Abstract
Proper positioning of nucleosomes in eukaryotic cells is determined by a complex interplay of factors, including nucleosome-nucleosome interactions, DNA sequence, and active chromatin remodeling. Yet, characteristic features of nucleosome positioning, such as gene-averaged nucleosome patterns, are surprisingly robust across perturbations, conditions, and species. Here, we explore how this robustness arises despite the underlying complexity. We leverage mathematical models to show that a large class of positioning mechanisms merely affects the quantitative characteristics of qualitatively robust positioning patterns. We demonstrate how statistical positioning emerges as an effective description from the complex interplay of different positioning mechanisms, which ultimately only renormalize the model parameter quantifying the effective softness of nucleosomes. This renormalization can be species-specific, rationalizing a puzzling discrepancy between the effective nucleosome softness of S. pombe and S. cerevisiae. More generally, we establish a quantitative framework for dissecting the interplay of different nucleosome positioning determinants.
INTRODUCTION
Eukaryotic DNA is condensed into chromatin in a hierarchy of spatial organization and compaction levels. This organization is dynamic on all scales, from the large-scale organization of the genome (Bonev and Cavalli, 2016; Fraser et al., 2015) down to the positions of individual nucleo-somes, the smallest subunit of chromatin, which consists of 147 base pairs of DNA wrapped around an octamer of histone proteins (Luger et al., 1997). Since nucleosomes restrict accessibility of other factors to DNA, their positioning is essential for gene regulation (Bai and Morozov, 2010; Korber and Barbaric, 2014; Lam et al., 2008; Teif et al., 2013; Venkatesh et al., 2013).
Genome wide mapping of nucleosome positions (Voong et al., 2017) reveals gene-specific nucleosome patterns, which survive the average over many cells inherent to these methods. Known determinants of nucleosome positions (Hughes and Rando, 2014; Struhl and Segal, 2013) include the DNA sequence (Kaplan et al., 2009; Segal et al., 2006), competition with other DNA binding proteins (Ozonov and van Nimwegen, 2013), ATP-dependent chromatin remodeling enzymes that can relocate, modify or evict nucleosomes (Bartholomew, 2014; Clapier et al., 2017; Mueller-Planitz et al., 2013; Zhou et al., 2016), RNA polymerase and the DNA replication machinery (Radman-Livaja et al., 2011; Weiner et al., 2010), and interactions between the nucleosomes themselves, which can partially invade each other, due to unwrapping of DNA from the histone core (Chereji and Morozov, 2014; Engeholm et al., 2009). However, a complete understanding of how gene specific nucleosome patterns emerge from this interplay remains elusive.
The complex gene-specific nucleosome patterns yield a simple, characteristic pattern when averaged over many genes (Yuan et al., 2005), with a depleted promoter region followed by a downstream array. The nucleosomes within this array display a degree of variability in their positions that increases with distance from the promoter, as reflected in the oscillatory nucleosome density with peaks of decaying amplitude and increasing widths, cf. Fig. 1. The qualitative shape of this consensus pattern is universal across species with the nucleosome peak to peak distance varying from 150 base pairs (bp) in the yeast S. pombe (Lantermann et al., 2010; Moyle-Heyrman et al., 2013) to more than 200 bp in humans (Schones et al., 2008; Valouev et al., 2011). Furthermore, the shape is surprisingly robust against a multitude of perturbations: introduction of foreign DNA into the genome (Hughes et al., 2012), substitution of remodeler-encoding genes by non-endogenous variants and removal of H1 linker histones (Hughes and Rando, 2016), reduction of the overall histone abundance (Celona et al., 2011; Gossett and Lieb, 2012), different growth conditions (Kaplan et al., 2009), and diamide stress (Weiner et al., 2015) all affect the oscillatory consensus pattern only mildly. The passage of RNA polymerase also appears to have only a minor influence on the gene-averaged nucleosome pattern (Bintu et al., 2011; Radman-Livaja et al., 2011; Weiner et al., 2010), given that the pattern is only weakly dependent on transcription rate in yeast (Chereji and Morozov, 2015; Weiner et al., 2010). However, some amount of transcription may be required to establish the pattern, since silent genes show very weak oscillations in Drosophila melanogaster (Chereji et al., 2016). Substantial changes in the gene-averaged nucleosome pattern have been observed shortly after replication (Fennessy and Owen-Hughes, 2016; Vasseur et al., 2016) and in some remodeler deletion strains, which display reduced positioning oscillations (Gkikopoulos et al., 2011; Ocampo et al., 2016).
