:
Conference on Low-Dimensional Topology and Applications to
Molecular Biology and
Biomedical Mathematics
May 4 - 5, 2007
DNA knots from bacteriophage P4 suggest a chromosomal organization with high writhe values
Javier Arsuaga
Mathematics Department, San Francisco State University
http://math.sfsu.edu/arsuaga/
Extraction of DNA from P4 phage capsids results in a large
proportion of highly knotted DNA circles. These knots are formed
inside the viral capsid and are believed to be driven by the effects
of the confinement. The sole effect of the confinement on the knot
formation predicts an increase of the knotting probability with
increasing chromosome length. However observations by Wolfson et al.
on P4 deletion mutants show that shorter viral chromosomes have
higher knotting probability than longer ones. We here show that
elevated writhe values can account for these differences on knotting
probabilities. First, by combining experimental and computational
methods, we provide evidence for a chiral arrangement of the DNA
inside the viral capsids. Next we perform computational
investigations on how the knotting probability is affected by the
writhe. We conclude that both the writhe and the confinement are
important contributors to the formation of P4 knots.
Topology of Spiral Waves in Excitable Media
Irma Cruz-White
Mathematics Department, Chipola College, Marianna, Florida
http://www.chipola.edu/instruct/math/cruz-white/cruz-white%201.htm
Waves of excitation occur naturally in biological, chemical, and physical
systems. Some examples of excitable media include biological media, including
heart muscle, nerve tissue and Dictyostelium amoebae aggregation, and chemical
media like the unstirred Belousov-Zhabotinsky reagent.
This talk will present necessary and sufficient mathematical conditions for the
formation and existence of spiral excitation wave patterns in 2-dimensional and
3-dimensional excitable media. The mathematical model for excitation waves in
excitable media is based on a phase map that assigns at a fixed time a reaction
phase to spatial locations, and computes the obstruction to extending waves
observed on the boundary of a manifold to waves on the entire manifold.
Modeling protein-DNA complexes using tangles
Isabel Darcy
Department of Mathematics, University of Iowa
http://www.math.uiowa.edu/~idarcy/
Protein-DNA complexes were first mathematically modeled using
tangles in Ernst and Sumners seminal paper, "A calculus for
rational tangles: applications to DNA recombination" (Math Proc
Camb Phil Soc, 1990). A tangle consists of arcs properly embedded in
a 3-dimensional ball. The protein is modeled by the 3D ball while
the segments of DNA bound by the protein can be thought of as arcs
embedded within the protein ball. This is a very simple model of
protein-DNA binding, but from this simple model, much information
can be gained. The main idea is that when modeling protein-DNA
reactions, one would like to know how to draw the DNA. For example,
are there any crossings trapped by the protein complex? How do the
DNA strands exit the complex? Is there significant bending? Tangle
analysis cannot determine the exact geometry of the protein-bound
DNA, but it can determine the overall entanglement of this DNA,
after which other techniques may be used to more precisely determine
the geometry. Software for solving 2-string tangle equations,
TopoICE (part of Rob Scharein's KnotPlot), will be demonstrated.
3-string tangle analysis of Mu transposase will be briefly
discussed.
Tutte Polynomials of Tensor Products of Signed Graphs and their Applications in Knot Theory
Yuanan Diao
Department of Mathematics and Statistics, University of North Carolina, Charlotte
http://www.math.uncc.edu/~ydiao/
It is well-known that the Jones polynomial of an alternating knot is
closely related to the Tutte polynomial of a special graph obtained
from a regular projection of the knot. Relying on the results of
Bollobás and Riordan, we introduce a generalization of Kauffman's
Tutte polynomial of signed graphs for which describing the effect of
taking a signed tensor product of signed graphs is very simple. We
show that this Tutte polynomial of a signed tensor product of signed
graphs may be expressed in terms of the Tutte polynomials of the
original signed graphs by using a simple substitution rule. Our
result enables us to compute the Jones polynomials of some large
non-alternating knots. The combinatorics used to prove our main
result is similar to Tutte's original way of counting "activities"
and specializes to a new, perhaps simpler proof of the known
formulas for the ordinary Tutte polynomial of the tensor product of
unsigned graphs or matroids.
