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J Biomed Biotechnol. 2005; 2005(2): 172–180. doi: 10.1155/JBB.2005.172. | PMCID: PMC1184052 |
Copyright © 2005 Hindawi Publishing Corporation Computational, Integrative, and Comparative Methods for the Elucidation
of Genetic Coexpression Networks Nicole E. Baldwin,1 Elissa J. Chesler,2 Stefan Kirov,3 Michael A. Langston,1* Jay R. Snoddy,3 Robert W. Williams,2 and Bing Zhang3 1Department of Computer Science, The University of Tennessee, Knoxville, TN 37996, USA 2Department of Anatomy and Neurobiology, The University of Tennessee, Memphis, TN 38163, USA 3Life Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received June 24, 2004; Revised September 12, 2004; Accepted September 14, 2004. This is an open access article distributed under the Creative
Commons Attribution License which permits unrestricted use,
distribution, and reproduction in any medium, provided the
original work is properly cited. Gene expression microarray data can be used for the assembly of
genetic coexpression network graphs. Using mRNA samples obtained
from recombinant inbred Mus musculus strains, it is
possible to integrate allelic variation with molecular and
higher-order phenotypes. The depth of quantitative genetic
analysis of microarray data can be vastly enhanced utilizing this
mouse resource in combination with powerful computational
algorithms, platforms, and data repositories. The resulting
network graphs transect many levels of biological scale. This
approach is illustrated with the extraction of cliques of
putatively coregulated genes and their annotation using gene
ontology analysis and cis-regulatory element discovery.
The causal basis for coregulation is detected through the use of
quantitative trait locus mapping. The purpose of this paper is to describe novel research combining
- emergent computational algorithms,
- high performance platforms and implementations,
- complex trait analysis and genetic mapping,
- integrative tools for data repository and exploration.
In this effort we employ huge datasets extracted from a panel of
recombinant inbred (RI) strains that were produced by crossing two
fully sequenced strains of C57BL/6J and DBA/2J mice [ 1].
The essential feature of these isogenic RI strains is that they
are a genetic mapping panel. They can therefore be used to
convert associative networks into causal networks. This is done
by finding those polymorphic genes that actually produce natural
endogenous variation in gene networks [ 2]. In this
regard, RI strains differ fundamentally from standard inbred
strains, knockout strains, transgenic lines and mutants. This
approach, termed quantitative trait locus (QTL) mapping, is
usually limited to a single continuously distributed trait such
as brain weight or neuron number [ 3], or a behavioral
trait such as open-field activity [ 4]. In this paper,
however, we map regulators of entire networks, clusters, and
cliques [ 5]. We employ combinatorial algorithms and graph theory to reduce the
high dimensionality of this megavariate data. Advances in clique
finding algorithms generate highly distiled gene sets, which we
interpret using novel, integrative bioinformatics resources. See
. Tools of choice include GeneKeyDB
[ http://genereg.ornl.gov/gkdb], WebQTL [ 6], and
GoTreeMachine (GOTM) [ 7]. Experimental design Microarrays provide an extraordinarily efficient tool to obtain
very large numbers of quantitative assays from tissue samples. For
example, using the Affymetrix M430 arrays one can obtain
approximately 45  000 measurements of relative mRNA abundance
from a whole tissue such as the brain or from a single cell
population, such as hematopoietic stem cells. In much of our
recent work we have used the Affymetrix U74Av2 array to estimate
the abundance of 12 422 transcripts from the mouse brain. The
design of our experiments is quite simple. We extract mRNA from
three litter-mate mice of the same strain and sex, pool the mRNA,
and hybridize the sample to the microarray. We do this three times
for each strain and sex (independent biological replicates). There
is no intentional experimental manipulation of the animals or
strains of mice. The essential feature to note in our experimental
design is that the isogenic strains of mice that we study are all
related yet genetically unique from one another. RI strains These related strains of mice collectively form what is called a
“mapping panel.” The strain set that we use is called the BXD
mapping panel because all of the 32 strains originate from the
same two original progenitor strains: C57BL/6J (the B strain) and
DBA/2J (the D strain). The 32 derivative BXD strains are genetic
mosaics of the two parental strains. If one were to pick one of
the 32 strains at random and examine a piece of one particular
chromosome, there would be an approximately 0.5 probability of
that piece having descended down through the generations from the
B or the D parental strain. If one looked at the same part of a
particular chromosome in all the 32 strains one would end up
with a vector of genotypes. For example, the tip of chromosome 1
of BXD strains 11 through 16 might read BDBBDD. Thus there are
2 32 or 4.29 × 10 9 possible combinations of these
vectors of genotypes. The chromosomes of individual BXD strains
actually consist of very long stretches of B-type or D-type
chromosomes. The average stretch is almost 50 million base pairs
long. The entire set of 32 BXD strains incorporate sufficient
recombinations between the parental chromosomes to encode a
total of about 2 11 locations across the mouse genome. This
means that in the best case one can only specify locations to
about 1.27 × 10 6bp. An amount 1.27 million bp will
typically contain 17 genes. (Of course, the locations of these
recombinations is close to a random Poisson process.) Unlike other recombinant cross progeny used for QTL analysis, all
of the BXD strains are fully inbred. To make these RI strains,
full siblings were mated successively for 20 generations to
produce each of the 32 strains. This has been an expensive
process that has made several strain sets available to the
research community by commercial suppliers (The Jackson
Laboratory, www.jax.org) or the originating
laboratory. Making fully RI strains from a crossbreeding between
the C57BL/6J and DBA/2J parental strains has taken about eight
years. These strains have been used for over 20 years for the
detection of QTLs in a wide range of phenotypes [ 8].
