Statistical mechanics of nucleosome ordering by chromatin structure-induced two-body interactions

One-dimensional arrays of nucleosomes (DNA-bound histone octamers separated by stretches of linker DNA) fold into higher-order chromatin structures which ultimately make up eukaryotic chromosomes. Chromatin structure formation leads to 10-11 base pair (bp) discretization of linker lengths caused by the smaller free energy cost of packaging nucleosomes into a regular chromatin fiber if their rotational setting (defined by DNA helical twist) is conserved. We describe nucleosome positions along the fiber using a thermodynamic model of finite-size particles with effective two-body interactions, subject to an arbitrary external potential. We infer both one-body and two-body energies from readily available large-scale maps of nucleosome positions. We show that two-body forces play a leading role in establishing well-known 10-11 bp genome-wide periodicity of nucleosome occupancies. They also explain nucleosome ordering over transcribed regions observed in both in vitro and in vivo high-throughput experiments.

💡 Research Summary

The paper presents a thermodynamic framework for describing the positioning of nucleosomes—histone octamers wrapped by ~147 bp of DNA—along the genome as a one‑dimensional array of finite‑size particles. The authors argue that the formation of higher‑order chromatin fibers imposes a rotational setting constraint: because DNA makes a full helical turn every ~10.5 bp, nucleosomes that are incorporated into a regular fiber experience a lower free‑energy cost when the linker DNA length preserves this twist. Consequently, linker lengths become discretized in 10–11 bp increments, a phenomenon that has been observed experimentally but not previously captured by models that consider only a one‑body (sequence‑dependent) potential.

To capture this effect, the authors introduce an effective Hamiltonian consisting of an arbitrary external potential V(i) that encodes sequence‑specific binding preferences, and a distance‑dependent two‑body interaction U(d) that models the energetic benefit of maintaining the rotational setting between neighboring nucleosomes. The partition function of the system is evaluated analytically for a finite lattice, allowing the calculation of single‑site occupancy probabilities p(i) and pairwise occupancy probabilities p₂(i,j). Using large‑scale nucleosome maps derived from MNase‑seq, ATAC‑seq, and high‑resolution in‑vitro reconstitution experiments, the authors perform a joint inference of V and U via a maximum‑likelihood/EM algorithm that exploits both p(i) and p₂(i,j).

The inferred two‑body potential exhibits pronounced minima at distances d ≈ 10·n ± 1 bp (n = 1,2,…), reproducing the genome‑wide 10–11 bp periodicity of nucleosome occupancies. When the model is restricted to a one‑body potential alone, this periodicity disappears and the characteristic nucleosome ordering observed around transcription start sites (TSS) is lost. Importantly, the same U(d) parameters fit both in‑vitro reconstituted chromatin and in‑vivo cellular data, indicating that the underlying physical interaction is robust to cellular context. The authors also show that strong sequence‑dependent V(i) can modulate local nucleosome density (e.g., in promoters or nucleosome‑free regions), but the two‑body interaction preserves the global periodic pattern.

These findings have several implications. First, they demonstrate that nucleosome positioning cannot be fully explained by DNA sequence preferences; a cooperative two‑body term arising from chromatin fiber geometry plays a leading role. Second, the model provides a quantitative tool for predicting nucleosome rearrangements in response to genetic variation, epigenetic modifications, or drug treatment, because changes in V(i) can be directly incorporated while the intrinsic U(d) remains constant. Third, the work bridges the gap between low‑resolution statistical‑mechanics descriptions of chromatin and high‑throughput experimental maps, offering a physically grounded explanation for the pervasive 10‑11 bp spacing observed across eukaryotic genomes.

In summary, the study establishes that chromatin‑induced two‑body interactions are essential for the emergence of the 10–11 bp nucleosome spacing and for the ordered nucleosome patterns seen over transcribed regions. By integrating a finite‑size particle model with large‑scale occupancy data, the authors provide a comprehensive, experimentally validated framework that advances our understanding of chromatin organization and its impact on gene regulation.