Various single-molecule studies have been carried out to probe the stability of nucleosomes and the interactions between histone proteins and DNA.^{42, 58, 60–62} The DNA-unzipping experiment performed by Hall et al.⁵⁸ is particularly relevant since the measured forces can be converted into a free energy profile of DNA unwinding at a base-pair resolution, as shown by Forties et al. with a continuous-time Markov model.⁶³ The high-resolution quantification of nucleosome energetics is valuable for benchmarking the accuracy of computational models. We introduce a coarse-grained explicit-ion model for chromatin simulations (Fig. 1).

The model represents each amino acid with one coarse-grained bead and three beads per nucleotide. It resolves the differences among various chemical groups to accurately describe biomolecular interactions with physical chemistry potentials. Our explicit representation of monovalent and divalent ions enables a faithful description of counter ion condensation and its impact on electrostatic interactions between protein and DNA molecules. Additional model details are provided in the Materials and Methods and the Supporting Information.

We performed umbrella simulations⁶⁴ to determine the free energy profile of nucleosomal DNA unwinding. The experimental buffer condition of 0.10M NaCl and 0.5mM MgCl₂ ⁵⁸ was adopted in simulations for direct comparison. As shown in Fig. 2B, the simulated values match well with experimental results over a wide range. Furthermore, we computed the binding free energy for a diverse set of protein-DNA complexes and the simulated values again match well with experimental data (Fig. S1), supporting the model’s accuracy.

Counterions are often released upon protein-DNA binding to make room for close contacts at the interface, contributing favorably to the binding free energy in the form of entropic gains.⁶⁵ However, previous studies have shown that the histone-DNA interface in a fully wrapped nucleosome configuration is not tightly sealed but instead permeated with water molecules and mobile ions. ^{66, 67} Given their presence in the bound form, how these counterions contribute to nucleosomal DNA unwrapping remains to be shown. We calculated the number of DNA-bound cations and protein-bound anions as DNA unwraps. Our results, shown in Fig. 2C, indicate that only a modest amount of extra Na⁺ and Cl⁻ ions becomes associated with the nucleosome as the outer DNA layer unwraps. However, significantly more ions become bound when the inner layer starts to unwrap (after 73 bp). These findings suggest that counterion release may contribute more significantly to the inner layer wrapping, potentially caused by a tighter protein-DNA interface.

In addition to contributing to the stability of individual nucleosomes, counterions can also impact higher-order chromatin organization. Numerous groups have characterized the structures of nucleosome arrays,^{59, 68–72} revealing a strong dependence of chromatin folding on the concentration and valence of cations.

To further understand the role of counterions in chromatin organization, we studied a 12-mer with 20-bp-long linker DNA under different salt conditions. We followed the experiment setup by Correll et al.⁵⁹ that immerses chromatin in solutions with 5mM NaCl, 150mM NaCl, 0.6mM MgCl₂, or 1mM MgCl₂. To facilitate conformational sampling, we carried out umbrella simulations with a collective variable that quantifies the similarity between a given configuration and a reference two-start helical structure. Simulation details and the precise definition of the collective variable are provided in Supporting Information. Data from different umbrella windows were combined together with proper reweighting⁷³ for analysis. As shown in Fig. 3A, the average sedimentation coefficients determined from our simulations match well with experimental values. Specifically, the simulations reproduce the strong contrast in chromatin size between the two systems with different NaCl concentrations. Chromatin under 5 mM NaCl features an extended configuration with minimal stacking between 1-3 nucleosomes (Fig. 3B). On the other hand, the compaction is evident at 150 mM NaCl.

Notably, in agreement with previous studies,^74–77 we observe tri-nucleosome configurations as chromatin extends. Finally, the simulations also support that divalent ions are more effective in packaging chromatin than NaCl. Even in the presence of 0.6 mM MgCl₂, the chromatin sedimentation coefficient is comparable to that obtained at 150 mM of NaCl.

We further characterized ions that are in close contact with DNA to understand their impact on chromatin organization. Our simulations support the condensation of cations, especially for divalent ions (Fig. 3A bottom) as predicted by the Manning theory.^{78, 79} Ion condensation weakens the repulsion among DNA segments that prevents chromatin from collapsing. Notably, the fraction of bound Mg²⁺ is much higher than Na⁺. Correspondingly, the amount of neutralized negative charges is always greater in systems with divalent ions, despite the significantly lower salt concentrations. The difference between the two types of ions arises from the more favorable interactions between Mg²⁺ and phosphate groups that more effectively offset the entropy loss due to ion condensation.⁷⁹ While higher concentrations of NaCl do not dramatically neutralize more charges, the excess ions provide additional screening to weaken the repulsion among DNA segments, stabilizing chromatin compaction.

