G.A. Arteca
,
Dpartement de Chimie etBiochimie,
LaurentianUniversity/Universit Laurentienne,
Ramsey Lake Road/Chemin du lac Ramsey,Sudbury, Ontario P3E 2C6
Canada
Configurational transitions during the mechanical stretching of
grafted homo- and heteropolymer single chains.
We consider here the problem of an external force acting on a grafted chain, under conditions that simulate the soft-steering of an atomic-force microscope or an optical tweezer. Such mechanical forces can elicit configurational transitions depending on the chains flexibility, its monomer-monomer interactions, the occurrence of topological defects, and the nature of the external bias (e.g., the stretching frequency) [1,2].
Ingrafted proteins and nucleic acids, single-molecule stretching experiments yield force-extension curves that depend on supercoiling, tertiary and secondary contacts, as well as the occurrence of native and nonnative folds[3]. Extracting this information requires computer simulations and appropriate models for the potential energy functions. In our case, we study these phenomena by using molecular dynamics and Monte Carlo simulations that incorporate a soft stretching caused by periodic perturbations of a terminal bond [1,4]. In this talk, we present recent results where this method is applied to study the stretching and relaxation behaviour of model homopolymers and copolymers in both linear and locally self-entangled (knotted)configurations. In particular, we follow the evolution of entanglement complexity as a function of external stretching frequency, chain length, composition, and the polymers initial configuration.
Some qualitative aspects of the transition mechanism can be probed with self-attracting wormlike chains [4]. Subtler changes associated with local interactions and topological defects require instead all-atom force fields. In the particular case of simple grafted loops with various (initial) transient knot topologies [5], we discuss the effect of chain composition on stability and loop relaxation. For helical oligopeptides [2], we show that even single mutations can affect the ability of the chain to resist an applied force.
[1] G.A. Arteca, Phys. Chem. Chem. Phys.
6
(2004) 3500.
[2] Z. Li and G.A. Arteca, Phys. Chem. Chem. Phys.
7
[3] T.R. Strick, M.-N.Dessinges, G. Charvin, N.H. Dekker, J.-F. Allemand, D. Bensimon,
& V. Croquette, Rep. Prog. Phys.
66
(2003) 1-45.
[4] J.M. Kneller, C. Elvingson, & G.A. Arteca, Chem.
Phys. Lett.
407
(2005) 384.
[5] G.A. Arteca, Chem. Phys. Lett., in press (2006).
Isabel Darcy
Iowa University USA.
Modeling protein-DNA complexes in 3D with TopoICE: Topological
Interactive Construction Engine.
TopoICE is part of Rob Scharein's Knotplot, software for visualizing and manipulating knots in 3d.
TopoICE can be used to model protein-DNA complexes.
Currently TopoICE solves 2-string tangle equations
modeling topoisomerases and recombinases.
We are adding the ability to solve n-string tangle equations to model a protein complex bound to n segments of DNA.
An n-stringt angle consists of n arcs properly embedded in a 3-dimensional ball. A protein-DNA complex can be modelled using tangles. The protein is modelled 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 modelling 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.
We are adding subroutines to help with geometrical modelling. This includes improved tangle energies and a subroutine that searches various looped configurations.
KnotPlot, which includes the TopoICE software, can be freely downloaded from www.knotplot.com/download. Manuals, example files, and a tangle primer can also be found on this website.
Tetsuo Deguchi
Ochanomizu University, Tokyo, Japan
Correlation and scattering functions of random knots.
We discuss the pair correlation function and the scattering function of such random polygons that have a fixed knot type, i.e. random knots. We evaluate the probability distribution function of distance between two nodes of a random knot through computer simulation.
We introduce an expression of the distribution function of distance between two nodes, and we show that it gives good fitting curves to the data. We then obtain analytic expressions of the pair correlation function and the scattering function of random knots. We derive asymptotic behaviors of the scattering function analytically in the large and the small wavelength limits, respectively.
(This work is in collaboration with A. Yao.)
Yuanan Diao
Department of Math and Stat UNC Charlotte
The ropelengths of physical knots
The ropelength of a knot is the minimum amount of rope needed to tie the knot (assuming the rope has a unit thickness). An overview of the knot ropelength problem will be given in this talk. Topics will cover the global minimum ropelength of non trivial knots, general lower and upper ropelength bounds for knots and links, and ropelength bounds for some knot and link classes.
