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\begin{document}

\title{Manual for the R \texttt{casper} package}
\date{}  %comment to include current date

\maketitle


We put a lot of time and effort in developing casper,
if you use casper for your research please do cite
our paper \cite{rossell:2014}.
Our support comes from research institutions and grants,
your citations is an easy way to ensure we continue having time to develop casper.


\section{Quick start}
\label{sec:quickstart}

The function \texttt{wrapKnown} runs the whole analysis pipeline for a single sample, 
starting from a sorted and indexed BAM file and reference transcriptome and
returning estimated log-expression in an ExpressionSet object.
\texttt{wrapKnown} also returns the following secondary output,
which may be ignored for routine analyses: processed reads (procBam object), path counts (pathCounts object)
and read start and fragment length distributions (readDistrs object).

The \texttt{wrapKnown} function needs an annotated transcriptome,
which can either come from the UCSC or any user-specified gtf file
({\it e.g.} as provided by the Gencode project or returned by the Cufflinks
RABT module).
To generate a transcriptome from the UCSC database you may use the following code,
changing \texttt{'hg19'} for your desired genome.

\small
\begin{verbatim}
library(GenomicFeatures)
genome='hg19'
genDB<-makeTranscriptDbFromUCSC(genome=genome, tablename=''refGene'')
hg19DB <- procGenome(genDB=genDB, genome=genome, mc.cores=6)
\end{verbatim}
\normalsize

To generate a transcriptome from a gtf file use the following code,
changing the file name for your gtf file (including the full path).

\small
\begin{verbatim}
genDB <- import('gencode.v18.annotation.gtf')
gencode18DB <- procGenome(genDB, genome='gencode18')
\end{verbatim}
\normalsize

Recall that the input BAM file should be indexed and sorted, and that this index is placed in the same directory as the corresponding BAM. 
The \texttt{samtools} ''index'' function can be used to generate such an index.

To call \texttt{wrapKnown} use the code below
(for information on the parameters please refer to the man page of the function).

\begin{verbatim}
  bamFile="/path_to_bam/sorted.bam"
  ans <- wrapKnown(bamFile=bamFile, mc.cores.int=4, mc.cores=3, genomeDB=hg19DB, readLength=101)
  names(ans)
  head(exprs(ans\$exp))
\end{verbatim}

After you run \texttt{wrapKnown} on all your bam files, you can easily combine
the expressions from all samples into a single \texttt{ExpressionSet} using
function \texttt{mergeExp}.
The code below contains an example to combine four samples.
Adding \texttt{'explCnts'} to the \texttt{keep} argument results in the number
of counts for each gene and each sample to be saved in the \texttt{fData}
of the combined \texttt{ExpressionSet}
(by default, only the total count across all samples is stored).
The function \texttt{quantileNorm} performs quantile normalization,
which is typically needed to take into account the different sequencing depth
in each sample.

\small
\begin{verbatim}
sampleNames <- c('A1','A2','B1','B2')
x <- mergeExp(A1$exp,A2$exp,B1$exp,B2$exp,sampleNames=sampleNames, keep=c('transcript','gene','explCnts'))
x$group <- c('A','A','B','B')
xnorm <- quantileNorm(x)
boxplot(exprs(x))
boxplot(exprs(xnorm))
\end{verbatim}
\normalsize


The remaining sections explain in more detail the model, functions and classes included in the package.
They are intended to help you obtain a better understanding of the methodology
underlying \texttt{casper} by obtaining expression estimates step-by-step.
However, in practice we recommend that you use \texttt{wrapKnown}, as it is
more efficiently implemented both in terms of memory requirements and
computational speed.



\section{Introduction}
\label{sec:intro}

The package \texttt{casper} implements statistical methodology to infer
gene alternative splicing from paired-end
RNA-seq data \citep{rossell:2014}.
In this section we overview the methodology
and highlight its advantages.
For further details, please see the paper.
In subsequent sections we illustrate how to use the package with a worked example.

\texttt{casper} uses a probability model to estimate expression at the variant level.
Key advantages are that \texttt{casper} summarizes RNA-seq data in a manner that is more informative
than the current standard, and that is determines the read non-uniformity
and fragment length (insert size) distribution from the observed data.
More specifically, the current standard is to record the number of reads overlapping with each exon
and connecting each pair of exons.
The fact that only pairwise connections between exons are considered disregards important information,
namely that numerous read pairs visit more than 2 exons.
While this was not a big issue with older sequencing technologies,
it has become relevant with current protocols which produce longer sequences.
For instance, in a 2012 ENCODE Illumina Hi-Seq dataset \cite{rossell:2014} found that roughly 2 out of 3 read pairs visited $\geq 3$ exons.