KNOTS: Knowledge-based, Neuroimaging, Optimality, Topology & Statistics
Ivo D. Dinov
Departments of Statistics and Neurology, UCLA
http://www.loni.ucla.edu/~dinov
In this talk I will discuss a variety of computational, biomedical and visualization
challenges. For many of these, I will point to the exceptional contributions of De Witt
Sumners and draw parallels between his theoretical work, its applications and relations to
different biological and neuroscientific efforts. In particular, I will summarize prior
efforts on knowledge-based statistical inference, canonical mapping, symbolic
visualization, optimal function estimation, topological shape characterization and limit
theorems for knot-type distributions. Finally, I will illustrate the interplay between all
these research directions and point how it may lead to novel approaches for shape
representation, modeling and analysis.
Generating 4-regular Hamiltonian Plane Graphs
Claus Ernst
Department of Mathematics, Western Kentucky University
http://www.wku.edu/~claus.ernst/
This talk describes the study of a special class of 4-regular plane
graphs which are Hamiltonian. These graphs are of special interest
in knot theory. An algorithm is presented that randomly generates
such graphs with n vertices with a fixed (and oriented)
Hamiltonian cycle in O(n) time. An exact count of the number of
such graphs with n vertices is obtained and the asymptotic growth
rate of this number is determined. Numerical evidence is presented
to show that the algorithm can be modified to generate these graphs
with a near uniform probability. This can be considered as a first
step in generating large random knots without bias.
Nanotechnology in dimension 4
Ronald Fintushel
Department of Mathematics, Michigan State University
http://www.mth.msu.edu/~ronfint/
One of the key problems in 4-manifold topology is to understand
whether "standard manifolds" admit exotic smooth structures, i.e.
given X, if there are manifolds homeomorphic but not diffeomorphic
to it. I will discuss joint work with Ron Stern and Doug Park on a
'reverse engineering' process which leads to infinite families of
such manifolds where X is S² x S² or CP² # 3(-CP²).
The unknotting number of a knot
Cameron Gordon
Department of Mathematics, University of Texas at Austin
http://www.ma.utexas.edu/text/webpages/gordon.html
The unknotting number u(K) of a knot K is one of the oldest and
most natural knot invariants, but is still quite mysterious. I will
discuss some results about u(K), especially those that make use of
the connection with Dehn surgery. In particular I will describe
joint work with John Luecke determining which generic
Conway-algebraic knots have u(K)=1. I will also discuss some open
questions.
Gin enzyme mechanism and rational 3-tangles
Hugo Cabrera Ibarra
Applied Mathematics and Computational Sciences Division, IPICyT
http://sipicyt.ipicyt.edu.mx/web/curriculaInvestigadoring_new2.Portada?p_cvePersonal=39
The Gin enzyme acts on DNA molecules by attaching to it and binding certain specific sites, micro-graphs of this act show the synaptic
complex as a blob with 3 loops coming out from it. This synaptic complex can be modeled as a 3-string tangle and, under the assumption
that it is a 3-braid, solutions to the mechanism can be obtained.
The writhe of polygonal open curves and its application as an RNA shape descriptor
Christian Laing
Department of Mathematics, Florida State University
http://www.math.fsu.edu/~claing
Given an edge-oriented polygonal graph in 3-dimensional space, we describe a method for computing the writhe
as the average of weighted
directional writhe numbers of the graph in a few directions. These directions are determined by the graph and
the weights are determined
by areas of path-connected open regions on the unit sphere. Within each open region, the directional writhe is
constant. We obtain a
closed formula which extends the formula for the writhe of a polygon in 3-dimensional space, including the
important special case of
writhe of embedded open arcs.
