Schedule for: 19w5226  The Topology of Nucleic Acids: Research at the Interface of LowDimensional Topology, Polymer Physics and Molecular Biology
Sunday, March 24  

16:00  17:30  Checkin begins at 16:00 on Sunday and is open 24 hours (Front Desk  Professional Development Centre) 
17:30  19:30 
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
20:00  22:00  Informal gathering (Corbett Hall Lounge (CH 2110)) 
Monday, March 25  

07:00  08:45 
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
08:45  09:00 
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) 
09:00  09:30 
De Witt Sumners: Scientific Applications of Topology ↓ This talk will give a brief overview of some of the scientific applications of lowdimensional topology, and some future directions for applied topology. (TCPL 201) 
09:30  10:00 
Alice Pyne: Untangling twisted DNA, one molecule at a time ↓ Alice Pyne 1*†, Agnes Noy 2†, Kavit Main 1, Lesley Mitchenall 3, Anthony Maxwell 3, Sarah Harris 4* DNA in the cell is tangled and twisted, adopts complex topologies, is decorated with a myriad of DNA binding proteins and is frequently maintained under superhelical stress (1). To process DNA, biology has evolved DNA packaging machines, such as eukaryotic histones and prokaryotic gyrases, that maintain order in this topological landscape. Unlike the crystalline, linear DNA that led to the discovery of its double helical nature (2), there is now a growing appreciation that the context of DNA in the wider genome is vital to its function (3). Understanding how DNA behaves in its cellular environment is currently as challenge of complexity, which can be enhanced by a better understanding of the fundamental properties of DNA. This includes how DNA responds to supercoiling and stress (4) to interact with other oligonucleotides and proteins. Here, we use a combination of highresolution AFM (5) and atomistic MD simulations (6) to provide unparalleled single molecule insight into the structure of DNA under superhelical stress down to the base pair level (Figure 1). We use DNA minicircles, only twice the persistence length of DNA, to probe the structure and function of supercoiled DNA(7). We observe that though the discrete, quantised nature of these minicircles may imply a homogenous population, the innate flexibility of DNA under superhelical stress results in a large heterogenous population. Furthermore, we show that DNA at this length scale is highly dynamic, and able to change its conformation even whilst tethered to a surface. We believe that it is this dynamic conformational heterogeneity which allows for supercoiled DNA to bind with a diverse range of substrates. We demonstrate that two distinct binding modalities, triplex formation and repressor factor binding, which either rigidify or bend DNA, are both able to be accommodated within the large conformational population of DNA minicircles. Through combining high resolution microscopy and atomistic simulations, we provide a toolkit to understand the structure and dynamics of supercoiled DNA. We observe conformational changes in DNA and associated binding proteins at subnanometre spatial resolution and sub second temporal resolution. These results improve our understanding of DNA structure, and how this may influence the interaction of DNA with itself, diagnostic therapeutics, and innate proteins  key in regulating gene expression. Figure 1: High resolution AFM and atomistic molecular dynamics simulations reveal the structure of supercoiled DNA to the base pair level. (UNABLE TO SHOW ON THIS WEBSITE)1. Bates AD, Maxwell A. DNA Topology. Oxford University Press; 2005. 2. Franklin RE, Gosling RG. Evidence for 2Chain Helix in Crystalline Structure of Sodium Deoxyribonucleate. Nature. 1953 Jul;172(4):156–7. 3. Fogg JM, Randall GL, Pettitt BM, Sumners DWL, Harris SA, Zechiedrich L. Bullied no more: when and how DNA shoves proteins around. Quarterly Reviews of Biophysics 4. Irobalieva RN, Fogg JM, Catanese DJ, Sutthibutpong T, Chen M, Barker AK, et al. Structural diversity of supercoiled DNA. Nat Comms. 2015 Oct 12;6:8440. 5. Pyne A, Thompson R, Leung C, Roy D, Hoogenboom BW. SingleMolecule Reconstruction of Oligonucleotide Secondary Structure by Atomic Force Microscopy. Small. 2014 Apr 17;10(16):3257–61. 6. Fogg JM, Kolmakova N, Rees I, Magonov S, Hansma H, Perona JJ, et al. Exploring writhe in supercoiled minicircle DNA. J Phys: Condens Matter. 2006 Apr 12;18(14):S145–59. 
