Toward Convergence of Experimental Studies and Theoretical Modeling of the Chromatin Fiber*

Understanding the structural organization of eukaryotic chromatin and its control of gene expression represents one of the most fundamental and open challenges in modern biology. Recent experimental advances have revealed important characteristics of chromatin in response to changes in external conditions and histone composition, such as the conformational complexity of linker DNA and histone tail domains upon compact folding of the fiber. In addition, modeling studies based on high-resolution nucleosome models have helped explain the conformational features of chromatin structural elements and their interactions in terms of chromatin fiber models. This minireview discusses recent progress and evidence supporting structural heterogeneity in chromatin fibers, reconciling apparently contradictory fiber models.

The 3 billion DNA base pairs of the human genome are densely packed within eukaryotic chromatin, a nucleoprotein complex in which the DNA is wrapped around nucleosomes (see Fig. 1). Nucleosomes and higher order chromatin structures serve essential cellular functions, including condensation of meters-long genomic DNA by several orders of magnitude to enable its packaging into the micrometer-sized cell nucleus and regulation of DNA-directed processes, such as transcription, replication, recombination, and repair through local and dynamic unfolding of chromatin.
Whether the higher order structure of nuclear chromatin is organized into a hierarchy of folding states (1) or non-hierarchical fractal geometry (2), previous landmark studies established the nucleosome as the repeating unit of chromatin (Figs. 1 and 2) (3). This basic unit consists of ϳ200 bp of DNA, 147 bp of which are wound around the outside of a spool composed of core histones (two each of histones H2A, H2B, H3, and H4) to form the nucleosome core; the remainder (linker DNA) joins adjacent nucleosomes. At low salt, arrays of nucleosomes adopt an extended "beads-on-a-string" conformation ( Fig. 1). In vitro, nucleosome arrays in solutions of physiological ionic strength form a compact higher order structure called the 30-nm diameter chromatin fiber. Determination of the detailed architecture of the chromatin fiber has occupied experimental and theoretical scientists for decades (4 -8).
In this minireview, we discuss recent progress in experimental analysis of chromatin folding, as well as specific molecular interactions and forces stabilizing compact chromatin. We include recent modeling studies that, consistent with experimental data, reveal a structural stability of internally heterogeneous chromatin fibers. These collective experimental and modeling approaches appear to reconcile contradictory chromatin fiber models by relating variable chromatin organization to functional roles.

Nucleosome Core Structure
The nucleosome core can be reconstituted from ϳ147-bp DNA fragments and histone octamers. Such particles have been crystallized and studied by x-ray diffraction ( Fig. 3) (e.g. Refs. 9 -14), revealing fine detail and differences from free DNA and providing solid anchors for modeling studies.
Each histone contains a well ordered domain responsible for the primary wrapping of DNA and tail regions, which make important points of contact between the protein and the DNA (see hypothetical model in Fig. 2). Specifically, an underwinding (10.2 versus ϳ10. 5 bp/turn) of the nucleosome-bound DNA superhelix lines up neighboring DNA grooves to form a channel through which the H3 and H2B N-terminal tail domains pass. The tails likely play key roles in regulating biological processes, such as transcription, that require a conformational change in higher order chromatin structures. Unfortunately, the tail domains are typically poorly ordered in crystal structures of nucleosome cores due to the high salt concentrations required for crystallization. Because of this structural uncertainty (15), for modeling purposes, these domains are typically approximated as flexible polymers.
Solution studies revealed that the nucleosome core structure is highly dynamic and undergoes spontaneous and reversible unwrapping of DNA segments from the histone surface (16,17); this flexibility provides accessibility and opportunity for interactions with DNA-binding proteins. In addition, it should be noted that inclusion of histone variants, e.g. centromeric histone H3 variant CENP-A (centromere protein A) (18,19), may significantly alter both the extent and stability of nucleosome DNA wrapping.

