Modeling nucleosome assembly dynamics

In our MD simulations, the DNA is modeled using the 3SPN2.C coarse grained model [26], where each nucleotide is represented by three beads centered on the base, sugar and phosphate groups. The histones are modeled according to the AICG2+ structure-based model [27], where each residue is represented by one bead centered on the Cα atom. This and similar coarse-grained models have been successfully applied to study complex processes such as nucleosome sliding [28–31], unwrapping [7,32,33], and assembly [34], at a fraction of the computational cost required for all-atom simulations [35–39]. Notably, the 3SPN2.C model can capture the sequence-dependent elasticity of DNA [26], and it has been shown to correctly predict the affinity of nucleosome formation for different sequences [33], making this model suitable to investigate the effect of sequence on the assembly kinetics. Further details on the coarse-grained model and its parametrization are provided in the Methods section.

Our simulation system consists of the 147-bp nucleosome-positioning 601 DNA [22], one histone H3/H4 tetramer, and two copies of histone H2A/H2B dimers. The 601 sequence was chosen for its widespread use in experiments and for their characteristic positioning motifs asymmetrically distributed around the nucleosome, allowing us to study their role during assembly. Specifically, 10-bp periodic A/T base steps are more frequent on the left side of 601 (called the beta side in Ref. [19], see Fig 1A), leading to enhanced nucleosome unwrapping on the right side relative to the left (i.e. asymmetric unwrapping) [11,40].

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Fig 1. Simulation of nucleosome assembly on the 601-sequence.

(A) The 601 DNA sequence. The left (0 to -73 bp) and right (0 to +73 bp) halves are shown on the first and second lines, relative to the dyad position at 0 bp (the left end), highlighting the locations of the A/T periodic motifs in red. (B) Representative nucleosome assembly simulation at 400 mM salt, showing the timeline of the sum of radii of gyration of the left and right halves of DNA (Rg_L+Rg_R) during the formation of a complete nucleosome starting from a tetrasome. In the 3 snapshots corresponding to tetrasome (left), hexasome (center) and nucleosome (right), the DNA is shown in grey, the H3/H4 tetramer in yellow, the H2A/H2B dimer binding to the left side of 601 in cyan, and the H2A/H2B dimer binding to the right in green (throughout the paper, the same color code is used). The left and right DNA entry/exits are indicated by the letters L and R respectively. (C) The same trajectory as in panel B (at 400 mM, solid line) projected on the two dimensions Rg_L and Rg_R, together with one successful trajectory at 200 mM (dotted line). The trajectories are superimposed on the free energy landscape at 400 mM calculated by the Markov state model.

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We performed nucleosome MD simulations for 10⁸ MD steps at 400 mM mono-valent ion concentration starting from the tetrasome conformation, in which the H3/H4 tetramer is bound on the 601 DNA at the optimal position and both of H2A/H2B dimers are unbound from the tetramer but bound on DNA at different positions depending on the simulation run (see the left snapshot in Fig 1B as an example). Out of 215 runs, we found only one trajectory that reached to the complete nucleosome, which is shown in Fig 1B and S1 Movie. As highlighted by the timeline of the sum of the radii of gyration of the left and right DNA sections (Rg_L+Rg_R), in this successful trajectory assembly occurs in two steps: firstly, the right half of DNA bends together with the binding of one H2A/H2B dimer to the tetramer (central cartoon in Fig 1B), then, the left half of DNA folds concomitantly with the assembly of the other H2A/H2B dimer to form the complete nucleosome (right cartoon). In Fig 1C, the same trajectory is projected on the 2-dimensional free energy landscape (obtained by the Markov state model as described below) along the left and right radii of gyration of DNA. The free energy landscape also shows that the left side of the 601 DNA bends more easily (i.e., with a lower free energy cost) than the right side; this asymmetry is discussed later in the results. Other trajectories stopped at various apparently metastable states, due to the limitation of the simulation time.

