Communication between Layers in Biological
Transcriptional Networks
by
Alex Tsankov
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Computer Science and Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2005
c Alex Tsankov. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute
publicly paper and electronic copies of this thesis document in whole or in
part.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Electrical Engineering and Computer Science
May 19, 2005
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Moe Win
Associate Professor, Massachusetts Institute of Technology
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pamela Silver
Professor, Harvard Medical School
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Arthur C. Smith
Chairman, Department Committee on Graduate Students
2
Communication between Layers in Biological Transcriptional
Networks
by
Alex Tsankov
Submitted to the Department of Electrical Engineering and Computer Science
on May 19, 2005, in partial fulfillment of the
requirements for the degree of
Master of Science in Computer Science and Engineering
Abstract
Chromatin-immunoprecipitation experiments in combination with microarrays (known as
ChIP-chip) have recently allowed biologists to map where proteins bind in the yeast genome.
The combinatorial binding of different proteins at or near a gene controls the transcription
(copying) of a gene and the production of the functional RNA or protein that the gene encodes. Therefore, ChIP-chip data provides powerful insight on how genes and gene products
(i.e., proteins, RNA) interact and regulate one another in the underlying network of the cell.
Much of the current work in modeling yeast transcriptional networks focuses on the regulatory effect of a class of proteins known as transcription factors (TF). However, other sets
of factors also influence transcription, including histone modifications and states (HS), histone modifiers (HM) and remodelers, nuclear processing (NP), and nuclear transport (NT)
proteins. In order to gain a holistic understanding of the non-linear process of transcription,
our work examines the communication between all five forementioned classes (or layers) of
regulators. We use vastly available rich-media ChIP-chip data for various proteins within
the five classes to model a multi-layered transcriptional network of the yeast species Saccharomyces cerevisiae. Following the introduction in Chapter 1, Chapter 2 describes the
non-trivial process of incorporating the different sources of data into a coherent set and normalizing the heterogeneous data to improve biological accuracy. Using the normalized data,
Chapter 3 finds biologically meaningful pairwise statistics between proteins, including filtered correlation coefficient, and mutual information p-values. It then combines the p-values
of the two complementary approaches in order to increase the reliability of our predictions.
Chapter 4 uncovers group-wise relationships between proteins using a novel semi-supervised
clustering algorithm that preserves information about elements of a cluster in order to better
capture group-wise dependencies. Throughout the theoretical analysis, we confirm various
known biological processes and uncover several novel hypotheses. Based on the developed
methodology, Chapter 5 builds a multi-layered transcriptional network and quantifies the
communication between levels in biological transcriptional networks.
3
Thesis Supervisor: Moe Win
Title: Associate Professor, Massachusetts Institute of Technology
Thesis Supervisor: Pamela Silver
Title: Professor, Harvard Medical School
4
Acknowledgments
The fruition of this project would not have been possible without the cumulative contributions of many people. I first want to express my gratitude for my advisor, Moe Win. You
are the most thoughtful professor I have ever me, and a boss I can truly call a friend. Thank
you for your trust and loyal support during both happy times and tough moments at MIT.
Thank you for being open-minded about my research direction and for your encouragements
as I first started learning biology. I also want to acknowledge Ae Suwansantisuk and Hyundong Shin for their valuable contributions in proofreading the final document. In addition,
I want to thank Professor Jaakola, Sourav Dey, and Obrad Scepanovic for their insights
during Machine Learning class.
This project required the bridging of two disciplines and would not have even taken place
without all the help form Pam Silver’s lab at Harvard Medical School (HMS). Thank you
Pam for bringing me into the family, granting me computing resources, and for taking a
chance on someone who knew nearly nothing about biology nine months ago. Mike, thank
you for your patience and experimental validations, without which the project would be
incomplete. And finally, I want to especially thank Jason Casolari, who’s creative insight
inspired this undertaking. Thank you for always being honest with me, even during times
when everything was foreign to me. Thank you for being my mentor in this new field,
for explaining what YPD meant every Sunday afternooon, and for guiding me with your
ingenious insight and biological intuition–I will never forget it.
In the end, I want to express my eternal gratitude towards my family. Thank you Dona,
for reaching out to me as your younger brother and for understanding me and accepting me
with all my flaws. Fetzi, thank you for becoming my best friend at a time when I had no
other friends. And Sparti, thank you for your unconditional love and care from the day I
was born. You are all a large part of who I am.
5
6
Contents
1 Introduction
13
1.1
Regulation of Transcription . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2
ChIP-chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2 Data Preparation
19
2.1
Data Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Data Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
Finding Missing P -values . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2.2
Evaluating Data Normalization: Correlation Analysis . . . . . . . . .
27
3 Pairwise Statistics
3.1
3.2
3.3
3.4
31
Filtered Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.1.1
Filtered Correlation Coefficient P -values . . . . . . . . . . . . . . . .
34
Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2.1
Mutual Information P -values . . . . . . . . . . . . . . . . . . . . . .
39
3.2.2
Evaluating Data Classification: Mutual Information Analysis . . . . .
41
Combining P -values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.3.1
Biological Predictions using Combined P -values . . . . . . . . . . . .
45
Minimized Mutual Information P -value . . . . . . . . . . . . . . . . . . . . .
49
4 Group-wise Relationships: PCA and Clustering
4.1
Principal Components Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
7
51
51
4.2
Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.2.1
K-means Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.2.2
Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.2.3
Semi-Supervised Clustering . . . . . . . . . . . . . . . . . . . . . . .
57
4.2.4
Biological Validation of Semi-Supervised Clustering . . . . . . . . . .
59
4.2.5
Active Processes and SIR2 . . . . . . . . . . . . . . . . . . . . . . . .
62
5 Network of the Nucleus
5.1
5.2
67
Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.1.1
Assigning Links: Second Pass . . . . . . . . . . . . . . . . . . . . . .
71
5.1.2
Network Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
Analysis of Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6 Conclusion
81
A Semi-Supervised Clustering based on KL-divergence
83
8
List of Figures
1-1 Layers in the yeast transcriptional network.
. . . . . . . . . . . . . . . . . .
15
2-1 A histogram of the binding profile of protein Polymerase III. . . . . . . . . .
24
2-2 Correlation coefficient distance matrix using the BR, N &P R, P V , and SD
data representations of a test set of proteins. . . . . . . . . . . . . . . . . . .
28
3-1 Venn diagram for the subsets of genes bound by proteins POL3 and TBP. . .
39
3-2 Filtered correlation coefficient, mutual information, and combined p-values for
a test set of proteins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4-1 Casolari et al. model for nuclear transport and validation using Principal
Component Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4-2 Potential pitfall in using hierarchical clustering. . . . . . . . . . . . . . . . .
57
4-3 Confirmation of known biological processes and new insight using semi-supervised
clustering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4-4 Active biological processes and new insight using semi-supervised clustering.
65
5-1 Multi-layered transcriptional network of highly significant interactions. . . .
73
A-1 Semi-supervised clustering of all nuclear transport and nuclear processing factors binding profiles using the KL distance metric. . . . . . . . . . . . . . . .
9
86
10
List of Tables
2.1
Total correlation coefficient distance at likely and unlikely interactions using
different data representations. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Mutual information evaluation at probable and unlikely interactions using
different data representations. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
74
Total links and percentage of links realized within and between the five layers
of the entire and the positive edge transcriptional network. . . . . . . . . . .
5.3
42
Comparison between various protein-protein interaction data sets and our
network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
29
76
Network topology statistics for the entire and the positive edge transcriptional
network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
78
12
Chapter 1
Introduction
Intricate communication between genes regulate a complicated network within and between
millions of cells that make our body function. The recent emergence of novel, data-rich experimental techniques that can simultaneously monitor the behavior of all genes within a cell
has enabled computational biologists to study gene networks quantitatively. Understanding
the process of transcription, or how genes are turned “on” and copied into more functional
forms, is central to modeling gene regulatory networks.
1.1
Regulation of Transcription
Ordered sequences of deoxyribonucleic acid (DNA) base pairs, protected within the nucleus
of each cell, encode the unique hereditary information of each individual. Genes are segments
of DNA located on different chromosomes (large bundles of continuous DNA) that encode a
unique product with a specific cellular function. During transcription, a gene is copied (or
expressed) to a more mobile strand of information, known as RNA. For most genes, their
RNA products are mere messengers (or mRNA) of the DNA code. The mRNA transports
the DNA’s information outside of the nucleus where it is translated into proteins, another
even more functional string composed of amino acid molecules. The final product of some
genes, however, is the RNA itself. Such strands execute specific tasks within the body in the
13
same manner as proteins. The repertoire of RNAs and proteins expressed from “on” genes
in each cell allows for the myriad of functions necessary for cell survival and, on a larger
scale, for the many different cell types found in complex organisms.
The transcription of genes is regulated by various RNAs and proteins, or products of other
genes. Hence, genes regulate each other’s activity through their respective products. Several
classes of proteins or factors, which we refer to as layers or levels, collaborate to regulate the
process of transcription. A set of proteins, known as transcription factors (TF), can enhance
or suppress how actively a gene g works to produce its corresponding protein by binding
to the DNA preceding g, or the promoter of g. Nuclear transport factors (NTs) represent
another layer of proteins that affects transcription by controling the flux of molecules (such
as TFs, mRNAs, etc) going in and out of the nucleus. Nuclear processing factors (NPs) also
control transcription by executing functions within the nucleus, such as the actual copying of
DNA to RNA or post-transcriptional processing of mRNAs. Chromatin, a macromolecular
complex consisting of DNA and proteins that is found in eukaryotes (nucleus-containing
organisms), also has profound affects on transcription. Chromatin is organized into packed,
inaccessible and unpacked, accessible regions of DNA by structural proteins such as histones;
therefore, histones have the potential to alter the expression state of genes. A fourth class
of proteins that regulate transcription consists of histone modifiers (HMs), which change
the accessibility of a gene’s DNA by adding or removing acetyl, methyl, or other molecular
groups on specific histone amino acids sites. We also include nucleosome remodelers in this
category, or protein complexes that displace and reposition macromolecules of eight histones
called nucleosomes. And finally, the actual histone states (HSs) in terms of the acetylation
and methylation levels on specific histone amino acids, form yet another layer of control in
transcription. For example, some proteins bind to specific patterns of histone acetylation
levels in order to regulate the nearby DNA.
The protein classes described above each contribute to the expression of a gene in unique
and situation-specific manners. Indeed, large research fields are devoted to the study of
each of the described gene/protein classes and how particular stimuli are able to alter the
14
transcriptional output of a gene. In order to better understand the non-linear process of
transcription, it is necessary to examine the genome-wide interplay of all of these transcriptional regulators both within particular fields, such as how modification of a histone at one
amino acid affects modification at other amino acids, and between fields of study, such as
the relationship between histone modification and recruitment of transcription factors to
particular genes. The budding yeast Saccharomyces cerevisiae offers a unique opportunity
to achieve this essential next step in understanding transcriptional control as it has a fully
sequenced and annotated genome as well as an extraordinary body of literature regarding
functions for many of these genes and their resulting RNAs or proteins. Figure 1 summarizes the different layers of organization known to regulate the transcriptional network in
Saccharomyces cerevisiae which we will model. In order to bettter understand the non-linear
process of transcription, we need information about where different classes of proteins bind
to in the genome.
Figure 1-1: The five layers in the yeast transcriptional network that we aim to model. Twosided arrows represent undirected functional connections between components within a level
as well as interplay of components between layers of organization.
15
1.2
ChIP-chip
A biochemical technique known as chromatin immunoprecipitation (ChIP) allows for the
chemical tethering of particular proteins to chromatin. The recent advent of ChIP in combination with microarrays (known as ChIP-chip or genomic location analysis) allows biologists
to adapt this technique to a genome-wide scale, identifying the DNA binding sites for particular proteins across the entire genome [10]. Each ChIP-chip experiment begins by harvesting
a yeast strain in a carefully prepared solution simulating a specific environmental condition.
Most ChIP-chip experiments use the standard YPD (Yeast extract Peptone Dextrose) solution containing the sugar dextrose, in order to simulate rich media conditions where the
yeast population grows exponentially with time. Next, addition of formaldehyde crosslinks
or “fixes” any proteins bound to DNA inside the normal growing cell, or in vivo. The inner
contents of the cell are then extracted and the strands of DNA along with the bound proteins
are sheared into fragments by sonication. Using an anti-body that binds specifically to the
protein i of interest, one can isolate all DNA fragments bound to the designated protein i by
precipitation (called immunoprecipitation). The DNA-(protein i) crosslink is then reversed
in order to release the DNA and digest the protein. The free DNA, previously bound by
protein i, is then amplified by Polymerase Chain Reaction (PCR) and labeled with a fluorescent dye (Cy5). In parallel, a sample of sheared DNA is taken as a control and is not
enriched for binding with protein i using immunoprecipitation. This sample is also amplified
using PCR and labeled using a different fluorescent dye (Cy3). Ultimately, the ratio of Cy5
and Cy3 fluorescence intensities at each gene will allow one to infer protein i’s predilection
for binding to g.
Microarray technology when coupled with ChIP enables biologists to simultaneously measure the binding preference of protein i at all genes in the genome relative to other genes in
just one experiment. One first needs to prepare a microarray matrix of wells, where each well
g contains a probe specific to gene g (for example the complementary DNA of gene g) . Next,
each microarray well is filled with a sample from both the pool of sheared DNA enriched
(Cy5) and unenriched (Cy3) for protein i. Inside each well g, the differently dyed fragments
16
originally sheared from gene g will competitively bind to the complementary DNA specific
to g during the process of hybridization. The array of wells can then be scanned using a
laser that detects the intensities of the fluorescent red (Cy5) and green (Cy3) dye [39]. The
ratio of intensities (Cy5/Cy3), also known as a binding ratio, is proportional to the ratio of
the number of gene g fragments bound by protein i to the number of g fragments randomly
resulting from the same process without immunoprecipitation. Each ChIP-chip experiment
is often repeated three or more times in order to reduce the inherent experimental noise
and the resulting binding ratios from each experiment are usually combined using weighted
averaging.
ChIP-chip provides powerful, in vivo measurements of how genes and gene products (i.e.
proteins, RNA) interact and regulate one another in the complex underlying network of a
cell. Much of the current work in modeling yeast networks focuses on the regulatory effect of
TFs [2]. However, recent publications [4,6,7,13–20] show that other sets of proteins (including
HMs, HSs, NPs, and NTs) also control gene expression. Hence, transcription has several
levels of organization. The next four chapters use ChIP-chip data to infer the underlying
communication between different layers in the transcriptional network of the yeast species
Saccharomyces cerevisiae.
Following the introduction in Chapter 1, four chapters build up the theory behind our
transcriptional network model. Chapter 2 describes the preliminary data preparation on
which we will base all future analysis. It first describes the integration of the different
ChIP-chip data sets into one coherent set and then explores the crucial issue of binding
data normalization and representation. Using a normalized data set, Chapter 3 finds biologically meaningful pairwise statistics between binding profiles of proteins, including filtered
correlation coefficient, mutual information, and combined p-values. The next chapter uncovers group-wise binding relationships between proteins using Principal Component Analysis
(PCA) and clustering. Based on the measures developed in Chapter 3, we introduce a novel
semi-supervised clustering algorithm that preserves information about elements of a cluster in order to better capture group-wise dependencies between proteins. Throughout the
17
theoretical analysis, we confirm various known biological processes and predict several novel
hypotheses. And finally, Chapter 5 combines the methodology developed in the previous
chapters in order to build a multi-layered transcriptional network of the nucleus. To the
best of our knowledge, our finalized network is the first attempt in the literature to quantify
the communication between layers in biological transcriptional networks.
18
Chapter 2
Data Preparation
This chapter explores the extremely important issue of data preparation. Our data comes
from several different sources and in various forms. The first section explains the nontrivial
procedure for integrating all the different sources of data into one coherent set. The second
part of the chapter attempts to reconcile the discrepancies between ChIP-chip experiments
done in different labs by normalizing the data.
2.1
Data Integration
Integrating the publicly available data sets proved to be a tedious but non-trivial part of
the project. We downloaded published genome-wide binding data for several sets of proteins
that may be involved in regulation of genes, including T F s [1, 17, 21], N T s [13, 14], N P s
[8, 14, 16, 23, 24, 28], HM s [4, 6, 7, 15, 20, 21, 25], and HSs [17, 18, 26, 27]. All of the ChIPchip experiments were performed on yeast strains grown in rich media conditions, where the
sugar dextrose and other nutrients are readily present so that the yeast can quickly multiply.
Since some genes only become active during specific environmental conditions, we would
ideally want ChIP-chip binding data for yeast grown in various media. However, from our
experience and as evidenced in [7], binding relationships in biological processes that are not
turned “on” during rich media conditions can often still be detected using our ChIP-chip
19
data. For example, our analysis in Section 4.2.4 captures the collaborative effect of GAL3
and GAL80, two transcription factors that regulate genes involved in breakdown of the sugar
galactose, using ChIP-chip data for yeast grown in dextrose-rich, YPD medium. In addition,
the authors in [7] found that the RSC nucleosome remodeling complex associates with many
genes involved in nitrogen regulation and non-fermentative carbohydrate metabolism using
YPD ChIP-chip data. In order to explain the relationship fully, more experiments under
different conditions were necessary, but the initial prediction came from looking at rich
media ChIP-chip data. Thus, we felt that usage of the vastly more complete YPD datasets
would still allow for the formation of hypotheses reflecting other growth conditions.
The data sets further differed in the microarray probes used to detect binding relationships, where some experiments used probes that target the open reading frames (ORF), or
regions of genes that code for RNA, while other experiments used probes for intergenic regions, or the promoter or control regions of genes. We used a gene-centric approach, where
each relevant probe was assigned to its corresponding gene(s). For ORF arrays, we simply
assigned the ChIP-chip information at each ORF to the corresponding gene.
For intergenic arrays, we assigned each DNA probe (or fragment) to the gene that it
most likely regulates. Biologists commonly refer to the DNA region preceding the site where
transcription starts as the upstream intergenic region of a gene, and the region following
the site where transcription ends as the downstream intergenic region of a gene. In yeast,
each intergenic region can control zero genes (e.g., telomeres at the end of chromosomes,
intergenic regions at the downstream end of two genes), a single gene, or two genes (e.g.,
intergenic regions at the upstream end of two genes). Moreover, there may exist several
probes for a single intergenic region. For example, small genes that encode tRNAs (a class
of RNAs that have a role in protein production) are often contained within the intergenic
region upstream of a longer gene; therefore, tRNA gene probes also measure the binding
of factors that regulate the longer gene. To further complicate matters, for some genes it
is still not known which intergenic region controls them. Hence, we needed to develop a
many-to-many mapping from intergenic fragments to the genes they might control. The
20
mapping algorithm uses the union of intergenic probe-gene assignment pairs as defined by
several authors [1, 6, 8, 9, 15, 19, 21, 23, 24, 27], where we included assignment pairs from at
least one author from each different lab when it was available. Moreover, when two or more
intergenic fragments mapped to the same gene, the probe that contains the most amount of
information was chosen. Since ChIP-chip experiments contain more information at the tails
of the binding distribution, we chose the most bound fragment for multiple probes that were
consistently bound and the least bound fragment for multiple probes that were consistently
not bound.
For each experiment, we obtained binding ratio (BR) data or the combination of BR
and p-value (PV) data. We first mapped the binding ratios and p-values to all annotated
genes for which data was available using the assignment algorithm described above. We then
integrated the data into a single BR matrix and a single P V matrix, where rows represent
factors, columns represent unique genes and entries represent the data. For example, the
(i, g)th entry of the matrix P V , P Vi,g , represents the p-value of factor i binding to gene g.
Missing data was annotated with NaNs.