Additional 45 strains have been generated recently [ 9]. Finding the genetic regulatory locus There are numerous genetic polymorphisms (allelic variants) that
exist between the two parental strains. As an illustrative
example, consider two alleles of a gene coding for a product that
is absolutely required to deposit pigmentation in the hair and
eye. Further let us assume that these two alleles act as a
digital switch: the B allele inherited from C57BL/6J is the
active form and the D allele inherited from DBA/2J is the
inactive form. In the D state, the mice are albinos; in the B state, they are
normally pigmented. The vector of this phenotype across the
strains might look PWWWWP (P, pigmented; W, white) for strains
11 through 16. A simple comparison of this vector of
phenotypes to the vector of genotypes on the tip of chromosome 1
(BDBBDD) clearly rules out this location since the vectors do not
match particularly well. A vector of genotypes on chromosome 7
at 77 million base pairs (Mb), however, is a perfect fit:
BDDDDB. Depending on the coding convention that we use, this will
give a correlation either of 1 or of −1. This is the central
concept of mapping simple one-gene (monogenic) traits to discover
one or more genotype vectors (markers) that have tight
quantitative associations with the phenotype vectors across a
large mapping panel. Recall however that our particular
32-strain genotype vector only provides enough resolution to get
us down to a genetic neighborhood containing about 17 genes. We
call this a genetic locus (sometimes called a gene
locus), although we have to remember that we cannot yet assert
which gene in this locus is actually the pigmentation switch. Up to this point we have considered a trait that can be easily
dichotomized. The vast majority of traits in which we are
interested, however, are spread continuously over a broad range
of values that often approximates a Gaussian distribution. These
traits are frequently controled by more than a single genetic
locus. Furthermore, environmental factors typically introduce a
complementary nongenetic source of variance to a trait measured
across a genetically diverse group of individuals. Consider, for
example, body weight. This is a classic example of a complex
highly variable population trait that is due to a multifactorial
admixture of genetic factors, environmental factors, and
interactions between genes and environment. Even a trait such as
the amount of mRNA expressed in the brains of mice and measured
using microarrays is a very complex trait. We refer the
interested reader to our previous work [ 5, 7] for
more information on this subject. The abundance of mRNA is
influenced by rates of transcription, rates of splicing and
degradation, stages of the circadian cycle, and a variety of other
environmental factors. Many of these influences on transcript
abundance exert their effects via the actions of other genes. QTL
mapping of mRNA abundance allows one to detect these genetic
sources of variation in gene
expression [ 5, 7, 10, 11]. A clique-centric approach Current high-throughput molecular assays generate immense numbers
of phenotypic values. Billions of individual hypotheses can be
tested from a single BXD RI transcriptome profiling experiment.
