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Organizers |
DNA knots reveal a chiral organization of DNA in phage capsids
by
Javier Arsuaga
Comprehensive Cancer Center, UCSF
Coauthors: Mariel Vazquez, Paul McGuirk, De Witt Sumners, Joaquim Roca
It is believed that all icosahedral bacteriophages package their double-stranded DNA genomes to near-crystalline density in similar fashion. Nevertheless despite numerous studies, the organization of DNA inside viruses such as lambda, T4, T7, P2, P4, and Phi29 is still unknown. We propose a new approach to this problem. We recently showed that most DNA molecules extracted from bacteriophage P4 are highly knotted due to the cyclization of the linear DNA molecule confined inside the viral capsid. Here we show that these knots provide information about the global arrangement of the DNA inside the phage. We analysed the viral distributions of DNA knots by high-resolution gel electrophoresis and performed Monte Carlo computer simulations of random knotting confined to spherical volumes. A rigorous proof of non-random packaging of the phage DNA is given by comparing the knot distributions obtained by both techniques. Furthermore, our results indicate that the packaging geometry of the DNA inside the viral capsid is writhe directed.
Date received: October 15, 2004
Topological Approaches to the Analysis of Spatiotemporal Organization in Chemical and Biological Systems
by
Leon Glass
Department of Physiology, McGill University, Montreal, Quebec, Canada
Chemical and biological systems often display complex spatiotemporal patterns that can be affected by stimuli delivered directly to the system. In this talk, I will discuss the use of phase maps to help analyze and predict unexpected dynamic behavior in chemistry and biology. In a phase map, each point in a manifold is labelled by a phase, i.e. a point on the unit circle. I assume that the phase maps are continuous except at a finite number of singular points (or perhaps singular lines, planes or hyperplanes). I first consider phase maps on two dimensional, compact, oriented manifolds (perhaps containing holes) that are continuous except at a finite number of singular points. I define the indices of the holes and the singular points. The sum of the indices is zero [1]. This result is applicable to excitable media (such as the Belousov-Zhabotinsky chemical reaction, or cardiac tissue), and helps us to understand the geometry of spiral waves propagating in inhomogeneous spherical shells [2].
As originally emphasized by Art Winfree, phase maps can also be useful to think about the resetting of limit cycle oscillations. A state X, in the basin of attraction of a limit cycle oscillation and a state Y, on the limit cycle oscillation have the same asymptotic phase if the distance between trajectories starting at X and Y at t=0, approach each other in the limit as t goes to infinity. If a perturbation of fixed magnitude to a limit cycle oscillation always leaves one in the basin of attraction of the limit cycle independent of the phase of delivery of the perturbation, then from the continuity of the phase maps, the resulting resetting curves will be continuous. Hence, numerical and wet-lab experimental observations of discontinuities in resetting experiments pose problems for interpretation. I discuss discontiuities in phase resetting curves that arise from perturbations given excitable waves circulating on a one-dimensional ring [3, 4] and from perturbations delivered to spontaneous cardiac oscillations [5].
Although they might seem abstract, these results pertain to practical matters, such as the initiation and termination of potentially fatal cardiac arrhythmias. By emphasizing qualitative topological aspects, I focus on results that should be testable for broad classes of systmes independent of the detailed equations of motion.
[1] L. Glass. Patterns of supernumerary limb regeneration. Science 198, 321-322 (1977).
[2] J. Davidsen, L. Glass, R. Kapral. Topological constraints on spiral wave dynamics in spherical shells with inhomogeneous excitability. Physical Review E (2004, In Press).
[3] L. Glass, M.E. Josephson. Resetting and annihilation of reentrant abnormally rapid heartbeat. Physical Review Letters 75, 2059-2063 (1995).
[4] T. Gedeon, L. Glass. Continuity of resetting curves for FitzHugh-Nagumo equations on the circle. In: Fields Institute Communications: Differential Equations with Applications to Biology, 225-236 (1998).
[5] T. Krogh-Madsen, L. Glass, E. Doedel, M.R. Guevara. Apparent discontinuities in the phase-resetting response of cardiac pacemakers. Journal of Theoretical Biology (2004, In Press).
