Nucleosome profiles can be well distinguished based on their coordinated positioning in WT

In order to compare nucleosome profiles over the gene body in WT conditions, we measured the pairwise Pearson cross-correlation of the MNase-seq data produced by [12, 18] for all protein-coding regions [39] using Eq 1. The Pearson correlation index is positive when the sequencing signals of both genes tend to change towards the same direction at the same position; and it is negative when one profile is likely to increase whereas the other one decreases. Therefore, it compares similarity of the distributional shape—i.e. whether genes are apt to contain nucleosomes at similar positions—and it does not take the scaling of the sequencing data into account. The entire arrangement for each gene is treated as an entity. For both replicates, we considered 1000 bp after and 200 bp before the +1 position (= 1200 bp, approximately the average size of a gene in Saccharomyces cerevisiae), containing 6–7 nucleosome dyads.

These Pearson coefficients were used as a distance metric to cluster nucleosome profiles into distinct partitions using k-mean clustering. In a nutshell, the algorithm divides a data set of m observations (the MNase-seq data) into k groups by minimising the variance within each cluster based on a distance metric (here, the pairwise Pearson indices over all genes). Therefore, genes within a group tend to have nucleosomes at comparable positions, whereas profiles of different groups are likely to be less similar. Using a silhouette criterion measurement—which compares the similarity of an object to its own cluster with the similarity to other clusters—we determined that the Pearson coefficients are most distinctly divided when k = 2 (i.e. when having two groups, Fig 1A). As the k-mean clustering algorithm aggregates profiles together with a large pairwise Pearson index, we expected that the distribution over all pairwise Pearson correlations between nucleosome arrangements from different clusters is minimal (in the following called inter-cluster correlation). We validated the significance of the obtained Pearson gene groups by creating 500 random clusters and calculating their p-value using a one-sided Kolmogorov–Smirnov (KS) test. We averaged the p-value over the KS test result over all 500 repetitions to smooth out random fluctuations (average p-value 0.0009, S1 Fig). This shows that nucleosomal arrays can be significantly separated into two groups using linear correlation of MNase-seq data between genes (Fig 1B and 1C).

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Fig 1. Pearson clusters and fPCA considering all protein-coding genes.

(A) The silhouette plot clearly indicates that the data can be best divided into two clusters, and creating more groups would only decrease the difference between each cluster. (B) and (C) display the profiles for each cluster. Large values are given in copper, low values are black, and the colour gradient in between is uniformly scaled. It is therefore a perceptually uniform representation. Both heatmaps are normalised independently, such that their respective largest value is displayed in the strongest copper hue and their lowest value in black. Unfortunately, it is difficult to quantify visually why these clusters were established. This is particularly true because the Pearson index measures only general trends in the profile, and it does not take the scaling into account. Each row represents a gene, and the x-axis shows the position along the coding region, with the +1 nucleosome defined to be at position 0 bp. The colour code represents MNase-seq amplitude, i.e. copper values show large MNase-seq signal values, whereas dark areas indicate a low amplitude. (D) The cartoon presents the hypothesised differences that could occur between the Pearson clusters. Due to the well-positioned nucleosomes and the wave-like structure of MNase-seq data, we presume that the Pearson correlation measures coordinated nucleosome positioning along the gene. If two profiles (orange and blue) are in two different clusters, this could indicate either a shift in the exact nucleosome positions (left); or a general trend in the MNase-seq signal amplitude, i.e. either increasing or decreasing (right). (E) Pearson clusters considering all genes are linearly separable with respect to their fPC scores. This indicates that two fPCs are sufficient to interpret the gene groups. We use the symmetric Jensen-Shannon (JS) distance to describe separability between the clusters along fPC1 and fPC2. The JS distance between the cluster distributions is much larger for fPC 2 than for fPC 1. Orange and blue indicate each one group, the dashed line symbolises the best linear separation using a SVM. The x-axis represents the score of the first fPC ζ¹, the y-axis gives the score for the second fPC ζ². Both axes are scaled to the same range, points outside the range (29) were included in the analysis but not plotted. (F) When analysing the effect of the major fPCs, they describe predominantly position-dependent scaling (transparent black lines, fPC 1) and collective nucleosome phasing (transparent black arrows, fPC 2). The second fPC in WT indicates an increasing or decreasing signal magnitude as a function of distance from the TSS, suggesting stronger or weaker presence (corresponding to panel D right). The mean is given as a dashed black line, a positive contribution—i.e. adding the fPC to the mean—is displayed in magenta, and a negative contribution—i.e. subtracting from the mean—is shown in green. Trends over the entire array are indicated by grey arrows. When exact positions were seemingly not affected by the fPC, we marked the positions with a grey vertical bar. See Methods for more information about how the plots were produced.

https://doi.org/10.1371/journal.pcbi.1011799.g001

It is difficult to straightforwardly determine how the k-mean clustering algorithm distinguishes between these two groups; yet the interpretation of the discriminating boundary could reveal important insights about the nucleosome positioning that is presumably imposed by chromatin remodelers. As the data by Ocampo et al. [12, 18] contains several mutants, we want to identify this discriminator repeatedly with the same mathematical framework to make the results comparable. Due to the nature of the Pearson correlation index, we can make the following assumptions. As nucleosomes are commonly well positioned in budding yeast, the MNase-seq data resembles a wave-like function with one peak approximately every 200 bp. Moreover, single histone complexes cannot overlap in a single cell. The Pearson correlation measures therefore the average phasing similarity of the entire nucleosome array of two genes. Differences in similarity come either from shifts in exact positioning (i.e. well-defined peaks, Fig 1D left) or from a change in the signal amplitude (i.e. increasing or decreasing MNase-seq magnitude over the profile or at particular locations, Fig 1D right). The clusters must be separated based on either of these two trends, or possibly a combination of them.

