Free energy spectroscopy reveals the mechanistic landscape of chromatin compaction

Abstract

Eukaryotic genomic DNA is repeatedly wrapped into nucleosome spools: the basic building block of chromatin. This organization regulates the physical accessibility of the genome to gene transcription, replication, and repair regulatory factors. Chromatin compaction is controlled by multivalent weak interactions, resulting in a complicated conformational landscape that remains challenging to characterize. This work reports a method for characterizing chromatin compaction, Free Energy Spectroscopy (FES), which is based on DNA nanotechnology and transmission electron microscopy. This method experimentally determines the chromatin compaction free energy landscape in terms of end-to-end distance and nucleosome stacking interactions. By deconvolving the free energy landscapes of partially and fully compact tetranucleosomes, FES revealed three separate mechanisms by which linker histones reshape the compaction energetics to condense chromatin. This study establishes FES as a method with the potential to help answer a broad range of mechanistic questions about genome and epigenome function.

Graphical Abstract

Graphical Abstract.

Introduction

Chromatin structural dynamics is a key regulator of accessibility for gene transcription, repair, and replication. For example, transcription regulatory complexes access DNA within open euchromatin but are excluded from compact heterochromatin [1, 2]. An important level of chromatin compaction is clusters of four nucleosomes, i.e. tetranucleosomes. They form in vivo to help form chromatin clutches that regulate transcription [3, 4] and in vitro within longer nucleosome arrays [5, 6, 7]. This tetranucleosome unit of chromatin is compacted by magnesium ions [8] and the linker histone H1, an abundant chromatin architectural protein that binds near the nucleosome dyad region [9]. H1 regulates tetranucleosome compaction by influencing the extent to which DNA is wrapped into nucleosomes [10, 11] and the stacking between nucleosomes [12, 13], which are two essential differences between euchromatin and heterochromatin.

The folded structure of the tetranucleosome has been investigated in vitro by X-ray crystallography [14], cryo-electron microscopy (cryo-EM) [5], cysteine crosslinking [12], and ensemble fluorescence [15]. These studies provide high-resolution information about static structures and have revealed that the primary folded structure consists of a zig-zag pattern with two nucleosome–nucleosome stacking interactions (NSI) between nucleosomes 1–3 and 2–4 [14]. Stopped flow [16] and single-molecule [6] fluorescence studies have provided kinetic information about the time to transition between different compacted states, while single-molecule force spectroscopy (smFS) has provided information about discrete free energy differences between distinct compacted states [17–19].

Importantly, chromatin is a continuum of conformational states, where the relative probability of open and compact chromatin states is determined by the free energy difference between each of the states through the Boltzmann distribution. A comprehensive description of chromatin compaction is a full free energy landscape describing the continuum of states, which has been probed with smFS measurements [17–22] and predicted by computational modeling [23–28]. In addition, an elegant DNA nanotechnology approach has been used to investigate nucleosome–nucleosome stacking [29, 30], which is an important component of chromatin compaction. While computational modeling has enabled prediction of continuous free energy landscapes, to date, experimental measurements have been limited to measuring free energy differences between discrete states. The ability to experimentally determine chromatin compaction free energy landscapes would open a new level of understanding of how chromatin controls genome accessibility. DNA nanotechnology has the potential to provide an approach to probe tetranucleosome structure and dynamics, including measuring the free energies associated with compaction, while being imaged by transmission electron microscopy (TEM) to elucidate the corresponding tetranucleosome conformation [31, 32].

Scaffolded DNA origami is a powerful approach in the field of DNA nanotechnology, which utilizes a long single-stranded DNA (ssDNA) scaffold and numerous short ssDNA oligonucleotides that base-pair together to construct functional nanostructures [33–35] that can be designed to be stable under physiological conditions [36]. DNA-based nanostructures have been used as probes to detect and quantify many biomolecular interactions [31]. For example, DNA-based nanomechanical devices have quantified DNA bending [37], DNA shearing [38], DNA–protein interactions [39], histone octamer stacking [29], and nucleosome unwrapping [36, 40]

In this work, we use the recently established set of nanocalipers, which were termed nanoscale DNA Force Spectrometers (nDFSs) and were used to apply forces to biomolecules [37, 38]. Here, we use these same devices to carry out free energy landscape measurements of tetranucleosomes. We refer to them with their original nDFS names for clarity with published literature, even though they are used to measure free energies in this study. These nDFS devices consist of two caliper arms, which are approximately 50 nm in length, and have an 8 × 3 cross-section of double-stranded DNA (dsDNA) helices with some helices removed in the middle layer. The two stiff arms of the nDFS nanocalipers are connected by both short (2 bases) and long (70 bases) ssDNA loops. By introducing additional DNA base pairing within the long ssDNA loops, the angular free energy landscape of the nanocalipers can be modulated. After incorporating a tetranucleosome between the two nanocaliper arms, the end-to-end distance, can be used as a natural “reaction coordinate” for the compaction free energy landscape, which relates to the angular free energy landscape through the arm length of the nanocaliper. Three different nanocaliper designs are used here: nDFS.A, nDFS.B, and nDFS.C [37], allowing for a wide range of tetranucleosome end-to-end distances to be probed.

By using these nDFS nanocalipers, we experimentally determine the compaction free energy landscapes of tetranucleosomes as a function of end-to-end distance. We find that the free energy landscape of magnesium compacted (Mg²⁺, 1 mM) tetranucleosomes has a minimum free energy plateau over a broad range of end-to-end distances. This indicates Mg²⁺ compacted tetranucleosomes have significant conformational freedom. Above about 30 nm, the free energy increases linearly by about 4 k_BT as the end-to-end distance is doubled, revealing the energetic cost for extending tetranucleosomes. We also determined the tetranucleosome free energy landscape at a lower Mg²⁺ concentration (0.25 mM) and found that the landscape compared well to a previously established coarse-grained molecular dynamics simulation model [24]. The human linker histone, H1.0, converts the free energy plateau to a 3 k_BT free energy well with a free energy minimum at an end-to-end distance of about 5 nm, revealing how H1.0 alters the compaction free energy landscape to regulate tetranucleosome compaction. Because the free energy landscapes are determined from TEM images, the number of nucleosome stacking interactions (NSIs) can also be determined for each particle. This allows for determining a two-dimensional free energy landscape in terms of the end-to-end distance and the stacking state. This multidimensional analysis reveals the energetic cost and changes in tetranucleosome conformational freedom as one nucleosome stacking interaction is disrupted. These results provide a new mechanistic picture of how tetranucleosomes are “opened” for chromatin regulatory factors. Overall, this approach for determining tetranucleosome compaction free energy landscapes, which we refer to as Free Energy Spectroscopy (FES), is positioned to provide important insight into the mechanisms of a wide range of chromatin compaction regulatory factors and more generally into conformational transitions of biomolecular complexes at the 10–100 nm scale.

Materials and methods

Preparation of nDFS devices

Methods for designing and fabrication of the nDFS devices (nDFS.A, nDFS.B, and nDFS.C) are described previously, and the design of the nDFS was previously shared on the public DNA nanostructure design repository nanobase.org (https://nanobase.org/structures/248) [37, 38]. nDFS.C is nDFS.C35 in the previous report [37]. The list of staples for all nDFS designs used was previously reported [37]. For folding of the structure, 20 nM of an M13MP18 8064 nt ssDNA scaffold was combined with 200 nM of each staple ssDNA in a ddH₂O solution containing 5 mM Tris, 5 mM NaCl, 1 mM EDTA, and 18 mM MgCl₂, at pH 8.0. The sequence of each staple was determined using the computer assisted design software caDNAno [41]. The folding reaction was carried out in a thermocycler (Bio-Rad, Hercules, CA) with a 15-min melting step at 70°C followed by an annealing step from 63°C to 57°C at a cooling rate of 1°C every 3 h.

Purification of nDFS devices

The nDFS devices were purified using previously described methods [37]. Briefly, the solution containing newly folded nDFS structures was mixed with equal volume PEG buffer (15% PEG MW8000, 200 mM NaCl, and 100 mM Tris) and placed into a standard tabletop centrifuge at 20 000 × g for 30 min. Then, the supernatant was removed, and the purified structures were resuspended into a buffer containing 100 mM HEPES pH 8.0, 1 mM MgCl₂, and 200 mM NaCl. The concentration of the purified nanocalipers was measured via absorption measurements at 260 nm from a NanoDrop, and the structures were diluted in the same buffer to 10 nM for future experiments.

Nucleosome template DNA preparation

Tandem repeats of the mp1 [42] nucleosome positioning sequence (NPS) were cloned into pUC19 and amplified through polymerase chain reaction (PCR) using Pfu DNA polymerase, which contains an endonuclease proofreading enzyme. The amplified DNA was purified using an anion exchange chromatography column (Gen-Pak^™ Fax – Waters Column) using high-performance liquid chromatography (abbreviated as HPLC, Agilent). Purified DNA was buffer-exchanged into 0.5× TE (5 mM Tris–HCl pH 8.0, 0.5 mM EDTA) by Amicon^® Ultra Centrifugal Filters (Mereck Millipore). 147 bp long buffer DNA used for tetranucleosome reconstruction was amplified out of the ampicillin gene of the pUC19 plasmid, and HPLC purified in the same manner as described above.