Starting from the same tetrasome conformations used at 400 mM, we also performed 215 simulations at 200 mM and 300 mM, finding respectively 20 and 7 cases of complete nucleosome assembly (a representative trajectory at 200 mM is shown in Fig 1C). We will discuss the salt concentration dependence in more details later in the results.

Kinetic landscape of nucleosome assembly via Markov state modeling

In order to characterize the complete landscape and kinetics of nucleosome assembly and disassembly, we generated a Markov state model (MSM) of nucleosome dynamics using 1000 independent simulation runs of 10⁸ MD steps each at 400 mM salt, starting from a diverse set of partially or fully assembled nucleosomes (215 of these are the same described in the previous section, see Methods for more details). The salt concentration of 400 mM was chosen so that neither the fully disassembled nor the fully assembled nucleosome conformations are too much preferred over the others, facilitating the characterization of the dynamics. To build an MSM, we first divide the conformational space into a set of M discrete microstates, each corresponding to a group of similar conformations, and then we learn from the unbiased MD trajectories the M x M transition probabilities to go from each state to another after a certain time interval, the so-called lag-time. Observing either complete assembly or disassembly events in a single trajectory is extremely rare, but the Markov state modeling approach allows us to combine many trajectories together to fully characterize the kinetics of the system, identifying the main long-lived conformations and the dominant pathways of nucleosome assembly. We generated an MSM with M = 400 microstates and a lag-time of 4x10⁶ MD steps (see Methods section for more details, and S1 Fig for the convergence of the MSM timescales).

In order to aid the interpretation of the MSM, we can use the PCCA clustering method [41] to further group the 400 microstates into a much smaller number of metastable basins. PCCA provides a hierarchical representation of the model by clustering the conformations into more and more basins by considering progressively faster relaxation processes within the system (associated to the eigenvalues and eigenvectors of the MSM). The simplest, yet meaningful, representation of our system can be obtained using just 4 basins. With this choice, the PCCA method automatically identifies the 4 states that are commonly observed in experiments: the tetrasome, two hexasomes, and the complete nucleosome (representative conformation in S2 Fig). S2 Fig shows that the 4 basins are well separated onto a 2-dimensional projection defined by the two slowest coordinates derived from time-lagged independent component analysis [42] (TICA, see Methods for their definition), further indicating the metastability of these 4 states.

However, we can extract more information from our MSM by characterizing a higher number of PCCA basins and the transition probabilities between them. We decided to consider 11 PCCA basins, since this is high enough to provide interesting insights into nucleosome dynamics and low enough to still make the model interpretable. We obtained the transition probabilities between these 11 basins from a coarse-grained hidden Markov state model derived from the full MSM [43]. Fig 2 summarizes the resulting kinetic landscape of nucleosome assembly and disassembly at 400 mM, displaying the identified 11 metastable basins and transition probabilities between these basins (in S3 Fig, we visualize the basins along various combinations of TICA projections). Among the 11 basins, we find one state corresponding to the fully assembled canonical nucleosome (N), seven hexasome states (H_X) and three tetrasome states (T, T_L and T_R). The three tetrasome states are distinguished by the positions of the unbound H2A/H2B dimers. In the T state one dimer is on the left side of the unwrapped DNA while the other dimer is on the right side, remaining close to the target H3/H4 interface. On the other hand, in the T_L and T_R states both unbound dimers interact solely with either the left or right side of the unwrapped DNA, respectively. The seven hexasome structures are divided into two types, depending on whether one H2A/H2B dimer is bound to the left (H_L0, H_L1 and H_L2) or the right (H_R0, H_R1, H_R2 and H_R3) binding interface of the H3/H4 tetramer (respectively wrapping the left or the right sides of the 601 DNA in the canonical nucleosome). Each of these two hexamer types can adopt different conformations depending on the location of the unbound H2A/H2B dimer. In the left hexasome H_L0, the unbound dimer is either far away or it interacts only weakly with nucleosomal DNA. In H_L1, the unbound dimer is located on the right side of the DNA relatively close to its optimal location in the complete nucleosome. In H_L2, the unbound dimer bridges two DNA gyres while facing outside of the nucleosome, in a configuration that cannot easily lead to assembly. H_R0, H_R1 and H_R2 are the respective right hexasome versions of those just described. Finally, in hexasome H_R3 a H2A/H2B dimer binds to the right interface of the H3/H4 tetramer, but it is stabilized by the wrapping of the left side of the 601 sequence. This metastable configuration is not found on the opposite side (there is no H_L3 basin), suggesting that it originates from the marked propensity of the left side of 601 to bend. Notably, in the H_R3 hexasome the DNA follows a right-handed super-helix, contrary to the canonical form that follows a left-handed one. For each of the 11 identified metastable states, we deposited as S1 Data 4 representative coarse-grained nucleosome configurations in PDB format (in S1 Data, which also contains the equilibrium probabilities and the transition probabilities between the basins). Researchers interested in investigating sub-nucleosome conformations in all-atom MD may use these as starting configurations following a previously-developed back mapping procedure [44].