Why are gene-averaged nucleosome patterns so universal (across species) and robust (against perturbations)? An important step towards addressing this fundamental question was made by Kornberg and Stryer (Kornberg and Stryer, 1988), who suggested that nucleosome patterns could be described by a simple model, known as the ‘Tonks gas’ in physics (Tonks, 1936). The model assumes that nucleosome arrangements along the DNA correspond to an equilibrium ensemble of extended, non-overlapping particles positioned at a certain average density along a one-dimensional substrate. A perturbation in this gas, created for instance by a repulsive barrier, then induces oscillatory patterns in the average positions of the adjacent particles, which seemed consistent with the ladders observed in gels after nuclease digestion (Kornberg and Stryer, 1988). Later, when whole-genome nucleosome mapping became feasible, the model could be compared to the average nucleosome patterns in the vicinity of promoters (Mavrich et al., 2008). It was shown that the patterns downstream of yeast promoters are in good agreement with the model, if the perturbation is assumed to be created by a mechanism that holds the first nucleosome in a fixed position, whereas the patterns upstream of the promoters are compatible with a repulsive barrier (Möbius and Gerland, 2010).
While this “nucleosome gas” model originally treated nucleosomes as hard particles with footprints that cannot overlap on DNA, subsequent analyses took into account the “softness” of nucleosomes arising from transient DNA unwrapping from the histone core (Chereji and Morozov, 2014; Möbius et al., 2013). With this addition, the nucleosome gas model was able to provide a unified effective description of the gene-averaged nucleosome patterns across different Ascomycota fungi (Möbius et al., 2013). However, while this model can describe the data, it does not suffice to explain it. A priori, the effectiveness of such a simple description even seems unreasonable, since it neither accounts for the sequence specificity of the histone-DNA interaction, nor the action of remodeling enzymes. The resolution of this conundrum is our primary goal here.
We first illustrate the surprising effectiveness of the simple nucleosome gas model by showing that it simultaneously describes two different properties of the nucleosome patterns, the gene averaged nucleosome density and the spacing distribution, in both S. cerevisiae and S. pombe. This provides the basis for our subsequent analysis of how the interplay of various nucleosome positioning mechanisms can produce the robust nucleosome gas features. Towards this end, we design a class of nucleosome positioning models that explicitly incorporate the effects of active remodeling and DNA sequence-dependent nucleosome stability. We leverage the modeling framework to demonstrate that many (but not all) of the considered mechanisms are compatible with the simple nucleosome gas model, in the sense that they merely modify (or “renormalize”) the apparent softness of the nucleosomes. As a consequence, the resulting gene-averaged patterns display a high degree of universality, because their qualitative shape is not produced by a fine-tuned balance of specific mechanisms, but emerges robustly from the interplay of different positioning mechanisms. We suggest that a particular renormalization effect in S. pombe can explain why nucleosomes have a different apparent softness in this species compared to S. cerevisiae and related species, even though the underlying histone proteins are highly conserved. Finally, we use our models to infer which nucleosome positioning scenarios are incompatible with the available data, and discuss which novel types of experimental data would provide more effective discrimination between different scenarios.
RESULTS
To study the interplay of nucleosome positioning mechanisms, one would ideally monitor the arrangements of groups of nucleosomes in individual cells over time. Such dynamical information would directly reveal the effects of positioning mechanisms, and different mechanisms could be disentangled with appropriate mutants. Instead, the established whole-genome nucleosome mapping techniques provide statistical information, due to the ensemble average over large numbers of cells that is inherent to these techniques (Fig. 1A). In the case of MNase-seq data, a sequencing read reflects the DNA footprint of a single nucleosome in one of these cells. In contrast, the chemical cleavage technique (Brogaard et al., 2012) yields sequencing reads of the DNA connecting the midpoints (dyads) of two nucleosomes. Both techniques permit the extraction of nucleosome density profiles, n(x), i.e. histograms of nucleosome dyad positions along the DNA (Fig. 1B). Gene-averaged nucleosome density profiles, 〈n(x)〉, yield the consensus nucleosome pattern for a set of genes, e.g. see Fig. 1C,D (left panels) for the S. cerevisiae and S. pombe patterns, respectively (here, x denotes the distance in basepairs to the +1 nucleosome, see caption and ‘Methods’). Such consensus patterns haven proven to be very reproducible, and are commonly used as quantitative signatures of the interplay between different nucleosome positioning mechanisms in different species or mutants (Gkikopoulos et al., 2011; Hughes and Rando, 2016; Tsankov et al., 2010; Zhang et al., 2011).