Nucleosome Array Folding in Vitro: The 30-nm Fiber
Nucleosome arrays are characterized by variable linker DNA lengths ranging from 10 to 90 bp. Thus, the nucleosome repeat lengths (NRLs) 2 vary from the shortest 155-bp NRL in fission yeast (20) to the longest NRL in echinoderm sperm (ϳ240 bp) (21). As mentioned above, arrays are further compacted in * This work was supported, in whole or in part, by National Institutes of Health Grants R01 GM055164 (to T. S.) and R01 GM52426 (to J. H.). This work was also supported by National Science Foundation Grants MCB-0316771 (to T. S.) and MCB-1021681 (to S. G.) and Philip-Morris International (to T. S.). 1 To whom correspondence should be addressed. E-mail: schlick@nyu.edu.
higher order chromosomal structures. In vitro, nucleosome arrays form a compact 30-nm chromatin fiber, whose exact structure (e.g. solenoid, zigzag, superbead, and others) has long been debated (Figs. 4 and 5) (3). Part of this structural uncertainty is due to variability in the native nucleosome arrays studied. A recent advance in understanding the internal organization of the chromatin fiber occurred with the construction of polynucleosome templates for precise positioning of histone octamers, initially from repeats of 5 S ribosomal DNA (22) and later from clone 601 DNA selected from random synthetic DNA sequences (23). Note that the strong positional sequences may create structural artifacts, so any generalizable conclusions require independent confirmation with native chromatin.
Studies using 601 repeats have provided structural insights for oligonucleosomes with relatively short NRLs (167 and 177 bp) folded in the presence of divalent cations. For example, in cross-linking studies using arrays of 601 nucleosomes with 177-bp repeats, Richmond and co-workers (24) produced compelling evidence for a zigzag model in which nucleosome i is closest in space to nucleosomes iϮ2 (see contact patterns in Fig. 5, A-C). Zigzag folding was also evident in a subsequent crystallographic study of a tetranucleosome (25) with a 167-bp NRL.
However, Rhodes and co-workers (26,27) produced strong evidence for the earlier interdigitated solenoid model (28) using electron microscopy of chromatin fibers reconstituted with longer NRLs (Ͼ177 bp) and with linker histone (LH) and divalent ions. More recently, cross-linking experiments combined with modeling have demonstrated an internal heterogeneity of compact 30-nm fibers (29). In particular, divalent ions appear to promote some bending of the linker DNAs, so compact structures for the chromatin fiber arise that combine mostly straight-linker (zigzag-like) fibers with a small percentage of bent linkers (Fig. 5C). Thus, bent-linker DNAs caused by divalent ions lead to a heteromorphic fiber form that combines features of both zigzag and solenoid models. On length scales much smaller than the persistence length (p b ), DNA can be considered straight. In eukaryotic cells, DNA wraps around a core of histone proteins to form the chromatin fiber. The fiber is shown in both the extended view and a hypothetical compact zigzag view (the "30-nm fiber") deduced from a modeling study (29). Chromosomes are made up of a dense chromatin fiber, shown here in the metaphase stage. For reference, we highlight in pink in all the DNA/protein views the hierarchical organizational unit preceding it. The length scale on the right indicates the level of compaction involved.  Shown are the tetranucleosome complex (25), a nucleosome core particle containing a poly(dA⅐dT) tract (red) (12), and a human centromeric nucleosome containing the centromere-specific histone H3 variant CENP-A (18). In single nucleosome particles, the histone proteins are colored by type (i.e. H3, purple; H4, silver; H2A, orange; H2B, blue). Arrows point to residues present in CENP-A but not in the canonical H3 histone.
The heterogeneous nature of the 30-nm fiber is consistent with earlier electron microscopy (30) and atomic force microscopy (31) imaging. Other evidence for an irregular zigzag structure was provided by single-molecule experiments combined with a low-resolution model to interpret force/extension curves from pulling experiments by optical tweezers (32). However, recent single-molecule force microscopy studies (33) subjecting 25-nucleosome arrays with two linker lengths (NRLs of 167 and 197 bp) to forces up to 4 piconewtons suggested a fundamental one-start solenoid organization for the longer array. Clearly, many chromatin configurations are viable in ambient conditions. In fact, this multiplicity of conformations can be explained by the sensitivity of fiber architecture to a large number of internal and external factors (LH, NRL, ionic conditions, and variations in NRL from one core to the next; see next section). Indeed, such a dependence makes the chromatin fiber infinitely interesting and suitable for performing a rich array of functions in the cell.