10:30  11:00 
Esaias J Janse van Rensburg: Thoughts on lattice knot statistics ↓ The study of lattice knots in the cubic lattice started in 1988 when the FrischWassermanDelbruck conjecture was proven by Sumners and Whittington (J Phys A Math Gen 21: L857–861, 1988), and also by Pippenger (Disc Appl Math 25: 273–278, 1989). In this talk I will review this model and its applications, and in particular focus on the numerical sampling techniques used to simulate lattice knots. These include the use of BFACF moves implemented in Metropolis style MC algorithms, or implemented in dynamic growth algorithms such as GAS to approximately enumerate lattice knots. I will discuss some long standing open problems in this model, and also present some results about the physical properties of lattice knots in confined spaces. (TCPL 201) 
11:00  11:30 
Jason Cantarella: Gaussian Network models for Topological Polymers ↓ Very recently, new advances in synthetic chemistry have enabled the synthesis of polymers with more graph types polymers with the structure of a tetrahedron, for instance, or a thetacurve, or a complete bipartite graph. So many new polymer topologies are being synthesized at such a rapid rate over the past few years (2016present) that chemists describe it as a "Cambrian explosion" of topological polymers. The new polymers seem to have fascinating and previously unseen properties, which hold out promise in technology, energy, and medicine.
In this talk, we present a model for these topological polymers as Gaussian random walks whose overall topology is constrained by any (arbitrary) graph G. It turns out to be the case that understanding the constraints comes down to a mixture of topology and linear algebra, and that the model ties into some rich mathematics connecting graph theory and chemistry from the 1990's. We'll present an algorithm for sampling configurations of these polymers, as well as some theoretical results. In particular, we'll give a nice formula for the expected radius of gyration of a topological polymer in terms of the eigenvalues of the graph Laplacian of the underlying graph G.
This talk represents joint work with Clayton Shonkwiler (mathematics, Colorado State University), Tetsuo Deguchi, and Erica Uehara (physics, Ochanomizu University) which was funded by the Simons Foundation. (TCPL 201) 
11:30  12:00 
Massa Shoura: Something Old, Something New, and Something Circular ↓ Our chromosomes are classically depicted as floppy pieces of spaghetti. However, there exits another flavor of DNA in a circular form. This class, which is called Extrachromosomal circular DNA (eccDNA), comprises products of genomic shuffling (or recombination) that cause somatic changes in gene length and sequence. These circular molecules are derived (or "shed") from linear chromosomal loci, expanding the diversity in coding and regulatory capacity within eukaryotic genomes (DNA) and transcriptomes (RNA). Using a brandnew multidisciplinary approach to investigate eccDNAmediated allelic diversity, I have identified various coding regions of eccDNA fromation, such as Titin and Mucin loci (genes that encode a muscle and a mucus protein, respectively). In order to systematically investigate the biological implications and mechanisms of eccDNA formation, this talk will focus on Titin eccDNAs as a prototype for eccDNAmediated chromosomal rearrangements. Titin is an extremely large protein that is responsible for the passive elasticity of muscle, functioning as a molecular spring. This single gene (TTN) is expressed in various isoforms, each with its own associated “spring constant" conferring a specific rigidity to cardiac muscle fiber. It has been assumed that Titin protein diversity results from complex mechanisms dependent code shuffling on the RNA level (alternative RNA splicing.) However, there is thus far no comprehensive alternativeRNAsplicing framework that accounts for the full spectrum of TTN diversity. Preliminary data we obtained suggest a novel mechanism of TTN diversity involving circularDNA excision that generates recombinant/shuffled TTN loci at the DNA level. Such findings have implications pertaining to the molecular mechanisms of heart disease (such as dilated cardiomyopathy) and therapeutics. (TCPL 201) 
12:00  13:00 
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