The chemical cleavage technique (Brogaard et al., 2012) also permits to extract another type of statistical information, the nucleosome spacing distribution 〈n2(d)〉, i.e. the distribution of distances d between the dyads of neighboring nucleosomes, shown in Fig. 1C,D (right panels). Importantly, the spacing distribution provides additional statistical information about the nucleosome configurations of individual cells, which is not contained in the nucleosome density pattern. This is illustrated in Fig. 1E, by showing two ensembles of nucleosome configurations that yield the same ensemble-averaged nucleosome density profiles 〈n(x)〉, but different spacing distributions 〈n2(d)〉. We therefore treat both of these statistical characteristics as quantitative signatures of the interplay between different nucleosome positioning mechanisms.
Robust statistical characteristics of nucleosome positioning
To illustrate the conundrum that motivated this study, we analyze the quantitative signatures 〈n(x)〉 and 〈n2(d)〉 in S. cerevisiae and S. pombe, based on existing chemical cleavage maps (Brogaard et al., 2012; Moyle-Heyrman et al., 2013). As seen in Fig. 1C,D (black dots), the data displays qualitatively similar signatures in these evolutionarily distant yeast species. The gene-averaged nucleosome density profiles show pronounced oscillations, with amplitudes that decay with increasing distance from the first nucleosome (the “+1” nucleosome) of the genes, while the spacing distributions display a cusp-like peak with a fine-grained substructure. However, on a quantitative level, the signatures differ significantly between the two species. The peak-to-peak distance in 〈n(x)〉 is much smaller in S. pombe (150 bp) than in S. cerevisiae (167 bp), and the oscillations persist over a longer range in S. pombe. Furthermore, the spacing distribution is considerably wider in S. cerevisiae than in S. pombe, and peaked at a larger average spacing.
S. cerevisiae is the best-studied representative from a group of Ascomycota fungi, for which the gene-averaged nucleosome density profiles 〈n(x)〉 have previously been analyzed with the “nucleosome gas” model (Möbius et al., 2013). This analysis revealed that the precise shape of 〈n(x)〉, as well as its species-to-species variation within this group, can be described within the same model. Within this model, the differences between the profiles of different species arise purely as a consequence of the different nucleosome packing densities, or, equivalently, different average linker lengths, as opposed to different nucleosome properties. A conceptually very similar model was also used to interpret the spacing distribution of S. cerevisiae, again finding surprisingly good agreement(Chereji and Morozov, 2014). The study showed that the model can even capture the fine-grained structure of 〈n2(d)〉, by taking into account that nucleosomal DNA tends to unwrap from the nucleosome core in discrete steps of one helical turn.
Here, we adopt the “nucleosome gas” model (Möbius et al., 2013), which describes the nucleosome properties by two effective biophysical parameters, see Fig. 1F. The softness of nucleosomes, i.e. their ability to invade each others DNA footprint (Engeholm et al., 2009), is taken into account in a coarse grained way by assuming an energetic cost εeff per base pair to unwrap DNA from the histone core. Furthermore, the reach of the repulsive interaction between nucleosomes is parameterized by the effective nucleosome footprint radius weff, corresponding to a maximal footprint of 2weff + 1 base pairs, which is expected to exceed the canonical 147 base pair length of nucleosomal DNA, due to steric constraints arising when two nucleosomes are very close. The resulting effective nucleosome-nucleosome interaction potential is derived by summing the statistical weights of all unwrapping states compatible with a given nucleosome separation d (measured dyad-to-dyad), see Supplement for a detailed description of all Methods.
In contrast to prior studies, we test whether the model is able to simultaneously describe both signatures, 〈n(x)〉 and 〈n2(d)〉, with the same parameter values. The red lines in Fig. 1C,D show the resulting best fit. For S. cerevisiae, the fit yields an effective interaction radius of weff = 82 bp and an effective unwrapping cost of εeff = 0.152 kBT/bp. This is consistent with the previous results for a group of Ascomycota fungi that included S. cerevisiae, where weff = 83 bp, εeff = 0.153 kBT/bp were obtained from the density patterns alone (Möbius et al., 2013). Here we find that the spacing distribution is compatible with the same nucleosome gas model and parameters. The visible discrepancy in 〈n2(d)〉 for large spacings is expected, since long sequencing reads are suppressed in the data, see Supplement.