Histone octamer preparation

Human histones (H2AK119C, H2B, H3C110A, and H4) were obtained from The Histone Source at Colorado State University. Histone octamer was refolded using a 1.3:1 molecular ratio of H2AK119C and H2B to H3C110A and H4 through double dialysis using unfolding buffer (7 M guanidine hydrochloride, 20 mM Tris–HCl pH 7.5, 10 nM Dithiothreitol (DTT) to refolding buffer (2 M NaCl, 10 mM Tris–HCl pH 7.5, 1 mM EDTA pH 8.0, 5 mM 2-Mercaptoethanol) [43, 44]. For gel imaging following native polyacrylamide gel electrophoresis (PAGE) of tetranucleosomes, the H2AK119C was Cy5 labeled at K119C using the maleimide-thiol reaction that was quenched with 10 mM DTT. The refolded histone octamer was purified using Fast Protein Liquid Chromatography (abbreviated FPLC, Cytiva) using a Superdex 200 (Cytiva). The fractions were analyzed by 16% SDS–PAGE gel. Selected fractions were concentrated to a volume of ∼100 μl using 30 K Amicon^® Ultra Centrifugal Filters (Merck Millipore).

Preparation of NeutrAvidin bound tetranucleosomes

Tetranucleosome arrays were reconstituted and purified as previously described [16]. Template tetranucleosome DNA containing a repeat of four mp1 [42] NPSs and 30 bp linker DNA was combined with buffer DNA in a 1:3 mass ratio with Cy5-labeled histone octamer. A mass ratio of total DNA to histone octamer at 1:0.8 was then prepared in a 50 μl volume in the reconstitution buffer (0.5× TE pH 8.0, 1 mM benzamidine hydrochloride, and 2 M NaCl). Tetranucleosomes were reconstituted by double dialysis where the 50 μl sample was loaded into a small dialysis chamber that was placed in a dialysis chamber containing 80 ml of the reconstitution buffer. This was dialyzed against 4 L of 0.5× TE pH 8.0, 1 mM benzamidine hydrochloride for 6 h and then against new buffer overnight. The tetranucleosome sample was purified on a 5% to 35% (w/v) sucrose gradient in 0.5× TE at 41 000 rpm for 16 h in a SW41 rotor at 4°C. Fractions were analyzed by PAGE, and then selected fractions based on the image from a typhoon scanner (Cytiva) were pooled, buffer exchanged, and concentrated with 30 K amicon filters. The purity of the tetranucleosome sample was then confirmed by 4% PAGE (Supplementary Fig. S1).

NeutrAvidin was attached to both biotin-labeled 5 prime ends of the tetranucleosomes by incubating with 40-fold molar excess of NeutrAvidin for 30 min at room temperature. Excess Neutravidin was purified away from NeutrAvidin bound tetranucleosomes on a sucrose gradient as described above for the first sucrose gradient purification. The purified Neutravidin labeled tetranucleosomes were analyzed by 4% PAGE (Supplementary Fig. S2). Glycerol was added to the sample at a 20% final concentration, which was then aliquoted, flash frozen, and stored at −80°C.

Optimization of glutaraldehyde crosslinking

Deposition of nucleosome arrays onto TEM grids disrupts nucleosomes and chromatin compaction, so mild glutaraldehyde (GA) crosslinking is used to maintain nucleosomes and chromatin compaction [5, 45, 46]. To determine the optimal GA crosslinking conditions for FES, conditions tested were based on previous reports [47].

To determine the minimal concentration of GA that prevents nucleosome disruption without inducing nucleosome stacking or tetranucleosome aggregation in decompacting conditions, free tetranucleosomes were diluted to 2.5 nM in decompacting buffer conditions (2 mM NaCl, 100 mM HEPES pH 8.0) and crosslinked for 1 min with 0.025%, 0.05%, 0.1%, 0.2%, 0.5%, 1%, 2%, and 4% GA. (Supplementary Fig. S3 and Table S1). Tetranucleosomes were then deposited on TEM grids and imaged. Above 0.5% GA, TEM imaging revealed that multiple tetranucleosomes were crosslinked together, while at 0.5% GA and below, individual tetranucleosomes were largely observed. The dot number distributions were then determined for 0.5% GA and below to quantify nucleosome stability and tetranucleosome compaction. At 0.5% GA and below, only 4 dot and 3 dots were observed. However, below 0.2% GA, the fraction of 3 dots increased. This is likely due to the disruption of single nucleosomes, since lowering the GA concentrations should not increase nucleosome stacking nor increase the NSI₁ state relative to the NSI₀ state. The subscript after NSI indicates the number of stacking interactions. Instead, below 0.2% GA, nucleosomes were disrupted, increasing arrays with only three nucleosomes. These TEM measurements under decompacting conditions revealed that GA concentrations between 0.2% and 0.5% stabilize single nucleosomes without significantly inducing tetranucleosome compaction or multimerization.

To determine the range of GA concentrations that maintained tetranucleosome compaction under compacting conditions (200 mM NaCl, 1 mM MgCl₂, and 100 mM HEPES pH 8.0), tetranucleosomes were diluted to 2.5 nM and then GA crosslinked for 1 min. GA concentrations of 0.02%, 0.04%, 0.06%, 0.08%, 0.1%, 0.2%, and 0.4% were tested. (Supplementary Fig. S4  and Table S2). 2.5 nM of nanocalipers were included to better replicate experimental conditions. NeutrAvidin was not included, so tetranucleosomes could not incorporate into the nanocalipers. At GA concentrations below 0.08% GA, a significant fraction of NSI₀ states were observed, which were not expected with 1 mM MgCl₂, while a similar fraction of NSI₂ and NSI₁ states was observed for 0.08% GA and above. Therefore, 0.2% GA is the minimal concentration that is needed to prevent deposition-induced nucleosome disruption and tetranucleosome decompaction, without inducing tetranucleosome compaction or multimerization.

Preparation of the nDFS device with a tetranucleosome and linker histone H1

Tetranucleosomes and nDFS devices were combined at equal molar ratio (5 nM TNucs and 5 nM nDFS Devices) in a buffer containing 10 mM HEPES pH 8.0, 200 mM NaCl, and 0.25 mM or 1 mM MgCl₂ and then incubated at room temperature for 2 h. For linker histone experiments, after the initial 2-h incubation, H1.0 (NEB) was added at a final concentration of 5, 10, 20, 40, and 80 nM and incubated at room temperature for 30 min. The sample was then crosslinked with 0.2% GA for 1 min by quenching the crosslinking with 100 mM final concentration of Tris–HCl pH 8.0. Each sample was then deposited on a TEM grid for imaging.

Transmission electron microscopy grid preparation, imaging, and analysis

TEM imaging samples were prepared as previously described [37]. Briefly, 10 μl of the final tetranucleosome sample was deposited on Formvar-coated copper TEM grids, stabilized with evaporated carbon film (Ted Pella; Redding, CA). The sample was incubated on the grid for 10 min. The tetranucleosome sample was then wicked off the grid with filter paper. The sample was stained by applying 6 μl of 2% uranyl formate (SPI, West Chester, PA) twice for 1 and 5 s, respectively. The staining solution was wicked off after each incubation with filter paper. TEM imaging was carried out at the OSU Campus Microscopy and Imaging Facility on an FEI Tecnai G2 Spirit TEM at an acceleration voltage of 80 kV at a magnification of 75 000× .

Single molecules (nDFS_only, TNuc_only, and nDFS_TNuc) were manually selected from the original TEM images at each H1 and Mg²⁺ condition using ImageJ. At 0.25 mM Mg²⁺, the particles were sufficiently spread out that both nDFS_TNuc and nDFS_only (i.e. nanocalipers with no TNucs incorporated) particles were collected from the same grid for nDFS.A and nDFS.B. For the other nDFS_only measurements, separate grids without tetranucleosomes were prepared for particle imaging. Example TEM images are shown in the Supplementary Information. Selected 100 × 100 pixel images containing a single device were filtered using the built-in bandpass filter in ImageJ. This filtered small structures up to 3 pixels (half of the nucleosome diameter) and large structures down to 90 pixels (twice the hinge arm length). A subset of images were analyzed both with and without the filter to confirm that image filtering did not alter the analysis (Supplementary Fig. S5). After filtering, only images that met the selection criteria (see below) were retained for further analysis. Representative 10 × 10 image galleries showing 100 randomly selected particles for each condition are shown in the Supplementary Information. The selection criteria include verifying that the hinges are folded properly and deposited on the surface in a clear side view orientation and contain one tetranucleosome that is bound to both hinge arms. A table reporting the number of “selected” and “excluded” particles is included in the Supplementary Information (Supplementary Table S3), and examples of images of particles that did not meet our selection criteria are provided in Supplementary Fig. S6. We measured the incorporation efficiency for each device (Supplementary Fig. S7 and Supplementary Table S4). After selection, end-to-end distances were measured manually in ImageJ by placing points at the inner edge of each hinge arm.

CSV files from ImageJ containing the location of each point were used to make end-to-end probability distributions, which were in turn used to determine free energy landscapes. The number of particles measured for each end-to-end distance bin is provided in Supplementary Table S5. Analysis and measurements were carried out in MATLAB. The stacking state of each tetranucleosome was classified manually by visual inspection of each particle. Each dataset was split according to the stacking state and then used to determine the probability distributions and free energy landscapes of each stacking state.

