Session W42: Focus Session: The Physics of Genome Folding I: Fractal Globules and Condensed Polymer States

How the genome folds

I describe Hi-C, a novel technology for probing the three-dimensional architecture of whole genomes by coupling proximity-based ligation with massively parallel sequencing. Working with collaborators at the Broad Institute and UMass Medical School, we used Hi-C to construct spatial proximity maps of the human genome at a resolution of 1Mb. These maps confirm the presence of chromosome territories and the spatial proximity of small, gene-rich chromosomes. We identified an additional level of genome organization that is characterized by the spatial segregation of open and closed chromatin to form two genome-wide compartments. At the megabase scale, the chromatin conformation is consistent with a fractal globule, a knot-free conformation that enables maximally dense packing while preserving the ability to easily fold and unfold any genomic locus. The fractal globule is distinct from the more commonly used globular equilibrium model. Our results demonstrate the power of Hi-C to map the dynamic conformations of whole genomes.   

Contact probability scaling of the Hilbert curve

Using Hi-C experiments, it has become possible to measure contact probability scalings for genomic polymers. However, the theoretical analysis of such scalings remains in its infancy. Here, we prove that contact probability scales with linear distance for lattice approximations of the Hilbert curve. These results point to the potential for new theoretical approaches to the study of contact probability, and shed light on the analysis behind the fractal globule, a recent model for the three-dimensional structure of the human genome.   

Territorial Polymers and Large Scale Genome Organization

Chromatin fiber in interphase nucleus represents effectively a very long polymer packed in a restricted volume. Although polymer models of chromatin organization were considered, most of them disregard the fact that DNA has to stay not too entangled in order to function properly. One polymer model with no entanglements is the melt of unknotted unconcatenated rings. Extensive simulations indicate that rings in the melt at large length (monomer numbers) $N$ approach the compact state, with gyration radius scaling as $N^{1/3}$, suggesting every ring being compact and segregated from the surrounding rings. The segregation is consistent with the known phenomenon of chromosome territories. Surface exponent $\beta$ (describing the number of contacts between neighboring rings scaling as $N^{\beta}$) appears only slightly below unity, $\beta \approx 0.95$. This suggests that the loop factor (probability to meet for two monomers linear distance $s$ apart) should decay as $s^{-\gamma}$, where $\gamma = 2 - \beta$ is slightly above one. The later result is consistent with HiC data on real human interphase chromosomes, and does not contradict to the older FISH data. The dynamics of rings in the melt indicates that the motion of one ring remains subdiffusive on the time scale well above the stress relaxation time.   

Diffusion of Particles in the Melt of Polymeric Rings and Diffusion of Proteins in the Cell Nucleus

Ring polymers in the melt are partially collapsed and partially segregated as revealed by both simulation and experiment. This behavior is qualitatively consistent with the arrangement of chromosomes in the cell nucleus which are found in distinct territories. Working under the hypothesis that a melt of nonconcatenated rings serves as a simple model for the packing of chromatin fibers in the nucleus of higher eukaryotes, we have investigated the dynamic behavior of a non-sticky spherical particle in polymer melts composed of rings using molecular dynamics simulation. Linear chains are also studied for comparison. In the case of rings such systems are thought to represent protein diffusion in the cell nucleus. The sub-diffusive motion of the particle is found to be independent of the polymer architecture and chain length but depends strongly on particle size. The long-time behavior suggests that these particles diffuse faster in the rings. We compare our results to existing models of protein diffusion.   

Spatial autocorrelation in fractal and knotted globules

The unknotted fractal globule is a model for the state of nuclear chromatin, and has properties that are distinct from those of knotted globules. One crucial property is a stronger correspondence between one-dimensional position along the polymer contour and three-dimensional position in the bulk. Here we introduce measures of spatial autocorrelation, such as Moran's I and Geary's C, and investigate spatial autocorrelation in both fractal and knotted globules. We show that fractal globules exhibit higher levels of spatial autocorrelation at both small and intermediate length scales. This autocorrelation may be relevant to the study of how proteins find their DNA targets inside the nucleus.   

In silico simulations of polymer condensation: the fractal globule as a metastable state

The fractal globule is a model for the metastable conformation of a long polymer after initial immersion in a poor solvent. Recent experimental findings describing the conformation of the human genome at the megabase scale are consistent with a fractal globule conformation. Here, we simulate the collapse of a polymer in a poor solvent using both molecular dynamics and Monte-Carlo simulations. We show that the statistics of the resulting configuration, such as radius of gyration, end-to-end distance, and contact probability, are independent of the approach used to simulate the condensation process. (* contributed equally).   

Fractal Globule as a model of DNA folding in eukaryotes

A recent study (Lieberman-Aiden et al., Science, 2009) observed that the structure of the genome, on the scale of a few megabases, is consistent with a fractal globule. The fractal globule is a quasi-equilibrium state of a polymer after a rapid collapse. First proposed theoretically in 1988, this structure had never been simulated. Fractal globule was seen as a state, in which each subchain is compact, and doesn't mix with other subchains due to their mutual unentanglement (topological constraints). We use GPU-assisted dynamics to create fractal globules of different sizes and observe their dynamics. Our simulations confirm that a polymer after rapid collapse has compact subchains. We measure the scaling of looping probability of a subchain with it's length, and observe the remarkably robust inverse proportionality. Dynamic simulation of the equilibration of this state show that it exhibits Rose type subdiffusion. Due to diffusion, fractal globule quickly degrades to a quasi-equilibrium state, in which subchains of a polymer are mixed, but topologically unentangled. We propose that separation of spatial and topological equilibration of a polymer chain might have implications in different fields of physics.   

High-order chromatin architecture shapes the landscape of chromosomal alterations in cancer

The rapid growth of cancer genome structural information provides an opportunity for a better understanding of the mutational mechanisms of genomic alterations in cancer and the forces of selection that act upon them. Here we test the evidence for two major forces, spatial chromosome structure and purifying (or negative) selection, that shape the landscape of somatic copy-number alterations (SCNAs) in cancer (Beroukhim et al, 2010). Using a maximum likelihood framework we compare SCNA maps and three-dimensional genome architecture as determined by genome-wide chromosome conformation capture (HiC) and described by the proposed fractal-globule (FG) model (Lieberman-Aiden and Van Berkum et al, 2009). This analysis provides evidence that the distribution of chromosomal alterations in cancer is spatially related to three-dimensional genomic architecture and additionally suggests that purifying selection as well as positive selection shapes the landscape of SCNAs during somatic evolution of cancer cells.