Biological pathway selection through Bayesian integrative modeling

Lingling Zheng; Xiao Yan; Sunil Suchindran; Holly Dressman; John P. Chute; Joseph Lucas

doi:10.1515/sagmb-2013-0043

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Biological pathway selection through Bayesian integrative modeling

Ein Erratum zu diesem Artikel finden Sie hier: https://doi.org/10.1515/sagmb-2014-0087

Lingling Zheng , Xiao Yan , Sunil Suchindran , Holly Dressman , John P. Chute und Joseph Lucas

Veröffentlicht/Copyright: 17. Juni 2014

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Aus der Zeitschrift Statistical Applications in Genetics and Molecular Biology Band 13 Heft 4

Abstract

Pathway analysis has become a central approach to understanding the underlying biology of differentially expressed genes. As large amounts of microarray data have been accumulated in public repositories, flexible methodologies are needed to extend the analysis of simple case-control studies in order to place them in context with the vast quantities of available and highly heterogeneous data sets. To address this challenge, we have developed a two-level model, consisting of 1) a joint Bayesian factor model that integrates multiple microarray experiments and ties each factor to a predefined pathway and 2) a point mass mixture distribution that infers which factors are relevant/irrelevant to each dataset. Our method can identify pathways specific to a particular experimental trait which are concurrently induced/repressed under a variety of interventions. In this paper, we describe the model in depth and provide examples of its utility in simulations as well as real data from a study of radiation exposure. Our analysis of the radiation study leads to novel insights into the molecular basis of time- and dose- dependent response to ionizing radiation in mice peripheral blood. This broadly applicable model provides a starting point for generating specific and testable hypotheses in a pathway-centric manner.

Keywords: Bayesian joint factor analysis; pathway analysis; gene expression; integrated analysis; radiation study

Corresponding author: Lingling Zheng, Duke Univeristy – Computational Biology and Bioinformatics, Durham, North Carolina 27708, USA, Tel.: 91 944 85 862, e-mail: lz35@duke.edu; catherine.ll.zheng@gmail.com

¹
The sparsity constant τ₀ refers to the percentage of binary numbers in the pathway selection matrix γ.
²
As opposed to “partial pathways,” “intact pathways” refer to those the entire gene set of which is activated/repressed under a particular experimental perturbation.

Acknowledgments

The authors thank three anonymous reviewers and associate editor for their careful reading and constructive comments. This work was supported by funding from the grant “R01 DK089705” and Biomedical Advanced Research and Development Authority (BARDA). The research and results contained in this article are the contributions of the authors and should not be interpreted as representing the official views or policies, either expressed or implied, of the sponsor.