In the following, we refer with coordinated positioning to the configuration of the entire nucleosome array, and consequently, to their behaviour with respect to the two separating trends of the k-mean clustering. Unfortunately, the Pearson coefficient measures only the average linear pairwise correlation over the entire profile, rather than taking position-dependent particularities into account. Therefore, simply extracting the boundary from the k-mean clusters does not explain whether the groups were established with respect to a shift or a change in amplitude (i.e. the previously determined discriminators). Instead, it is possible to investigate how the clusters distribute with respect to the data itself; or, alternatively, with respect to a different description using dimensionality reduction methods. By evaluating the major differences between the two groups of genes, we can interpret the separating clustering boundary and link it to particular properties along the nucleosome profile.

Conventional approaches apply dimensionality reductions like PCA to visually analyse clustering distributions. However, using PCA would implicitly mean that we assume independence between every position along the gene. By using the Pearson correlation measurement, we treat every profile as a single entity, which would be violated by the independence conjecture. This also contradicts the fundamental assumption of the barrier model where the positioning of earlier nucleosomes affect later phasing. Instead, we understand the arrangement as the result of a coordinated process. We assume that the MNase-seq signal along each gene can be described as a single (unknown) continuous function, which can be approximated by a mixture of a finite number of known simpler functions (so-called basis functions). In this study, we used 20 B-Splines to represent the MNase-seq data along each gene, which were subsequently averaged to a mean profile. This permitted the application of fPCA to determine the two best-characterising functional Principal Components (fPCs) that describe each nucleosome arrangement. It incorporates specific assumptions about the spatial relationship in the distribution through the basis functions, which is the crucial difference between conventional PCA and fPCA. To be more precise, the establishment of the MNase-seq distribution is understood as a stochastic process with a mean behaviour. Each considered nucleosome array can be regarded as a realisation of this stochastic process with a deviance from the expected average distribution. Instead of defining a data representation for every gene individually, fPCA determines how the mean profile needs to be transformed to approximate a particular gene. This transformation is found by combining the basis functions over all coding regions to more complex functions that are orthonormal to each other and describe the most variance along the data (i.e. the fPCs, Eq 4). These functions transform the mean by adding them to the average profile with a gene-specific scaling factor (i.e. for the j-th fPC of the i-th gene). Consequently, every nucleosome array can be also described exclusively by the factors together with the respective fPCs, and we can evaluate how the two Pearson clusters distribute with respect to these factors.

Interestingly, the two clusters—which were independently obtained by classical hierarchical k-mean clustering of Pearson coefficients—are visually neatly separated by using only the first two fPCs, indicating that they are sufficient to quantify the difference between the two sets of genes (Fig 1E). In fact, the separating boundary is almost exclusively dependent on the second fPC, whilst it is seemingly independent of the first. This is slightly less clear for the B replicate, although still distinct (S2(B) Fig). Using our previous considerations about how the algorithm establishes the two clusters, we intuited that the second fPC describes coordinated nucleosome phasing along the gene body. By analysing the effect of the second fPC on the function shape, we conclude that the clusters are determined based on the downstream presence of nucleosomes (corresponding to the right cartoon in Fig 1D). We found that the first fPC largely represents amplitude scaling at a given position, as it does not influence the location of the peak (Fig 1F). The analysis shows that position-dependent amplitude scaling and coordinated arrangement are the best two independent functional descriptors for the MNase-seq data. We show and discuss the effect of the 10 major fPCs that describe most variance in S3 Fig. Despite the fact that the ratio of explained variance is not high (21.4% and 11.5% for fPC1 and fPC2, respectively), they are completely sufficient to distinguish between the Pearson correlation groups and permit an interpretation of the linear separating boundary between the clusters.

FPCA reveals size-dependent Rsc8-mediated phasing of nucleosome positions

Since the smallest genes are ≈300 bp long, the 1000 bp window after the +1 position can contain much more than the actual length of the coding region. In order to analyse how nucleosome phasing is affected by the gene size, we repeated the fPCA considering exclusively small (≤ 1000 bp, ≈26.7%) or large genes (>1000 bp, ≈73.3%). Consequently, the mean as well as the two fPCs changed, whilst we kept their allocation to the previously determined Pearson clusters the same (in the following also referred to as all-gene clusters). If coordinated positioning is substantially affected by the length of the transcribed region, we expected that the factors of the two major fPCs should exhibit a changed behaviour with respect to the linear separability. We can confirm that the linear separation is preserved for large genes, although the boundary becomes slightly sloped (S2(C) and S2(D) Fig). The fPCs for only large genes are almost identical to the all-gene fPCs (S5 Fig). We therefore presume that the clusters can be still largely separated by the second fPC. We also considered a possible impact of the downstream NDR by analysing exclusively very large genes (≥ 3000 bp, ≈11.5%). Once again, the boundary was clearly visible (S2(G) and S2(H) Fig). We concluded that the MNase-seq distribution over the first 6–7 nucleosomes of all genes larger than 1000 bp can be best clustered by the collective positioning, and it can be surmised that phasing within the gene body is only negligibly affected by the downstream NDR or nucleosomes outside the 1000 bp window.

However, the neat separation between the two clusters fully vanished for small genes (Fig 2A, for replicate B S2(E) Fig). Almost all data points belong to the same group, although both are present. We want to remind that clusters were established using all coding regions, whereas the functional representation depends now exclusively on genes smaller than 1000 bp. The newly determined fPCs include overlapping positioning inside and outside the gene body due to their varying size (S4 Fig). The fact that the clusters are not separable indicates that coordinated nucleosome phasing disappears after the Transcription Termination Site (TTS), and we hypothesised that the arrangement is strictly limited to the gene body. Indeed, the second small-gene fPC indicates well-defined positioning only for up to the +2 nucleosome (≈300 bp), and the function loses quickly its frequent wave-like shape thereafter (Fig 2D). The two major fPCs for small genes are not sufficient anymore to separate the all-gene clusters, which are discriminated by the presence of downstream nucleosomes. To verify our hypothesis of gene-size dependent phasing, we divided the regions into small and large genes before performing the Pearson clustering. When considering exclusively small genes, the two Pearson groups become linearly separable again, which is—in accordance with our hypothesis—predominantly determined by the size (S6 Fig). This shows that the nucleosome arrangement is strictly limited to the gene body.