\texttt{casper} summarizes the data by recording the {\it exon path} followed by each pair,
and subsequently counts the number of exons following each path.
For instance, suppose that the left end visits exons 1 and 2, while the right end visits exon 3.
In this case we would record the path \texttt{1.2-3},
and count the number of reads which also visit the same sequence of exons.
Table \ref{tab:sampledata} illustrates how these summaries might look like for a gene with 3 exons
(examples in subsequent sections show counts for experimental data).
The first row indicates there are 210 sequences for which both ends only overlap with exon 1.
The second and third rows contain counts for exons 2 and 3.
The fourth row indicates that for 90 sequences the left end visited exon 1 and the right end exon 2.
The fifth row illustrates the gain of information associated to considering exon paths:
we have 205 pairs where the left end visits exons 1-2 and the right end visits exon 3.
These reads can only have originated from a variant that contains the three exons in the gene,
and hence are highly informative.
If we only counted pairwise connections, this information would be lost
(and incidentally, the usual assumption that counts are independent would be violated).
The sixth row indicates that 106 additional pairs visited exons 1-2-3, but they did so in a different manner
(now it's the right end the one that visits two exons).
In this simplified example rows 5 and 6 give essentially the same information and could be combined,
but for longer genes they do provide different information.

\begin{table}
\begin{center}
\begin{tabular}{|c|r|r|r|} \hline
 Path & Number of read pairs & $P(\mbox{path} \mid v_1)$ & $P(\mbox{path} \mid v_2)$ \\ \hline
 1-1 &   210    & 0.2  & 0.35 \\
 2-2 &   95     & 0.1  & 0.25 \\
 3-3 &   145    & 0.15 & 0    \\
 1-2 &   90    & 0.1  & 0.4  \\
 1.2-3 & 205    & 0.2  & 0    \\
 1-2.3 & 106    & 0.1  & 0    \\
 2-3 &   149    & 0.15 & 0    \\
\hline
\end{tabular}
\end{center}
\caption{Exon path counts is the basic data fed into \texttt{casper}. Counts are compared to the probability of observing each path under each considered variant.}
\label{tab:sampledata}
\end{table}

Now suppose that the gene has two known variants: the full variant $v_1$ (i.e. using the 3 exons) and the variant $v_2$ which only contains exons 1 and 2.
The third column in Table \ref{tab:sampledata} shows the probability that a read pair generated from $v_1$ follows each path,
and similarly the fourth column for $v_2$.
These probabilities are simply meant as an example, in practice \texttt{casper} estimates these probabilities precisely
by considering the fragment length distribution and possible read non-uniformity.
Notice that read pairs generated under $v_2$ have zero probability of following any path that visits exon 3,
as $v_2$ does not contain this exon.
Further, the proportion of observed counts following each path is very close to what one would expect if all reads came from $v_1$,
hence intuitively one would estimate that the expression of $v_1$ must be close to 1.
From a statistical point of view, estimating the proportion of pairs generated by each variant can be viewed
as a mixture model where the aim is to estimate the weight of each component (i.e. variant) in the mixture.

A key point is that, in order to determine the probability of each path, one would need to know the
distribution of fragment lengths (i.e. outer distance between pairs)
and read starts (e.g. read non-uniformity due to 3' biases).
These quantities are in general not known, and in our experience reports from sequencing facilities are oftentimes inaccurate.
Further, these distributions may differ substantially from simple parametric forms that are usually assumed
({\it e.g.} the fragment length distribution is not well approximated by a Normal or Poisson distribution).
Instead, \cite{rossell:2014} proposed estimating these distributions non-parametrically from the observed data.
In short, these distributions are estimated by selecting reads mapping to long exons (fragment size) 
or to genes with a single known transcript (read start).
There are typically millions of such reads, therefore the estimates can be obtained at a very high precision.
Examples are shown in subsequent sections (note: the illustration uses a small subset of reads, in real applications the
estimates are much more precise). 