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Fig 2. Markov state model of nucleosome assembly.

We show the 11 metastable basins identified by PCCA clustering from the MSM at 400 mM, indicating their corresponding representative structures, and the transitions probabilities among the basins. The nodes of the tetrasome (T, T_L, T_R), left hexasome (H_L0, H_L1, H_L2), right hexasome (H_R0, H_R1, H_R2, H_R3), and nucleosome (N) basins, are respectively colored in yellow, blue, cyan and red. The areas of the nodes are proportional to their equilibrium populations. The transition probabilities between basins are represented by the thickness of arrows and the numbers associated (e.g., 3e-2 stands for 3x10^-2).

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Asymmetric dynamics of 601 nucleosomes

Our MSM also reveals a clear asymmetry between the left and right side of the 601 sequence. For example, the left hexasome state H_L1 has an equilibrium probability about 20 times higher than its right counterpart H_R1, consistent to what observed in past experiments on hexasomes [45]. Kinetics itself is also highly asymmetric: binding always occurs with a higher rate on the left side of 601, whereas unbinding occurs with a higher rate on the right (Fig 2), as observed in past experimental studies [19]. For instance, a tetrasome T transitions into a left hexasome H_L1 after 4x10⁶ MD steps with probability of ~3x10^-3, one order of magnitude greater than the probability to observe a T to H_R1 transition (~2x10^-4). Similarly, the rate of nucleosome formation is ten times faster when this last assembly step involves H2A/H2B binding on the left side (i.e., from H_R1 to N).

What is the molecular origin of this kinetic asymmetry? As shown in Fig 1A, the left side of 601 contains periodically spaced A/T base steps at most 5+10n positions from the dyad, whereas on the right side this periodic pattern is much weaker. The importance of the periodic motifs in nucleosome folding stems from the intrinsic bending of the DNA helix, which lowers the energy cost of wrapping around histones [33,40,46]. Indeed, when the histone octamer is assembled (state N), the left side of the 601 DNA has a much higher probability to be fully wrapped than the right side: Fig 3A shows how unwrapping DNA from the left side of 601 (the A/T-rich one) costs ~4 additional k_BT of energy compared to unwrapping from the right side (the A/T-poor one), similar with what found in another coarse-grained study [46]. Furthermore, even the distributions of H2A/H2B dimer positions along DNA when they are unbound from the tetramer display strong differences (Fig 3B): the left distribution has two sharp peaks separated by 10 base pairs, one of which corresponds to the location found in the left hexasome and complete nucleosome states, while H2A/H2B positioning on the right side of 601 is much more uniform. This shows that the sequence motifs on the left side of 601 (highlighted again in Fig 3B) facilitate the positioning of the H2A/H2B dimer so that it is poised for a successful binding with the H3/H4 tetramer.

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Fig 3. Asymmetric dynamics in 601 nucleosomes.

