Topological Constraints and Finite-Size Effects in Quantitative Polymer Models of Chromatin Organization

Abstract

Polymer physics simulations have provided a versatile framework to quantitatively explore the complex mechanisms driving chromosome organization. However, simulating whole chromosomes over biologically relevant time scales at a high resolution often constitutes a computationally intensive task, while genes or other regions of biological interest may typically only span a small fraction of the full chromosome length. Conversely, only simulating the subchromosomal region of interest might provide an oversimplistic or even wrong description of the mechanism controlling the 3D organization. In this work, we characterize what should be the minimal length of the chromosome to be simulated in order to correctly capture the properties of a given restricted region. In particular, since the physics of long, topologically constrained polymers may significantly deviate from those of shorter chains, we theoretically investigate how chromosomes being a long polymer quantitatively affects the structure and dynamics of their subsegments. We show that increasing the total polymer length impacts on the topological constraints acting on the system and thus influences the compaction and mobility of subchains. Depending on the entanglement properties of the system, we derive a phenomenological relation defining the minimal total length to account for to maintain a correct topological regime. We finally detail the implications of these conclusions in the case of several biological systems.

Transcription regulates the spatio-temporal dynamics of genes through micro-compartmentalization

“…Previous biophysical models have already addressed some aspects of Pol II-mediated phase separation via attractive 35,48 or active forces 86 , focusing on large-scale inter-gene condensation, but never investigating intra-gene organization nor explicitly accounting for the transcription dynamics. By assuming self-attractive, short-range interactions between genomic loci bound to Pol II 35,48 , our approach is able to recapitulate qualitatively the overall augmentation of intra-gene contacts associated to an enrichment of Pol II density inside gene body and to longer genes, consistent with a standard cooperative coil-globule transition observed for finite-size chains 71,87–89 . Our model suggests that limiting the number of possible interactions per Pol II-bound region to low values (e.g., 2 or 3) allows to align quantitatively our predictions with experiments, leading to percolated but less condensed 3D domains 90,91 .…”

DNA replication and polymer chain duplication reshape the genome in space and time

“…For example, we simulate chromatin as a topologically-constrained, self-avoiding walk, excluding chain crossing and thus possible effects of Topoisomerases that may act in vivo. In particular, this might have possible consequences on the polymer brush effect and the relative catenation of sister chromatids by regulating the degree of entanglements [58]. However, our framework is modular and offers the possibilities to integrate various biologically-relevant ingredients and to refine and contextualize our generic conclusions to specific biological systems by implementing more realistic 1D replication dynamics and probing their 3D consequences.…”

“…For example, we simulate chromatin as a topologically-constrained, self-avoiding walk, excluding chain crossing and thus possible effects of Topoisomerases that may act in vivo . In particular, this might have possible consequences on the polymer brush effect and the relative catenation of sister chromatids by regulating the degree of entanglements [58].…”

“…Chromatin is described as a self avoiding walk composed of L beads and dynamically evolving on a SxSxS fcc lattice with periodic boundary conditions via a Kinetic Monte Carlo algorithm, as previously described [21, 58]. Briefly, in addition to excluded volume, the polymeric chain is subjected to a standard potential to account for the chain bending rigidity: with θ i being the angle between monomers I − 1, i and i + 1 and κ the bending modulus in kT units.…”

“…S = 16, a ssuming a typical base-pair density of 0.007 bp/nm 3 (6% volumic fraction initially) in yeast nuclei. κ = 2.65 kT , assuming a Kuhn length for chromatin of 100 nm [21, 63, 58]. Similarly to [21, 64, 65], time mapping between the simulation Monte Carlo time step ( MCS ) and real time is done by computing the Mean Squared Displacement ( MSD ) of a monomer.…”