Finally we highlight a more technical issue. 
By default \texttt{casper} uses a prior distribution which,
while being essentially non-informative, it pushes the estimates away from the boundaries (e.g. variants with 0 expression)
and thus helps reduce the estimation error.
The theoretical justification lies in the typical arguments in favor of pooling that stem from Stein's paradox and related work.
Empirical results in \cite{rossell:2014} show that, by combining all the features described above, \texttt{casper} may reduce the estimation error by a factor of 4
when compared to another popular method.
Currently, \texttt{casper} implements methods to estimate the expression for a set of known variants.
We are in the process of incorporating methodology for de novo variant searches,
and also for sample size calculations, i.e. determining the sequencing depth, read length or the number of patients needed for a given study.



\section{Aligning reads and importing data}
\label{sec:import}

The input for \texttt{casper} are BAM files containing aligned reads.
There are several software options to produce BAM files.
TopHat \citep{trapnell:2009} is a convenient option, as it is specifically designed to map
reads spanning exon junctions accurately.
As an illustration, suppose paired end reads
produced with the Illumina platform
are stored
in the FASTQ files \texttt{sampleR1.fastq} and \texttt{sampleR2.fastq}.
The TopHat command to align these reads into a BAM file is:

\footnotesize
\begin{verbatim}
> tophat --solexa1.3-quals -p 4 -r 200 /pathToBowtieIndexes/hg19 
sampleR1.fastq sampleR2.fastq
\end{verbatim}
\normalsize

The option \texttt{--solexa1.3-quals} indicates the version of quality scores
produced by the Illumina pipeline and \texttt{-p 4} to use 4 processors.
The option \texttt{-r} is required by TopHat for paired-end reads and indicates the average fragment size.
The fragment size is around 200-300 for many experiments,
so any value of \texttt{-r} in this range should be reasonable.
After importing the data into R,
one can use the \texttt{casper} function \texttt{getDistrs}
to estimate the fragment size distribution (see below).
This can be used as a check that the specified \texttt{-r} was reasonable.
In our experience, results are usually robust to moderate miss-specifications of \texttt{-r}.

BAM files can be read into R using the \texttt{Rsamtools} package \citep{rpkg:Rsamtools}.
For the sake of computational speed,
in this vignette we will use data that has already been imported in a previous session.
The data was obtained from the RGASP1 project at \\
ftp://ftp.sanger.ac.uk/pub/gencode/rgasp/RGASP1/inputdata/human\_fastq.

We used reads from replicate 1 and lane 1 in sample K562\_2x75.
In order for the vignette to compile quickly here we illustrate
the usage of the package by selecting the reads mapping
to 6 genes in chromosome 1 (see Section \ref{sec:preprocess}).
The code required to import the data into Bioconductor is provided below.
It is important to add the option \texttt{tag='XS'},
so that information on whether the experiment was stranded or not is imported.

\footnotesize
\begin{verbatim}
> library(Rsamtools)
> what <- scanBamWhat(); what <- what[!(what %in% c('seq','qual'))]
> flag <- scanBamFlag(isPaired=TRUE,hasUnmappedMate=FALSE)
> param <- ScanBamParam(flag=flag,what=what,tag='XS')
> bam0 <- scanBam(file='accepted_hits.bam',param=param)[[1]]
\end{verbatim}
\normalsize


\section{Pre-processing the data for analysis}
\label{sec:preprocess}

We start by obtaining and processing genome annotation data.
Here we illustrate our package with a few selected genes obtained
from the human genome version hg19.
The commands that one would use to store the full annotated genome into \texttt{hg19DB} is

\begin{verbatim}
genome='hg19'
genDB<-makeTranscriptDbFromUCSC(genome=genome, tablename="refGene")
> hg19DB <- procGenome(genDB=genDB, genome=genome, mc.cores=6)
\end{verbatim}

We load the imported BAM file and processed human genome annotation.
\texttt{K562.r1l1} was imported using \texttt{scanBam}
and is a list containing read-level information such as read identifier,
chromosome and alignment position, position of the matched paired end etc.
\texttt{hg19DB} is an object of class \texttt{annotatedGenome}
and contains information regarding genes, transcripts, exons etc.
It also indicates the genome version that was used to create the genome
and the creation date.

\footnotesize
<<process1>>=
library(casper)
data(K562.r1l1)
names(K562.r1l1)
data(hg19DB)
hg19DB
head(sapply(hg19DB@transcripts,length))
@ 
\normalsize

The lengths displayed above indicate the number of transcripts per island.