Free energy measurements and weighted average computation

Free energy landscape measurements for nDFS_only and nDFS_TNuc were computed using , where P (ℓ) is the probability for a given ℓ, while P_max is the probability for the most likely value of ℓ. The free energy landscape of the tetranucleosome is then determined by calculating the difference between the ∆G (ℓ) for nDFS_TNuc and nDFS_only:

.

Therefore, the free energy landscape of the tetranucleosome was calculated using:

.

The offset resulting from the fact that the most likely end-to-end distance is different for nDFS_only, and nDFS_TNuc is not included because it does not provide additional physical insight. The uncertainty in the probability was computed using the square root of the number of bin counts. Standard uncertainty propagation of the formula above leads to

For the H1.0 measurements, we found that H1.0 does not significantly alter the nDFS.C_only probability distribution (Supplementary Fig. S8B). Therefore, the nDFS.C_only probability distribution in the absence of H1.0 was used for determining the compaction free energy landscape ∆∆G (ℓ)_TNuc at each H1.0 concentration.

There is a constant free energy offset between the free energy measurements from each of the devices. To compute this offset, the following loss function, V, was minimized:

Here, , , and are the measured free energies from nDFS.A, nDFS.B, and nDFS.C at end-to-end distance . The purpose of this loss function was to find two constant offsets and that shifted the free energy curves for the measurements from nDFS.A and nDFS.B such that it minimized the difference between datapoints from each nanocaliper measurement but also considered the uncertainty of each measurement. This offset was determined before the weighted average measurement was computed.

The weighted average mean for each end-to-end distance from the three independent measurements from each nanodevice was computed with

where n represents each nanocaliper design (e.g. nDFS.C). This is weighted by the inverse of the uncertainty for each nanocaliper at each end-to-end distance.

To compute the weighted uncertainty, we added in quadrature the standard deviation (δ_std) from the three independent measurements at each bin center and the weighted uncertainty ) as determined by the individual uncertainties from each of the three measurements:

We repeated this same process to determine the weighted end-to-end free energy landscape for NSI₂ and NSI_1, where the free energies were calculated from the probability distributions for tetranuclesomes in each stacking state.

To compute the ensemble free energy difference between NSI₁ and NSI₂, we used ∆G_{NSI (2–1)} = −k_BT ln [P (NSI₁)/P (NSI₂)], where P (NSI_i) is the probability that a tetranucleosome has i nucleosome stacking interactions. We used the normalized square root of the counts for each NSI state as the uncertainty and propagated the uncertainty so that

Coarse grained modeling of free energy landscape

To model the free energy landscape, the 3SPN.2C DNA model [48] was employed, which represents each nucleotide with three interaction sites, and the SMOG2 protein model [49] was used to describe the histone components. Additional protein–DNA interactions were incorporated through electrostatic and Lennard–Jones potentials [24]. Two-dimensional umbrella sampling simulations were performed. In addition to the end-to-end distance, ℓ, which served as a collective variable (CV) and facilitates direct comparison with experimental data, a second variable, D_p, was introduced. This additional coordinate, which was used in a previous study [24], enables sampling of multiple chromatin unfolding pathways. Specifically, D₁₃ and D₂₄ represented the distances between the centers of mass (COMs) of nucleosomes 1 and 3, and nucleosomes 2 and 4, respectively. Their difference, defined as D_p = D₁₃ − D₂₄, served as the second collective variable.

Applying an explicit bias along the D_p axis allowed the simulations to selectively probe distinct unfolding mechanisms: one in which the interaction between nucleosomes 1 and 3 was disrupted (D_p > 0), another where the contact between nucleosomes 2 and 4 broke (D_p < 0), and a symmetric pathway where both interactions were lost simultaneously (D_p ∼ 0). This targeted control over pathway selection enhances the overlap between adjacent umbrella windows and improves the convergence of the resulting free energy profile.

A total of 66 GPU-accelerated MD simulations using OpenMM [50] as the molecular mechanics software along with CG model implementations from OpenABC [51] were performed. The DNA was prepared with the same sequence as that used by our experimental setup. Each nucleosome in the tetranucleosome had 11 DNA base pairs centered at the dyad and non-disordered regions of the histone core treated as a rigid body to prevent nucleosome sliding and histone dissociation. Electrostatic and solvent interactions, represented by the Debye–Hückel potential, were calculated at 300 K with a 200 mM ionic strength solution. Due to the approximate nature of this potential, the effects of ions, particularly Mg²⁺, are only captured in a simplified manner. A more accurate treatment would require explicit simulations [26]; however, the increased computational cost of explicit ions exceeds the feasibility for the tetranucleosome system investigated herein. The 66 MD simulations were run in NVT systems with Langevin dynamics via OpenMM’s Langevin Middle Integrator for a total of 2 μs at a 10 fs timestep with collective variables and structures reported every 10 ps. Each simulation had a unique set of CV centers maintained by an added restraining potential given by 0.5k [CV (x) – CV₀)]², where k is a force constant, CV (x) is the simulated tetranucleosome’s current CV values, and CV₀ is the specific CV center assigned to a given simulation [24]. CV centers were selected such that each had a spacing of ∼7.5 nm along both CV coordinates, within a set of bounding lines which ensured that at ℓ = 2 nm, |D_p| ≤ 5 nm and at ℓ = 80 nm, |D_p| ≤ 25 nm. These bounding lines ensured that the restraining potentials maintained physically reasonable sets of inter-nucleosome distances, ensuring numerical stability and allowing a reduced number of total simulations compared to a naïve 2D grid. Prior to running the simulations, each system had a short warm-up simulation where CV restraints were slowly applied over 1 ns. In addition to CV restraints, the warm-up simulation also included restraints to maintain the desired D₁₃ and D₂₄ distances; these additional restraints are released after the warm-up steps have been completed.

FastMBAR [52] was used to solve the MBAR equations and “de-bias” each completed simulation, that is, calculate the energy of each state in the full statistical ensemble as generated by the combined set of simulations. With the de-biased energies, the free energy landscape of the tetra-nucleosome with respect to the set of collective variables was computed.

Results

DNA nanocalipers directly measure the compaction free energy landscape of tetranucleosome compaction

To investigate tetranucleosome free energy landscapes, we developed a DNA nanotechnology approach (Fig. 1), building on previous measurements of single nucleosomes [36, 40] and histone octamers [29]. We used a set of nDFS nanocalipers: nDFS.A, nDFS.B, and nDFS.C, which were previously developed to study dsDNA bending and nucleosomal DNA wrapping [37]. Both the nDFS nanocalipers [37] and tetranucleosome arrays [16, 42] were prepared and validated as previously described. The tetranucleosome DNA template contained a 4× repeat of a high-affinity NPS, mp1 [42], which is a variant of the Widom 601 NPS [53], which were spaced by 30 bp of linker DNA and flanked by 50 bp of DNA. Biotins were included at each 5-prime end of the tetranucleosome DNA and the end of each nanocaliper arm. NeutrAvidin was then used to attach a single tetranucleosome between the two nanocaliper arms (Fig. 1).

Figure 1.

Free energy measurement strategy. (A) Schematic of tetranucleosome incorporation into the nDFS devices, including DNA origami nanocaliper folding and tetranucleosome reconstitution. (Inset) TEM image of a properly incorporated Mg²⁺ compacted tetranucleosome into a nanocaliper. (B) Schematic representation showing the process for computing the tetranucleosome free energy as a function of end-to-end distance from free energy measurements. (C and D) End-to-end probability distributions (bars) and corresponding free energy landscapes (open circles) from (C) nanocalipers with tetranucleosomes (nDFS.C_TNuc) and (D) free nanocalipers (nDFS.C_only). (E) End-to-end tetranucleosome free energy landscape. The error bars of the free energy values were propagated from the uncertainties of the probability measurements, which were estimated by the square root of the number of events in each bin.

For the initial proof-of-principle measurement, we used nDFS.C because its end-to-end distance distribution overlapped with the typical size of compacted tetranucleosomes (Supplementary Fig. S9), which we anticipated would help incorporation efficiency. Tetranucleosomes were integrated into the nDFS device by combining them at an equimolar concentration at room temperature in physiological ionic conditions (200 mM NaCl and 1 mM MgCl₂, Fig. 1 and Supplementary Fig. S10) and then imaged using negative stain TEM. These integration conditions resulted in 68 ± 3% of the nDFS.C devices having a tetranucleosome properly integrated (Supplementary Fig. S7). Fixation of samples with conditions used in previous TEM studies of chromatin [47] prevented the disruption of nucleosome stacking without inducing tetranucleosome compaction (Supplementary Figs S3 and  S4) or altering the end-to-end distance distribution of the nDFS.C device (Supplementary Fig. S8). Regions of interest within TEM images containing a single nDFS.C device were selected and analyzed with ImageJ to quantify the end-to-end distance distribution of the nDFS.C device with and without tetranucleosomes (see Materials and methods for details).