As previously explained, a binding ratio for protein i and gene g represents a weighted
average of ratios of the number of immunoprecipitated DNA fragments from g enriched
with protein i to the number of control (unenriched) DNA fragments from g that occur at
random. The p-value for the binding ratio of protein i at gene g measures the probability of
erroneously deciding that protein i binds to g when the null hypothesis (i does not bind to
g) is in fact true. Hence, small p-values correspond to large binding ratios. The error model
used for calculating the p-values varies as chosen by the authors of each paper. Since we
are integrating binding data from various papers that use different ChIP-chip experimental
protocols and different error models, we needed to derive a normalized representation of the
binding data in order to compare binding profiles across papers.
21
2.2
Data Normalization
To normalize the binding data, we first sorted the protein-gene binding interactions for each
protein (i.e., row-wise in our matrices) from most bound to least bound, as defined by the
authors of each paper. We then mapped the sorted binding data to uniformly spaced discrete
points in the interval [0, 1], where 1 and 0 correspond to the most and least bound genes
for each protein, respectively. Hence we transformed the information in the BR and P V
matrices to a percentile rank P R matrix of the same dimensions. For example, P Ri,g = 0.95
means that gene g is in the top 5% of genes most bound by protein i. Moreover, since
each protein binds to a varying number of genes, each author usually defines a strength
of binding threshold, above which all protein-gene interactions are classified as bound. We
further normalized the P R matrix by subtracting the strength of binding threshold from
each entry, making all interactions classified as bound positive and all interactions classified
as unbound negative. We called this matrix the normalized N data matrix. By thresholding
N at 0, we also easily derived a bound/unbound B data representation matrix, where all
bound interactions were mapped to 1 and all unbound interactions to 0.
Each data representation has inherent advantages and disadvantages that may prove
more biologically useful or less informative depending on the nature of the analysis. The
bound/unbound B representation of the data seems most natural for representing the presence or absence of a biological interaction. However, ChIP-chip data contains significant
experimental noise that varies from lab to lab and protein to protein; therefore, the classified bound/unbound data contains a large number of false positives and false negatives. The
data set from [1] estimates the false positive rate to be around 4-6% and the false negative
rate around 24%, for a binding threshold at p-value = 0.001. The percentile rank P R matrix removes discrepancies between the different averaging techniques used to derive binding
ratios BR and the different error models used to calculate p-values P V . The P R matrix
measures the pairwise binding strengths of gene-protein interactions consistently across data
from different papers; however, it contains no information about which percentile determines significant binding for a given factor and the number of gene targets vary greatly for
22
different proteins. The normalized N matrix improves the P R matrix by subtracting the
percentile threshold that each author used to classify interactions as bound and unbound.
This achieves a soft, unclassified representation than the B matrix, where the more negative
or positive an entry in the N matrix is, the more confident one can be that the gene-protein
interaction is unbound or bound, respectively. While the P R and the N matrices measure
binding strength relative to other interactions, they fail to capture the absolute strength of
binding. To remedy this last shortcoming, we derive the SD matrix in the next section by
finding the missing p-values in our data sets.
2.2.1
Finding Missing P -values
We consider the P V matrix as the most reliable source of information about making decisions
on the presence or absence of a binding interaction. Further, nearly all of data sets with pvalues base their calculation on adaptations of the single array error model [11]. Specifically,
the data sets from [13, 14, 16] use a two-sided model, which can easily be converted to the
equivalent right-sided single array model by the following conversion:
1
BRi,g > 1 : P Vi,g|1−sided = P Vi,g|2−sided
2
1
BRi,g ≤ 1 : P Vi,g|1−sided = 1 − P Vi,g|2−sided .
2
(2.1)
(2.2)
Reconciling these discrepancies provides reliable, consistent p-values for two thirds of the
proteins in the data. We need to find the missing p-values for the other third based on
the available BR information in order to obtain a complete P V matrix. Figure 2-1 shows
the binding disribution of protein Polymerase III (POL3) across the natural logarithm of
its binding ratios. The figure reveals that the distribution of binding ratios consists of two
heterogeneous classes: i) binding ratios measured at genes that are not bound by POL3
and ii) binding ratios at genes that are bound by POL3, congregated at the right tail of
the distribution. We need to model the distribution of the unbound genes in order to find
23
appropriate p-values.
Figure 2-1: A histogram of the binding profile of protein Polymerase III: the number of genes
bound by Pol III versus the natural logarithm of the binding ratios.
Mathematically, let Xi,g denote random variables describing the natural logarithm of the
binding ratios of protein i (POL3 in our example) at genes g and lets assume that the binding
of i at each gene g, or Xi,g , is independent and identically distributed (i.i.d.). Since there is
inherent noise in ChIP-chip experiments, it makes sense to use random variables to model
our data. Following the convention of using capital letters for random variables and lowercase
letters for their realizations, we use xi,g to denote the measured outcomes of random variable
Xi,g at each gene g. Since random variables Xi,g are identically distributed at all genes g, let
Xi denote the underlying binding tendency of protein i that we want to describe. Therefore,
Figure 2-1 shows a representative, empirical example of the estimated distribution of Xi , or
P̂Xi (xi ). In order to extract p-values, we want to estimate the distribution of the natural
logarithm of binding ratios at all genes g under the null hypothesis H0 that protein i does not
bind to g, or the noise distribution P̂Xi |H0 (xi |H0 ). ChIP-chip experiments introduce various,
independent sources of multiplicative noise, which corresponds to several independent sources
of additive noise in the logarithm domain. Therefore, the Central Limit Theorem makes it
24
reasonable to assume that the distribution of the logarithm of binding ratios for the class
of unbound genes is Gaussian [39]. Since logarithm of binding ratios in our data result
from weighted averages of logarithm of binding ratios from single independent ChIP-chip
experiments and since linear combinations of independent Gaussian random variables is also
Gaussian, we can reasonably model PXi |H0 (xi |H0 ) using a Gaussian distribution.
The Gaussian noise distribution PXi |H0 (xi |H0 ) is uniquely defined by its mean and variance. We use the peak or mode of the overall distribution P̂Xi (xi ) (e.g., the peak at
log(BR) = 0 in Figure 2-1) to estimate the mean of the noise distribution, µ̂Xi |H0 . This
follows from the fact that the alternative hypothesis that protein i binds to gene g, H1 , is
is significantly less likely than the null hypothesis (on average, about 4% of gene targets are
classified as bound, or Pr(H1 ) ≈ 0.04 and Pr(H0 ) ≈ 0.96). Since the overall distribution
decomposes as follows,
PXi (xi ) = Pr(H0 )PXi |H0 (xi |H0 ) + Pr(H1 )PXi |H1 (xi |H1 )
(2.3)
≈ 0.96PXi |H0 (xi |H0 ) + 0.04PXi |H1 (xi |H1 ) ,
the peak of the unbound distribution should roughly correspond to the observed mode of the
overall distribution P̂Xi (xi ). Moreover, since the noise distribution is Gaussian, the mostly
unaffected peak of PXi |H0 (xi |H0 ) corresponds to the mean µ̂Xi |H0 . Hence, we can write
µ̂Xi |H0 = modeg∈Gi (xi,g ) ,
(2.4)
where Gi is the set of all genes with measured binding information for protein i. In order
to estimate the variance of the noise distribution, σˆ2 Xi |H0 , we consider genes with a binding
ratio smaller than µ̂Xi |H0 to be extremely unlikely targets of protein i. Hence, we use Ui to
denote the set of genes to the left of the mode of P̂Xi (xi ), or the genes the make up the left
side of the unbound distribution (the distribution to the left of the peak at log(BR) = 0
in Figure 2-1). Due to the symmetry of Gaussian distributions, we estimate the variance
25
of PXi |H0 (xi |H0 ) only using the left side of the noise distribution. The maximum likelihood
estimator for variance of Gaussian random variables states that
1 X
σˆ2 Xi |H0 =
(xi,g − µ̂Xi |H0 )2 ,
|Ui | g∈U
(2.5)
i
where |Ui | denotes the number of elements in set Ui , or the cardinality of Ui . The p-value
P Vi,g corresponds to the probability that just as extreme or more extreme of an observation
could occur if we assume the null hypothesis H0 that the factor i does not bind to gene g.
To calculate the missing p-value for observation xi,g , or P Vi,g , we integrate our estimated
noise distribution over the interval [xi,g , ∞):
Z
∞
Z
∞
P̂Xi |H0 (xi |H0 )dxi =
P Vi,g =
xi,g
xi,g
1
q
exp[
2π σˆ2 Xi
−(xi − µ̂Xi )2
]dxi .
2σˆ2 X
(2.6)
i
After obtaining a complete P V matrix, we wanted to normalize the P V matrix so that we
can assume a nearly Gaussian binding distribution for entries across rows for in the following
analysis. Using the inverse of the Gaussian Q-function, we obtained the normalized SD
matrix, where each entry SDi,g represents the number of standard deviations of confidence
in rejecting the null hypothesis that protein i binds to gene g. Mathematically,
∞
−w2
1
√ exp[
]dw
2
2π
z
= Q−1 (P Vi,g ) .
Z
Q(z) =
(2.7)
SDi,g
(2.8)
Since each row in the SD matrix approximates a Gaussian distribution with zero mean and
unit variance, the SD matrix, just like the N matrix, fixes the problem in normalizing the
data set so that binding levels can be compared across different experiments. Moreover,
since the entries in the SD matrix represent standard deviation of confidence in rejecting
the null hypothesis, the SD matrix preserves the absolute strength of binding of gene-protein
interactions in a continuous manner, which the N matrix fails to capture. In the following
26
section we explore the advantages of the SD data matrix in relation to correlation analysis.
2.2.2
Evaluating Data Normalization: Correlation Analysis
We use the measure of correlation coefficient distance to gauge the performance of the various data representations in relation to correlation analysis. We define correlation coefficient
distance as one minus the correlation coefficient; therefore, a distance of 0 represents full
positive linear dependence and a distance of 1 denotes no linear relationship [?]. Figure 2-2
illustrates the concepts from the previous paragraph in Section 2.2.1. It shows four correlation coefficient distance matrices using the BR, N &P R, P V , and SD data representations of
a test set of proteins. Note that since the N and P R rows are simply shifted versions of one
another and since correlation analysis is invariant to scales and shifts, the two matrices produce an identical correlation coefficient distance matrix, which we denote as resulting from
N &P R. Two known biological processes are shown: RSC2, RSC3, RSC8, and STH1 are
all components of the RSC nucleosome remodeling complex [7] and MLP1, MLP2, NIC96,
and KAP95 all play role in nuclear transport [13]. We would expect the members of each
biological processes to share similar binding profiles. This is shown by small correlation distance values in the two squares along the diagonal. The BR matrix clearly performs worse
than the P V matrix in confirming the known relationships, corroborating the fact that enrichment ratios contain less reliable information than p-values. The N and P R matrices are
ranked version of the P V matrix, so the three, naturally, share very similar results. Since
correlation coefficient analysis assumes Gaussian binding profiles, the SD matrix improves
on the P V matrix even though (2.8) shows that the two are mere transformations of each
other. Moreover, the SD matrix outperforms the N matrix not only because its row vector
entries approximate a normal distribution, but also because it preserves information about
the absolute strength of binding.
The performance of the different data representation in correlation analysis also extends
to the whole data. We generated a test set of 1447 probable interactions, most of which were
confirmed or suggested by published literature. Therefore, the known relationships discussed
27
(a) BR correlation coefficient distance.
(b) N &P R correlation coefficient distance.
(c) P V correlation coefficient distance.
(d) SD correlation coefficient distance.
Figure 2-2: Correlation coefficient distance matrix using the (a) BR, (b) N &P R, (c) P V ,
and (d) SD data representations of a test set of proteins. We define correlation coefficient
distance as one minus the correlation coefficient; therefore, a distance of 0 represents full
positive linear dependence and a distance of 1 denotes no linear relationship [?]. Protein
names are listed on the x and y axes with an “o” or i at the end of the names signifying that
the data came from an ORF or intergenic array, respectively. Two known biological processes
are shown, namely RSC2-RSC3-RSC8-STH1 [7] and MLP1-MLP2-NIC96-KAP95 [13]. The
SD matrix performs the best in confirming the known relationships followed closely by the
P V and N &P R matrices, while the BR matrix clearly performs the worst.
28
Matrix Total Distance at Probable Interactions
BR
978.5
log(BR)
974.3
N &P R
945.8
PV
966.3
SD
879.7
Total Distance at Unlikely Interactions
5496.9
5415.7
5341.4
5763.9
5374.5
Table 2.1: Total correlation coefficient distance for a test set of likely and unlikely interactions. A smaller total distance at known interactions should correspond to a smaller false
negative rate for a given data representation, while a large total distance at unlikely interactions should lead to a smaller false positive rate. The SD matrix performs best in normalizing
the data since it has the lowest total distance at known interactions and a comparable total
distance to all the other matrices at unlikely interactions. The P V , N , and P R matrices do
slightly worse but still better than both the log(BR), and BR data representations.
in Figure 2-2 represent a subset of this test set. We also created a test set of 5926 extremely
unlikely interactions. Table 2.1 shows the computed total correlation coefficient distance for
a test set of likely and unlikely interactions. A smaller total distance at known interactions
should correspond to a smaller false negative rate for a given data representation, while a
large total distance at unlikely interactions should lead to a smaller false positive rate. The
SD matrix again performs best in normalizing the data since it has the lowest total distance
at known interactions and a comparable total distance to all the other matrices at unlikely
interactions. The P V , N , and P R matrices do slightly worse but still better than both the
log(BR), and BR data representations. Note that the log(BR) data representation has a
near Gaussian distribution for entries across rows. The fact that it slightly outperforms the
BR matrix shows that the Gaussian data distribution accounts for some but not all of the
improvement in normalizing the data. These results substantiate our decision to use the SD
matrix for the correlation analysis that follows. The following chapter utilizes the different
data representations in building a statistical method for detecting pairwise relationships
between proteins.
29
30
Chapter 3
Pairwise Statistics
With an integrated and normalized data set, this chapter tries to find pairwise binding relationships between two proteins. We introduce two biologically meaningful pairwise measures
of binding dependencies: filtered correlation coefficient and mutual information. We also find
consistent methods for evaluating the p-values of the two analyses. In the end, we combine
the p-values from the two complementary pairwise approaches in order to reduce the number
of false positive and false negative biological predictions and increase the reliability of our
overall analysis.
3.1
Filtered Correlation Coefficient
This section introduces the pairwise measure of filtered correlation coefficient, an extension of
the standard correlation analysis commonly encountered in biology. This technique extracts
the binding relationships between two factors with great accuracy, by isolating the analysis
on the pertinent dimensions in ChIP-chip data. To estimate the filtered correlation coefficient
between two proteins i and j, we use the SD matrix (as justified in Section 2.2.2). As in
Section 2.2.1, we consider binding of factors i and j at genes g, denoted as Xi,g and Xj,g ,
as i.i.d. random variables with measured outcomes xi,g and xj,g , respectively. Moreover, we
again denote the underlying binding tendency of the two factors i and j as random variables
31
Xi and Xj . Since the rows of the SD data representation closely resemble samples from a
Gaussian distribution, we assume that Xi and Xj are jointly Gaussian. For factors i and j, let
Gi and Gj represent the set of all genes for which we have binding information, respectively.
Moreover, we use Fi and Fj to denote the filtered sets of genes bound by proteins i and j,
respectively, and Fi,j = Fi ∪ Fj to represent the overall filtered set of genes, or the set of
genes classified as bound by at least one of the two factors. Using the SD data matrix, the
following equations find the means, filtered variances and filtered covariance of the binding
tendencies of two proteins using maximum-likelihood (ML) estimators for jointly Gaussian
random variables as shown below:
µ̂Xi =
µ̂Xj =
σˆ2 Xi =
σˆ2 Xj =
σ̂Xi ,Xj =
1 X
xi,g
|Gi | g∈G
i
1 X
xj,g
|Gj | g∈G
j
1 X
(xi,g − µ̂Xi )2
|Fi,j | g∈F
i,j
X
1
(xj,g − µ̂Xj )2
|Fi,j | g∈F
i,j
X
1
(xi,g − µ̂Xi )(xj,g − µ̂Xj ) .
|Fi,j | g∈F
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
i,j
Note that the estimates of the means, µ̂Xi and µ̂Xj , consider all genes for which we have
binding information, while the estimates of the variances, σˆ2 Xi and σˆ2 Xj , and covariance,
σ̂Xi ,Xj , consider only the genes classified as bound by at least one of the two factors. We
call these estimates the filtered variances and filtered covariance of binding profiles Xi and
Xj . We again use the ML estimator for jointly Gaussian random variables to estimate the
filtered correlation coefficient between binding profiles Xi and Xj , or ρ̂Xi ,Xj , as follows:
32
ρ̂Xi ,Xj = q
σ̂Xi ,Xj
.
(3.6)
σˆ2 Xi σˆ2 Xj
The difference between the filtered correlation coefficient and the standard correlation
coefficient is that the estimates of the filtered variances and covariance of binding profiles Xi
and Xj consider just a filtered subset of genes. As we would expect, the filtered correlation
coefficient still retains some of the important properties of the correlation coefficient. For
example, 0 ≤ |ρ̂Xi ,Xj | ≤ 1, with ρ̂Xi ,Xj = 0 if and only if two data vectors have no linear
dependence (uncorrelated) and ρ̂Xi ,Xj = 1 if and only if one data vector is a shifted and
scaled version of the other. This directly follows from Schwartz’s inequality, which states
that
1 X
|
(xi,g − µ̂Xi )(xj,g − µ̂Xj )| ≤
|Fi,j | g∈F
i,j
s
(
1 X
1 X
(xi,g − µ̂Xi )2 )(
(xj,g − µ̂Xj )2 ) ,
|Fi,j | g∈F
|Fi,j | g∈F
i,j
i,j
(3.7)
or equivalently that |σ̂Xi ,Xj | ≤
q
σˆ2 Xi σˆ2 Xj , where the equality holds true if and only if at
every g, xi,g = αxj,g + β for some choice of constants α and β.
In addition, the filtered correlation coefficient is also shift and scale invariant. This is
a particularly useful property for our analysis, since it normalizes for the fact that some
binding profiles may vary more than others. To prove that this property still holds, let
Xi0 = aXi + b and Xj0 = cXj + d represent linear transformations of random variables Xi
and Xj with transformed observations x0i,g = axi,g + b and x0j,g = cxj,g + d, respectively. The
estimates for the means, filtered variances, and filtered covariance change as follows:
33
µ̂Xi0 =
1 X
a X
1 X 0
xi,g =
(axi,g + b) =
(xi,g ) + b = aµ̂Xi + b
0
|Gi |
|Gi | g∈G
|Gi | g∈G
0
(3.8)
1
|G0j |
(3.9)
g∈Gi
µ̂Xj0 =
σˆ2 Xi0 =
σˆ2 Xj0 =
σ̂Xi0 ,Xj0 =
X
i
x0j,g =
g∈G0j
1
0
|Fi,j
|
1
0
|Fi,j
|
X
i
1
c
(cxj,g + d) =
(xj,g ) + d = cµ̂Xj + d
|Gj | g∈G
|Gj | g∈G
X
j
j
(x0i,g − µ̂Xi0 )2 =
1
(axi,g + b − aµ̂Xi − b)2 = a2 σˆ2 Xi (3.10)
|Fi,j | g∈F
(x0j,g − µ̂Xj0 )2 =
1
(cxj,g + d − cµ̂Xj − d)2 = c2 σˆ2 Xj (3.11)
|Fi,j | g∈F
0
g∈Fi,j
X
X
X
i,j
0
g∈Fi,j
X
i,j
1 X 0
(xi,g − µ̂Xi0 )(x0j,g − µ̂Xj0 ) = acσ̂Xi ,Xj .