Appendices

updated distribution for loadings βFor factor k, let xg, d, j*=xg, d, j−μg, d−∑l≠kβg, lfl, d, j, so that xg, d, j∗~N(βg, kfk, d, j, θg, d). Therefore, Pr(βg, k|zg, k=1,−) ∝∏d=1D∏j=1NdN(xg, d, j*|βg, kfk, d, j, θg, d)N(βg, k|mg, k, ϕg, k) ∝∏d=1D∏j=1Ndexp(−(xg, d, j*−βg, kfk, d, j)22θg, d)exp(−(βg, k−mg, k)22ϕg, k) ∝exp(−βg, k22(∑d∑jfk, d, j2θg, d+1ϕg, k)+βg, k(∑d∑jxg, d, j*fk, d, jθg, d+mg, kϕg, k)) ~N(ηβ, υβ) where υβ=1/(∑d∑jfk, d, j2θg, d+1ϕg, k), ηβ=υβ(∑d∑jxg, d, j*fk, d, jθg, d+mg, kϕg, k).if z_{g, k}=0, then β_{g, k}=0.
updated distribution for m_{g, k}Pr(mg, k|−)∝N(βg, k|mg, k, ϕg, k)N(mg, k; u0, n0)∝exp(−(βg, k−mg, k)22ϕg, k)exp(−(mg, k−u0)22n0)~N(um, nm) where nm=1/(1n0+1ϕg, k), um=nm(u0n0+βg, kϕg, k).u₀ and n₀ are respectively the prior mean and variance.
updated distribution for φ_{g, k}Pr(ϕg, k|−)∝N(βg, k|mg, k, ϕg, k)Ga(ϕg, k−1; c0, d0)∝exp(−(βg, k−mg, k)22ϕg, k)ϕg, k−0.5ϕg, k−(c0−1)exp(−d0ϕg, k)~Ga(ϕg, k−1; cϕ, dϕ) where c_{_φ}=c₀+0.5, d_{_φ}=d₀+0.5×(β_{g, k}–m_{g, k})². c₀ and d₀ are the prior shape and rate parameters of the gamma distribution, respectively.
updated distribution for factor scores Ff_{k, d, j}=0 if γ_{k, d,·} =0, otherwise Pr(fk, d, j|γk, d,⋅=1,−)∝Pr(xd, j|fk, d, j,−)Pr(fk, d, j|yk, d, j, γk, d,⋅)∝N(xd, j*; βkfk, d, j, θd)N(fk, d, j; 0, 1)∝exp(−0.5×(xd, j*−βkfk, d, j)Tθd−1(xd, j*−βkfk, d, j))exp(−fk,d,j22)∝N(fk, d, j; ηf, υf) where υf=1/(1+βkTθd−1βk), ηf=υf×βkTθd−1xd, j*.
updated distribution for γ_{k, d}, Pr(γk, d,⋅|−) ∝∏j∫Pr(xd, j|γk, d,⋅, yk, d, j,−)Pr(yk, d, j|γk, d,⋅)Pr(γk, d,⋅|πk)dyk, d, j=∏j∫N(xd, j*; βkγk, d,⋅ yk, d, j, θd)N(yk, d, j; 0, 1)((1−πk)δ0+πkδ1)dyk, d, j=(1−πk)δ0∏jN(xd, j*; 0, θd)+πk∏j∫|θd|−122πexp(−0.5×(xd, j*−βkyk, d, j)T θd−1(xd, j*−βkyk, d, j))12πexp(−yk, d, j22)dyk, d, j∝(1−πk)δ0+πk∏j∫12πexp(−0.5×(yk, d, j2(βkTθd−1βk+1)−2×yk, d, jβkTθd−1xd,j*))dyk, d, j=(1−πk)δ0+πk(Vk, d)Ndexp(∑jMk, d, j22Vk, d)∏j∫exp(−(yk, d, j−Mk, d, j)22Vk, d)12πVk, ddyk, d, j where V_{k, d}=υ_f, M_{k, d, j}=η_f. Therefore, sampling γ_{k, d,·} =1 with posterior odds π^k1−π^k=πk1−πk(Vk, d)Ndexp(∑jMk, d, j22Vk, d) Otherwise, γ_{k, d,·} =0
updated distribution for π_kPr(πk|−)∝∏d∏jPr(γk, d,⋅|πk)Pr(πk|ρ)=(1−πk)N−SπkS(((−ρ)Beta(α0, κ0)+ρBeta(κ0, α0))∝(1−πk)N−SπkS(1−ρ)πkα0−1(1−πk)κ0−1+(1−πk)N−SπkSρπkκ0−1(1−πk)α0−1=(1−ρ)Beta(πk; S+α0, N−S+κ0)B(S+α0, N−S+κ0)+ρBeta(πk; S+κ0, N−S+α0)B(S+κ0, N−S+α0) Where S=Σ_dΣ_jI(γ_{k, d,·} =1), N is the total sample size. Therefore, sampling π_k from Beta(π_k; S+κ₀, N–S+α₀) with posterior oddsρ^1−ρ^=ρ1−ρB(S+κ0, N−S+α0)B(S+α0, N−S+κ0) Otherwise, π_k is sampled from Beta(π_k; S+α₀, N–S+κ₀)·κ₀ and α₀ are the prior shape parameters of the beta distribution.
updated distribution for ρPr(ρ|−)∝∏kPr(πk|ρ)Pr(ρ)=(1−ρ)K−QρQBeta(e0, l0)∝(1−ρ)K−QρQρe0−1(1−ρ)l0−1~Beta(ρ; Q+e0, K−Q+l0) where Q is the number of times π_k is sampled from Beta(π_k;S+κ₀, N–S+α₀). e₀ and l₀ are the shape parameters of the beta distribution.
updated distribution for noise variance θPr(θg, d|−)∝∏jN(xg, d, j−μg, d; βgfd, j, θg, d)Ga(θg, d−1; a0, b0)∝exp(−0.5×∑j=1Nd(xg, d, j−μg, d−βgfd, j)2θg, d)θg, d−Nd2θg, d−(a0−1)exp(−b0θg, d)~Ga(θg, d−1; aθ, bθ) where aθ=a0+Nd2, bθ=b0+∑j=1Nd(xg, d, j−μg, d−βgfd, j)22.a₀ and b₀ are the prior shape and rate parameters of the gamma distribution, respectively.
updated distribution for sample mean μPr(μg, d|−)∝∏jN(xg, d, j−μg, d; βgfd, j, θg, d)N(μg, d; m0, s0)∝exp(−0.5×∑j=1Nd(xg, d, j−μg, d−βgfd, j)2θg, d)exp(−(μg, d−m0)22s0)~N(μg, d; mμ, sμ) where sμ=1/(1s0+Ndθd), mμ=sμ×(m0s0+θg, d−1×∑j=1Nd(xg, d, j−βgfd, j)).m₀ and s₀ are respectively prior mean and variance parameters.

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The online version of this article (DOI 10.1515/sagmb-2013-0043) offers supplementary material, available to authorized users.

Article note

Availability: The source code for this model is written in MATLAB, and has been made publicly available in http://linglingzheng.com/code/

Data: The radiation data has been published, and can be found in GEO with Series record GSE52403 in http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE52403.

Published Online: 2014-6-17

Published in Print: 2014-8-1

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Bayesian joint factor analysis; pathway analysis; gene expression; integrated analysis; radiation study

Biological pathway selection through Bayesian integrative modeling

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