In addition, geometric measures involving combinations of writhe and average crossing numbers of subcurves can
be computed to obtain a set
of features for the purpose of shape characterization. The function of nucleic acids and proteins are
determined to a very large degree by
the 3-dimensional shape of the biomolecule. We apply these writhe-based shape descriptors to RNA tertiary
structures based on the
polygonal carbon-phosphate backbone of the RNA. Also clustering techniques such as multiple discriminant and
principal component analysis
methods are used in the RNA shape classification. Among tRNA structures a differentiation between mesophilic
and thermophilic species is
obtained, and also, a clear distinction between tRNA and diverse ribozymes is produced.
S¹-category of three manifolds
José Carlos Gómez Larrañaga
Centro de Investigacion en Matematicas
We will introduce and discuss results and conjectures about the S¹-category of three manifolds.
DNA Knotting: Biological Consequences and Resolution
Jennifer K. Mann
Department of Mathematics, Florida State University
http://www.math.fsu.edu/~jmann
Cellular DNA knotting is driven by DNA compaction,
topoisomerization, supercoiling-promoted strand collision, and DNA
self-interactions resulting from transposition, site-specific
recombination, and transcription. Type II topoisomerases are the
ubiquitous, essential enzymes that interconvert DNA topoisomers to
resolve knots. These enzymes pass one DNA helix through another by
creating an enzyme-bridged transient break in the DNA. Explicitly
how type II topoisomerases recognize their substrate and decide
where to unknot DNA is unknown. We investigate the physiological
effects of DNA knotting, the biophysics of knotting/unknotting, and
the unknotting mechanism of human topoisomerase IIα.
How many knots are enough?
Kenneth Millett
Department of Mathematics, University of California, Santa Barbara
http://www.math.ucsb.edu/~millett/KM.html
This question addresses the problem of estimating the number of
distinct topological knot types and their proportion in the space of
(equilateral) polygonal knots with a fixed number of edges. For
very small numbers of edges, one knows the number of knot types and
can estimate their proportion but, for larger numbers of edges, only
rough estimates are available. Estimates derive from Monte Carlo
explorations of the (equilateral) polygonal knot space and an
analysis using the HOMFLY polynomial as a surrogate for the
topological knot type. As a consequence, one is interested in
knowing how large a sample of knots is needed to give a good
estimate of the number of topological knot types as reflected by
distinct HOMFLY polynomials. Some theoretical and experimental
efforts concerning this question will be discussed.
Densely Ordered Braid Subgroups
Dale Rolfsen
Department of Mathematics, University of British Columbia
http://www.math.ubc.ca/~rolfsen/
Dehornoy proved in the early 1990's that the braid groups
Bn can be given an ordering which is invariant under
left-multiplication. Dehornoy's remarkable ordering is discrete -- every braid has a unique predecessor and successor. Nevertheless,
certain subgroups of
Bn, under the same ordering are densely ordered, meaning any two elements of the subgroup have a
subgroup element strictly between them. Examples are the commutator subgroup [
Bn,
Bn] and the kernel
of the
Burau representation (for those
n for which it is unfaithful). This seemingly paradoxical result is joint work with Adam Clay.
Measurement of Cortical Thickness
David Rottenberg
Department of Neurology, University of Minnesota
http://www.neurovia.umn.edu/home/dar/
Measurements of cortical thickness have important biomedical
applications including the study of normal cortical development
during infancy and childhood; abnormal cortical development in
genetic disorders; changes associated with normal aging and
neurodegenerative diseases; and the effects of drugs, X-irradiation,
and neurotoxins on the central nervous system. Measurements of
cortical thickness may also serve as surrogate biomarkers in
therapeutic trials.
Regional estimates of cortical thickness-which varies from two to
five millimeters-can be obtained from high-resolution T1-weighted
MRI volumes. This process requires (i) extraction of the brain from
the cranial vault, (ii) correction of the brain volume for image
intensity nonuniformity, (iii) isolation of the cerebral
hemispheres, (iv) extraction of the white-matter surface as a
triangular mesh, (v) extraction (or definition) of the pial surface
as a triangular mesh, (vi) ensuring simple connectivity and
topological correctness of surface meshes, and (vii) computation of
the distance between the white-matter and pial surfaces. These
operations are associated with a variety of methodological errors
and uncertainties. Nevertheless, neuroscientifically useful
measurements have been obtained in normals subjects and in patients
with developmental and degenerative disorders.