0
|Fi,j |
0
(3.12)
g∈Fi,j
Substituting the new estimates for filtered variances and covariance into 3.6 we see that the
filtered correlation coefficient of the scaled and shifted versions of the data is the same as
that of the original binding profiles Xi and Xj :
σ̂Xi0 ,Xj0
ρ̂Xi0 ,Xj0 = q
σˆ2 Xi0 σˆ2 Xj0
=q
acσ̂Xi ,Xj
a2 σˆ2
Xi
c2 σˆ2
= ρ̂Xi ,Xj .
(3.13)
Xj
Filtered correlation coefficient is a simple but powerful measure of binding relationships
between two factors. It isolates the analysis on the pertinent dimensions (or genes) of the
ChIP-chip data and predicts biological interaction between factors with great accuracy, as
shown in the following sections and chapters. Moreover, it can compare vastly different
data sets in a consistent manner. We explore the issue of significance in filtered correlation
analysis next.
3.1.1
Filtered Correlation Coefficient P -values
Ultimately, we want to use our filtered correlation coefficient to make decisions on whether
two factors are linearly related. Hence, under our null hypothesis H0 the binding profiles of
two factors are not linearly related (i.e., ρXi ,Xj = 0) and under our alternative hypothesis
34
H1 they are linearly related (i.e., ρXi ,Xj 6= 0). For sample size n and filtered correlation
coefficient ρ̂Xi ,Xj , we want to evaluate the probability that we reject the null hypothesis
when it is actually true, or the p-value. To evaluate the significance of our filtered correlation
coefficient, we use the test statistic
√
ρ̂Xi ,Xj n − 2
T =
.
1 − ρ̂2Xi ,Xj
(3.14)
Assuming that binding profiles Xi and Xj are jointly Gaussian, [35] shows that T is a
Student-T random variable of n−2 degrees of freedom. Further, T results from a generalized
likelihood ratio test and hence defines the optimal decision boundary [35]. Let tρ̂,n−2 denote
one positive outcome of the random variable T for a given ρ̂Xi ,Xj = ρ̂ and n − 2 . Also, let
xi = [xi,g1 . . . xi,g|G| ] and xj = [xj,g1 . . . xj,g|G| ] represent the row vectors of binding data for
proteins i and j across all genes g in the set G, following the convention of using boldface
to represent vectors. Since, tρ̂,n−2 only depends on n − 2 and the estimated value ρ̂ found
using xi and xj , tρ̂,n−2 is completely determined by xi and xj . We represent the filtered
correlation coefficient p-value for a given tρ̂,n−2 , or for a given xi and xj , as pvF CC (xi , xj ).
To find this quantity, we need to find the probability that test statistic t can have a value
more extreme than tρ̂,n−2 . This corresponds to integrating the distribution of T , PT (t), over
the two disjoint intervals [−∞, −tρ̂,n−2 ] ∪ [tρ̂,n−2 , ∞] where T exceeds the outcome tρ̂,n−2 .
Due to the symmetry of the distribution of Student-T random variable T , we can write
Z
∞
pvF CC (xi , xj ) = P r(T ≥ tρ̂,n−2 ) = 2
PT (t)dt .
(3.15)
tρ̂,n−2
Note that we implicitly assume that both highly positive and negative values of ρ̂ are significant here. If we wanted to simply consider positive/negative correlation coefficients as
significant, we would need to remove the factor of two and only integrate over the interval
corresponding to the right/left tail of T ’s distribution.
The filtered correlation coefficient measures how consistently two binding profiles fluctuate above and below their respective means. It represents a normalized measure of the linear
35
relationship between two data vectors. However, two random entities can have no linear
relationship (i.e., ρXi ,Xj = 0) but still dependent on each other in a non-linear fashion. We
explore a more general notion of probabilistic dependence between random variables in the
next section.
3.2
Mutual Information
Mutual information is a general measure of the probabilistic dependence between two random
variables. As in the filtered correlation analysis, we again consider binding profiles Xi and
Xj of proteins i and j as random variables, but now their outcomes only take on NXi and
NXj discrete values. The entropy of binding profile Xi measures the amount of uncertainty in
predicting the observations of Xi and forms the basis for calculating the mutual information.
Given a probability mass distribution of binding profile Xi , PXi (Xi = xi,r ) for r = 1, . . . , NXi ,
the entropy of X is define as
N Xi
X
PXi (Xi = xi,r ) log PXi (Xi = xi,r )
H(Xi ) = −
r=1
N Xi NXj
XX
=−
PXi ,Xj (Xi = xi,r , Xj = xj,s ) log PXi (Xi = xi,r ) .
(3.16)
r=1 s=1
Suppose now that we observe the binding profile Xj , which might be related to Xi . The
conditional entropy of Xi given Xj measures the amount of uncertainty that Xi contains
with prior knowledge of Xj and can be calculated as follows:
36
N
Xj
X
H(Xi |Xj ) = −
PXj (Xj = xj,s )H(Xi |Xj = xj,s )
s=1
N Xj
NXi
X
X
=−
PXj (Xj = xj,s )
PXi |Xj (Xi = xi,r |Xj = xj,s ) log PXi |Xj (Xi = xi,r |Xj = xj,s )
s=1
NXi N Xj
r=1
XX
=−
PXi ,Xj (Xi = xi,r , Xj = xj,s ) log PXi |Xj (Xi = xi,r |Xj = xj,s ) .
(3.17)
r=1 s=1
The amount that the uncertainty in Xi decreases with the observation of Xj ; hence, the
entropy reduction H(Xi ) − H(Xi |Xj ) corresponds to the amount of information that Xj
contains about Xi . Combining (3.16) and (3.17), the mutual information between random
binding profiles Xi and Xj takes the form:
I(Xi ; Xj ) = H(Xi ) − H(Xi |Xj ) = H(Xj ) − H(Xj |Xi )
N
=
NXi Xj
X
X
PXi ,Xj (Xi = xi,r , Xj = xj,s )(log PXi |Xj (Xi = xi,r |Xj = xj,s ) − log PXi (Xi = xi,r ))
r=1 s=1
NXi N Xj
=
XX
PXi ,Xj (Xi = xi,r , Xj = xj,s ) log
r=1 s=1
PXi ,Xj (Xi = xi,r , Xj = xj,s )
.
PXi (Xi = xi,r )PXj (Xj = xj,s )
Note that the second equality in the first line shows that mutual information is symmetric,
or I(Xi ; Xj ) = I(Xj ; Xi ). This can easily be verified by observing that trading the places of
all the Xi and Xj in (3.18) results in no change. Another property of the mutual information
is that it is non-negative, or I(Xi ; Xj ) ≥ 0.
In order to estimate the mutual information between the binding profiles Xi and Xj of
proteins i and j, we need to estimate the marginal and joint probability mass functions of
Xi and Xj based on the data. Using the P V and P R data representations (justification
discussed in Section 3.2.2), we classify each data entry as a 1 or a 0, signifying the presence
or absence of an interaction, respectively. Hence, we model our binding profiles as Bernoulli
37
(3.18)
random variables with i.i.d. random samples Xi,g and Xj,g and outcomes xi,g and xj,g , where
xi,g , xj,g ∈ {0, 1} at all genes g. Let Gi,j denote the set of all genes with binding observations
for both proteins i and j, and let Gi,j have w gene members. We can use maximum likelihood
estimators for the parameters of Bernoulli random variables to estimate the marginal and
joint mass distributions using
P̂ (Xi = 1) =
1
w
X
g∈{g:xi,g =1}
1=
v
w
(3.19)
w−v
P̂ (Xi = 0) = 1 − P̂ (Xi = 1) =
w
X
1
u
P̂ (Xj = 1) =
1=
w
w
(3.20)
(3.21)
g∈{g:xj,g =1}
w−u
P̂ (Xj = 0) = 1 − P̂ (Xj = 1) =
w
X
1
h
P̂ (Xi = 1, Xj = 1) =
1=
w
w
(3.22)
(3.23)
g∈{g:xi,g =1,xj,g =1}
P̂ (Xi = 1, Xj = 0) =
P̂ (Xi = 0, Xj = 1) =
P̂ (Xi = 0, Xj = 0) =
1
w
1
w
1
w
X
1=
v−h
w
(3.24)
1=
u−h
w
(3.25)
1=
w−v−u+h
,
w
(3.26)
g∈{g:xi,g =1,xj,g =0}
X
g∈{g:xi,g =0,xj,g =1}
X
g∈{g:xi,g =0,xj,g =0}
where h denotes the number of genes bound by both proteins, v the number of genes bound
by protein with binding profile Xi and u the number of genes bound by protein with profile
Xj . Setting NXi = NXj = 2 and substituting our estimated distributions in (3.18) gives us
ˆ i ; Xj ).
the estimated mutual information between binding profiles Xi and Xj , I(X
Mutual information provides a more general framework for measuring the dependence
between two random variables. Moreover, the following sections and chapters demonstrate
that it is a very natural and biologically meaningful measure of protein dependence in ChIPchip data. We ultimately want to use our estimate of I(Xi ; Xj ) in order to make decisions
38
on whether two proteins with binding profiles Xi and Xj participate in the same biological
process. In the next section, we introduce a method for finding p-values for the mutual
information analysis.
3.2.1
Mutual Information P -values
P -values allow us to make unbiased decisions about biological dependence based on mutual
information. In this scenario, under the null hypothesis H0 the two factors have no binding
dependence (i.e., I(Xi ; Xj ) = 0). The p-value measures the probability that an estimated
mutual information of a given significance or greater can occur at random. Note that our
estimate of the mutual information in (3.19)-(3.26) depends only on four parameters, namely
ˆ i ; Xj ) maps to the Venn diagram shown in Figure 3-1, where
h, u, v, and w. Hence, each I(X
Xi is the binding profile of the TATA-Box Protein (TBP) and Xj is the binding profile of
POL3.
Figure 3-1: Venn diagram for the subsets of genes bound by proteins POL3 and TBP: w
denotes the number of genes with observed binding information about both POL3 and TBP,
v is the number of genes bound by TBP, u is the number of genes bound by POL3, and
h is the number of genes bound by both. The classification of bound was chosen based on
estimated p-values (see Section 2.2.1) below the threshold 0.001.
39
Given a superset of w genes and two subsets of u and v genes, the probability of the
two subsets having an overlap of h elements at random has a hypergeometric distribution
PH|U,V,W (h|u, v, w). Hence, we can calculate the probability of estimating a mutual informaˆ i ; Xj ) = îh,u,v,w at random using
tion I(X
w−v v
u−h h
w
u
ˆ i ; Xj ) = îh,u,v,w |H0 ) = PH|U,V,W (h|u, v, w) =
Pr(I(X
.
(3.27)
The denominator in (3.27) represents the number of ways to choose u − h non-overlaping
elements from the complement of the set containing v objects, times the number of ways to
choose the h overlapping elements from the set with v objects. The product hence represents
the number of ways to choose a subset of u elements from a superset of w objects, such
that exactly h of them overlap with a pre-designated subset of v elements. The numerator
computes all the possible ways of choosing a subset of u elements from a set of size w,
normalizing (3.27) in order to obtain a probability.
ˆ i ; Xj ) = îh,u,v,w to hypergeometric
The mapping from mutual information estimate I(X
probabilities is not one to one. For example, the parameters (h = 1, u = 2, v = 2, w = 20)
and (h = 300, u = 600, v = 600, w = 6000) have the same îh,u,v,w but hypergeometric
probabilities of 0.1895 and 2.3028 × 10−165 , respectively. This exaggerated example shows
that the mutual information estimate does not take into account the sample size w in judging
the significance of the dependence between two random variables, unlike the corresponding
hypergeometric probability PH|U,V,W (h|u, v, w). Since a larger value of w should increase
ˆ i ; Xj ), random variable H when conditioned on U = u,
the confidence in our estimate I(X
V = v, and W = w is a more appropriate test statistic for evaluating mutual information
significance. As before, outcomes h, u, v, and w are completely determined by discrete
vectors of binding data xi and xj for proteins i and j. To find the p-value, or the probability
of randomly estimating a mutual information of a positive binding relationship equally or
more significant than îh,u,v,w , we sum over the right tail of the hyper-geometric test statistic
H|U = u, V = v, W = w for the outcome h as follows:
40
min (u,v)
pvM I (xi , xj ) = Pr(H ≥ h|u, v, w) =
X
r=h
w−v v
u−r
r
w
u
.
(3.28)
As before, we express our mutual information p-value pvM I (xi , xj ) as a function of the
binding data xi and xj . The hyper-geometric p-values above only evaluate the significance of
synergistic binding between two factors, while mutual information can capture both positive
and negative binding relationships. To evaluate the p-value of a negative binding relationship,
or the probability of having an overlap of h or smaller at random, we would need to sum
from 0 to h in (3.28). However, due to the high sensitivity at h = 0 and due to the large
amount of false positives and false negatives in our binding data, this evaluation proves
unreliable. Opposing binding relationships can evince interesting biological phenomena, as
well, but it becomes clear later why such relationships complicate future analyses and their
biological interpretation. Hence, we avoid using mutual information in finding negative
binding relationships and will exclusively use the filtered correlation coefficients for that
purpose. Now that the we have developed a framework for evaluating the significance of
our mutual information analysis, we can substantiate our decision for using the P V and P R
ˆ i ; Xj ).
data representations in classifying the data and finding I(X
3.2.2
Evaluating Data Classification: Mutual Information Analysis
This section gauges how accurately the different data representations classify their binding
information into 0s and 1s, in order to estimate the mutual information and find p-values as
in Sections 3.2 and 3.2.1. We generated a test set of 1447 probable protein-protein binding
dependencies, most of which were confirmed or suggested by published literature. We also
created a test set of 20707 extremely unlikely relationships. In order to compare the different
matrices in an unbiased manner, we designated reasonable thresholds for classifying the data
sets into a similar number of bound (1s) and unbound (0s) gene-protein interactions. Table
3.1 lists the data representations, their corresponding threshold test and the number of
41
Data
Matrix
BR
PR
N
PV
N + PR
PV + PR
Classification
Total Entries
Hit Rate (captured
Test for Binding
Classified Bound
/probable links)
BR ≥ 1.9
81450
738/1447
P R ≥ .965
77377
1402/1447
N ≥ −.002
79639
1409/1447
P V ≤ .015
80948
1445/1447
N ≥ −.002 or P R ≥ .99
80215
1414/1447
P V ≤ .015 or P R ≥ .99
81461
1446/1447
Total Noise at
Unlikely Links
5283.9
3784.5
1460.5
1021.1
1394.7
1021.0
Table 3.1: Hit rate (capture/probable links) at probable interactions and total noise at unlikely interactions based on mutual information estimates using different data representations
(see text).
entries classified as bound (all around 80,000) in the first three columns. The classified data
for each data representation was then used to find mutual information p-values, pvM I , as
in Section 3.2.1. The next column of Table 3.1 enumerates the hit rate, or the number of
relationships captured at a significance level of 10−10 divided by the total number of probable
links in our test set. The last column lists the total noise at unlikely links, computed by
accumulating the total significance at interactions in second test set. The significance of an
interaction between proteins i and j was calculated by replacing pvC with pvM I in (4.4). A
higher hit rate should correspond to fewer false negatives for a given data representation,
while less total noise at unlikely relationships should lead to fewer false positives.
The P V (equivalent to thresholding on SD) matrix performs best in individually classifying the data, since it has the highest hit rate at known interactions and the lowest total
noise at unlikely interactions. The N and P R matrices perform slighly worse in confirming
probable relationships. However, the N and P R matrices introduce 43% and 271% more
noise than the P V matrix, respectively, which will undoubtedly lead to more false positive
claims. Since some proteins have very few gene targets at the chosen thresholds, the classified, binary data would provide little information about their behavior. In order to use
mutual information to compare these proteins with the rest, the last two rows in Table 3.1
classify the data using a combination of the top two data representations (P V and N ) and
the P R data representation, guaranteeing that at least the top 1% of most immunoprecipitated gene probes is considered bound for each factor. The number of entries classified
42
as bound increase negligibly, since most publications already considered the strongest 1%
of gene-protein interactions for each protein as bound. Creating a buffer of bound targets
slightly improves the hit rate of both the individual N and P V matrix at no expense of
added noise. These results substantiate our decision to use the combination of the P V and
P R matrices for classifying the data in order to estimate the mutual information p-values.
Now that we have a method for evaluating the significance of both our filtered correlation
coefficient and mutual information analyses, we can combine the evidence from the two
approaches in order to improve the reliability of our biological predictions.
3.3
Combining P -values
The p-value calculations for both the filtered correlation coefficient and mutual information
estimation test similar null hypotheses. In Section 3.1.1, we test the null hypothesis that
two binding profiles Xi and Xj are not linearly dependent (ρx,y = 0) while in the last section
we considered the null hypothesis that the two profiles are not probabilistically dependent
(I(Xi ; Xj ) = 0). Ultimately, we aim to find out whether two proteins belong in the same
biological process. Hence we want to combine the p-values from the two analyses in order to
incorporate both sources of evidence in testing the overall null hypothesis that two proteins
with binding profiles Xi and Xj do not share a biological function.
We combine p-values using Fisher’s method. It assumes that the p-values result from
independent studies that use continuous random variables as test statistics to challenge similar null hypotheses. Since we use the same data source to estimate the filtered correlation
coefficient and mutual information, our studies are not completely independent. Moreover,
the hyper-geometric test statistic is not continuous. Despite these approximations, the authors in [37] and Section 3.3.1 demonstrate how combining p-values using Fisher’s method
improves biological prediction.
In order to derive Fisher’s method for combining p-values for synergistic binding relationships, we only consider positive correlations as significant for the following. Hence, in this section we calculate our p-values for the filtered correlation coefficients using a one-sided test, or
43
using (3.15) without the factor of 2. Let p1 = Pr(T ≥ tρ̂,n−2 ) and p2 = Pr(H ≥ h|u, v, w) represent two p-value observations that result from test statistics T and H|U = u, V = v, W = w
and observations (ρ̂, n − 2) and h, respectively. Using the assumption of independent studies, the joint probability of observing events T ≥ tρ̂,n−2 and H ≥ h|u, v, w under the null
hypothesis equals
Pr(T ≥ tρ̂,n−2 , H ≥ h|u, v, w) = Pr(T ≥ tρ̂,n−2 ) Pr(H ≥ h|u, v, w) = p1 p2 .
(3.29)
From the equation above, it seems natural to consider the observation of p-value pairs
(p1 = 0.01, p2 = 0.01) and (p1 = 0.1, p2 = 0.001) with equivalent products p1 p2 as evenly
significant sources of evidence for rejecting the null hypothesis H0 . Intuitively, this refers to
having two equally strong sources of evidence for dependence between two proteins or having
a slightly stronger and a slightly weaker source of evidence. Hence, we want to make decisions
about the strength of our combined evidence using the test statistic K = P1 P2 , where P1 and
P2 are the p-values resulting from the filtered correlation coefficient and mutual information
estimations, respectively. Since P1 and P2 depend on assumed independent random variables
T and H, they are themselves independent random variables. The distributions of p-values P1
and P2 resulting from a continuous test statistics is exactly uniform on the interval [0, 1] [37].
Although, our test statistic for mutual information is discrete, assuming a continuous uniform
distribution for P2 is just an approximation that only slightly affects the accuracy of our
evaluation [37].
To find the overall p-value for a given p-value product p1 p2 , we need to evaluate the
probability of randomly arriving at a p-value product more significant than the observation
k2 = p1 p2 . This corresponds to the region in probability space where p1 p2 ≤ k2 . Hence, we
need to find the volume of the region p1 p2 ≤ k2 over the joint distribution of P1 and P2 .
