Modeling of chromosome intermingling by partially overlapping uniform random polygons

Abstract

During the early phase of the cell cycle the eukaryotic genome is organized into chromosome territories. The geometry of the interface between any two chromosomes remains a matter of debate and may have important functional consequences. The Interchromosomal Network model (introduced by Branco and Pombo) proposes that territories intermingle along their periphery. In order to partially quantify this concept we here investigate the probability that two chromosomes form an unsplittable link. We use the uniform random polygon as a crude model for chromosome territories and we model the interchromosomal network as the common spatial region of two overlapping uniform random polygons. This simple model allows us to derive some rigorous mathematical results as well as to perform computer simulations easily. We find that the probability that one uniform random polygon of length n that partially overlaps a fixed polygon is bounded below by 1 − O(1/√n). We use numerical simulations to estimate the dependence of the linking probability of two uniform random polygons (of lengths n and m, respectively) on the amount of overlapping. The degree of overlapping is parametrized by a parameter [Formula: see text] such that [Formula: see text] indicates no overlapping and [Formula: see text] indicates total overlapping. We propose that this dependence relation may be modeled as f (ε, m, n) = [Formula: see text]. Numerical evidence shows that this model works well when [Formula: see text] is relatively large (ε ≥ 0.5). We then use these results to model the data published by Branco and Pombo and observe that for the amount of overlapping observed experimentally the URPs have a non-zero probability of forming an unsplittable link.

Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics

“…This makes an obvious analogy with the melt of nonconcatenated rings; indeed, if the rings in the simulated melt are shown in different colors, the image of a political map emerges 20 . This strongly suggests topological properties of chromatin chains as a likely mechanism behind chromosome segregation, similar to segregation of non-concatenated rings in the melt [21][22][23] . Why the topology of the chromatin fibers is restricted is a somewhat open question, but it might be that the DNA ends are attached to the nuclear envelope, or simply that the cell lifetime is not long enough for reptation to develop [24][25][26] .…”

Uncovering Proximity of Chromosome Territories using Classical Algebraic Statistics

“…These models are based on different properties of the genome that can obtained from basic physical priciples such as the radial organization of chromosomes using overlapping sphere or ellipsoid packings [17,28,48] , gene density [30] or DNA decondensation processes [42] or through the folding of chromatin fibers [4,8,33] Acknowledgements This work is partially supported by NSF grant DMS-1217324 and NIH grant RO1-GM109457 to J. Arsuaga. We want to thank J. L. García from Centro de Investigació del cancer de La Universidad de Salamanca (Spain) for sharing Figure 1.…”

Chromosome Territory Organization within the Nucleus

“…The development of quantitative models of the CT structure, and their integration into the functional nuclear organization, are still limited by a lack of quantitative, high-resolution data regarding the size distribution and possible hierarchies of chromatin fibers, loops, and domains [160][161][162][163][164]. On the way towards a full understanding of the dynamic, nuclear organization in living cells, it is important to recognize artifacts of fixation and subsequent treatment protocols.…”