Generating equilateral random polygons in confinement

Abstract

One challenging problem in biology is to understand the mechanism of DNA packing in a confined volume such as a cell. It is known that confined circular DNA is often knotted and hence the topology of the extracted (and relaxed) circular DNA can be used as a probe of the DNA packing mechanism. However, in order to properly estimate the topological properties of the confined circular DNA structures using mathematical models, it is necessary to generate large ensembles of simulated closed chains (i.e., polygons) of equal edge lengths that are confined in a volume such as a sphere of certain fixed radius. Finding efficient algorithms that properly sample the space of such confined equilateral random polygons is a difficult problem. In this paper we propose a method that generates confined equilateral random polygons based on their probability distribution. This method requires the creation of a large database initially. However, once the database has been created, a confined equilateral random polygon of length n can be generated in linear time in terms of n. The errors introduced by the method can be controlled and reduced by the refinement of the database. Furthermore, our numerical simulations indicate that these errors are unbiased and tend to cancel each other in a long polygon.

Probability Theory of Random Polygons from the Quaternionic Viewpoint

“…Orlandini and Whittington [18] provide an overview of the standard methods, many of which depend on establishing Markov chains on equilateral closed polygon space and then iterating the chain until the resulting distribution on polygon space converges. Moore and Grosberg [17] discuss some potential difficulties with these iterative methods and give a method for explicitly sampling random equilateral closed polygons for small numbers of edges by computing the conditional probability distribution of the (n C 1) st edge based on the choice of the first n edges (see [6,7,8] for conditional probability methods applied to the even more difficult problem of sampling equilateral closed polygons confined to a sphere, and [20] for an alternate approach to generating ensembles of equilateral closed polygons). These conditional probability algorithms are somewhat challenging to implement and fairly slow, requiring high-precision arithmetic and scaling with O.n 3 /.…”

Generating equilateral random polygons in confinement II

“…A new approach using quaternions has been reported recently [3]. In this paper, we continue our earlier research on equilateral random polygons that are confined inside a sphere of fixed radius [5,6]. The motivation of such an equilateral random polygon model is the well known fact of the highly compact packing of genomic material (long DNA chains) inside living organisms observed in macromolecular self-assembly processes in the complex network of interactions that take place in every organism.…”

“…We also give an outline of the general idea in using the conditional probability density functions of confined equilateral random walks (conditioned on that its end points are fixed) to generate a confined equilateral random polygon. These results are either well known results or have been established in [5,6]. But they are essential for this paper to be self-contained and are helpful for the reader to understand our methods and arguments.…”

“…In [5] and [6], we have introduced and studied two models of confined equilateral random polygons. In both models, the polygons are "rooted" at the center of the confining sphere.…”

Computational investigation of DNA packing in confinement

“…Packaging is modeled by randomly generating polygons which fit inside the sphere. In a 'local' model, obeying confinement only matters when the segment might actually breach it [2]. A 'global' model biases the generation of every segment to avoid a future breach of the boundary [3].…”