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Phase separated domains (PSDs) are ubiquitous in cell biology, representing nanoregions of high molecular concentration. PSDs appear at diverse cellular domains, such as neuronal synapses but also in eukaryotic cell nucleus, limiting the access of transcription factors and thus preventing gene expression. We develop a generalized cross-linker polymer model, to study PSDs: we show that increasing the number of cross-linkers induces a polymer condensation, preventing access of diffusing molecules. To investigate how the PSDs restrict the motion of diffusing molecules, we compute the mean residence and first escaping times. Finally, we develop a method based on mean-square-displacement of single particle trajectories to reconstruct the properties of PSDs from the continuum range of anomalous exponents. We also show here that PSD generated by polymers do not induces a long-range attracting field (potential well), in contrast with nanodomains at neuronal synapses. To conclude, PSDs can result from condensed chromatin organization, where the number of cross-linkers controls molecular access., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Papale, Holcman. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Chromatin of mammalian nucleus folds into discrete contact enriched regions such as Topologically Associating Domains (TADs). Folding hierarchy and internal organization of TADs is highly dynamic throughout cellular differentiation, and are correlated with gene activation and silencing. To account for multiple interacting TADs, we developed a parsimonious randomly cross-linked (RCL) polymer model that maps high frequency Hi-C encounters within and between TADs into direct loci interactions using cross-links at a given base-pair resolution. We reconstruct three TADs of the mammalian X chromosome for three stages of differentiation. We compute the radius of gyration of TADs and the encounter probability between genomic segments. We found 1) a synchronous compaction and decompaction of TADs throughout differentiation and 2) high order organization into meta-TADs resulting from weak inter-TAD interactions. Finally, the present framework allows to infer transient properties of the chromatin from steady-state statistics embedded in the Hi-C/5C data.

Chromatin moves with subdiffusive and spatially constrained dynamics within the cell nucleus. Here, we use single-locus tracking by time-lapse fluorescence microscopy to uncover information regarding the forces that influence chromatin movement following the induction of a persistent DNA double-strand break (DSB). Using improved time-lapse imaging regimens, we monitor trajectories of tagged DNA loci at a high temporal resolution, which allows us to extract biophysical parameters through robust statistical analysis. Polymer modeling based on these parameters predicts chromatin domain expansion near a DSB and damage extrusion from the domain. Both phenomena are confirmed by live imaging in budding yeast. Calculation of the anomalous exponent of locus movement allows us to differentiate forces imposed on the nucleus through the actin cytoskeleton from those that arise from INO80 remodeler-dependent changes in nucleosome organization. Our analytical approach can be applied to high-density single-locus trajectories obtained in any cell type., (Copyright © 2017 The Author(s). Published by Elsevier Inc. All rights reserved.)

Is it possible to extract the size and structure of chromosomal territories (confined domain) from the encounter frequencies of chromosomal loci? To answer this question, we estimate the mean time for two monomers located on the same polymer to encounter, which we call the mean first encounter time in a confined microdomain (MFETC). We approximate the confined domain geometry by a harmonic potential well and obtain an asymptotic expression that agrees with Brownian simulations for the MFETC as a function of the polymer length, the radius of the confined domain, and the activation distance radius ε at which the two searching monomers meet. We illustrate the present approach using chromosome capture data for the encounter rate distribution of two loci depending on their distances along the DNA. We estimate the domain size that restricts the motion of one of these loci for chromosome II in yeast.

We study the first passage time for a polymer, that we call the narrow encounter time (NETP), to reach a small target located on the surface of a microdomain. The polymer is modeled as a freely joint chain (beads connected by springs with a resting non zero length) and we use Brownian simulations to study two cases: when (i) any of the monomer or (ii) only one can be absorbed at the target window. Interestingly, we find that in the first case, the NETP is an increasing function of the polymer length until a critical length, after which it decreases. Moreover, in the long polymer regime, we identified an exponential scaling law for the NETP as a function of the polymer length. In the second case, the position of the absorbed monomer along the polymer chain strongly influences the NETP. Our analysis can be applied to estimate the mean first time of a DNA fragment to a small target in the chromatin structure or for mRNA to find a small target.

