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\title{pwOmics - \\
Pathway-based Integration of time-series Omics Data using public database knowledge}
\author{Astrid Wachter\\
Medical Statistics, University Medical Center Göttingen, Germany}


\begin{document}
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\tableofcontents
\section{Introduction}
Characterization of biological processes can be performed in great detail with
the increased generation of omics data on different functional levels of
the cell. Especially interpretation of time-series omics data measured in
parallel with different platforms is a complex but promising task, needing
consideration of time-independent combination of omics data and additionally
time-dependent signaling analysis. As each measurement technique shows a certain
bias and has natural limitations in identifying full signaling responses
\cite{yeger-lotem_bridging_2009}, such cross-platform analysis is an up-to-date
approach in order to connect biological implications on different signaling
levels. Using diverse data types is expected to provide a deeper understanding
of global biological functions and the underlying complex processes
\cite{Kholodenko2012}.\\
This is why computational data analysis tools for interpretation of data from
proteomics and transcriptomics measurements in parallel are needed.\\
\textit{pwOmics} is a tool for pathway-based level-specific data comparison and
analysis of single time point or time-series omics data measured in parallel.
It provides individual analysis workflows for the different omics data sets
(see Figure~\ref{fig:analysis}) and in addition enables consensus analysis of
omics data as shown in the workflow overview in Figure~\ref{fig:workflow}.
\begin{figure}[htbp]
\centering
\includegraphics [angle=270, width = 320pt]{updown.pdf}
\caption{\textit{pwOmics} downstream and upstream analysis.}
\label{fig:analysis}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics [width = 360pt]{workflow.pdf}
\caption{\textit{pwOmics} workflow overview.}
\label{fig:workflow}
\end{figure}

Up to this point analysis is restricted to human species. In future an expansion
of the package is possible dependent on available online open access database
information.


\section{Databases}
\label{sec:Databases}
As \textit{pwOmics} is a package for data integration based on prior pathway
and transcription knowledge data, it is necessary to define the databases to
work with. Three different kinds of databases are necessary to do all analyses
steps:\\
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item Pathway databases:\\ The user can choose from Biocarta 
\cite{nishimura_biocarta_2001}, Reactome \cite{milacic_annotating_2012, 
croft_reactome_2014}, PID \cite{schaefer_pid:_2009} from the National 
Cancer Institute (NCI) and KEGG \cite{kanehisa_data_2014, 
kanehisa_kegg:_2000}.\\
\item Protein-protein interaction (PPI) database:\\ STRING 
\cite{franceschini_string_2013}.\\
\item Transcription factor (TF) - target gene databases:\\ The user can choose
from ChEA \cite{lachmann_chea:_2010}, Pazar \cite{portales-casamar_pazar_2009,
portales-casamar_pazar:_2007} and/or decide to specify an own file e.g. based
on a commercial database.\\
\end{enumerate}

The pathway database information is used to identify the pathways of the
differentially abundant proteins in the downstream analysis as well as upstream
protein regulators of TFs in the upstream analysis. The PPI database STRING
\cite{franceschini_string_2013} was chosen to define the protein net for the
consensus analysis. The TF - target gene databases information is necessary for
the TF identification in pathways in the downstream analysis. Additionally the
upstream TFs of differentially expressed genes/transcripts are identified in the
upstream analysis based on this information.\\
In downstream analysis the pathway gene set information is used, whereas in the
upstream analysis also the pathway topology information is exploited.\\

The database information is downloaded internally via \textit{STRINGdb} and
\textit{AnnotationHub} \cite{Morgan} package. In case the author is interested
also in the metadata of the pathway database and TF - target database it can be
received by

<<mychunk1,  eval=TRUE, echo=TRUE>>=
library(pwOmics)
library(AnnotationHub)
ah = AnnotationHub()
#pathway databases
pw = query(ah, "NIH Pathway Interaction Database")
pw[1]
#TF-target databases
chea = query(ah, "ChEA")
chea[1]
pazar = query(ah, "Pazar")
pazar[1]
@

In case you want to use TF - target gene information which is not part of the
mentioned databases but e.g. part of a commercial database, a
user-specified file can be used for the analysis. This file should be a `.txt'
file with first column transcription factors and second column target gene
symbols without a header, e.g.:\\

\begin{center}
\fbox{\parbox[c]{3cm}{
REST    SUPT7L\\
REST	SUV420H1\\
REST	SUV420H2\\
REST	SVOP\\
REST	SYCN\\
REST	SYN2\\
REST	SYP\\
REST	SYPL2\\
REST	SYT1\\
...}}
\end{center}

The STRING PPI-information is downloaded automatically while processing and
analyzing the data: The \textit{STRINGdb} package \cite{franceschini_string_2013}
is used here.