Since P1 and P2 were shown to be independent and uniformly distributed random variables,
their joint distribution PP1 ,P2 (p1 , p2 ) is a 2-dimensional unit cube defined on {(p1 , p2 ) : p1 ∈
[0, 1], p2 ∈ [0, 1]}. The combined p-value from the two analyses for observed p-value product
44
k2 , pvC (k2 ), follows from the integration below:
ZZ
pvC (k2 ) =
=
PP1 ,P2 (p1 , p2 )dp1 dp2
p1 p2 ≤k2 , 0≤p1 ,p2 ≤1
Z 1
Z k2
k2
1dp1 +
dp1 = p1 |k02 +k2 ln p1 |1k2 =
p
1
k2
0
k2 − k2 ln k2 .
(3.30)
Since p1 = pvF CC (xi , xj ) and p2 = pvM I (xi , xj ) for two proteins i and j with binding data
vectors xi and xj , the combined p-value for the binding relationship between proteins i and
j only depends on xi and xj and can be equivalently denoted as pvC (xi , xj ). The equation
in (3.30) only depends on the number of dimensions, 2, and the product of our observed
p-values, k2 . The extension for combining p-values from n independent experiments, also
depends only on n and the observation kn = p1 p2 · · · pn . Now, the integration takes place
over the n dimensional region p1 p2 · · · pn ≤ kn of the joint distribution PP1 ,...,Pn (p1 , . . . , pn ).
Again from independence, the joint distribution is an n-dimensional unit hypercube defined
on {(p1 , . . . , pn ) : p1 ∈ [0, 1], . . . , pn ∈ [0, 1]} and the combined p-value is
pvC (kn ) = kn
n−1
X
(− ln kn )r
r=0
r!
.
(3.31)
The following section mitigates the problem of assuming independent analyses and illustrates how combining p-values can improve biological prediction.
3.3.1
Biological Predictions using Combined P -values
To show the benefit of combining p-values we present an example of a biological prediction
that is representative of the rest of the pairwise protein analysis. Figure 3-2 shows filtered
correlation coefficient, mutual information, and combined p-values for a test set of proteins.
The filtered correlation coefficient and mutual information p-values were created using onesided models that only considered positive relationships as significant. The scale on the right
is in units of log10 (pv) ranging from -2 (or p-value = .01) to -20 (or p-value = 10−20 ). Protein
45
names are listed on the x and y axis with and “o” or “i” at the end of the names signifying
that the data came from an ORF or intergenic array, respectively. Since combining p-values
integrates evidence from the two complementary pairwise analyses, it should lead to fewer
errors in deciding whether two factors have a significant binding dependence.
The first 3 factors, RSC3, RSC8, and RSC, are components of the RSC nucleosome remodeling complex. Since these proteins carry out tasks as part of the same complex, we
would expect a high degree of similarity in their binding profiles, as shown in [7]. Indeed,
each subfigure shows a p-value ≤ 10−20 for the interactions within the entire complex (dark
blue box on the top left). For individual analyses, we consider a p-value of 10−10 as significant for a pairwise interaction. However, since we are combining two studies that are not
fully independent, we consider a p-value of 10−20 as significant for the combined p-values. If
our two analyses were completely dependent on one another, the p-values testing the same
null hypothesis should be identical and a significance of 10−10 would translate to a significance of 10−20 when the p-values are combined. However, since our pairwise studies are not
fully dependent on one another, a p-value threshold of 10−20 for combined p-values is more
stringent than a p-value threshold of 10−10 for a single analysis.
The next two proteins LRP1 and RRP6, are involved in mRNA degradation and surveillence, respectively. In [16], the authors show that LRP1 depends on RRP6 for recruitment
to a large fraction of its gene targets. Therefore, we would expect similarity in their binding
profiles and both analyses capture this interaction with a low p-value. Curiously, protein
PRP20 also shows high similarity with the LRP1-RRP6 biological process. PRP20, also
known as RanGEF in humans, binds preferentially to inactive genes [13]. This leads us to
a novel biological hypothesis. It is known that the exosome (a complex of proteins which
also contains LRP1 and RRP6) controls protein synthesis by degrading the mRNAs of genes
that are improperly formed, processed, or transcribed at a given time. For example, we
would not want the GAL proteins, involved in galactose breakdown, to be synthesized when
the cell is grown in a dextrose-rich YPD medium. Moreover, [13] shows that PRP20 binds
to genes that should be turned off during a given condition, such as GAL genes in YPD.
46
(a) Filtered correlation coefficient p-values for
a test set of proteins.
(b) Mutual information p-values for test a set
of proteins.
(c) Combined p-values for test a set of proteins.
Figure 3-2: Filtered correlation coefficient (a), mutual information (b), and combined (c)
p-values for a test set of proteins. The filtered correlation coefficient and mutual information
p-values were created using one-sided models that only considered positive relationships as
significant. Protein names are listed on the x and y axis with an “o” or “i” at the end of
the names signifying that the data came from an ORF or intergenic array, respectively. The
scale on the right is in units of log10 (pv) ranging from -2 (or p-value = .01) to -20 (or p-value
= 10−20 ).
47
The fact that the exosome binds to a similar set of gene targets as PRP20 suggests that
inactive genes are actually not completely turned off. It may be possible that the process of
transcription is leaky, in which case inactive genes that should not be transcribed in a given
condition may be erroneously transcribed. Indeed, there is a growing body of evidence in
the transcription field to suggest that RNA polymerase is promiscuous in its induction of
transcription at non-ORF sites within the genome, the functional consequences of which are
still unclear. This suggests that the exosome may have a role in quality control at inactive
genes, binding to them in order to readily degrade their unwanted mRNA.
The last 2 proteins, Polymerase III (POL3) and the TATA-Box Protein (TBP), also associate significantly for all three p-value calculations. In a recent paper [28], the authors show
that the TBP preferentially associates with gene targets of POL3 in various environmental
conditions. Although, the POL3 and TBP experiments were done in different laboratories,
our normalized analyses confirm the results in [28].
After identifying connections within 3 biological processes, we consider the interplay between the three complexes. The authors in [7] show that the RSC nucleosome remodeling
complex has a preference for POL3 gene targets. Hence, we should expect significant interaction in the top right corner of Figures 3-2(a), 3-2(b), and 3-2(c). The filtered correlation
coefficient analysis finds all 6 interactions significant (p-value ≤ 10−10 ) but the mutual information and combined p-value calculation finds 5/6 relationships as significant (p-value
≤ 10−10 and p-value ≤ 10−20 , respectively), making an error on RSC3-POL3. The filtered
correlation coefficient analysis additionally predicts interactions LRP1-POL3 and LRP1TBP but no relationships between its recruitment factor RRP6 and POL3 or TBP. Due to
this discrepancy and due to the lack of literature supporting this interaction, we consider
these two predictions as likely false positives. The mutual information and the combined
p-value do not make this probable mistake. In summary, the filtered correlation makes 2 potentially false positive claims while the mutual information and combined p-values make one
false negative error. Moreover, if we reduce the combined p-value threshold for significance
to a less conservative and more accurate 10−18 (since our complementary analyses are not
48
completely dependent), we obtain 0 errors.
This example accurately represents the overall trend that correlation coefficient and mutual information tests will generate a higher number of false positives and false negatives,
respectively. This is because mutual information makes a hard decision on classifying the
data as 0s and 1s prior to the analysis and cannot capture relationships when a significant
number of errors are made in the classification, resulting in false negatives. In contrast,
the filtered correlations cost function considers a soft and continuous version of the data in
terms of standard deviations of confidence and does capture these relationships. However,
this also leads to considering unbound genes as partially bound and creates a higher number
of false positives. Combining the two complementary analyses leads to fewer errors or to a
more optimal operational point on an ROC curve of false negative rate versus false positive
rate. We use this concept in building a more reliable network of our nucleus in Chapter 5.
The next section attempts to reduce the false positive rate of predictions based on mutual
information.
3.4
Minimized Mutual Information P -value
The previous section revealed that mutual information p-values are highly sensitive to the
chosen cutoff for classifying the binding data into 1s and 0s. To alleviate this problem,
this section introduces minimized mutual information p-values. This method allows for
the binding cutoffs to take on several values from an allowable set and finds the cutoffs
that minimize the mutual information p-value between two proteins. Section 3.2.2 shows
that transforming data into 0s and 1s seems most appropriate when using the P V and P R
matrices. Let A = {0.001, 0.005, 0.01, 0.02} denote the allowable set of p-value cutoffs for
thresholding the P V matrix. Moreover, let ci , cj ∈ A denote the p-value cutoffs of proteins
i and j for transforming continuous row vectors xi and xj from the P V matrix to binary
vectors, respectively. For each combination of cutoffs ci , cj ∈ A, lets also consider the
strongest 1% of gene-protein interactions for proteins i and j as bound. Then the minimized
mutual information p-value, pvM M I , between binding vectors xi and xj takes the form
49
pvM M I (xi , xj ) =
min
ci ∈A,cj ∈A
pvM I (xi , xj ) ,
(3.32)
where pvM I is the mutual information p-value previously calculated in (3.28). The minimized
mutual information p-value mitigates the sensitivity in the mutual information analysis due
to hard classification of the data. However, since this analysis allows for the same protein to
have different bound cutoffs, it is only suited for making decisions on pairwise interactions
between two proteins and does not allow for unbiased group-wise comparisons. Hence, we will
use the minimized mutual information p-values for making decisions about adding links in a
network between two proteins (Chapter 5) but refrain from using it for inferring group-wise
relationships between proteins (Chapter 4).
Filtered correlation coefficient, mutual information, and combined p-values allow us to
judge the strength of pairwise interactions between two proteins. The following chapter
introduces clustering and PCA, which can evince group-wise relationships between factors
that regulate transcription.
50
Chapter 4
Group-wise Relationships: PCA and
Clustering
Biological processes depend on the collaborative interaction of several proteins. In the previous chapter we developed pair-wise statistics in order to describe the relationship between
two proteins. This chapter introduces Principal Component Analysis (PCA) and clustering,
two methods that can evince group-wise dependencies between proteins. The two techniques
verify known biological mechanisms and uncover novel cellular processes.
4.1
Principal Components Analysis
PCA is a common technique for data reduction and visualization. It determines the directions
of the greatest variance within a data matrix by calculating the orthonormal eigenvectors
associated with the largest eigenvalues of the scatter matrix of the data [36]. The principal
components, usually referred to as scores, are row vectors of the same dimensionality as row
vectors of the data matrix. The data can be modeled by taking linear combinations of the
scores; the associated weights with each score are called loads. Since most of the variance
within a data set can usually be captured with a small number of principal components,
PCA exploits the common information within the data in order to reduce its dimensionality.
51
One can choose the number of principal components one wishes to calculate; more principal
components can characterize the data better at the cost of having to keep track of more
information. For visualization purposes, two or three principle components are often used.
For a given choice of data representation, let xi = [xi,g1 . . . xi,g|G| ] denote a row vector of
binding data for protein i across all genes g in the set G. If performing PCA on the binding
profiles of each protein i, we can write
xi = µ̂i 1 +
N
X
wni sn .
(4.1)
n=1
The PCA equation (4.1) uses N principal components, with row vector scores s1 , . . . , sN and
i
} specific to each xi , where µ̂ is a scalar that is the mean across the
scalar loads {w1i , . . . , wN
elements of xi , and 1 is row a vector of all ones.
Figure 4-1 shows how PCA can provide useful biological insight. The large isolation
of protein Prp20 and Nup100 from the rest of the nuclear pore related proteins in Figure
4-1(b) supports a model introduced in [13]. Casolari et al. shows that Prp20, known as
RanGEF in humans, binds preferentially to transcriptionally inactive genes. RanGEF is also
known to catalyze the unloading of transcription factors (TF) transported into the nucleus by
nuclear import proteins, such as Kap. The left part of Figure 4-1(a) illustrates how RanGEF
converts RanGDP to RanGTP which in turn unloads the TF transported into the nucleus by
Kap. Hence, the authors suggest that Prp20 may contribute to transcriptional activation by
facilitating the release of transcription factors at genes requiring fast activation. Accordingly,
induction of said genes leads to experimentally confirmed loss of RanGEF association and
a gain in binding at the nuclear pore on the periphery of the nucleus, as illustrated on the
right part of Figure 4-1(a). Thus, Prp20 and the rest of the nuclear pore proteins should
bind to very few genes in common at any one time as shown by the great separation between
the two in Figure 4-1(b). Further, the results suggest that the other aberrant nuclear factor,
Nup100, also has a role that is orthogonal to the workings of the majority of the nuclear
pore proteins. The following section discusses another method for finding biological processes
using ChIP-chip data.
52
(a) Casolari et al. Model for Nuclear Transport (picture from [13])
Principal Component Scatter Plot with Colored Clusters
100
1
2
3
Nup100
Second Principal Component
80
60
40
20
0
−20
−50
Prp20
0
50
100
150
200
250
First Principal Component
(b) Principal Component Analysis Validates the Model
Figure 4-1: The subfigures show (a) Casolari et al. model for nuclear transport and (b) its
validation using Principal Component Analysis (PCA) [13]. Figure (b) plots the loads on
the first two principal components. The colors denote clusters constructed using hierarchical
clustering based on correlation coefficient distance, as defined in Section 2.2.2, and a cutoff
for three clusters.
53
4.2
Clustering
Clustering of genomic data can evince group-wise relationships between proteins or genes. We
first introduce K-means and hierarchical clustering and then adapt the algorithms in order
to use the pair-wise distance metrics developed in the previous chapter. We then develop a
novel semi-supervised clustering algorithm that preserves information about elements within
each cluster in order to better capture group-wise dependencies between proteins.
4.2.1
K-means Clustering
K-means clustering is one of the most commonly encountered algorithms in biology, due to
its ease of implementation. K-means clustering uses the expectation maximization (EM)
algorithm to partition (cluster) N elements into K disjoint subsets Sk , k = 1, . . . , K. Given
an appropriate choice of data representation, let xi denote a row vector of binding data for
protein i, as in Section 4.1. In order to find groups of related proteins, we represent binding
profiles as elements that we want to cluster. The K-means algorithm finds K disjoint subsets
Sk , k = 1, . . . , K that minimize the overall cost function
J=
K X
X
d2 (xi , ck ) ,
(4.2)
k=1 i∈Sk
where d(xi , ck ) is the distance between binding profile xi and centroid ck of subset Sk . The
algorithm usually uses Euclidean distance for d and defines ck as the mean of all the elements
in Sk :
ck =
1 X
xi .
|Sk | i∈S
(4.3)
k
The user initiates the clustering by designating the K underlying groups of objects. Then
the K centroids are initialized, usually by assigning each element to one of the K clusters at
random and using (4.3). The algorithm then iterates between (i) assigning all the elements to
subset Sk with the “closest” (in terms of distance d) centroid ck (E-step) and (ii) reestimating
54
the cluster centroids based on the new assignments using (4.3) (M-step). The procedure stops
once no further change in assignments occurs.
We adapted the K-means algorithm in order to incorporate the pairwise statistical analysis from Chapter 3. The following formula converts the combined p-values from Section 3.3
to a non-negative distance, where a small p-value corresponds to a small distance between
the binding profiles xi and xj of two proteins i and j:
d(xi , xj ) = − log10 (1 − pvC (xi , xj )) .
(4.4)
K-means clustering (as any EM based algorithm) does not guarantee global minimization
of the cost function in (4.2), and its performance heavily depends on the initialization of the
K centroids. Due to this problem, the K-means algorithm often fails to identify known
protein clusters (e.g., the RSC complex). Other EM based partitioning algorithms, such as
fuzzy membership clustering, would also suffer in performance due to random initialization;
therefore, the next section introduces a different family of algorithms based on hierarchical
clustering.
4.2.2
Hierarchical Clustering
Hierarchical clustering has wide applicability in biology, as well. In addition, hierarchical
partitioning does not require a priori knowledge of the number of clusters and can be easily
visualized using a tree structure (called dendrogram), making it often preferable to K-means
clustering. Hierarchical clustering is subdivided into agglomerative methods, which proceed
by series of fusions of the N objects into groups, and divisive methods, which separate N
elements successively into finer subsets.
We adapted the agglomerative hierarchical clustering algorithm in order to incorporate
the pairwise statistical analysis from Chapter 3. As in the previous section, we use binding
profiles as elements and define all the pairwise distances between elements using (4.4). At
the start, the algorithm treats each element as a cluster, and proceeds for N − 1 iterations.
At each iteration, the algorithm links the two most similar clusters (represented as linking
55
two child nodes to a parent node on a dendrogram) until all N elements are unified into
one partition. Hierarchical clustering algorithms can define distance (or similarity) between
clusters in various ways. The following equations describe several common distances for
linking two clusters Ck and Cl :
Nearest Neighbor :
d(Ck , Cl ) =
Farthest Neighbor :
d(Ck , Cl ) =
Average :
d(Ck , Cl ) =
d(xi , xj )
(4.5)
max d(xi , xj )
(4.6)
min
i∈Ck ,j∈Cl
i∈Ck ,j∈Cl
XX
1
d(xi , xj ) .
|Ck ||Cl | i∈C j∈C
k
(4.7)
l
Hierarchical clustering based on combined p-values from Section 3.3 confirms known biological processes with great accuracy, especially when using the average distance for linking
clusters. However, hierarchical clustering only considers pairwise relationships between binding profiles. Figure 4-2 uses an example to illustrate the potential pitfalls with clustering
solely on pairwise distances. The Venn diagrams on the left and on the right describe two
hypothetical relationships between three factors (proteins). Each circle or oval represents
the subset of genes classified as bound by the six different factors. In the left scenario, each
possible pair of factors (i.e., A↔B, A↔C, and B↔C) have a number of genes that they bind
to in common. But when considering the group-wise relationship (i.e., A↔B↔C), we find
no common intersection in bound genes between all three factors. In contrast, the scenario
on the right shows that factors X, Y, and Z share a large common intersection and thus
interact in a pairwise as well as group-wise manner. Since both sets of three factors have
equivalent pairwise distances, hierarchical clustering cannot distinguish any difference between the two scenarios. However, the much stronger group-wise interaction of the scenario
on the right makes it more likely for factors X-Y-Z to share a common biological process
than for factors A-B-C. In the next section, we develop a novel semi-supervised clustering
algorithm that preserves information about elements within each cluster in order to better
capture group-wise dependencies between proteins.
56
Figure 4-2: The Venn diagrams on the left and on the right describe two hypothetical relationships between three factors. Since both groups of three factors have equivalent pairwise
overlaps in bound genes, hierarchical clustering cannot distinguish any difference between the
two scenarios (see text). However, the much stronger group-wise interaction of the scenario
on the right makes it more likely for factors X-Y-Z to share a common biological process
than for factors A-B-C.
4.2.3
Semi-Supervised Clustering
Semi-supervised clustering derives its name from the fact that it retains information about
elements within each cluster as it partitions the objects. In order to preserve information
about the elements of cluster Ck the algorithm maintains two vectors, fk and xk . Vector
fk = [fk,g1 . . . fk,g|G| ] records the fraction of elements that bind to each gene g in the set
G. Vector xk = [xk,g1 . . . xk,g|G| ] represents the averaged binding profile at gene g ∈ G for
all members within the partition, based on the SD data representation. When merging two
clusters Ck and Cl into Co , the resulting two vectors for cluster Co are weighted combinations
of the vectors for Ck and Cl :
1
(|Ck |xk + |Cl |xl )
|Ck | + |Cl |
1
fo =
(|Ck |fk + |Cl |fl ) .
|Ck | + |Cl |
xo =
(4.8)
(4.9)
Note that merged vector fo still maintains the fraction of genes bound by the elements of
57
the new cluster and xo still represent the average binding profile of all the joined objects.