Using a novel theoretical approach, we study the mean first-encounter time (MFET) between the two ends of a polymer. Previous approaches used various simplifications that reduced the complexity of the problem, leading, however, to incompatible results. We construct here for the first time a general theory that allows us to compute the MFET. The method is based on estimating the mean time for a Brownian particle to reach a narrow domain in the polymer configuration space. In dimension two and three, we find that the MFET depends mainly on the first eigenvalue of the associated Fokker-Planck operator and provide precise estimates that are confirmed by Brownian simulations. Interestingly, although many time scales are involved in the encounter process, its distribution can be well approximated by a single exponential, which has several consequences for modeling chromosome dynamics in the nucleus. Another application of our result is computing the mean time for a DNA molecule to form a closed loop (when its two ends meet for the first time).

We review here recent progress to link the nuclear organization to its function, based on elementary physical processes such as diffusion, polymer dynamics of DNA, chromatin and the search mechanism for a small target by double-stranded DNA (dsDNA) break. These physical models and their analysis make it possible to compute critical rates involved in cell reorganization timing, which depend on many parameters. In the framework of polymer models, various empirical observations are interpreted as anomalous diffusion of chromatin at various time scales. The reviewed theoretical approaches offer a framework for extracting features, biophysical parameters, predictions, and so on, based on a large variety of experimental data, such as chromosomal capture data, single particle trajectories, and more. Combining theoretical approaches with live cell microscopy data should unveil some of the still unexplained behavior of the nucleus in carrying out some of its key function involved in survival, DNA repair or gene activation. [ABSTRACT FROM AUTHOR]

We study the mean first time that two monomers, located on the same polymer, encounter in a confined microdomain. Approximating the confined geometry by a harmonic potential well, we obtain an asymptotic expression for the mean first encounter time (MFETC) as a function of the radius ε around one monomer. By studying the end-to-end distance of the polymer in a ball using the Edwards’ formalism, we derive an other estimation of the MFETC. We validate the asymptotic formulas using Brownian simulations and derive their range of validity in terms of the polymer length. We apply the present models to compute the mean time for a gene located far away from a promoter site to be activated during looping in confined genomic territories. [ABSTRACT FROM AUTHOR]

Chromatin inside the cell nucleus consists of the DNA and its hierarchy of interacting molecules, which can be modeled as a complex polymer. To describe the chromatin dynamic, we develop and analyze here a polymer model that accounts for long-range interactions and not just those between the closest neighbors as in the Rouse polymer model. Our construction of the polymer model allows us to recover the local interaction between monomers from the anomalous diffusion exponent, which can be directly measured experimentally. We compute asymptotically for this polymer model the cross-correlation function for a given monomer and the mean time for a loop to be formed. Finally, we discuss some possible applications for interpretation of chromosome capture data. [ABSTRACT FROM AUTHOR]

We present a new approach to investigating the dynamics of loop formation in a very crude polymer model and estimate the mean first time for the two ends to meet. This time depends on the number of monomers N. We obtain analytical formulas when N=3, 4 in dimension two and an asymptotic formula when N is large. Our analysis is confirmed by stochastic simulations. [ABSTRACT FROM AUTHOR]

The time for a DNA sequence to find its homologous counterpart depends on a long random search inside the cell nucleus. Using polymer models, we compute here the mean first encounter time (MFET) between two sites located on two different polymer chains and confined locally by potential wells. We find that reducing tethering forces acting on the polymers results in local decondensation, and numerical simulations of the polymer model show that these changes are associated with a reduction of the MFET by several orders of magnitude. We derive here new asymptotic formula for the MFET, confirmed by Brownian simulations. We conclude from the present modeling approach that the fast search for homology is mediated by a local chromatin decondensation due to the release of multiple chromatin tethering forces. The present scenario could explain how the homologous recombination pathway for double-stranded DNA repair is controlled by its random search step. [ABSTRACT FROM AUTHOR]