A nonparametric empirical Bayes approach to joint modeling of multiple sources of genomic data

Abstract

With the rapid accumulation of various high-throughput genomic and proteomic data, one is compelled to develop new statistical methods that can take advantage of existing multiple sources of data. In our motivating example, a chromatin-immunoprecipitation (ChIP) microarray experiment was conducted to detect binding target genes of a broad transcription regulator, leucine responsive regulatory protein (Lrp) in E. coli. In addition, a cDNA microarray dataset is avail-able to compare gene expression of the wild type with that of a mutant with the Lrp gene deleted in E. coli. It is biologically reasonable to assume that the genes with altered expression are more likely to be regulated by Lrp than those with no expres-sion change. Hence we aim to borrow information in the gene expression data to increase statistical power to detect the binding targets of Lrp. We propose a novel joint model for protein-DNA binding data and gene expression data; under mild modeling assumptions, it is shown that the method is optimal, equivalent to a joint likelihood ratio test. We compare the joint modeling with two existing methods of combining separate analyses. We adopt a nonparametric empirical Bayes (EB) method to draw statistical inference in the joint model; in particular, we propose a new method, maximum likelihood conditional on the binding data, to estimate two prior probabilities for the expression data, which are non-identifiable based on the expression data alone. We use simulated data to demonstrate the improved performance of the joint modeling over other approaches. Application to the Lrp data also shows better performance of the joint modeling than that of analyzing the binding data alone.

Full text

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Statistica Sinica 18(2008), 709-729
A NONPARAMETRIC EMPIRICAL BAYES APPROACH TO
JOINT MODELING OF MULTIPLE SOURCES
OF GENOMIC DATA
Wei Pan1, Kyeong S. Jeong2, Yang Xie3and Arkady Khodursky1
1University of Minnesota,2University of California, Los Angeles
and3University of Texas Southwestern Medical Center
Abstract: With the rapid accumulation of various high-throughput genomic and
proteomic data, one is compelled to develop new statistical methods that can
take advantage of existing multiple sources of data. In our motivating example,
a chromatin-immunoprecipitation (ChIP) microarray experiment was conducted to
detect binding target genes of a broad transcription regulator, leucine responsive
regulatory protein (Lrp) in E. coli. In addition, a cDNA microarray dataset is avail-
able to compare gene expression of the wild type with that of a mutant with the Lrp
gene deleted in E. coli. It is biologically reasonable to assume that the genes with
altered expression are more likely to be regulated by Lrp than those with no expres-
sion change. Hence we aim to borrow information in the gene expression data to
increase statistical power to detect the binding targets of Lrp. We propose a novel
joint model for protein-DNA binding data and gene expression data; under mild
modeling assumptions, it is shown that the method is optimal, equivalent to a joint
likelihood ratio test. We compare the joint modeling with two existing methods
of combining separate analyses. We adopt a nonparametric empirical Bayes (EB)
method to draw statistical inference in the joint model; in particular, we propose
a new method, maximum likelihood conditional on the binding data, to estimate
two prior probabilities for the expression data, which are non-identifiable based on
the expression data alone. We use simulated data to demonstrate the improved
performance of the joint modeling over other approaches. Application to the Lrp
data also shows better performance of the joint modeling than that of analyzing
the binding data alone.
Key words and phrases: ChIP-chip, computational biology, false discovery rate,
gene expression, Lrp, microarray.
1. Introduction
High-throughput biotechnologies, such as microarrays, have generated large
amounts and various types of genomic and proteomic data. A widespread use
of microarray experiments is to monitor genome-wide gene expression. Gene
expression or transcription is the process of genetic information flow from DNA
sequence to messenger RNA (mRNA). Although any cell of an organism contains

