Semi-nonparametric Modeling of Topological Domain Formation from Epigenetic Data

Abstract

Hi-C experiments capturing the 3D genome architecture have led to the discovery of topologically-associated domains (TADs) that form an important part of the 3D genome organization and appear to play a role in gene regulation and other functions. Several histone modifications have been independently suggested as the possible explanations of TAD formation, but their combinatorial effects on domain formation remain poorly understood at a global scale. Here, we propose a convex semi-nonparametric approach called nTDP based on Bernstein polynomials to explore the joint effects of histone markers on TAD formation as well as predict TADs solely from the histone data. We find a small subset of modifications to be predictive of TADs across species. By inferring TADs using our trained model, we are able to predict TADs across different species and cell types, without the use of Hi-C data, suggesting their effect is conserved. This work provides the first comprehensive joint model of the effect histone markers on domain formation.

Appendix

$R(f^{p}_{m})$ can be written more explicitly as in (18) according to [19]:

$$\begin{aligned} \frac{\partial ^{2} f^{p}_{m}(x,\mathbf {w^{p}_{m}})}{\partial x^{2}} = A(A-1) \sum _{i=0}^{A-2} (w^{p}_{m}[i+2]-2w^{p}_{m}[i+1]+w^{p}_{m}[i]) \genfrac(){0.0pt}0{A-2}{i} x^{i}(1-x)^{A-2-i} \end{aligned}$$

(18)

which turns $R(f^{p}_{m})$ into (19):

$$\begin{aligned} \int _{0}^{1} \left( \frac{\partial ^{2} f^{p}_{m}(x)}{\partial x^{2}}\right) ^{2} dx = A^{2}(A-1)^{2} \sum _{i=0}^{A} \sum _{j=i}^{A} (w^{p}_{m}[i] w^{p}_{m}[j]) \nonumber \\ \Bigg ( \sum _{q=\overline{e}_{i}}^{\min (i,2)} \sum _{r=\overline{e}_{j}}^{\min (j,2)} (-1)^{q+r} \genfrac(){0.0pt}0{2}{q} \genfrac(){0.0pt}0{2}{r} T^{i-q}_{j-r}(x) \Bigg ) \end{aligned}$$

(19)

where $\overline{e}_{p} = \max (0,2-A+p)$, $T^{i-q}_{j-r}(x)$ is defined below and $\beta (i+j-q-r+1,2A-3-i-j+q+r)$ is the beta function:

$$\begin{aligned} T^{i-q}_{j-r}(x) = \genfrac(){0.0pt}0{A-2}{i-q} \genfrac(){0.0pt}0{A-2}{j-r} \underbrace{\int _{0}^{1} x^{i-q}(1-x)^{A-2-i+q} x^{j-r}(1-x)^{A-2-j+r} dx}_{\beta (i+j-q-r+1, 2A-3-i-j+q+r)} \end{aligned}$$

(20)

$R(f^{p}_{m})$ is convex which follows from semidefiniteness of the resulting polynomial.