RNA-seq experiments typically contain some very short RNA sequences,
which can be due to RNA degradation.
The function \texttt{rmShortInserts} removes all sequences with insert size
({\it i.e.} distance between start of left-end and start of right-end)
below a user-specified level. We remove reads with insert sizes below 100bp.
We then use \texttt{getDistrs} to estimate the fragment length distribution
and the read start distribution.

\footnotesize
<<process2>>=
bam0 <- rmShortInserts(K562.r1l1, isizeMin=100)
distrs <- getDistrs(hg19DB,bam=bam0,readLength=75)
@ 
\normalsize

We visualize the fragment length distribution.
The resulting plot is shown in Figure \ref{fig:plotprocess1}, left panel.
Notice there few fragments shorter than 140bp.
Given the reduced number of reads in our toy data the estimate is not accurate,
and hence we overlay a smoother estimate (blue line).

\footnotesize
<<plotprocess1>>=
plot(distrs, "fragLength")
@ 
\normalsize

We produce a histogram to inspect the read start distribution.
The histogram reveals that reads are non-uniformly distributed
along transcripts (Figure \ref{fig:plotprocess1}, right panel).
Rather, there is a bias towards the 3' end.

\footnotesize
<<plotprocess2>>=
plot(distrs, "readSt")
@ 
\normalsize

\setkeys{Gin}{width=0.45\textwidth} 
\begin{figure}
\begin{center}
\begin{tabular}{cc}
<<label=plotprocess1,fig=TRUE,echo=FALSE>>=
<<plotprocess1>>
@ &
<<label=plotprocess2,fig=TRUE,echo=FALSE>>=
<<plotprocess2>>
@ 
\end{tabular}
\end{center}
\caption{Left: fragment length distribution; Right: read start distribution}
\label{fig:plotprocess1}
\end{figure}

As a final pre-processing step, we use the function \texttt{procBam}
to divide each read pair into a series of disjoint intervals.
The intervals indicate genomic regions that the read
aligned to 
consecutively, {\it i.e.} with no gaps.

\footnotesize
<<procBam>>=
pbam0 <- procBam(bam0)
pbam0
head(getReads(pbam0))
@ 
\normalsize

The resulting object \texttt{pbam0} is a list
with element \texttt{pbam} of type \texttt{RangedData}
and \texttt{stranded} indicating whether the RNA-seq experiment was stranded or not.


\section{Estimating expression for a set of known variants}
\label{sec:knownvar}
 
In order to obtain expression estimates,
we first determine the exons visited by each read,
which we denominate the {\it exon path},
and count the number of reads following 
the same exon path.

\footnotesize
<<pathCounts>>=
pc <- pathCounts(pbam0, DB=hg19DB)
pc
head(pc@counts[[1]])
@ 
\normalsize

The output of \texttt{pathCounts} is a 
named integer vector counting exon paths. 
The names follow the format ".exon1.exon2-exon3.exon4.", with
dashes making the split between exons visited by left and
right-end reads correspondingly.
For instance, an element in \texttt{pc} named
\texttt{.1314.1315-1315.1316.}
indicates the number of reads for which the left end
visited exons 1314 and 1315 
and the right end visited exons 1315 and 1316.
The precise genomic coordinates of each exon are stored in
the annotated genome.
 
The function \texttt{calcExp} uses the exon path counts,
read start and  fragment length distributions and genome annotation
to obtain RPKM expression estimates.
Expression estimates are returned in an \texttt{ExpressionSet} object,
with RefSeq transcript identifiers as \texttt{featureNames}
and the internal gene ids used by \texttt{hg19DB} stored as feature data.

\footnotesize
<<calcExp>>=
eset <- calcExp(distrs=distrs, genomeDB=hg19DB, pc=pc, readLength=75, rpkm=FALSE)
eset
head(exprs(eset))
head(fData(eset))
@ 
\normalsize

When setting \texttt{rpkm} to \texttt{FALSE}, \texttt{calExp} returns relative
expression estimates for each isoform.
That is, the proportion of transcripts originating from each variant,
so that the estimated expressions add up to 1 for each island.
If you would prefer relative expressions that add up to 1 within each gene
you can use function \texttt{relexprByGene}.
When setting \texttt{rpkm} to \texttt{TRUE},
expression estimates in reads per kilobase per million (RPKM)
are returned instead.