The end-to-end distance (ℓ), measured as the distance between the inner tip of each caliper arm of each device (Fig. 1A), was determined for 546 nDFS.C nanocalipers with properly incorporated tetranucleosomes (nDFS.C_TNuc) and 635 nDFS.C nanocalipers that were not incubated with tetranucleosomes (nDFS.C_only) (Supplementary Table S6). The ensemble of ℓ values was used to determine the end-to-end distance probability distribution, P (ℓ), (Fig. 1C and D). By assuming that the ensemble is in thermal equilibrium, the free energy landscape as a function of ℓ can be calculated using the Boltzmann probability, ∆G = −k_BT ln (P (ℓ) / P_ref). P (ℓ) is the probability for a given end-to-end distance ℓ, while P_ref is the probability of the reference state (Fig. 1C and D). Since we are only interested in differences in free energy, the choice of P_ref does not influence the shape of the landscape, and the position of ∆G = 0 does not provide physical insight. We therefore plot ∆G without indicating ∆G = 0 where the graph’s free energy tick marks indicate a step in ∆G of 1 k_BT.

When the tetranucleosome is properly attached to a nanocaliper, visual inspection of the images reveals that the nucleosomes are not in direct contact with the nanocaliper beyond the biotin–Neutravidin–biotin interactions (Supplementary Fig. S10). This implies that free energy landscapes of the tetranucleosome alone and the nanocaliper alone are additive, where the sum equals the free energy landscape of the nanocaliper–tetranucleosome complex. Therefore, the free energy landscape of the tetranucleosome alone can be calculated from the difference between the free energies of the nanocaliper–tetranucleosome complex and the nanocaliper alone, i.e.  (Fig. 1B). Using this approach to determine the tetranucleosome compaction free energy landscape ∆∆G (ℓ)_TNuc, we find that the tetranucleosome free energy does not change significantly for end-to-end distances ℓ < ~30 nm, while above 30 nm the free energy rises linearly (Fig. 1C–E). Again, since we are only interested in differences in free energy, the position of ∆∆G = 0 does not provide physical insight into chromatin compaction. Therefore, we plot ∆∆G where the free energy tick marks indicate a change in ∆∆G of 1 k_BT without indicating a zero value. These results demonstrate that DNA origami nanocalipers can be used as a probe to measure the continuous free energy landscape of the tetranucleosome in terms of its end-to-end distance ℓ. To our knowledge, this is the first experimental measurement of a chromatin compaction free energy landscape.

Combined measurements with different DNA nanocalipers improve the precision and dynamic range of the tetranucleosome free energy landscape

While the nDFS.C nanocaliper enabled the measurement of the free energy landscape of tetranucleosome compaction, the range of end-to-end distances for which the free energy can be accurately measured is limited (Fig. 1E). This is because only end-to-end distances that are thermally accessible can be observed and quantified. We are using about 500 conformation measurements to quantify a free energy landscape with a single device and require at least three events in an end-to-end distance bin to be included in the probability distribution. Therefore, detectable free energy differences are limited to about k_BT ln (500/3) ∼ 5 k_BT. To expand the end-to-end distance range of the measurement, we used multiple nDFS devices, each with a different free energy landscape, to bias the tetranucleosome to varying ranges of conformations. This approach is analogous to umbrella sampling, an approach widely used in molecular dynamics simulations of bimolecular complexes [54, 55], where a known energy potential (typically harmonic) is added to bias conformational distributions. This allows for low probability states to be sufficiently populated for quantification. The applied energy potential is then removed after the simulation. By applying multiple energy potentials where each bias over a different range of states, free energy landscapes over a wider range of conformations can be quantitatively studied and reconstructed.

Here, we used two additional nDFS devices, nDFS.A and nDFS.B, which were designed with larger mean opening angles [37] by modifications to the vertex within the nDFS nanocaliper design (Supplementary Fig. S11). The nDFS.A and nDFS.B free energy landscapes have minima at ℓ = 55 and 65 nm, respectively, which are larger than 25 nm, the end-to-end distance ℓ for the nDFS.C free energy minimum (Fig. 2D and Supplementary Fig. S12). The interquartile ranges of nDFS.C, nDFS.A, and nDFS.B are 20.6 to 41.3 nm, 45.7 to 58.1 nm, and 56.5 to 66.7 nm, respectively (Supplementary Fig. S13 and Table S7). Hence, these three probes sample overlapping yet distinct end-to-end distance ranges. This should allow nDFS.A and nDFS.B to probe tetranucleosome conformations corresponding to larger end-to-end distances. We carried out free energy measurements of tetranucleosomes with both nDFS.A and nDFS.B, as we did with nDFS.C. We found the percentage of nDFS.A and nDFS.B devices that contained properly integrated tetranucleosomes were 45 ± 2% and 27 ± 2%, respectively (Supplementary Fig. S7 and Table S4). These are lower than the integration efficiency into nDFS.C, which is expected since the tetranucleosome end-to-end distance ℓ is less likely to match the end-to-end distance ℓ of either nDFS.A and nDFS.B than of nDFS.C.

Figure 2.

The application of multiple nanocaliper designs increases the precision and dynamic range of the free energy measurement. (A–C) End-to-end distance probability distributions for nanocalipers are shown in gray for each nanocaliper design (nDFS.C_only, nDFS.A_only, nDFS.B_only). End-to-end distance probability distributions of nanocalipers with tetranucleosomes (nDFS.C_TNuc, nDFS.A_TNuc, nDFS.B_TNuc) are shown in (A) cyan, (B) magenta, and (C) green, respectively. Insets show examples of images used to determine each probability distribution. The black scale bar in (C) is 50 nm. (D) End-to-end distance tetranucleosome free energy landscapes from each nanocaliper design. The error bars of the free energy values were propagated from the uncertainties of the probability measurements, which were estimated by the square root of the number of events in each bin. (E) End-to-end distance tetranucleosome free energy landscapes determined from the weighted average of the three independent free energy measurements using the three different nanocalipers in 1 mM Mg²⁺ (open gray circles) and 0.25 mM Mg²⁺ (open gray triangles). The error bars of the free energies were determined as described in the Materials and methods section. The free energy landscape derived from molecular dynamics simulations of the coarse-grained model is plotted as the shaded gray area, which represents the 95% confidence interval. The P-value and reduced chi-squared () between the model and the experimental data are 0.04 ( = 4.06) and 0.21 ( = 1.53) for the 1 and 0.25 mM data, respectively.

We collected TEM images of the nDFS.A and nDFS.B nanocalipers alone (nDFS.A_only and nDFS.B_only) and with tetranucleosomes (nDFS.A_TNuc and nDFS.B_TNuc) (Supplementary Table S6). Regions of interest containing individual nanodevices were selected (Supplementary Figs S14–S17) and then analyzed to determine end-to-end distance probability distributions P (ℓ) for each device (Fig. 2A–C, insets) as described above. Visual inspection of each probability distribution P (ℓ) reveals the integration of tetranucleosomes into nDFS.A and nDFS.B resulted in a shift in tetranucleosome conformations corresponding to larger end-to-end distances ℓ compared to nDFS.C. Furthermore, the probability distributions P (ℓ) with nDFS.A_TNuc and nDFS.B_TNuc spans a much wider range of end-to-end distances ℓ than the probability distributions P (ℓ) with nDFS.C_only, which is consistent with wider angle distributions of nDFS.A and nDFS.B. These observations indicate that nDFS.A and nDFS.B extend the tetranucleosome end-to-end distances ℓ more than nDFS.C. We have included a table with the counts for each end-to-end distance bin for each of the devices (Supplementary Table S5).

We separately determined the free energy landscape ∆∆G (ℓ)_TNuc using nDFS.A and nDFS.B, as previously described for nDFS.C (Fig. 2D). As overall offsets to the free energy landscape are arbitrary, a constant free energy offset needs to be subtracted from each landscape to compare the three free energy landscape ∆∆G (ℓ)_TNuc measurements from the three different nanocalipers. To determine the offset, a loss function with two constant offsets for ∆∆G (ℓ)_TNuc from nDFS.A and nDFS.B was minimized, where the uncertainty of the measurements for each device was taken into account (see Materials and methods for details). The three independent free energy landscape measurements show good agreement within the overlapping ranges of end-to-end distance ℓ, indicating that this method can reliably measure the free energy landscape ∆∆G (ℓ)_TNuc.

Now that we have established that the free energy landscape ∆∆G (ℓ)_TNuc can be measured by three distinct nanocalipers, we combined these three measurements (Fig. 2E) by computing the weighted average of ∆∆G (ℓ)_TNuc at each value of ℓ (see Materials and methods for details). This increased the range of end-to-end distances ℓ to extend from 5 to 85 nm, while having a free energy precision of about 1 k_BT over a range of about 4 k_BT. The resolution of the end-to-end distance is set by our bin size of 5 nm. This bin size was used as a good tradeoff between spatial resolution and maintaining a smooth probability distribution, given the total number of measurements. This bin size is larger than the precision of an individual end-to-end distance measurement of a nanocaliper, which we estimate to be about 1 nm (Supplementary Fig. S18). We quantified the improved precision from using the two additional nanocalipers: nDFS.A and nDFS.B, which increases both the range of end-to-end distances probed and the total number of measurements. To do this, we computed the mean uncertainty for each FES measurement with only one type of nanocaliper and compared it to the weighted average measurement (Supplementary Tables S8 and S9), which quantifies the uncertainty of each free energy measurement for each nanocaliper at each end-to-end distance bin. We found that the mean uncertainty reduces from ∼0.5 k_BT for an individual nanocaliper measurement to ∼0.3 k_BT for the weighted average measurement. This quantitatively reveals the impact of using multiple nanocalipers on the precision of the free energy landscape measurement. The improved precision and range is also illustrated by visual comparison in Supplementary Fig. S12.

