# Poster Session and Refreshments<br/><br/><br/><br/>

Friday, April 9, 2010 - 4:10pm - 6:10pm

Lind 400

**Tangle tabulation**

Danielle Washburn (The University of Iowa)

Like knots, tabulating tangles is done by crossing number. Tangles are similar to knots, but contain strings whose endpoints are nailed down on the boundary of a 3-dimensional ball. The crossing number is the minimal number of crossings needed to draw the diagram of a knot (tangle). We will discuss some basic concepts common between knots and tangles, how to code this and issues that have arisen. Finally, we will introduce why we are interested in tabulating tangles: math biology.**Three dimensional reconstructions of knotted particles**

John Collins (San Francisco State University)

Single Particle Reconstructions are commonly used to elucidate

structures of different types of small particles with precision

approaching that of x-ray techniques. Such reconstructions

assume a homogeneity of data or a small heterogeneous

collection of homogeneous subgroups. Current models of the

packing of small capsid viruses like bacteriophages suggest a

spooling type model with some non-uniformity resulting in small

numbers of knots. Three dimensional reconstructions have been

used to justify different models but such assumptions assuming

uniform homogeneity of a data set disregard slight variations

which could be present. We present a look at the single

particle reconstruction process as a whole as well as a view of

two different sets of data. We ask, what happens if single

particle reconstruction is used to reconstruct a single model

from a data set in which each particle is very similar but no

two are exactly alike?.**Direct entropy calculations for discrete wormlike chains**

Stefan Giovan (University of Texas at Dallas)

Joint work with Stephen D. Levene^{*†}.

The thermodynamic properties of a semi-flexible linear polymer,

DNA for example, are examined using discrete wormlike chains

(dWLCs) as a model. Monte Carlo ensembles of dWLCs were

generated to investigate the effect of excluded volume on the

configurational entropy of the chain, SC, which is calculated

based on the Schlitter approximation. We examined the

dependence of absolute and relative entropies on the cylinder

diameter and also practical aspects of this approach such as

fluctuations in computed SC values as a function of ensemble

size. Future applications include estimating the free energy

of DNA looping in complex nucleoprotein assemblies.

Departments of Molecular and Cell Biology^{*}and Physics^{†}

University of Texas at Dallas

Richardson, TX 75080**Symmetry-breaking in cumulative measures of shapes of polymer models**

Vy Tran (University of St. Thomas)

In a thermally agitated environment, randomly generated polygons are used to model the conformations of fluctuating polymer chains. To characterize the shapes of these polygons, we created 3D density plots of the vertex distributions of families of random 6 edge polygons. The distributions give a measure of the shapes of the polygons, and our symmetry-breaking alignment procedure is not only able to reveal their average bulk shape, but also distinguish between different knot topologies and chirality. We looked at the family of 6 edge polygons, separating them by knot type, and we also looked at 6 edge open chains.**Modeling local knots in proteins caused by random crossing changes**

John Froehlig (The University of Iowa)

Proteins are linear chains of amino acids. Proteins are composed of secondary structure units called alpha helices and beta sheets, which are energetically stable, and random coils, which are not. Many diseases are caused by protein folding disorders. Local knots in proteins are much rarer than is expected for a long polymer. As of 2006, only thirty-nine proteins out of 9,553 proteins with determined structures contain local knots. Eighteen of those thirty-nine proteins contain shallow knots which can deform to the unknot with the removal of five to ten residues from the N- and C-termini, which are the ends of the protein. The most complicated knot in proteins with known structures is the 5_2 knot found in ubiquitin hydrolase (pdb code 1xd3).

The purpose of this project is to engineer knots into proteins with known structures that currently do not contain knots. The archive for all current structures of proteins and nucleic acids is the Protein Data Bank (PDB). For the purpose of this project, the central carbon atom will signify the amino acid. We will use a program called KnotPlot to graph the coordinates of each alpha carbon and join the two termini. From here, we will perform crossing changes in random coils on the chain and determine whether this creates a local knot that is not the unknot.