For S. pombe, the simultaneous fit provides an even better description of the density profiles and the spacing distribution, see Fig. 1D. However, the corresponding parameter values, weff = 75 bp and εeff = 0.236 kBT/bp, deviate markedly from those for S. cerevisiae and the previously studied group of Ascomycota fungi. Fig. S1A-C shows that the fitting parameters are well constrained for both species, aided by the inclusion of the spacing distribution in the fit. As an additional test of the consistency of the model, we also reverse engineered the nucleosome-nucleosome interaction directly from the experimentally measured spacing distributions, finding good agreement with our best fit model, see Supplement and Fig. S1D,E.
Taken together, these results support the nucleosome gas model as an effective description of gene averaged in vivo nucleosome data across evolutionarily distant yeast species. This implies that the statistical characteristics of the nucleosome gas are robust features of nucleosome positioning. How does this robustness arise despite the underlying complexity of nucleosome positioning mechanisms? In the remainder of this article, we seek to resolve this conundrum. A second question triggered by the above results is why the effective nucleosome stiffness εeff is so different between S. pombe and S. cerevisiae. We will see that these questions are indeed related: the nucleosome properties observed in the gene average are not the bare biophysical nucleosome properties, because they are renormalized by additional positioning determinants – and this renormalization can be different in different species.
To address these questions, we probe the effect of several simplified, but biologically motivated nucleosome positioning mechanisms in the context of the nucleosome gas model. We use the general approach outlined in Fig. 2. First, we compute the gene-averaged nucleosome density 〈n(x)〉 and the internucleosomal spacing distribution 〈n2(d)〉 for a modified nucleosome gas model that features an additional positioning determinant, e.g. DNA sequence specificity or a remodeling mechanism. Then, we fit 〈n(x)〉 and 〈n2(d)〉 by only the nucleosome gas model. Thereby we determine two things. First, the goodness of the fit, which measures how compatible the specific positioning determinant is with an effective description by the nucleosome gas model. Second, we determine to what extent the effective parameters εeff and weff differ from the bare parameters ε and w. In the following, we apply this framework to several different positioning mechanisms.
DNA sequence specific positioning leads to effective nucleosome softening
We first ask how gene-specific nucleosome positioning encoded in the DNA sequence affects gene-averaged patterns. We use energy landscapes to account for DNA sequence specificity (Fig. 3B,C). Various models for such energy landscapes have been developed (Tolkunov and Morozov, 2010), e.g. based on DNA elasticity (Eslami-Mossallam et al., 2016; Morozov et al., 2009; Tolstorukov et al., 2007), machine learning (Kaplan et al., 2009), or simple rules (van der Heijden et al., 2012). Here, our goal is not to assess the predictive power of these models, but rather to describe their generic effects on gene-averaged data. We will find that in this context the main parameter is the “positioning power” of energy landscapes, which scales with their standard deviation σ (Fig. 3C). Specifically, we use a DNA elasticity based model (Morozov et al., 2009). Exemplary landscapes u(x) for S. cerevisiae genes are shown in Fig. 3C (cyan). We consider only genes longer than 2500 bp to avoid interference with positioning effects at gene ends.
We compute the nucleosome density and spacing distribution on each of the obtained landscapes as follows: we fix a barrier particle at the consensus position of the first nucleosome and compute the downstream densities and spacing distributions exactly using the transfer matrix method for given values of the nucleosomes properties w and ε (Ssupplement). The resulting nucleosome densities (Fig. 3C, black) are strongly influenced by the energy landscapes and thus differ between genes. From these, we compute 〈n(x)〉 and 〈n2(d)〉, where 〈〉 is the average across genes (Fig. 3D, black). We point out that it is crucial to first compute the nucleosome density on each gene and then take the average, not vice versa:
Indeed, the gene-averaged landscape 〈u(x)〉 is almost flat (σ = 0.056 kBT for our set of 914 long genes in S. cerevisiae, Suppl. Fig. S3E), which indicates that there are no sequence encoded positioning clues in coding regions on average according to the used landscape model.
Next, we determine to what extent the nucleosome gas model is compatible with the gene averaged density and spacing distribution by fitting the model to 〈n(x)〉 and 〈n2(d)〉 simultaneously (Fig. 3D, red, see Suppl. Fig. S2 for details). We find that the gene averaged quantities can be surprisingly well reproduced by this simple model with renormalized nucleosome properties εeff and weff. We also find that nucleosomes in the gene average appear softer than the bare particles, εeff < ε, while the maximal footprint size is almost unaffected, weff ≈ w. The degree of effective nucleosome softening depends on the average landscape amplitude σ (Fig. 3E). We determine a realistic value of σ from the peakedness of the nucleosome density on single S. cerevisiae genes to be σS·cer = 1.56 kBT (green dashed line in Fig. 3E, see also Suppl. Fig. S3A).