With these improvements, distinct regions within the free energy landscape ∆∆G (ℓ)_TNuc are better resolved. At small values of end-to-end distance ℓ, 5–30 nm, the free energy landscape ∆∆G (ℓ)_TNuc is constant, implying a free energy plateau. This indicates that within this range of ℓ, there is no free energy difference between these levels of tetranucleosome compaction, implying it can fluctuate freely over a range of compaction. At larger values of ℓ, >30 nm, the free energy increases by 4 k_BT, implying that it is energetically costly to extend the tetranucleosomes over these larger end-to-end distances.

Overall, these results demonstrate that this DNA nanotechnology-based method can be used to determine a continuous free energy landscape for characterizing tetranucleosome compaction. The approach leverages the tunability of the DNA origami nanocalipers where measurements with different vertex designs are combined to provide significantly greater precision and dynamic range. This technique directly measures a spectrum of free energies through probabilities, which is why we refer to this approach as FES. The measured free energy landscapes are rich with mechanistic details, including revealing that tetranucleosomes have distinct regions of compaction, which is important for understanding how chromatin compaction regulates genome accessibility. Finally, these results provide a critical baseline for measurements of chromatin regulatory factors and how they function to regulate chromatin compaction.

The measured compaction free energy landscape is consistent with an independently determined coarse-grained computational model

To validate our measured free energy landscape, we computed the free energy profiles of the tetranucleosome using umbrella sampling simulations based on coarse-grained protein–DNA models [48, 49]. This modeling framework has been previously validated in studies of nucleosome DNA unwrapping [56], the force–extension behavior of chromatin fibers [57], and the folding pathways of tetra-nucleosome assemblies [24, 27]. In addition to the end-to-end distance, the umbrella sampling simulations incorporate a collective variable that enables the exploration of multiple tetranucleosome unfolding pathways, including both unstacking and unwrapping (see Materials and methods).

Our simulations exhibit good agreement with the magnesium compacted tetranucleosome free energy landscape for end-to-end distances exceeding 25 nm (Fig. 2E). Representative images from the simulations that correspond to these end-to-end distances are shown in Supplementary Fig. S19. At distances of <25 nm, the computed free energy profile diverges from the experimental one. One plausible explanation for this divergence is the implicit treatment of divalent ions in our simulations, which does not account for the effect of divalent ions such as Mg²⁺. Under compact configurations, increased interactions of divalent ions with chromatin may further attenuate electrostatic repulsion among DNA segments, thereby reducing the energetic barrier [26].

We therefore determined the tetranucleosome free energy landscape with 4-fold less Mg²⁺ (0.25 mM). Since Mg²⁺ helps stabilize the nDFS structures, we confirmed by TEM imaging that the nDFS structures remained well folded during the experiment in 0.25 mM Mg²⁺ (Supplementary Fig. S20). We then used nDFS.C, nDFS.A, and nDFS.B to determine the tetranucleosome free energy landscape at 0.25 mM Mg²⁺ (Fig. 2E and Supplementary Figs S21–S24). We find that at distances of <25 nm the free energy landscape increases similarly to the simulation results. To quantitatively compare the data to the coarse-grained model, we determined the reduced chi-squared () and P-value for both the 1 mM Mg²⁺ ( = 4.06 with a P-value = 0.04) and the 0.25 mM Mg²⁺ ( = 1.53 with a P-value = 0.21) data sets. This quantitatively shows that not only does the 0.25 mM Mg²⁺ data agree better with the model, but that the 0.25 mM Mg²⁺ data are statistically consistent with the model. Given that the model parameters were derived independently of the experimental data, this agreement strongly supports the quantitative reliability of our approach in probing the energy landscape of chromatin folding.

The linker histone H1.0 compacts tetranucleosomes by locally biasing the free energy landscape to the higher compacted states through converting the short-range plateau to an energy well

After establishing the FES method for investigating magnesium-mediated tetranucleosome compaction, we used FES to investigate the mechanisms by which the linker histone H1 compacts chromatin. We focused on the human H1.0 [58] because it is an extensively studied linker histone isoform [59]. H1.0 binds near the nucleosome dyad [60], which increases DNA wrapping into the nucleosome [61] and promotes chromatin compaction [61–63]. This allows H1.0 to function as a key regulator of transcription by limiting genome accessibility in compact chromatin and suppressing binding of transcription regulatory factors [64, 65]. While smFS studies have investigated linker histone-mediated chromatin compaction [18, 19], the impact of linker histones on the energetics of chromatin compaction remains enigmatic.

To determine the impact of H1.0 on the compaction free energy landscape ∆∆G (ℓ)_TNuc, nDFS.C was used initially because it has the highest resolution at small ℓ, and H1.0 is anticipated to impact the free energy landscape at small ℓ since it compacts chromatin. We carried out FES measurements with tetranucleosomes incubated for 30 min at room temperature with increasing concentrations of H1.0 (5, 10, 20, 40, and 80 nM) (Supplementary Figs S25–S29 and Table S10). We measured the probability distribution P (ℓ) of nDFS.C_TNuc at each H1.0 concentration (Fig. 3). There is a noticeable shift toward smaller ℓ, implying that H1.0 converts tetranucleosomes toward more compacted states. We then determined the compaction free energy landscape ∆∆G (ℓ)_TNuc for increasing H1.0 concentrations and observed the formation of a free energy well over a range of ℓ from 5 to 25 nm, with the minimum free energy at the 5 nm bin. The formation of this free energy well appears at 10 nM of H1.0 and saturates at about 40 nM where the free energy well is ∼3 k_BT deep. These results show that H1.0 increases the compaction of tetranucleosomes by converting the 5–20 nm range of the free energy plateau into a free energy well with a minimum at an end-to-end distance ℓ around 5 nm.

Figure 3.

H1 reshapes the tetranucleosome compaction free energy landscape by converting the short end-to-end distance plateau to an energy well. (A–F) End-to-end distance probability distributions of nDFS.C nanocalipers alone (nDFS.C_only) and with tetranucleosomes (nDFS.C_TNuc,H1) are shown in grey and cyan, respectively. The H1.0 concentrations (0, 5, 10, 20, 40, and 80 nM) used in the measurements are listed in the upper left of each plot. Resulting free energy curves are plotted in black closed circles. The error bars of the free energy values were propagated from the uncertainties of the probability measurements, which were estimated by the square root of the number of events in each bin.

While the FES measurements with nDFS.C revealed the impact of H1.0 on tetranucleosome compaction free energies at small end-to-end distances ℓ, states above ℓ ∼ 60 nm were so rarely populated that free energies could not be determined. Therefore, to measure the compaction free energy landscape ∆∆G (ℓ)_TNuc over a wider range of end-to-end distances ℓ, we repeated the FES measurements with nDFS.A and nDFS.B focusing on the subsaturating and saturating H1.0 concentrations of 10 and 40 nM, respectively (Fig. 4A–C; Supplementary Figs S30–S33, S34A–C, and Supplementary Table S10). Similar to nDFS.C, both 10 and 40 nM of H1.0 significantly compacted the tetranucleosome where the nDFS.A_TNuc and nDFS.B_TNuc states occupied lower values of ℓ. The compaction free energy landscape ∆∆G (ℓ)_TNuc was determined for each nDFS (Fig. 4D and Supplementary Fig. S34D), which extended the free energy measurements out to a value of ℓ ∼ 85 nm. We then combined the three measurements of tetranucleosomes with 10 and 40 nM of H1.0 as was done without H1.0 (Fig. 2E and Supplementary Fig. S34E) to determine a final compaction free energy landscape ∆∆G (ℓ)_TNuc with improved free energy resolution and increased range of ℓ (Fig. 4E). Comparing the compaction free energy landscapes ∆∆G (ℓ)_TNuc with 10 and 40 nM of H1.0 (Fig. 4E, black) to no H1.0 (Fig. 4E, gray) confirms that as H1.0 is increased, the free energy plateau converts to a free energy well, while for larger end-to-end distances H1.0 has less of an impact on ∆∆G (ℓ)_TNuc. The free energy well has a depth of 3 k_BT, which increases the probability of these localized states by about 10-fold. Interestingly, this change in free energy is much smaller than the binding free energy of H1.0 to nucleosomes [66]. Overall, these results reveal that H1.0 compacts the tetranucleosome by converting the free energy plateau that already prefers compacted configurations into a free energy well that further localizes the energetically favorable compacted states with end-to-end distances of about 5 nm.

Figure 4.