QTL mapping, however, tends to be highly focused on small sets of
traits and genes. Many public users of our data resources approach
the data with specific questions of particular gene-gene and/or
gene-phenotype relationships [ 12]. These high-dimensional
datasets are best understood when the correlated phenotypes are
determined and analyzed simultaneously. Data reduction via
automated extraction of coregulated gene sets from transcriptome
QTL data is a challenge. Given the need to analyze efficiently
tens of thousands of genes and traits, it is essential to develop
tools to extract and characterize large aggregates of genes,
QTLs, and highly variable traits. There are advantages of placing our work in a graph-theoretic
framework. This representation is known to be appropriate for
probing and determining the structure of biological networks
including the extraction of evolutionarily conserved modules of
coexpressed genes. See, for example, [ 13, 14, 15]. A major
computational bottleneck in our efforts to identify sets of
putatively coregulated genes is the search for cliques, a classic
graph-theoretic problem. Here a gene is denoted by a vertex, and a
coexpression value is represented by the weight placed on an edge
joining a pair of vertices. Clique is widely known for its
application in a variety of combinatorial settings, a great number
of which are relevant to computational molecular biology.
See, for example, [ 16]. A considerable amount of effort has
been devoted to solving clique efficiently. An excellent survey
can be found in [ 17]. In the context of microarray analysis, our approach can be viewed
as a form of clustering. A wealth of clustering approaches has
been proposed. See [ 18, 19, 20, 21, 22] to list just a few.
Here the usual goal is to partition vertices into disjoint
subsets, so that the genes that correspond to the vertices within
each subset display some measure of homogeneity. An advantage
clique that holds over most traditional clustering methods is that
cliques need not be disjoint. A vertex can reside in more than one
(maximum or maximal) clique, just as a gene product can be
involved in more than one regulatory network. There are recent
clustering techniques, for example those employing factor analysis
[ 23], that do not require exclusive cluster membership for
single genes. Unfortunately, these tend to produce biologically
uninterpretable factors without the incorporation of prior
biological information [ 24]. Clique makes no such demand.
Another advantage of clique is the purity of the categories it
generates. There is considerable interest in solving the dense
k-subgraph problem [ 25]. Here the focus is on a cluster's
edge density, also referred to as clustering coefficient,
curvature, and even cliquishness [ 26, 27]. In this respect,
clique is the “gold standard.” A cluster's edge density is
maximized with clique by definition. The inputs to clique are an undirected graph G with n
vertices, and a parameter k ≤ n. The question asked is
whether G contains a clique of size k, that is, a subgraph
isomorphic to Kk, the complete graph on K vertices. The
importance of Kk lies in the fact that each and every pair of
its vertices is joined by an edge. Subgraph isomorphism, clique
in particular, is  -complete. From this it follows that
there is no known algorithm for deciding clique that runs in time
polynomial in the size of the input. One could of course solve
clique by generating and checking all  candidate
solutions. But this brute force approach requires O(nk) time,
and is thus prohibitively slow, even for problem instances of
only modest size. Our methods are employed as illustrated in .
We will concentrate our discussion on the classic maximum clique
problem. Of course we also must handle the related problem of
generating all maximal cliques once a suitable
threshold has been chosen, which is itself often a function of
maximum clique size. There are a variety of other issues dealing
with preprocessing and postprocessing. Although we do not
explicitly deal with them in the present paper, they are for the
most part quite easily handled and are dwarfed by the
computational complexity of the fundamental clique problem at the
heart of our method. Fixed-parameter tractability The origins of fixed-parameter tractability (FPT) can be
traced at least as far back as the work done to show, via the
graph minor Theorem, that a variety of parameterized problems are
tractable when the relevant input parameter is fixed. See, for
example, [ 28, 29]. Formally, a problem is FPT if it has an
algorithm that runs in O(f(k)nc), where n is the problem
size, k is the input parameter, and c is a constant
independent of both n and k [ 30]. Unfortunately, clique
is not FPT unless the  hierarchy collapses. (The
 hierarchy, whose lowest level is FPT, can be viewed
as a fixed-parameter analog of the polynomial hierarchy, whose
lowest level is  .) Thus we focus instead on clique's
complementary dual, the vertex cover problem. Consider
 , the complement of G. (  has the
same vertex set as G, but edges present in G are absent in
 and vice versa.) As with clique, the inputs to
vertex cover are an undirected graph G with n vertices, and a
parameter k ≤ n. The question now asked is whether G
contains a set C of k vertices that covers every edge in G,
where an edge is said to be covered if either or both of its
endpoints are in C. Like clique, vertex cover is  -complete. Unlike clique, however, vertex cover is also FPT.