To define similarity between clusters, we again use the distance based on combined p-values
in (4.4). Although there are many choices for how to define similarity, we wanted to incorporate the robust measures we derived in the previous chapter. Appendix A develops the
algorithm with a more theoretically intuitive but less biologically meaningful distance based
on Kullback-Leibler(KL) divergence.
To find combined p-values for the relationship between two clusters, the algorithm evaluates the filtered correlation coefficient and mutual information but incorporates the groupwise dependence by modifying how to select the pertinent sets. For filtered correlation, let
Fk and Fl represent the filtered subsets of genes for clusters Ck and Cl , respectively, and
let Fk,l = Fk ∪ Fl again denote the filtered subset of genes over which we will estimate
the variances and covariance of two partitions using (3.1) - (3.5). The filtered subsets Fk
and Fl no longer consist of the set of genes bound by one factor but now represent the
sets of genes bound by a fraction of objects within a cluster. Mathematically, letting fthresh
represent the fraction of proteins that need to bind to a gene in the filtered subsets, we
define Fk = {g : fk,g ≥ fthresh } and Fl = {g : fl,g ≥ fthresh }, respectively. Having defined Fk,l = Fk ∪ Fl , the algorithm uses the averaged binding vectors xk and xl in (3.1) (3.5) to compute the means, filtered variances, and filtered covariance of clusters Ck and
Cl . Next, using (3.6), (3.14), and (3.15), the algorithm derives the filtered correlation coefficient p-value between the two partitions. For finding mutual information p-values, we
can now assign u = |Fk |, v = |Fl |, h = |Fk ∩ Fl |, and w equal to the number of genes with
binding information for both clusters. Using these quantities, we can use (3.28) to find the
mutual information p-value between two clusters. And finally, we again combine p-values
using (3.30).
Similar to hierarchical clustering, the algorithm treats each element as a cluster at the
start and proceeds for N − 1 iterations. At each iteration, the algorithm links the two most
similar partitions, based on the combined p-value distance in (4.4), until all N elements
are unified into one partition. However, the new method for finding the combined p-values
58
between clusters incorporates the group-wise dependence of elements, rectifying the potential
problem with using hierarchical clustering. The next section validates the algorithm using
several known biological processes.
4.2.4
Biological Validation of Semi-Supervised Clustering
In order to evaluate the performance of the new algorithm, we clustered the SD data representation using a bound cutoff of P V ≤ .015 ∨ P R ≥ .99, where ∨ denotes logical or,
consistent with our results in Sections 2.2.2 and 3.2.2. Moreover, we set fthresh = 0.5 in order
to consider genes bound by half or more of the factors in a partition as representative of the
biological process of the partition. We first clustered the whole data set and then extracted
the produced clusters into separate plots to facilitate visualization. Figure 4-3 illustrates
how the algorithm performs on a selected group of biological processes using tree structures
or dendograms. The horizontal branches on the dendrogram represent the merging of two
similar objects at an iteration in the algorithm, while the length of the vertical branches
corresponds to the similarity distance (as in (4.4)) of the fused nodes. Unlike K-means,
semi-supervised clustering does not decide the final number of clusters but allows the user
to choose a significant threshold distance for a cluster. In Figure 4-3, we chose a distance of
less than 303 as significant, which corresponds to a combined p-value of 10−20 in our implementation. For visualization purposes, however, we color coded clusters of distance ≤ 253
or of combined p-value ≤ 10−70 .
Figure 4-3(a) shows the clustering of four known biological processes using different colors.
It again illustrates the significant association of components of the RSC nucleosome remodeling complex (RSC, RSC1, RSC2, RSC3, RSC8, RSC9, and STH1) and of components
of the Polymerase III transcriptional machinery (POL3, TBP, BRF, TRF4, and RPC34).
It also demonstrates the significant clustering between transcription factors STE12, DIG1,
and TEC1. In [5], the authors show that STE12 activates two different sets of genes under
different conditions. In the presence of a nutrient limiting agent butanol, binding of STE12,
DIG1, and TEC1 activates genes necessary for growth of filaments, or structures that expand
59
(a) Validation of the RSC, POL3, STE12, and GAL biological processes.
(b) Clustering of the MCM/ORC, SIR, and LRP1 biological processes.
Figure 4-3: Confirmation of known biological processes and new insight using semi-supervised
clustering (see text). The two dendrograms above show a subset of the results from clustering
the whole data set using the SD data representation and a bound cutoff of P V ≤ .015∨P R ≥
.99. We set fthresh = 0.5 in order to consider genes bound by half or more of the factors
in a partition as representative of the biological process of the partition. The horizontal
branches signify the merging of two similar objects, while the length of the vertical branches
corresponds to the similarity distance (as in (4.4)) of the fused nodes. For visualization
purposes, different colors represent clusters of distance ≤ 253 or of combined p-value ≤ 10−70 .
60
the size of the cell and facilitate storage of nutrients. During pheromone exposure, binding of
STE12 and DIG1 (no TEC1) induces genes involved in mating. Although our data monitors
the binding of the three factors under no treatment in YPD, the clustering still evinces the
strong binding dependence between STE12 and DIG1, and their auxiliary relationship with
TEC1. Moreover, Figure 4-3(a) also confirms the significant interdependence between the
POL3 machinery and RSC complex (p-value ≤ 10−40 ). It further suggests a previously unknown relationship between the STE12 cluster and the POL3 and RSC biological processes.
And finally, the right most partition shows significant similarity between transcription factors GAL3 and GAL80. As discussed in Section 2.1, the two transcription factors regulate
the expression of genes that breakdown the sugar galactose. Similar to the STE12 analysis,
using YPD data again uncovers the general relationship between the two galactose factors.
However, we do not believe that the GAL transcription factors share significant functions
with the previous three processes (p-value ≥ 10−5 ).
Figure 4-3(b) displays a dendogram of several other known and hypothesized biological
mechanisms. The rightmost colored tree corroborates our hypothesis that LRP1, RRP6,
and PRP20 may synergistically bind at inactive genes. As discussed in Sections 3.3.1 and
4.1, LRP1 and RRP6 may bind to inactive genes in order to readily degrade their unwanted
mRNA, while PRP20 may bind at inactive genes to facilitate fast activation [13]. Moreover,
the association LRP1 and RRP6 with mRNA export factor YRA1, supports the results
in [16] that demonstrate coupling of mRNA production quality assurance to mRNA export.
Further, the similar relationship between NUP100 and PRP20 suggests that NUP100 may
also have a role that is orthogonal to the workings of the majority of the nuclear pore factors,
as predicted in the PCA analysis. However, this cluster does not seem to share significant
commonality with the rest of the colored trees on the figure.
The other three colored clusters in Figure 4-3(b) also provide validation of our analysis
and new biological insight. During normal growing conditions, a yeast cell periodically cycles
through several phases of growth. If a mother cell has sufficient nutrients in its environment
it may decide to divide into two daughter cells, in a phase of the cell cycle called mitosis. To
61
make sure that the daughter cells inherit the genetic information of the mother cell, prior to
mitosis and during the S phase of the cell cycle, the mother cell makes two identical copies
of all of its genetic material in a process known as DNA replication. In [8], the authors show
that the Origin Recognition Complex (ORC) and the MiniChromosome Maintenance (MCM)
proteins bind together at origins of replication, or sites along the chromosomes where DNA
replication is initiated. The two left most colored trees confirm the significant association of
components of the ORC (ORC1, ORC1-6) and MCM (MCM3, MCM4, and MCM7) complex
at both intergenic and open reading frame (ORF) regions of the genome, denoted by an “i”
and “o” at the end of each protein name, respectively. Although DNA replication only takes
place during the S phase of the cell cycle, and despite the fact that our unsynchronized
ChIP-chip data samples cells during all phases of growth, the semi-supervised clustering still
uncovers the relationship described in [8]. The next colored partition confirms the significant
association between components of the SIR complex at both intergenic and ORF regions [21].
Moreover, the SIR complex associates significantly (p-value ≤ 10−35 ) with the MCM/ORC
cluster, suggesting a mutual dependence between the two processes. The binding profile of
SIR2o also has a close relationship with RAP1o, a TF involved in transcriptionally active
processes, which we will explore in more detail in the next section.
4.2.5
Active Processes and SIR2
Transcriptionally active processes, or simply active processes in our context, refer to biological mechanisms that occur at highly expressed genes. For example, the model introduced
in Section 4.1 [13] of how highly expressed genes bind to the nuclear pore describes an active process. To gauge how actively a gene is expressed in YPD, we downloaded data for
the number of mRNAs produced per hour, or the transcriptional frequency of a gene, as
measured in [12]. The results from [12] show that only about 21% of the genes produce
more than 6 mRNAs/hr. Therefore, we define genes with transcriptional frequency in the
top 20% as active. To determine whether a factor participates in active processes, we again
calculate filtered correlation coefficient, mutual information, and combined p-values between
62
the bound set of a factor and the set of active genes. Mathematically, we let A represent the
set of active genes and let Fi represent the set of genes classified as bound for a potential
active factor i. Then we define Fi,j = Fi ∪ A in the filtered correlation coefficient equations
(3.1) - (3.5), and use (3.6), (3.14), and (3.15) in succession to derive p-values. For finding
mutual information p-values, we can now assign u = |A|, v = |Fi |, h = |A ∩ Fi |, and w equal
to the number of genes with observed information for both data sets. Using these quantities,
we can use (3.28) to find the mutual information p-value. And finally, we again combine
p-values using (3.30) and consider a factor as active if it has a combined p-value ≤ 10−20 or
a mutual information p-value ≤ 10−10 (justified in Section 5.1).
In order to explore group-wise relationships between active factors, we categorized all
proteins as active or non-active using the stringent criterion above. Figure 4-4 displays the
results of clustering only the active factors using the SD data representation and a bound
cutoff of P V ≤ .015 ∨ P R ≥ .99 once again. Note that all the factors included have a
significant similarity distance of ≤ 283, or cluster combined p-value ≤ 10−40 . The results
provide interesting biological validation and exciting new insight. Figure 4-4 includes all the
nuclear proteins considered to have a preference for binding to active genes in [13] (CSE1,
NUP116, NIC96, MLP1, MLP2, XPO1, NUP2, KAP95, and NUP60) and in [14] (THO2,
NPL3, and HMT1), justifying the above criterion for finding active factors. Moreover, the
tight clustering of the active nuclear pore factors once again validates the model in [13],
which suggests that these factors share a biologically active mechanism. The large similarity
between nuclear proteins THO2, NPL3, and HMT1 also seems to confirm the results in
[14]. Yu et al. introduced a model showing that methyltransferase HMT1 plays a role in
disassociating transcriptional elongation protein THO2 and mRNA export factor NPL3.
Moreover, nonfunctional (mutated) HMT1 leads to failed export of mRNA produced at a
selected active gene. Since all three factors bind preferentially to highly expressed genes, they
seem to share an active biological process, whereby HMT1 dependent dissociation of THO2
and NPL3 may prove necessary for export of mRNAs produced by active genes. Further,
their close relationship with the nuclear pore proteins implies that this mechanism might
63
also occur at the periphery of the nucleus.
The red tree on the left part of Figure 4-4(b) illustrates another active biological process.
Histone acetyltransferases (HATs) ESA1 and GCN5 have a predilection for binding to active
genes [4], while data H3i, H2Bi, and iNuclsm measuring nucleosome depletion also overlaps
significantly with promoters of active genes [27]. Moreover, transcription factors FHL1 and
RAP1 regulate protein biosynthesis and other active processes [38]. The nearby congregation
of RAP1, nucleosome depletion, and HATs ESA1 and GCN5 seems to support a conjecture
mentioned in [27]. Berstein et al. showed that RAP1 is necessary but not sufficient for
the mechanism that displaces nucleosomes at highly expressed genes. Hence, the authors
conjectured that other factors, including ESA1 and GCN5, might be required along with
RAP1 in order to reposition nucleosomes at induced genes. Moreover the large similarity
between this process and the nuclear pore factors as seen in Figure 4-4(a) again implies that
this active process takes place at the periphery of the nucleus.
Surprisingly, Figure 4-4(b) also uncovers a great similarity between the binding of SIR2
at ORF regions, denoted as SIR2o, and other active proteins. The SIR complex does not
seem to bind to DNA directly but interacts with other factors, such as RAP1 or deacetylation
at histone 4, in order to access the nearby genes. This explains the association of SIR2 with
active factor RAP1 in Figure 4-3(b). However, SIR2 is a histone deacetylase that plays an
important role in silencing transcription at telomeres (or ends of chromosomes) and matingtype gene loci HML and HMR [21]. Hence, the designation of SIR2o as an active factor
and its significant binding similarity with all other active factors is unexpected. In Figure
4-4(a), SIR2o associates with several active transcription factors, including GAT3, FHL1,
RAP1, and YAP5. It also has a significant binding similarity with semi-active (combined
p-value ≤ 10−10 instead of ≤ 10−20 ) transcription factors PDR1, RGM1, and SMP1. These
transcription factors (TFs) regulate various different biological pathways, including biosynthesis, environmental response, and metabolism. However, all the enumerated TFs bind
preferentially to nucleosome depleted promoters in a functionally cooperative manner [27].
This may suggest a role for SIR2 in nucleosome displacement. Moreover, SIR2 deacetylates
64
(a) Active cluster, part I
(b) Active cluster, part II
Figure 4-4: Active biological processes and new insight using semi-supervised clustering.
Figures 4-4(a) and 4-4(a) display the results of clustering all active factors (see text), using
the SD data representation and a bound cutoff of P V ≤ .015 ∨ P R ≥ .99 once again. We
again set fthresh = 0.5 in order to consider genes bound by half or more of the factors in
a partition as representative of the biological process of the partition. All members of the
cluster associate significantly (combined p-value ≤ 10−40 ), but for visualization purposes
different colors represent clusters of distance ≤ 253 or of combined p-value ≤ 10−70 .
65
lysine 16 on histone 4, which is generally associated with active processes. Finding the full
extent of SIR2o’s role in active processes, and why it only appears to take place at ORF and
not intergenic regions may uncover exciting and new biological insight.
Figure 4-4 also includes several other factors with proposed roles in active processes. First,
histone deacetylase HOS2 was shown to play a role in deacetylation at active genes in [20] and
clusters tightly with the nuclear pore factors in Figure 4-4(a). Second, [6] shows that histone
methyltransferase SET1, also in Figure 4-4(a), adds three methyl groups at lysine 4 of histone
3 during transcription (i.e., at active genes), which may serve as memory that a gene was
recently turned “on”. Third, Santos-Rosa et al. also shows an intimate connection between
SET1 and nucleosome remodeling complex protein ISW1, shown in Figure 4-4(b) [25]. The
authors show that ISW1 recruitment at active genes is dependent upon methylation by
SET1. Further, the paper illustrates that proper transcription at selected genes depends
upon the collaborative function of both ISW1 and SET1. And finally, [17] shows that high
acetylation level at histone 3 lysine 18 at both intergenic and ORF regions correlates with
gene activity. Figure 4-4(b) includes the three factors, designated as H3K18o, H3K18no,
and H3K18ni, where “o”, “no”, and “ni” at the end refer to ORF, normalized ORF, and
normalized intergenic regions [17]. Our clustering analysis captures the collective relationship
between all the mentioned active factors for the first time in the literature. In the next
chapter, we use the theory developed thus far to build a multi-layered transcriptional network
of the nucleus.
66
Chapter 5
Network of the Nucleus
This chapter builds on the methodology developed in the previous chapters in order to build
a network of the nucleus. Our holistic approach is the first attempt in the literature to
quantify the communication between layers in eukaryotic transcriptional networks. Analysis
of the finalized network will hopefully unveil a general view of the non-linear process of
transcription. The next section describes the model we used to determine the interplay
between various proteins.
5.1
Network Model
In order to build a transcriptional network of the nucleus, we represent each factor as a
node (or oval) and use the pairwise statistics developed in Chapter 3 to find significant
binding relationships between factors, represented as edges (or links) between nodes. Using
the convention of italicizing sets but not members of sets, we again define the following
five categories of proteins: Transcription Factors (T F ), Nuclear Transport proteins (N T ),
Nuclear Processing proteins (N P ), Histone Modifiers and Nucleosome Remodelers (HM ),
and Histone State (HS) in terms of acetylation levels, methylation levels, and nucleosome
distribution. We use an undirected network model, since our ChIP-chip data cannot evince
directionality in binding dependencies, or whether factor i causes factor j to bind to gene g.
67
Therefore, an edge between two factors i and j contains zero or two arrows in our model,
lacking information about directionality.
Our network model consists of two types of edges—a positive and a negative edge representing a significant synergistic and an opposing binding relationship, respectively. Since our
mutual information p-values only evaluate the significance of positive binding relationships,
decisions on extending negative edges will solely depend on correlation analysis. Hence, we
use the two sided integration in (3.15) to evaluate the significance of finding both extremely
positive or negative binding relationships. To avoid ambiguity when using correlation analysis, we set the p-value between proteins i and j to one if ρ̂Xi ,Xj < 0 when considering positive
edges and if ρ̂Xi ,Xj ≥ 0 when considering negative edges.
For simplicity, we first consider protein-protein interactions, excluding all factors within
the HS level. Mathematically, let νi and νj denote the nodes (or vertices) for proteins i and
−
j, respectively. Then +
i,j and i,j represent a positive and a negative binding relationship
−
between i and j, respectively. Moreover, +
i,j and i,j can only equal to 1 or 0, corresponding
to a presence or absence of a significant binding relationship between i and j. We add a
positive edge (+
i,j = 1) between proteins i and j from all layers except the set HS if their
respective binding profiles xi and xj have a combined p-value pvC ≤ 10−20 as in (3.30) or a
minimized mutual information p-value pvM M I ≤ 10−10 as in (3.32):
i∈
/ HS, j ∈
/ HS :
−10
+
) ∨ (pvC (xi , xj ) ≤ 10−20 ) .
i,j = 1 if (pvM M I (xi , xj ) ≤ 10
(5.1)
The decision to use the or condition (denoted by the symbol ∨) above is substantiated by our
observation in Section 3.3.1 that mutual information analysis leads to very few false positive
claims at a significance level of 10−10 . In addition, the test incorporates the information
from the filtered correlation analysis in the evaluation of the combined p-value. Since filtered
correlation analysis leads to more false positives, it needs to combine its evidence with that
of the mutual information analysis and meet a stricter significance threshold to create an
edge.
68
Negative edges in the network prove much more complicated to assign and interpret.
Section 3.2.1 discusses the inherent difficulty with using mutual information p-values for
finding opposing binding relationships. Moreover, it is possible to consider the overlap
between extremely bound and unbound sets of genes but the biological interpretation of
significant overlap in that scenario is not clear. Hence, it seems reasonable to solely base
our decision on assigning negative edges in the network using filtered correlation coefficient
p-values. To assign a negative link (−
i,j = 1) between the binding profiles xi and xj of the
proteins i and j not in the set HS, we use a filtered correlation coefficient p-value pvF CC
threshold of 10−10 :
i∈
/ HS, j ∈
/ HS :
−10
−
).
i,j = 1 if (pvF CC (xi , xj ) ≤ 10
(5.2)
Data from the HS class of factors requires particular care. For proteins, it seems natural
to classify binding data at a gene as 0s and 1s, representing an absence or presence of
association with a gene’s DNA. However, ChIP-chip data of histone acetylation levels, for
example, can indicate gradations of acetylation (i.e., hypo, medium, hyperacetylation, etc.).
Hence, it seems more natural to consider the continuum of data for members of the HS layer.