Entanglement Complexity of Self-Avoiding Walk
Systems
Chris Soteros
Department of Mathematics and Statistics, University of Saskatchewan
http://math.usask.ca/~soteros/
Sumners and Whittington (1988) investigated questions about
knottedness of a closed curve embedded in the three dimensional
integer lattice Z³ (i.e. self-avoiding polygons on the simple
cubic lattice). They proved that all but exponentially few
sufficiently long self-avoiding polygons are knotted. Using a
self-avoiding polygon to model a ring polymer, this proved the long
standing Frisch-Wasserman-Delbruck conjecture. Motivated by other
questions regarding the entanglement complexity of polymers, the
theoretical approach developed by Sumners and Whittington has been
applied and extended to answer many other questions about the
entanglement complexity of embeddings in Z³. However, at least as
many open questions still remain. One such problem is determining
"good" measures for the entanglement complexity of dense polymer
systems. In 2000, Orlandini et al proposed such a measure based on
taking random tubular sections from the system and using linking
numbers. In this talk, this measure is explored theoretically by
studying the entanglement complexity of a system of self-avoiding
walks confined to an infinite rectangular tube in Z³.
Topological analysis of enzymatic actions
Mariel Vazquez
Mathematics Department, San Francisco State University
http://math.sfsu.edu/vazquez/
DNA topology is the study of geometrical (supercoiling) and
topological (knotting) properties of DNA loops and circular DNA
molecules. Multiple cellular processes, such as DNA replication and
transcription, alter the topology of DNA. Controlling these changes
is key to ensuring stability inside the cell.
Site-specific recombinases and topoisomerases are enzymes able to
change the topology of circular DNA by breaking the DNA molecules
and introducing one or more crossing changes. Mathematical analysis
of such changes may provide relevant information about the possible
enzymatic pathways. I will talk about the analysis of site-specific
recombinases using topological and computational methods.
Arithmetic for Topologists
Stu Whittington
Department of Chemistry, University of Toronto
http://www.chem.utoronto.ca/~swhittin/
There are many interesting questions about the counting embeddings
of graphs with various topological constraints in lattices in three
dimensions. De Witt Sumners has made important contributions to
this field and I shall talk about three such problems. These are
(i) the Frisch-Wasserman-Delbruck conjecture, (ii) embeddings of
collections of circles with given link type, and (ii) almost
unknotted embeddings of theta graphs. I shall also discuss some
open problems in the area.
DNA Entanglement and Resolution: A Matter of Life, Death, and Evolution
Lynn Zechiedrich
Department of Molecular Virology and Microbiology, Baylor College of Medicine
http://www.bcm.edu/labs/zechiedrich/?PMID=1623
DNA must be long enough to encode for the complexity of an organism,
yet thin and flexible enough to fit within the cell. The combination
of these properties greatly favors DNA collisions, which can tangle
the DNA. Despite the well-accepted propensity of cellular DNA to
collide and react with itself, it is not clear what the
physiological consequences are. When cells are broken open, the
classified knots have all been found to be the mathematically
interesting twist knots. These remarkable knots can have very high
knotting node numbers (complexity), but can be untied in only one
strand passage event. We used the Hin site-specific recombination
system to tie twist knots in plasmids in
E. coli cells to
assess the effect of knots on the function of a gene. Knots block
DNA replication and transcription. In addition, knots promote DNA
rearrangements at a rate four orders of magnitude higher than an
unknotted plasmid. These results show that knots are potentially
toxic, and may help drive genetic evolution. The enzymes that untie
knots are the type-2 topoisomerases. How they carry out their
function to unknot and not knot DNA is largely unknown. Although
domains of type-2 topoisomerases have been crystallized and the
atomic structures solved, no complete, intact, active enzyme
structure is known and no co-crystals with DNA have been obtained.
We used electron cryomicroscopy (CryoEM) to generate the first
three-dimensional structure of any intact, active type-2
topoisomerase. Our data suggest a simple one-gate mechanism for
enzyme function.