The crucial observation here is this: a vertex cover of size k
in  turns out to be exactly the complement of a
clique of size n − k in G. Thus, we search for a minimum vertex
cover in  , thereby finding the desired maximum
clique in G. Currently, the fastest known vertex cover algorithm
runs in O(1.2852 k + kn) time [ 31]. Contrast this with
O(nk). The requisite exponential growth (assuming
 ≠  ) is therefore reduced to a mere
additive term. Kernelization, branching, parallelization, and load balancing The initial goal is to reduce an arbitrary input instance down to
a relatively small computational kernel, then decomposing
it so that an efficient, systematic search can be conducted.
Attaining a kernel whose size is quadratic in k is relatively
straightforward [ 32]. Ensuring a kernel whose size is linear
in k has until recently required much more powerful and
considerably slower methods that rely on linear programming
relaxation [ 33, 34]. In [ 35], we introduced and analyzed a new technique, termed
crown reduction. A crown is an ordered pair
( I, H) of subsets of vertices from G that satisfies the
following criteria: (1)  is an independent set
of G, (2) H = N(I), and (3) there exists a matching M on the
edges connecting I and H such that all elements of H are
matched. H is called the head of the crown. The
width of the crown is |H|. This notion is depicted in
. Theorem (see[35]). Any graph G can be decomposed into a crown (I, H) for which H
contains a minimum-size vertex cover of G and so that |H| ≤
3k. Moreover, the decomposition can be accomplished in
O(n5/2) time. The problem now becomes one of exploring the kernel efficiently. A
branching process is carried out using a binary search tree.
Internal nodes represent choices; leaves denote candidate
solutions. Subtrees spawned off at each level can be explored in
parallel. The best results have generally been obtained with
minimal intervention, in the extreme case launching secure shells
(SSHs) [ 36]. To maintain scalability as datasets grow in size
and as more machines are brought on line, some form of dynamic
load balancing is generally required. We have implemented such a
scheme using sockets and process-independent forking. Results on
32–64 processors in the context of motif discovery are
reported in [ 37]. Large-scale testing using immense genomic
and proteomic datasets are reported in [ 38]. SAMPLE COMPUTATIONAL RESULTS We are now able to solve real, nonsynthetic instances of clique
on graphs whose vertices number in the thousands. (Just imagine a
straightforward O(nk) algorithm on problems of that size!) To illustrate, we recently solved a problem on Mus
musculus neurogenetic microarray data with 12 422 vertices
(probe sets). With expression values normalized to [0, 1] and the
threshold set at 0.5, the clique we returned (via vertex cover)
denoted a set of 369 genes that appear experimentally to be
coregulated. This took a few days to solve even with our best
current methods. Yet solving it at all was probably unthinkable
just a short time ago. After iterating across several threshold
choices, a value of 0.85 was selected for detailed study. For
this graph, G, the maximum clique size is 17. Because it is
difficult to visualize G, we employ a clique intersection graph,
CG, as follows. Each maximal clique of size 15 or more in G
is represented by a vertex in CG. An edge connects a pair of
vertices in CG if and only if the intersection of the
corresponding cliques in G contains at least 13 members. CG
is depicted in , with vertices representing
cliques of size 15 (in green), cliques of size 16 (in black),
and cliques of size 17 (in red). One rather surprising result is
that the gene found most often across large maximal cliques is
Veli3 (aka Lin7c). This appears not to be due to
some so-called “housekeeping” function, but instead because the
relatively unstudied Veli3 is in fact central to
neurological function [ 39, 40]. CLIQUES OF HIGHLY CORRELATED TRANSCRIPTS AND BEHAVIORAL PHENOTYPES We can infer that coexpression of genes in mice of common genetic
background is due to a shared regulatory mechanism, because the
correlation is between trait means from different lines of mice,
rather than from within an experimental group. Clique membership
alone does not tell us anything about the basis of common genetic
regulation. By combining clique data with QTL analysis, the
regulatory loci underlying the shared genetic mediation of gene
expression can be identified. This allows us to determine the
impact of genetic variability in gene expression on other
biological processes. Using the aforementioned stringent
correlation threshold of 0.85, the most highly connected
transcript identified was that of Veli3. One maximal
clique of seventeen highly associated transcript abundances
includes several nuclear proteins. A single principal component of
these transcripts accounts for 95% of the total genetic sample
variance. No single QTL can be found for the members of this clique, but a
multiple QTL mapping analysis reveals an interacting pair of loci
on chromosomes 12 and X, at markers D12Mit46
(29.163 Mb) and DXMit117 (110.670 Mb). See
, which shows the results from a search for
pairs of genetic loci that modulate expression of a clique.