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Fig 2. Nucleosome phasing is strictly limited to the gene body, which is maintained by Rsc8 but antagonised by Chd1.

The cluster distribution plots in panels A-C show the distribution of both gene groups with respect to the small-gene fPCs of WT, rsc8-depleted cells, and chd1Δ strains. Orange and blue indicate the two clusters, and the black dashed line shows the separating boundary determined by a linear SVM. The histograms present the cluster distribution with respect to each axis. Panels D-F display the transformation of the average small-gene nucleosome profile by the two major fPCs for WT, rsc8 depletion, and chd1Δ, respectively. The dashed black line as well as the solid lines in magenta and green display the mean, a positive contribution of the fPC, and a negative contribution. Turquoise arrows indicate the effect on the +1, dark blue arrows on the +4, and orange arrows on the +6 position. (A) When plotting the cluster distribution with respect to small-gene fCPs in WT, the linear separability is lost. (B) The fPCs of the rsc8-depleted strain maintain the linear separability, despite the fact that the groups were established for all genes. As we interpret the Pearson clusters as similarity in positioning between genes of 1000 bp mediated by chromatin remodelers, it possibly suggests that positioning outside coding regions influences nucleosomes inside and vice versa. (C) Whilst most mutants that were rsc8 depleted could discriminate between the all-gene clusters using small-gene fPCs, this separability is lost again in rsc8-depleted chd1Δ, revealing partly antagonistic roles to maintain gene-specific phasing for Rsc8 and Chd1. (D) The effect of two fPCs sheds light on why the Pearson groups are not linearly separable in WT using small-gene fPCs. The distribution of the second fPC changes its regular wave-like form to much broader peaks and valleys after the +2 nucleosome, which corresponds to approximately the size of the smallest genes in budding yeast. (E) Nucleosome positioning in rsc8-depleted conditions is clearly visible along the entire considered region, despite the included genes being smaller. This suggests that gene-specific nucleosome arrangement cannot be maintained. It is of note that the phasing also changes for the +1 nucleosome, and the NDR can be seemingly not conserved. (F) On the other hand, rsc8-depleted chd1Δ loses the regular wave-like shape of its second fPC after the +2 nucleosome to form broader peaks, indicating the presence of gene-specific nucleosome profiles as in WT conditions. All axes are scaled to the same size for each strain; shapes and amplitudes are therefore comparable (see Methods for more details).

https://doi.org/10.1371/journal.pcbi.1011799.g002

The data produced by [12, 18] contain two replicates for chd1Δ, isw1Δ, and isw2Δ cells as well as rsc8-depleted strains, together with their combinations as double, triple, and quadruple mutants. In order to analyse how gene-size dependent nucleosome phasing alters in varying contexts, we compared the small-gene fPCs in mutant and WT conditions. Surprisingly, the separation of the all-gene clusters was clearly visible for the fPCs of small coding regions in rsc8-depleted strains (Fig 2B). Indeed, the average MNase-seq profile exhibits phased peaks along the entire 1000 bp-window (Fig 2E), and nucleosome positioning continued outside the gene boundaries (S4 Fig). The linear separability of the all-gene clusters using small-gene fPCs can be found in almost all mutants which are depleted of Rsc8 (S7 Fig), with the sole exception of Rsc8-depleted chd1Δ strains (Fig 2C, replicate B S8(B) Fig). Here, the groups cannot be visually separated by ζ¹ and ζ², and the determined fPCs resemble small-gene fPCs in WT conditions (Fig 2F, replicate B S8(D) Fig). This indicates that the gene-specific boundaries for nucleosome phasing can be re-established, and the second fPC loses its wave-like shape again after the +2 position (S4 Fig). Consequently, we hypothesise that Chd1 and Rsc8 have partially antagonistic roles for maintaining chromatin organisation that distinguishes transcribed from non-transcribed regions. Taken together, this analysis exhibits strictly constrained and Rsc8-mediated nucleosome organisation within coding regions.

Nucleosome phasing changes in remodeler mutants

We were particularly interested in how nucleosome remodeler complexes affect coordinated phasing. To remove the gene size-dependent bias from the clustering and the established fPCs, we applied the Pearson clustering to exclusively large genes (>1000 bp) for all strains and determined their two major fPCs (S2C, S2D and S5 Figs). We can confirm that the created groups for all mutants were again significant (i.e. average p-value of a KS test with 500 random partitions was lower than 5%, see S9 Fig). Interestingly, the Pearson clusters were always visually separated by using solely the first two fPCs, although some strains exhibited a larger overlap between the groups than others (S10 and S11 Figs for replicate A and B, respectively). This suggests that coordinated phasing in all mutants can be represented by considering only the two fPCs that describe the most variance, and including more fPCs is not necessary in order to interpret the discriminating function.

The respective contribution of the two major fPCs to separate the clusters varied between the strains, suggesting that fPCA is sufficiently sensitive to capture strain-dependent consequences (S10 and S11 Figs for replicate A and B, respectively). This caused the slope of the discriminating boundary to tilt. Therefore, the transformations of the mean distribution (i.e. fPCs and their factors ) changed for these strains. This indicates that they had not only a global effect on the average MNase-seq profile, but also caused a gene-specific disruption of the nucleosome positioning. We deemed those strains particularly important that altered the gene-specific collective behaviour of the entire nucleosome array with respect to the WT. To clarify this point, we provide an example for the impact of a fictional mutant that only affects the mean but not the ensemble of nucleosomes as a whole in S12 Fig. We determined the slope for all strains using a linear SVM. As aforementioned, the boundary is tilted when only considering large genes in WT conditions (Fig 3A), and the two available replicates differ slightly. The observed deviation between replicates was used as a reference for the anticipated variability in the data. By using Eq 7, we determined chromatin remodeler-deficient strains that had a sufficiently different linear boundary with respect to the WT.

