Statistical Inference on Three-Dimensional Structure of Genome by Truncated Poisson Architecture Model

Abstract

In recent years, next generation sequencing technology, coupled with an assay that is capable of detecting genome-wide chromatin interactions, has produced a massive amount of data and led to a greater understanding of long-range, or spatial, gene regulation mechanisms. Hence, the traditional one-dimensional linear view of a genome, which is especially prevalent in statistical and mathematical modeling, is inadequate in many genomic studies. Instead, it is essential, in studying genomic functions, to estimate the three-dimensional (3D) structure of a genome. The availability of genome-wide interaction data necessitates the development of analytical methods to recover the underlying 3D spatial chromatin structure, but challenges abound. One particular issue is the excess of zeros, especially with higher resolution, or inter-chromosomal, data. This leads to questions concerning the appropriateness of using the Poisson distribution to model such data. In this article, we introduce a truncated Poisson Architecture Model (tPAM) to directly model sequencing counts with many zeros. We carried out an extensive simulation study to evaluate tPAM and to compare its performance with an existing method that uses the Poisson distribution to model the counts. We applied tPAM to reconstruct the underlying 3D structures of two data sets, one of human and one of mouse, to demonstrate its utility. The analysis of the human data set considered chromosomes 14 and 22 jointly, thereby illustrating tPAM’s capability of analyzing inter-chromosomal data. On the other hand, the mouse analysis was focused on a region on chromosome 2 to evaluate tPAM’s performance for recovering structure with loci in different topologically associated domains.

Appendices

Appendices

1.1 A. Isometric Transformation

To make $\varOmega$ uniquely estimable, instead of incorporating the restrictions on $\varOmega$ into prior, we employed a group of isometric (distance preserving) mappings. Suppose we sample $\varOmega ^t$ at iteration t. For simplicity, we let $\varOmega$ denote the transformed one throughout the rest of this appendix.

• Step 1. $\mathbf {p}_1 \rightarrow (0,0,0)$.

  To place $\mathbf {p}_1^t$ at the origin (0, 0, 0), we apply a translation operation $\mathscr {R}_\tau$ such that

  $$\begin{aligned} \mathscr {R}_{\tau }: {} \mathbf {p}_i^t \rightarrow \mathbf {p}_i^t - \mathbf {p}_1^t. \end{aligned}$$

  (18)

  Let $\varOmega =\{\mathbf {p}_1,\ldots ,\mathbf {p}_n\}$ be the translated architecture.

• Step 2. $\mathbf {p}_n \rightarrow (p^x_n,0,0)$ with $p^x_n>0$.

  • a. $\mathbf {p}_n \rightarrow (p^x_n,0,p^z_n)$.

    To place $\mathbf {p} _{n}$ on the xz-plane, we apply a rotation operation $\mathscr {R}_{\mathring{z}}$ with associated matrix $R_{\mathring{z}}$, clockwise-rotation matrix on $\mathbf {p} _{n}$ about the z-axis, sending it to the xz-plane:

    $$\begin{aligned} R_{\mathring{z}}=\left[ \begin{array}{ccc} \cos \phi _1 &{} \sin \phi _1 &{} 0 \\ -\sin \phi _1 &{} \cos \phi _1 &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right] , \end{aligned}$$

    where,

    $$\begin{aligned} \cos \phi _1 = p^{x}_n /\sqrt{(p_n^{x})^2+(p_n^{y})^2},\\ \sin \phi _1 = p^{y}_n /\sqrt{(p_n^{x})^2+(p_n^{y})^2}. \end{aligned}$$

    Let $\varOmega =\{\mathbf {p}_1,\ldots ,\mathbf {p}_n\}$ be the rotated architecture.

  • b. $\mathbf {p}_n \rightarrow (p^x_n,0,0)$.

    To place $\mathbf {p} _{n}$ on the x-axis, we apply a rotation operation $\mathscr {R}_{\mathring{y}}$ with associated matrix $R_{\mathring{y}}$, a clockwise-rotation matrix around the y-axis:

    $$\begin{aligned} R_{\mathring{y}}=\left[ \begin{array}{ccc} \cos \phi _2 &{} 0 &{} \sin \phi _2 \\ 0 &{} 1 &{} 0 \\ -\sin \phi _2 &{} 0 &{} \cos \phi _2 \end{array} \right] , \end{aligned}$$

    where

    $$\begin{aligned} \cos \phi _2 = p^{x}_n /\sqrt{(p_n^{x})^2+(p_n^{z})^2},\\ \sin \phi _2 = p^{z}_n /\sqrt{(p_n^{x})^2+(p_n^{z})^2}. \end{aligned}$$

    Let $\varOmega =\{\mathbf {p}_1,\ldots ,\mathbf {p}_n\}$ be the rotated architecture.

• Step 3. $\mathbf {p}_2 \rightarrow (p^x_2,0,p^z_2)$ with $p^z_2>0$.