Since most proteins are linear strands and not closed loops, it is not generally possible to talk about mathematical knots in proteins. We will start with choosing a method to close the gap between the N- and C-termini. As a positive control, we will start by using proteins with local knots to see if the algorithm works. We will then perform crossing changes caused by changing random coils. We will create a protocol to do these functions using KnotPlot, and then write a program to do this automatically so that we can discover all places that can be knotted.**Braid indices in a class of closed braids**

A long-standing problem in knot theory concerns the additivity of crossing numbers of links under the connected sum operation. It is conjectured that if A and B are links, then Cr(A#B)=Cr(A)+Cr(B), but so far this has been proved only for certain classes of links. One such class is the zero-deficiency links, which includes some alternating links and some non-alternating, such as torus knots. In this paper the known realm of zero-deficiency links is expanded to include some cases of links represented by alternating closed braids. It is shown that for a link L represented by a reduced, alternating, k-string closed braid diagram D having at most three sequences of consecutive crossings between each pair of adjacent strings, the braid index of L is k. This result makes use of a well-known property of the HOMFLY polynomial, which provides a lower bound for the braid index of a link. It is then seen that the deficiency of L is zero. It seems likely that this result can be extended to more complex alternating closed braids.**The local and global shape of regular embedded polygons:**

Theoretical and experimental

Laura Zirbel (University of California)

We consider $\mathcal{P}_n$, the space of equilateral, n-sided polygons embedded in $\mathbb{R}^3$. There are several descriptions of the global shape of a polygon $P \in \mathcal{P}_n$, including the convex hull volume, miniball radius, asphericity and radius of gyration. We sought a description of shape that was sensitive to both local and global behavior, and to look at average trends over both the whole population of $\mathcal{P}_n$, as well as ﬁnding the average over sub populations of a speciﬁc knot type.

We developed two such descriptions. For a given $P \in \mathcal{P}_n$, we find the average of the squared distance between vertex $i$ and $i+k$ for all $1 \leq i \leq n$. We call this the Average Squared End to End Distance of length $k$ of $P$.

Similarly, we find the squared radius of gyration for all sub-segments of $P$ of length $k$, and we call the average of these values the Average Squared Radius of Gyration of length $k$ of $P$.

We determine the theoretical averages of these values, taken over all of $\mathcal{P}_n$, in terms of $n$ and $k$. In addition, we examine specific examples of embedded polygons, to determine the effect of knotting of these descriptions of shape.**Polygonal cable links**

Rolland Trapp (California State University)

Given a polygonal knot we present an efficient construction of polygonal

cables of the knot. The construction is applied to polygonal unknots to

obtain results about stick numbers of torus knots. In particular, we show

that (2,q) torus links can be constructed with about two-thirds q sticks.

This is used to show that for q greater than 14, minimal stick representatives

of (2,q) torus links are supercoiled. Finally we show that for 2p the stick number of (p,q) torus links is 4p.**Symmetries of knots and links**

Matt Mastin (University of Georgia)

Two links are equivalent if, roughly speaking, one can be physically deformed into the other. However, we have a choice as to what information we are keeping track of. For example, if we label the components of a link we could ask whether or not the components can be permuted. A labeling of components could arise naturally in application, for example if the components are different polymers. The poster describes a method of recording all of the symmetry information of links as a certain group. We also preview an upcoming paper in which the symmetry groups for prime links through 8 crossings are computed and discuss future directions including the tabulation of composite links.**Table of rational links and their invariants**

Isabel Darcy (The University of Iowa)Guanyu Wang (The University of Iowa)

Joint work with Thomas LeHew and Joe Eichholz.

We are creating a

webpage which will allow users to create

tables of links, knots and their invariants.

Our plan is to provide a platform which

visualizes the information about knots and

links on a table that will satisfy most types

of users. The best platform for this plan is a

webpage. The webpage would be separated

into three main components. First, the

actually html document that the user will

see and interact with to properly generate

the specific table of knots/links that the

user would like to see. Second, a series of

scripts will be set up to take in the input

provided to retrieve the information needed

from the database and then output the

information in an easy to read table for the

user to see. Last, a component specific to

our project is the desire to allow others to

contribute to the table. Livingston and

Cha’s KnotInfo is an outstanding webpage

for creating knot tables. However, for our

project, we put more emphasis on links and

their relevant invariants. It will also handle

composition of links, orientation, and

mirror images.**Engineering multiple site-specific modifications in**

supercoiled DNAs

Anusha Bharadwaj (University of Texas at Dallas)

Joint work with Matthew R. Kesinger^{*}, Massa J. Shoura^{*},

Alexandre Vetcher^{*}, and Stephen D. Levene^{*†}.

Biological processes such as DNA recombination, replication,

and gene expression involve specific interactions between one

or more DNA-binding proteins and multiple protein-binding sites

along a single DNA molecule. Such interactions lead to the

formation of a topologically closed DNA loop between

protein-recognition sites, whose energetics depends on the

structure and the flexibility of the intervening DNA, the

degree of supercoiling, and the binding of additional proteins

such as HU and Fis in bacterial systems or histones and HMG

proteins in the case of eukaryotic cells. We present here a

novel technique for incorporating multiple modifications such

as covalently attached fluorescent probes to multiple defined

sites within covalently closed DNA molecules. Applications of

this technology include the use of two- and three-color FRET to

investigate effects of DNA supercoiling on*lac*-repressor DNA

interactions both*in vitro*and*in vivo*.