Giovanni Dietler
Laboratoire de Physique de la Matiere Vivante Institut de Physique de la Matiere Complexe
Ecole Polytechnique Fdrale de Lausanne, BSP CH-1015 Lausanne Switzerland
Numerical Simulation of Gel Electrophoresis of DNA Knots in Weak and
Strong Electric Fields
Gel electrophoresis allows to separate knotted DNA (nicked circular) of equal length according to the knot type. At low electric fields, complex knots being more compact, drift faster than simpler knots. Recent
experiments have shown that the drift velocity dependence on the knot type is inverted when changing from low to high electric fields. We present a computer simulation on a lattice of a closed, knotted, charged DNA chain drifting in an external electric field in a topologically restricted medium. Using a Monte Carlo algorithm,the dependence of the electrophoretic migration of the DNA molecules on the knot type and on the electric field intensit is investigated. The results are in qualitative and quantitative agreement with electrophoretic experiments done under conditions of low and high electric fields.
Erika Ercolini
Study of the Fractal Properties of DNA Knots by Atomic Force Microscopy
.
E. Ercolini1, J.Adamčk1, F. Valle, J. Roca, and G. Dietler
DNA samples consisting of a heterogeneous mixture ofDNA knots were studied by Atomic Force Microscopy (AFM). DNA knots were irreversibly adsorbed on freshly cleaved mica exposed to 3- aminopropyltriethoxysilane vapors. The DNA contours were analyzed using a box counting algorithm,giving the knot mass as a function of the box size L. The relation between mass and size is given by Mass ~ Ldf, where df is the fractal dimension of the knot which is in turn related to the scaling exponent by ν=1/df.
This relationship is complicated by the presence of a persistence length of DNA (about 45 nm) which introduces a crossover from a rigid rod behavior to a Self Avoiding Walk behavior. At present, the data
indicate that the fractal dimension is of the order of df =1.7, implying a scaling exponent ν = 0.58, very
close to the renormalization group prediction forlinear self-avoiding polymers in 3 dimensions. The same
kind of study has been done also for DNA knots imaged after deposition onto freshly cleaved mica from a
solution containing Mg+2 ions.
Claus Ernst
Department of Mathematics, Western Kentucky University, Bowling Green,KY 42101.
The Total Curvature of Thick Knots
We study the total curvature of a knot when it is embedded on the cubic lattice. Let K be a knot or link with a lattice embedding of minimum total curvature tau(K) among all possible lattice embeddings of K. Then there exist constants a and b such that a Cr(K)^(1/2)<= tau(\K) <= b Cr(K)$.Furthermore we show that the powers of Cr(K) in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal rope length for a given knot type. Finally, we discuss and compare the differences between the total curvatures of smooth thick knots and lattice knots.
Alexander Grosberg
Physics Department,University of Minnesota, Minneapolis, MN.
Results and insights from simulation of knots: from simple models to protein evolution.
The plan is to present several closely connected results. First, we discuss the cross-over length separating knotted and unknotted regimes in a polymer loop with no excluded volume. Second,we show that the
analogy with self-avoiding statistics, which seems to exist on one side of the cross-over,fails on the other. Third, we show the dependence of the cross-over length on the distribution of segments,including blow up
for quasi-Lorentz distributed segments. Forth, we show the systematic comparison of knots statistics in the compact homopolymers and in protein data bank. We have found few complex knots (including 5_2 in
one protein),their presence in our opinion only confirms the rule that most of knots for some reason have been eliminated (or avoided) by evolution.
Buks Janse van Rensburg
York University,Canada.
Lattice models of vesicles.
In this talk I shall briefly review what we know about lattice models of surfaces and vesicles. While there were significant activity in this area in the early to mid 1990s, not much progress was made since. The basic model ofa lattice vesicle is a closed plaquette surface in the lattice, in three or higher dimensions. The existence of growth constants and free energies in these models are known in some cases,and can be determined using arguments analogous to those in models of self-avoiding walks. The topology of vesicles introduces a complicating factor; and one may consider vesicles with the topology of a handle-body which may be embedded in topogically distinct ways in three dimensions. In these models even the most basic questions have not been answered, and solutions appear to remain outside the scope of current techniques. More progress have been made in models of surfaces with boundaries, which may be knotted. I shall review some of these results. Much of this work was done in collaboration with Stu Whittington.
