The mammalian genome is organized into submegabase-sized chromatin domains (CDs) including topologically associating domains, which have been identified using chromosome conformation capture-based methods. Single-nucleosome imaging in living mammalian cells has revealed subdiffusively dynamic nucleosome movement. It is unclear how single nucleosomes within CDs fluctuate and how the CD structure reflects the nucleosome movement. Here, we present a polymer model wherein CDs are characterized by fractal dimensions and the nucleosome fibers fluctuate in a viscoelastic medium with memory. We analytically show that the mean-squared displacement (MSD) of nucleosome fluctuations within CDs is subdiffusive. The diffusion coefficient and the subdiffusive exponent depend on the structural information of CDs. This analytical result enabled us to extract information from the single-nucleosome imaging data for HeLa cells. Our observation that the MSD is lower at the nuclear periphery region than the interior region indicates that CDs in the heterochromatin-rich nuclear periphery region are more compact than those in the euchromatin-rich interior region with respect to the fractal dimensions as well as the size. Finally, we evaluated that the average size of CDs is in the range of 100–500 nm and that the relaxation time of nucleosome movement within CDs is a few seconds. Our results provide physical and dynamic insights into the genome architecture in living cells.

Dynamic chromatin behavior plays a critical role in various genome functions. However, it remains unclear how chromatin behavior changes during interphase, where the nucleus enlarges and genomic DNA doubles. While the previously reported chromatin movements varied during interphase when measured using a minute or longer time scale, we unveil that local chromatin motion captured by single-nucleosome imaging/tracking on a second time scale remained steady throughout G 1 , S, and G 2 phases in live human cells. This motion mode appeared to change beyond this time scale. A defined genomic region also behaved similarly. Combined with Brownian dynamics modeling, our results suggest that this steady-state chromatin motion was mainly driven by thermal fluctuations. Steady-state motion temporarily increased following a DNA damage response. Our findings support the viscoelastic properties of chromatin. We propose that the observed steady-state chromatin motion allows cells to conduct housekeeping functions, such as transcription and DNA replication, under similar environments during interphase.

20of the 3D genome in cell nuclei. Here, we describe a 4D simulation method, PHi-C (Polymer 21 dynamics deciphered from Hi-C data), that depicts dynamic 3D genome features through 22 polymer modelling. This method allows for demonstrations of dynamic characteristics of 23 genomic loci and chromosomes, as observed in live-cell imaging experiments, and provides 24 physical insights into Hi-C data. 25 Genomes consist of one-dimensional DNA sequences and are spatio-temporally organized 26 within the cell nucleus. Contact frequencies in the form of matrix data, measured using genome-27 wide chromosome conformation capture (Hi-C) technologies, have uncovered three-dimensional 28 (3D) features of average genome organization in a cell population 1, 2 . Moreover, live-cell imaging 29 experiments can reveal dynamic chromatin organization in response to biological perturbations 30 within single cells 3, 4 . Bridging the gap between these different sets of data derived from population 31 and single cells is a challenge for modelling dynamic genome organization 5, 6 . 32Several modelling methods have been developed to reconstruct 3D genome structures and 33 predict Hi-C data 7, 8 . In addition, there has been development of bioinformatic normalization 34 techniques in Hi-C matrix data processing to reduce experimental biases 9-11 . However, the mean-35 ing of a contact matrix as quantitative probability data has not been discussed; moreover, a four-36 dimensional (4D) simulation method to explore dynamic 3D genome organization remains lacking. 37Here, we introduce PHi-C, a method that can overcome these challenges by polymer mod-38 elling from a mathematical perspective and at low computational cost. PHi-C is a method that 39 2 deciphers Hi-C data into polymer dynamics simulations ( Fig. 1a, https://github.com/ 40 soyashinkai/PHi-C). PHi-C uses Hi-C contact matrix data generated from a hic file through 41 JUICER 12 as input ( Supplementary Fig. 1a). PHi-C assumes that a genomic region of interest at 42 an appropriate resolution can be modelled using a polymer network model, in which one monomer 43 corresponds to the genomic bin size of the contact matrix data with attractive and repulsive interac-44 tion parameters between all pairs of monomers described as matrix data (Methods, Supplementary 45 Note). Instead of finding optimized 3D conformations, we can utilize the optimization procedure 46 ( Supplementary Fig. 1b,c) to obtain optimal interaction parameters of the polymer network model 47 by using an analytical relationship between the parameters and the contact matrix. We can then 48 reconstruct an optimized contact matrix validated by input Hi-C matrix data using Pearson's cor-49 relation r. Finally, we can perform polymer dynamics simulations of the polymer network model 50 equipped with the optimal interaction parameters. 51First, we evaluated PHi-C's theoretical assumption about chromosome contact. Here, we 52 started with a simple polymer model called the bead-spring model, in which the characteristic 53 length...

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of L 1 (m) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arcsine distribution for time averages of non-L 1 (m) functions. Here, we provide another distributional behavior of time averages of non-L 1 (m) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-L 1 (m) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-L 1 (m) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-L 1 (m) function in the one-dimensional intermittent maps.

Genomes are spatiotemporally organized within the cell nucleus. Genome-wide chromosome conformation capture (Hi-C) technologies have uncovered the 3D genome organization. Furthermore, live-cell imaging experiments have revealed that genomes are functional in 4D. Although computational modeling methods can convert 2D Hi-C data into population-averaged static 3D genome models, exploring 4D genome nature based on 2D Hi-C data remains lacking. Here, we describe a 4D simulation method, PHi-C (polymer dynamics deciphered from Hi-C data), that depicts 4D genome features from 2D Hi-C data by polymer modeling. PHi-C allows users to interpret 2D Hi-C data as physical interaction parameters within single chromosomes. The physical interaction parameters can then be used in the simulations and analyses to demonstrate dynamic characteristics of genomic loci and chromosomes as observed in live-cell imaging experiments. PHi-C is available at https://github.com/soyashinkai/PHi-C.