(A) Free energy curves of DNA wrapping for the left (in red) and right (in blue) halves of assembled nucleosome at 400 mM. Only the structural ensemble corresponding to the state N, i.e., complete nucleosome, in the MSM (Fig 2) is used. The free energy at the complete wrapping of 70 bp is set to zero. (B) For the tetrasome states including T, T_R and T_L, we plot the probability of the positions of the unbound H2A/H2B dimers on the left (in red) and the right (in blue) sides of the 601 DNA. The DNA position corresponds to the base pair of the phosphate group that is closest to residue ARG29 on the H2A histone (as described in Methods). On the left side the dimer has a much higher probability to be localized at the same position as when bound to the H3/H4 tetramer (indicated by the black dotted line), facilitating successful assembly. On top of the graph, we align each side of the 601 sequence to the DNA position axis (left side in red, right side in blue).

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Assembly pathways and salt dependence

The analysis of transition pathways [23] (S4 Fig) in the coarse-grained MSM of the 11 long-lived states shows that two pathways make up most of the successful assembly transitions between the tetrasome T and the full nucleosome N: 66% of the pathways go through the more stable left hexasome H_L1 (T➔H_L1➔N), while 19% go through the less stable right hexasome H_R1 (T➔H_R1➔N). Therefore, all other hexasome conformations (H_L0, H_L2, H_R0, H_R2 and H_R3) do not significantly contribute to successful assembly and may be therefore considered off-pathway kinetic traps.

The above landscape of nucleosome assembly and disassembly was obtained at an intermediate ionic strength (400 mM). However, an interesting question is how the assembly and disassembly processes are affected by changes in salt concentration. As described before, our assembly simulations from the tetrasome configurations at 200 mM, 300 mM, and 400 mM salt resulted in a complete nucleosome assembly in 20, 7 and 1 cases out of 215 runs, suggesting that low salt favors assembly. This is reasonable, since nucleosome are stabilized by protein-DNA electrostatic interactions. However, it does not explain the observation that the optimal procedure to assemble nucleosomes in vitro is by slowly decreasing the salt concentration from high to low values [25], and otherwise chromatin remodelers are required to successfully complete assembly [16]. One possibility is that at low ionic strength the kinetic landscape becomes more rugged, and the system struggles to escape from off-pathway traps. Indeed, Fig 4A indicates that although lower salt concentration increases the probability to reach the complete nucleosome, it also significantly increases the probability to reach the kinetic traps identified from the MSM (H_L0, H_L2, H_R0, H_R2 and H_R3).

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Fig 4. Salt-dependent assembly kinetics and escape from metastable states.

(A) The time-dependent probability that, starting from a tetrasome, the system reaches either a full nucleosome (left panel) or one of the kinetic traps (right panel), which are defined as all the off-pathway conformations corresponding to the H_R0, H_L0, H_R2, H_L2 and H_R3 basins. (B) The escape kinetics from the H_R2 kinetic trap. The left panel plots the timeline of the slowest characteristic coordinate, the first coordinate derived from time-lagged independent component analysis [42] (TICA 1, see Methods section for its definition) for a representative trajectory for each salt concentration. The right panel shows, as a function of time, the probability to remain the initial state, defined as the TICA 1 interval from -4 to +4, obtained from 100 trajectories (see Methods for more details). (C) The escape from the partially opened right hexasome identified during the successful hexamer-to-nucleosome assembly events. In order to highlight the conformational change, we indicate with two dotted red lines the wider angle between the unbound H2A/H2B dimer and the neighboring H3/H4 dimer, relative to that found in the fully assembled nucleosome. The left and right panels are the same plot as (B) except that the initial state is defined as the TICA 1 interval from -0.5 to 0.5. The time required to escape from these trapped states increases at low salt.

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To further study this effect, we run for each ionic strength 100 10⁷-steps simulations starting from a H_R2 hexasome conformation, one of the traps that do not participate in the assembly, where the unbound H2A/H2B dimer bridges both left and right sides of DNA. Consistently with our hypothesis, the time required to escape from the initial state increases as we lower the salt concentration (Fig 4B). This analysis was performed by plotting the MD trajectories along the slowest TICA coordinate (see Methods for more details).