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Fig 3. The fPCs, their gene specific scores, and the discriminating boundary explain collective phasing and how this changes in chd1Δ with respect to WT conditions.

The figure shows the cluster distribution with respect to , the impact of the determined fPCs, and the location-specific impact of the separating boundary for WT (i.e. panels A-C) and chd1Δ conditions (i.e. panels D-F). Panels A and D show the fPC scores of WT and chd1Δ strains, respectively. For the latter, the boundary slope changed notably (black dashed line). As indicated by the fCPs in panels B and E for WT and chd1Δ, respectively, the functional description of the data changes. Indeed, the second fPC of chd1Δ abates quickly after the +1, with a strong effect on the effect of the +2 (grey arrows). The dashed black line as well as the solid lines in magenta and green indicate the mean, a positive contribution of the fPC, and a negative contribution, respectively. When exact positions were seemingly not affected by the fPC, we marked the positions with a grey vertical bar. General trends are given in grey arrows along the gene. The location-specific impact of the separating boundary is given in panel C for WT and panel F for chd1Δ strains. Interestingly, despite the median distributions of the clusters (blue and orange) are clearly different with respect to the +1 and +2 in WT conditions, later positions are much more important for allocating a profile to a particular group (grey areas, mean in black). Whilst this is also true for chd1Δ, the importance of later nucleosomes is even more accentuated, whereas the influence of the +1 and +2 positions are further decreased. All axes are scaled to the same size for each strain; shapes and amplitudes are therefore comparable (see Methods for more details).

https://doi.org/10.1371/journal.pcbi.1011799.g003

We provide three different perspectives on the data. Firstly, the cluster distribution with respect to the factors together with the slope highlight mutants that particularly disrupt gene-specific collective nucleosome phasing. In the following, clusters are always indicated using the colours orange and blue. Secondly, analysing the transformation of the two major fPCs of the mean unlocks an additional understanding of the variance present in the data and allow quantifying the general impact of chromatin remodeler deficiency. Here, we plot a positive contribution to the mean in magenta and a negative contribution in green. Lastly, the location-specific effect of the discriminator links spatial properties to the Pearson gene clusters, which describe likely similarity of nucleosome positioning mediated by chromatin remodelers. The impact of the discriminator is in the following given as a grey area around the mean, indicating more important regions when the margin is larger. The median profile of each cluster using the determined fCPs will be given again in orange and blue. This allows a comprehensive analysis of the impact of gene deletions or rsc8 depletion with respect to the WT.

We can determine the importance of particular positions to separate the clusters as follows. The slope of the SVM indicates the contribution of each fPC to separate the clusters. For example, a 0° angle shows that the descriminator can be solely described by the second fPC; 45° suggest an equal contribution of both fPCs to separate the clusters; and 90° indicate that collective nucleosome phasing is exclusively dependent on the first fPC. Consequently, by linearly combining both fPCs together as implied by the slope (Eq 9), we can evaluate which positions along the profile are particularly important for the classification. Indeed, understanding the separating boundary is not straightforward. Although the median profiles for each profile can differ substantially at some positions, this variance might be less important for separating and interpreting the clusters (e.g. the +2 nucleosome in WT conditions, Fig 3C). Reciprocally, whilst the median profiles for both groups can be very similar, the variance over all considered genes at this locus could be much larger and therefore play an important role for the classification (e.g. the +3 position in WT strains Fig 3C).

We identified 5 mutants—namely chd1Δ, isw2Δchd1Δ, rsc8-depleted chd1Δ, isw1Δisw2Δ, and rsc8-depleted isw2Δchd1Δ—that evoked notable changes considering the experimental variability between replicates (Figs 3–5). For a correct interpretation of the results, it is crucial to highlight that this does not imply that other mutants had no effect on the profile. Rather, this suggests that the considered mutation caused a gene-specific change of nucleosome phasing regulated by chromatin remodelers, which we assume is represented by the deviance of the stochastic process (i.e. the variance to describe the MNase-seq profiles). Other gene deletions can have other impacts that do not disrupt the gene-specific collective positioning. All measurements are given in Table 1.

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Table 1. SVM boundary slopes for both replicates.

The first two rows give the boundary slope for replicate A and B, respectively. Mean μ is the mean slope for both. The s value represents our significance measurement defined in Eq 7. Noteworthy changes of the boundary slope are marked in green (bold), all others are red. The s-value in WT is per definition equal 0.

https://doi.org/10.1371/journal.pcbi.1011799.t001

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Fig 4. The fPCs, their gene specific scores, and the discriminating boundary explain changing collective phasing in double mutants.