\section{Example dataset}
The example dataset used here for demonstration purposes is the one presented
in \cite{Waters2012}, which comprises the mitogenic response of human 
mammary epithelial cells to epidermal growth factor (EGF). This dataset includes
whole genome time course microarray data measured with NimbleGen whole genome
60-mer oligonucleotide arrays (Design Version 2003\_10\_27) at time points
0, 1, 4, 8, 13, 18 and 24 hr after EGF stimulation.
The complementary proteomics data was measured with LC-FTICR (Fourier-transform
ion cyclotron resonance-mass spectrometry coupled with advanced capillary
liquid chromatography) at time points 0.25, 1, 4, 8, 13, 18 and 24 hours after
EGF stimulation. Preprocessing of data was done as described in
\cite{Waters2012} resulting in lists of significant genes and proteins for
each time point as log10 expression ratios relative to the time 0 hr controls.


\section{Data pre-processing}
\label{sec:Datapre}

\textit{pwOmics} is a package for secondary data analysis, i.e. it needs already
pre-processed data as input for the analysis. The input required is\\
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item a list of all protein IDs measured,\\
\item a list of all gene/transcript IDs measured,\\
\item a list of differentially abundant proteins + log fold changes,\\
\item a list of differentially expressed genes/transcripts + log fold changes.\\
\end{enumerate}
The IDs need to be gene symbols, both for protein and gene/transcript data.
In case time-series data is analyzed inputs 3. and 4. needs to be specified for
each time point. It is absolutely necessary, that all proteins and
genes/transcript in inputs 3. and 4. are part of the lists of all protein IDs
and all gene/transcript IDs, respectively.
\\

The OmicsData object is the format used for data analysis in \textit{pwOmics}
package. It contains a list of four main elements:\\
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item OmicsD - here the omics data set, its description and the results are
stored\\
\item PathwayD - here the chosen pathway databases and the generated Biopax
model is stored\\
\item TFtargetsD - here the chosen TF-target gene databases and the combined
TF-target gene information is stored\\
\item Status - The status variable equals `1' in case not all information needed
for the analysis is read in yet and `2' after identification of the first
upstream/downstream signaling levels.
As the enrichment step is not necessarily part of the analysis and dependent on
the pathway database and the TF-target gene database the identification of
signaling molecules in further levels might not be successful, the status
variable is not used in the further analysis.\\
\end{enumerate}
Thus \textit{pwOmics} reads in the omics data set provided by the user to the
first element of the OmicsData object and further on stores all the results in
this part as well.\\

This is why the user has to provide the omics data set in a special format: A
list needs to be generated with a protein list named `P' as first element and a
gene/transcript list named `G' as second element. These lists contain as first
element a data frame with all (unique) protein IDs and gene/transcript IDs in
the first column, respectively, and as second element a list with data frames
for each time point of measurement. The data frames have two columns with the
first one containing the differentially abundant/expressed proteins or
genes/transcripts as gene symbols and the second column containing the
corresponding log fold changes, e.g.:\\
<<mychunk2,  eval=TRUE, echo=TRUE, results=hide>>=
data(OmicsExampleData)
OmicsExampleData
@
Generated as in the following example:
\begin{verbatim}
OmicsExampleData = list(P = list(allPIDs, 
                   list(PIDstp0.25, PIDstp1, PIDstp4, PIDstp8, 
                        PIDstp13, PIDstp18, PIDstp24)), 
                        G = list(allGIDs, 
                   list(GIDstp1, GIDstp4, GIDstp8, GIDstp13, 
                        GIDstp18, GIDstp24)))
\end{verbatim}

<<mychunk3,  eval=TRUE, echo=TRUE>>=
head(OmicsExampleData$P[[2]][[1]])
@




In case the user only wants to analyze omics data from a single time point just
one data frame has to be specified.\\
The time points do not have to be the same for protein and gene/transcript data
and need to be specified when reading in the omics data set separately via the
`tp\_prots' and `tp\_genes' parameters of the `readOmics' function. 