Date received: September 15, 2004
A Khovanov-type homology theory for graphs
by
Laure Helme-Guizon
George Washington University
Coauthors: Yongwu Rong and Józef Przytycki, George Washington Universsity
In recent years, there have been a great deal of interests in Khovanov homology theory. For each link L in S3, Khovanov defines a family of homology groups whose "graded" Euler characteristic is the Jones polynomial of L. These groups were constructed through a categorification process which starts with a state sum of the Jones polynomial, constructs a group for each term in the summation, and then defines boundary maps between these groups appropriately for each positive integer n. It is natural to ask if similar categorifications can be done for other invariants with state sums.
In the first part of this talk, we establish a homology theory that categorifies the chromatic polynomial for graphs. We show our homology theory satisfies a long exact sequence which can be considered as a categorification for the well-known deletion-contraction rule for the chromatic polynomial. This exact sequence helps us to compute the homology groups of several classes of graphs. In particular, we point out that torsions do occur in the homology for some graphs. This is joint work with Yongwu Rong, George Washington University, Washington DC, USA
The second part of this talk will discuss for which graphs these homology groups have torsion. This is joint work with Yongwu Rong and Józef Przytycki, George Washington University, Washington DC, USA.
Date received: November 11, 2004
Does every triangle admit a periodic billiard path?
by
Patrick Hooper
SUNY Stony Brook and Yale University
Coauthors: Rich Schwartz
I will discuss the current state of this open question. It has long been known that every acute triangle admits a periodic billiard path. The story is somewhat similar for right triangles. For obtuse triangles, the question seems more difficult. Often however, periodic billiard paths can be found by computer experimentation. Motivated by these experiments, Rich Schwartz has shown every triangle with angles less that 100 degrees admits a periodic billiard path. I will discuss this result and obstacles to extending this result further. The program we are developing is available on Rich Schwartz's website: http://www.math.umd.edu/~res/
Date received: November 6, 2004
Topological questions related to DNA self-assembly of graph structures
by
Natasha Jonoska
Department of Mathematics, University of South Florida
Coauthors: Masahico Saito
Recent years have seen the advent of DNA-based computatioin as well as development of DNA nanotechnology. All these are based mainly on DNA self-assembly of three dimensional objects, robust devices and periodic arrays. Main building blocks in these assemblies are branched junction molecules. Arbitrary non-regular graphs can also be constructed by using junction molecules as vertices and duplex DNA molecules as edge connectors. Many NP-complete problems such as 3-SAT can be solved by graph self-assembply in one biostep (one-pot reaction). There are several mathematical problems that arise from these assemblies such as the minimal number of DNA strands needed for construction of the graph. We address this question by considering the DNA strands that make up the graphh as boundaries of a compact orientable surface (2-dimensional manifold) such that the graph is topologically embedded (as 1-complex) as deformation retract. We will also discuss DNA splicing and present a topological model for studying splicing processes.
Date received: October 1, 2004
The Effects of Topology of Evolutionary Tree on Predicting Protein Interaction Specificity
by
Raja Jothi
National Center for Biotechnology Information (NCBI/NLM/NIH)
Coauthors: Teresa Przytycka (NIH/NLM/NCBI)
To perform their function in the cell, proteins need to interact with each other. Predicting protein-protein interactions plays an important role in understanding protein functionalities within the cell. Proteins, in general, can be divided into groups of evolutionarily related proteins called families. It has been observed, that if members of one protein family interact with members of another protein family then the evolution of the two families is correlated. It is also believed that if two families of proteins co-evolved, it is highly likely that those two families interact. Several computational approaches that compare the evolutionary trees of proteins in order to predict potential interactions have been proposed in recent years. A much harder problem is to predict interaction specificity (matching members of one family to specific members of the other family), a largely unsolved problem. In this paper, we study on the effectiveness of topology of evolutionary tree (number of automorphisms in particular) on predicting protein interaction specificities.