The figure shows the cluster distribution with respect to , the impact of the determined fPCs, and the location-specific impact of the separating boundary for all double mutants, in particular isw2Δchd1Δ (i.e. panels A-C), rsc8chd1Δ (i.e. panels D-F), and isw1Δisw2Δ (i.e. panels G-I). The linear separation of the cluster distribution with respect to factors indicate a notable gene-specific change for the three mutants in panels A, D, and G. The two clusters are given in orange and blue, and the SVM boundary is depicted by the black dashed line. Whilst isw2Δchd1Δ and isw1Δisw2Δ require both fPCs to linearly separate the Pearson clusters, rsc8chd1Δ is almost exclusively dependent on the second fPC, which means this mutant decreased the slope tilt. This can be better understood when analysing the two fPCs and their effect on the mean ((B) for isw2Δchd1Δ, (E) for rsc8chd1Δ, and (H) for isw1Δisw1Δ). The solid lines in magenta and green in these plots indicate a positive contribution of the fPC and a negative contribution, respectively, whereas the black dashed line depicts the mean. Grey arrows along the gene suggest general trends. Grey vertical bars suggest positions that remain largely unperturbed by the fPC. Grey arrows pointing to a single peak suggest remarkable properties. Interestingly, whilst the first fPC of the isw2Δchd1Δ and isw1Δisw2Δ strains shows a similar transformation of the mean, the second fPC indicates a different behaviour, particularly with respect to the +2 nucleosome. As suggested by the fact that clusters in the rsc8chd1Δ mutant are exclusively dependent on the second fPC, the first fPC explains only the average profile amplitude and does not contain any information about collective phasing. The location-specific effect of the linear separator for each mutant is given in (C), (F), and (I). The grey areas indicate the importance of each position to determine the clusters, whose median profile is shown as a blue and orange dashed line. The mean is depicted in black. Although the impact on the grouping of the +1 and +2 position in isw2Δchd1Δ conditions is similar to the isw1Δisw2Δ strain, the latter is seemingly particularly dependent on the +3 and +4 nucleosome. Positions thereafter become less important, which keep having a strong impact on clustering in isw2Δchd1Δ. As expected rsc8chd1Δ is exclusively dependent on the second fPC. Interestingly, the entire profile seems to be influential for classifying genes, with the largest impact allocated to the first two nucleosomes. All axes are scaled to the same size for each strain; shapes and amplitudes are therefore comparable (see Methods for more details).

https://doi.org/10.1371/journal.pcbi.1011799.g004

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Fig 5. The fPCs, their gene specific scores, and the discriminating boundary explain changing collective phasing in isw2Δrsc8chd1Δ.

The two clusters are given in orange and blue. The figure shows the fPC scores ζ of the isw2Δchd1Δrsc8 mutant and their separating boundary (black dashed line, A). The slope decreases with respect to the WT, making the gene groups almost solely dependent on the second fPC. Both fPCs transform the mean in a similar way as the double mutant rsc8chd1Δ (compare panel B with Fig 4E). The dashed black line as well as the solid lines in magenta and green indicate the mean, a positive contribution of the fPC, and a negative contribution, respectively. As expected, the separating boundary discriminate between the two clusters largely following the second fPC (C). The grey areas show the importance of each position to determine the clusters, whose median profile is shown as a blue and orange dashed line. The mean is depicted in black. All axes are scaled to the same size for each strain; shapes and amplitudes are therefore comparable.

https://doi.org/10.1371/journal.pcbi.1011799.g005

Most single mutants had only a small or negligible effect on the collective nucleosome phasing along transcribed regions, with the exception of chd1Δ (Fig 3D–3F). Indeed, the boundary was notably tilted with respect to WT conditions (Fig 3D). This suggests that the functional composition of the MNase-seq signal changed. In fact, the amplitude of the second fPC decreases more quickly along the gene body in chd1Δ mutants, and the variance of the peak at the +2 position strongly diminished (Fig 3E). When interpreting the effect of the discriminating boundary, we observe that the +1 and +2 nucleosomes only exhibit a small importance for establishing the clusters, whereas the impact of the NDR and later nucleosomes increased (Fig 3F). Consequently, the +1 position remains largely unaffected. As Chd1 is responsible for nucleosome spacing along genes and is particularly involved in maintaining chromatin integrity during Pol II elongation, it is intuitive that the chd1-deletion influences phasing within the gene body. This outcome shows the clear effect of chromatin maintenance by Chd1 after the +2 nucleosome, whilst leaving the +1 position well preserved.

The double mutant isw2Δchd1Δ exhibited also a noteworthy shift of the separating boundary (Fig 4A), yet with different results to the chd1Δ single mutant. The second fPC seemingly preserves its wave-like shape (Fig 4B). This indicates that nucleosome presence is less perturbed, and peaks are comparatively well positioned. Similar to the chd1Δ single mutant, both of the fPCs strongly contribute to distinguish between the Pearson clusters. The discriminating function exhibits similar local effects as the chd1Δ strain, but the positions after the +2 nucleosome clearly indicate an additional shift which contributes to the separation (Fig 4C). Interestingly, rsc8-depleted chd1Δ significantly decreases the slope tilting (instead of accentuating it), therefore making coordinated phasing almost exclusively dependent on the second fPC (Fig 4D). This can be better understood when analysing their respective effects (Fig 4E). The first fPC solely explains average signal amplitude (which is not measured by the Pearson correlation index) and hence contains almost no information about coordinated positioning. As expected, the local effect of the discriminating boundary follows the trend described by fPC2 (Fig 4F). The second fPC also indicates that the NDR before the +1 cannot be maintained (see arrow in Fig 4E and the grey area in NDR and +1 position in Fig 4F), which is in line with other studies [12, 40]. Remarkably, all nucleosome positions along the entire array seem to be important for the classification—particularly the first two—which is not the case for the other two double mutants. It should be noted that not all double mutants that include chd1Δ show a similarly notable tilting of the slope as the single mutant. This could possibly mean that these double mutants have opposing effects, although it is difficult to give a clear indication with the variation between only two replicates. We found an interesting behaviour for isw1Δisw2Δ (Fig 4G). The effect of the second fPC hints that the positioning of the +2 is strongly impacted, and following phasing becomes inharmonious (Fig 4E). The +1 is kept well positioned. The first fPC, on the other hand, resembles the first fPC of the isw2Δchd1Δ mutant, with a minor difference at the +3 nucleosome (compare Fig 4E with Fig 4B). Indeed, when analysing the location-specific properties of the separating function (Fig 4I), nucleosome profiles in the isw1Δisw2Δ strain seem to be clustered particularly with respect to a shift at the +3 and +4 position. This shift is apparently slightly corrected thereafter and becomes less important. Whilst seemingly similar, a shift in the isw2Δchd1Δ strain after the +2 position remains important for the entire arrangement to determine the gene groups (compare Fig 4I with Fig 4C). This indicates that Chd1 and Isw1 contribute differently to nucleosome phasing in isw2Δ conditions, with the effect of Isw1 being possibly more confined. Taken together, these results show that double mutants can have varying and non-linear effects.