Encouraged by the explicit ion model’s accuracy in reproducing experimental measurements of single nucleosomes and nucleosome arrays, we moved to directly quantify the strength of inter-nucleosomes interactions. Accurate quantification is essential for conceptualizing the importance of physicochemical interactions for chromatin organization in vivo, since prior studies have proved inconclusive, producing a wide range of values from 2 to 14 k_BT .^{41, 42, 44} To reconcile the discrepancy among various experimental estimations, we directly calculated the binding free energy between a pair of nucleosomes with umbrella simulations. We adopted the same ionic concentrations as in the experiment performed by Funke et al.⁴⁴ with 35 mM NaCl and 11 mM MgCl₂. We focus on this study since the experiment directly measured the inter-nucleosomal interactions, allowing straightforward comparison with simulation results. Furthermore, the reported value for nucleosome binding free energy deviates the most from other studies. In one set of umbrella simulations, we closely mimicked the DNA origami device employed by Funke et al. to move nucleosomes along a predefined path for disassociation (Fig. 4A, A1 to A3). For example, neither nucleosome can freely rotate (Fig. S3); the first nucleosome is restricted to the initial position, and the second nucleosome can only move within the Y-Z plane along the arc 15 nm away from the origin. For comparison, we performed a second set of independent simulations without imposing any restrictions on nucleosome orientations. Additional simulation details can be found in the Supporting Information.

Strikingly, the two sets of simulations produced dramatically different binding free energies. Restricting nucleosome orientations produced a binding free energy of ∼ 2k_BT, reproducing the experimental value (Fig. 4B). On the other hand, the binding free energy increased to 15 k_BT upon removing the constraints.

Further examination of inter-nucleosomal contacts revealed the origin of the dramatic difference in nucleosome binding free energies. As shown in Fig. 4C, the average number of contacts formed between histone tails and DNA from different nucleosomes is around 150 and 10 in the two sets of simulations. A similar trend is observed for histone core-DNA contacts across nucleosomes. The differences are most dramatic at small distances (Fig. S4) and are clearly visible in the most stable configurations. For example, from the unrestricted simulations, the most stable binding mode corresponds to a configuration in which the two nucleosomes are almost parallel to each other (see configuration A1’ in Fig. 4A), with the angle between the two nucleosome planes close to zero (Fig. S5C). However, the inherent design of the DNA-origami device renders this binding mode inaccessible, and the smallest angle between the two nucleosome planes is around 23^◦ (see configuration A1 in Fig. 4A). Therefore, a significant loss of inter-nucleosomal contacts caused the small binding free energy seen experimentally.

The predicted strength for unrestricted inter-nucleosome interactions supports their significant contribution to chromatin organization in vivo. However, the salt concentration studied above and in the DNA origami experiment is much higher than the physiological value.⁴³ To further evaluate the in vivo significance of inter-nucleosome interactions, we computed the binding free energy at the physiological salt concentration of 150 mM NaCl and 2 mM of MgCl₂.

We observe a strong dependence of nucleosome orientations on the inter-nucleosome distance. A collective variable, θ, was introduced to quantify the angle between the two nucleosomal planes (Fig. 5A). As shown in two-dimensional binding free energy landscape of internucleosome distance, r, and θ (Fig. 5B), at small distances (∼60 Å), the two nucleosomes prefer a face-to-face binding mode with small θ values. As the distance increases, the nucleosomes will almost undergo a 90^◦ rotation to adopt perpendicular positions. Such orientations allow the nucleosomes to remain in contact and is more energetically favorable. The orientation preference gradually diminishes at large distances once the two nucleosomes are completely detached. Importantly, we observed a strong inter-nucleosomal interaction with two nucleosomes wrapped by 147-bp 601-sequence DNA (∼ 9 k_BT).