Although mutual information analysis extends to continuous distributions, finding p-values
for this scenario that are consistent with the previously developed pairwise statistics proves
more difficult. Hence, we decided to use standard correlation coefficient p-values (using
ChIP-chip data at all genes) to quantify the relationships between two factors within the
HS layer. The p-values for standard correlation coefficient estimates, pvSCC , can also be
assigned using (3.6), (3.14), and (3.15) in succession. Mathematically, let i and j represent
two factors from the HS layer with binding profiles xi and xj . We extend positive and
negative links in our network model if the p-value pvSCC ≤ 10−20 :
i ∈ HS, j ∈ HS :
−20
+
) ∧ (ρ̂Xi ,Xj ≥ 0)
i,j = 1 if (pvSCC (xi , xj ) ≤ 10
(5.3)
i ∈ HS, j ∈ HS :
−20
−
) ∧ (ρ̂Xi ,Xj < 0) .
i,j = 1 if (pvSCC (xi , xj ) ≤ 10
(5.4)
69
In the above equations, ∧ stands for a logical and, requiring that positive and negative
interactions have a positive and negative estimate of the filtered correlation coefficient, respectively. Now let us consider relationships between one protein not in the set HS and one
factor from the set HS. Ultimately, we want to make statements such as genes bound by
transcription factor i have high acetylation. In order to use filtered correlation coefficient
analysis in this context, we need to let the non-HS factor determine the pertinent set of
dimensions or genes. Letting Fi denote the set of genes bound by the non-HS protein i, we
define the filtered set Fi,j = Fi in (3.1) - (3.5) and use (3.6), (3.14), and (3.15) in succession
to derive p-values. Mutual information p-value analysis again proves difficult to extend in
this context. For non-HS protein i and HS factor j, we assign positive and negative links
in our network using the following thresholds on our filtered correlation coefficient p-value
pvF CC :
i∈
/ HS, j ∈ HS :
−4
+
i,j = 1 if (pvF CC (xi , xj ) ≤ 10 ) ∧ ρ̂Xi ,Xj ≥ 0
(5.5)
i∈
/ HS, j ∈ HS :
−4
−
i,j = 1 if (pvF CC (xi , xj ) ≤ 10 ) ∧ ρ̂Xi ,Xj < 0 .
(5.6)
Using a lower threshold for assigning links in the scenario above is necessary due to the lower
dimensionality of the data. Since, our filtered set now only contains the set of genes defined
as bound by a single factor, it often proves impossible to achieve such high significance
thresholds as in the previous analyses. Moreover, we used an empirical, non-parametric
method for calculating p-values for the above scenario in order to check if they matched our
analytical p-value derivation in Section 3.1.1. The algorithm first finds the filtered correlation
for the genes bound by the non-HS factor, g ∈ Fi , as described above. Then it picks 10000
random sets of genes of the same size, |Fi |, finds the filtered correlation coefficient for each
random set and counts the n filtered correlation coefficients that exceed that of factor i.
Then, the non-parametric p-value equals
n
.
10000
Both approaches achieve very similar p-
values, proving the reliability of our method for finding p-values introduced in Section 3.1.1.
The next section considers how we can exploit clustering in order to uncover other significant
70
interactions omitted by our initial thresholds.
5.1.1
Assigning Links: Second Pass
The thresholding approach for finding significant binding relationships introduced in the
previous section has an inherent sensitivity to the chosen cutoffs. To alleviate this problem
and to minimize the chance of false negatives, this section describes a second pass in our
algorithm for assigning links. As Section 4.2.4 illustrated, clustering ChIP-chip data at a
stringent significance cutoff (combined p-value ≤ 10−70 ) evinces various known biological
complexes. Moreover, we noticed in Section 3.3.1 that we can infer the binding relationship
between POL3 and RSC3 based on the interactions between POL3 and other components
of the RSC complex (i.e. RSC8, RSC). Generally, we can deduce that protein i’s binding
profile xi has a significant relationship with protein j1 ’s binding profile xj1 contained in
cluster Cj , xj1 ∈ Cj , if the combined binding relationship between xi and all the binding
vector components of Cj is significant. Using (3.31), we can find the total p-value for the
binding relationship between xi and the cluster Cj , or pvT (xi , Cj ), by combining the pvalues for the individual binding dependencies between xi and each component of Cj , where
Cj = {xj1 , . . . , xj|Cj | }:
|Cj |−1
pvT (xi , Cj ) = pvC (pv(xi , xj1 ) · · · pv(xi , xj|Cj | )) = k|Cj |
X (− ln k|Cj | )r
.
r!
r=0
(5.7)
Equation 5.7 only depends on the number of p-values combined, |Cj |, and the product of
Q|Cj |
pv(xi , xjm ), as in (3.31). However, (5.7) incorporates p-values
the p-values k|Cj | = m=1
based on binding relationships between independent data sets xi , xj1 , . . . , xj|Cj | . In this
context, our use of (3.31) satisfies the independent studies assumption and we do not have
to double the p-values as in Section 3.3.1. Moreover, (5.7) does not specify which type
of p-values to combine. For a possible positive or negative link, our network algorithm
incorprates the combined pvC (xi , xj ) or filtered correlation coefficient pvF CC (xi , xj ) p-values,
respectively. For extending a potential positive or negative link, we consider a total p-value
71
pvT (xi , Cj ) ≤ 10−20 or ≤ 10−10 as significant, respectively. With the additional evidence that
protein i’s binding profile associates significantly with the biologically related components
of Cj , the threshold for significant binding relationship between i and j1 should decrease.
Repeating this process for every possible combination of i and j1 , where j1 ∈ Cj , the second
pass of our network algorithm assigns positive and negative edges as follows:
−20
] ∧ [(pvM M I (xi , xj1 ) ≤ 10−7.5 ) ∨ (pvC (xi , xj1 ) ≤ 10−15 )]
+
i,j1 = 1 if [pvT (xi , Cj ) ≤ 10
−10
−
] ∧ [(pvF CC (xi , xj1 ] ≤ 10−7.5 )] .
i,j1 = 1 if [pvT (xi , Cj ) ≤ 10
(5.8)
The first pass of the network algorithm found 4692 positive and 2629 negative binding
relationships, while the second pass uncovered 806 and 574 more significant binding relationships, respectively. The larger number of positive links reflects the fact that significant
positive binding relationships are easier to find using ChIP-chip data than significant negative binding relationships. Actual visualization of all the significant links proves unwieldy.
Therefore, Figure 5-1 displays only highly significant positive binding relationships that have
a minimized mutual information p-value ≤ 10−20 or combined p-value ≤ 10−40 . Different colored ovals represent factors from the various levels and lines between ovals denote significant
synergistic interactions between factors. Inspecting the graph, one can notice neighborhoods
of nodes that share common biological processes. For example, the MCM-ORC cluster occupies the top left corner of Figure 5-1, with the SIR complex directly to its right. Continuing
to the right from the SIR complex, one can observe a congregation of nuclear transport
proteins, factors from the active cluster, the POL3 machinery, and the RSC complex at the
middle right end of the figure. Having built a network of the nucleus, the next section verifies
the ability of our network model to predict protein-protein interactions by comparing it with
several previous protein-protein studies.
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Figure 5-1: Network of highly significant positive binding relationships that have a minimized mutual information p-value ≤ 10−20 or combined p-value ≤ 10−40 . Colored ovals
represent factors from the various levels and lines between ovals denote significant synergistic interactions between factors. Different colors represent the five layers: yellow = T F ,
green = HS, red = HM , blue = N P , and pink = N T . Graph rendering was performed
with Pajek (http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/pajekman.htm).
73
Data Set
Interactions
Yu et al.
395
Gavin et al.
462
Ho et al.
137
Ito et al.
43
Uetz et al.
17
Overlap
P-Value
134
7.42×10−46
129
3.11×10−34
45
9.32×10−16
8
0.0280
4
0.0521
Table 5.1: Comparison between various protein-protein interaction data sets and our network. The first column lists the source of the data. The second column records the number
of predictions from each data set between proteins considered in this work. The third column
lists the overlap in predicted significant interactions between the five studies and our network.
The last column finds p-values for the significance in the overlap using the hypergeometric
test statistic discussed in Section 3.2.1 (see text).
5.1.2
Network Comparison
Numerous previous large scale and small scale methods have predicted protein-protein interactions with varied success. For comparison, we downloaded data from five protein-protein
interaction studies [30–34]. Ito et al. and Uetz et al. used an experimental technique called
Yeast Two-Hybrid (Y2H) to infer associations between proteins, while Gavin et al. and Ho
et al. used a more accurate mass spectrometry method. Yu et al. incorporated the significant predictions from the above data sets along with interactions found using several
small scale studies. Small scale studies generally have less noise in their predictions than
the high-throughput methods in [31–34]; therefore, the Yu et al. data set is probably most
reliable.
Table 5.1 demonstrates the results of the comparison. The first column lists the source of
the data. The second column records the number of predictions from each data set between
proteins considered in this work. The third column lists the overlap in predicted significant
interactions between the five studies and our network. The last column finds p-values for
the significance in the overlap using the hypergeometric test statistic discussed in Section
3.2.1. The p-values were calculated by setting u equal to entries in the second column, h to
entries in the third column, v = 3771, or the number of significant positive protein-protein
binding relationships in our network (i.e., excluding edges with factors in the HS layer), and
w = 43956, or all possible protein-protein relationships that could be predicted.
74
The table above shows that the three most reliable data sets overlap significantly with
the predictions made in our network. The Y2H data does not have enough predictions in
common with our data set in order to accurately determine the significance in the overlap.
Moreover, the Y2H technique is considered to have the most amount of noise. Generally,
we do not expect a full overlap between the methods since they target different biological
mechanisms. Our method finds biological complexes that associate at or near DNA but does
not capture other cellular protein-protein interactions. For example, most of the above data
sets show that protein Hrp1 and Nab2 form a complex but our analysis demonstrates that
the two proteins have very different binding profiles. Moreover, our method can capture
binding dependencies between proteins that consistently bind to the same gene targets and
hence participate in similar biological processes, but that do not necessarily bind at the same
time or associate with one another. And finally, it is generally agreed upon that the data
sets above have a large amount of noise. In summary, this comparison validates the ability of
our method to find protein-protein interactions predicted in previous studies. Having fully
specified the network structure enables us to infer the underlying communication between
layers in the eukaryotic transcriptional network in the next section.
5.2
Analysis of Network Topology
This section introduces several statistical methods for quantifying the interplay between the
layers in our yeast transcriptional network. Table 5.2 shows the number of links within and
between the five categories of factors (namely T F , HS, HM , N P , and N T ) for the entire
and the positive edge network. Each entry within the table represents the total number of
links and the percentage of possible links realized (in parenthesis) between factors from the
layers listed in the corresponding entry in the first row and column. For example, the first
entry in Table 5.2(a) shows that there are 3659 links between pairs of transcription factors,
which represents 17.2% of the total possible links that could be realized between all pairs
of TFs. Percentage of links realized represents a normalized measure of the connectedness
between two layers, which accounts for the number of factors in each layer. Given two layers
75
Layers
TF
HS
HM
NP
NT
TF
HS
HM
3659(17.2%) 1018(9.28%) 737(7.58%)
1018(9.28%) 616(44.7%) 776(31.2%)
737(7.58%) 776(31.2%) 198(18.3%)
409(7.32%) 236(16.5%) 208(16.4%)
171(5.16%) 276(32.5%) 100(13.3%)
NP
409(7.32%)
236(16.5%)
208(16.4%)
118(33.6%)
97(22.5%)
NT
171(5.16%)
276(32.5%)
100(13.3%)
97(22.5%)
95(79.2%)
(a) Total links and percentage of links captured for the entire network.
Layers
TF
HS
TF
2099(9.84%) 602(5.49%)
HS
602(5.49%) 513(37.2%)
HM
519(5.33%)
323(13%)
NP
328(5.87%) 151(10.6%)
NT
108(3.26%) 138(16.3%)
HM
519(5.33%)
323(13%)
161(14.9%)
178(14%)
85(11.3%)
NP
328(5.87%)
151(10.6%)
178(14%)
113(32.2%)
93(21.5%)
NT
108(3.26%)
138(16.3%)
85(11.3%)
93(21.5%)
87(72.5%)
(b) Total links and percentage of links captured for the positive edge network.
Table 5.2: Total links and percentage of links realized within and between the five layers
of the (a) entire and the (b) positive edge transcriptional network. Each entry within each
table represents the total number of links and the percentage of possible links realized (in
parenthesis) between factors from the layers listed in the corresponding entry in the first
row and column. For example, the first entry in Table (a) shows that there are 3659 links
between pairs of transcription factors, which represents 17.2% of the total possible links that
could be realized between all pairs of TFs.
with n and m factors, respectively, and k mutual interactions the inter-level percentage of
k
links realized is simply 100 nm
%. Moreover, if a layer with n elements has l interactions
2l
within its layer, the intra-level percentage of links realized is 100 n(n−1)
%.
Table 5.2(a) measures the level of connectedness or communication within and between
layers. By examining the percentage of links realized, we see that most of the communication
occurs within layers. The diagonal in Table 5.2(a) shows that the five layers have 17.2%,
44.7%, 18.3%, 33.6%, and 79.2% of intra-connectedness. Moreover, inspecting the entries in
row 1 of Table 5.2(a) from left to right, we see that the percentages gradually decrease, with
TFs being most highly associated with HSs, followed by HMs, NPs, and NTs. These results
confirm the current thought in biology that close collaboration between TFs, HSs, and HMs
76
induces specific classes of genes while NPs and NTs carry out more general mechanisms
related to transcription. However, there are still a number of interactions between the
levels, illustrating that significant amount of communication between layers of the yeast
transcriptional network.
Several other network statistics can also quantify the behavior of the different classes of
proteins. For, example we can count the number of links stemming from each node, or the
number of degrees, and find a distribution of degrees for nodes within a layer. Computing
the average of that distribution measures the average level of connectivity for members from
each layer. Table 5.3 shows the average number of degrees for proteins within the five layers
in both the entire and the positive edge network. Members of the HS level seem most
promiscuous in their association with other factors, but, overall, all layers have a similar
number of average degrees.
Clustering coefficient is a common method for measuring the cliquishness of each node
in the network, or how much the node’s nearest neighbors (adjacent nodes) interact with
one another. Finding the average clustering coefficient for all proteins within a particular
class measures the tendency for cliquishness in each layer. Let node i of layer L have ki
nearest neighbors. Moreover, let its ki adjacent nodes have ci connections between them
out of a possible
ki (ki −1)
2
links. Then the clustering coefficient at node i, or the fraction of
links realized between i’s nearest neighbors becomes
2ci
.
ki (ki −1)
Finally, the average clustering
coefficient for all factors i in layer L, CCL , takes the form:
CCL =
1 X
2ci
|L| i∈L ki (ki − 1)
(5.9)
Table 5.3 shows the average clustering coefficient for proteins within the five layers in both
the entire and the positive edge network. The preference for cliquish interaction between
nearest neighbors of nodes seems strongest amongst members of the N T class of proteins.
Given that almost all nuclear transport factors considered seem to associate with members of
the nuclear pore, this measure seems consistent with biological mechanism of the N T class.
In summary, the network topology statistics discussed in this section confirm that the
77
Layer Avg. Number Avg. Clustering
Name
of Degrees
Coefficient
TF
47.2
0.458
HS
68.9
0.427
HM
47.3
0.495
NP
43.9
0.539
NT
52.1
0.631
All
50.5
0.473
(a) Network topology statistics for the entire
network.
Layer Avg. Number Avg. Clustering
Name
of Degrees
Coefficient
TF
27.8
0.526
HS
42.3
0.482
HM
30.4
0.539
NP
36.1
0.575
NT
37.4
0.676
All
31.4
0.532
(b) Network topology statistics for the positive
edge network.
Table 5.3: Network topology statistics for (a) the entire and (b) the positive edge transcriptional network.
78
T F , HM , and HS levels seem most tightly coupled. However, over a thousand significant interactions also occur between factors outside of these subsets, demonstrating that eukaryotic
transcription depends on the intricate interplay between all five layers presented.
79
80
Chapter 6
Conclusion
Genome-wide binding data from ChIP-chip experiments provides a wealth of information
about protein-gene interactions and about the underlying workings of eukaryotic cells. In
order to gain a holistic understanding of the non-linear process of transcription, our work
examines the communication between various classes of regulators in the yeast specie Saccharomyces cerevisiae. We used ChIP-chip data to quantify the interplay between five categories
of factors that affect transcription: histone states, histone modifiers and nucleosome remodelers, transcription factors, nuclear processing proteins, and nuclear transport factors.
Following the introduction in Chapter 1, Chapter 2 described the non-trivial process of
incorporating the different sources of ChIP-chip data into a coherent set. Due to the heterogeneity of the obtained data, this chapter further discussed the need for data normalization
and showed the benefits of a normalized data set. Chapter 3 used the processed data to
find pairwise statistics that test binding dependencies between two proteins. Combining the
complementary pairwise measures of filtered correlation coefficient and mutual information
p-values reduced the false positive and false negative rates in our biological predictions and
increased the reliability of our analysis. Next, Chapter 4 uncovered group-wise relationships
between factors using Principal Component Analysis (PCA) and clustering. PCA allowed us
to visualize subsets of the data in 2-D. Based on the developed pairwise measures, Chapter
4 also introduced a novel semi-supervised clustering algorithm that preserves information
81
about elements of a cluster in order to better capture group-wise dependencies between proteins. And finally, Chapter 5 combined the methodology developed in the previous chapters
in order to build a multi-layered transcriptional network of the nucleus.
Throughout the theoretical analysis, we validated various known biological processes that
occur in dextrose rich conditions. Moreover, our analysis and previous literature [7] showed
that usage ChIP-chip experiments in rich media YPD conditions can still provide insight
about biological processes in other growth environments. Further, the fact that our data
does not synchronize cells at a particular phase of growth, does not preclude formulation of
hypothesis about biological processes that occur only in a specific phase of the cell cycle,
as shown with the MCM-ORC complex in Section 4.2.4. Our theoretical analysis also uncovered several novel biological hypothesis. In particular, Chapter 3 suggests that proteins
LRP1 and RRP6 might participate in a mechanism that ensures quality control at falsely
transcribed genes, by binding to inactive genes in order to readily degrade unwanted production of their mRNAs. Moreover, Chapter 4 hypothesized that SIR2, long thought to silence
the transcription of DNA, might have a role in transcriptional activation at the ORF region
of genes. And finally, Chapter 5 quantified the communication between layers in biological
transcriptional networks for the first time in the literature. The network topology statistics
discussed in Chapter 5 confirm that the T F , HM , and HS levels seem most tightly coupled. However, over a thousand significant interactions also occur between factors outside of
these three categories, demonstrating that eukaryotic transcription depends on the intricate
interplay between all five layers presented.
82
Appendix A
Semi-Supervised Clustering based on
KL-divergence
Semi-supervised clustering is more theoretically intuitive when objects are represented using
distibutions. We chose to define the probability distribution of object i as the normalized
binding profile of binding ratios across all genes, or normalized rows in the BR matrix.
Specifically, if xi = [xi,g1 . . . xi,g|G| ] denotes a row vector from the BR matrix for protein i
across all genes g in the set G, the normalized binding profile (or binding distribution) for
protein i is
xi,g
.
g∈G xi,g
P (g|i) = P
(A.1)
We implemented a semi-supervised clustering algorithm that can perform hierarchical
partitioning of distributions as defined in (A.1). We treat the binding distribution P (g|i) for
each protein i as an object (or node) and merge objects into clusters (aggregations of nodes)
based on a distance metric using the Kullback-Leibler divergence.