Among the triple and quadruple mutants, the only one that changed notably the clustering boundary is isw2Δchd1Δrsc8 (Fig 5A). Once again, tilting is decreased. The effect of the fPCs and the separating boundary is almost identical to the chd1Δrsc8 mutant, suggesting that isw2Δ does not have a strong effect on the phenomenon (Fig 5B and 5C). However, it should be mentioned that the variability between the two replicates is considerably large, as the two clusters can be only neatly separated in replicate B, whereas replicate A exhibits a great overlap. Whilst the result in the latter replicate could suggest that more fPCs are necessary to interpret the gene groups, the results for replicate B indicate that sufficient information is preserved in the first two fPCs. More replicates would be needed to provide an answer. We also want to highlight that mutants with more than two gene deletions exhibited less clear nucleosome peaks, and a straightforward interpretation of the Pearson correlation with respect to the two discriminating trends (compare with cartoon in Fig 1D) could be difficult. The results for these strains should be taken with a pinch of salt.

Taken together, these outcomes show that remodeler mutants have varying effects on nucleosome positioning. Whilst most mutations do not notably alter the gene-specific nucleosome coordination with respect to the WT, we identified 5 mutants that exhibited a strong effect on phasing. Interestingly, most of them include chd1Δ, which indicates an important role of Chd1 for local arrangement within the gene body. Using fPCA to visualise the Pearson clusters permits the clear and position-specific quantification of the induced impact among varying strains.

Pol II presence correlates with nucleosome organisation in chd1Δ mutants

In order to assess an interdependence of nucleosome organisation with other genomic properties, we compared the two Pearson clusters to Pol II levels, Sth1 occupancy, AT ratio over the entire gene, as well as upstream NDR length and orientation of the upstream NDR (i.e. tandem or divergent). We also included Mediator presence as a large protein complex with transient interactions predominantly at the NDR. All of these factors were clustered into two equally-sized groups where possible. For example, Pol II presence along the gene was evenly separated into transcribed regions with high and low Pol II occupancy. Interdependence was measured by training a simple neural network with no hidden neurons using Hebbian learning [41]. Consequently, we assessed which nuclear groups (e.g. high or low Pol II presence) corresponded to which Pearson clusters. We want to stress that we did not aim to find a predictive model. Rather, this approach allowed us to measure a direct correlation between similarity of nucleosome phasing and other genomics properties. The initial k-mean clustering did not impose a constraint on the group size, and they could therefore differ in the number of genes they contained. To remove any prediction bias, we forced the clusters to be of the same size. Genes in the larger group with closest Pearson coefficient to all distributions in the smaller group changed the cluster. We found the analysis for WT conditions non-conclusive, and correlations varied between A and B replicate (S13 Fig top). Whilst A was slightly correlated with the AT sequence content (Fig 6A and 6B), this trend disappeared for B, and it might in both replicates rather correspond to the fPC orthogonal to the cluster boundary (S16 Fig). Overall, we were surprised that none of the investigated properties could indicate a clear interdependence with nucleosome phasing in WT (Fig 6A).

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Fig 6. Remodeler deletions have varying effects on the interdependence with other genomic properties.

The orange bars in panels A, C, and E show the ratio of correct predictions, and blue bars are wrong guesses. As we distinguish between two clusters, the dashed black line at 0.5 indicates random guessing. The dashed grey line with black edging in panels B, D, and F display the linear boundary for the Pearson clusters. Panels A and B: WT conditions are seemingly correlated with the sequence composition. However, the results are different for the B replicate, and therefore non-conclusive. All possible correlations are surprisingly low. Panels C and D: chd1Δ mutants increase particularly their dependence on Pol II and other transcription-coupled properties, such as Mediator presence. Surprisingly, the mutant showed also an increased interdependence on NDR length. Panels E and F: despite the Rsc8-mediated gene limits, there is no correlation with coordinated nucleosome phasing and the size of transcribed regions or NDR length. Although Sth1 indicates a slightly increased interdependence, this cannot be confirmed when plotting the groups with respect to the Pearson clusters. This is in line with our hypothesis that positioning in different regions interfere, and therefore, nucleosome localisation become increasingly independent from region-specific factors.

https://doi.org/10.1371/journal.pcbi.1011799.g006

The correlation between positioning and other nuclear properties changed among the mutants (S13 Fig). The effect is particularly clear for chd1Δ (Fig 6C), as there is a strong interdependence between phasing and Pol II (Fig 6D), Mediator presence, and NDR size (S13 Fig). As aforementioned, Chd1 maintains, among others, chromatin integrity during Pol II elongation. The correlation is therefore in line with our previous conclusions and the function of Chd1. The established link between Pol II presence and nucleosome organisation remains conserved in all strains with a Chd1 gene deletion, except isw1Δchd1Δrsc8. This is similarly true for the correlation with Mediator occupancy and NDR length. There was also a slight correlation to Sth1 and AT ratio in cells containing chd1Δ, which were, despite being weak, still notably stronger than in WT. The results are in agreement with the effects of Chd1 on chromatin maintenance during gene expression.

Due to the Rsc8-mediated nucleosome organisation within transcribed regions, we were wondering whether there is an increased interdependence to NDR length or gene size. We can report that there is no correlation with NDR size in any rsc8-depleted strain (Fig 6E). This is in line with our hypothesis that Rsc8 decouples processes at different genes. However, rsc8 mutant cells exhibited a slightly increased correlation with Sth1. By looking at the separation with respect to the Pearson cluster boundary, we find that there is no noticeable impact (Fig 6F). The results indicate that any correlation with region-specific properties is lost, which is likely due to the interference of nucleosome positioning of various regions.

isw1Δ single mutant did exhibit only a slightly increased correlation with Pol II, Sth1 and Mediator presence as well as AT ratio. isw2Δ might as well show a weak correlation with Pol II occupation. Each of the replicates of their double mutant indicates different correlations, and it is therefore difficult to tell whether transcription-related factors influence nucleosome phasing in the isw1Δisw2Δ strain. However, none of them indicate any strong interaction, suggesting that—on a global scale—these effects might be negligible in comparison to the WT (S13 Fig).