Molecular Interactions and Factors Stabilizing Compact Chromatin
LHs Define the Path of Linker DNA-LHs play a crucial role in compacting 30-nm fibers (34, 35) by shielding the negatively charged linker DNA and thereby allowing close apposition of these segments in the folded fiber. Typical LHs have a tripartite structure, with extended positively charged C-and N-terminal domains protruding from a central globular domain. Mapping with micrococcal nuclease (29) or hydroxyl radicals (36) provided strong evidence for positioning of LH at the dyad axis with symmetric protection of ϳ11 bp of each linker DNA. This symmetry and the ability of the globular domain alone to bring two linker DNA segments in close juxtaposition to form linker DNA stem motifs are consistent with energy-minimized models (37,38).
Although the C-and N-terminal domains of LH are disordered in solution, modeling suggests that folding of the C terminus upon contacting the nucleosome contributes to the linker stem formation. On the basis of a striking homology between the C-terminal domain of LH and the HMG box fold motif, Bharath et al. (39) predicted that the folded C-terminal domain could induce formation of linker DNA stems by kinking inward and then diverging at the C-terminal domain. More recent analysis of experimental data and modeling studies have provided a nanoscale model of the LH-induced stem structure (38). Condensation consistent with folding of the C-terminal domain upon H1 binding to the nucleosome was recently con-  Thus, LH refines the path of linker DNA within the chromatin fiber by forming rigid stems that reduce the separation angle of entering/exiting DNA (35,42). Additionally, LH shields the  (72). The center images also show the individual histone tail beads. Color coding is as described above. E, effect of various dynamic LH binding mechanisms on the chromatin unfolding mechanism for 24-core 209-bp fibers as revealed from stretching simulations mimicking single-molecule pulling experiments at monovalent salt conditions of 0.15 M (43). Panel 1, resulting force extension curves for fibers with one LH rigidly fixed to each core (blue curve) versus LHs that bind/unbind dynamically (average concentration of 0.8 LH/core; red curve) with added divalent cations. Dynamic LH binding/unbinding dramatically decreases the fiber stiffness and the forces needed for unfolding with respect to fibers with fixed LH, improving the agreement with experiments (33) significantly. pN, piconewtons. Panel 2, images representing unfolding intermediates at different forces along the dynamic LH curve in panel 1. Intermediates reveal "superbead-on-a-string" structures in which compact clusters coexist with extended fiber regions. Panel 3, effect of fast and slow dynamic LH binding/unbinding during chromatin fiber unfolding without divalent ions (43). The slow-rebinding LH molecules cause a more dramatic softening effect than a pool of fast-rebinding LH, whereas fast LH rebinding promotes formation of superbead-on-a-string conformations with compact clusters. Together, fast-and slow-binding LH pools provide facile fiber unfolding through heteromorphic superbead conformations. electrostatic repulsion of the linker DNAs and promotes linker DNA bending. Still, recent single-molecule force studies (33) suggested a stabilizing rather than structure-determining effect of LH on higher order chromatin. This difference may reflect the intrinsic ability of LH to rearrange under the applied force and thus different organization of static and dynamic chromatin (Fig. 5E) (43).
Histone Tails Bridge Nucleosomes-The N-terminal domains of all core histones and the C-terminal domain of histone H2A lack a defined secondary structure when free in solution and are highly mobile when assembled into nucleosomes (15). These histone tails are highly positively charged but make only marginal contributions to individual nucleosome structure. Still, they are required for formation of higher order chromatin structures (44,45), probably due to mediation of internucleosome interactions. A possible molecular mechanism of intrinsic nucleosome core interactions emerged from an x-ray analysis of nucleosome core crystals where the histone H4 N-terminal tail from one nucleosome contacts an acidic patch formed by histones H2A and H2B in a neighboring nucleosome (9). Evidence for H4 tail-H2A interactions was provided by cross-linking studies, suggesting both intra-and inter-array contacts (46,47).
Recent cross-linking studies demonstrated that the same tail domains of core histones can alternate between intra-and inter-array interactions (47-49); we designate these as cis-and trans-interactions, respectively (see hypothetical model in Fig.  2). The inclusion of core histone variants (50) or mutations (51) altering the histone octamer surface can dispatch internucleosome interactions toward either cis-interactions and intra-array folding or trans-interactions and inter-array oligomerization.