\footnotesize
<<relIsoformExpr>>=
eset <- calcExp(distrs=distrs, genomeDB=hg19DB, pc=pc, readLength=75, rpkm=TRUE)
head(exprs(eset))
@ 
\normalsize

Let $\hat{\pi}_{gi}$ be the estimated relative expression for transcript $i$ within gene $g$,
$w_{gi}$ the transcript width in base pairs,
$n_g$ the number of reads overlapping with gene $g$
and $\sum_{}^{} n_g$ the total number of reads in the experiment.
The RPKM for transcript $i$ within gene $i$ is computed as
\begin{align}
  r_{gi}= 10^9 \frac{ \hat{\pi}_{gi} n_g}{w_{gi} \sum_{}^{} n_g}
\label{eq:rpkm}
\end{align}



\section{Plots and querying an annotatedGenome}
\label{sec:plots}

\texttt{casper} incorporates some functionality to plot splicing variants and estimated expression levels.
While in general we recommend using dedicated visualization software such as IGV \citep{robinson:2011},
we found useful to have some plotting capabilities within the package.

We start by showing how to extract information from an \texttt{annotatedGenome} object.
Suppose we are interested in a gene with Entrez ID$=27$.
We can obtain the known variants for that gene with the function \texttt{transcripts},
and the chromosome with \texttt{getChr}.
We can also find out the island identifier that \texttt{casper} assigned to that gene
(recall that \texttt{casper} merges multiple genes that have some overlapping exons into a single gene island).

\footnotesize
<<plot1>>=
tx <- transcripts(entrezid='27', genomeDB=hg19DB)
tx
getChr(entrezid='27',genomeDB=hg19DB)
islandid <- getIsland(entrezid='27',genomeDB=hg19DB)
islandid
getChr(islandid=islandid,genomeDB=hg19DB)
@ 
\normalsize

Once we know the islandid, we can plot the variants with \texttt{genePlot}.
The argument \texttt{col} can be set if one wishes to override the default rainbow colours.

<<plot2>>=
genePlot(islandid=islandid,genomeDB=hg19DB)
@ 

Figure \ref{fig:plot2} shows the resulting plot.
The plot shows the identifiers for all transcripts
in the gene island, and exons are displayed as boxes.
The x-axis indicates the genomic position in bp.
For instance, the last three variants have a different transcription end site
than the rest, indicated by their last exon being different.
Similarly, the first variant has an alternative transcription start site.

\setkeys{Gin}{width=0.6\textwidth} 
\begin{figure}
\begin{center}
<<label=plot2,fig=TRUE,echo=FALSE>>=
<<plot2>>
@
\end{center}
\caption{Transcripts for gene with Entrez ID$=27$}
\label{fig:plot2}
\end{figure}

It can also be useful to add the aligned reads and estimated expression to the plot.
This can be achieved by passing the optional arguments \texttt{reads} (the object returned by \texttt{procBam})
and \texttt{exp} (the object returned by \texttt{calcExp}).

<<plot3>>=
genePlot(islandid=islandid,genomeDB=hg19DB,reads=pbam0,exp=eset)
@ 

Figure \ref{fig:plot3} shows the plot.
Black segments correspond to pairs with short insert size (i.e. where both ends are close to each other, by default up to
\texttt{maxFragLength}=500bp).
They indicate the outer limits of the pair
(i.e. left-most position of the left read and right-most position of the right read).
Red/blue segments indicate pairs with long insert sizes.
The red lines indicates the gapped alignments and the discontinuous blue lines simply fill in the gaps, so that they are easier to visualize.
By staring at this plot long enough, one can make some intuitive guesses
as to which variants may be more expressed.
For instance, many reads align to the left-most exon, which suggests that variant NM\_00136001 is not highly expressed.
Accordingly, \texttt{casper} estimated expression for this variant is lowest.
There are few reads aligning to the exons to the right-end (which may be partially explained by the presence of a 3' bias).
The last variant does not contain several of these genes, and hence has the highest estimated expression.
Of course, inspecting the figure is simply meant to provide some intuition,
to quantify alternative splicing \texttt{casper} uses precise probability calculations.

