Estimating the 2D Self-Ligation Read Pair Distribution.

We assume that ChIA-PET linker tags have been removed from the read pair sequences, that read pairs that are known to have resulted from chimeric ligation events because they contain two different linker tags have been removed, and that the remaining linkerless read pairs have been aligned to the reference genome. Let R be the set of all aligned read pairs such that each read pair r_i ∈ R is represented by the pair of genomic coordinates to which the ends of the read pair align. We assume that the coordinates for each read pair are ordered so that if $r_i=⟨r_i^{(1)},r_i^{(2)}⟩$, then $r_i^{(1)}\le r_i^{(2)}$. We also assume that each read pair has an associated label according to the chromosome strands to which the ends align. There are four possible strandedness labels given the imposed ordering on the read pair ends. They are ++, -+, +-, and –. As mentioned above, all self-ligation read pairs have strand orientation -+, but not all -+ read pairs were produced by self-ligation.

A distribution estimated from all -+ read pairs would not accurately model the distribution of self-ligation read pairs because self-ligation read pairs are much more likely to align within a short distance than inter-ligation read pairs. This is because the fragment length distribution induced by fragmentation limits the distance between which the ends of self-ligation read pairs may align whereas there is no constraint on the distance between which the ends of inter-ligation read pairs may align. To more accurately estimate the distribution of self-ligation read pairs, we weight the contribution of each -+ read pair by the estimated likelihood that the read pair was produced by self-ligation according to the distance between the aligned locations of the read pair ends.

Let z_i indicate whether -+ read pair r_i was produced by self-ligation or inter-ligation and d(r_i) be the distance between the aligned locations of the ends of -+ read pair r_i. The likelihood that -+ read pair r_i was produced by self-ligation according to d(r_i) can be expressed in terms of quantities that can be estimated from the data (1)

Pr(d(r_i)) for all -+ read pairs can be estimated by applying an unweighted kernel approach (2)

N₋₊ is the total number of -+ read pairs and K₁ is a standard univariate Gaussian distribution. The bandwidth h₋₊ is a parameter that controls the trade-off between fitting the training data and discovering a smooth estimate. To choose an appropriate h₋₊ we use a least-squares cross-validation approach that minimizes the integrated square error (ISE) of $\widehat{\text{Pr}}(x)$. (3)

The $ISE(\widehat{\text{Pr}}(d(r)=x))$ can be approximately minimized by minimizing for all -+ read pairs [22] (4)

We cannot estimate Pr(d(r_i)|z_i = self) directly for the same reason that we cannot estimate the self-ligation read pair distribution directly. We can estimate Pr(d(r_i)|z_i = inter) directly because all non -+ read pairs are produced by inter-ligation. We also apply an unweighted kernel approach to estimate this distribution (5)

We choose an appropriate h_non−+ by approximately minimizing the $ISE(\widehat{\text{Pr}}(d(r)=x|z=inter))$.

Given estimates for Pr(d(r_i)) and Pr(d(r_i)|z_i = inter), we can estimate Pr(d(r_i)|z_i = self) by assuming that Pr(d(r_i)) is a mixture of the distributions Pr(d(r_i)|z_i = self) and Pr(d(r_i)|z_i = inter) (6)

By rearranging the terms in this equation we can obtain (7)

The final missing component is Pr(z_i = self) = 1 − Pr(z_i = inter). We assume that the average number of read pairs with each of the three strand orientations other than -+ is a good estimator for the number of -+ read pairs that were produced by inter-ligation. We use this information to estimate Pr(z_i = inter) (8)

This allows us to estimate the self-ligation read pair distribution using a weighted kernel approach weighted by Pr(z = self|d(r_i)) (9) where in this case K₂ is a bivariate standard Gaussian distribution with no correlation between the dimensions. To choose an appropriate bandwidth h_self we approximately minimize $ISE(\widehat{\text{Pr}}(r=⟨x,y⟩|z=self))$ by minimizing (10)

Estimating the 1D Marginal Distribution of Protein Occupancy.

