%&pdflatex
\documentclass[fleqn,a4paper]{article}
\usepackage{a4wide}
\usepackage{Sweave}
\usepackage{amsmath, amssymb}
\usepackage{charter}
\renewcommand{\sfdefault}{phv}
\usepackage[sf]{titlesec}

\newcommand{\Dir}{\operatorname{Dir}}
\newcommand{\B}{\operatorname{B}}
\newcommand{\E}{\operatorname{E}}
\newcommand{\Beta}{\operatorname{Beta}}
\newcommand{\MAP}{\mathrm{MAP}}
\newcommand{\Mult}{\operatorname{Mult}}
\newcommand{\Bin}{\operatorname{Bin}}
\renewcommand{\bold}{\textbf}

%\VignetteIndexEntry{An R package for detecting low frequency variants in deep sequencing experiments}
%\VignetteDepends{deepSNV}

\begin{document}

\title{Calling subclonal mutations with \texttt{deepSNV}}
\author{Moritz Gerstung and Niko Beerenwinkel}

\maketitle

\section{Introduction}
This package provides algorithms for calling single nucleotide variants in deep sequencing experiments of polyclonal samples. 
The package uses a clonal control experiment for
estimating the local error rate and tests whether the observed nucleotide frequencies differ between test and control. 
The basic model is a binomial model for the counts $X_{i,j}$ and $Y_{i,j}$ of nucleotide $j$ at  position $i$, in the test and the control experiment, respectively:
\begin{align}
X_{i,j} &\sim \operatorname{Bin}(n_i, p_{i,j})\cr
Y_{i,j} &\sim \operatorname{Bin}(m_i, q_{i,j}).
\end{align}
Here $n_i$ and $m_i$ denote the coverage in the two experiments, and $p_{i,j}$ and $q_{i,j}$ are learned from the data.
The presence of an SNV in the test experiment amounts to testing the hypothesis $H_1: p_{i,j} > q_{i,j}$ against the null-hypothesis $H_0:  p_{i,j} = q_{i,j}$.
The deepSNV algorithm uses likelihood ratio test with a $\chi^2_1$-distribtution. 

As an alternative to the binomial distribution, a beta-binomial model can be used that has a global parameter of overdispersion: 
\begin{align}
X_{i,j} &\sim \operatorname{BB}(n_i, \alpha, p_{i,j}) \cr
Y_{i,j} &\sim \operatorname{BB}(m_i, \alpha, q_{i,j}).
\end{align}
The parameter $\alpha$ defines a parameter that quantifies the overdispersion of the model, shared across sites and nucleotides. This parametrization is equivalent to 
setting $\beta_{i,j} = \alpha (1-p_{i,j})/p_{i,j}$. For small $p_{i,j}$, one obtains a variance of $\E[X_{i,j}] \approx n_i p_{i,j} + (n_i p_{i,j})^2/\alpha$.

All parameters are determined by a maximum likelihood criterion, where for $p_{i,j}$ (and similarly for $q_{i,j}$) a methods-of-moments approximation 
is used, $\alpha/(\alpha+\hat\beta_{i,j}) = X_{i,j}/n_i$.
The binomial model arises from the beta-binomial model in the limit $\alpha \rightarrow \infty$.

To achieve a higher specifitiy the test is performed on both strands separately, and the resulting p-values are combined into a single one using either
the product, average, or maximum as a statistic and their corresponding distributions under a uniform for computing a joint p-value. For more information and for citing
the \texttt{deepSNV} package please use:
\begin{itemize}
\item
<<echo=FALSE, results=tex>>=
	print(citation("deepSNV"), style="LaTeX")
@
\end{itemize}


\section{Working example}
In this example, we load some real world data. The data of length 1,512 nt were sequenced with a Roche 454 Junior sequencer at about 500x coverage. They consist of 
a mixture of two HIV clones at 10\% and 90\%(test) and a clonal control. The data were aligned to the HXB2 reference genome with novoalign, and can be downloaded
from the authors' website, or attached . We first load the package and define the genomic \texttt{region} of interest:
<<>>=
    library(deepSNV)
	regions <- data.frame(chr="B.FR.83.HXB2_LAI_IIIB_BRU_K034", start = 2074, stop=3585)
@
Now the data can be loaded from the remote .bam files with the \texttt{deepSNV} command (not run)
\begin{Schunk}
\begin{Sinput}
> # HIVmix <- deepSNV(test = "http://www.bsse.ethz.ch/cbg/software/deepSNV/data/test.bam", 
> #                   control = "http://www.bsse.ethz.ch/cbg/software/deepSNV/data/control.bam", 
> #                   regions=regions, q=10)
\end{Sinput}
\end{Schunk}
The \texttt{data.frame regions} contains the genomic region to be parsed from the two files by the method \texttt{deepSNV}. The additional parameter \texttt{q=10} specifies
that only nucleotides with PHRED higher than 10 are counted. As this might fail in the absence of a running internet connection, we load the resulting object that comes along with the deepSNV package:
<<>>=
	data(HIVmix) # Attach the data instead, as it could fail in routine checks without internet connection.
	show(HIVmix)
@
The counts are stored in the slots \texttt{test} and \texttt{control}:
<<>>=
	control(HIVmix)[100:110,]
	test(HIVmix)[100:110,]
@
Uppercase nucleotides are from the reference strand, lowercase nucleotides from the reverse. Also note the strand bias. 