<<mychunk4,  eval=TRUE, echo=TRUE>>=
data_omics = readOmics(tp_prots = c(0.25, 1, 4, 8, 13, 18, 24), 
                       tp_genes = c(1, 4, 8, 13, 18, 24), 
                       OmicsExampleData,
                       PWdatabase = c("biocarta", "kegg", "nci",
                                      "reactome"),
                       TFtargetdatabase = c("chea", "pazar"))
@

If data from a single timepoint measurement should be analyzed the user simply
assigns the experiment number `1' for these parameters:\\
\begin{verbatim}
#for single time point data set:
omics = list(P = list(allPIDs, list(PIDs_1)),
             G = list(allGIDs, list(GIDs_1)))
data_omics = readOmics(tp_prots = c(1), 
                    tp_genes = c(1), 
                    OmicsExampleData,
                    PWdatabase = c("biocarta", "kegg", "nci", 
                                      "reactome"), 
                    TFtargetdatabase = c("chea", "pazar"))
\end{verbatim}
Additionally the selected databases have to be specified.\\

The stored information can be easily accessed via the following functions:
<<mychunk5,  eval=FALSE, echo=TRUE>>=
getOmicsTimepoints(data_omics)
head(getOmicsallProteinIDs(data_omics))
head(getOmicsallGeneIDs(data_omics))
head(getOmicsDataset(data_omics, writeData = FALSE)[[1]])
@


\section{Individual analysis}

As shown in Figure~\ref{fig:analysis} the analysis is based on an individual
analysis of the proteomic and the genomic/transcriptomic data. The downstream
analysis and upstream analysis are described in the following subsections.

Prior to that the database information has to be read in. In a first step 
the TF- target information can be made accessible to the OmicsData object by:

\begin{verbatim}
data_omics = readTFdata(data_omics)
\end{verbatim}

Via the `TF\_target\_path' parameter the path of the user-specified file can be
given. This information can be used additionally to the selected database
content.

Secondly, the `readPWdata' function takes the OmicsData object with the
provided information about the omics data set and the path of the prepared
`.RData' files from the pathway databases (see section~\ref{sec:Databases})
and automatically generates the corresponding genelists of the pathway data
if `loadgenelists = FALSE'. In this step the automatic definition of internal
differing IDs for different pathway databases is necessary, which are stored
in a new biopax model in the OmicsData object.
\begin{verbatim}
data_omicsPW = readPWdata(data_omics, 
               loadgenelists = FALSE)
\end{verbatim}
As the process of generating genelists with these IDs can take some time - 
especially for rather big databases such as Reactome 
\cite{milacic_annotating_2012, croft_reactome_2014} - the genelists for the
different databases are automatically stored in the working directory and can
be reused in another analysis when the corresponding path containing these files
is given to the `readPWdata' function as loadgenelists parameter.
\begin{verbatim}
data_omics = readPWdata(data_omics, 
             loadgenelists = "Genelist_reactome.RData")
\end{verbatim}
Automatically the information of the selected databases and/or the corresponding
user-specified file are merged. The file format (if this option is used) should
be exactly as specified in section~\ref{sec:Databases}.\\ 


\subsection{Downstream analysis}
The downstream analysis is starting with the provided proteomic data (either
single time point data or time-series data). The first step is the 
identification of the pathways in which the differentially abundant proteins 
play a role. \textit{pwOmics} performs this searching step on the basis of the 
provided proteomic data set and the selected pathway database(s). 

After reading in these information the user can follow the workflow for 
downstream analysis and identify the pathways in which the differentially 
abundant proteins are present:
\begin{verbatim}
data_omics = identifyPWs(data_omics)
\end{verbatim}
In a next step pathway enrichment can be conducted. The user can specify the 
multiple testing correction method as well as the significance level for this 
step. In case of few identified pathways this might result in too few pathways
for further analysis. In this case the enrichment step should be skipped.
\begin{verbatim}
data_omics = enrichPWs(data_omics, "BH", alpha = 0.05)
\end{verbatim}
Following the workflow the next step is the identification of the transcription
factors in these (enriched) pathways, which is done with the information
provided by the chosen TF-target gene database. The user can choose if only the
enriched pathways or all pathways should be considered for further analysis:
\begin{verbatim}
data_omics = identifyPWTFTGs(data_omics, only_enriched = FALSE)
\end{verbatim}
For use of this function the working directory should contain the previously
generated genelists.\\