Histone Post-translational Modifications Directly Alter Nucleosome Interactions and Chromatin Folding-Post-translational modifications (PTMs) of histone proteins have emerged as an important mechanism for modulating chromatin structure and function (52,53). Although many of the histone PTMs affect chromatin structure and function by recruiting additional chromatin-remodeling or architectural factors consistent with the histone code hypothesis (54), histone acetylation and phosphorylation alter the protein charge and chemical properties of the amino acid side group and may directly affect internucleosome interaction and higher order folding.
For example, acetylation of Lys-16 in histone H4 was sufficient to destabilize salt-dependent folding of a nucleosome fiber (55). Surprisingly, charge alterations could not reproduce the action of Lys-16 acetylation; this suggests a highly specific interaction of the N-terminal domain with the histone octamer surface (56), perhaps dependent on formation of ␣-helical secondary structure by the histone H4 N-terminal tail (57). The other sites of histone N-terminal tail acetylation can also directly alter the folding and compaction in chromatin (58) and apparently act in a less specific manner in both cis-and transinteractions (49). These modifications are likely functionally redundant as demonstrated by mutagenesis in vivo (59) and histone tail swapping experiments (60). This non-specificity is consistent with primarily electrostatic interactions of the N-terminal tails with other histones and DNA (61).
Besides acetylation, residue substitution experiments in which histone H4 was trimethylated at Lys-20 and probed by array sedimentation showed increased compaction of nucleosome arrays (11). Although the mechanism is unclear, the position of Lys-20 in the histone H4 tail (residues 16 -20) may affect the internucleosome interaction with a surface of histone H2A. Moreover, ubiquitylation of histone H2B causes a notable disruption of a chromatin fiber structure through a mechanism distinct from histone acetylation (62).
Although the above studies were conducted on nucleosome arrays devoid of LH, other experiments have shown that histone PTMs can also override LH-stabilized fiber compaction. For example, citrullination of arginines in histone H3 and H4 N-terminal tails (63) and in vitro acetylation of chromatin fibers (64) or acetylation mimics within the H4 tail domain (49) override the strong stimulation of condensation of nucleosome arrays caused by LHs. Thus, histone acetylation may alter the secondary structure of the core histone tail domains in a way that does not simply abolish internucleosome interactions but actively interferes with other stabilizing chromatin interactions (65).
Effect of Nucleosome Positioning and NRL on Chromatin Compaction-Although NRL variations were initially proposed to alter chromatin fiber diameter proportionally to linker DNA length (21), more recent ultrastructural studies suggest a stepwise increase in the chromatin diameter from ϳ33 nm for chromatin with 30 -60-bp linkers to ϳ42 nm for 70 -90-bp linkers (26). The observed structural transitions were explained by topological variations of linker folding from the helical ribbon configuration to a crossed zigzag configuration (66), as well as by polymorphic chromatin models where the nucleosome linkers were tangentially oriented in the fiber and did not cross the fiber axis (67). For short NRLs typical of yeast and neuronal cells (167 bp), a smaller diameter of ϳ21 nm with a zigzag morphology (68) was obtained, consistent with nucleosome arrangement in the tetranucleosome x-ray crystal structure (25).
Linker length variations were also predicted to have a strong effect on the chromatin higher order structure via altered internucleosome orientations (30,31,69,70). However, experiments on nucleosomes with NRLs varied in a fixed pattern mimicking natural variations (207 Ϯ 2) showed similar folding to uniform arrays of the same length (29). Consistent with these findings, modeling work has suggested that whereas short linkers are too rigid and long linkers are too flexible, medium values (in the range of 200 -210 bp) can adopt variable conformations to optimize overall fiber compaction (Figs. 2 and 5D) (71,72).

Modeling Approaches
Challenges and Approaches-As discussed in a recent modeling and simulation perspective (73), our computing power and algorithms have improved markedly over the past decade, rendering problems of greater scientific significance solvable with enhanced confidence and accuracy. Although all-atom simulations of nucleic acids have steadily increased in accuracy, scope, and length (e.g. microsecond simulations of solvated B-DNA dodecamer (73)), coarse-grained models are required to simulate macromolecular chromatin systems that are too large for atomic models and highly dynamic. Creation of such models is required to resolve key functional components of the molecular system while approximating others. Ultimately, modeling the various folded states of the chromatin fiber requires multiscale methods to bridge the resolution among different spatial and temporal scales.