Finally, we determine how stiff individual nucleosomes are “truly”, given that they appear softened by energy landscapes. To do so, we start from the nucleosome stiffness which fits the S. cerevisiae data, and from the above estimate for the landscape amplitude, i.e. from the point (Fig. 3E, green star). We then trace back the renormalization flow and find the bare nucleosome stiffness (green circle). Our results thus suggest that this is the true average energetic cost of unwrapping one bp of DNA from a nucleosome in S. cerevisiae.
In order to show that the observed robustness and parameter renormalization are not a peculiarity of the chosen energy landscape model we repeated the above steps with uncorrelated landscapes, where for every lattice site an energy is drawn at random from a Gaussian distribution with standard deviation σ. The renormalization flow is very similar (Suppl. Fig. S3D) which indicates that the effective nucleosome softening is a generic phenomenon of averaging over nucleosome positioning landscapes.
We have found that positioning by gene specific energy landscapes is “renormalizable”, i.e. the gene averaged quantities can be reproduced without landscapes by altered nucleosome properties. We point out that it is easy to come up with landscapes that do not belong to this renormalizable class. For example, by specific variations in the landscape amplitude σ or by introducing specific correlations we obtain gene averaged patterns that are not reproduced by the nucleosome gas model (Fig. S4). We thus conclude that S. cerevisiae energy landscapes are random enough to be renormalizable.
Remodeler effects: directional sliding
Like DNA sequence, nucleosome remodeling enzymes (“remodelers”) are a crucial determinant of nucleosome positioning. They are required for proper nucleosome spacing (Krietenstein et al., 2016; Lieleg et al., 2015; Zhang et al., 2011), and are known to relocate nucleosomes along the DNA using energy from ATP hydrolysis (Flaus and Owen-Hughes, 2011; Mueller-Planitz et al., 2013). It is therefore remarkable that the emerging in vivo patterns are well described by the equilibrium nucleosome gas model. Here, we rationalize this agreement by showing that many remodeling mechanisms are indeed “renormalizable”.
We first ask how directional sliding of nucleosomes by remodelers affects positioning. Our minimal model is guided by two observations. First, in vitro reconstitution experiments result in stereotypic nucleosome array patterns only in the presence of ATP (Zhang et al., 2011). This indicates that remodelers mobilize nucleosomes that are otherwise kinetically frozen, which is also supported by direct observations of strongly increased DNA unwrapping rates upon remodeler binding (Singh et al., 2018). Second, remodelers can displace nucleosomes in successive steps in the same direction (Blosser et al., 2009). Here, we consider two types of remodelers, upstream (U) and downstream (D) sliders, which preferentially move nucleosomes upstream/downstream along the DNA (they could represent the same protein binding nucleosomes in different orientations). Both bind and unbind nucleosomes with rates and , respectively (Fig. 4A). Remodeler bound nucleosomes perform a biased random walk with a rate parameter r and a bias parameter γ, namely ru,d = re±γ/2 (see Fig. 4A and Supplement). The parameters r and γ account for general mobilization and directionality, respectively. We measure the remodeler effectiveness by its processivity p, the mean displacement during its dwell time in the absence of interactions with other particles (Supplement). Nucleosome interactions alter the binding and sliding rates, and simulations are performed with a kinetic Monte Carlo scheme (Supplement).
We find that nucleosome sliding strongly reduces the characteristic oscillations in the nucleosome density n(x) and broadens the spacing distribution n2(d) (Fig. 4B, see Fig. S5 for a parameter sweep). Yet, both quantities are excellently compatible with the nucleosome gas model (red vs black lines in Fig. 4B). The effective nucleosome stiffness εeff decreases for increasing processivity p (Fig. 4C). This is expected, since our sliding remodelers promote nucleosomes invading each other. The softening is more pronounced when the processivity is altered via the bias parameter γ instead of the rate parameter r because the bias is more important in overcoming interactions with neighboring nucleosomes (see Supplement). In conclusion, we found that remodeler mediated directional nucleosome sliding is excellently compatible with the nucleosome gas model and leads to effective nucleosome softening.