13:00  14:00 
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) 
14:00  14:30 
Harrison Chapman: New questions for knot diagrams ↓ The Markov chain Monte Carlo sampler for knot diagrams allows us to randomly sample many more knot diagrams of far larger size than prior rejection sampling methods. We will discuss some new questions and results about about the relationship between knot diagram models and space curve models, with an eye towards how effectively the purely topological knot diagram model can explain and predict the behavior of biopolymers. (TCPL 201) 
14:30  15:00 
Wilma Olson: Contributions of nucleotide sequence and Lac repressor geometry to DNA looping events ↓ In addition to the genetic message, DNA base sequence carries a multitude of structural and energetic signals important to its biological packaging and function. One way in which these signals enter is through the looping of DNA, mediated by proteins that attach to specific, widely separated elements along the chain molecule. Subtleties in the disposition of base pairs become particularly significant in these systems, dictating how the DNA folds and whether the ends of the loops lie in correct register with the binding sites on protein. The doublehelical repeat has a profound effect on the lengths and types of loops most likely to anchor against protein as do deformations of the protein itself. Our group has been exploring the effects of individual base pairs and Lac repressor geometry on the ease and types of DNA loops adopted by specific chain constructs, with the goal of accounting for largescale, sequencedependent behaviors observed in singlemolecule and gene expression studies. The looped configurations are obtained using an optimization procedure that fixes the positions of terminal base pairs and allows for elastic deformations of the intervening residues from a set of rigidbody parameters describing the equilibrium rest state. Treatment of proteinmediated DNA looping at this level makes it possible to decipher the competition among the many factors that contribute to the responses of short pieces of DNA to imposed spatial constraints. The optimization allows us to investigate the effects of protein and DNA on loop formation at levels unattainable with random sampling techniques. Thus we can investigate rare supercoiled states and precise anchoring conditions, as opposed to the approximately closed structures of arbitrary topology and substantially higher energy captured in numerical simulations. (TCPL 201) 
15:00  15:30  Coffee Break (TCPL Foyer) 
15:30  18:00 
Poster Session and Discussions ↓ Poster Presentations with links to abstracts
(TCPL Foyer and 201)

17:30  19:30 
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
Tuesday, March 26  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
Rasika Harshey: A Dynamic E. coli Genome: Widespread DNA Contacts Revealed by Monitoring Mu Transposition. ↓ David Walker and Rasika M. Harshey*, Department of Molecular Biosciences, University of Texas at Austin, Austin, TX 78712 (TCPL 201) The problem
of compacting genomes while still performing cellular processes is common to
all life forms. The current view for E.
coli is a compacted genome with wellorganized domains where replication
initiates and terminates. Some studies have suggested that these and other
regions are sequestered or compartmentalized such that there is little to no
interaction between them. We have exploited the high efficiency and promiscuity
of phage Mu transposition to directly measure the in vivo rates of interactions between genomic loci, and have
developed new tools for
analyzing the proximity of loci across the genome. We
observe widespread contacts between all regions of the E. coli chromosome, revealing a dynamic, effectively
uncompartmentalized genome. We detect longrange interactions
between several genes in different gene families such
as dna and rrna, implicating
spatial
proximity of many distantly coregulated genes for the first time in a
prokaryote. We also see a higher interaction between the two halves
of the chromosome during replication, consistent with the deduced proximity and
higher mobility of the chromosomal arms as they segregate during replication. Our work advances a new view of genome organization in E. coli. 