A visual representation of the data can be obtained with the \texttt{plot} method:
<<HIV, fig=TRUE>>=
	plot(HIVmix)
@

One realizes that there are many variants nicely separated by the test at the topleft corner, althought the noise level
also extends along the diagonal to similar frequencies. Grey dots have a $P$-values smaller than 0.05.

Significant SNVs are tabularized with the \texttt{summary} command:

<<>>=
	SNVs <- summary(HIVmix, sig.level=0.05, adjust.method="BH")
	head(SNVs)
	nrow(SNVs)
	min(SNVs$freq.var)
@
We chose a significance level of \texttt{sig.level=0.05} and Benjamini-Hochberg correction for multiple testing (\texttt{adjust.method="BH"}). The test selected \Sexpr{nrow(SNVs)} variants. 
This compares to 
<<>>=
	sum(RF(test(HIVmix), total=T) > 0.01 & RF(test(HIVmix), total=T) < 0.95)
@
candidate variants with frequencies above 0.01! In this experiment we also know the truth from direct Sanger sequencing of the clones before pooling. Load the data
and study the confusion matrix with:
<<>>=
	data(trueSNVs, package="deepSNV")
	table(p.adjust(p.val(HIVmix), method="BH") < 0.05, trueSNVs)
@
So \Sexpr{table(p.adjust(p.val(HIVmix), method="BH") < 0.05, trueSNVs)[4]} of \Sexpr{sum(trueSNVs)} SNVs could be recovered by the experiments.

%We may want to learn a Dirichlet prior, and see how many variants we obtain:
%<<>>=
%	experiment.prior <- estimateDirichlet(experiment)
%	show(slot(experiment.prior,"dirichlet.prior"))
%	SNVs.prior <- significantSNV(experiment.prior, adjust.method="BH")
%	sum(apply(SNVs.prior[,c("pos","ref","SNV")],1,paste, collapse="") %in% apply(SNVs[,c("pos","ref","SNV")],1,paste, collapse=""))
%	table(p.adjust(experiment.prior@p.val, method="BH") < 0.05, trueSNVs)
%@

%Now let's test the null model. To this and compute the test with \texttt{alternative="less"}:
%<<>>=
%	p.null <- deepSNV(experiment, alternative="less")@p.val
%@
%$P$-values of the consensus are set to \texttt{NA}. Now visualize the data
%
%<<p-val, fig=TRUE>>=
%    n <- sum(!is.na(p.null))
%    qqplot(p.null, seq(1/n,1, length.out=n), log="xy", pch=16, xlab="P-value", ylab="CDF")
%    abline(0,1)
%@
%
%The $P$-values are a bit too pessimistic, because many positions with count 0 have $P$-value of exactly 1, which causes the CDF to jump. But more 
%importantly, the CDF returns to the diagonal for small values, therefore providing accurate measures here.

\section{Normalization}
We want to further assess the null model with experimental data from two homogeneous replicates. In particular we want to analyze whether the empirical distribution 
of the p-values is uniform. The data we study comes from two phiX sequences sequenced on separate runs on a GAII$_x$.
<<phiX, results=hide>>=
	data(phiX, package="deepSNV")
	jpeg("deepSNV-phiX.jpg", 2.5, 2.5, units="in", res=600, pointsize=7)
	par(mar=c(4,4,1,1), bty="n")
	plot(phiX, cex.min=.5)
	dev.off()
@
(We use a jpg device to reduce the plot size.) From the plot in Figure~\ref{fig:phiX}A it appears that there is a systematic bias between the two experiments, likely because they were sequenced in different runs. 
We therefore normalize with:
<<phiN, results=hide>>=
    phiN <- normalize(phiX, round=TRUE)
	jpeg("deepSNV-phiN.jpg", 2.5, 2.5, units="in", res=600, pointsize=7)
	par(mar=c(4,4,1,1), bty="n")
    plot(phiN, cex.min=.5)
	dev.off()
@
\begin{figure}
\begin{tabular}{ll}
A) & B) \cr
\includegraphics[width=.49\linewidth]{deepSNV-phiX} &
\includegraphics[width=.49\linewidth]{deepSNV-phiN}
\end{tabular}
\caption{Scatterplot for two phiX experiments before (A) and after (B) normalization.}
\label{fig:phiX}
\end{figure}

