Statistical model of intra-chromosome contact maps

The statistical properties of intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) are described in the framework of the hierarchical crumpling model of heteropolymer chain with quenched disorder in the primary sequence. We conjecture that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding, and show that the existence of quenched primary structure coupled with hierarchical folding can induce the observed fine structure of intra-chromosome contact maps. The analysis of DNA folding in human genome based on the genome-wide chromosome conformation capture method [1, 2] provides a comprehensive information on spatial contacts between genomic parts and imposes essential restrictions on available 3D genome structures. The Hi-C maps obtained in the experiments on various organisms and tissues [2–8] (some examples are shown in the Fig.1A-C) define the average contact probability, P(s) between two particular parts of genome separated by the (genomic) distance s along a chain. These maps display very rich structure in a broad interval of scales, where P(s) ∼ 1/s (see the Fig.1D) in consistency with the so-called fractal (or " crumpled ") globule [9] model of polymer chain packing. The latter is characterized by two main features: (i) it has almost no knots, and (ii) it is self-similar in a wide range of scales, forming a fractal space-filling structure. Both these properties are essential for genome folding: fractal organization makes genome tightly packed at all scales, while the lack of knots ensures easy and independent opening and closing of genomic domains, necessary for transcription [10, 11]. In 3D this packing results in a space-filling conformation of a chain with fractal dimension D f = D = 3. The Hi-C contact probability, P i,j , between two ge-nomic units i and j depends in a complicated way on a combination of structural and energetic factors. However , in a fractal globule with D f = 3 the average contact probability, P(s) = (N − s) −1 N −s i=0 P i,i+s , between two units separated by the genomic distance s = |i − j| should decay as P(s) ∼ s −1. This probability can be naively estimated as P(s) ∼ 1/R D (s), where R(s) is the typical size occupied by the s–fragment. For curves with the fractal dimension D f , one has R(s) ∼ s 1/D f , hence, P(s) ∼ s −D/D f , and for …