Interestingly, the rsc8-depleted isw2Δ indicated a strong correlation with Sth1 and Mediator presence as well as NDR length. The effect was observable in almost all strains that contained the double mutation with the exception of the quadruple mutant (S13 Fig). Taken together, this could indicate an impact along the gene body and the promoter region in strains that contain rsc8-depleted isw2Δ.

Surprisingly, combining two factors together (e.g. Pol II presence and AT ratio) to predict Pearson clustering did not increase accuracy. Instead, one factor dominated the correlation measurement, e.g. Pol II presence for chd1Δ strains. This could possibly suggest that—despite several factors showing increased interdependence—they can be reduced to a main influencing factor (which is not necessarily one of the tested properties).

Taken together, the results indicate a strong interdependence between local genomic properties—such as presence of large protein complexes or NDR length—and strains containing chd1Δ. This supports our hypothesis of Chd1 being responsible for local nucleosome coordination.

Data treatment

MNase sequencing reads were taken from [18] and [12] (GEO accession numbers GSE69400 and GSE73428, respectively) and treated as in our previous study [56]. To be precise, reads from Fastq files were trimmed with trim_galore (v0.6.5) [57] and cutadapt (v3.1) [58]. Subsequently, they were mapped on the Saccharomyces cerevisiae genome (University of California at Santa Cruz [UCSC] version sacCer3) using bowtie2 (v2.3.4.3) [59]. Files were converted with samtools (v1.9) [60] and deeptools (v3.5.0) [61]. Read counts were normalised in Reads Per Million (RPM) of mapped reads. We used the option --MNase of bamCoverage so that only the mononucleosome fragments were kept. This means that fragments shorter than 130 bp and longer than 200 bp were removed from analysis. Mediator and Sth1 ChIP-seq were taken from [56] (ArrayExpress accession number E-MTAB-12198). We used Pol II ChIP-seq from our previous study [62].

Following [12, 18], we retrieved positioning profiles along the coding regions 200 bp before and 1000 bp after the +1 nucleosome. Genes on the Crick strand were inverted. Consequently, all data is calibrated such that the +1 position is at 200 bp. The profile of genes for which the +1 position is known were considered as in [18].

Measuring profile correlation and clustering

The pairwise Pearson correlation of MNase-seq distributions for each gene was determined using equation (1) Here, x and y denote two genes, and symbolise their respective average MNase-seq value along the coding region, and n = 1200 is the length of the considered region. Eq 1 ranges between -1 and 1, and indicates whether the two gene profiles tend to change into the same (positive Pearson correlation) or opposite directions (negative Pearson indices).

Genes were divided using the k-mean clustering implementation in MATLAB with the correlation distance metric. To define the optimal number of k-mean clusters, we used the silhouette criterion measurement [63, 64]. For all analysed strains, the highest silhouette value occurs at 2 groups, suggesting that in order to divide the profiles into classes with respect to their Pearson indices, the optimal number of clusters is 2 (Fig 1A). Therefore data were grouped in two clusters (Fig 1B and 1C).

Cluster significance was validated by comparing the inter-cluster correlation (i.e. each pairwise Pearson index between a profile in cluster 1 and a profile in cluster 2) between the determined and random clusters. It is expected to be minimal for the gene groups found by the k-mean clustering. Therefore, we applied the KS test implemented in scipy with the hypothesis that the inter-cluster correlation of the found gene sets is significantly lower than for random groups. It should be noted the number of all pairwise comparisons can be quite large. It is a known problem that small differences are marked as being significant by the KS significance test when sample sizes are too large. Therefore, we sub-sampled randomly 500 pairwise inter-cluster correlations for the determined and the random clusters, respectively. It should be mentioned that this sometimes led to very similar sub-samples, and the p-value was for some few runs fairly high. We therefore averaged the p-value over for the KS test over all 500 random clusters and used the conventional p-vale threshold of 0.05 (i.e. 5%) to reject the null hypothesis. To be more precise, we required that the average probability of finding random clusters that have a similarly low inter-cluster correlation was lower than 5%. This was the case for all tested mutants.

Functional principal component analysis

Functional clustering in a Hilbert space H can be achieved by fPCA. It applies—similar to PCA in Euclidean space—a functional dimensionality reduction in H to investigate the dominant mode in functional data. Instead of relying on values in discrete dimensions, fPCA uses a given number of basis functions (e.g. B-splines or Wavelet) to create the eigenfunction basis that accounts for most functional variation. Despite the fact that MNase-seq data is stored in a discrete array (i.e. one value per bp), we can nevertheless find a functional approximation over a range using a given choice of basis functions. It should be noted that this implicitly smooths out high frequencies in the signal. We presume that nearby values in MNase-seq data possess a strong interdependence, therefore justifying a smoothed and continuous functional representation of the high dimensional data. In this study, we apply B-splines as a basis to represent the nucleosome array (Fig 8). We use the Python library scikit-fda to determine the fPCs and the corresponding weights explaining the distribution [65]. Here, we describe briefly the underlying principles of the method.

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Fig 8. Representing the MNase data array as a composition of B-spline base functions in WT conditions.