The results of the downstream analysis can be easily accessed by the following
functions:
\begin{verbatim}
getDS_PWs(data_omics)
getDS_TFs(data_omics)
getDS_TGs(data_omics)

#Access biopax model generated newly on basis of selected 
#pathway databases:
getBiopaxModel(data_omics)
\end{verbatim}



\subsection{Upstream analysis}
The upstream analysis is starting with the provided gene/transcript data (either
single time point data or time-series data). It first of all identifies the
upstream TFs of the differentially expressed genes/transcripts. This step is
done with the provided/selected information of TF-target gene pairs.

Given this information, the identification of upstream TFs can be done:
\begin{verbatim}
data_omics = identifyTFs(data_omics)
\end{verbatim}

Similarly as in the downstream analysis also in the upstream analysis an optional
enrichment step can be conducted, but here on the TF level.
\begin{verbatim}
data_omics = enrichTFs(data_omics, "BH", alpha = 0.05)
\end{verbatim}

Upstream of the (enriched) TFs the regulator proteins can be identified with the
following function:
\begin{verbatim}
data_omics = identifyRsofTFs(data_omics, only_enriched = FALSE,
             noTFs_inPW = 1, order_neighbors = 10)
\end{verbatim}
Again, the user can specify if only the enriched TFs or all TFs should be
considered for further analysis. The identification of upstream regulators
is done in the following way:\\
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item Identification of the pathways the previously identified TFs are part of.\\
\item Selection of pathways according to the user-specified parameter
`noTFs\_inPW': Only those pathways are considered in further analysis with at
least this number of TFs in the pathway. Minimum number of TFs in the pathway
is 2.\\
\item Upstream regulators are identified for these TFs. This is done by finding
first for each TF the pathway neighborhood according to the user-specified
parameter `order\_neighbors'. This parameter specifies the order of the
identified pathway neighborhood. Then the intersection of all identified
neighborhoods for all TFs in a pathway is determined. The resulting pathway
node set is defined here as the set of regulator proteins.\\
\end{enumerate}

In case the pathways under consideration do not have pathway components in the
downloaded biopax model, this will be indicated with a warning. This warning can
be ignored by the user in regard to the following analysis steps.

The results of the upstream analysis can be accessed with the following functions:
\begin{verbatim}
getUS_TFs(data_omics)
getUS_PWs(data_omics)
getUS_regulators(data_omics)
\end{verbatim}


\section{Consensus analysis}

The consensus analysis combines the results from upstream and downstream analysis 
by constituting in particular the comparative analysis of the results of the
two different data sets. The intersection analysis simply compares the results
of the separate upstream and downstream analysis. The static consensus analysis
enables setting up static consensus graphs for each time point measured in
parallel. Finally, the dynamic consensus analysis provides the user with one
final dynamic consensus graph obtained from the data changes over time
based on dynamic bayesian network inference. The dynamic consensus analysis is
self-evidently only conductable with time-series data sets measured for proteome
and genome/transcriptome data in parallel.