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710LING CHEN AND ROBERT W. JERNIGAN
all the necessary DNA information for gene expression, obviously, at any given
moment, not all the genes in the cell are equally expressed, or even expressed
at all: some genes are switched on while others are off, and those switched on
may have different expression levels. Gene expression largely depends on phys-
iological and environmental conditions of the cell at the moment. Biologically,
a fundamental question is how gene expression is regulated. A general mecha-
nism is through some regulatory proteins called transcription factors (TFs): a
TF binds to one or more specific DNA subsequences in a gene’s promoter re-
gion, called binding sites or motifs, then works with other TFs to stimulate
or inhibit the expression of the gene. Although significant progress has been
made in understanding a few specific examples, as well as the general aspect of
gene transcription regulation, it remains largely unknown which TFs regulate
which genes. The biological goal of this work is to discover the target genes of
a TF, not necessarily its binding sites or motifs. The most popular approach to
identifying bindings sites of a TF is by computationally predicting motifs (i.e.,
specific DNA sequences that a TF binds to) through DNA sequence alignment
(e.g., Liu, Neuwald and Lawrence (1999)). However, presence of a motif in a
gene’s control region does not necessarily imply that the TF indeed binds to
the site in vivo. A new application of microarray technology is to identify in
vivo genome-wide binding locations of a TF via chromatin-immunoprecipitation
(ChIP) (e.g., Ren et al. (2000)). Because the resulting DNA-protein binding
data are in the usual format of cDNA microarray gene expression data, it is
technically possible to apply any of many existing statistical methods of de-
tecting differential gene expression to binding data (Lee et al. (2002)), see Pan
(2002) and Smyth, Yang and Speed (2003) for reviews on statistical analysis of
gene expression data. Due to high noise in microarray data and typically few
replicates, any statistical method being applied may yield results with relatively
high false positives or high false negatives. To maximize statistical power, we can
take advantage of the existence of other sources of data, and thus their contained
information. One source is gene expression data. In our motivating example,
on the one hand we have a DNA-protein binding data set to detect the binding
targets of a broad transcription regulator, leucine responsive regulatory protein
(Lrp), on the other hand microarray experiments were done to survey gene ex-
pression changes for the wild type, as compared to a mutant with the Lrp gene
knocked-out (Tani, Khodursky, Blumenthal, Brown, and Matthews (2002)). It
is intuitively reasonable that, in such experiments, the genes with expression
changes are more likely to be regulated by the Lrp than those with no altered ex-
pression. Therefore, it is natural to conduct a combined analysis of DNA binding
data and expression data.
A simple way is to analyze DNA binding data and expression data sepa-
rately with each resulting in a list of genes, then take an intersection of the two

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JOINT MODELING OF MULTIPLE SOURCES OF GENOMIC DATA711
lists and identify the common set of genes as the targets (Ren et al. (2000)).
This approach can reduce the false positive number, but is likely to yield a high
false negative number: many target genes may be missing from the common
set; it is even possible that there is no intersection between the two lists of the
genes. Existing integrated analysis strategies include regressing one source of
data on the other (e.g., Conlon, Liu, Lieb and Liu (2003), Zhao, Wu and Sun
(2003), Gao, Foat and Bussemaker (2004) and Sun, Carroll and Zhao (2006)),
using one source to validate the other (e.g., von Mering et al. (2002)), sequen-
tial methods of using one source to generate hypotheses or priors for the fol-
lowing analysis on the second source of data (Liu, Brutlag and Liu (2002) and
Xie, Pan, Jeong and Khodursky (2007)), and combining (e.g., taking an intersec-
tion of) the results of separate analyses on individual sources of data (Ren et al.
(2000) and Xiao and Pan (2005)). A potentially more powerful, though more
challenging, approach is to jointly model the two sources of data to improve
the statistical power for new discoveries, as advocated and demonstrated by
Holmes and Bruno (2000) for integrating DNA motif-finding and expression pro-
file clustering.
Here we propose a novel joint model for protein-DNA binding data and gene
expression data such that information in gene expression data is combined to
nonparametrically infer the binding targets of a TF; to our knowledge, this is the
first endeavor to do so in the literature. The basic idea is to exploit the correlation
between TF binding and altered expression of a gene. We point out a connection
of our proposal with the likelihood ratio test, thus establishing the optimality of
our proposal. We extend a nonparametric empirical Bayes (EB) approach, pro-
posed by Efron, Tibshirani, Storey and Tusher (2001) for gene expression data
alone, to the joint model; in particular, we propose a novel conditional likelihood
method to estimate two key mixing parameters for the expression data. That
are not identifiable based on the expression data alone. We compare our pro-
posal with other methods based on separate analyses of the two types of data:
an intersection method, and an analog of Fisher’s method to combine the results
from separate analyses. Because the two sources of data support two different
hypotheses, we clarify why such combined analyses are meaningful. Using both
simulated and real data, we demonstrate that the joint modeling improves over
the other methods.
2. Joint Modeling of Binding Data and Expression Data
2.1. Data and analysis goal
We assume throughout that we have DNA binding data of a TF (e.g., Lrp) as
the primary data, with secondary data drawn from a gene expression experiment