The KL-divergence represents how different two distributions are. For the normalized
binding profiles, it is defined as
83
D(P (g|Ck )kP (g|Ck ∪ Cl )) =
X
g∈G
P (g|Ck ) log
P (g|Ck )
,
P (g|Ck ∪ Cl )
(A.2)
where Ck and Cl represent two clusters and Ck ∪ Cl denotes the merged partition of Ck and
Cl . In information theoretic terms, the KL-divergence represents the information lost by
joining Ck and Cl into a combined distribution for Ck ∪ Cl . Note that the KL-divergence is
not a distance metric; it does not satisfy the triangle inequality nor is it commutative. For
the semi-supervised clustering algorithm, we define the following distance measure based on
the KL-divergence:
d(Ck , Cl ) = |Ck |D(P (g|Ck )kP (g|Ck ∪ Cl )) + |Cl |D(P (g|Cl )kP (g|Ck ∪ Cl )) .
(A.3)
The above equation (A.3) can intuitively be interpreted as the total information lost in
combining the distributions of Ck and Cl into P (g|Ck ∪ Cl ), where the weights |Ck | and
|Cl | account for the number of nodes in each cluster. When a partition Ck contains just a
single protein, |Ck | = 1. Note that this distance, which we refer to as the KL-distance, is
commutative.
The hierarchical clustering proceeds by initially treating all objects as clusters and then
successively merging partitions that are “closest”, in terms of the KL distance in (A.3).
When objects are merged into clusters, the new clusters preserve information about their
component nodes in their distribution. The combined distribution is a weighted average of
the constituent distributions:
P (g|Ck ∪ Cl ) =
1
(|Ck |P (g|Ck ) + |Cl |P (g|Cl )) .
|Ck | + |Cl |
(A.4)
The performance of the clustering algorithm is illustrated in Figure A-1. Figure A-1
partitions the nuclear processing and nuclear transport factors and shows similar results as
the PCA analysis in Section 4.1. The related NTs cluster together once again. In addition, HMT1 (a protein that adds methyl groups to other proteins), THO2 (a protein that
84
starts transcription), and YRA1 (a protein that exports mRNA from the nucleus) also cluster
tightly with the NTs in [13], suggesting a common regulatory mechanism. Moreover, proteins
RRP6, an mRNA surveillance factor, and mRNA export factor NPL3 cluster together, supporting the results in [16] that demonstrate coupling of mRNA quality assurance to mRNA
export. The algorithm also captures the relationship between LRP1 and PRP20 discussed
in section 3.3.1. However, when the algorithm is run on the entire data, it fails to find
commonality between several known biological processes. In the end, the pairwise statistics
discussed in Chapter 3 prove much more biologically meaningful than KL-divergence.
85
1.2
Distance
1
0.8
0.6
0.4
0.2
Genes
0
YAL002W
YAL001C
YAL004W
YAL003W
YAL005C
YAL008W
YAL007C
YAL009W
YAL011W
YAL010C
YAL013W
YAL012W
YAL016W
YAL015C
YAL014C
YAL017W
YAL018C
YAL019W
YAL020C
YAL023C
YAL022C
YAL021C
YAL025C
YAL024C
YAL028W
YAL027W
YAL026C
YAL030W
YAL029C
YAL033W
YAL032C
YAL031C
YAL035C−A
YAL034C
YAL035W
YAL036C
YAL038W
YAL037W
YAL039C
YAL041W
YAL040C
YAL043C−A
YAL042W
YAL043C
YAL046C
YAL044C
YAL048C
YAL047C
YAL053W
YAL051W
YAL049C
YAL055W
YAL054C
YAL058C−A
YAL056W
YAL058W
YAL060W
YAL059W
YAL062W
YAL061W
YAL063C
YAL064W
YAL065C
YAL068C
YAL067C
YAR003W
YAR002W
YAR007C
YAR008W
YAR009C
YAR015W
YAR014C
YAR010C
YAR019C
YAR018C
YAR027W
YAR023C
YAR020C
YAR029W
YAR028W
YAR033W
YAR031W
YAR044W
YAR042W
YAR035W
YAR053W
YAR050W
YAR062W
YAR061W
YAR060C
YAR066W
YAR064W
YAR068W
YAR070C
YAR069C
YAR073W
YAR071W
YAR075W
YBL001C
YBL002W
YBL004W
YBL003C
YBL005W−A
YBL005W
YBL005W−B
YBL007C
YBL006C
YBL009W
YBL008W
YBL011W
YBL010C
YBL012C
YBL013W
YBL014C
YBL016W
YBL015W
YBL019W
YBL018C
YBL017C
YBL020W
YBL021C
YBL024W
YBL023C
YBL022C
YBL026W
YBL025W
YBL027W
YBL029W
YBL028C
YBL031W
YBL030C
YBL032W
YBL033C
YBL036C
YBL035C
YBL034C
YBL038W
YBL037W
YBL041W
YBL040C
YBL039C
YBL043W
YBL042C
YBL044W
YBL045C
YBL046W
YBL049W
YBL047C
YBL050W
YBL051C
YBL054W
YBL053W
YBL052C
YBL056W
YBL055C
YBL059W
YBL058W
YBL057C
YBL060W
YBL061C
YBL063W
YBL062W
YBL067C
YBL066C
YBL064C
YBL069W
YBL068W
YBL072C
YBL071C
YBL070C
YBL075C
YBL074C
YBL079W
YBL078C
YBL076C
YBL081W
YBL080C
YBL083C
YBL082C
YBL085W
YBL084C
YBL086C
YBL088C
YBL087C
YBL090W
YBL089W
YBL091C
YBL092W
YBL093C
YBL095W
YBL094C
YBL096C
YBL098W
YBL097W
YBL099W
YBL100C
YBL101W−B
YBL101W−A
YBL101C
YBL102W
YBL103C
YBL106C
YBL105C
YBL104C
YBL109W
YBL107C
YBL113C
YBL112C
YBL111C
YBR002C
YBR001C
YBR003W
YBR004C
YBR006W
YBR005W
YBR007C
YBR009C
YBR008C
YBR012W−A
YBR010W
YBR011C
YBR012W−B
YBR013C
YBR016W
YBR015C
YBR014C
YBR018C
YBR017C
YBR020W
YBR019C
YBR022W
YBR021W
YBR023C
YBR024W
YBR025C
YBR028C
YBR027C
YBR026C
YBR030W
YBR029C
YBR033W
YBR031W
YBR036C
YBR035C
YBR034C
YBR038W
YBR037C
YBR041W
YBR040W
YBR039W
YBR043C
YBR042C
YBR046C
YBR045C
YBR044C
YBR048W
YBR047W
YBR050C
YBR049C
YBR051W
YBR053C
YBR052C
YBR054W
YBR055C
YBR056W
YBR058C
YBR057C
YBR060C
YBR059C
YBR063C
YBR062C
YBR061C
YBR064W
YBR065C
YBR067C
YBR066C
YBR070C
YBR069C
YBR068C
YBR072W
YBR071W
YBR075W
YBR074W
YBR073W
YBR076W
YBR077C
YBR078W
YBR080C
YBR079C
YBR082C
YBR081C
YBR084C−A
YBR083W
YBR085W
YBR084W
YBR086C
YBR087W
YBR088C
YBR089W
YBR091C
YBR090C
YBR093C
YBR092C
YBR094W
YBR096W
YBR095C
YBR098W
YBR097W
YBR100W
YBR099C
YBR103W
YBR102C
YBR101C
YBR104W
YBR105C
YBR106W
YBR108W
YBR107C
YBR110W
YBR109C
YBR114W
YBR112C
YBR111C
YBR117C
YBR115C
YBR119W
YBR118W
YBR122C
YBR121C
YBR120C
YBR124W
YBR123C
YBR127C
YBR126C
YBR125C
YBR129C
YBR128C
YBR131W
YBR130C
YBR134W
YBR133C
YBR132C
YBR136W
YBR135W
YBR139W
YBR137W
YBR140C
YBR142W
YBR141C
YBR146W
YBR145W
YBR143C
YBR148W
YBR147W
YBR149W
YBR150C
YBR153W
YBR152W
YBR151W
YBR155W
YBR154C
YBR158W
YBR157C
YBR156C
YBR160W
YBR159W
YBR162W−A
YBR161W
YBR162C
YBR163W
YBR164C
YBR165W
YBR166C
YBR168W
YBR167C
YBR169C
YBR171W
YBR170C
YBR175W
YBR173C
YBR172C
YBR176W
YBR177C
YBR178W
YBR180W
YBR179C
YBR182C
YBR181C
YBR184W
YBR183W
YBR187W
YBR186W
YBR185C
YBR189W
YBR188C
YBR192W
YBR191W
YBR193C
YBR194W
YBR195C
YBR198C
YBR197C
YBR196C
YBR200W
YBR199W
YBR202W
YBR201W
YBR203W
YBR205W
YBR204C
YBR207W
YBR206W
YBR210W
YBR209W
YBR208C
YBR212W
YBR211C
YBR215W
YBR214W
YBR213W
YBR217W
YBR216C
YBR220C
YBR218C
YBR224W
YBR222C
YBR221C
YBR225W
YBR227C
YBR228W
YBR230C
YBR229C
YBR233W
YBR231C
YBR235W
YBR234C
YBR237W
YBR236C
YBR238C
YBR240C
YBR239C
YBR242W
YBR241C
YBR243C
YBR244W
YBR245C
YBR246W
YBR248C
YBR247C
YBR250W
YBR249C
YBR252W
YBR251W
YBR253W
YBR255W
YBR254C
YBR257W
YBR256C
YBR259W
YBR258C
YBR260C
YBR263W
YBR261C
YBR267W
YBR265W
YBR264C
YBR268W
YBR269C
YBR271W
YBR270C
YBR274W
YBR273C
YBR272C
YBR276C
YBR275C
YBR279W
YBR278W
YBR280C
YBR282W
YBR281C
YBR285W
YBR284W
YBR283C
YBR287W
YBR286W
YBR289W
YBR288C
YBR290W
YBR293W
YBR291C
YBR295W
YBR294W
YBR297W
YBR296C
YBR298C
YBR299W
YBR300C
YBR301W
YCL004W
YCL002C
YCL005W
YCL006C
YCL008C
YCL007C
YCL011C
YCL010C
YCL009C
YCL014W
YCL016C
YCL019W
YCL018W
YCL017C
YCL020W
YCL022C
YCL024W
YCL023C
YCL025C
YCL028W
YCL027W
YCL030C
YCL029C
YCL032W
YCL031C
YCL033C
YCL034W
YCL035C
YCL036W
YCL038C
YCL037C
YCL039W
YCL041C
YCL042W
YCL043C
YCL047C
YCL045C
YCL044C
YCL048W
YCL049C
YCL051W
YCL050C
YCL052C
YCL055W
YCL054W
YCL057W
YCL056C
YCL058C
YCL061C
YCL059C
YCL063W
YCL064C
YCL066W
YCL065W
YCL067C
YCL069W
YCL068C
YCR001W
YCL074W
YCL073C
YCR003W
YCR002C
YCR006C
YCR005C
YCR004C
YCR008W
YCR007C
YCR010C
YCR009C
YCR012W
YCR011C
YCR013C
YCR015C
YCR014C
YCR016W
YCR018C
YCR017C
YCR019W
YCR020C
YCR020C−A
YCR023C
YCR021C
YCR024C−A
YCR024C
YCR026C
YCR025C
YCR030C
YCR028C
YCR027C
YCR033W
YCR032W
YCR034W
YCR036W
YCR035C
YCR038C
YCR037C
YCR041W
YCR040W
YCR039C
YCR043C
YCR042C
YCR045C
YCR044C
YCR048W
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YDL190C
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YDL194W
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YDL201W
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YDR343C
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YDR350C
YDR349C
YDR348C
YDR352W
YDR351W
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YDR353W
YDR355C
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YDR357C
YDR358W
YDR360W
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YDR362C
YDR361C
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YDR368W
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YDR370C
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YDR374C
YDR377W
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YDR378C
YDR380W
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YDR382W
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YDR384C
YDR383C
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YDR387C
YDR389W
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YDR391C
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YDR397C
YDR399W
YDR398W
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YDR400W
YDR402C
YDR403W
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YDR407C
YDR409W
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YDR410C
YDR414C
YDR413C
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YDR415C
YDR420W
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YDR422C
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YDR423C
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YDR430C
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YDR431W
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YDR464W
YDR463W
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YDR465C
YDR469W
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YDR467C
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YDR472W
YDR474C
YDR473C
YDR476C
YDR475C
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YDR479C
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YDR481C
YDR483W
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YDR484W
YDR486C
YDR485C
YDR488C
YDR487C
YDR489W
YDR492W
YDR490C
YDR494W
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YDR496C
YDR495C
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YDR500C
YDR501W
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YDR505C
YDR503C
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YDR507C
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YEL051W
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YEL052W
YEL053C
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YEL055C
YEL054C
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YEL060C
YEL062W
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YEL065W
YEL064C
YEL066W
YEL069C
YEL067C
YEL071W
YEL070W
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YEL075C
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YEL076C−A
YEL076C
YEL076W−C
YEL077C
YER002W
YER001W
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YER005W
YER004W
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YER006W
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YER009W
YER008C
YER012W
YER011W
YER010C
YER014W
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YER016W
YER015W
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YER017C
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YER028C
YER027C
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YER033C
YER037W
YER036C
YER040W
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YER046W
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YER047C
YER049W
YER051W
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YER053C
YER052C
YER056C
YER055C
YER054C
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YER057C
YER060W
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YER060w−A
YER061C
YER063W
YER062C
YER066W
YER065C
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YER100W
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YER104W
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YER110C
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YER113C
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YFR025C
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YFR041C
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YGL049C
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YGL051W
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YGL056C
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YGL095C
YGL094C
YGL098W
YGL097W
YGL101W
YGL100W
YGL099W
YGL103W
YGL102C
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YGL104C
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YGL107C
YGL111W
YGL110C
YGL114W
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YGL141W
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YGL149W
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YGL147C
YGL151W
YGL150C
YGL153W
YGL152C
YGL154C
YGL156W
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YGL160W
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YGL161C
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YGL187C
YGL186C
YGL190C
YGL189C
YGL192W
YGL191W
YGL193C
YGL195W
YGL194C
YGL198W
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YKL162C
YKL164C
YKL166C
YKL165C
YKL169C
YKL168C
YKL167C
YKL171W
YKL170W
YKL173W
YKL172W
YKL175W
YKL174C
YKL176C
YKL179C
YKL178C
YKL182W
YKL181W
YKL180W
YKL184W
YKL183W
YKL185W
YKL187C
YKL186C
YKL189W
YKL188C
YKL191W
YKL190W
YKL194C
YKL193C
YKL192C
YKL195W
YKL196C
YKL201C
YKL199C
YKL197C
YKL204W
YKL203C
YKL205W
YKL207W
YKL206C
YKL208W
YKL209C
YKL210W
YKL211C
YKL212W
YKL214C
YKL213C
YKL216W
YKL215C
YKL217W
YKL219W
YKL218C
YKL221W
YKL220C
YKR001C
YKL224C
YKL222C
YKR003W
YKR002W
YKR005C
YKR004C
YKR008W
YKR007W
YKR006C
YKR010C
YKR009C
YKR013W
YKR012C
YKR011C
YKR015C
YKR014C
YKR016W
YKR017C
YKR020W
YKR019C
YKR018C
YKR021W
YKR022C
YKR025W
YKR024C
YKR026C
YKR028W
YKR027W
YKR030W
YKR029C
YKR031C
YKR034W
YKR035C
YKR037C
YKR036C
YKR041W
YKR039W
YKR038C
YKR042W
YKR043C
YKR044W
YKR046C
YKR045C
YKR047W
YKR048C
YKR051W
YKR050W
YKR049C
YKR053C
YKR052C
YKR055W
YKR054C
YKR058W
YKR057W
YKR056W
YKR060W
YKR059W
YKR062W
YKR061W
YKR063C
YKR064W
YKR065C
YKR067W
YKR066C
YKR068C
YKR070W
YKR069W
YKR072C
YKR071C
YKR074W
YKR076W
YKR075C
YKR078W
YKR079C
YKR080W
YKR082W
YKR081C
YKR084C
YKR083C
YKR086W
YKR085C
YKR087C
YKR089C
YKR088C
YKR091W
YKR090W
YKR093W
YKR092C
YKR094C
YKR096W
YKR095W
YKR097W
YKR099W
YKR098C
YKR101W
YKR100C
YKR104W
YKR103W
YKR102W
YKR106W
YKR105C
YLL002W
YLL001W
YLL004W
YLL003W
YLL005C
YLL006W
YLL007C
YLL008W
YLL010C
YLL009C
YLL012W
YLL011W
YLL014W
YLL013C
YLL015W
YLL019C
YLL018C
YLL021W
YLL020C
YLL024C
YLL023C
YLL022C
YLL026W
YLL025W
YLL029W
YLL028W
YLL027W
YLL032C
YLL031C
YLL033W
YLL034C
YLL035W
YLL038C
YLL036C
YLL040C
YLL039C
YLL043W
YLL042C
YLL041C
YLL046C
YLL045C
YLL049W
YLL048C
YLL050C
YLL052C
YLL051C
YLL054C