\subsection{Intersection analysis}
The intersection analysis of \textit{pwOmics} is a simple comparative analysis
of the results of upstream and downstream analysis. Thus, it enables a
comparison of single time point data and time-series data, the latter also for
non-corresponding time points in the different data sets. The comparison is
possible on the three different functional levels considered in this package:
On the proteome level, the transcription factor level and gene/transcript level.
\begin{verbatim}
getProteinIntersection(data_omics, 
                       tp_prot = 4, 
                       tp_genes = 4)
getTFIntersection(data_omics, 
                  tp_prot = 4,
                  tp_genes = 1)
getGenesIntersection(data_omics,
                     tp_prot = 4,
                     tp_genes = 13)
\end{verbatim}
These functions not only enable a comparison of the same timepoints on the
distinct levels, but for time-series data sets also for non-matching time
points:\\
With the time resolution of measuring omics data in most cases being pre-defined
by expected signaling changes and financial limitations the potential in the
interpretation of the results is strongly confined to the experimental design
decisions. Thus, measured signaling changes, which naturally consist of a
superposition of diverse time-scales of transcriptional and translational
processes and comprehend diverse frequency patterns \cite{yosef_impulse_2011},
are dependent on the sampling. This means for some of the signaling axes it
might be the case, that 
\begin{itemize}
\item changes are not detected at all as their rate is too high,
\item hopefully most are represented in the data and 
\item some might be so slow that their change is not considered significant and
thus are excluded from analysis.
\end{itemize}
As the corresponding signaling changes are not expected to be seen at the same
time point in proteome data and gene/transcript data it is necessary to enable
also the comparison of non-corresponding time points.\\
The possibility to compare such time points naturally cannot account for the
changes not captured during measurement, however, it gives the possibility to
consider also the time needed for regulatory control mechanisms in the
interpretation of the measurement results - even if this shows considerable
variations as well.\\
In case the user wants to compare the corresponding time points on the three
levels simultaneously he can do so by using the following function:
\begin{verbatim}
gettpIntersection(data_omics)
\end{verbatim}

\subsection{Static consensus analysis}

The static consensus analysis goes one step ahead and integrates the results
gained from the comparative analysis of the corresponding time points to a
consensus net for each time point. The change of this consensus net over time
gives a first insight into the changes seen statically at the different time
points. However, the static consensus nets do not yet include information
gathered over time - as it is the case in the dynamic consensus analysis
(see section~\ref{sec:DynCons}). This is why the static consensus analysis
is also applicable for single time point measurements.\\

The static consensus analysis is conducted by generation of a Steiner tree
\cite{Kleinberg_AlgorithmDesign_2006} on basis of matching proteins and TFs
identified in downstream and upstream analysis for each corresponding time
point. The underlying graph used is the protein-protein-interaction (PPI)
graph from the STRING database reduced to the connected nodes.
The matching proteins and TFs are considered as terminal nodes and are
connected via a shortest path-based approximation of the Steiner tree
algorithm \cite{Takahashi_Steiner_1980, Sadeghi2013} across the reduced
PPI-STRING-graph. Subsequently knowledge of TF-target gene pairs from the
chosen database is used to expand the graph with matching genes/transcripts
from both upstream and downstream analysis. In case the consensus graph contains
corresponding proteins and genes/transcripts, feedback loops are added automically.
\begin{verbatim}
consensusGraphs = staticConsensusNet(data_omics)
\end{verbatim}


\subsection{Dynamic consensus analysis}
\label{sec:DynCons}

Unlike the static consensus analysis, the dynamic consensus analysis takes into
consideration also the signaling changes over time by applying dynamic bayesian
network inference. The packages used for the dynamic consensus analysis are
\textit{longitudinal} \cite{Opgen_longi_2006, rainer_opgen-rhein_inferring_2006} 
to adjust the format of the data and the actual network inference part is done
via the \textit{ebdbNet} \cite{Rau_ebdbnet_2010} package. 
This package includes an iterative empirical Bayesian procedure with a Kalman
filter estimating the posterior distributions of the network parameters.
The defined prior structure of the network is used for a straightforward
estimation of hyperparameters via an expectation maximization (EM)-like
algorithm and the dimension of the hidden states are determined via the
singular value decomposition (SVD) of a block-Hangel matrix.\\

The nodes included into the network inference step are nodes which are part of
the static consensus graphs from corresponding time points of the two different
measurement types, i.e.\\
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item proteins identified in upstream and downstream analysis at the same
time points,\\
\item Steiner nodes identified via static consensus analysis,\\
\item TFs identified in upstream and downstream analysis at the same time
points and\\
\item genes/transcripts identified in upstream and downstream analysis at the
same time points.\\
\end{enumerate}


To apply dynamic network inference a reasonable number of measurements needs
to be available. As in most cases of parallel protein and gene/transcript
measurements only very few corresponding time steps are available it is
necessary to artificially introduce additional time steps. This is done by
generating smoothing splines applied on the log fold changes provided by the
user under the simplifying assumption of a gradual change of signaling between
the different time points.\\

