web page of these pictures and I put them together in groups because there were many pictures that looked the same.  I sent the web page to Joanna who noticed that the similar pictures were coming from proteins that performed common functions in different organisms.  These families of proteins had diverged over hundreds of millions of years of evolution, yet the knotting patterns stayed the same.  That suggests that the knot is there for a reason. For a knot guy like me, that was pretty cool.  Our results are in a paper titled “Conservation of complex knotting and slipknotting patterns in proteins” published in the Proceedings of the National Academy of Sciences in 2012 (with Joanna Sulkowska, Ken Millett, Jose Onuchic, and Andrzej Stasiak).”

For those very interested in the topic, Rawdon directs readers to visit their website KnotProt  that has all the information you would ever want about knotted proteins.

For a general audience, Rawdon thinks his work is appealing because of all of the places where knotting appears in the sciences.  Here are just a few examples:

  • A large number of cancer drugs attack topoisomerases, enzymes that are necessary for DNA untangling during replication.
  • The antibiotic Ciprofloxacin Hydrochloride, which also targets topoisomerases, is used to treat anthrax exposure.
  • Recently, Chunfeng Zhao, M.D., of the Mayo Clinic proposed that surgeons use a new type of knot for certain types of surgeries.
  • It has been proposed that there are subatomic particles, called glueballs, which form tight knots.

As for Pi, it is not clear cut whether Rawdon uses it in his work.  He says, “Yes and no.  I think a lot about knots made out of tubes and any time you are you dealing with anything round, Pi is lurking in the shadows. Plus, I teach math, so I probably say Pi and mean the number, as opposed to the dessert, more than the average person.  But I do not think about Pi every day.  Still, each March 14, I take a moment to recognize Pi.

Tomorrow, to celebrate, the Math and Actuarial Science Club at the University of St. Thomas will buy some pies, as they do every year on Pi Day.  Since Pi Day is on the weekend this year, they are celebrating on Friday, March 13.  There are t-shirts and a pie-eating contest.  It promises to be a special event.

This image comes from a paper that was just published and is freely available for downloading: Eric Rawdon, Ken Millett, and Andrzej Stasiak, Subknots in ideal knots, random knots, and knotted proteins Scientific Reports 5:8928 (2015) http://www.nature.com/srep/2015/150310/srep08928/full/srep08928.html In the paper, we are trying to understand what sort of simpler knots lie inside more complicated knots.  These disks are our way of encoding the information.  But in and of themselves, I find them very beautiful.

This image comes from a paper that was just published and is freely available for downloading:
Eric Rawdon, Ken Millett, and Andrzej Stasiak, Subknots in ideal knots, random knots, and knotted proteins Scientific Reports 5:8928 (2015)
In the paper, we are trying to understand what sort of simpler knots lie inside more complicated knots. These disks are our way of encoding the information. But in and of themselves, I find them very beautiful.


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