YLL053C
YLL055W
YLL057C
YLL056C
YLL058W
YLL060C
YLL061W
YLL063C
YLL062C
YLL066C
YLL064C
YLR002C
YLR001C
YLL067C
YLR004C
YLR003C
YLR005W
YLR006C
YLR007W
YLR009W
YLR008C
YLR011W
YLR010C
YLR013W
YLR012C
YLR014C
YLR015W
YLR016C
YLR017W
YLR019W
YLR018C
YLR021W
YLR020C
YLR023C
YLR022C
YLR025W
YLR024C
YLR026C
YLR028C
YLR027C
YLR031W
YLR030W
YLR029C
YLR033W
YLR032W
YLR036C
YLR035C
YLR034C
YLR038C
YLR037C
YLR040C
YLR039C
YLR041W
YLR043C
YLR042C
YLR045C
YLR044C
YLR048W
YLR047C
YLR046C
YLR050C
YLR049C
YLR052W
YLR051C
YLR055C
YLR054C
YLR053C
YLR057W
YLR056W
YLR060W
YLR059C
YLR058C
YLR063W
YLR061W
YLR064W
YLR066W
YLR065C
YLR068W
YLR067C
YLR070C
YLR069C
YLR072W
YLR071C
YLR073C
YLR075W
YLR074C
YLR077W
YLR079W
YLR078C
YLR081W
YLR080W
YLR084C
YLR083C
YLR082C
YLR086W
YLR085C
YLR088W
YLR087C
YLR091W
YLR090W
YLR089C
YLR092W
YLR093C
YLR096W
YLR095C
YLR094C
YLR099C
YLR097C
YLR100W
YLR103C
YLR102C
YLR104W
YLR105C
YLR107W
YLR106C
YLR109W
YLR108C
YLR110C
YLR113W
YLR114C
YLR116W
YLR115W
YLR117C
YLR119W
YLR118C
YLR124W
YLR121C
YLR120C
YLR125W
YLR126C
YLR128W
YLR127C
YLR129W
YLR131C
YLR130C
YLR133W
YLR132C
YLR135W
YLR134W
YLR136C
YLR138W
YLR137W
YLR143W
YLR142W
YLR141W
YLR145W
YLR144C
YLR147C
YLR146C
YLR150W
YLR149C
YLR151C
YLR153C
YLR152C
YLR156W
YLR155C
YLR154C
YLR158C
YLR157C
YLR159W
YLR160C
YLR162W
YLR161W
YLR163C
YLR164W
YLR165C
YLR167W
YLR166C
YLR168C
YLR169W
YLR170C
YLR171W
YLR173W
YLR172C
YLR175W
YLR174W
YLR177W
YLR176C
YLR180W
YLR179C
YLR178C
YLR182W
YLR181C
YLR186W
YLR185W
YLR183C
YLR188W
YLR187W
YLR191W
YLR190W
YLR189C
YLR193C
YLR192C
YLR195C
YLR194C
YLR197W
YLR196W
YLR198C
YLR200W
YLR199C
YLR204W
YLR203C
YLR205C
YLR207W
YLR206W
YLR208W
YLR210W
YLR209C
YLR212C
YLR211C
YLR215C
YLR213C
YLR217W
YLR216C
YLR218C
YLR220W
YLR219W
YLR223C
YLR222C
YLR221C
YLR224W
YLR225C
YLR226W
YLR228C
YLR227C
YLR231C
YLR229C
YLR232W
YLR233C
YLR234W
YLR237W
YLR236C
YLR238W
YLR239C
YLR241W
YLR240W
YLR242C
YLR243W
YLR244C
YLR246W
YLR245C
YLR247C
YLR249W
YLR248W
YLR251W
YLR250W
YLR253W
YLR252W
YLR254C
YLR257W
YLR256W
YLR258W
YLR260W
YLR259C
YLR263W
YLR262C
YLR264W
YLR265C
YLR268W
YLR267W
YLR266C
YLR271W
YLR270W
YLR274W
YLR273C
YLR272C
YLR275W
YLR276C
YLR279W
YLR278C
YLR277C
YLR282C
YLR281C
YLR283W
YLR284C
YLR285W
YLR287C
YLR286C
YLR287C−A
YLR288C
YLR289W
YLR291C
YLR290C
YLR293C
YLR292C
YLR297W
YLR296W
YLR295C
YLR299W
YLR298C
YLR301W
YLR300W
YLR303W
YLR305C
YLR304C
YLR307W
YLR306W
YLR308W
YLR310C
YLR309C
YLR313C
YLR312C
YLR315W
YLR314C
YLR316C
YLR318W
YLR317W
YLR320W
YLR319C
YLR322W
YLR321C
YLR323C
YLR324W
YLR325C
YLR326W
YLR328W
YLR327C
YLR330W
YLR329W
YLR332W
YLR336C
YLR333C
YLR340W
YLR338W
YLR342W
YLR341W
YLR345W
YLR344W
YLR343W
YLR347C
YLR346C
YLR350W
YLR349W
YLR348C
YLR352W
YLR351C
YLR353W
YLR355C
YLR354C
YLR357W
YLR356W
YLR360W
YLR359W
YLR362W
YLR361C
YLR363C
YLR366W
YLR364W
YLR369W
YLR368W
YLR367W
YLR371W
YLR370C
YLR372W
YLR373C
YLR375W
YLR374C
YLR376C
YLR378C
YLR377C
YLR381W
YLR380W
YLR379W
YLR383W
YLR382C
YLR386W
YLR385C
YLR384C
YLR388W
YLR387C
YLR390W
YLR389C
YLR394W
YLR393W
YLR392C
YLR396C
YLR395C
YLR399C
YLR398C
YLR397C
YLR400W
YLR401C
YLR405W
YLR404W
YLR403W
YLR407W
YLR406C
YLR409C
YLR408C
YLR412W
YLR411W
YLR410W
YLR413W
YLR414C
YLR417W
YLR415C
YLR418C
YLR420W
YLR419W
YLR422W
YLR421C
YLR423C
YLR425W
YLR424W
YLR427W
YLR426W
YLR430W
YLR429W
YLR431C
YLR432W
YLR433C
YLR435W
YLR434C
YLR436C
YLR438W
YLR437C
YLR439W
YLR441C
YLR440C
YLR443W
YLR442C
YLR446W
YLR445W
YLR449W
YLR448W
YLR447C
YLR451W
YLR450W
YLR454W
YLR453C
YLR452C
YLR456W
YLR455W
YLR458W
YLR457C
YLR459W
YLR461W
YLR460C
YLR462W
YLR463C
YLR464W
YLR466W
YLR465C
YML001W
YLR467W
YML003W
YML002W
YML004C
YML005W
YML006C
YML007W
YML008C
YML010W
YML009C
YML011C
YML013C−A
YML012W
YML014W
YML013W
YML015C
YML017W
YML016C
YML020W
YML019W
YML018C
YML022W
YML021C
YML024W
YML023C
YML027W
YML026C
YML025C
YML029W
YML028W
YML031W
YML030W
YML032C
YML034W
YML035C
YML035C−A
YML036W
YML037C
YML039W
YML038C
YML040W
YML041C
YML042W
YML045W
YML043C
YML046W
YML047C
YML048W−A
YML048W
YML049C
YML051W
YML050W
YML052W
YML054C
YML053C
YML055W
YML056C
YML058C−A
YML057W
YML058W
YML060W
YML059C
YML062C
YML061C
YML063W
YML065W
YML064C
YML067C
YML066C
YML070W
YML069W
YML068W
YML072C
YML071C
YML074C
YML073C
YML077W
YML076C
YML075C
YML079W
YML078W
YML082W
YML081W
YML080W
YML085C
YML083C
YML087C
YML086C
YML088W
YML092C
YML091C
YML094W
YML093W
YML095C−A
YML096W
YML095C
YML098W
YML097C
YML100W−A
YML100W
YML099C
YML102W
YML101C
YML104C
YML103C
YML106W
YML105C
YML107C
YML109W
YML108W
YML112W
YML111W
YML110C
YML113W
YML114C
YML117W
YML116W
YML115C
YML117W−A
YML118W
YML119W
YML120C
YML121W
YML124C
YML123C
YML126C
YML125C
YML127W
YML129C
YML128C
YML131W
YML130C
YML132W
YMR001C
YML133C
YMR003W
YMR002W
YMR005W
YMR004W
YMR007W
YMR006C
YMR008C
YMR010W
YMR009W
YMR012W
YMR011W
YMR013C
YMR014W
YMR015C
YMR018W
YMR017W
YMR016C
YMR020W
YMR019W
YMR022W
YMR021C
YMR025W
YMR024W
YMR023C
YMR027W
YMR026C
YMR028W
YMR030W
YMR029C
YMR031W−A
YMR031C
YMR033W
YMR032W
YMR034C
YMR035W
YMR036C
YMR038C
YMR037C
YMR040W
YMR039C
YMR041C
YMR043W
YMR042W
YMR044W
YMR046C
YMR045C
YMR048W
YMR047C
YMR050C
YMR049C
YMR052W
YMR051C
YMR053C
YMR054W
YMR055C
YMR059W
YMR058W
YMR056C
YMR061W
YMR060C
YMR064W
YMR063W
YMR062C
YMR066W
YMR065W
YMR068W
YMR067C
YMR070W
YMR069W
YMR071C
YMR072W
YMR073C
YMR075C−A
YMR075W
YMR074C
YMR077C
YMR076C
YMR079W
YMR078C
YMR080C
YMR083W
YMR081C
YMR085W
YMR084W
YMR087W
YMR086W
YMR088C
YMR090W
YMR089C
YMR093W
YMR092C
YMR091C
YMR094W
YMR095C
YMR096W
YMR098C
YMR097C
YMR100W
YMR099C
YMR102C
YMR101C
YMR106C
YMR105C
YMR104C
YMR108W
YMR107W
YMR109W
YMR111C
YMR110C
YMR113W
YMR112C
YMR115W
YMR114C
YMR116C
YMR118C
YMR117C
YMR119W−A
YMR119W
YMR123W
YMR121C
YMR120C
YMR125W
YMR124W
YMR129W
YMR128W
YMR126C
YMR130W
YMR131C
YMR134W
YMR133W
YMR132C
YMR135W−A
YMR135C
YMR136W
YMR137C
YMR140W
YMR139W
YMR138W
YMR142C
YMR141C
YMR144W
YMR143W
YMR145C
YMR147W
YMR146C
YMR149W
YMR148W
YMR152W
YMR151W
YMR150C
YMR153C−A
YMR153W
YMR155W
YMR154C
YMR156C
YMR158W
YMR157C
YMR161W
YMR160W
YMR159C
YMR163C
YMR162C
YMR165C
YMR164C
YMR167W
YMR166C
YMR168C
YMR170C
YMR169C
YMR173W
YMR172W
YMR171C
YMR173W−A
YMR174C
YMR177W
YMR176W
YMR175W
YMR179W
YMR178W
YMR181C
YMR180C
YMR184W
YMR183C
YMR182C
YMR186W
YMR185W
YMR189W
YMR188C
YMR187C
YMR191W
YMR190C
YMR194W
YMR193W
YMR192W
YMR196W
YMR195W
YMR198W
YMR197C
YMR200W
YMR199W
YMR201C
YMR203W
YMR202W
YMR206W
YMR205C
YMR204C
YMR208W
YMR207C
YMR211W
YMR210W
YMR209C
YMR213W
YMR212C
YMR215W
YMR214W
YMR217W
YMR216C
YMR218C
YMR220W
YMR219W
YMR223W
YMR222C
YMR221C
YMR225C
YMR224C
YMR228W
YMR227C
YMR226C
YMR230W
YMR229C
YMR232W
YMR231W
YMR234W
YMR233W
YMR235C
YMR237W
YMR236W
YMR238W
YMR240C
YMR239C
YMR241W
YMR243C
YMR244C−A
YMR244W
YMR246W
YMR245W
YMR247C
YMR251W
YMR250W
YMR251W−A
YMR253C
YMR252C
YMR255W
YMR256C
YMR259C
YMR258C
YMR257C
YMR261C
YMR260C
YMR263W
YMR262W
YMR264W
YMR267W
YMR265C
YMR269W
YMR268C
YMR272C
YMR271C
YMR270C
YMR274C
YMR273C
YMR277W
YMR276W
YMR275C
YMR278W
YMR279C
YMR281W
YMR280C
YMR285C
YMR283C
YMR282C
YMR288W
YMR287C
YMR290W−A
YMR289W
YMR290C
YMR292W
YMR293C
YMR294W−A
YMR294W
YMR295C
YMR297W
YMR296C
YMR298W
YMR299C
YMR302C
YMR301C
YMR300C
YMR304W
YMR303C
YMR306W
YMR305C
YMR308C
YMR310C
YMR309C
YMR312W
YMR311C
YMR313C
YMR315W
YMR314W
YMR316C−B
YMR316W
YMR317W
YMR319C
YMR318C
YMR322C
YMR321C
YMR323W
YNL001W
YNL002C
YNL004W
YNL003C
YNL006W
YNL005C
YNL008C
YNL010W
YNL009W
YNL014W
YNL012W
YNL016W
YNL015W
YNL017C
YNL019C
YNL018C
YNL021W
YNL020C
YNL022C
YNL025C
YNL024C
YNL027W
YNL026W
YNL028W
YNL030W
YNL029C
YNL032W
YNL031C
YNL034W
YNL033W
YNL035C
YNL036W
YNL037C
YNL040W
YNL039W
YNL038W
YNL042W
YNL041C
YNL045W
YNL044W
YNL046W
YNL048W
YNL047C
YNL050C
YNL049C
YNL053W
YNL052W
YNL051W
YNL054W
YNL055C
YNL056W
YNL059C
YNL058C
YNL061W
YNL062C
YNL063W
YNL064C
YNL067W
YNL066W
YNL065W
YNL069C
YNL068C
YNL072W
YNL071W
YNL070W
YNL073W
YNL074C
YNL077W
YNL076W
YNL075W
YNL078W
YNL079C
YNL081C
YNL080C
YNL083W
YNL085W
YNL084C
YNL087W
YNL086W
YNL088W
YNL090W
YNL089C
YNL092W
YNL091W
YNL094W
YNL093W
YNL095C
YNL097C
YNL096C
YNL099C
YNL098C
YNL102W
YNL101W
YNL100W
YNL103W
YNL104C
YNL107W
YNL105W
YNL108C
YNL111C
YNL110C
YNL113W
YNL112W
YNL117W
YNL115C
YNL114C
YNL119W
YNL120C
YNL123W
YNL122C
YNL121C
YNL124W
YNL125C
YNL128W
YNL127W
YNL126W
YNL129W
YNL130C
YNL132W
YNL131W
YNL135C
YNL134C
YNL133C
YNL136W
YNL137C
YNL138W
YNL141W
YNL139C
YNL142W
YNL144C
YNL147W
YNL146W
YNL145W
YNL149C
YNL148C
YNL152W
YNL151C
YNL155W
YNL154C
YNL153C
YNL157W
YNL156C
YNL158W
YNL160W
YNL159C
YNL162W
YNL161W
YNL165W
YNL163C
YNL166C
YNL168C
YNL167C
YNL171C
YNL169C
YNL172W
YNL175C
YNL173C
YNL177C
YNL176C
YNL178W
YNL181W
YNL180C
YNL183C
YNL182C
YNL187W
YNL186W
YNL185C
YNL189W
YNL188W
YNL191W
YNL190W
YNL193W
YNL195C
YNL194C
YNL197C
YNL196C
YNL201C
YNL200C
YNL199C
YNL202W
YNL203C
YNL206C
YNL205C
YNL204C
YNL208W
YNL207W
YNL210W
YNL209W
YNL214W
YNL213C
YNL211C
YNL216W
YNL215W
YNL218W
YNL217W
YNL219C
YNL220W
YNL221C
YNL223W
YNL222W
YNL227C
YNL225C
YNL224C
YNL228W
YNL229C
YNL232W
YNL231C
YNL230C
YNL234W
YNL233W
YNL237W
YNL236W
YNL235C
YNL239W
YNL238W
YNL241C
YNL240C
YNL243W
YNL242W
YNL244C
YNL246W
YNL245C
YNL247W
YNL249C
YNL248C
YNL250W
YNL251C
YNL253W
YNL252C
YNL254C
YNL256W
YNL255C
YNL258C
YNL257C
YNL261W
YNL260C
YNL259C
YNL262W
YNL263C
YNL266W
YNL265C
YNL264C
YNL268W
YNL267W
YNL272C
YNL271C
YNL270C
YNL273W
YNL274C
YNL275W
YNL276C
YNL281W
YNL279W
YNL278W
YNL282W
YNL283C
YNL287W
YNL286W
YNL284C
YNL289W
YNL288W
YNL290W
YNL292W
YNL291C
YNL293W
YNL294C
YNL296W
YNL295W
YNL299W
YNL298W
YNL297C
YNL300W
YNL301C
YNL304W
YNL302C
YNL305C
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YNL307C
YNL311C
YNL310C
YNL308C
YNL312W
YNL313C
YNL314W
YNL315C
YNL317W
YNL316C
YNL318C
YNL320W
YNL319W
YNL321W
YNL323W
YNL322C
YNL326C
YNL325C
YNL327W
YNL328C
YNL331C
YNL330C
YNL329C
YNL333W
YNL332W
YNL336W
YNL335W
YNL334C
YNL338W
YNL339C
YNR003C
YNR002C
YNR001C
YNR004W
YNR005C
YNR006W
YNR007C
YNR010W
YNR009W
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YNR012W
YNR011C
YNR015W
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YNR013C
YNR017W
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YNR020C
YNR021W
YNR022C
YNR024W
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YNR027W
YNR026C
YNR025C
YNR028W
YNR029C
YNR030W
YNR032W
YNR031C
YNR034W
YNR033W
YNR037C
YNR036C
YNR035C
YNR038W
YNR039C
YNR040W
YNR041C
YNR044W
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YNR042W
YNR046W
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YNR048W
YNR047W
YNR049C
YNR051C
YNR050C
YNR054C
YNR053C
YNR052C
YNR057C
YNR055C
YNR059W
YNR058W
YNR060W
YNR062C
YNR061C
YNR063W
YNR064C
YNR068C
YNR067C
YNR066C
YNR070W
YNR069C
YNR072W
YNR071C
YNR073C
YNR075W
YNR074C
YNR076W
YOL001W
YOL004W
YOL003C
YOL002C
YOL006C
YOL005C
YOL008W
YOL007C
YOL009C
YOL011W
YOL010W
YOL013C
YOL012C
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YOL016C
YOL017W
YOL018C
YOL020W
YOL019W
YOL021C
YOL023W
YOL022C
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YOL029C
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YOL031C
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YOL039W
YOL038W
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YOL041C
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YOL063C
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YOL075C
YOL079W
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YOL077C
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YOL080C
YOL083W
YOL082W
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YOL086C
YOL089C
YOL088C
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YOL091W
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YOL093W
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YOL097C
YOL096C
YOL095C
YOL099C
YOL098C
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YOL101C
YOL103W
YOL102C
YOL104C
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YOL105C
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YPL234C
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YPL236C
YPL239W
YPL238C
YPL240C
YPL242C
YPL241C
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YPL244C
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YPL249C
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YPL253C
YPL252C
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YPL255W
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YPL257W
YPL256C
YPL258C
YPL260W
YPL259C
YPL262W
YPL261C
YPL265W
YPL264C
YPL263C
YPL267W
YPL266W
YPL270W
YPL269W
YPL268W
YPL271W
YPL272C
YPL274W
YPL273W
YPL277C
YPL279C
YPL278C
YPL280W
YPL281C
YPR001W
YPL283C
YPL282C
YPR002W
YPR003C
YPR006C
YPR005C
YPR004C
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YPR068C
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YPR074C
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YPR076W
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YPR079W
YPR078C
YPR082C
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YPR084W
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YPR085C
YPR089W
YPR088C
YPR090W
YPR092W
YPR091C
YPR094W
YPR093C
YPR097W
YPR095C
YPR098C
YPR101W
YPR100W
YPR103W
YPR102C
YPR106W
YPR105C
YPR104C
YPR108W
YPR107C
YPR109W
YPR111W
YPR110C
YPR113W
YPR112C
YPR116W
YPR115W
YPR114W
YPR118W
YPR117W
YPR119W
YPR120C
YPR124W
YPR122W
YPR121W
YPR127W
YPR125W
YPR129W
YPR128C
YPR130C
YPR132W
YPR131C
YPR135W
YPR134W
YPR133C
YPR137W
YPR136C
YPR139C
YPR138C
YPR140W
YPR143W
YPR141C
YPR145W
YPR144C
YPR149W
YPR148C
YPR147C
YPR150W
YPR151C
YPR154W
YPR153W
YPR152C
YPR156C
YPR155C
YPR158W
YPR157W
YPR160W
YPR159W
YPR161C
YPR163C
YPR162C
YPR165W
YPR164W
YPR166C
YPR168W
YPR167C
YPR171W
YPR169W
YPR172W
YPR174C
YPR173C
YPR175W
YPR176C
YPR178W
YPR177C
YPR179C
YPR180W
YPR181C
YPR184W
YPR183W
YPR182W
YPR185W
YPR186C
YPR187W
YPR188C
YPR189W
YPR191W
YPR190C
YPR192W
YPR193C
YPR196W
YPR194C
YPR197C
YPR198W
YPR199C
YPR202W
YPR201W
YPR200C
YPR204W
YPR203W
Hmt1 Mlp1 Xpo1 Mlp2 Nup60 Cse1 Nup116 Yra1 Nup2 Nic96 Kap95 Nsp1 Nup84Nup145 Tho2 Rsc9 Npl3 Rrp6 Nup100 Nab2 Lrp1 Prp20 Hrp1
Proteins
Hmt1 Mlp1 Xpo1 Mlp2 Nup60 Cse1 Nup116 Yra1 Nup2 Nic96 Kap95 Nsp1 Nup84 Nup145 Tho2 Rsc9 Npl3 Rrp6 Nup100 Nab2 Lrp1 Prp20 Hrp1
Proteins
Figure A-1: Semi-supervised clustering of all nuclear transport and nuclear processing factors binding profiles [13–16] using the KL distance metric. The upper plot illustrates the
dendrogram. The vertical length of each branch is proportional to the distances between
clusters. The lower plot illustrates the vectors of binding ratios of each factor on a scale from
0 to 2. The binding vectors are arranged in the same order as their corresponding protein
in the dendrogram.
86
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