Stephen Levene
Institute of Biomedical Sciences and Technology and Department of Molecular and Cell Biology, University of Texas at Dallas.
The Lives of DNA Loops, Knots, and Catenanes in 2.5 Dimensions as Revealed by Atomic-force Microscopy
Stephen D.Levene
,* Alexandre A.Vetcher,* Michael Seeligson,* Travis Thompson,* Alexander Y. Lushnikov,Yuri L. Lyubchenko, Robert G. Scharein, and Isabel K.Darcy
Mechanisms of proteins that change DNA topology, such as recombinases and topoisomerases,often involve activities at two widely separated sites along the DNA contour.
This mode of interaction requires the DNA segment between the sites to form a loop, the global, average geometry of which is strongly dependent on the local DNA conformation at the protein-bound sites. Using atomic-force microscopy (AFM) we have investigated looped structures formed by the site-specific recombinases Flp and Cre acting on nicked plasmid DNAs.
In conjunction with the AFM analysis,simulations of looped DNA structures in solution lead to the conclusion that the four-stranded Holliday-junction intermediate in the recombination reaction is likely to be a non-planar, approximately tetrahedral structure. This solution structure deviates strongly from the square-planar intermediate found in Flp and Cre co-crystalstructures. Extensions of our approach to analyzing the polymer topology of other looped DNA structures as well as the properties of DNA knots and catenanes will be discussed.
Davide Marenduzzo
University of Warwick, UK.
Probing the softness of a knot: measuring entropic forces
acting on a polymer close to a wall and between two approaching loops.
We consider a polymer ring with fixed topology -- be it linear, unknot, trefoil or that of a figure-eight knot -- and find the entropic repulsive force which is needed to place a polymer close to a solid wall varies with the
positioning of the chain origin, which is kept pinned. This force is purely entropic in origin as it simply arises from the loss of available structures close to the wall. We show that with present day micromanipulation experiments it should be possible to determine whether the chain is linear or closed via an accurate measurement of these entropic ''approach curves'', and we discuss the possibility to characterize knot topology via the same procedure.
As a second simulated experiment, we measure the entropic force which two unlinked loops feel as they approach each other along a given axis, and again probe the feasibility in principle of a single molecule experiment to determine the topology of a closed chain by approaching to a control loop -- of known topology.
In both cases the calculations are performed with random ring, but we discuss how our results change when self-avoidance is included. These experiments may be thought of as a way to determine how the ''softness'' of a looped chain depends on its knot type.
This is a joint work with Enzo Orlandini.
Ralf Metzler
NORDITA,Blegdamsvej 17, 2100 Copenhagen OE, Denmark.
Localization of DNA Knots in the Dilute Phase and Confined in Nanochannels.
Both from a polymer physics and a molecular biology point of view the localization properties of knots are of interest: Does the knot region of a knotted polymer, containing all topological entanglements, localize into a
small region ofthe remaining simply connected ring polymer, or is it rather spread over the entire chain (delocalized)? This question is closely related to the understanding of the enzyme topoisomerase II that is able to detect and remove knots from DNA molecules. We proved that knots confined to motion in (pseudo)two dimensions localize tightly. By recent experimental techniques, this localization behaviour can be
studied by singleDNA knot imaging through AFM. Apart from the (almost) equilibrated 2D structure of the knot it is also possible to produce a projection of the 3D structure,and from this learn about the properties of the knot in 3D. I will report on recent
results from these measurements. In a second part I will introduce how
knots are formed in DNA molecules that are threaded in nanochannels,
and present a phenomenological model to explain the dynamics of the knot.
The formation of knots during this process is important with respect to
applications such as DNA analysis in nanochannels.
Cristian Micheletti
SISSA, Trieste,Italy.
Knotting of random ring polymers in confined spaces
.