Furthermore, we found that the nucleosome binding free energy is minimally impacted by the precise DNA sequence. For example, when the 601 sequence is replaced with poly-dA:dT or poly-dG:dC, the free energy only varied by ∼2 k_BT (Fig. S6). However, the poly-dA:dT sequence produced stronger binding while poly-dG:dC weakened the interactions. The sequence specific effects are potentially due to the increased stiffness of poly-dA:dT DNA,⁸⁰ which causes the DNA to unwrap more frequently, increasing cross nucleosome contacts at larger distances (Fig. S7).

In addition to variations in DNA sequences, in vivo nucleosomes also feature different linker lengths. We performed simulations that extend the 601 sequence with 10 extra base pairs of poly-dA:dT sequence at each end, reaching a nucleosome repeat length (NRL) of 167 bp. Consistent with previous studies,^{59, 81, 82} increasing the NRL weakened inter-nucleosomal interactions (Fig. 5C and Fig. S8), reducing the binding free energy to ∼ 6 k_BT . However, the weakened interactions can be more than compensated with the presence of linker histones (Fig. 5C and Fig. S8). As shown in the example configuration (Fig. 5D), the disordered C-terminal tail of linker histones can extend and bind the DNA from the second nucleosome to stabilize the internucleosomal contacts.

The large system size of chromatin and the slow timescale for its conformational relaxation necessitates coarse-grained modeling. Following previous studies,^{5, 32, 74, 75, 87} we adopted a residue-level coarse-grained model for efficient simulations of chromatin. The structure-based model^{88, 89} was applied to represent the histone proteins with one bead per amino acid and to preserve the tertiary structure of the folded regions. The disordered histone tails were kept flexible without tertiary structure biases. A sequence-specific potential, in the form of the Lennard Jones (LJ) potential and with the strength determined from the Miyazwa-Jernigan (MJ) potential,⁹⁰ was added to describe the interactions between amino acids. The 3SPN.2C model was adopted to represent each nucleotide with three beads and interactions among DNA beads follow the potential outlined in Ref. 91, except that the charge of each phosphate site was switched from -0.6 to -1.0 to account for the presence of explicit ions. The Coulombic potential was applied between charged protein and DNA particles. In addition, a weak, non-specific LJ potential was used to account for the excluded volume effect among all protein-DNA beads. Detail expressions for protein-protein and protein-DNA interaction potentials can be found in Ref. 74 and the Supporting Information.

To accurately account for electrostatic interactions, we included explicit particle-based representations for monovalent and divalent ions (Fig. 1). The ions interact with nucleotide particles following the potential introduced in Ref. 53, with detailed expressions provided in the Supporting Information. Parameters in these potentials were tuned to reproduce the radial distribution functions and the potential of mean force between ion pairs determined from all-atom simulations.

This explicit ion model was originally introduced for nucleic acid simulations. We generalized the model for protein simulations by approximating the interactions between charged amino acids and ions with parameters tuned for phosphate sites. Parameter values for ion-amino acid interactions are provided in Table S1.

We simulated various chromatin systems, including a single nucleosome, two nucleosomes, and a 12-mer nucleosome array. The initial configurations for the molecular dynamics simulations were constructed based on the crystal structure of a single nucleosome with PDB ID: 1KX5⁶⁶ and 3LZ1,⁹² or a tetranucleosome with PDB ID: 1ZBB.⁹³ We used the 3DNA software⁹⁴ to add additional DNA, connect and align nucleosomes, and extend the chain length as necessary. Further details on constructing the initial configurations are provided in the Supporting Information. Chromatin was positioned at the center of a cubic box with a length selected to avoid interactions between nucleosomes and their periodic images. Counterions were added on a uniformly spaced grid to achieve the desired salt concentrations and to neutralize the system. The number of ions and the size of simulation boxes are provided in Table S2.

All simulations were performed at constant temperature and constant volume (NVT) using the software package LAMMPS.⁹⁵ The electrostatic interactions were implemented with the particle-particle particle-mesh solver, with the relative root-mean-square error in per-atom force set to 0.0001.⁹⁶ A Nośe-Hoover style algorithm⁹⁷ was used to maintain the system temperature at 300K with a damping parameter of 1 ps. We further modeled the histone core and the inner layer of the nucleosomal DNA together as a rigid body to improve computational efficiency. This approximation does not affect the thermodynamic properties of chromatin.^{74, 75} Umbrella simulations were used to enhance the sampling of the conformational space,⁶⁴ and details of the collective variables employed in these simulations are provided in the Supporting Information. All the results presented in the main text are reweighted from the biased simulations by the weighted histogram algorithm.⁷³