The application of FES with three separate nanocalipers allows for the impact of H1.0 on the tetranucleosome compaction free energy landscape to be probed over a range of 80 nm. (A–C) End-to-end probability distributions for free nanocalipers are shown in gray for each nanocaliper design (nDFS.C_only, nDFS.A_only, nDFS.B_only). Nanocalipers with tetranucleosomes and 40 nM H1.0: nDFS.C_{TNuc,40nM H1}, nDFS.A_{TNuc,40nM H1}, and nDFS.B_{TNuc,40nM H1}, are shown in (A) cyan, (B) magenta, and (C) green, respectively. (D) End-to-end distance tetranucleosome free energy landscapes from each nanocaliper design. The error bars of the free energy values were propagated from the uncertainty of the probability measurements, which were estimated by the square root of the number of events in each bin. (E) End-to-end distance tetranucleosome free energy landscapes determined from the weighted average of the three independent free energy measurements using the three different nanocalipers in 1 mM Mg²⁺ with 0 nM H1.0 (open gray circles), 10 nM H1.0 (closed dark gray square), and 40 nM H1.0 (closed black filled circles). The error bars of the free energies were determined as described in the Materials and methods section.

Nucleosome-nucleosome stacking, a second reaction coordinate, is stabilized by H1.0 and disrupted by the DNA nanocaliper

Up to this point, these measurements have considered the full ensemble of tetranucleosome conformations as a function of one parameter, the end-to-end distance ℓ. Each value of ℓ includes an ensemble of different tetranucleosome conformations. Visual inspection of the TEM images of tetranucleosomes alone and within each of the nDFS devices revealed that each tetranucleosome can be classified as 1-, 2-, 3-, or 4- dot states (Fig. 5A and Supplementary Fig. S35). The number of dots is related to the number of NSI since two stacked nucleosomes appear as a single dot in TEM images, as previously reported [12]. This stacking interaction is mediated through interactions between the histone octamer acidic patch and the H4 N-terminal tail [67, 68], which is critical for the formation of compact chromatin [69, 70].

Figure 5.

Subclassifying by the number of nucleosome stacking interactions provides insight into the nucleosome stacking free energies. (A) Example images of single nanocalipers with a tetranucleosome that has three, two, one, or zero stacking interactions based on the number of observed dots. The scale bar is 50 nm. (B) Nucleosome stacking interaction distributions without (open bar plot) and with (closed bar plot) 40 nM H1.0. Tetranucleosomes not incorporated into devices (TNuc_only) are shown in gray, while tetranucleosomes incorporated into nDFS.C, nDFS.A, and nDFS.B are shown in cyan, magenta, and green, respectively. The error bars are determined from the square root of the number of events in each bin. (C) The free energy difference between the state with two nucleosome stacking interactions and the state with 1 nucleosome stacking interaction. The error bars were determined by propagating the uncertainties from the probability measurements in (B).

The 4-dot state is consistent with all four nucleosomes being separately visible, implying there are no stacked nucleosomes (Fig. 5A). The 3-dot state is consistent with one pair of nucleosomes stacked, which appears as a single dot that is often brighter than the other two single nucleosome dots (Fig. 5A and Supplementary Fig. S35). Since tetranucleosomes with 30 base pair linker DNA are reported to compact into a 2-start geometry where the nth nucleosome stacks with the n + 2nd nucleosome [12, 14], the 3-dot state likely involves stacking between either the first and third nucleosomes or the second and fourth nucleosome, with the other pair being unstacked. The 2-dot state is consistent with two pairs of stacked nucleosomes in the conformation that was solved by x-ray crystallography [14] and cryo-EM [5], where the first and third nucleosomes stack and the second and fourth nucleosomes stack (Fig. 5A and Supplementary Fig. S35). Finally, the rarely observed 1-dot state most likely contains three stacking interactions so that there is no space between any of the four nucleosomes (Fig. 5A). Therefore, we refer to the 4-, 3-, 2-, and 1-dot states with the number of nucleosome stacking interactions, NSI₀, NSI₁, NSI₂, and NSI₃, respectively.

We classified each tetranucleosome image from four different conditions, without or within an nDFS device and without or with 40 nM H1.0 (see Materials and methods for details). For each condition, the probability of each stacking state was then determined (Fig. 5B and C). The NSI₂ state was dominant for all conditions, with probabilities ranging from 0.67 ± 0.04 to 0.90 ± 0.03 (Fig. 5B and Supplementary Table S11). The NSI₁ state was the second most common, with probabilities ranging from 0.08 ± 0.01 to 0.29 ± 0.02 (Fig. 5B). Both the NSI₀ and NSI₃ states were rare, occurring at probabilities of 0.037 ± 0.009 and 0.240 ± 0.006, respectively, or less (Fig. 5B and Supplementary Table S11).

The relative probabilities can be converted to free energy differences via the Boltzmann distribution, ∆G_Si,Sj = −k_BT ln [P (NSI_i)/P (NSI_j)]. We decided to focus on the NSI₁ and NSI₂ states since these were the most probable, providing sufficient sampling. We find that the free energy cost between the NSI₂ and the NSI₁ state, ∆G_S1,S2, is 1.6 ± 0.1 k_BT and 2.4 ± 0.1 k_BT without and with H1.0, respectively (Fig. 5C and Supplementary Table S12). In order to compare to previous studies, the free energy difference ∆G_S1,S2 can be related to the nucleosome–nucleosome unstacking free energy by accounting for the two partially compacted states that are included in the NSI₁ state. Since these states are likely to be equally probable, an additional factor of 2 needs to be included for the relative probability for two specific nucleosomes to be unstacked. Therefore, a factor of k_BT ln2 needs to be added to ∆G_S1,S2 to determine the unstacking free energy, ∆G_stacking, which is 2.3 ± 0.1 k_BT and 3.1 ± 0.1 k_BT without and with H1.0, respectively. These nucleosome–nucleosome stacking free energy differences are in line with multiple reports of nucleosome stacking free energy [17, 19, 29] and are at the low end of the full range of reported free energies of about 2 to 20 k_BT [6, 18, 20, 22]. These studies were done over a range of ionic conditions, linker geometries, and nucleosome repeat numbers, which likely explains this order-of-magnitude range of nucleosome stacking free energy values. For example, a coarse-grained model found the nucleosome stacking free energy to be about 2.7 k_BT [24] when linkers (and thus their mutual repulsion) were included, while the same model determined the stacking between two separate mononucleosomes to be about 15 k_BT [71]. Interestingly, nDFS.C does not impact these free energy differences (Fig. 5C and Supplementary Table S12). However, nDFS.A and nDFS.B reduce the free energy differences between the NSI₁ and NSI₂ states, which is expected since both nanocalipers significantly shift the end-to-end distance distributions to larger values (Figs 2A–C and 4A–C). Overall, these results demonstrate that the number of nucleosome stacking interactions is a second reaction coordinate probed by the nDFS experiments, which is complementary to end-to-end distance ℓ, and provides additional information about tetranucleosome compaction.

A multidimensional framework for understanding the energetics of chromatin opening

Now that we have established measurements of the tetranucleosome end-to-end distance free energy landscape  and the nucleosome stacking free energy (NSI₁ versus NSI₂), we combined these parameters to provide a multidimensional free energy analysis: ∆∆G (ℓ,NSI_i)_TNuc. The two primary mechanisms for increasing the end-to-end distance of the tetranucleosome are nucleosome-nucleosome unstacking and nucleosomal DNA unwrapping. By removing nucleosome unstacking as a variable, the contribution of nucleosome unwrapping can be inferred. To achieve this, we divided the data into subsets based on the nucleosome stacking state (Supplementary Figs S36 and  S37). We only included the NSI₁ and NSI₂ states because the NSI₀ and NSI₃ states did not occur often enough to determine an energy landscape for these states alone. Importantly, ∆∆G (ℓ,NSI_i)_TNuc measurements that compare the NSI₁ versus NSI₂ states are from the same dataset, so differences in free energy can be meaningfully compared. This implies that the comparison between the NSI₁ and NSI₂ free energy landscapes provides the free energy difference between a partially stacked and fully stacked tetranucleosome for a given end-to-end distance.

We first focused on the compaction free energy landscape ∆∆G (ℓ,NSI_i)_TNuc in the absence of H1.0 for the two stacking states, NSI₁ versus NSI₂, which we refer to as and ∆∆G (ℓ,NSI₂)_TNuc;H1=0nM, (Fig. 6A and Supplementary Fig. S38). We find that the NSI₁ free energy landscape ∆∆G (ℓ,NSI₁)_TNuc;H1=0nM is generally higher in free energy as compared to the NSI₂ free energy landscape ∆∆G (ℓ,NSI₂)_TNuc;H1=0nM. This is because the probability of the NSI₁ state is about 5-fold lower than the NSI₂ state. Since the NSI₂ state is the most probable, the shape of the NSI₂ free energy landscape  (Fig. 6A black circles, Supplementary Fig. S38) is similar to the overall free energy landscape  (Fig. 2E), where the free energy is constant for ℓ < 30 nm, then increases for ℓ > 30 nm. In contrast, the NSI₁ free energy landscape ∆∆G (ℓ,NSI₁)_TNuc;H1=0nM is distinct from the overall free energy landscape ∆∆G (ℓ)_TNuc in that it has a shallow free energy well for the end-to-end distance range, 30 nm < ℓ < 45 nm. As the end-to-end distance ℓ is reduced below 30 nm, the NSI₁ free energy landscape  increases, and since the NSI₂ free energy landscape is constant for this range of ℓ, the NSI₂ state is strongly preferred in this range of end-to-end distances ℓ. Finally, for ℓ > 45 nm, the NSI₁ and NSI₂ free energy landscapes overlap, i.e. ∼ . This implies that in the absence of H1.0, the NSI₁ and NSI₂ states are equally probable for these large end-to-end distances.