Chromosomes 12, 19, and X are shown. Likelihood
ratio statistics for multiple QTL models are plotted on the
pseudocolor scale. The upper left triangle shows fit results for
an interaction model, and the lower right triangle shows fit
results for a model containing both additive and interaction
effects of the two loci. The joint model including markers on 12
and X is significant ( P < .05) by permutation analysis. A D allele
at both loci results in low levels of the phenotype and a B allele
at both loci results in a high level of the phenotype. The
chromosome 12 locus is the physical location of two clique
members: B cell receptor associated protein Bcap29, and myelin
transcription factor 1-like protein, Myt1l. An interesting
functional and positional candidate at the Chromosome X locus is
integral membrane protein 1, Itm1. While this is not a member of
the clique we are analyzing here, it does frequently cooccur
along with Veli3 in many maximal cliques. In addition to tens of thousands of transcript traits, we have
assembled a database of over 600 organismic phenotypes,
including many morphometric traits. An understanding of the
genetic control of these phenotypes can help explain their
evolution. In the present example, we have found the previously
mentioned clique to associate with behavioral and metabolic
phenotypes. This clique correlates with both midbrain iron levels,
and locomotor behavior. Interestingly, one of the clique members,
Gs2na (GS2 nuclear autoantigen), that, at 46.048 Mb
on chromosome 12, is a little too far afield to be a positional
QTL candidate gene, is a striatin family member and the negative
correlation we observe between clique expression and locomotor
behavior is consistent with literature reports of locomotor
impairment associated with decreased levels of striatin [ 41].
At this point research becomes hypothesis driven; indeed, the
result of this collaborative analysis is a simple testable
hypothesis, extracted from many billions of data relations. We are
now in the process of evaluating the hypothesis that genetic
variation in iron metabolism influences expression of the
Veli3 clique members in the brain and consequently
affects locomotor activity. INTEGRATIVE GENOMIC DATA MINING GeneKeyDB High-throughput, high volume data like these gene expression data
from genetically variant mice should be examined in a biological
context. The subsets of interesting genes must be analyzed, in
part, by using existing information that describe the role these
genes play in biological processes. When computing and navigating
these data in terms of graphs and networks, we need to have a way
to manipulate various kinds of metadata about sets of genes and
gene products. We have developed several such tools for genes and gene products
that are discovered from the clique and QTL data analysis. Most
gene-centered data resources that are generally available for
retrieving metadata about genes are displayed and manipulated in a
one-gene-at-a-time format (eg, Entrez Gene). We have developed a
lightweight data mining environment that allows the automated
integration of various types of data about sets of genes. This
environment is called GeneKeyDB. This system includes
metadata from GenBank, Entrez Gene, Ensembl, and several other
well-established biological databases. GeneKeyDB uses a relational
database backend to facilitate interactions between tools and
data. Among other functions, GeneKeyDB automatically converts the
different database identifiers from these different databases. It
can, for example, start with GenBank cDNA identifiers, locate the
“sequence feature” information from genome sequence data
entries, and assist in retrieving sequences for detailed analyses.
GeneKeyDB can also obtain various kinds of homologs, functional
annotation, or other attributes of genes and gene products.
Furthermore, it serves as a repository for results that are
created by our computational tools. We have devised two types of computational analyses that are
supported by the underlying GeneKeyDB system. GoTreeMachine Gene ontology (GO) produces structured, precisely defined,
common, controled vocabulary for describing the roles of genes
and gene products [ 42]. GO has been used
frequently in the functional profiling of high-throughput data.