This assumption, however, has to be applied consciously and carefully, as there
might be higher frequency signaling components superimposed (see for a
comprehensive analysis of temporal dynamics of gene expression
\cite{yosef_impulse_2011}).
In theory a signal has to be sampled 2 times its maximal frequency in order to
be able to reconstruct it exactly from time discrete measurements
(Nyquist-Shannon sampling theorem \cite{Shannon_theorem_1949, 
Nyquist_theorem_1928}). This means only exact interpretation of those signaling
axes are possible that have a frequency which is smaller than half of the
sampling frequency. However, under certain preconditions on signal structure
and the sampling operator reconstruction of the original signal can be done with
a lower sampling rate \cite{DBLP:journals/tit/BlumensathD09}.
This is an interesting starting point for a more comprehensive dynamic analysis
of the expected signals and the sampling needed for an extensive data mining of
omics data sets measured in parallel, but exceeds the scope of this package.
\\

The number of time points generated additionally via smoothing splines is based
on simulation results of \textit{ebdbNet} analysis for median area under the
curve (AUC) values of receiver operating characteristic (ROC) curves: In their
results it was shown that a plateau at around 50 to 75 time points was reached.
Thus in \textit{pwOmics} 50 time points are predicted with smoothing splines in
order to apply dynamic bayesian network inference on omics data sets measured
in parallel.\\

After generation of these time points a block-Hankel matrix of autocovariances
is constructed based on the time series abundance/expression data. For this the
user needs to provide the \textit{laghankel} parameter, specifying the maximum
relevant time lag to be used in constructing the block-Hankel matrix.
With a singular value decomposition (see function `hankel' of \textit{ebdbNet}
package) the number of hidden states can be determined. Here, the user can
specify the \textit{cutoffhankel} parameter to choose the cutoff to determine
the desired percent of total variance explained by the singular values. 
Additional parameters on convergence criteria and iterations performed can be
specified. For further details the user is referred to \cite{Rau_ebdbnet_2010}.
\begin{verbatim}
library(ebdbNet)
library(longitudinal)
dynInferredNet = dynamicConsensusNet(data_omics, 
                 laghankel = 3, 
                 cutoffhankel = 0.9)
\end{verbatim}

\section{Time profile clustering}
An additional analysis option is clustering of co-regulation patterns over time.
It provides information about the signaling molecules with common dynamic
behaviour and thus allows to draw conclusions in terms of signaling chronology.
Time profile clustering is performed as soft clustering based on the
\textit{Mfuzz} package \cite{Mfuzz2012}. The advantage of this clustering method
is that a protein, TF or gene/transcript can be assigned to several clusters,
thus reducing the sensitivity to noise and the information loss hard clustering
exhibits. It is implemented as fuzzy c-means algorithm \cite{Hathaway_Pattern1986} 
and iteratively optimizes the objective function to minimize the variation of
objects within the clusters. The user needs to provide a `min.std' threshold
parameter if proteins or genes/transcripts with a low standard deviation
should be excluded. In addition the maximum number of cluster centers which
should be tested in the `minimum distance between cluster centroid test' has
to by given. This number is used as initial number to determine the data-specific
maximal cluster number based on the number of distinct data points. For more
details see \cite{Mfuzz2012} and \cite{schwammle_simple_2010}.
\begin{verbatim}
library(Mfuzz)
fuzzyClusters = clusterTimeProfiles(dynConsensusNet, 
                                    min.std = 0, 
                                    ncenters = 12)
\end{verbatim}


\section{Visualization}
To complement the results from the different comparisons and analyses
(accessible via the `get...' functions) the \textit{pwOmics} package provides
visualization functions for the different analyses.
The consensus graphs of the static analysis for one or more corresponding time
points can be plotted with the following function
(see Figure~\ref{fig:Consgraph}):
\begin{verbatim}
plotConsensusGraph(consensusGraphs, data_omics)
\end{verbatim}  

\begin{figure}[htbp]
\centering
\includegraphics{ConsensusGraphs_1.pdf}
\caption{\textit{pwOmics} static consensus graph: Time point 1 hr.}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics{ConsensusGraphs_2.pdf}
\caption{\textit{pwOmics} static consensus graph: Time point 4 hrs.}
\end{figure}