  To place $\mathbf {p}_{2}$ on the xz-plane, we apply a counter-clockwise rotation about the x-axis $\mathscr {R}_{\mathring{x}}$ with associated matrix $R_{\mathring{x}}$:

  $$\begin{aligned} R_{\mathring{x}}=\left[ \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} \cos \phi _3 &{} -\sin \phi _3 \\ 0 &{} \sin \phi _3 &{} \cos \phi _3 \end{array} \right] , \end{aligned}$$

  where

  $$\begin{aligned} \cos \phi _3 = p^{z}_2 /\sqrt{(p_2^{y})^2+(p_2^{z})^2},\\ \sin \phi _3 = p^{y}_2 /\sqrt{(p_2^{y})^2+(p_2^{z})^2}. \end{aligned}$$

  Let $\varOmega =\{\mathbf {p}_1,\ldots ,\mathbf {p}_n\}$ be the rotated architecture.

• Step 4. $\mathbf {p}_3 \rightarrow (p^x_3,p^y_3,p^z_3)$ such that $p^y_3>0$.

  To satisfy $p _{3}^{y}>0$, if $p_3^{y} < 0$, reflect $\mathbf {p}$ as

  $$\begin{aligned} \mathscr {R}_{rfl}: {} p_i^y \rightarrow -p_i^y. \end{aligned}$$

  (19)

Let transformation $\mathscr {I}$ be the composite of the five isometric transformations, $\mathscr {R}_{\tau }$, $\mathscr {R}_{\mathring{z}}$, $\mathscr {R}_{\mathring{y}}$, $\mathscr {R}_{\mathring{x}}$, and $\mathscr {R}_{rfl}$ in the following way: $\mathscr {I}\equiv R_{rfl} \mathscr {R}_{\mathring{x}} \mathscr {R}_{\mathring{y}}\mathscr {R}_{\mathring{z}}\mathscr {R}_{\tau }$. Then $\mathscr {I}$ is an isometric (distance-preserving) transformation and the transformed coordinates satisfy the following estimability conditions on $\mathbf {p}$ : $\mathbf {p}_{1}=(0,0,0)$, $\mathbf {p}_{2}=(p_{2}^{x},0,p _{2}^{z})$ with $p_2^{z}>0$, $\mathbf {p}_{3}=(p_{3}^{x},p_{3}^{y},p_{3}^{z})$ with $p_{3}^{y}>0$, and $\mathbf {p}_{n}=(p_n^x,0,0)$ with $p_{n}^{x}>0$.

1.2 B. Leapfrog Method for Hamiltonian MCMC

In the second stage of Hamiltonian MCMC, we simultaneously update $(\mathbf {p}_i, \mathbf {v}_i)$ to obtain a proposal vector $(\mathbf {p}_i^*, \mathbf {v}_i^*)$ using a leapfrog method which involves a leap scale $\varepsilon$ and a repetition number L:

• (1) For each of x, y, z, update $v_i^{x}, v_i^{y}, v_i^{z}$ as

  $$\begin{aligned} v_i^{(.)} \leftarrow v_i^{(.)} + \frac{1}{2}\varepsilon \frac{d \log p(p_i^{(.)} |\mathbf {y},\vartheta _{-\mathbf {p}_i})}{d p_i^{(.)}}. \end{aligned}$$

  (20)

• (2) Repeat the following updates $L-1$ times:

  $$\begin{aligned} v_i^{(.)} \leftarrow v_i^{(.)} + \frac{1}{2}\varepsilon \frac{d \log p(p_i^{(.)} |\mathbf {y},\vartheta _{-\mathbf {p}_i})}{d p_i^{(.)}},\;\;\;\;\; p_i^{(.)} \leftarrow p_i^{(.)}+\varepsilon v^{(.)}_i. \end{aligned}$$

  (21)

• (3) Update $v_i^{x}, v_i^{y}, v_i^{z}$ as

  $$\begin{aligned} v_i^{(.)} \leftarrow v_i^{(.)} + \frac{1}{2}\varepsilon \frac{d \log p(p_i^{(.)} |\mathbf {y},\vartheta _{-\mathbf {p}_i})}{d p_i^{(.)}}. \end{aligned}$$

  (22)

• (4) The updated $\mathbf {p}_i$ and $\mathbf {v} _i$ constitute a proposal vector $(\mathbf {p}_i^*, \mathbf {v}_i^*)$.

  In the leapfrog method, the essential quantities to evaluate are

  $$\begin{aligned} \frac{d \log p(p_i^x |\mathbf {y},\vartheta _{-\mathbf {p}_i})}{d p_i^x} = \sum _{j \ne i }\left( y_{ij}-\lambda _{ij}\frac{e^{\lambda _{ij}}}{e^{\lambda _{ij}}-1}\right) \alpha _1\frac{p_i ^x-p_j^x}{\delta _{ij}^2},\end{aligned}$$

  (23)
  $$\begin{aligned} \frac{d \log p(p_i^y |\mathbf {y},\vartheta _{-\mathbf {p}_i})}{d p_i^y} = \sum _{j \ne i }\left( y_{ij}-\lambda _{ij}\frac{e^{\lambda _{ij}}}{e^{\lambda _{ij}}-1}\right) \alpha _1\frac{p_i ^y-p_j^y}{\delta _{ij}^2},\end{aligned}$$

  (24)
  $$\begin{aligned} \frac{d \log p(p_i^z |\mathbf {y},\vartheta _{-\mathbf {p}_i})}{d p_i^z} = \sum _{j \ne i }\left( y_{ij}-\lambda _{ij}\frac{e^{\lambda _{ij}}}{e^{\lambda _{ij}}-1}\right) \alpha _1\frac{p_i ^z-p_j^z}{\delta _{ij}^2}. \end{aligned}$$

  (25)