During the successful assembly trajectories at 200 mM and 300 mM salt, we also found evidence of new metastable states that are not clearly captured by our MSM at 400 mM. In these conformations the nucleosome is already very compact but partially opened: due to the absence of DNA bending at super-helical location +/-3 (i.e., 3 DNA helical turns away from the dyad), one or both H2A/H2B dimers are unbound from the tetramer and oriented at a wider angle relative to the canonical conformation (see snapshot in Fig 4C). During our trajectories, we found this angle to vary between ~10 and ~50 degrees (the angle was computed based on the orientation of the longest helices in histones H2A and H4). PDB coordinates of representative partially opened conformations (part of the H_L, H_R or T states) have been deposited in S1 Data together with the other metastable states defined from the MSM. To further investigate their properties, we run 100 10⁷-steps simulations starting from the partially opened right hexasome at 200, 300, and 400 mM salt. Fig 4C shows that, similarly to the H_R2 state, the escape from the partially opened state becomes slower as we decrease the salt concentration, presumably because this conformation is stabilized by electrostatic interactions of the H2A C-terminal and H2B N-terminal histone tails bridging the two DNA gyres. Furthermore, at 300 mM we find that starting from a partially opened hexasome conformation the probability to transition either into the complete nucleosome or into a more open hexasome is about 50%, similarly to a transition state (within 5x10⁶ MD steps, we observe complete assembly events in 269 runs out of 400 starting from the left partially opened state, and in 156 out of 400 from the right one). Finally, we note that all the evidence points to these partially opened nucleosomes being the same as those identified in a recent FRET study [19]. In both experiments and in our simulations, 1. the angle between the open H2A/H2B dimer and the tetramer very similar (a value of ~20 degrees was suggested from experiments), 2. the stability of these states decreases with increasing ionic strength, and 3. they are intermediates along the nucleosome assembly pathways. In order to test the accuracy of our predictions, it is also useful to compare the transition rates between complete nucleosomes and opened states, which are also available from FRET experiments [19]. In principle, one MD timestep in our simulations equals 15 fs; however, coarse-graining smoothens the free energy landscape, speeding up the dynamics, so we map 1 MD timestep to 30 ps, based on the speedup observed in our previous work on nucleosome sliding using the same model [29] (a similar speedup was also observed in coarse-grained simulations of protein folding [47]). Experimentally, at low salt concentration, the characteristic times for the transitions (inverse rates) from the complete nucleosome to the partially opened state and back have been estimated to be respectively >50 ms and >3 ms, with an equilibrium constant of 0.06. From the simulations at 300 mM, we estimated the inverse rates to be respectively ~5x10⁸ MD steps = 15 s and ~10⁶ MD steps = 30 ms (see Methods), consistent with the lower bounds from experiments, and an equilibrium constant of ~0.002, somewhat underestimating the stability of the partially opened state.

In order to explore whether the disassembly mechanism is also affected by the salt concentration, we run additional 10⁸-step simulations starting from the complete nucleosome at 500, 600, 700, and 800 mM salt (30 trajectories each). As expected, nucleosomes become unstable at high salt, with frequent DNA unwrapping and dimer unbinding. Interestingly, disassembly pathways appear qualitatively different at intermediate (400 mM) and high (800 mM) salt: while in the former case H2A/H2B unbinding from the tetramer proceeds cooperatively with the unwrapping of DNA, in the latter case DNA unwrapping anticipates histone unbinding (Fig 5). This trend is in qualitative agreement with the experimental literature: the cooperative transition is consistent with the recent results obtained by FRET at intermediate salt [19], while the non-cooperative behavior was suggested by a combination of SAXS and FRET at very high 1.9 M salt concentration [11].

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Fig 5. Salt-dependent nucleosome disassembly pathways.