We assume that the self-ligation read pair distribution is the result of the convolution of the marginal distribution of protein occupancy and a distribution that models DNA fragmentation which we will refer to as the read spread function (RSF). If we let q be the genomic location occupied by the protein, (11)

Simultaneously deconvolving the marginal distribution of protein occupancy and the RSF from the self-ligation read pair distribution is an example of a blind deconvolution problem. This problem commonly arises in the context of image processing. It is often the case that a camera will systematically blur the images that it captures because of flaws in its lens. This blurring process is modeled as a convolution of the distribution of light that enters the camera lens with a point spread function (PSF) that is induced by the flaws in the lens. The PSF specifically describes the effect that the lens flaws will have on a theoretical point source of light. In our case, the RSF describes the manner in which self-ligation read pairs are likely to be distributed given the theoretical occupancy of the protein at a genomic location.

If we assume at first that the RSF is known, the marginal distribution of protein occupancy can be approximately recovered using a standard approach known as Richardson-Lucy (RL) deconvolution [23, 24]. The RL algorithm iteratively applies the following EM-like update (12)

RL deconvolution has been shown empirically to converge to a maximum-likelihood estimate for Pr(q = u) and preserves the non-negativity and sum of the initial guess Pr₀(q = u). To extend RL deconvolution to the blind case, we take an approach similar to that proposed in [25] and alternate the updates described by Eq 12 with the following updates (13)

The overall procedure then entails going back and forth between updating $\widehat{\text{Pr}}(q=u)$ for several iterations while holding $\widehat{RSF}(⟨x-u,y-u⟩)$ fixed and then updating $\widehat{RSF}(⟨x-u,y-u⟩)$ for several iterations while holding $\widehat{\text{Pr}}(q=u)$ fixed. Despite the unconstrained nature of the blind deconvolution approach, the recovered RSF conforms to our expectations. The RSF in Fig 2 is typical of what is recovered from RNA PolII ChIA-PET data. Given a location bound by the protein, we would expect the most likely alignment of the ends of self-ligation read pairs to be roughly equidistant to the occupied location with the distance from the occupied location determine by the degree of fragmentation. The typical RSF that we estimate has the greatest value along the line through the origin that is perpendicular to the identity line. Points along this line reflect self-ligation read pairs that align equidistantly to the occupied location which is represented by the origin in the RSF. The distance of the peak in the RSF from the origin reflects the most likely fragment size generated by the sonication step. Thus, the RSF that we recover using our blind deconvolution approach conforms to our expectations and provides useful information about the fragmentation step of the ChIP procedure.

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Fig 2. A typical read spread function estimated from RNA PolII ChIA-PET data.

https://doi.org/10.1371/journal.pone.0122420.g002

Efficiently estimating the genome-wide protein occupancy distribution.

RL blind deconvolution works well for deconvolving the protein occupancy distribution for regions of the genome that are on the order of megabases in size. However, the time that it would take to deconvolve the full genome-wide distribution of protein occupancy is impractical. Based on observations made about typical RSFs estimated by RL blind deconvolution from portions of real ChIA-PET datasets, we devised a highly efficient procedure that achieves a level of accuracy comparable to full RL blind deconvolution. The observation we made was that typical RSFs estimated by RL blind deconvolution from portions of real datasets are unimodal and sharply peaked. This implies that the RSF can be approximated by a function with all of its mass at the peak of the RSF. This approximation allows for a very efficient deconvolution procedure. If the peak of the estimated RSF is at ⟨−λ, λ⟩, we estimate the protein occupancy distribution as (14)

In summary, to estimate the marginal distribution of protein occupancy from a full genome-wide ChIA-PET dataset we first estimate the genome-wide self-ligation read pair distribution. We then apply RL blind deconvolution to a 5 megabase region of the genome to obtain a good estimate for the RSF. Finally, we identify the peak of the estimated RSF and estimate the distribution of RNA PolII occupancy as in (Eq 14).

Estimating the 2D Joint Distribution of Protein Occupancy.