Departments of Molecular and Cell Biology^{*}and Physics^{†}

University of Texas at Dallas

Richardson, TX 75080**Energetics of DNA tangling in complex**

nucleoprotein assemblies

Mary Padberg (The University of Iowa)Gregory Witt (The University of Iowa)

Tangle analysis, a branch of mathematical knot

theory, in conjunction with difference topology experiments has become

a powerful emerging approach for the analysis of complex nucleoprotein

assemblies containing DNA loops. A tangle consists of strings properly embedded in a 3-dimensional ball. The protein complex can be thought of as a 3D ball while the DNA segments bound by the protein complex can be thought of as strings embedded within the ball. At present, tangle analysis can only provide information about 2-dimensional diagrams representing the topology of DNA bound within a protein complex. Many DNA geometries can be consistent with a particular topological solution, however, limiting the value of tangle analysis in deducing biological mechanism and function. In addition, many problems of interest do not yield unique tangle solutions. Information about the relative energies of

geometric solutions is badly needed to evaluate the plausibility of a

particular mathematical solution both physically and biologically. We

will demonstrate preliminary software for determining likely DNA

geometries consistent with protein-bound DNA topologies.**Solving a system of four tangle equations**

Lauren Beaumont (The University of Iowa)Dianne Smith (The University of Iowa)

A tangle consists of strings properly embedded within a 3-dimensional ball. Solutions of tangle equations have proven quite useful when applied to recombinases. Recombinases are enzymes that cut DNA strands and interchange the ends, changing the topology of the DNA. The recombinase action will be mathematically modeled by replacing the zero tangle with the tangle t/w, resulting in a new DNA product. If we model experiments involving two topologically different substrates and/or two topologically different products, we have a corresponding system of four tangle equations. Given a1, a2, b1, b2, z1, z2, v1, and v2, we are solving the following system of four tangle equations for t/w:

N(j1/p1 + 0/1) = N(a1/b1)

N(j1/p1 + t/w) = N(z1/v1)

N(j2/p2 + 0/1) = N(a2/b2)

N(j2/p2 + t/w) = N(z2/v2).**High school level introduction to knots**

Jeffrey Hunt (The University of Iowa)

Joint work with Bruce MacTaggart.

Our educational lesson plans focus on elementary properties of knots and are meant to be a mini-unit in basic knot theory for high school students. There are five total lessons in our introductory unit with a summative assessment on the fifth day. The lessons address knot notation, basic definitions, knot equivalence, and knot arithmetic. There are also various activities with hands-on manipulatives for modeling knots and activities involving the program KnotPlot. We believe that since knot theory is a relatively new field of both mathematics and biology it is important to generate interest with younger mathematics students.**Minimal step number of cubic lattice knots in thin slabs**

Robert Scharein (San Francisco State University)

We present provisional data on the minimal step number of cubic lattice knots confined to a thin slab. In particular, we investigate thin slabs of thickness 1, 2 and 3. For most knot types, several ergodicity classes are found, often with dramatically different minimal step numbers. We discuss the number of distinct minimal step embeddings found within each class. We show that in the case of the 1-slab, arbitrarily high step number representatives for each knot type may be found that are irreducible within the 1-slab. Finally, we examine recurring patterns across the entire database of minimal step knots, both in thin slabs and for the unconstrained case.**Invariance of the sign of the average space writhe of free and**

confined knotted polygons

Juliet Portillo (San Francisco State University)

Our group studies topological properties of DNA molecules in

solution. We consider highly compacted models of knotted DNA,

such as DNA extracted from P4 phages. Circular DNA molecules

are modeled as self-avoiding polygons (SAPs) in

three-dimensional space. Using different Monte Carlo

algorithms, we sample the space of knotted SAPs and study

knotting probabilities. To better understand how DNA knotting

is affected in confined environments, we generate knotted

configurations confined inside small spheres. Writhe is a

geometric invariant that measures the entanglement complexity

of a given configuration. A comparison of the writhe of

confined versus free knots suggests that the sign of the

average writhe is invariant for each chiral knot type under

varying polygonal lengths on the simple cubic lattice and in

R3. We propose that the sign of the average space writhe is a

robust measure of knot chirality.