Remodeler effects: nucleosome attraction and spacing
Next, we address remodeler mediated nucleosome attraction and spacing. In vitro (Lieleg et al., 2015; Zhang et al., 2011) and in vivo (Celona et al., 2011; Gossett and Lieb, 2012) experiments have shown that in the presence of remodelers nucleosome spacing does not change even when the nucleosome abundance is strongly reduced. This is incompatible with the nucleosome gas model, which predicts that the average nucleosome spacing increases for reduced density. Thus, remodeler mediated attraction was proposed to maintain a fixed spacing in gene-coding regions (Lieleg et al., 2015; Möbius et al., 2013).
We implement attraction towards a fixed spacing by superimposing the repulsive soft-core nu-cleosome interaction with an attractive potential well centred around the S. cerevisiae spacing of d0 = 167 bp (Fig. 5A). Its standard deviation σ parametrizes the remodeler’s spacing accuracy and the depth a the spacing activity (see Supplement).
We find that high nucleosome spacing accuracy (σ = 2 bp) results in a strong peak in the density n(x) and especially in the spacing distribution n2(d)(Fig. 5B, top panels, see Fig. S6 for a parameter sweep of a). Consequently, the compatibility with the nucleosome gas model, which lacks this spacing peak, is not good. For a more sloppy nucleosome spacing (σ = 6 bp), however, the compatibility is better (Fig. 5B, bottom panels). We find that for accurate and sloppy spacing alike the effective particle stiffness εeff increases with the remodeler activity (Fig. 5C). This is expected, since nucleosomes are preferentially kept at a fixed distance instead of invading each other. In conclusion, we found that a somewhat sloppy remodeler mediated nucleosome attraction towards a preferred spacing can be effectively described by the nucleosome gas model with increased nucleosome stiffness.
ATtrapping explains unusually strong oscillations in theS. pombenuclesome density
We have shown that gene averaged nucleosome patterns can be modified by a multitude of mechanisms which potentially differ between species. This puts us in a position to ask why nucleosomes appear much stiffer in S. pombe than in S. cerevisiae (Fig. 1C,D), even though the underlying histones are conserved. We suggest that the large effective stiffness results from an S. pombe specific positioning mechanism, namely nucleosome trapping on AT-rich sequences. We motivate our hypothesis by the observation from Moyle-Heyrman et al. (Moyle-Heyrman et al., 2013) that in S. pombe nucleosomes are preferentially located on AT rich sequences within coding regions. This correlation is even more pronounced in our alignment of genes by their +1 nucleosome (Fig. 6A), but still completely absent in S. cerevisiae (Fig. 6B).
To corroborate our hypothesis we dissect the correlations between nucleosome positions and AT content in more detail (Fig. 6C-F). We subdivide our set of long genes (> 2500 bp) in four clusters using k–means clustering (other numbers of clusters yield the same conclusions). First, we perform the clustering on the nucleosome density and check for correlations in AT content. We find that two clusters that differ in peak to peak distance (black and magenta lines in Fig. 6C), also show changes in the AT content. Furthermore, genes with weakly pronounced density peaks (pale green line) also show weak oscillations in the AT content. S. cerevisiae shows similar clusters in the nucleosome density but none of the features in the AT content (Fig. 6D). Next, we perform the clustering on the AT content and check correlations in nucleosome density. We find that in S. pombe shifted AT peaks are accompanied by shifted nucleosome patterns (black and magenta lines in Fig. 6E). In S. cerevisiae no genes with pronounced AT periodicity are found (Fig. 6F).
We conclude that nucleosome positions and AT content correlate strongly in S. pombe and we thus posit an energy landscape that preferentially positions nucleosomes on AT rich regions (Fig. 7B). Its period λ = 150 bp and decay length l0 = 555 bp are derived from the AT content oscillations (Fig. S7B).
We point out that the sequence based model for energy landscapes used above predicts no AT positioning on average on S. pombe genes (Fig. S3E). While it will be interesting to determine if other sequence based models do so, or if an indirect mechanism like remodeling has to be invoked, this question is irrelevant for our current goal to show that AT trapping can explain the large effective nucleosome stiffness in S. pombe. Here, we account for the combined effect of DNA sequence specificity and AT trapping by superimposing sequence derived landscapes and the trapping landscape. Examples are shown in Fig. 7B.