Date received: October 12, 2004
Topological constancy in the perception of Lissajous figures
by
Paul C. Kainen
Dept. of Math., Georgetown University
A phenomenon involving mathematical psychophysics is described and an interpretation is proposed which involves the KAM-theory of dynamical systems. Psychophysics studies various quantitative relationships of visual inputs to perceptual output. Mathematical, in this context, means that the input has a simple and explicit mathematical aspect and/or that perception is interpreted using an explicit mathematical model.
The display of a dot of light on a screen with a regular type of linear or rotational oscillation produced by two or more independently controllable mirrors or other deflection devices will cause the percept of a "trace figure" when the ratio between the periods of distinct oscillations is a low-order fraction. Such figures were first studied by Jules Antoine Lissajous and Nathaniel Bowditch and they can have a topological aspect, like a flexible wire moving in 3-dimensional space. It is the apparent constancy of the figure, under distorted frequency ratios and irregularity of oscillation, which is the phenomenon to be studied. In a reverse Doppler fashion, such distortion gives rise to writhing, twisting, and spinning of the figure whose fundamental topology nevertheless remains invariant up to a certain limit of distortion.
A brief description of portions of the Kolomogorov-Arnold-Moser (KAM) theory is given and a possible connection with topological constancy of Lissajous figure perception is considered. It is argued that the veridical perception of a topology as well as of changes in geometry (i.e., shape) show that object constancy does not suffice as an explanation.
Date received: October 24, 2004
Analyzing Mu Transposase's Mechanism Using Tangle Coloring
by
Junalyn Navarra-Madsen
Texas Woman's University, P. O. Box 425886, Denton, TX 76204
Coauthors: Isabel K. Darcy
Colorability as knot or link invariant can be applied to n-string tangles. An n-string tangle, T, with k crossings will generate a system of linear equations with k x (k+n) coefficient matrix. Tangle T is colorable if and only if there is a nontrivial solution to the linear system of equations obtained after coloring T. The n x 2n lower right hand corner submatrix of the standard echelon form of the matrix obtained after coloring T is an invariant of T. Tangle coloring will be utilized in the analysis the mechanism of Mu transposase.
Date received: October 13, 2004
An introduction to Khovanov homology categorification of skein modules
by
Jozef H. Przytycki
George Washington University
Coauthors: Marta M. Asaeda (U.Iowa), Adam S. Sikora (SUNY, Buffalo)
Khovanov homology offers a nontrivial generalization of the Jones polynomials of links in R3 (and of the Kauffman bracket skein modules of some 3-manifolds). We define Khovanov homology of links in R3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to construction of Khovanov homology and utilize the fact that one works with unoriented diagrams (unoriented framed links) in which case there is a long exact sequence of Khovanov homology. Khovanov homology, over the field Q, is a categorification of the Jones polynomial (i.e. one represents the Jones polynomial as the generating function of Euler characteristics). However, for integral coefficients Khovanov homology almost always has torsion. We define Khovanov homology, Hi, j, k for links in products of surfaces and an interval and in twisted I-bundles over unorientable surfaces (excluding RP2). We show how to stratify this homology so that (in the product case, F×I) it categorifies the Kauffman bracket skein module (KBSM) of F×I. That is, for any link L in F×I we can recover coefficients of L in the standard basis B(F) of the KBSM of F×I. In other words if L = Sumb ab(A) b where the sum is taken over all basic elements, b in B(F), then each coefficient ab(A) can be recovered from polynomial Euler characteristics of the stratified Khovanov homology. In the case of unorientable F we are able to recover coefficients ab(A) only partially. We propose another basis of the KBSM for which categorification seems to be possible even for unorientable F (we use cores of Möbius bands several times even if they intersect one another).
Date received: November 13, 2004
A configuration space approach to the chromatic polynomial, after Eastwood and Huggett
by
Yongwu Rong
George Washington University
This talks is motivated by two pieces of recent work, both realizing the chromatic polynomial of graphs using certain homology groups, but using very different approaches.
The first is a Khovanov type categorification for the chromatic polynomial, due to Laure Helme-Guizon and the speaker. This work will be presented at the same conference by L. Helme-Guizon.