The dynamic consensus analysis result can be visualized as follows
(see Figure~\ref{fig:DynConsgraph}):
\begin{verbatim}
plotdynConsensusNet(dynInferredNet, sig.level = 0.65)
\end{verbatim}
Here, the parameter `sig.level' is the significance level used as cutoff for
plotting edges in the network and has to be specified in the range between
0 and 1. Furthermore the user can indicate if unconnected nodes should be
removed and provide additional \textit{igraph} \cite{igraph2006}
layout parameters.\\
\begin{figure}[htbp]
\centering
\includegraphics{dynConsensusNet_2.pdf}
\caption{\textit{pwOmics} dynamic consensus graph.}
\label{fig:DynConsgraph}
\end{figure}

However, as the user can access the networks easily \textit{tkplot} from the
\textit{igraph} R package is a nice interactive graph drawing alternative.
In addition plot parameters can be easily changed as the result networks
are of class `igraph'.

In order to plot the results from time profile clustering 
(see Figure~\ref{fig:cluster}) the following function can be used:
\begin{verbatim}
plotTimeProfileClusters(fuzzyClusters) 
\end{verbatim}
The different colours represent the different clusters. The legend is only shown
if the number of genes and proteins is not too large. Otherwise the user can
easily access this information by having a look to the output of the
`clusterTimeProfiles' function which provides information about cluster centers,
the number of data points in each cluster of the closest hard clustering,
cluster indices, and additional parameters explained in detail in the `mfuzz'
documentation. In the legend the attachments `\_g' and `\_p', respectively,
indicate, if the node originally derives from protein or gene/transcript 
measurements.

\begin{figure}[htbp]
\centering
\includegraphics{TimeProfiles.pdf}
\caption{\textit{pwOmics} time profile clusters.}
\label{fig:cluster}
\end{figure}

\newpage
\section{Session Information}
\begin{itemize}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item R Under development (unstable) (2015-03-11 r67980),
Platform: x86\_64-unknown-linux-gnu (64-bit),
Running under: Ubuntu precise (12.04.5 LTS)\\
\item Locale: LC\_CTYPE=en\_US.UTF-8, LC\_NUMERIC=C,\\
LC\_TIME=en\_US.UTF-8, LC\_COLLATE=en\_US.UTF-8,\\
LC\_MONETARY=en\_US.UTF-8, LC\_MESSAGES=en\_US.UTF-8,\\
LC\_PAPER=en\_US.UTF-8, LC\_NAME=C, LC\_ADDRESS=C,\\
LC\_TELEPHONE=C, LC\_MEASUREMENT=en\_US.UTF-8,\\
LC\_IDENTIFICATION=C\\
\item Attached base packages: tcltk, parallel, stats, graphics, grDevices,
utils, datasets, methods, base\\
\item Other attached packages: ebdbNet\_1.2.3, Mfuzz\_2.27.1, DynDoc\_1.45.0,
widgetTools\_1.45.0, e1071\_1.6-4, Biobase\_2.27.3, BiocGenerics\_0.13.11,
longitudinal\_1.1.11, corpcor\_1.6.7, igraph\_0.7.1, pwOmics\_0.99.2\\
\item Loaded via namespace (and not attached): Rcpp\_0.11.5, XVector\_0.7.4,
BiocInstaller\_1.17.7, GenomeInfoDb\_1.3.19, plyr\_1.8.1,\\
AnnotationHub\_1.99.82, tkWidgets\_1.45.0, class\_7.3-12, bitops\_1.0-6,\\
biomaRt\_2.23.5, digest\_0.6.8, RSQLite\_1.0.0, shiny\_0.11.1, DBI\_0.3.1,\\
rBiopaxParser\_2.5.0, stringr\_0.6.2, httr\_0.6.1, S4Vectors\_0.5.23,\\
gtools\_3.4.2, caTools\_1.17.1, IRanges\_2.1.44, stats4\_3.2.0,\\
data.table\_1.9.4, R6\_2.0.1, AnnotationDbi\_1.29.24, XML\_3.98-1.1,\\
RJSONIO\_1.3-0, gdata\_2.13.3, reshape2\_1.4.1, STRINGdb\_1.7.0,\\
gplots\_2.16.0, htmltools\_0.2.6, GenomicRanges\_1.19.54, mime\_0.3,\\
interactiveDisplayBase\_1.5.6, xtable\_1.7-4, httpuv\_1.3.2,\\ 
KernSmooth\_2.23-14, RCurl\_1.95-4.5, chron\_2.3-45\\
\end{itemize}

\printbibliography

\end{document}