09:30  10:00 
Nathan Clisby: Recent advances in the Monte Carlo sampling of polymer configurations ↓ I will discuss work in progress to extend the use of efficient data structures from lattice selfavoiding walks to the Monte Carlo sampling of configurations for other polymer models. In particular, I will discuss the sampling of dense polymer configurations (in fact, Hamiltonian paths) via socalled backbite moves, and the sampling of continuum polymers in the selfavoiding walk universality class via an efficient implementation of the pivot algorithm. (TCPL 201) 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
Mario Nicodemi: Models of Polymer Physics for the 3D Structure of Chromosomes ↓ Principled approaches from polymer physics are important to make sense of
the complexity of experimental data on chromosome 3D architecture and to
explain their underlying molecular mechanisms. I discuss first the current
picture of the spatial organisation of our DNA across genomic scales at
the single cell level, as emerging from technologies such as microscopy,
HiC, SPRITE or GAM [1]. Next, I discuss how different models of polymer
physics can help understanding the origin of the patterns in the data and
the underlying folding mechanisms [2,3,4]. Finally, I show that polymer
physics can be used to predict the impact of large mutations (Structural
Variants) on chromosome structure, in particular on how the network of
contacts between genes and regulators is rewired, hence enabling the
identification of their pathogenic potential [5,6]. 
11:00  11:30 
Uta Ziegler: Geometric measures of knots in extreme confinement ↓ We investigate nearlyequilateral polygons generated under extreme confinement. The knot types (as much as possible) and some geometric measurements were computed for each polygon.
The goal is to compare the topological and geometric measures from these extremely confined polygons with previously report measurements of equilateral polygons in more moderate confinement. Preliminary report about joint work done in collaboration with Claus Ernst and Eric Rawdon. (TCPL 201) 
11:30  12:00 
Nicholas Beaton: Knotting statistics for polygons in lattice tubes ↓ I will discuss recent work with Chris Soteros and Jeremy Eng on the probabilities of different knot types for selfavoiding polygons in narrow tubes of the cubic lattice. Polygons in a tube can be characterised by a finite transfer matrix, and this allows for the derivation of pattern theorems, calculation of growth rates and exact enumeration. We also develop a static Monte Carlo method which allows us to sample polygons of a given size directly from a chosen Boltzmann distribution. (TCPL 201) 
12:00  13:30  Lunch (Vistas Dining Room) 
13:30  14:00 
Radmila Sazdanovic: TDA and machine learning approaches to the colored Jones polynomial ↓ Data can often be encoded by a graph or a point cloud. Homology groups of associated simplicial complex, and other tools of Topological Data Analysis (TDA) can be used to compare data sets and detect properties and patterns in the data. We focus on the theoretical results that can be used to address computational challenges caused by the complexity and size of the data. For example, we show that the (local) bottleneck distance between zigzag persistence modules restricted to an interval provides a lower bound on the bottleneck distance and analyze persistence of metric wedge sums and metric gluings. This is joint work with Henry Adams, Michael Adamaszek, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Bei Wang, Yusu Wang, and Lori Ziegelmeier. (TCPL 201) 
14:00  14:30 
Yuanan Diao: Braid Index Bounds Ropelength From Below ↓ For an unoriented link $K$, let $L(K)$ be the ropelength of $K$. It is known that when $K$ has more than one component, different orientations of the components of $K$ may result in different link types and hence different braid indices. We define the largest braid index among all braid indices corresponding to all possible orientation assignments of $K$ the absolute braid index of $K$ and denote it by $\textbf{b}(K)$. In this talk, we show that there exists a constant $a>0$ such that $L(K)\ge a \textbf{b}(K) $ for any $K$, i.e., the ropelength of any link is bounded below by its absolute braid index (up to a constant factor). In particular, the ropelength of the $(2n,2)$ torus link is of the order of $O(n)$. (TCPL 201) 