The second, the main focus of this talk, is a generalized configuration space construction due to Eastwood and Huggett. They construct, for each positive integer k, and each each graph G, a topological space whose Euler characteristic is P_G(k) where P is the chromatic polynomial. Their construction is natural in the sense that the homology groups of the spaces satisfy a long exact sequence which yileds the well-known deletion-contraction rule.
We will present the work of Eastwood and Huggett and discuss some relations with our own approach.
Date received: November 14, 2004
The Topology of Evolutionary Biology
by
Peter F. Stadler
University Leipzig
The search spaces in combinatorial chemistry as well as the sequence spaces underlying (molecular) evolution are conventionally thought of as graphs. Recombination, however, implies a non-graphical structure of the combinatorial search spaces. Central notions in evolutionary biology are intrinsically topological. This claim is maybe most obvious for the discontinuities associated with punctuated equilibria.
We explore a framework based on generalized topological spaces (defined in terms of a subset of Kuratowski's closure axioms) and show that concepts of immediate biological interest correspond to natural properties of the topology of the underlying spaces. While the topological structure of genotype space is defined by well-known genetic operators, phenotype space inherits its structure from genotype space by virture of the 'genotype-phenotype map' that encapsulates the processes of development and morphogenesis.
The concepts of a 'phenotypic character' and a notion of homology in the spirit of Lewontin's definition as quasi-independent variational units can be derived rigorously from the topological framework.
Date received: September 29, 2004
Topological analysis of enzymatic actions: DNA link formation by Xer recombination and DNA unknotting by type II topoisomerase.
by
Mariel Vazquez
Mathematics Department, U.C. Berkeley
Coauthors: De Witt Sumners, Sean D. Colloms and Javier Arsuaga
DNA topology is the study of geometrical (supercoiling) and topological (knotting) properties of DNA loops and circular DNA molecules. Virtually every reaction involving DNA is influenced by DNA topology, or has topological effects. Site-specific recombinases and topoisomerases are enzymes able to change the topology of circular DNA by breaking the DNA and introducing one or more crossing changes. Mathematical analysis of such changes may provide relevant information about the possible enzymatic pathways, and about DNA conformation at the moment of double-stranded break induction. In this talk I will discuss some of the problems that I am currently interested in, and the topological tools used in their analyses.
First I will talk about Xer recombination and how we applied, and extended, the tangle model for site-specific recombination to propose a unique enzymatic mechanism. I will then present the Java applet TangleSolve that makes the tangle model easily accessible to the interested molecular biologist. Finally, if time permits I will talk about our recent work on DNA unknotting by type II topoisomerase.
Date received: October 13, 2004
Knot Invariants in Protein Structure Comparison
by
Elena Zotenko
NIH/NLM/NCBI and University of Maryland College Park
Coauthors: Teresa M. Przytycka (NIH/NLM/NCBI)
Given a query protein the ability to identify all structurally similar proteins is of primary importance in the study of protein evolution and function. As the number of protein structures grows there is a need to develop screening methods that will perform quick yet accurate filtering of the database before a more computationally expensive protein structure comparison method is applied.
Our long term goal is to develop such screening method for VAST (Vector Alignment Search Tool), a protein structure comparison method used at NCBI. Average crossing number, a geometric invariant based on the writhing number of a knot, was used by several researchers to provide a global descriptor of protein shape [1, 2]. We suggest a screening method that uses average crossing number to detect locally similar substructures. Local similarities are then efficiently combined to detect similar protein structures.
In this talk I will give an overview of protein structure comparison problem and previous work on using average crossing number to compare protein structures. Then I will describe our screening method and the ability of average crossing number to detect locally similar substructures.
References:
[1] G.A. Arteca and O. Tapia. Characterization of fold diversity among proteins with the same number of amino acid residues. J.Chem.Inf.Comput.Sci., 39:642649, 1999.
[2] P. Rogen and B. Henrik. A new family of global protein shape descriptors. Mathematical Biosciences, 182:167181, 2003.
Date received: October 11, 2004
Copyright © 2004 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.