We project a set of representative disassembly trajectories along the DNA radius of gyration (Rg, the vertical axis) and a coordinate capturing dimer-tetramer contact formation (the horizontal axis, see Methods for the precise definition), at either 400 mM (in brown) or 800 mM salt (in cyan). While at intermediate salt concentration disassembly occurs in a single step involving dimer unbinding from the tetramer cooperatively with the stretching motion of DNA, at high salt concentration dimer unbinding follows a significant unwrapping of DNA from the histone octamer.

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Coarse-grained model

To enable the complete study of nucleosome assembly and disassembly, we utilize coarse-grained molecular modeling: we map each amino-acid to a single bead [27], and each nucleotide to three beads corresponding to base, sugar, and phosphate groups [26]. The functional potentials and force-field parameters are chosen according to the AICG2+ structure-based model for histones [27] and the 3SPN2.C sequence-dependent model for DNA [26]. The flexible histone tails (residue ids 1–32 for H3, 1–23 for H4, 1–14 and 121–128 for H2A, 1–26 for H2B) are modeled according to a sequence-dependent statistical local potential specifically designed for disordered regions, which was derived by Boltzmann inversion from the distribution of angles and dihedral angles observed in the non-structured subset of the Protein Data Bank [59]⁠. The potential of the remaining folded regions is based on the native structures of the first copies of each H3/H4 and H2A/H2B dimers present in the nuclesome crystal structure with PDB id 1KX5 (using a single reference for each pair of dimers ensures the symmetry of the histone octamer, which is important to study the asymmetry induced by DNA sequence). Whenever two protein residues i and j are within a certain cutoff distance in the native structure, the coarse-grained potential includes a contact interaction that favors the binding of the two residues [27]:

Where ε_ij is a contact-dependent energy determined from the all-atom reference structure and r₀^ij is the native distance between the residues. In addition, the protein potential also includes bond, angle and dihedral interactions with equilibrium values set according to the native structure, and non-native excluded volume interactions proportional to the 12^th power of the distance.

Experimentally, in the absence of DNA or at high salt the histone octamer breaks into a H3/H4 tetramer and two H2A/H2B dimers [11,19,60], but using the default AICG2+ potential we do not observe histone octamer breakage under these conditions. This happens because the default AICG2+ scaling factor defining the overall strength of the residue-residue interactions was optimized for studying the folding of single-domain proteins, but in our case the attraction between the H3/H4 and H2A/H2B dimers is over-estimated. In order to find a suitable scaling factor for the native interactions between H3/H4 and H2A/H2B residues, we scanned different values performing MD simulation at either low salt (200 mM), at which we expect our nucleosomes to be stable, or high salt (800 mM), at which we expect disassembly. We find that scaling factors lower than 0.2 lead to unstable nucleosomes even at low salt, whereas for values higher than 0.4 we do not observe disassembly at even at high salt. Rescaling the dimer-tetramer interactions by a factor of 0.3 reproduces the expected behavior of nucleosomes: out 30 independent simulation runs of 10⁸ MD steps starting from a fully assembled nucleosome, we observed no disassembly events at 200 mM salt, whereas we observed disassembly in 4 out of 30 runs at 800 mM; as shown by the MSM reported in the Results section, nucleosome assembly is reversible at intermediate 400 mM salt, with comparable equilibrium populations of tetrasomes and complete nucleosomes. This critical concentration at which nucleosomes disassemble is somewhat lower than the value observed in experiments, which is around 600 mM [19]; however, taking into account the high computational cost of the parameterization, we regard the agreement between experiments and simulations satisfactory, and sufficient to provide reliable insights into the nucleosome assembly dynamics. We use the dimer-tetramer scaling factor of 0.3 in all simulations reported in this study.