14:30  14:50 
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL 201) 
15:00  15:30  Coffee Break (TCPL Foyer) 
15:30  18:00 
Poster Session and Discussion ↓ Poster Presentations with links to abstracts
(TCPL Foyer and 201)

17:30  19:30  Dinner (Vistas Dining Room) 
19:30  20:00  Michele Klingbeil (TCPL 201) 
20:00  20:30 
Koya Shimokawa: 3dimensional topology and polycontinuous pattern ↓ Block copolymers produce spherical, cylindrical, lamellar and bicontinuous patterns as microphaseseparated structures. Typical examples of bicontinuous patterns are Gyroid, Dsurface and Psurface. A mathematical model of such a structure is a triply periodic noncompact surface embedded in the 3dimensional space which divides it into two possibly disconnected submanifolds. We will consider the case where submanifolds are open neighborhood of networks. Here a network means an infinite graph embedded in the 3dimensional space. In this case the bicontinuous pattern is uniquely determined by networks up to isotopy. We say such a bicontinuous pattern is associated to a network. On the other hand, for example triblockarm starshaped molecules yields a tricontinuous pattern. One mathematical model of such a tricontinuous (resp. polycontinuous) pattern is a triply periodic noncompact multibranched surface (or more generally polyhedron)
dividing it into 3 (resp. several) possibly disconnected noncompact submanifolds. We assume that each submanifold is the open neighborhood of three (resp. several) networks. We call such a multibranched
surface a tricontinuous pattern(resp. polycontinuous pattern). The relation between polycontinuous patterns and networks is not obvious in this case. Two different polycontinuous patterns are associated to
one network and vice versa. In this talk we will give a condition for polycontinuous patterns to give the same network. We will also show that two polycontinuous patterns can be related by a finite sequence
of moves and discuss further applications. (TCPL 201) 
Wednesday, March 27  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
Makkuni Jayaram: Role of DNA Topology in Biological Machines and Evolutionarily Related Biological Functions: Chemical Chirality in SiteSpecific DNA Recombination ↓ Difference topology: The DNA path within a highorder
DNAprotein assembly can be revealed from the topology of DNA knots and links
formed by the action of Flp or Cre
sitespecific recombinase on target sites placed close to (and flanking) the
assembly. This ‘analysis is experimentally simple, and has broad applicability.
Choice between alternative reaction
mechanisms: Recombination product topologies resulting from a fixed
synaptic topology can help distinguish between alternative modes of site
arrangement and strand exchange by a given recombinase enzyme. Conserved biological functions revealed
by conserved topology of DNA loci in vivo: DNA topology can reveal evolutionary
relationships between two diverged DNA loci that perform analogous but distinct
biological functions. The unusual positive chromatin writhe at the centromeres
of yeast chromosomes is shared by the partitioning locus of an extrachromosomal
circular DNA element present in the yeast nucleus. This finding lends credence
to the possible origin of these loci from a common ancestor that once directed
the segregation of both the chromosomes and the plasmid during cell division. Chemical chirality in DNA recombination:
The active site of the Flp/Cre
recombinase contains two conserved positively charged sidechains (ArgI and ArgII) responsible for
compensating the negative charge on the nonbrridging
oxygen atoms of the scissile phosphate group. When one of the oxygens is
replaced by a neutral group, one of the arginine residues becomes dispensable.
The reactivity of DNA substrates containing substitutions at either of the two
oxygens with Flp/Cre
lacking either of the two arginines reveals the
chirality of Argoxygen interactions, and defines the
stereochemical course of the recombination reaction. 