\setkeys{Gin}{width=0.6\textwidth} 
\begin{figure}
\begin{center}
<<label=plot3,fig=TRUE,echo=FALSE>>=
<<plot3>>
@
\end{center}
\caption{Transcripts for gene with Entrez ID$=27$}
\label{fig:plot3}
\end{figure}

%\section{De novo variant discovery}
%\label{sec:denovo}
%
%\texttt{casper} implements a Bayesian model selection strategy to detect novel variants and quantify their expression.
%The basic idea is to first expand the annotated genome by adding new exons (or groups of exons, suggesting a new transcripts) 
%suggested by the data.
%Second, for each gene we consider all the variants that can be formed by including/excluding each exon,
%and report the posterior probability that a given set of variants is expressed.
%Conditional on their being expressed, we also estimate their expression level.
%Denote a set of variants of interest by $\bm{\nu}$ and by $\bm{\pi}$ their relative expression.
%We perform inference on the set of expressed variants using the posterior probability
%$P(\bm{\nu} \mid {\bf y})$ conditional on the observed data ${\bf y}$,
%and estimate their expression from $P(\bm{\pi} \mid \bm{\nu}, {\bf y})$.
%Finally, we use these results to estimate the expression of each variant,
%either by selecting the most probably set of variants or by 
%perform modeling averaging, {\it i.e.} combine the estimated $\bm{\pi}$
%under different sets of variants $\bm{\nu}_1,\bm{\nu}_2,\ldots$ according the their posterior probabilities $P(\bm{\nu}_i \mid {\bf y})$.
%
%We start by
%expanding the annotated genome by detecting new exons and transcripts suggested by the data.
%We then obtain path counts based on the expanded genome.
%
%<<createDenovoGenome>>=
%hg19denovo <- createDenovoGenome(pbam0,DB=hg19DB,readLen=75)
%hg19denovo
%pcnovo <- pathCounts(pbam0,DB=hg19denovo)
%@ 
%
%Second, we must set the prior probability that a set of variants is expressed.
%We recommend setting these probabilities based on the annotated genome,
%so that we assign high prior probability to combinations of variants that
%are consistent with those in the genome.
%For instance, genes with more exons tend to have a larger number of variants,
%but variants missing many exons are typically not expressed.
%
%<<modelPrior>>=
%mprior <- modelPrior(hg19DB)
%mprior
%@ 
%
%We can visualize the prior probabilities with \texttt{plotPriorAS}.
%Setting \texttt{type='nbVariants'} displays the prior probability that
%a gene with $E$ exons has $1,2,\ldots,2^{E}-1$ expressed variants.
%Figure \ref{fig:plotprior1} displays these probabilities for $E=9$
%(there are too few genes in \texttt{hg19DB} to obtain precise estimates for other $E$ values).
%The figure also shows the observed proportion in the annotated genome \texttt{hg19DB}.
%
%\footnotesize
%<<plotprior1>>=
%plotPriorAS(mprior, type='nbVariants',exons=9)
%@ 
%\normalsize
%
%
%\setkeys{Gin}{width=0.45\textwidth} 
%\begin{figure}
%\begin{center}
%<<label=plotprior1,fig=TRUE,echo=FALSE>>=
%<<plotprior1>>
%@
%\end{center}
%\caption{Prior distribution on the number of variants for genes with 2, 3, 4 and 5 exons}
%\label{fig:plotprior1}
%\end{figure}
%
%We can also display the prior distribution on the number of exons
%contained in an expressed variants by setting \texttt{type='nbExons'}.
%
%\footnotesize
%<<plotprior2>>=
%plotPriorAS(mprior, type='nbExons',exons=9)
%@ 
%\normalsize
%
%\setkeys{Gin}{width=0.45\textwidth} 
%\begin{figure}
%\begin{center}
%<<label=plotprior2,fig=TRUE,echo=FALSE>>=
%<<plotprior2>>
%@
%\end{center}
%\caption{Prior distribution on the number of exons in a variant for genes with 2, 3, 4 and 5 exons}
%\label{fig:plotprior2}
%\end{figure}
%
%The function \texttt{calcDenovo} computes posterior probabilities and estimated expression for each set of variants.
%For a gene with $E$ exons, there's $2^E-1$ possible variants and hence $2^{2^E-1}-1$ possible sets of expressed variants.
%It is computationally unfeasible to enumerate such a super-exponential number of models.
%By default, \texttt{calcDenovo} performs this exhaustive enumeration only for genes with up to 5 exons.
%For genes with $\geq 6$ exons its uses an MCMC scheme combined with systematic search strategies
%that attempts to explore the sets of variants wit highest posterior probability.
%
%
%<<calcDenovo>>=
%fit1 <- calcDenovo(distrs=distrs, genomeDB=hg19denovo, pc=pcnovo, readLength=75, islandid="4")
%fit1
%fit1[['4']]
%@ 

\bibliographystyle{plainnat}
\bibliography{references} 

\end{document}