As summarized in Fig. 4 (and in the indicated references), several groups have developed different models with various levels of complexity and involving different simulation techniques. In most of these models, nucleosome geometries are simplified by using a few key variables (e.g. nucleosome positions and charges and angular orientation between the wrapped and existing/entering DNA), water is treated implicitly, and configurations are generated by Monte Carlo sampling or via analytic formulations that sample various parameter ranges. Model validation is performed using available experimental data, such as salt-induced compaction of oligonucleosomes to reproduce experimental sedimentation coefficients and nucleosome packing ratios, diffusion behavior of oligonucleosomes, salt-dependent extension of histone tails measured by the tail-to-tail diameter of the core and radii of gyration for mononucleosomes, LH orientations, and internucleosome interaction patterns (see Ref. 61 for a summary of such model validation details and available experiments). Overall, such modeling is particularly useful for probing structural and energetic effects as a function of certain parameters or conditions like the ionic salt concentration, NRL, and the presence of LHs. Simulations can also suggest specific configurations to help interpret single-molecule pulling experiments by associating specific force versus extension data points with fiber conformations, as shown in Fig. 5.
Effect of Ionic Conditions, NRL, and LH on Fiber Architecture-As an example of these parameter dependences, Wong et al. (67) showed the dependence of fiber width on the linker DNA length and the orientation of LHs. Modeling of simplified coarse-grained nucleosome models by Rippe and co-workers (74) reinforced the large effect of the linker length and nucleosome twist angles on the extent of fiber compaction. Mesoscale modeling combined with experimental cross-linking studies that measured nucleosome interaction patters complemented by EM visualization (29) revealed a compaction pattern as LH and divalent ions are added: an open, disordered, zigzag organization for chromatin fibers without LH rearranges to form compact zigzag forms with LHs under monovalent ion conditions. With divalent ions, further compaction arises by bending a small portion of the linker DNAs to form a heteromorphic architecture (Fig. 5C).
In our model (61,72), each nucleosome histone octamer (without protruding tails and with the wound DNA) is treated as an irregularly shaped electrostatic charged object with point charges parameterized to reproduce the atomistic electric field. Each flexible histone tail is modeled as a chain of spherical beads representing five amino acids each, and LHs are modeled using three beads representing the C-terminal, N-terminal, and globular domains (Fig. 5D). The linker DNA connecting the nucleosomes is treated using the wormlike chain elastic approximation for DNA. The total energy consists of bending, stretching, torsion, excluded volume, and electrostatic contributions (43,61).
Studies investigating the folding of the chromatin fiber as a function of NRL, with and without LH, suggested that short-tomedium NRL fibers (173-209 bp) with LH condense into irregular zigzag structures and that solenoid features are viable only for longer NRLs (218 -226 bp) (Fig. 5D) (71,72). These studies suggest that medium NRL favors chromatin compaction throughout the cell cycle, unlike short and long NRL fibers: short NRL arrays fold into narrow fibers, whereas long NRLs do not easily lead to high packing ratios due to possible linker DNA bending. Furthermore, the histone tails influence fibers with medium NRL more easily. The small compaction effect of LH in short linker fibers is consistent with experiments (33,68).