09:30  10:00 
Allison Moore: Sitespecific recombination and the band surgery along knots ↓ Sitespecific recombinases mediate DNA recombination at sites that are directly or inversely repeated. We model circular DNA as knots or links. In this context, sitespecific recombination is modeled as band surgery, a topological operation that transforms a knot into a new knot or link. We will discuss the differences in this model when the sites are directly or inversely repeated, and mention some recent work in the latter case. In particular, we prove some topological obstructions to the existence of noncoherent band surgeries relating pairs of knots. This work is joint with Vazquez. (TCPL 201) 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
Kai Ishihara: First steps of unlinking pathways ↓ Unlinking of replication catenanes in E. coli is largely achieved by a type II topoisomerase (TopoIV). However, the replication catenanes are also resolved by a sitespecific recombinase (XerCDdif recombination) without type II topoisomerase. In previous research we have characterized unlinking pathways under the assumption that iterative enzyme actions reduce the topological complexity stepwise. In this talk we focus on first steps of the unlinking pathways. (TCPL 201) 
11:00  11:30 
Carolina Medina Graciano: When can a link be obtained from another using crossing exchanges and smoothings? ↓ Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if $L$ is a prime link with crossing number at most five, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus $T_{2,m}$ and all twist knots. (TCPL 201) 
11:30  12:00  Mariel Vazquez (TCPL 201) 
12:00  13:30  Lunch (Vistas Dining Room) 
13:30  17:30  Free Afternoon (Banff National Park) 
17:30  19:30  Dinner (Vistas Dining Room) 
Thursday, March 28  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
Stephen Levene: Kinetic Pathways of Topology Simplification by TypeII Topoisomerases in Knotted Supercoiled DNA ↓ Riccardo Ziraldo^{1,†},
Andreas Hanke^{2,}*^{,†} and Stephen D. Levene^{1,3,4}* Departments of ^{1}Bioengineering, ^{3}Biological
Sciences, and ^{4}Physics, University of Texas at Dallas, Richardson,
TX 75080, USA ^{2} Department of Physics and Astronomy,
University of Texas Rio Grande Valley, Brownsville, TX 78520, USA 
09:30  10:00 
Agnes Noy: Modelling DNA under proteinbinding, stretching and torsional stress ↓ In vivo, DNA is subjected to torsional and stretching stress due to different cellular processes like protein binding, transcription and replication, that are capable of distorting DNA structure beyond its helicoidal shape. On one hand, DNA is subjected to a supercoiling stress that coils the fiber around itself and opens the double helix facilitating the formation of melting bubbles. On the other hand, different proteins like the TATAbinding protein and the recombinase enzyme RecA can overstretch DNA up till 40% beyond its contour length. In this workshop I will show which are the effect of these disturbing factors on the molecule of DNA through the use of molecular dynamics simulations at atomic level. For analysing DNA dynamics due to thermal fluctuation, my team is developing SerraNA, which is a program that calculates the overall elastic constants of DNA from ensembles obtained by molecular simulations [1]. In addition, we also performed simulations on overstretching DNA within the biological regime: up to 40% on extension and within the in vivo range of supercoiling density) exhibit the capacity of the molecule to present melting bubbles independently to torsional stress, even on positively supercoiled DNA [2]. In parallel, torsionallystressed DNA minicircles are being studied by a combination of highresolution AFM experiments and atomistic simulations, describing for the first time the structural details of supercoiled DNA at a subhelical scale [3]. Finally, the DNAbending IHF protein is embedded on the same type of DNA minicircles for analysing its effect on the global and local shape [4]. 1. Noy et al, Phys Rev Lett, (2012), 109, 228101. 2. Shepherd, Noy and Leake et al, In preparation 3. Pyne, Noy, Main, Velasco, Mitchenall, Cugliandolo, Piperakis, Stevenson, Hoogenboom,
Bates, Maxwell and Harris, In preparation 4. Watson, Guilbau, Yoshua, Fogg, Zechiedrich, Leake and Noy, In preparation 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
Tetsuo Deguchi: Rouse Dynamics of Topological Polymers through Gaussian Random Embeddings and Comparison with Experiments ↓ Recently, polymers of complex chemical connectivity expressed with graphs have been synthesized in experiments [13]. It is indeed marvelous that even such polymers expressed with K_{3,3} bipartite graph have been produced [2]. We call polymers with nontrivial structures in chemical connectivity topological polymers. We also call polymers with nontrivial topology of spatial graphs as embeddings in three dimensions topological polymers [4]. The Rouse dynamics of a polymer plays an important role in the dynamical aspects of the polymer in dilute solution, and also in melts if the molecular weight is small [5]. In this talk, we formulate the Rouse dynamics of the topological polymer with a given graph. We derive several physical consequences of the model. In particular, we compare the experimental data of Size