Left shows the raw data, each colour depicting one profile over a gene. Center gives the smoothed profiles after representing the data as B-splines. Right displays the average profile using the functional composition.

https://doi.org/10.1371/journal.pcbi.1011799.g008

FPCA presumes that the functional data represents a stochastic process X(t) with expected value μ(t) = E[X(t)] and orthonormal eigenfunctions ϕⁱ(t), i = 1, 2…. Intuitively, ϕⁱ(t) describes the most variation in X orthogonal to all ϕ^j, j < i. This allows the iterative determination of the eigenfunctions in the functional data. It should be emphasised that in this study the process is defined in space rather than describing temporal data. We follow nevertheless the convention by denoting the independent variable as t. By using the Kosambi–Karhunen–Loève theorem, any stochastic process can be represented as an infinite linear combination ϕⁱ(t). Consequently, we can describe the stochasticity in X(t) via (2) ζ^k is the autocovariance operator (3)

To provide some intuition, it is presumed that the entire data set can be explained via an average behaviour μ(t). Variability to μ(t) for each gene is expressed by ϕ^k(t) together with a factor ζ^k. ζ^k can be loosely compared to a normal correlation measurement, i.e. ζ^k increases when ϕ^k(t) and ∫(X(t) − μ(t)) follow the same trend. If they describe opposing behaviours—for example ϕ^k(t) decreases when ∫(X(t) − μ(t)) increases—ζ^k becomes negative.

It is commonly justified to approximate Eq 2 as a finite sum (4)

It should be noted that ϕⁱ(t), i = 1, 2, … is a basis of the functional space in H.

This understanding of the underlying process permits the application of fPCA. A smoothed representation with the basis functions (e.g. B-splines) fulfilling Eq 4 can be obtained using L² regularisation. To reduce the dimensionality to K, we keep only the first K components (i.e.ϕⁱ(t)) that represent the dominant mode of variation in X by setting the first component to (5) and the following K − 1 components to (6)‖ϕ‖ is the square norm, i.e. . It should be emphasised that ϕ^k can differ by a factor of -1 due to the square norm, and consequently, the operator ζ^k (Eq 3) can be either positive or negative depending on ϕ^k. This means that the slope of the cluster-dividing boundary can be pointing upwards or downwards and still describe the same functional composition.

We exemplified the impact of the first two fPCs to analyse the consequences on nucleosome phasing in chromatin remodeler-deficient cells (see for example Figs 1, 2 and 4). It should be noted that the fPCs were amplified to highlight their functional contribution. We set the scaling factor to ζ¹ = ζ² = 20 in all figures that demonstrate their effects (i.e. magenta shows the effect of the fPC multiplied by 20 and added to the mean, and green depicts the fPC multiplied by 20 and subtracted from the mean). The determined factors were predominantly distributed in ζ^{1, 2} ∈ [−20, 20] for all strains and replicates, and most of them were in fact much lower. Therefore, we limited the scaling of the axes for ζ¹ and ζ² to [−20, 20] for all plots that show the cluster distribution with respect to the factors. Therefore, all figures and axes were directly comparable. The few outliers that were outside this range were incorporated into the analysis despite of being not shown in those plots. In most cases, only 1 or 2 genes were outside these ranges. However, large genes for isw1Δ replicate A and isw2Δrsc8chd1Δ replicate B had 71 and 72 outliers, respectively. Nonetheless, this value amounts to less than 2%/ of all considered protein-coding profiles for these setups.

In order to validate that nucleosome arrangements tend to separate along the first two fPCs when using a different clustering method, we repeated grouping using WARD with an Euclidean distance metric on the profiles. Clustering was implemented using the AgglomerativeClustering class of the scikit-learn Python package. In contrast to the Pearson correlation index, the Euclidean distance does not have an upper bound, and clustering was more sensitive to outliers. We removed 29 genes who had an absolute fPC score larger than 20 for either of the two fPCs. Despite the fact that the boundary is less neat than when using the k-mean clustering, our results suggest that the WARD gene groups tend to separate along the first two fPCs as well (S14 Fig).

Quantifying the cluster boundary

Long genes were linearly separable with respect to the Pearson coefficient clusters in all WT and mutant conditions. The boundary was determined using a linear SVM. We ignored the prediction error and the intercept of the linear boundary, and instead considered only the slope differences between the two replicates. As aforementioned, the sign of the slope m does not matter, and we consider therefore only |m|. To quantify the variability in the two replicates, we introduce the following measurement (7) denotes the average over the absolute slopes of both replicates. We defined a change as notable when s(i)>1, which implies that the mean variability between WT and mutant is larger than the variability within the replicates, i.e. (8)

As we consider only two replicates, we restrain from using the word significant as much as possible and use noteworthy or notable instead.

The slope of the boundary m indicates the contribution of each fPC to describe the discriminator between the clusters. As m shows the change of ζ² over one unit of ζ¹, we can determine the separating boundary by (9)

The impact of ϕ^/ can be visualised by multiplying a scaling factor which is followed by addition to and subtraction from the mean. In this study, we used a factor of ζ^/ = 5 to create the grey bands in the plots that show the effect of the separating function.

Measuring interdependence between nucleosome phasing and other nuclear properties

In order to analyse interdependence of nucleosome positioning with other nuclear properties, we divided all factors into two equally sized cluster using the median wherever possible. For example, the half with the smaller NDRs was assigned to group -1, whereas the larger half was group 1. This split was performed after filtering for the size (i.e. large or small genes). The analysis aimed to find a correlation between nuclear factor group and Pearson cluster. To remove any bias with respect to the group size, we forced both Pearson clusters to contain the same number of genes.

We used a simple feedforward network with no hidden neurons and a single output neuron whose activation indicated the predicted Pearson cluster. The number of input neurons varied between 1 and 2, depending on whether we considered a multivariate interdependence. The group of the nuclear factor (i.e. -1 or 1) was set as input neuron activation. This was weighted and summed together with all other input values. The activation function of the output was a modified sign function, which returned 0 when negative and 1 when positive. Therefore, if the weighted sum over the input was lower than or equal to 0, the output would be 0, and 1 otherwise.

Weights were trained using a Hebbian-like learning method [41]. In order to avoid any confusion, we name Pearson cluster 0 and nuclear factor group -1 low cluster, whereas we define group 1 in both cases to be the high cluster. The weight was defined to be the average number of genes where the nuclear factor group and Pearson coefficient cluster where both low or both high; minus the average number where one of them was low whilst the other high. The implementation as a neural network allowed the straightforward extension to compare interdependence with several factors at the same time using the same method.