Histones and DNA interact via excluded volume, Debye–Hückel electrostatics, and hydrogen bond interactions. The hydrogen bonds are modeled according to a distance- and angle-dependent potential based on nucleosome crystal structures, and was calibrated based on nucleosome structural stability and DNA unwrapping [28]. The size of the beads to define excluded volume interactions, and the parameter settings for the hydrogen bonds are the same used in our previous works [28–30]. The charges on the histone residues are determined via the RESPAC method [61], via which we optimized the coarse-grained Debye–Hückel electrostatic potential against the one obtained by solving the full Poisson-Boltzmann equation around the all-atom histone core structures, similarly to what done in other coarse-grained models of the nucleosome [62] (the optimization was performed at 100 mM salt, but the resulting charges are very robust this choice). The RESPAC procedure was applied to the globular part of the H3/H4 tetramer and the two H2A/H2B dimers, whereas for the flexible tails we used the standard integer charges on the Lysine, Arginine, Aspartic and Glutamic acids residues only.

Simulation setup

All reported molecular dynamics (MD) simulations have been performed with the software CafeMol [63] by integrating the equations of motion using Langevin dynamics at 300 K with a timestep of 0.3 CafeMol time units (corresponding to 15 fs). During the MD simulations, we apply harmonic restraints on the structured region of the two H3 histones (0.001 kcal/mol/Ų) and on the phosphate beads of the DNA dyad residues (bp id 73, 0.1 kcal/mol/Ų), limiting large-scale sliding of DNA relative to the H3/H4 tetramer (but not the unbound H2A/H2B dimers). The nucleosome sliding mechanism was already analyzed in previous studies [28,29]⁠, and especially for the 601 positioning sequence considered here can be extremely slow. In our simulations, the H3/H4 is always placed at its optimal experimental position on the DNA sequence; preventing its sliding allows to avoid exploring less favorable positions and to focus solely on the assembly dynamics, which is our main focus. Finally, the system is enclosed into a spherical volume with a radius of 20 nm by a harmonic potential. CafeMol input files to run nucleosome assembly simulations are provided in the S2 Data.

The starting conformations for the 1000 MD simulation runs at 400 mM used to generate the MSM have been obtained during high-salt (800 mM) simulations starting from complete nucleosomes, and then selecting 1000 timeframes that include tetrasomes (215 conformations), left and right hexasomes (105 and 118 conformations, respectively), and nucleosomes (562 conformations). To quantify the salt-dependence of the nucleosome kinetics, we also performed, at both 200 and 300 mM salt, 215 10⁸-steps simulation runs starting from the 215 tetrasome conformations. To analyze the escape from 2 metastable states (either the H_R2 hexasome or the partially opened state) as a function of salt concentration, we run 100 independent 10⁷-steps trajectories starting from the same conformation (either H_R2 or partially opened H_R1) at 200, 300, and 400 mM salt. In order to characterize the time necessary to escape from these states (analysis in Fig 4), we projected the molecular positions onto the slowest TICA motions [42] within each basin. The TICA projection was performed separately for the H_R2 and partially opened metastable basins based on the positions of H2A residues 46 to 73 of the unbound dimer and the left side of DNA (the one interacting with the unbound H2A/H2B dimer) during the simulations at 300 mM salt. The inverse transition rate from the partially opened state to the complete nucleosome at 300 mM was estimated from the 100 simulations starting from the partially opened state by fitting an exponential curve to the survival probability to remain in the initial state before transitioning into the complete nucleosome, while the inverse transition rate for the opposite process was obtained from 100 10⁷-MD-steps simulations starting from the complete nucleosome, during which we observed two transitions into the partially opened state.

Markov state modeling and analysis of metastable states

To generate the Markov state model of nucleosome assembly at 400 mM salt from our 1000 MD simulations, we first have to cluster the configurations into discrete states based on a distance between configurations defined on a suitable space. To this aim, we project the particle positions onto a series of collective variables that capture all the relevant modes of nucleosome dynamics: H2A/H2B dimer binding to DNA and to H3/H4, DNA wrapping, and DNA handedness. Since the two H2A/H2B dimers are identical, we generated variables that are symmetric under an exchange of these two dimers. In principle, the two H3/H4 binding interfaces are also identical, but the 601 DNA that is wrapped around the tetramer is not symmetric, so we do want to distinguish between the left and right H3/H4 interfaces, corresponding respectively to the dimer binding the left or the right side of the 601 DNA. We defined 5 types of collective variables (for a total of 4+4+2+2+1 = 13 collective variables):