Figure 6.

Multidimensional tetranucleosome compaction free energy landscape reveals the energetics for different nucleosome stacking states and how H1.0 reshapes these energy landscapes. (A and B) End-to-end distance free energy landscapes for partially (NSI₁, gray) and fully (NSI₂, black) compacted tetranucleosomes without (A) and with (B) 40 nM H1.0. The error bars of the free energies were determined as described in the Materials and methods section. (C and D) Transition rate model of tetranucleosome compaction based on the multidimensional free energy landscape for tetranucleosomes without (C) and with (D) 40 nM H1.0. The relative size of the arrows for a given transition relates to relative probabilities between states defined by both end-to-end distance ranges and nucleosome stacking interactions. Red arrows in (D) show state transitions that change by H1.0.

We then repeated this analysis with 40 nM H1.0 to determine the impact of this linker histone on the NSI₁ and NSI₂ free energy landscapes (Fig. 6B and Supplementary Fig. S39). As without H1.0, we find that with H1.0 the NSI₁ free energy landscape ∆∆G (ℓ,NSI₁)_TNuc;H1=40nM is generally higher than the NSI₂ free energy landscape ∆∆G (ℓ,NSI₂)_TNuc;H1=40nM, which is expected since the NSI₂ state is about 10-fold more probable than the NSI₁ state with 40 nm H1.0 (Fig. 5C). The shape of the NSI₂ free energy landscape with 40 nM H1.0 ∆∆G (ℓ,NSI₂)_TNuc;H1=40nM is similar to the overall free energy landscape ∆∆G (ℓ)_TNuc with 40 nM H1.0 (Fig. 4E), where H1.0 converts the small end-to-end distance ℓ plateau into a free energy well with a minimum at ℓ ∼ 5 nm. This is expected since the probability of the NSI₂ state with 40 nM H1.0 is about 0.9. Interestingly, for ℓ < 40 nm with 40 nM H1.0, the NSI₁ free energy landscape  has a different shape than the NSI₂ free energy landscape . In this region, the free energy remains essentially constant, forming a free energy plateau. For ℓ > 40 nm,  and  overlap, which implies that for this range of end-to-end distances, the NSI₁ and NSI₂ states are equally probable, as is observed without H1.0.

In addition, the free energy difference between the NSI₁ and NSI₂ states can be determined from their landscapes by summing the Boltzmann weights of each bin along the end-to-end distance reaction coordinate for each state, separately. The ratio of these sums is the relative probability of the NSI₁ and NSI₂ states: P (NSI₁)/P (NSI₂). The free energy difference between NSI₁ and NSI₂ landscapes (LS) is then calculated with ΔG_{S1,S2_LS} = −k_BT ln [P (NSI₁)/P (NSI₂)]. We find ΔG_{S1,S2_LS} to be 1.5 0.2 k_BT and 2.30.4 k_BT for tetranucleosomes without H1.0 and with H1.0, respectively. These results agree with the free energy difference directly measured above for the free tetranucleosomes, ΔG_{S1,S2_direct}, with and without H1.0 (Fig. 5; Supplementary Fig. S40 and Table S12). Since the measurements of ΔG_{S1,S2_LS} and ΔG_{S1,S2_direct} are based on different data, measurements, and analysis, their agreement provides additional confidence in our free energy measurements and approach.

The impact of H1.0 for small end-to-end distances (Fig. 6) can be understood by H1 suppressing DNA unwrapping, resulting in the free energy well for the NSI₂ state, so fully compacted NSI₂ tetranucleosomes are the preferred state. However, when the tetranucleosome partially decompacts into the NSI₁ state, there is only one stacking interaction. This additional conformational freedom allows for the tetranucleosome to explore a wide range of end-to-end distances with little free energy cost, resulting in a free energy plateau (Fig. 6). Interestingly, with or without H1.0, for end-to-end distances above about 50 nm, NSI₁ and NSI₂ have the same free energy landscape and relative probabilities (Fig. 6). Since the number of nucleosome stacking interactions does not change for a given NSI state, this suggests the free energy cost for increasing end-to-end distances for each NSI state is mostly due to nucleosome unwrapping. For the NSI₂ state, only the outer nucleosomes will partially unwrap, while for the NSI₁ state, three nucleosomes will partially unwrap. Therefore, while NSI₁ has one less stacking interaction, which costs some free energy, the unstacking results in additional degrees of freedom such that the inner nucleosome can unwrap. This appears to provide additional entropy for larger end-to-end distances, compensating for the lost free energy of stacking. Overall, these results demonstrate the power of this FES approach, where free energy landscapes of low probability states that are hidden in the overall free energy landscape, but biologically important, can be observed and quantified.

Discussion

In this work, we utilized a set of DNA origami nanocalipers to implement a methodology, FES, which experimentally determines multidimensional free energy landscapes of chromatin compaction. This work builds off previous nanocaliper measurements of histone octamer stacking interactions [29]. In particular, this work leverages the ability to tune mechanical properties of DNA origami devices to probe free energies spanning molecular distances over tens of nanometers, and the application of a set of calipers enables quantification of a broader free energy landscape with improved precision. Free energy landscapes provide detailed and mechanistic information about the energetics of chromatin compaction, which has been investigated by smFS [17–19, 21, 22] and computational methods [23–27]. In this study, we determined multidimensional free energy landscapes as a function of both the end-to-end distance ℓ and the number of nucleosome stacking interactions NSI. We show that by using multiple nanocalipers with different free energy landscapes, continuous tetranucleosome end-to-end distance free energy landscapes can be determined with a spatial resolution of 5 nm over a range of 5−85 nm and a free energy precision of <1 k_BT over a range of about 7 k_BT. By including the NSI coordinate, tetranucleosome free energy landscapes with a specific number of nucleosome stacking interactions can also be determined (Fig. 6A and B, and Supplementary Figs S38 and  S39).

A model for understanding tetranucleosome compaction free energy landscapes

These multidimensional free energy landscapes can be interpreted through the ratio of transition rates between states with different end-to-end distances ℓ and nucleosome stacking interactions NSI (Fig. 6C), where higher probability states have larger rates into the state than out of the state. Using this framework, it is possible to gain insight into the mechanisms that regulate chromatin compaction, including nucleosome-nucleosome stacking and nucleosomal DNA unwrapping. States with one nucleosome stacking interaction NSI₁ are in the upper row, while states with two nucleosome stacking interactions NSI₂ are in the lower row. The relative size of the arrows represents the relative state probabilities. For NSI₂ without H1.0, states with ℓ < 40 nm have the same free energy and are therefore equally probable. This implies the transition rates between states with different end-to-end distances ℓ over this range are the same (Fig. 6C, ℓ < 20 nm and 20 nm < ℓ < 40 nm). For ℓ > 40 nm, the free energy increases, and the probability decreases, as the end-to-end distance ℓ increases. This indicates that the transition rates to smaller end-to-end distances ℓ are greater than to larger ℓ. In contrast, for NSI₁ without H1.0, the free energy has a minimum between 30 nm < ℓ < 50 nm, implying that these are the highest probability states. This indicates that the transition rate to states with ℓ between 30 and 50 nm is larger than the rates out of this range of ℓ. Importantly, the lower free energy of NSI₂ relative to NSI₁ for ℓ < 50 nm, implies that NSI₂ is more probable for this range of ℓ and that transitions from NSI₂ to NSI₁ are energetically unfavorable. For ℓ > 50 nm, the free energies are the same, implying that NSI₁ and NSI₂ are equally probable and that the rates between states are essentially the same.

A model of how linker histone H1.0 facilitates chromatin compaction

This model can also help understand the influence of H1.0 on the compaction free energy landscape and the mechanisms by which H1.0 compacts chromatin (Fig. 6C and D). Importantly, these FES studies of H1.0 confirm previous studies of H1.0-mediated chromatin compaction. Linker histones are well established to increase nucleosomal DNA wrapping [60, 72, 73], which compacts chromatin [5, 62, 74, 75], facilitates chromatin condensate formation [76, 77], and reduces DNA accessibility to DNA-binding complexes [64, 65, 78]. These FES studies can be understood by the established functions of linker histones. In addition, these FES studies provide insight into how linker histone suppression of nucleosome unwrapping leads to chromatin compaction, which can be understood through the model in Fig. 6C and D. There are three H1.0 induced changes in the free energy landscapes: (i) H1.0 induces a free energy well of about 3 k_BT at small values of (<30 nm) for fully stacked (NSI₂) tetranucleosomes. (ii) H1.0 eliminates the free energy well of about 1 k_BT around intermediate values of (20–60 nm) for partially stacked (NSI₁) tetranucleosomes. (iii) H1.0 increases the free energy difference between the NSI₁ and NSI₂ states.

