20 research outputs found
Physical Links: Defining and detecting inter-chain entanglement
Fluctuating filaments, from densely-packed biopolymers to defect lines in structured fluids, are prone to become interlaced and form intricate architectures. Understanding the ensuing mechanical and relaxation properties depends critically on being able to capture such entanglement in quantitative terms. So far, this has been an elusive challenge. Here we introduce the first general characterization of non-ephemeral forms of entanglement in linear curves by introducing novel descriptors that extend topological measures of linking from close to open curves. We thus establish the concept of physical links. This general method is applied to diverse contexts: equilibrated ring polymers, mechanically-stretched links and concentrated solutions of linear chains. The abundance, complexity and space distribution of their physical links gives access to a whole new layer of understanding of such systems and open new perspectives for others, such as reconnection events and topological simplification in dissipative fields and defect lines
Knotted vs. Unknotted Proteins: Evidence of Knot-Promoting Loops
Knotted proteins, because of their ability to fold reversibly in the same topologically entangled conformation, are the object of an increasing number of experimental and theoretical studies. The aim of the present investigation is to assess, on the basis of presently available structural data, the extent to which knotted proteins are isolated instances in sequence or structure space, and to use comparative schemes to understand whether specific protein segments can be associated to the occurrence of a knot in the native state. A significant sequence homology is found among a sizeable group of knotted and unknotted proteins. In this family, knotted members occupy a primary sub-branch of the phylogenetic tree and differ from unknotted ones only by additional loop segments. These "knot-promoting" loops, whose virtual bridging eliminates the knot, are found in various types of knotted proteins. Valuable insight into how knots form, or are encoded, in proteins could be obtained by targeting these regions in future computational studies or excision experiments
Simulations of knotting in confined circular DNA
The packing of DNA inside bacteriophages arguably yields the simplest example of genome organisation in living organisms. As an assay of packing geometry, the DNA knot spectrum produced upon release of viral DNA from the P4 phage capsid has been analyzed, and compared to results of simulation of knots in confined volumes. We present new results from extensive stochastic sampling of confined self-avoiding and semi-flexible circular chains with volume exclusion. The physical parameters of the chains (contour length, cross section and bending rigidity) have been set to match those of P4 bacteriophage DNA. By using advanced sampling techniques, involving multiple Markov chain pressure-driven confinement combined with a thermodynamic reweighting technique, we establish the knot spectrum of the circular chains for increasing confinement up to the highest densities for which available algorithms can exactly classify the knots. Compactified configurations have enclosing hull diameter about 2.5 times larger that the P4 calliper size. The results are discussed in relation to the recent experiments on DNA knotting inside the capsid of a P4 tailless mutant. Our investigation indicates that confinement favours chiral knots over achiral ones, as found in the experiments. However, no significant bias of torus over twist knots is found, contrary to the P4 results. The result poses a crucial question for future studies of DNA packaging in P4: is the discrepancy due to the insufficient confinement of the equilibrium simulation or does it indicate that out-of-equilibrium mechanisms (such as rotation by packaging motors) affect the genome organization, hence its knot spectrum in P4
Entanglement complexity of lattice ribbons
We consider a discrete ribbon model for double-stranded polymers where
the ribbon is constrained to lie in a three-dimensional lattice. The
ribbon can be open or closed, and closed ribbons can be orientable or
nonorientable. We prove some results about the asymptotic behavior of
the numbers of ribbons with n plaquettes, and a theorem about the
frequency of occurence of certain patterns in these ribbons. We use this
to derive results about the frequency of knots in closed ribbons, the
linking of the boundary curves of orientable closed ribbons, and the
twist and writhe of ribbons. We show that the centerline and boundary of
a closed ribbon are both almost surely knotted in the infinite-n limit.
For an orientable ribbon, the expectation of the absolute value of the
linking number of the two boundary curves increases at least as fast as
root n, and similar results hold for the twist and writhe
THE WRITHE OF A SELF-AVOIDING WALK
The writhe of a self-avoiding walk in a three-dimensional space is the
average over all projections onto a plane of the sum of the signed
crossings. We compute this number using a Monte Carlo simulation. Our
results suggest that the average of the absolute value of the writhe of
self-avoiding walks increases as n(alpha), where n is the length of the
walks and alpha almost-equal-to 0.5. The mean crossing number of walks
is also computed and found to have a power-law dependence on the length
of the walks. In addition, we consider the effects of solvent quality on
the writhe and mean crossing number of walks
THE WRITHE OF A SELF-AVOIDING POLYGON
We discuss the writhe of a self-avoiding polygon on a lattice, as a
geometrical measure of its entanglement complexity. We prove a rigorous
result about the dependence of the absolute value of the writhe on the
number n of edges in the polygon, and use Monte Carlo methods to
estimate the distribution of the writhe both for all polygons with n
edges and for the subset of polygons that are trefoils
The writhe of knots in the cubic lattice
The writhe of a knot in the simple cubic lattice (Z(3)) can be computed
as the average linking number of the knot with its pushoffs into four
non-antipodal octants. We use a Monte Carlo algorithm to generate a
sample of lattice knots of a specified knot type, and estimate the
distribution of the writhe as a function of the length of the lattice
knots. If the expected value of the writhe is not zero, then the knot is
chiral. We prove that the writhe is additive under concatenation of
lattice knots and observe that the mean writhe appears to be additive
under the connected sum operation. In addition we observe that the mean
writhe is a linear function of the crossing number in certain knot
families
KNOTTING AND SUPERCOILING IN CIRCULAR DNA - A MODEL INCORPORATING THE EFFECT OF ADDED SALT
We consider a model of a circular polyelectrolyte, such as DNA, in which
the molecule is represented by a polygon in the three-dimensional simple
cubic lattice. A short-range attractive force between nonbonded monomers
is included (to account for solvent quality) together with a screened
Coulomb potential (to account for the effect of added salt). We compute
the probability that the ring is knotted as a function of the number of
monomers in the ring, and of the ionic strength of the solution. The
results show the same general behavior as recent experimental results by
Shaw and Wang [Science 260, 533 (1993)] and by Rybenkov, Cozzarelli, and
Vologodskii [Proc. Natl. Acad. Sci. U.S.A. 90, 5307 (1993)] on the knot
probability in circular DNA as a function of added salt. In addition, we
compute the writhe of the polygon and show that this also increases as
the ionic strength increases. The writhe computations model the
conformational behavior of nicked circular duplex DNA molecules in salt
solution
Novel display of knotted DNA molecules by two-dimensional gel electrophoresis.
We describe a two-dimensional agarose gel electrophoresis procedure that improves the resolution of knotted DNA molecules. The first gel dimension is run at low voltage, and DNA knots migrate according to their compactness. The second gel dimension is run at high voltage, and DNA knots migrate according to other physical parameters such as shape and flexibility. In comparison with one-dimensional gel electrophoresis, this procedure segregates the knotted DNA molecules from other unknotted forms of DNA, and partially resolves populations of knots that have the same number of crossings. The two-dimensional display may allow quantitative and qualitative characterization of different types of DNA knots simply by gel velocity