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712LING CHEN AND ROBERT W. JERNIGAN
comparing a wild type against a mutant with partial or full loss of function of
the TF (e.g., the deletion of the TF gene), both in the format of cDNA array
data. Specifically, suppose that M1ijand M2ijare the log ratios of the intensi-
ties of the two channels for gene i on array j for the binding data and expression
data, respectively, i = 1,...,G, j = 1,...,n1, in the binding experiment, and
j = 1,...,n2in the expression experiment. Suppose that necessary data normal-
ization has been accomplished. The central goal is to identify the genes bound
by the TF with E(M1ij) ?= 0. With the expression data alone, we can only detect
differentially expressed (DE) genes with E(M2ij) ?= 0. Note that the expression
data cannot give unambiguous evidence for DNA-protein binding because a DE
gene may or may not be bound by the TF.
For our purposes, for each gene i we construct two test statistics, Xiand Yi
based on the binding data and the expression data respectively. Although any
test statistic can in principle be used, in this paper we consider the use of the
SAM statistic, a regularized t-statistic (Tusher, Tibshirani and Chu (2001)), due
to its simplicity and good performance (Xie, Jeong, Pan, Khodursky and Carlin
(2004)). Specifically, suppose that the sample mean and the sample variance for
gene i are¯
Xi=¯M1i/(S1i+ s10), where s10= median(S11,...,S1n1) is used to stabilize the
denominator. There are Bayesian justifications for the use of s10(Baldi and Long
(2001), Wright and Simon (2003) and Cui, Hwang, Qiu, Blades and Churchill
(2004)). Similarly we define Yifor i = 1,...,G.
M1i=?n1
j=1M1ij/n1and S2
1i=?n1
j=1(M1ij−¯
M1i)2/(n1− 1). Then
2.2. A joint model
Now we propose a joint model for (Xi,Yi), the test statistics calculated from
the two types of data, respectively. Define Bias the indicator of whether gene
i is indeed bound by the TF. Our goal is to identify all the genes (i.e., i’s) with
Bi= 1. We assume that Bi∼ Bern(π), independently.
Our joint model consists of two mixture models for binding data and ex-
pression data, respectively. First, we specify a mixture model for the binding
data:
f(Xi) = (1 − π)f0(Xi) + πf1(Xi),(1)
where f0is the distribution of Xifor the genes with no binding, f1is that for the
bound genes, and π is a prior probability of any gene’s being bound by the TF.
Second, we specify two mixture models for the conditional distribution of Yi:
f(Yi|Bi= 1) = p1g1(Yi) + (1 − p1)g0(Yi),
(2)
f(Yi|Bi= 0) = p0g1(Yi) + (1 − p0)g0(Yi).

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JOINT MODELING OF MULTIPLE SOURCES OF GENOMIC DATA 713
This implies a two-component mixture model for the marginal distribution of Yi:
g(Yi) = (1 − πg)g0(Yi) + πgg1(Yi), (3)
where g0is the distribution of Yifor genes with no expression changes, while g1
is for genes with altered expression, and πg= πp1+ (1 − π)p0.
The main motivation of the model is the following. Intuitively, if a gene is
regulated by the TF, there is a higher chance (probability p1) that the gene is
DE with a distribution g1; otherwise, there is only a smaller chance (probability
p0) that the gene’s expression level will be changed. The difference between p1
and p0measures the amount of binding information contained in the expression
data; in particular, if p1= p0, then the binding and expression are completely
independent. Note that, because binding is neither a sufficient nor a necessary
condition for expression change, the above mixture model takes account of the
possibility of a non-bound gene’s having expression change.
We also assume that, conditional on the binding status Bi, a gene’s bind-
ing statistic Xi and expression statistic Yi are independent. Hence, the joint
distribution of Xiand Yiis
f(Xi,Yi) = f(Xi|Bi= 1)f(Yi|Bi= 1)π + f(Xi|Bi= 0)f(Yi|Bi= 0)(1 − π),
where f(Xi|Bi= 1) = f1(Xi) and f(Xi|Bi= 0) = f0(Xi). Using Bayes Theorem,
we have the posterior probability of gene i being a target as
Pr(Bi= 1|Xi,Yi) =πf(Xi|Bi= 1)f(Yi|Bi= 1)
f(Xi,Yi)
, (4)
which is used to draw inference on whether gene i is a binding site of the TF.
Remark 1. Optimality. From (3), we have
Pr(Bi=1|Xi,Yi)=
π
π+(1−π)f(Xi|Bi=0)
f(Xi|Bi=1)
f(Yi|Bi=0)
f(Yi|Bi=1)
=
π
π+
1−π
LRT(Bi|Xi)LRT(Bi|Yi)
,
where LRT(Bi|Xi) = f(Xi|Bi= 1)/f(Xi|Bi= 0) and LRT(Bi|Yi) = f(Yi|Bi=
1)/f(Yi|Bi= 0) are the two likelihood ratio test (LRT) statistics for testing H0i:
Bi= 0 vs H1i: Bi= 1 based on the binding data Xiand expression data Yi,
respectively. On the other hand, the corresponding LRT statistic based on both
the binding and expression data is
LRT(Bi|Xi,Yi) =f(Xi,Yi|Bi= 1)
f(Xi,Yi|Bi= 0)= LRT(Bi|Xi)LRT(Bi|Yi),