To account for AT trapping within the nucleosome gas model we first compute 〈n(x)〉 and 〈n2(d)〉 with AT trapping. On single genes the effect of AT trapping is rather weak (Fig. S7C), but when averaged over many genes the resulting oscillations are considerably strengthened (black vs grey line in Fig. 7D). Next we fit the gene averaged density and spacing distribution by the nucleosome gas model and determine the effective nucleosome parameters from the fit. We find that AT trapping renormalizes the nucleosome stiffness ε to larger values, εeff > ε (black line in Fig. 7E). This effective stiffening reaches εeff = 0.23 kBT (green star), which is close to the S. pombe value of εeff = 0.236 kBT (green solid line), at a trapping strength of A = 1.33 kBT (greed dashed line). This trapping strength is surprisingly low compared to the typical oscillations of sequence based landscapes (compare the bold and thin cyan lines in Fig. 7C) reflecting that nucleosomes guide each other by repelling reach other. The much slower decay of the density oscillations (compare black to grey line in Fig. 7D) is completely consistent with the much more pronounced oscillations in experimental S. pombe data compared to S. cerevisiae (Fig. 1C,D). We also observe that the effective nucleosome size parameter weff does not drop to the S. pombe value of 75 bp in the range suggested by the ε renormalization. We come back to this in the discussion.
We here assumed a nucleosome density whose oscillation period matches the S. pombe value of d = 150 bp even without the trapping landscape, the only effect of which was to result in more persistent oscillations in the gene body. For completeness, we also consider the case that the native density oscillation period equals the S. cerevisiae value of dA=0 = 167 bp. We find that cranking up the AT trapping imposes its period of 150 bp on the density and, for sufficiently strong trapping, again leads to effective nucleosome stiffening (thin grey line in Fig. 7E).
We conclude that nucleosome trapping on AT-rich sequences, which is motivated by strong correlations between nucleosome density and AT-content in S. pombe, can explain why nucleosomes appear much stiffer (i.e. positioning oscillations are more pronounced) in S. pombe than in S. cerevisiae even though the underlying histones are conserved.
DISCUSSION
Towards a quantitative understanding of nucleosome positioning we here asked how different molecular mechanisms interact. We particularly focussed on how such mechanisms are reflected in gene-averaged positioning data. The essence of gene-averaged nucleosome patterns can be understood by the surprisingly simple model of statistical positioning against a barrier (Kornberg and Stryer, 1988; Mavrich et al., 2008). We showed that a refined version of statistical positioning, where DNA unwrapping is taken into account via a soft-core nucleosome interaction, accurately describes the characteristic oscillations in gene coding regions as well as the distribution of inter-nucleosomal spacings. This left us with an apparent contradiction: while gene-specific positioning preferences are not required to reproduce gene averaged patterns, such preferences are very clearly present and indeed shape gene-specific nucleosome patterns (Fig. 1B). We therefore investigated how gene-specific positioning alters gene-averaged patterns. We found that it changes the gene-averaged patterns quantitatively but not qualitatively. This was true independently of whether gene-specific positioning was generated from a DNA elasticity model or randomly, which indicates that the compatibility of strong positioning on individual genes and the emergence of simple nucleosome gas features in the gene average are a generic phenomenon.
Going beyond the qualitative finding that specific positioning mechanisms can be compatible with the nucleosome gas model in the gene average, we investigated their quantitative effects. Our general approach was to first model the nucleosome gas together with some specific positioning mechanism, and to then reproduce the obtained patterns with only the nucleosome gas model (Fig. 2). We found that the resulting effective nucleosome properties, in particular the effective energy required for unwrapping DNA from the histone core (the “nucleosome stiffness”), is renormalized by the additional positioning mechanisms. We would thus argue that Kornberg and Stryer’s original conclusion, that “the sequence-specificity cannot bee too great” (Kornberg and Stryer, 1988) should be modified to “the trace of sequence-specificity in the gene-average is a renormalization of the effective nucleosome properties”.
We found that different positioning mechanisms alter the effective nucleosome properties in different ways: While DNA sequence preferences (Fig. 3) and remodeler mediated nucleosome sliding (Fig. 4) lead to an effective softening, the setting of a preferred spacing (Fig. 5) or positioning along genes (Fig. 7) makes nucleosomes appear stiffer. We suggested that the latter can resolve the apparent conflict that DNA unwrapping seems particularly hard in S. pombe even though the underlying histones are conserved.
The picture that emerges from our results is that the salient feature of gene averaged nucleosome patterns, the decaying oscillations, emerges robustly from steric exclusion on DNA. The quantitative aspects of these oscillations, however, reflect the additional positioning mechanisms, which can be specific to species, cell types and conditions.