• For each H3/H4 tetramer interface, the sum and the maximum value of the contacts with the two H2A/H2B dimers. A dimer-tetramer contact is defined as c = 1/(1+d/σ), where σ = 1 nm and d is the distance between the center of mass of H2B residues 73–100 and the center of mass of H4 residues 65–92 (corresponding to the two interfacial regions). Therefore, a contact takes a value close to 1 when a H2A/H2B dimer binds the H3/H4 tetramer in its native configuration, whereas it is close to 0 when the H2A/H2B dimer is far from the H3/H4 tetramer. We first compute the 4 contacts between each of the 2 tetramer interfaces (t1 and t2) with the two dimers (d1 and d2): c_t1,d1, c_t1,d2, c_t2,d1, and c_t2,d2. Then, we define the 4 collective variables: c_t1,d1+c_t1,d2; c_t2,d1+c_t2,d2; max(c_t1,d1,c_t1,d2); and max(c_t2,d1,c_t2,d2).
• For each side of the DNA (left, residue ids 1–73, and right, ids 75–147), the sum and the maximum value of the contacts with the two H2A/H2B dimers. A DNA-dimer contact is defined as 1/(1+d/σ), where σ = 1nm, and d is the distance between residue ARG29 on H2A (in the canonical nucleosome this residue binds the DNA at SHL +/-4.5) and the closest DNA phosphate group. The sum of the DNA-dimer contacts enables us to distinguish between the various tetrasome conformations where two dimers can bind either the same (states T_L and T_R in Fig 2 of the main text) or two different DNA regions (state T). The maximum value of the contact distinguishes strong or weak binding of the H2A/H2B dimer to the DNA.
• For each side of the DNA (left and right), the H2A/H2B dimer location along the DNA, which is defined as the residue id of the DNA phosphate group (on the 5’ strand for either side) that is closest to the H2A ARG29 residue on the dimer closest to the considered DNA side. For example, for the location on the left side of the DNA, if the first H2A/H2B dimer is closer than the second one (based on the minimum distance above), we consider the base pair id of the phosphate group closest to the first dimer. This variable enables us to capture the sliding of the H2A/H2B dimers along the DNA when they are unbound from the H3/H4 tetramer.
• For each side of the DNA, the number of base pairs wrapped around the histones, where a base pair is considered unwrapped if the center of mass of its base groups is displaced radially from the nucleosome super-helix axis by more than 6 Å.
• The handedness of the nucleosome, which is captured by the component of the distance vector between the centers of mass of base pairs with ids -35 and +35 (relative to the dyad id 0) along the nucleosome super-helix axis. Positive values correspond to canonical left-handed nucleosomes, whereas negative ones correspond to right-handed conformations such as those in the metastable state H_R3 in Fig 2 of the main text.

Starting from these 13 collective variables, we aggregate all 1000 MD trajectories and perform a further dimensionality reduction using time-lagged independent component analysis (TICA) [42] as implemented in PyEMMA 2 [64]⁠ using a lag-time of 10⁶ MD steps. We then select the top 8 slowest coordinates (which explain 95% of the kinetic variance in the data) and cluster the conformations into 400 discrete states using k-means [65]⁠. The MSM was then generated by PyEMMA 2 [64] using a lag-time of 4x10⁶ MD steps. Further increasing the lag-time does not lead to significant changes in the implied time scales of the MSM, indicating the robustness of the results (S1 Fig). The TICA co-ordinates are also ideally suited for visualization purposes: in S2 Fig we show the first two coordinates are sufficient to separate the system into 4 regions corresponding to nucleosomes, left and right hexasomes and tetrasomes (more TICA coordinates can further separate into more metastable basins with faster transitions among them, S3 Fig).