Chromatin looping allows proteins to simultaneously occupy two genomic locations [26]. Inter-ligation read pairs can be thought of as samples from a joint distribution of protein occupancy with positional noise introduced by fragmentation. We make several assumptions about this process. We assume that the inter-ligation read pairs are based on independent samples from the joint distribution of protein occupancy. We associate the lower coordinate protein location q⁽¹⁾ with the lower coordinate end of the read pair r⁽¹⁾ and the higher coordinate protein location q⁽²⁾ with the higher coordinate end of the read pair r⁽²⁾. (15) (16) (17)

The last equality reflects an assumption that we make that the location occupied by the protein is independent of the read pair end that it is not associated with. We will demonstrate that these terms are non-zero in only a relatively small window around their associated read pair end and that the non-associated read pair end has minimal effect on the manner in which we compute these terms. We transform the first term within the sum into quantities that we can compute using Bayes’ Theorem (18)

We assume that we can obtain $\text{Pr}(r_i^{(1)}|q^{(1)}=u)$ by marginalizing the RSF that was estimated during the blind deconvolution step. For read pair ends that align to the—strand (19)

Correspondingly, for read pair ends that align to the + strand (20)

Pr(q⁽¹⁾ = u) is the distribution of protein marginal occupancy that was estimated in the previous step. The prior read distribution $\text{Pr}(r_i^{(1)})$ reflects any factors that might influence the alignment of reads to locations in the genome. Such factors might include the uniqueness of the sequence around that location in the genome and bias in the library preparation or sequencing for the sequence around that location. We assume that $\text{Pr}(r_i^{(1)})$ is uniform in this work. However, future work may be improved by utilizing a more informative prior distribution.

We also transform the second term within the sum in (Eq 17) using Bayes’ Theorem (21) (22)

The approximation in (Eq 22) incorporates assumptions to simplify all terms involved. We assume that $r_i^{(2)}$ only depends on the location of protein occupancy that it is associated with, and hence $\text{Pr}(r_i^{(2)}|q^{(1)}=u,q^{(2)}=v)\approx \text{Pr}(r_i^{(2)}|q^{(2)}=v)$ which we obtain by marginalizing the estimated RSF. We next assume that q⁽¹⁾ and q⁽²⁾ are independent. This is clearly not true, since otherwise we would have no need of estimating their joint distribution. But, since $\text{Pr}(r_i^{(2)}|q^{(2)}=v)$ is only non-zero in a relatively small range around v, the purpose of Pr(q⁽²⁾ = v|q⁽¹⁾ = u) is mainly to fine tune the probability that q⁽²⁾ = v if $r_i^{(2)}$ falls within that range. We expect the locations of peaks of Pr(q⁽²⁾ = v|q⁽¹⁾ = u) to roughly agree with peaks of Pr(q⁽²⁾ = v) if they exist, and so we assume that we can swap one for the other in this case. Finally, we assume that $r_i^{(2)}$ is independent of the location of protein occupancy that it is not associated with, allowing us to substitute $\text{Pr}(r_i^{(2)})$ for $\text{Pr}(r_i^{(2)}|q^{(1)}=u)$.

These transformations allow us to write the estimated joint distribution of protein occupancy as (23)

Germ^X: Estimating the Conditional Distribution of Protein Occupancy with a Set of Locations X.

In many situations we are interested in estimating the joint occupancy of a protein with a set of genomic locations X. For example, when analyzing RNA PolII ChIA-PET data, a common query might be to detect regions that are jointly occupied by RNA PolII along with a location from set of annotated transcription start sites (TSSs). If we define TSS to be a set of annotated TSSs, we refer to Germ^TSS as the process of estimating Pr(q = ⟨u, v⟩|R_inter) only for v ∈ TSS.

Evaluating the Significance of Portions of Estimated Distributions of Marginal and Joint Protein Occupancy.

Once we have estimated distributions of marginal and joint protein occupancy from ChIA-PET data we evaluate the significance of the estimated protein occupancy within a given region or the joint occupancy within a given pair of regions. We describe our approach as applied to a marginal distribution of protein occupancy and then extend the approach to joint distributions. Given a genomic region reg of size w base pairs, let $p={\sum }_{u\in reg}\widehat{\text{Pr}}(q=u)$. If we let Z ∼ Binomial(N_self, p) and $Y\sim \mathrm{\text{Binomial}}(N_{self},\frac{w}{M})$ where M is the size of the mappable genome, we then evaluate the significance of the protein occupancy within reg as Pr(Y > Z). In other words, we calculate the probability that more self-ligation read pairs would be associated with reg according to a uniform distribution of protein occupancy than would be associated with reg according to the estimated distribution of protein occupancy.