Recent modeling studies that mimic chromatin stretching experiments have proposed interesting roles for LH in fiber compaction under various dynamic mechanisms. A wealth of chromatin pulling experiments as summarized recently (75,76) has emphasized the need for further mechanistic and structural interpretation of the force versus extension curves. For example, what factors stiffen the chromatin fiber, and what do the force versus extension curves imply regarding chromatin structure? Our studies focused on analyzing the stretching response of chromatin fibers as a function of the NRL and LH presence, including various binding mechanisms for LH. Indeed, LHs are known to be dynamic, with different binding ratios (77,78). Because fiber resistance to stretching decreases markedly with dynamic compared with static LHs due to possible stem rearrangements in the former (Fig. 5E, panel 1), we have suggested that dynamic LH binding may be an essential mechanism to soften chromatin fibers and allow unfolding at typical forces corresponding to natural molecular motors (Fig. 5E) (43). Moreover, among the dynamic LHs, pools of fast-and slowbinding LHs may cooperatively induce fiber unfolding at low forces: lower binding affinity softens fibers due to stem destabilization, whereas higher binding affinity promotes superbead constructs that combine nucleosome clusters with stretched fiber regions. The combination may offer both flexibility and selective DNA exposure (Fig. 5E) (43). These results thus suggest how modeling can help identify the factors that stiffen or soften chromatin fibers, as well as propose conformations and pathways linked to experimental force versus extension curves. In particular, our conformations are consistent with an overall zigzag fiber arrangement (Fig. 5E) rather than solenoid interpretation (33).

Tertiary and Higher Order Structures, in Situ Structures, and Functional Connections
Higher Order Structures beyond the Chromatin Fiber-Folding of the nucleosome array into the 30-nm fiber is far from achieving the 5 orders of spatial compaction realized by the chromosomes near the end of the cell cycle (Fig. 1). Various looping, scaffolding, wrapping, and specific contacts with other proteins and possibly RNA have been suggested for this higher folding to occur.
In vitro, in the presence of divalent cations in excess of 2 mM, as well as in situ, in interphase eukaryotic cells, chromatin fibers self-interact to form tertiary chromatin structures (1,6,45). This type of chromatin compaction may account for the in situ chromatin fibers that are partially interdigitated (79) and in vitro chromatin structures with diameters of Ͼ30 nm (80,81). In vitro, the formation of such structures is promoted by the same factors as the secondary structure: divalent cations, interactions between core histone N-terminal tails (46,82), LHs (83), and heterochromatin architectural factors such as MENT, MeCP2, and Sir3 (84,85). Modeling shows that longer nucleosome repeats typical of terminally differentiated cells promote lateral self-association of the chromatin fibers and nucleosome interdigitation (74).
Various ultrastructural studies, such as cryo-EM tomography (79), cryo-EM (86), electron spectroscopic imaging (87), and small-angle x-ray scattering (88,89), have not yet provided a uniform view of chromatin organization in the interphase state. These techniques revealed 30-nm fibers in a number of cells with condensed chromatin, such as Echinodermata sperm, chicken erythrocytes, and mouse retina (79,88,90). However, chromatin in proliferating cells showed thicker (100-nm diameter) "chromonema fibers," with none or few 30-nm structures detected (91)(92)(93). Even in the most condensed heterochromatic areas, no regular fibers of 30 nm in diameter were observed in the nuclei for interphase cells (94), thus raising the possibility that the 30-nm fiber is not a predominant structure in the interphase nucleus.
Dynamics of Higher Order Chromatin Folding-Although earlier structural models suggested a hierarchy of highly ordered static structures for eukaryotic chromatin, recent studies have revealed chromatin organization to be highly dynamic. This mobility includes the spontaneous reversible unfolding of the DNA segments from nucleosomes (16,17) and oligonucleosome arrays (95), the transient association of chromatin architectural proteins like heterochromatin protein HP1 (96) with chromatin, and the accessibility of condensed chromatin inside metaphase chromosomes for extrachromosomal proteins (97). In particular, as discussed above in connection with Fig. 5E, LH is highly dynamic, binding to or unbinding from the nucleosomes rapidly (78,98,99), and LH binding dynamics have been suggested to affect fiber compaction (100). Modeling has suggested that fast-and slow-binding populations of LH (77) merge optimally in heterogeneous forms of fibers to make chromatin more amenable to molecular motors (Fig. 5E, panel 3) (43).
The dynamic nature of chromatin necessitates developing new structural and modeling approaches to capture transient structural intermediates of chromatin folding in vitro and in vivo and to relate them to existing nucleosome structures, energetically favorable molecular models, and physical maps and bioinformatic analyses of native chromatin structures. This convergence of experiments and modeling will undoubtedly help translate the linear organization of the nucleosomes and their interaction in three-dimensional space to the hierarchy of chromatin and DNA conformations associated with key regulatory processes via modulation of chromatin accessibility to nuclear proteins and mediation of epigenetic interactions.