Stochastic simulations are used to characterize the knotting distributions of random ring polymers confined in spheres of various radii. The approach is based on the use of multiple Markov chains and reweighting techniques, combined with effective strategies for simplifying the geometrical complexity of ring conformations without altering their knot type. By these means we extend previous studies and characterize in detail how the the probability to form a given prime or composite knot behaves in terms of the number of ring segments, N, and confining radius, R. For 50 < N < 450 we show that the probability of forming a composite knot rises significantly with the confinement, while the occurrence probability of prime knots are, in general, non-monotonic functions of 1/R. The dependence of other geometrical indicators, such as writhe and chirality, in terms of R and N is also characterized.It is found that the writhe distribution broadens as the confining sphere narrows.
Kenneth C. Millett
Department ofMathematics,University of California Santa Barbara.
Scaling Behavior, Equilibrium Lengths, and Probabilities of Knotted Polymers.
Akos Dobay,Kenneth C. Millett, Michael Piatek, Eric Rawdon, and Andrzej Stasiak
Previous work on
radius of gyration and average crossing number has shown that polymers with afixed topology exhibit different scaling behaviors compared to the averages of
polymers with unrestricted topology. We show that this difference in the scaling behavior also occurs for a
wide range of other spatial characteristics such as total torsion and total
curvature or the radius of the smallest sphere containing the polymer.
In the finite-length regime, these relatively small polymers forming different knot types show distinct scaling profiles with respect
to the given characteristics.
For each knot type, the equilibrium length with respect to the characteristic is defined to be the number of edges of an equilateral polygonal model for the polymer at which the value of the
characteristic is the same as the average for all such polygons.
number has been seen to be correlated to physical properties of macromolecules for the radiusof gyration and average crossing number thereby provoking questions for other spatial characteristics.
We explore
the extent to which these equilibrium lengths are universal
for all spatial characteristics versus being clustered with a class of characteristics.
We determine the chain length for which the probability of forming a given knot type reaches its maximum and compare it with various equilibrium lengths.
Sergei Nechaev
LPTMS, Universite' Paris Sud, 91405 Orsay Cedex, France,
Landau Institutefor Theoretical Physics, 117334 Moscow, Russia .
Statistics of random walks on braid groups and growth of random heaps
The theory of knots since works of Artin, Alexander and others, is traditionally connected to the braid group Bn .
For example, Jones polynomials and a whole subsequent progress in knot theory
is based on representations of braid groups and Hecke algebras. In the talk
we consider topological and statistical properties of random walks on braid group Bn . We introduce a concept ofthe "locally free group", LFn , which is simultaneously a sub-group
and a factor-group of the braid group. Using the locally free group we obtain abilateral approximation of the number of nonequivalent words in the braidgroups and semi-groups. We show that the uniform random walk on locally freegroup and semi-group can be viewed as a standard ballistic deposition process.Hence, some morphological characteristics of a growing "heap" (forexample, the expectation of the density of local maxima and its variation) havedirect connection to statistics of random walk on braid groups and semigroups.We also discuss statistical and topological properties of a bunch of randomlywoven lines in 2+1 dimensions.
Olivier Pierre-Loiuis
CNRS/Laboratoire de Spectrometrie Physique,
Universite' J. Fourier, Grenoble, France,
Stiff Knots.
We report on the geometry and mechanics of stiff knotted strings, i.e.
knotted strings whose shape is dictated by the bending curvature energy.
Stiff knots, such as loose knots with nylon strings, are ordinary ob jects in
everyday life. Both the energy and the tension of stiff knots are found to
depend on the type of knot via a knot invariant called the bridge number.
Moreover, we show that braid localization is a general feature of stiff strings
entanglements. Finally, we observe braid localization on simple stiff knots
from Monte Carlo simulations.
This is a joint work with R. Gallotti.
Renzo Ricca
Mathematics Dept., U. Milano-Bicocca, via Cozzi 53,20125 Milano, ITALY
and Mathematics Dept., University College London, Gower St,. London WC1E 6BT.
Twist and fold modelling of supercoiled filaments.