Exclusion Chromatography (SEC) of some topological polymers with their theoretical estimates of the meansquare radius of gyration obtained by the Gaussian method [6]. We argue physical backgrounds of SEC data and discuss how they are consistent. After reviewing the method for constructing Gaussian random configurations of a topological polymer, i.e. Gaussian random graph embeddings [6], we derive the normal coordinates and modes for the topological polymer. Here we remark that while the MoorePenrose generalized inverse matrix has been addressed for general Gaussian molecules about three decades ago [7], it seems that important consequences were found later independently [8]. Moreover, it has not been known until quite recently how to generate Gaussian random configurations of a given topological polymer [6]. In fact, we can calculate any physical quantity at least numerically by taking the ensemble averages over generated configurations. It is quite nontrivial to generate such random walks that satisfy the constraints of all independent loops in the graph. The results of this talk are obtained in collaboration with Jason Cantarella, Clayton Shonkwiler, and Erica Uehara. 1) Topological Polymer Chemistry: Progress of cyclic polymers in synthesis, properties and functions, Y. Tezuka ed., World Scientific, Singapore, 2013. 2) T. Suzuki, T. Yamamoto and Y. Tezuka. J. Am. Chem. Soc., 2014, 136, 10148–10155. 3) Y. Tezuka. Acc. Chem. Res., 2017, 50, 2661–2672. 4) E. Uehara and T. Deguchi, J. Chem. Phys. 2016, 145, 164905. 5) M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1986. 6) J. Cantarella, T. Deguchi, C. Shonkwiler and E. Uehara, in preparation. 7) B. E. Eichinger, Macromolecules, 1980, 13, 111. 8) E. Estrada and N. Hatano, Chem. Phys. Let. 2010, 486, 166–170. 
11:00  11:30 
Eleni Panagiotou: A study of the effects of entanglement and chain architecture in polymers ↓ Biopolymers, like proteins, DNA and RNA, attain secondary and tertiary structures which affect their mechanics and function. In some cases, the secondary structure is local and in other cases it is global, affecting both their local and global topology, and even their global architecture. We use tools from topology to examine the effects of local and global entanglement on protein folding kinetics. Using Field Theoretic simulations, we examine the role of global changes in molecular architecture in polymer solutions. (TCPL 201) 
11:30  12:00 
Natasa Jonoska: Algebraic structures related to DNA origami ↓ "DNA origami", introduced by Rothemund in 2006, significantly facilitated the construction of 100x100nm 2D DNA nanostructures. The method typically involves combining an M13 singlestranded cyclic viral molecule called (scaffold) with 200250 short (staple) strands to produce about 100nm diameter 2D shapes. We classify rectangular DNA origami structures according to their scaffold and staples organization by associating a graphical representation to each scaffold folding. Inspired by well studied TemperleyLieb algebra, we identify basic modules that form the structures. The graphical description is obtained by `gluing' basic modules one on top of the other. To each module we associate a symbol such that gluing of molecules corresponds to concatenating the associated symbols. Every word corresponds to a graphical representation of a DNA origami structure. A set of rewriting rules defines equivalent words that correspond to the same graphical structure. We propose two different types of basic module structures and corresponding rewriting rules. For each type, we provide the number of all possible structures through the number of equivalence classes of words. (TCPL 201) 
12:00  13:30  Lunch (Vistas Dining Room) 
13:30  14:00 
Kenneth Millett: Gordian Knotted Structures ↓ According to the legend of Phrygian Gordium, Alexander the Great cut the Gordian Knot and eventually went on to rule Asia thereby fulfilling the ancient prophecy. While there are several versions of the precise nature of the Gordian Knot and Alexander’s action, an explicit mathematical treatment and the reasons for its contemporary importance will be described. The first example of such a simple Gordian Knotted Structure supported by a rigorous mathematical analysis will be presented. (TCPL 201) 
14:00  14:30 
Erica Flapan: Possible pathways for protein knot folding ↓ How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. In this talk, we introduce a new theory of knot folding that could describe a pathway for the formation of all currently known protein knot types and predict knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific configurations from our theory. In particular, our approach confirms Taylor's twisted hairpin theory of knot folding for the \(3_1\)knotted proteins and the $4_1$knotted KARI's as special cases, and presents an alternative folding mechanism for the $4_1$knotted phytochromes and the $5_2$ and $6_1$knotted proteins. (TCPL 201) 
14:30  15:00 
Claus Ernst: Computing the braid index using only a knot diagram and knots where the braid index equals the bridge index ↓ We talk about how to read the braid index of certain families of alternating knots from a minimal knot diagram. In addition we talk about a class of knots where the braid index equals the bridge index.