These findings imply H1.0 alters three sets of relative transition rates between different levels of chromatin compaction, which are highlighted by the red arrows in Fig. 6D. H1.0 converts the NSI₂ landscape with a free energy plateau with end-to-end distances ℓ of < ~30 nm to a free energy well with a minimum at the end-to-end distance ℓ of about 5 nm. This implies that H1.0 compacts fully stacked (NSI₂) tetranucleosomes by localizing the lowest free energy (most probable) states from a wide range of end-to-end distances ℓ to a small range of end-to-end distances around 5 nm. Interestingly, this free energy well is about 3 k_BT, which is significantly smaller than the linker histone binding free energy to nucleosomes [66], suggesting that H1.0 does not need to dissociate for the tetranucleosome to decompact. For NSI₁, the free energy well within the range of 30 nm <ℓ <50 nm, is converted to a constant free energy plateau for ℓ < 50 nm. This implies that H1.0 compacts partially stacked (NSI₁) tetranucleosomes by shifting their most probable large end-to-end distances that are within the free energy well when H1.0 is absent to a plateau when H1.0 is bound, allowing the tetranucleosome to explore more compacted states. In addition to these changes in the NSI₁ and NSI₂ free energy landscapes, H1.0 reduces the free energy of NSI₂ relative to NSI_1, increasing the probability of the NSI₂ state (Fig. 5), which further contributes to tetranucleosome compaction. This demonstrates how deconvolving the NSI₁ and NSI₂ free energy landscapes with FES identified and quantified three separate mechanisms by which the increased nucleosome wrapping by H1.0 compacts tetranucleosomes.

Interestingly, while H1.0 increases the free energy difference between fully and partially stacked tetranucleosomes by about 1 k_BT, it significantly biases both the fully and partially stacked states to smaller end-to-end distances. This is consistent with H1.0 binding near the nucleosome dyad and thus not being located to directly impact nucleosome stacking, which is mediated by H4 N-terminal tail interactions with the histone octamer acidic patch [67, 68]. H1.0 decreases nucleosome unwrapping fluctuations [64, 65], which is consistent with H1.0 having a limited range of ℓ that impacts the free energy landscape and having the largest impact for small values of ℓ. We conclude that changes in the NSI₁ and NSI₂ free energy landscapes are largely due to an increase in the nucleosome unwrapping free energies. These additional conclusions further highlight how multidimensional FES provides new mechanistic insight into how H1.0 compacts tetranucleosomes by directly disentangling its impact on nucleosome stacking and unwrapping.

Free energy spectroscopy is synergistic with single-molecule force spectroscopy

smFS is a powerful method for probing chromatin mechanics [21, 79] that changes the end-to-end distance of the chromatin fiber with a known amount of force. smFS controls either the end-to-end distance (optical trap) or applied force (magnetic tweezers) of the chromatin molecule that is attached to two surfaces through DNA handles to determine the tether’s force versus extension response. Previous smFS studies report a force plateau at about 4 pN, which is due to the mechanical disruption of chromatin compaction via changes in nucleosome–nucleosome stacking and nucleosome wrapping [17–19, 21, 22]. By modeling these smFS measurements [20, 80, 81], which take into account the known force response of the DNA tether and the amount of mechanical work done during a smFS measurement, the free energy of nucleosome-nucleosome stacking and nucleosome unwrapping can be inferred. A number of nucleosome stacking free energies have been reported (2–20 k_BT). This range of values is likely due to a variety of nucleosome array sizes, buffer conditions, and linker lengths.

In contrast to smFS, FES applies much lower forces with the nanocalipers. The average force applied by the tetranucleosome as a function of ℓ can be determined by differentiating the free energy with respect to ℓ (Supplementary Fig. S41). Since the system is highly overdamped, this force is equal and opposite to the average force applied on the tetranucleosome. Without H1.0, the plateau region of the free energy landscape (ℓ between 5 and 25 nm) has a slope of −0.03 ± 0.06 pN, implying an average force of zero, while the free energy landscape for ℓ between 30 and 75 nm can be approximated as a line with a slope of 0.35 ± 0.01 pN, implying an average force of about 0.35 pN (Supplementary Fig. S41). With H1.0, the average force on the tetranucleosomes within the H1.0 induced free energy well is ∼0.5 pN (slope = 0.50 ± 0.03 pN for ℓ between 5 and 25 nm), while for ℓ > 30 nm, the average force on the tetranucleosome is ∼0.4 pN (slope = 0.40 ± 0.02 pN for ℓ between 30 and 75 nm Supplementary Fig. S41). This implies that the FES relies on forces of less than a pN to bias thermally induced changes in the end-to-end distance instead of controlling the end-to-end distance with larger forces, as is done with smFS.

In addition, the nanocaliper and tetranucleosome are directly coupled to each other with only 50 bp of DNA at each end. This allows for the probability distribution of the tetranucleosome end-to-end distance within the nanocaliper to be measured directly. This is in contrast to smFS, which typically relies on longer DNA handles to attach chromatin to bead and glass surfaces. Since the nanocaliper free energy landscape is known, the tetranucleosome end-to-end distance free energy landscape is determined without modeling, while smFS models the DNA handles to remove their contribution to end-to-end distance. This allows FES to directly quantify end-to-end distance free energy landscapes model-free and determine free energy landscapes of specific conformational states (NSI₁ versus NSI₂). Overall, FES and smFS are complementary tools that provide synergistic information about chromatin compaction.

Free energy spectroscopy is complementary to computational studies

Computational studies have provided significant insight into chromatin energetics [25, 57, 82] which include tetranucleosome arrays [24, 27, 83, 84]. These studies indicate that tetranucleosomes are highly dynamic and explore a large range of conformations over a free energy range of about 5 k_BT, which is consistent with the FES measurements. However, computational studies typically do not use the same reaction coordinates as are used in FES, which makes quantitative comparisons challenging. By using a published and independently developed coarse-grained model [24], we were able to show quantitative agreement between this independently developed model and the FES measured free energy landscape. Computational studies have also investigated the impact of linker histones within nucleosome arrays [80, 85]. While these studies focused on longer nucleosome arrays, they suggest that linker histones introduce a free energy well of a few k_BT for small end-to-end distances as observed in the FES measurements. Future studies that combine these modeling approaches with FES have the potential to provide important mechanistic insight into the function and regulation of chromatin compaction.

Current limitations to free energy spectroscopy

There are limitations to this FES methodology, which include the inability to determine the transition time between states, the limited spatial resolution and energy range due to the size of the dataset, and the manual image analysis. The current nanocaliper designs explore up to ∼80 nm, so quantifying free energy at larger end-to-end distances, e.g. for longer nucleosome arrays, would likely require new nanocaliper designs. In addition, the mild GA crosslinking could impact the FES measurements by adding additional interactions within the tetranucleosome. However, multiple observations suggest GA crosslinking does not significantly impact the FES measurements. (i) GA crosslinking does not significantly affect the end-to-end distribution of the free nanocaliper. (ii) The nucleosome stacking energy measured by FES is consistent with previous measurements. (iii) The end-to-end distance free energy landscape at 0.25 mM Mg²⁺ agrees with coarse-grained molecular dynamics simulations. Future improvements to the FES methodology could include developing alternative imaging approaches that increase image dataset size and that do not require GA crosslinking.

Conclusion

FES is well-positioned to be used to investigate the numerous factors that control chromatin compaction and regulate eukaryotic genome transcription, repair, and replication. The factors include histone post-translational modifications, histone variants, linker histone isoforms, DNA linker length that impacts nucleosome spacing, histone modifying complexes, ATP-dependent chromatin remodeling complexes, chromatin architectural factors, and transcription factors. Furthermore, FES has the potential to investigate the structural dynamics of other biomolecular complexes, such as RNA folding and its regulation by RNA-binding proteins. Finally, given the ease of obtaining the DNA scaffold and DNA oligonucleotides used to fold nanocalipers, and the general availability of TEM, FES is positioned to become a broadly used method for investigating factors that regulate chromatin states within both euchromatin and heterochromatin.

Supplementary data

Supplementary data is available at NAR online.

Conflict of interest

A patent application has been filed for FES. Bin Zhang serves as a paid consultant for Donaldson Company, Inc. This consulting activity is entirely unrelated to the subject matter of the submitted work.

Funding

This work was funded by the U.S. National Science Foundation, Division of Molecular and Cellular Biosciences (#2411725 to M.G.P. and C.E.C.; #2042362 to B.Z.), and the National Institutes of Health (R35 GM139564 to M.G.P.; R35 GM133580 to B.Z.). We acknowledge resources from the Campus Microscopy and Imaging Facility (RRID:SCR_025078) and the OSU Comprehensive Cancer Center Microscopy Shared Resource, The Ohio State University. This facility is supported in part by grant P30 CA016058, National Cancer Institute, Bethesda, MD. Research reported in this publication was supported by the Office of the Director, National Institutes of Health under award S10 OD023582. This work used Bridges-2 at PSC through allocation BIO240299 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by U.S. National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296. Funding to pay the Open Access publication charges for this article was provided by the National Science Foundation (#2411725).

Data availability

Unique and stable reagents generated in this study are available upon request. TEM images have been deposited at Mendeley data and are publicly available as of the day of publication. The MatLab code for converting the final free energy measurements from three separate probability distributions is available on Zenodo: https://doi.org/10.5281/zenodo.17210278. The code used for running the molecular dynamics simulations is available on Zenodo: https://doi.org/10.5281/zenodo.17210278. Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.