Competing kinematic models for twist and fold
mechanisms of supercoiled filaments are presented and analysed here byexamining the coiling process of a circular filament in terms of curvature,writhing and twist deformation. The analysis is performed by using a simple
thin filament model of circular cross-section under conservation of linking
number. For simplicity, elastic energy contents are evaluated by using
only first quadratic contributions from bending and twist terms. Coil formation
is examined by using
different time-dependent evolutions that mimic transition of twist to writhe. Effects on the writhing mechanism are compared in relation to the number of
inflexional states that originate during the process and it is shown that this influences significantly the localization of elastic energy and the subsequent
relaxation of torsional energy to bending energy. Inflexional configurations are shown to be consistent with the generic behaviour
discovered by Moffatt and Ricca (1992). The intrinsic(internal) twist of the filament fibers about the central axis is shown to release most of its twist during passage through inflexional configuration,with torsion effects becoming a dominant factor afterwards.
This analysis complements a previous study (Ricca,1995) of the elastic relaxation of highly twisted supercoiled filaments and has also interesting consequences for estimates on the efficiency of filament packing into small volumes. These results find useful applications in modeling natural phenomena such as DNA packing in cell biology and provide useful information on structural complexity of tangled filamentary systems(Ricca, 2005).
Moffatt, H.K. and Ricca, R.L. 1992 Helicity and the Calugareanu invariant. Proc. R. Soc.
Lond. A 439, 411-429. [Also in: 1995 Knots and Applications (ed. L.H. Kauffman),
pp. 251-269. World Scientific, Singapore.]
Ricca, R.L. 1995 The energy spectrum of a twisted flexible string under elastic
relaxation. J. Phys. A: Math. & Gen. 28,2335-2352.
Ricca, R.L. 2005 Structural complexity. In Encyclopedia of Nonlinear Science (ed. A.
Scott),pp. 885-887. Routledge, New York and London.
Andrzej Stasiak
University ofLausanne, Laboratory of Ultrastructural Analysis, Lausanne.
Natural classification of knots.
AlessandroFlammini & Andrzej Stasiak.
One of the principal objectives of knot theory is to provide a simple way of classifying and ordering all the knot types. Probably, the simplest approach consists of ordering the knots according to their rope length i.e. the length/diameter
ratio of the shortest possible cylindrical tube that can form a given knot (1).However, such a simple ordering of knots does not convey the information about relationships between different knots and does not divide them into different groups or families. We propose here a natural classification of knots based on their intrinsic position in the knot space that is defined by an unique set of knots to which a given knot can be converted by individual intersegmental passages (2). In an analogy to the classification of elementary particles we characterize various knots using a set of simple quantum numbers that can be determined upon an inspection of minimal crossing diagram of a knot. These numbers include: crossing number, 3-D writhe, number of topological domains and the average relaxation value.
1. Katritch,V., Bednar, J., Michoud, D.,Scharein, R. G., Dubochet, J. & Stasiak, A. Geometry and physics of knots. Nature, 384, 142-145, 1996.
2. Flammini, A.,Maritan, A., & Stasiak, A. Simulations of action of DNA topoisomerases to investigate boundaries and shapes of spaces of knots. Biophysical Journal, 87,2968-2975, 2004.
John M Sullivan.
Inst. f.Mathematik, MA 3-2, TU Berlin,Str. des 17. Juni, 136 ,10623 Berlin,Germany.
Ropelength and Distortion of Knots using Essential Secants
Following Kuperberg, we define which subarcs of a knotted curve are (topologically)essential, creating the knottedness. Using a result of Denne---that knots haveessential alternatine quadrisecants---we can show the ropelength of any knot isat least 15.66, within 5 percent of the known upper bound for the trefoil.
By considering the shortest essential arc of a knot, we can show that the
Gromov distortion is at least 3.99, more than twice the value for an
unknotted circle.
This is largely joint
work with Elizabeth Denne and YuananDiao.
De Witt Sumners
Department ofMathematics, Florida State University.
DNA Knots Reveal Chiral Packing of DNA in Phage Capsids.
Bacteriophages are viruses that infect bacteria. They pack their double-stranded DNA genomes to near-crystalline density in viral capsids and achieve one of the highest levels of DNA condensation found in nature. Despite numerous studies some essential properties of the packaging geometry of the DNA inside the phage capsid are still unknown.
Although viral DNA is linear double-stranded with sticky ends, the linear viral DNA quickly becomes cyclic when removed from the capsid, and for some viral DNA the observed knot probability is an astounding 95%. This talk will discuss comparison of the observed viral knot spectrum with the simulated knot spectrum, concluding that the packing geometry of the DNA inside the capsid is non-random and writhe-directed.