The results are shown in Figure 1B. The points now symmetrically scatter around the diagonal and all p-values are within the expected range:

<<pval, fig=TRUE>>=
	p.norm <- p.val(phiN)
    n <- sum(!is.na(p.norm))
    qqplot(p.norm, seq(1/n,1, length.out=n), log="xy", type="S", xlab="P-value", ylab="CDF")
	p.val <- p.val(phiX) 
    points(sort(p.val[!is.na(p.val)]), seq(1/n,1, length.out=n), pch=16, col="grey", type="S", lty=2)
    legend("topleft", c("raw data", "normalized data"), pch=16, col=c("grey", "black"), bty="n", lty=3)
    abline(0,1)
@

%\includegraphics[width=.6\linewidth]{pval}\\
After normalization the cumulative distribution of the p-values is close to the diagonal, even for the smallest values. Hence the p-values accurately
measure the probability of type-1 errors.

\section{Overdispersion}

In some situations, the variance of the binomial model is too small, for example for templates with long repeats or heavy PCR amplification for target selection. 
An alternative model is the beta-binomial distribution that allows for a larger variance.

We load a data-set from two deep sequencing experiments of four genes extracted from a metastatic renal cell carcinoma with sequenced on separate lanes
of a GAII$_x$:
<<>>=
	data("RCC", package="deepSNV")
	show(RCC)
@
<<RCC-bin, results=hide>>=
	jpeg("deepSNV-RCC-bin.jpg", 2.5, 2.5, units="in", res=600, pointsize=7)
	par(mar=c(4,4,1,1), bty="n")
	plot(RCC, cex.min=.5) 
	dev.off()
@
\begin{figure}
\begin{tabular}{ll}
A) & B) \cr
\includegraphics[width=.49\linewidth]{deepSNV-RCC-bin} &
\includegraphics[width=.49\linewidth]{deepSNV-RCC-bb} 
\end{tabular}
\caption{Scatterplot of the RCC data in a binomial (A) and beta-binomial model (B).}
\label{fig:RCC}
\end{figure}

We see that a binomial model was used to generate the data. An inspection of the plot in Figure \ref{fig:RCC}A, shows a long noise tail where apparently the dispersion is underestimated 
causing some false positives. We use a beta-binomial model instead and conservatively estimate the dispersion factor on both sides with the argument \texttt{alternative="two.sided"}:
<<>>=
RCC.bb = estimateDispersion(RCC, alternative="two.sided")
@
<<RCC-bb, results=hide>>=
	jpeg("deepSNV-RCC-bb.jpg", 2.5, 2.5, units="in", res=600, pointsize=7)
	par(mar=c(4,4,1,1), bty="n")
	plot(RCC.bb, cex.min=.5) 
	dev.off()
@

The plot is shown in Figure\ref{fig:RCC}B. The log-likelihood of the two models are:
<<>>=
	RCC.bb@log.lik 
	RCC@log.lik
    RCC.bb@log.lik - RCC@log.lik
	log(4*nrow(test(RCC)))
@
Note that the difference is larger than $\log(n)$, the difference in BIC of the two models.


If we compare the number of called SNVs, we find
<<>>=
	summary(RCC, adjust.method="bonferroni")[,1:6]
@

compared to
<<>>=
	tab <- summary(RCC.bb, adjust.method="bonferroni")[,1:6]
	tab
@

A closer inspection will show that the variants with a negative change in frequency are all known SNPs on chromosome 3, which drop in frequency due to loss of heterozygousity in the tumor. 
The remaining variants have positive frequencies. The first is a deletion of \Sexpr{paste(tab[1,1],":",tab[1,2],tab[1,3], sep="")} on \Sexpr{round(100*tab[1,6],2)}\% of the alleles that truncates the VHL protein. 
The second is a C>G conversion in the 5$'$-UTR of the \emph{VHL} gene at \Sexpr{paste(tab[2,1],":", tab[2,2], tab[2,3], sep="")} in \Sexpr{round(100*tab[2,6],2)}\% of the alleles. 
The third variant is likely to be an alignment artifact resulting from imperfect alignments of the deletion of \Sexpr{paste(tab[1,1],":",tab[1,2],tab[1,3], sep="")}.

\section{sessionInfo()}
<<echo=FALSE, results=tex>>=
    toLatex(sessionInfo())
@

\end{document}