We extend this approach to evaluating the significance of pairs of regions according to a joint distribution of protein occupancy. Given a pair of regions reg_a and reg_b, let $p_{joint}={\sum }_{u\in re{g}_a}{\sum }_{v\in re{g}_b}\widehat{\text{Pr}}(q=⟨u,v⟩|R_{inter})$, $p_a={\sum }_{u\in re{g}_a}\widehat{\text{Pr}}(q=u)$, and $p_b={\sum }_{u\in re{g}_b}\widehat{\text{Pr}}(q=u)$. If we then let Z ∼ Binomial(N_inter, p_joint) and Y ∼ Binomial(N_inter, p_a p_b), we then evaluate the significance of the joint protein occupancy of the regions reg_a and reg_b as Pr(Y > Z).

Significance evaluation for Germ^X.

The estimate $\widehat{\text{Pr}}(q=⟨u,v⟩|R_{inter})$ for v ∈ X that is obtained by applying Germ^X is void of mass for much of its domain. This is because not enough inter-ligation read pairs can be sequenced to fully explore this space given current technologies. Without considering the mass that is missing from the estimate of $\widehat{\text{Pr}}(q=⟨u,v⟩|R_{inter})$, the significance of portions of the distribution for which mass is estimated will be overestimated. To remedy this issue, we introduce a method for estimating how much mass is missing from the estimate of $\widehat{\text{Pr}}(q=⟨u,v⟩|R_{inter})$ in order to more accurately evaluate the significance of portions of this distribution. We assume an ordering on the v_i ∈ X and let $t_i={\sum }_u\widehat{\text{Pr}}(q=⟨u,v_i⟩|R_{inter})$ and $m_i=\widehat{\text{Pr}}(q=v_i)$. If we assume that there is some amount of mass τ_i that is missing from t_i, then we can find a setting of the τ_i such that $\frac{t_i+{\tau }_i}{{\sum }_i{t}_i+{\tau }_i}=\frac{m_i}{{\sum }_i{m}_i}$. However, there are many valid settings of the τ_i and larger values of the τ_i will cause portions of the estimated distribution to be evaluated as less significant.

To choose an appropriate setting of the τ_i we introduce a procedure that allows us to choose τ_i large enough to avoid overestimating the significance of portions of the estimated distribution. We first choose a set of candidate regions for each v_i ∈ X which we will evaluate for significance based on $\widehat{\text{Pr}}(q=⟨u,v⟩|R_{inter})$. We do this by setting a threshold f and adding a region reg to the set for v_i if $\forall u\in reg,\widehat{\text{Pr}}(⟨u,v_i⟩|R_{inter})>f$. We then identify an i_max such that ∀i, t_imax ≥ t_i. We choose some c > 1 and set τ_imax = (c − 1)t_imax. We hold τ_imax fixed and apply an iterative procedure to find settings for τ_i (i ≠ i_max) such that $\frac{t_i+{\tau }_i}{{\sum }_i{t}_i+{\tau }_i}=\frac{m_i}{{\sum }_i{m}_i}$. For each iteration, we cycle through i ≠ i_max and compute (24)

Once this converges, we evaluate the significance of the regions defined using the threshold f in the following way. For a region reg in the set for v_i we let $p=\frac{{\sum }_{u\in reg}\widehat{\text{Pr}}(⟨u,v_i⟩|R_{inter})}{t_i+{\tau }_i}$ and $p\prime ={\sum }_{u\in reg}\widehat{\text{Pr}}(u)$. If we then let Z ∼ Binomial(N_inter, p) and Y ∼ Binomial(N_inter, p′), the significance of the estimated joint protein occupancy of v_i and reg is Pr(Y > Z). We evaluate the significance of the regions in the sets for all v ∈ X and identify the regions that have an associated Pr(Y > Z) less than some threshold such as 0.05. We call these regions significant. For each region, we also note the number of read pairs in R_inter that contributed to p for that region. If the ratio of the number of significant regions supported by only one read pair to the total number of significant regions is greater than some target threshold, such as 0.1, we increase c and begin the process of finding a new set of τ_i. If there are too few significant regions supported by one read pair with Pr(Y > Z) < 0.05 we reduce c and find new τ_i. In this manner we search for c that achieves a target fraction of weakly supported jointly occupied regions within the set of all regions that evaluate as significant.