Lattice Knots in a Slab

In this paper the number and lengths of minimal length lattice knots confined to slabs of width $L$, is determined. Our data on minimal length verify the results by Sharein et.al. (2011) for the similar problem, expect in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when it is squeezed to a minimal state is determined

Minimal knotted polygons in cubic lattices

An implementation of BFACF-style algorithms on knotted polygons in the simple cubic, face centered cubic and body centered cubic lattice is used to estimate the statistics and writhe of minimal length knotted polygons in each of the lattices. Data are collected and analysed on minimal length knotted polygons, their entropy, and their lattice curvature and writhe

The Compressibility of Minimal Lattice Knots

The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our results show that the compressibility is generally not monotonic, but in some cases increases with pressure. Differences of the compressibility for different knot types show that topology is a factor determining the compressibility of a lattice knot, and differences between the three lattices show that compressibility is also a function of geometry.Comment: Submitted to J. Stat. Mec

Partially directed paths in a wedge

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by $Y = \pm pX$, and an asymmetric wedge defined by the lines $Y= pX$ and Y=0, where $p > 0$ is an integer. We prove that the growth constant for all these models is equal to $1+\sqrt{2}$, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when $p=1$. From these we find asymptotic formulas for the number of partially directed paths of length $n$ in a wedge when $p=1$. The functional recurrences are solved by a variation of the kernel method, which we call the ``iterated kernel method''. This method appears to be similar to the obstinate kernel method used by Bousquet-Melou. This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi $\theta$-functions, and have natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT

Salivary gland-specific <i>P. berghei</i> reporter lines enable rapid evaluation of tissue-specific sporozoite loads in mosquitoes

Malaria is a life-threatening human infectious disease transmitted by mosquitoes. Levels of the salivary gland sporozoites (sgs), the only mosquito stage infectious to a mammalian host, represent an important cumulative index of &lt;i&gt;Plasmodium&lt;/i&gt; development within a mosquito. However, current techniques of sgs quantification are laborious and imprecise. Here, transgenic &lt;i&gt;P. berghei&lt;/i&gt; reporter lines that produce the green fluorescent protein fused to luciferase (GFP-LUC) specifically in sgs were generated, verified and characterised. Fluorescence microscopy confirmed the sgs stage specificity of expression of the reporter gene. The luciferase activity of the reporter lines was then exploited to establish a simple and fast biochemical assay to evaluate sgs loads in whole mosquitoes. Using this assay we successfully identified differences in sgs loads in mosquitoes silenced for genes that display opposing effects on &lt;i&gt;P. berghei&lt;/i&gt; ookinete/oocyst development. It offers a new powerful tool to study infectivity of &lt;i&gt;P. berghei&lt;/i&gt; to the mosquito, including analysis of vector-parasite interactions and evaluation of transmission-blocking vaccines

Punctured polygons and polyominoes on the square lattice

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area generating functions. We show that the critical point is unchanged by a finite number of punctures, and that the critical exponent increases by a fixed amount for each puncture. The increase is 1.5 per puncture when enumerating by perimeter and 1.0 when enumerating by area. A refined estimate of the connective constant for polygons by area is given. A similar set of results is obtained for finitely punctured polyominoes. The exponent increase is proved to be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure

Knotting probabilities after a local strand passage in unknotted self-avoiding polygons

We investigate the knotting probability after a local strand passage is performed in an unknotted self-avoiding polygon on the simple cubic lattice. We assume that two polygon segments have already been brought close together for the purpose of performing a strand passage, and model this using Theta-SAPs, polygons that contain the pattern Theta at a fixed location. It is proved that the number of n-edge Theta-SAPs grows exponentially (with n) at the same rate as the total number of n-edge unknotted self-avoiding polygons, and that the same holds for subsets of n-edge Theta-SAPs that yield a specific after-strand-passage knot-type. Thus the probability of a given after-strand-passage knot-type does not grow (or decay) exponentially with n, and we conjecture that instead it approaches a knot-type dependent amplitude ratio lying strictly between 0 and 1. This is supported by critical exponent estimates obtained from a new maximum likelihood method for Theta-SAPs that are generated by a composite (aka multiple) Markov Chain Monte Carlo BFACF algorithm. We also give strong numerical evidence that the after-strand-passage knotting probability depends on the local structure around the strand passage site. Considering both the local structure and the crossing-sign at the strand passage site, we observe that the more "compact" the local structure, the less likely the after-strand-passage polygon is to be knotted. This trend is consistent with results from other strand-passage models, however, we are the first to note the influence of the crossing-sign information. Two measures of "compactness" are used: the size of a smallest polygon that contains the structure and the structure's "opening" angle. The opening angle definition is consistent with one that is measurable from single molecule DNA experiments.Comment: 31 pages, 12 figures, submitted to Journal of Physics

Confinement of knotted polymers in a slit

We investigate the effect of knot type on the properties of a ring polymer confined to a slit. For relatively wide slits, the more complex the knot, the more the force exerted by the polymer on the walls is decreased compared to an unknotted polymer of the same length. For more narrow slits the opposite is true. The crossover between these two regimes is, to first order, at smaller slit width for more complex knots. However, knot topology can affect these trends in subtle ways. Besides the force exerted by the polymers, we also study other quantities such as the monomer-density distribution across the slit and the anisotropic radius of gyration.Comment: 9 pages, 6 figures, submitted for publicatio