We have developed a web-based tool, GOTM, for the analysis and
visualization of sets of genes using GO hierarchies
[ 7]. Besides being a stand-alone functional profiling
tool, GOTM can work with other computational tools for gene set
centered integrated analysis. GOTM has been employed in various
ways in this respect. This includes WebQTL's use of GOTM to narrow
down candidate gene lists and generate functional profiles for
genes in a relevance network or genes correlated to complex
phenotypes. We use ontology analysis to evaluate the functional significance
of the cliques found by our graph algorithms, and prioritize the
cliques for further study. depicts
a clique of size eight that was detected within the gene
coexpression network constructed using the microarray data from
the RI mouse lines. The five green vertices denote genes that
belong to the GO functional category of “DNA binding.” The red
vertices denote genes that either have no annotation or are
annotated as function “unknown.” If we randomly pick five genes
from all annotated genes on the microarray, the expected frequency
of genes in the category DNA binding is only 0.9. The chance of
finding all five genes in the category DNA binding is P = .00051
as calculated by the hypergeometric test implemented in GOTM. Out
of the 5227 maximal cliques we generated, ontology analysis has
detected a total of 342 of them that are significantly enriched
in one or more GO categories (P < .01). The clique shown in
has a P value less than .001, and is one
of several cliques we are studying. Note the presence of the gene
Veli3. Batch sequence analysis We are also deploying integrative methods that attempt to predict
cis-regulatory elements (CREs) in the upstream regions
from sets of genes that are putatively coregulated. These CREs
are thought to be the DNA sites to which protein regulatory
transcription factors preferentially bind in promoters or
enhancers and exert regulatory control of gene expression. We are
combining a number of analyses to look at sets of CREs that are
found in a subset of genes that seems to show strong coregulation
and are consistent with the clique and QTL data. We have assembled a pipeline, batch sequence analysis (BSA), that
can retrieve the sequence data for the target genes and their
orthologous counterparts in other chordate organisms. This
pipeline carries out a number of processes that enable us to use
both coregulated gene sets and phylogenetic footprinting in an
integrated pipeline to identify putative CREs. An important
advantage of the pipeline is its ability to define the
evolutionarily conserved non-coding sequences, which are thought
to contain most of the CREs [ 43]. This should substantially
reduce the noise levels. BSA can be carried out in a high
throughput, automated process because of the underlying GeneKeyDB
infrastructure. BSA is routinely using both multiple Em for motif
elicitation (MEME) and motif alignment and search tool (MAST) as
part of the sequence analysis, but other motif finding and
searching methods are under development. A set of CRE motifs can
be found in cliques or other interesting subsets of genes with
motif searches (like MEME). We can then use MAST or similar
searching tools to take those sets of putative CREs to do a global
search for all possible targets in a database that contains
promoter sequences from all human, mouse, and rat genes. The
latter step could help define new genes that are targets of a gene
regulatory network that were not initially identified. The BSA
pipeline stores its results in the GeneKeyDB relational database. SUMMARY AND DIRECTIONS FOR FUTURE RESEARCH Our current work demonstrates the use of clique to extract signal
from large genetic correlation matrices. We also employ
genome-scale tools to interpret the shared molecular function,
biological process, cellular localization, and sequence motifs of
clique members. Despite what has been accomplished in the BXD
lines, the size of existing RI strain sets limits the power and
resolution of this technique. The Complex Trait Consortium plans
to expand this set with the development of a 1024 RI strain
panel [ 44]. The creation of this resource will greatly
increase the depth of our analysis. The breadth of the analysis
can be expanded almost indefinitely. Although the work we have
described here has been restricted to the analysis of gene
expression microarray phenotypes, any attribute of these strains
that can be measured can in principle be incorporated into the
genetic correlation matrix. We already have a wealth of data on
microscale and macroscale biological phenotypes ranging from
cellular responses to behavior. Novel high-throughput molecular
phenotypes will greatly expand this collection. To accomodate such
vast increases in data dimensionality, we are currently in the
process of porting our codes to supercomputers at Oak Ridge
National Laboratory (ORNL) (Tennessee, USA). These are difficult
tasks indeed, given the many novel features of our algorithms.
Great care is required to manage processor and memory resources.
Load balancing can be especially problematic [ 37]. Initial
targets include a 256-node SGI Altix and a 256-node Cray X1.
In the longer term, we aim to employ the tremendously more
powerful machines now under construction and awarded to ORNL in
the recent competition to build the nation's next leadership-class
computing facility for science. We believe that with our
algorithms and these platforms we can solve the problem instances
previously considered hopelessly out of reach. This research has been supported in part by the National Science
Foundation under grants CCR–0075792 and CCR–0311500; by the
National Institute of Mental Health, National Institute on Drug
Abuse, and the National Science Foundation under award
P20–MH–62009; by the National Institute on Alcohol Abuse and
Alcoholism under INIA grants P01–Da015027, U01–AA013512–02,
U01–AA13499, and U24–AA13513; by the Office of Naval Research
under grant N00014–01–1–0608; and by the Department of Energy
under contracts DE–AC05–00OR33735 and DE–AC05–4000029264.
We wish to thank Drs. Lu Lu
and Kenneth Manly for helping develop
datasets, analytic tools, and the WebQTL website,
www.WebQTL.org . We also wish to express
our appreciation to Drs. Mike Fellows and Henry Suters for
helping develop fast kernelization alternatives and to Dr. Faisal
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