These knots seem to have particular physical properties as demonstrated by
several experiments. They undergo a periodic motion when sedimenting in a viscous fluid as knotted deformable closed
chains, they also seem to achieve a circular minimum as elastic knots when their bending energy (curvature) is minimized. (TCPL 201) 
15:00  18:00  Discussion (TCPL 201) 
15:00  15:30  Coffee Break (TCPL Foyer) 
17:30  19:30  Dinner (Vistas Dining Room) 
Friday, March 29  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
Javier Arsuaga: A liquid crystal model for DNA packing in bacteriophages ↓ A modeling framework of genome packing inside bacteriophage capsids is developed and tested. The main assumptions of the model are the chromonic columnar hexagonal structure of confined (ds)DNA and its high resistance to bending. The model has the capability to predict pressure and ejection force as well as the size of the disordered core. It also accounts for the phase transition from solid to fluidlike states as the concentration of DNA in the capsid decreases. (TCPL 201) Work in collaboration with C. Calderer, L. Hiltner, M. Vazquez and S. Walker 
09:30  10:00  Andrew Rechnitzer: Collapsing Hopf links and WangLandau simulations (TCPL 201) 
10:00  10:30 
Sarah Harris: TORC : A computational language for designing supercoilingdriven gene control in synthetic DNA circuits ↓ To exploit the potential of synthetic biology, we need engineering approaches, including computational tools and techniques, that work with, rather than against, biological complexity. Computational approaches have already provided many important advances in synthetic biology, allowing small `logic circuits' to be programmed and implemented in a cell's genetic makeup. In order to scale up the power of these approaches, we need a computational toolkit that is based on the full complexity of DNA processes. It is tempting to simplify genetics to a simple passive `code' along with genetic binary `switches' to control its expression. But the physical behaviour of the DNA itself is also implicated in gene regulation. Transcription generates positive supercoiling ahead of the transcription machinery and negative supercoiling behind, due to the topological changes required to separate the double helical DNA strands. Transcription induced supercoiling can both up and downregulate expression of distal genes, however, in general negative supercoiling promotes transcription by destabling the DNA duplex, and facilitating the formation of the open RNA polymerase complex. This information transfer along the DNA strands can be modulated by DNA writhing to form plectonemic supercoils, by DNA binding proteins capable of removing (such as topoisomerases) or absorbing (such FIS) superhelical stress, or can be modified by bending (e.g. IHF) and bridging proteins (e.g. LacI) that stabilise plectonemes and by environmental or metabolic changes, such as the salt concentration or amount of ATP available. This behaviour is an inherent part of the information storage abilities of genomic DNA, and offers powerful new programming and control mechanisms for synthetic biology, but is not yet captured in any synthetic biology toolkit. Our vision is of a complete computational toolkit based on DNA writhe and twist properties, their information processing capabilities and their programmatic modulation. The toolkit, which we dub `TORC', would include a logic for reasoning, a language for expression, an interpreter for testing, and an API for synthetic biology. TORC would thereby provide both an engineering development approach for advanced synthetic biology, and a scientific language and logic for modelling biological genetics. (TCPL 201) 
10:30  11:00  Coffee Break (TCPL Foyer) 
11:00  11:30  Closing Discussion (TCPL 201) 
11:30  12:00 
Checkout by Noon ↓ 5day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk  Professional Development Centre) 
12:00  13:30  Lunch from 11:30 to 13:30 (Vistas Dining Room) 