Our findings suggests that the quantitative features of nucleosome positioning data can be used to unravel the importance of specific positioning mechanisms. This requires some care, though. First, we have seen that some mechanisms lead to effective nucleosome stiffening and others to softening, and different contributions could thus cancel. Furthermore, different mechanisms can produce similar parameter renormalizations. For example, softening is caused by gene averaged DNA sequence effects as well as by remodeler mediated sliding, and stiffening could come from either a preferred nucleosome spacing (a two-particle quantity) or from preferred positioning along the genome (a single-particle quantity). With respect to the anomalously large unwrapping energy in S. pombe, which we suggested to stem from trapping on AT rich regions, alternative appealing scenarios are altered interactions due to the lack of the linker histone H1 or altered remodeling due to the lack of the ISWI remodeler in S. pombe. Further targeted studies are required to disentangle those effects.
In spite of the above mentioned cautionary notes our quantitative approach can greatly help in disentangling the interplay of positioning determinants. For example, we have demonstrated that the conjectured nucleosome spacing by remodelers must be either somewhat fuzzy, or can not be a dominant mechanism in vivo (Fig. 5). Furthermore, the correlation between AT content and nucleosome density in S. pombe suggested an AT trapping mechanism.
We point out that gene averaged patterns reveal information that would be hard to gain from single genes or specific loci, as illustrated by the above examples of AT trapping and fuzzy spacing. Of course, gene-specific patterns carry a great amount of additional information, which can be explored in the future. This will build on the here developed tools of quantitative simulation of specific remodeling mechanisms together with nucleosome interactions and sequence specificity.
Finally, we address experimental ramifications of our results. First, we suggest that the additional information revealed by chemical cleavage data over conventional MNase maps, namely the distribution of internucleosomal spacings (Fig. 1E), is extremely valuable: much more sensitive to mechanisms affecting the spacing than might be apparent from the nucleosome density patterns (Fig. 5B). Furthermore, it provides information that is simply not contained in the density patterns (Fig. 1E). In addition, the strength of oscillations in gene-averaged patterns depends on the pronouncedness of the barrier (no barrier, no oscillations). The barrier likely results from an interplay of sequence (Segal et al., 2006), remodeling (Krietenstein et al., 2016) and transcription (Chereji et al., 2016) and may vary across species and conditions. This again highlights the importance of the spacing distribution as a source of information about spacing mechanisms, since it does not rely on a barrier but rather reflects the nucleosome-nucleosome interaction directly. Measuring it could be of particular relevance in in vitro reconstitution experiments (Zhang et al., 2011), especially with purified remodelers (Krietenstein et al., 2016) in order to unravel their spacing activity.
Furthermore, trans-species experiments have proven useful: nucleosomes on K. lactis sequences cloned into S. cerevisiae were spaced with the shorter S. cerevisiae spacing, indicating that trans acting factors determined the spacing (Hughes et al., 2012). Similarly, replacing the endogenous gene for the remodeler Chd1 in S. cerevisiae by its K. lactis ortholog led to slightly increased nucleosome spacing (Hughes and Rando, 2016). Similar experiments with S. pombe might reveal the mechanism behind the correlation of nucleosome density and AT content.
In conclusion, we have established a framework for the quantitative understanding of how various mechanisms interact in nucleosome positioning. This approach can be particularly helpful in addressing a gap in current knowledge about chromatin remodeling enzymes: while single molecule experiments have revealed basic relocation steps (Flaus and Owen-Hughes, 2011; Mueller-Planitz et al., 2013) it is largely unclear how such relocations are combined to shape characteristic nucleosome positioning patterns. An iterative procedure of identifying candidate relocation mechanisms from single molecule experiments, exploring their single gene or genome wide effects in simulations as done here, and comparison to knockout or reconstitution experiments can unravel the building blocks of nucleosome positioning. Furthermore, an extremely interesting perspective is to unravel the relation between 1D nucleosome positioning and the 3D organization of chromatin. With Micro-C (Hsieh et al., 2015), which produces nucleosome positioning maps as well as 3D contact frequencies, questions about the 1D-3D relation might become addressable by joint experimental and modeling efforts.
Author contributions
Conceptualization: J.N. and U.G.; Investigation: J.N., M.W., B.O., W.M., and U.G.; Writing Original Draft: J.N. and U.G., with contributions from M.W., B.O., W.M.; Funding Acquisition: U.G.
Declaration of Interests
The authors declare no competing interests.
Acknowledgements
We acknowledge helpful discussion with Philipp Korber. MW is supported by SFB863 and a member of the Graduate School of Quantitative Biosciences Munich.
Footnotes
↵* Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA