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\title{Pairwise Sequence Alignments}
\author{Patrick Aboyoun \\
  Gentleman Lab \\
  Fred Hutchinson Cancer Research Center \\
  Seattle, WA}
\date{\today}
\maketitle

\tableofcontents

\section{Introduction}

In this document we illustrate how to perform pairwise sequence alignments
using the \Rpackage{Biostrings} package through the use of the
\Rfunction{pairwiseAlignment} function. This function aligns a set of
\Rfunarg{pattern} strings to a \Rfunarg{subject} string in a global, local, or
overlap (ends-free) fashion with or without affine gaps using either a fixed
or quality-based substitution scoring scheme. This function's computation time
is proportional to the product of the two string lengths being aligned.


\section{Pairwise Sequence Alignment Problems}

The (Needleman-Wunsch) global, the (Smith-Waterman) local, and (ends-free)
overlap pairwise sequence alignment problems are described as follows. Let
string $S_i$ have $n_i$ characters $c_{(i,j)}$ with
$j \in \left\{1, \ldots, n_i\right\}$. A pairwise sequence alignment is a
mapping of strings $S_1$ and $S_2$ to gapped substrings ${S'}_1$ and ${S'}_2$
that are defined by

\begin{eqnarray*}
{S'}_1 & = & g_{\left(1,a_1\right)}c_{\left(1,a_1\right)} \cdots g_{\left(1,b_1\right)}c_{\left(1,b_1\right)}g_{\left(1,b_1+1\right)}\\
{S'}_2 & = & g_{\left(2,a_2\right)}c_{\left(2,a_2\right)} \cdots g_{\left(2,b_2\right)}c_{\left(2,b_2\right)}g_{\left(2,b_2+1\right)}
\end{eqnarray*}
\begin{tabbing}
  where \= \\
  \> $a_i, b_i \in \{1, \ldots, n_i\}$ with $a_i \leq b_i$ \\
  \> $g_{(i,j)} = 0$ or more gaps at the specified position $j$ for aligned string $i$ \\
  \> $length({S'}_1) = length({S'}_2)$
\end{tabbing}

Each of these pairwise sequence alignment problems is solved by maximizing the
alignment \textit{score}. An alignment score is determined by the type of
pairwise sequence alignment (global, local, overlap), which sets the
$[a_i, b_i]$ ranges for the substrings; the substitution scoring scheme, which
sets the distance between aligned characters; and the gap penalties, which is
divided into opening and extension components. The optimal pairwise sequence
alignment is the pairwise sequence alignment with the largest score for the
specified alignment type, substitution scoring scheme, and gap penalties.
The pairwise sequence alignment types, substitution scoring schemes, and gap
penalties influence alignment scores in the following manner:

\begin{description}
  \item{Pairwise Sequence Alignment Types:  }
  The type of pairwise sequence alignment determines the substring ranges to
  apply the substitution scoring and gap penalty schemes. For the three primary
  (global, local, overlap) and two derivative (subject overlap, pattern overlap)
  pairwise sequence alignment types, the resulting substring ranges are as
  follows:
  \begin{description}
    \item{Global - } $[a_1, b_1] = [1, n_1]$ and $[a_2, b_2] = [1, n_2]$
    \item{Local - } $[a_1, b_1]$ and $[a_2, b_2]$
    \item{Overlap - }
      $\left\{[a_1, b_1] = [a_1, n_1], [a_2, b_2] = [1, b_2]\right\}$ or
      $\left\{[a_1, b_1] = [1, b_1], [a_2, b_2] = [a_2, n_2]\right\}$
    \item{Subject Overlap - } $[a_1, b_1] = [1, n_1]$ and $[a_2, b_2]$
    \item{Pattern Overlap - } $[a_1, b_1]$ and $[a_2, b_2] = [1, n_2]$
  \end{description}
  \item{Substitution Scoring Schemes:  }
  The substitution scoring scheme sets the values for the aligned character
  pairings within the substring ranges determined by the type of pairwise
  sequence alignment. This scoring scheme can be fixed for character
  pairings or quality-dependent for character pairings. (Characters that align
  with a gap are penalized according to the ``Gap Penalty'' framework.)
  \begin{description}
    \item{Fixed substitution scoring - }
    Fixed substitution scoring schemes associate each aligned character
    pairing with a value. These schemes are very common and include awarding
    one value for a match and another for a mismatch, Point Accepted Mutation
    (PAM) matrices, and Block Substitution Matrix (BLOSUM) matrices.
    \item{Quality-based substitution scoring - }
    Quality-based substitution scoring schemes derive the value for the aligned
    character pairing based on the probabilities of character recording errors
    \cite{Malde:2008}.
    Let $\epsilon_i$ be the probability of a character recording error.
    Assuming independence within and between recordings and a uniform
    background frequency of the different characters, the combined error
    probability of a mismatch when the underlying characters do match is
    $\epsilon_c = \epsilon_1 + \epsilon_2 - (n/(n-1)) * \epsilon_1 * \epsilon_2$,
    where $n$ is the number of characters in the underlying alphabet (e.g. in
    DNA and RNA, $n = 4$). Using $\epsilon_c$, the substitution score is given by
    $b * \log_2(\gamma_{(x,y)} * (1 - \epsilon_c) * n + (1 - \gamma_{(x,y)}) * \epsilon_c * (n/(n-1)))$,
    where $b$ is the bit-scaling for the scoring and $\gamma_{(x,y)}$ is the
    probability that characters $x$ and $y$ represents the same underlying
    letters (e.g. using IUPAC, $\gamma_{(A,A)} = 1$ and $\gamma_{(A,N)} = 1/4$).
  \end{description}  
  \item{Gap Penalties:  }
  Gap penalties are the values associated with the gaps within the substring
  ranges determined by the type of pairwise sequence alignment. These penalties
  are divided into \textit{gap opening} and \textit{gap extension} components,
  where the gap opening penalty is the cost for adding a new gap and the gap
  extension penalty is the incremental cost incurred along the length of the
  gap. A \textit{constant gap penalty} occurs when there is a cost associated
  with opening a gap, but no cost for the length of a gap (i.e. gap extension
  is zero). A \textit{linear gap penalty} occurs when there is no cost
  associated for opening a gap (i.e. gap opening is zero), but there is a cost
  for the length of the gap. An \textit{affine gap penalty} occurs when both
  the gap opening and gap extension have a non-zero associated cost.
\end{description}


\section{Main Pairwise Sequence Alignment Function}

The \Rfunction{pairwiseAlignment} function solves the pairwise sequence
alignment problems mentioned above. It aligns one or more strings specified in
the \Rfunarg{pattern} argument with a single string specified in the
\Rfunarg{subject} argument.

<<options,echo=FALSE>>=
options(width=72)
@ 

<<main1>>=
library(Biostrings)
pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede")
@

The type of pairwise sequence alignment is set by specifying the \Rfunarg{type}
argument to be one of \texttt{"global"}, \texttt{"local"}, \texttt{"overlap"},
\texttt{"global-local"}, and \texttt{"local-global"}.

<<main2>>=
pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede",
                  type = "local")
@

The gap penalties are regulated by the \Rfunarg{gapOpening} and
\Rfunarg{gapExtension} arguments.

<<main3>>=
pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede",
                  gapOpening = 0, gapExtension = 1)
@

The substitution scoring scheme is set using three arguments, two of which are
quality-based related (\Rfunarg{patternQuality}, \Rfunarg{subjectQuality}) and
one is fixed substitution related (\Rfunarg{substitutionMatrix}). When the
substitution scores are fixed by character pairing, the
\Rfunarg{substituionMatrix} argument takes a matrix with the appropriate
alphabets as dimension names. The \Rfunction{nucleotideSubstitutionMatrix}
function tranlates simple match and mismatch scores to the full spectrum of
IUPAC nucleotide codes.

<<main4>>=
submat <-
  matrix(-1, nrow = 26, ncol = 26, dimnames = list(letters, letters))
diag(submat) <- 0
pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede",
                  substitutionMatrix = submat,
                  gapOpening = 0, gapExtension = 1)
@

When the substitution scores are quality-based, the
\Rfunarg{patternQuality} and \Rfunarg{subjectQuality} arguments
represent the equivalent of $[x-99]$ numeric quality values for the
respective strings, and the optional \Rfunarg{fuzzyMatrix} argument
represents how the closely two characters match on a $[0,1]$
scale. The \Rfunarg{patternQuality} and \Rfunarg{subjectQuality}
arguments accept quality measures in either a \Rclass{PhredQuality},
\Rclass{SolexaQuality}, or \Rclass{IlluminaQuality} scaling. For
\Rclass{PhredQuality} and \Rclass{IlluminaQuality} measures $Q \in [0,
  99]$, the probability of an error in the base read is given by
$10^{-Q/10}$ and for \Rclass{SolexaQuality} measures $Q \in [-5, 99]$,
they are given by $1 - 1/(1 + 10^{-Q/10})$. The
\Rfunction{qualitySubstitutionMatrices} function maps the
\Rfunarg{patternQuality} and \Rfunarg{subjectQuality} scores to match
and mismatch penalties. These three arguments will be demonstrated in
later sections.

The final argument, \Rfunarg{scoreOnly}, to the \Rfunction{pairwiseAlignment}
function accepts a logical value to specify whether or not to return just the
pairwise sequence alignment score. If \Rfunarg{scoreOnly} is \Robject{FALSE},
the pairwise alignment with the maximum alignment score is returned. If more
than one pairwise alignment has the maximum alignment score exists, the first
alignment along the subject is returned. If there are multiple pairwise
alignments with the maximum alignment score at the chosen subject location,
then at each location along the alignment mismatches are given preference to
insertions/deletions. For example, \code{pattern: [1] ATTA; subject: [1] AT-A}
is chosen above \code{pattern: [1] ATTA; subject: [1] A-TA} if they both have
the maximum alignment score.

<<main5>>=
submat <-
  matrix(-1, nrow = 26, ncol = 26, dimnames = list(letters, letters))
diag(submat) <- 0
pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede",
                  substitutionMatrix = submat,
                  gapOpening = 0, gapExtension = 1, scoreOnly = TRUE)
@

\subsection{Exercise 1}
\begin{enumerate}
\item Using \Rfunction{pairwiseAlignment}, fit the global, local, and overlap
pairwise sequence alignment of the strings \Robject{"syzygy"} and
\Robject{"zyzzyx"} using the default settings.
\item Do any of the alignments change if the \Rfunarg{gapExtension} argument
is set to \Robject{-Inf}?
\end{enumerate}

[Answers provided in section \ref{sec:Answers1}.]


\section{Pairwise Sequence Alignment Classes}

Following the design principles of Bioconductor and R, the pairwise sequence
alignment functionality in the \Rpackage{Biostrings} package keeps the end-user
close to their data through the use of five specialty classes:  
\Rclass{PairwiseAlignments}, \Rclass{PairwiseAlignmentsSingleSubject},
\Rclass{PairwiseAlignmentsSingleSubjectSummary}, \Rclass{AlignedXStringSet}, and
\Rclass{QualityAlignedXStringSet}. The \Rclass{PairwiseAlignmentsSingleSubject}
class inherits from the \Rclass{PairwiseAlignments} class and they both
hold the results of a fit from the \Rfunction{pairwiseAlignment} function, with
the former class being used to represent all patterns aligning to a single
subject and the latter being used to represent elementwise alignments between a
set of patterns and a set of subjects.

<<classes1>>=
pa1 <- pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede")
class(pa1)
@

and the \Rfunction{pairwiseAlignmentSummary} function holds the results of a
summarized pairwise sequence alignment.

<<classes2>>=
summary(pa1)
class(summary(pa1))
@

The \Rclass{AlignedXStringSet} and \Rclass{QualityAlignedXStringSet} classes
hold the ``gapped'' ${S'}_i$ substrings with the former class holding the
results when the pairwise sequence alignment is performed with a fixed
substitution scoring scheme and the latter class a quality-based scoring
scheme.

<<classes3>>=
class(pattern(pa1))
submat <-
  matrix(-1, nrow = 26, ncol = 26, dimnames = list(letters, letters))
diag(submat) <- 0
pa2 <-
  pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede",
                    substitutionMatrix = submat,
                    gapOpening = 0, gapExtension = 1)
class(pattern(pa2))
@

\subsection{Exercise 2}
\begin{enumerate}
\item What is the primary benefit of formal summary classes like
\Rclass{PairwiseAlignmentsSingleSubjectSummary} and \Rclass{summary.lm} to end-users? 
\end{enumerate}

[Answer provided in section \ref{sec:Answers2}.]


\section{Pairwise Sequence Alignment Helper Functions}

Tables \ref{table:helperfuns1}, \ref{table:helperfuns1} and
\ref{table:alignfuns} show functions that interact with objects of class
\Rclass{PairwiseAlignments}, \Rclass{PairwiseAlignmentsSingleSubject}, and
\Rclass{AlignedXStringSet}. These functions should be used in preference to
direct slot extraction from the alignment objects.

\begin{table}[ht]
\begin{center}
\begin{tabular}{l|l}
\hline
Function                    & Description \\
\hline
\Rfunction{[}               & Extracts the specified elements of the alignment object \\
\Rfunction{alphabet}        & Extracts the allowable characters in the original strings \\
\Rfunction{compareStrings}  & Creates character string mashups of the alignments \\
\Rfunction{deletion}        & Extracts the locations of the gaps inserted into the pattern for the alignments \\
\Rfunction{length}          & Extracts the number of patterns aligned \\
\Rfunction{mismatchTable}   & Creates a table for the mismatching positions \\
\Rfunction{nchar}           & Computes the length of ``gapped'' substrings \\
\Rfunction{nedit}           & Computes the Levenshtein edit distance of the alignments \\
\Rfunction{indel}           & Extracts the locations of the insertion \& deletion gaps in the alignments \\
\Rfunction{insertion}       & Extracts the locations of the gaps inserted into the subject for the alignments \\
\Rfunction{nindel}          & Computes the number of insertions \& deletions in the alignments \\
\Rfunction{nmatch}          & Computes the number of matching characters in the alignments \\
\Rfunction{nmismatch}       & Computes the number of mismatching characters in the alignments \\
\Rfunction{pattern}, \Rfunction{subject} & Extracts the aligned pattern/subject \\
\Rfunction{pid}             & Computes the percent sequence identity \\
\Rfunction{rep}             & Replicates the elements of the alignment object \\
\Rfunction{score}           & Extracts the pairwise sequence alignment scores \\
\Rfunction{type}            & Extracts the type of pairwise sequence alignment \\
\hline
\end{tabular}
\end{center}
\caption{Functions for \Rclass{PairwiseAlignments} and \Rclass{PairwiseAlignmentsSingleSubject}
  objects.}
\label{table:helperfuns1}
\end{table}

\begin{table}[ht]
\begin{center}
\begin{tabular}{l|l}
\hline
Function                    & Description \\
\hline
\Rfunction{aligned}         & Creates an \Rclass{XStringSet} containing either ``filled-with-gaps'' or
                              degapped aligned strings \\
\Rfunction{as.character}    & Creates a character vector version of \Rfunction{aligned} \\
\Rfunction{as.matrix}       & Creates an ``exploded" character matrix version of \Rfunction{aligned} \\
\Rfunction{consensusMatrix} & Computes a consensus matrix for the alignments \\
\Rfunction{consensusString} & Creates the string based on a 50\% + 1 vote from the consensus matrix \\
\Rfunction{coverage}        & Computes the alignment coverage along the subject \\
\Rfunction{mismatchSummary} & Summarizes the information of the \Rfunction{mismatchTable} \\
\Rfunction{summary}         & Summarizes a pairwise sequence alignment \\
\Rfunction{toString}        & Creates a concatenated string version of \Rfunction{aligned} \\
\Rfunction{Views}           & Creates an \Rclass{XStringViews} representing the aligned region along the subject \\
\hline
\end{tabular}
\end{center}
\caption{Additional functions for \Rclass{PairwiseAlignmentsSingleSubject} objects.}
\label{table:helperfuns2}
\end{table}

The \Rfunction{score}, \Rfunction{nedit}, \Rfunction{nmatch},
\Rfunction{nmismatch}, and \Rfunction{nchar} functions return numeric vectors
containing information on the pairwise sequence alignment score, number of
matches, number of mismatches, and number of aligned characters respectively.

<<helper1>>=
submat <-
  matrix(-1, nrow = 26, ncol = 26, dimnames = list(letters, letters))
diag(submat) <- 0
pa2 <-
  pairwiseAlignment(pattern = c("succeed", "precede"), subject = "supersede",
                    substitutionMatrix = submat,
                    gapOpening = 0, gapExtension = 1)
score(pa2)
nedit(pa2)
nmatch(pa2)
nmismatch(pa2)
nchar(pa2)
aligned(pa2)
as.character(pa2)
as.matrix(pa2)
consensusMatrix(pa2)
@

The \Rfunction{summary}, \Rfunction{mismatchTable}, and
\Rfunction{mismatchSummary} functions return various summaries of the pairwise
sequence alignments.

<<helper2>>=
summary(pa2)
mismatchTable(pa2)
mismatchSummary(pa2)
@

\begin{table}[ht]
\begin{center}
\begin{tabular}{l|l}
\hline
Function                    & Description \\
\hline
\Rfunction{[}               & Extracts the specified elements of the alignment object \\
\Rfunction{aligned}, \Rfunction{unaligned} & Extracts the aligned/unaligned strings \\
\Rfunction{alphabet}        & Extracts the allowable characters in the original strings \\
\Rfunction{as.character}, \Rfunction{toString} & Converts the alignments to character strings \\
\Rfunction{coverage}        & Computes the alignment coverage \\
\Rfunction{end}             & Extracts the ending index of the aligned range \\
\Rfunction{indel}           & Extracts the insertion/deletion locations \\
\Rfunction{length}          & Extracts the number of patterns aligned \\
\Rfunction{mismatch}        & Extracts the position of the mismatches \\
\Rfunction{mismatchSummary} & Summarizes the information of the \Rfunction{mismatchTable} \\
\Rfunction{mismatchTable}   & Creates a table for the mismatching positions \\
\Rfunction{nchar}           & Computes the length of ``gapped'' substrings \\
\Rfunction{nindel}          & Computes the number of insertions/deletions in the alignments \\
\Rfunction{nmismatch}       & Computes the number of mismatching characters in the alignments \\
\Rfunction{rep}             & Replicates the elements of the alignment object \\
\Rfunction{start}           & Extracts the starting index of the aligned range \\
\Rfunction{toString}        & Creates a concatenated string containing the alignments \\
\Rfunction{width}           & Extracts the width of the aligned range \\
\hline
\end{tabular}
\end{center}
\caption{Functions for \Rclass{AlignedXString} and \Rclass{QualityAlignedXString} objects.}
\label{table:alignfuns}
\end{table}

The \Rfunction{pattern} and \Rfunction{subject} functions extract the aligned
pattern and subject objects for further analysis. Most of the actions that can
be performed on \Rclass{PairwiseAlignments} objects can also be performed
on \Rclass{AlignedXStringSet} and \Rclass{QualityAlignedXStringSet} objects as
well as operations including \Rfunction{start}, \Rfunction{end}, and
\Rfunction{width} that extracts the start, end, and width of the alignment
ranges.

<<helper3>>=
class(pattern(pa2))
aligned(pattern(pa2))
nindel(pattern(pa2))
start(subject(pa2))
end(subject(pa2))
@

\subsection{Exercise 3}
For the overlap pairwise sequence alignment of the strings \Robject{"syzygy"}
and \Robject{"zyzzyx"} with the \Rfunction{pairwiseAlignment} default settings,
perform the following operations:
\begin{enumerate}
\item Use \Rfunction{nmatch} and \Rfunction{nmismath} to extract the number
of matches and mismatches respectively.
\item Use the \Rfunction{compareStrings} function to get the symbolic
representation of the alignment.
\item Use the \Rfunction{as.character} function to the get the character string
versions of the alignments.
\item Use the \Rfunction{pattern} function to extract the aligned pattern and
apply the \Rfunction{mismatch} function to it to find the locations of the
mismatches.
\item Use the \Rfunction{subject} function to extract the aligned subject and
apply the \Rfunction{aligned} function to it to get the aligned strings.
\end{enumerate}

[Answers provided in section \ref{sec:Answers3}.]


\section{Edit Distances}

One of the earliest uses of pairwise sequence alignment is in the area of text
analysis. In 1965 Vladimir Levenshtein considered a metric, now called the 
\textit{Levenshtein edit distance}, that measures the similarity between two
strings. This distance metric is equivalent to the negative of the score of a
pairwise sequence alignment with a match cost of 0, a mismatch cost of -1, a
gap opening penalty of 0, and a gap extension penalty of 1.

The \Rfunction{stringDist} uses the internals of the
\Rfunction{pairwiseAlignment} function to calculate the Levenshtein edit
distance matrix for a set of strings. 

There is also an implementation of approximate string matching using
Levenshtein edit distance in the \Rfunction{agrep} (approximate grep) function
of the \Rpackage{base} R package. As the following example shows, it is
possible to replicate the \Rfunction{agrep} function using the
\Rfunction{pairwiseAlignment} function. Since the \Rfunction{agrep} function is
vectorized in \Rfunarg{x} rather than \Rfunarg{pattern}, these arguments are
flipped in the call to \Rfunction{pairwiseAlignment}.

<<editdist1>>=
agrepBioC <-
function(pattern, x, ignore.case = FALSE, value = FALSE, max.distance = 0.1)
{
  if (!is.character(pattern)) pattern <- as.character(pattern)
  if (!is.character(x)) x <- as.character(x)
  if (max.distance < 1)
    max.distance <- ceiling(max.distance / nchar(pattern))
  characters <- unique(unlist(strsplit(c(pattern, x), "", fixed = TRUE)))
  if (ignore.case)
    substitutionMatrix <-
      outer(tolower(characters), tolower(characters), function(x,y) -as.numeric(x!=y))
  else
    substitutionMatrix <-
      outer(characters, characters, function(x,y) -as.numeric(x!=y))
  dimnames(substitutionMatrix) <- list(characters, characters)
  distance <-
    - pairwiseAlignment(pattern = x, subject = pattern,
                        substitutionMatrix = substitutionMatrix,
                        type = "local-global",
                        gapOpening = 0, gapExtension = 1,
                        scoreOnly = TRUE)
  whichClose <- which(distance <= max.distance)
  if (value)
    whichClose <- x[whichClose]
  whichClose
}
cbind(base = agrep("laysy", c("1 lazy", "1", "1 LAZY"), max = 2, value = TRUE),
      bioc = agrepBioC("laysy", c("1 lazy", "1", "1 LAZY"), max = 2, value = TRUE))
cbind(base = agrep("laysy", c("1 lazy", "1", "1 LAZY"), max = 2, ignore.case = TRUE),
      bioc = agrepBioC("laysy", c("1 lazy", "1", "1 LAZY"), max = 2, ignore.case = TRUE))
@

\subsection{Exercise 4}
\begin{enumerate}
\item Use the \Rfunction{pairwiseAlignment} function to find the
Levenshtein edit distance between \Robject{"syzygy"} and \Robject{"zyzzyx"}.
\item Use the \Rfunction{stringDist} function to find the Levenshtein edit
distance for the vector
\Robject{c("zyzzyx", "syzygy", "succeed", "precede", "supersede")}.
\end{enumerate}

[Answers provided in section \ref{sec:Answers4}.]


\section{Application:  Using Evolutionary Models in Protein Alignments}

When proteins are believed to descend from a common ancestor, evolutionary
models can be used as a guide in pairwise sequence alignments. The two most
common families evolutionary models of proteins used in pairwise sequence
alignments are Point Accepted Mutation (PAM) matrices, which are based on
explicit evolutionary models, and Block Substitution Matrix (BLOSUM) matrices,
which are based on data-derived evolution models. The \Rpackage{Biostrings}
package contains 5 PAM and 5 BLOSUM matrices (\Robject{PAM30} \Robject{PAM40},
\Robject{PAM70}, \Robject{PAM120}, \Robject{PAM250}, \Robject{BLOSUM45},
\Robject{BLOSUM50}, \Robject{BLOSUM62}, \Robject{BLOSUM80}, and
\Robject{BLOSUM100}) that can be used in the \Rfunarg{substitutionMatrix}
argument to the \Rfunction{pairwiseAlignment} function.

Here is an example pairwise sequence alignment of amino acids from Durbin, Eddy
et al being fit by the \Rfunction{pairwiseAlignment} function using the
\Robject{BLOSUM50} matrix:

<<lkblo>>=
data(BLOSUM50)
BLOSUM50[1:4,1:4]
nwdemo <- 
  pairwiseAlignment(AAString("PAWHEAE"), AAString("HEAGAWGHEE"), substitutionMatrix = BLOSUM50,
                    gapOpening = 0, gapExtension = 8)
nwdemo
compareStrings(nwdemo)
pid(nwdemo)
@

\subsection{Exercise 5}
\begin{enumerate}
\item Repeat the alignment exercise above using \Robject{BLOSUM62}, a gap
opening penalty of 12, and a gap extension penalty of 4.
\item Explore to find out what caused the alignment to change.
\end{enumerate}

[Answers provided in section \ref{sec:Answers5}.]


\section{Application:  Removing Adapters from Sequence Reads}

Finding and removing uninteresting experiment process-related fragments like
adapters is a common problem in genetic sequencing, and pairwise sequence
alignment is well-suited to address this issue. When adapters are used to
anchor or extend a sequence during the experiment process, they either
intentionally or unintentionally become sequenced during the read process.
The following code simulates what sequences with adapter fragments at either
end could look like during an experiment.

<<adapter1>>=
simulateReads <-
function(N, adapter, experiment, substitutionRate = 0.01, gapRate = 0.001) {
  chars <- strsplit(as.character(adapter), "")[[1]]
  sapply(seq_len(N), function(i, experiment, substitutionRate, gapRate) {
    width <- experiment[["width"]][i]
    side <- experiment[["side"]][i]
    randomLetters <-
      function(n) sample(DNA_ALPHABET[1:4], n, replace = TRUE) 
    randomLettersWithEmpty <-
      function(n)
      sample(c("", DNA_ALPHABET[1:4]), n, replace = TRUE,
             prob = c(1 - gapRate, rep(gapRate/4, 4)))
    nChars <- length(chars)
    value <-
      paste(ifelse(rbinom(nChars,1,substitutionRate), randomLetters(nChars), chars),
            randomLettersWithEmpty(nChars),
            sep = "", collapse = "")
    if (side)
      value <-
        paste(c(randomLetters(36 - width), substring(value, 1, width)),
                sep = "", collapse = "")
    else
      value <-
        paste(c(substring(value, 37 - width, 36), randomLetters(36 - width)),
                sep = "", collapse = "")
    value
  }, experiment = experiment, substitutionRate = substitutionRate, gapRate = gapRate)
}

adapter <- DNAString("GATCGGAAGAGCTCGTATGCCGTCTTCTGCTTGAAA")
set.seed(123)
N <- 1000
experiment <-
  list(side = rbinom(N, 1, 0.5), width = sample(0:36, N, replace = TRUE))
table(experiment[["side"]], experiment[["width"]])
adapterStrings <-
  simulateReads(N, adapter, experiment, substitutionRate = 0.01, gapRate = 0.001)
adapterStrings <- DNAStringSet(adapterStrings)
@

These simulated strings above have 0 to 36 characters from the adapters
attached to either end. We can use completely random strings as a baseline for
any pairwise sequence alignment methodology we develop to remove the adapter
characters.

<<adapter2>>=
M <- 5000
randomStrings <-
  apply(matrix(sample(DNA_ALPHABET[1:4], 36 * M, replace = TRUE),
               nrow = M), 1, paste, collapse = "")
randomStrings <- DNAStringSet(randomStrings)
@

Since edit distances are easy to explain, it serves as a good place to start
for developing a adapter removal methodology. Unfortunately given that it is
based on a global alignment, it only is useful for filtering out sequences that
are derived primarily from the adapter.

<<adapter3>>=
## Method 1:  Use edit distance with an FDR of 1e-03
submat1 <- nucleotideSubstitutionMatrix(match = 0, mismatch = -1, baseOnly = TRUE)
randomScores1 <-
  pairwiseAlignment(randomStrings, adapter, substitutionMatrix = submat1,
                    gapOpening = 0, gapExtension = 1, scoreOnly = TRUE)
quantile(randomScores1, seq(0.99, 1, by = 0.001))
adapterAligns1 <-
  pairwiseAlignment(adapterStrings, adapter, substitutionMatrix = submat1,
                    gapOpening = 0, gapExtension = 1)
table(score(adapterAligns1) > quantile(randomScores1, 0.999), experiment[["width"]])
@

One improvement to removing adapters is to look at consecutive matches anywhere
within the sequence. This is more versatile than the edit distance method, but
it requires a relatively large number of consecutive matches and is susceptible
to issues related to error related substitutions and insertions/deletions.

<<adapter4>>=
## Method 2:  Use consecutive matches anywhere in string with an FDR of 1e-03
submat2 <- nucleotideSubstitutionMatrix(match = 1, mismatch = -Inf, baseOnly = TRUE)
randomScores2 <-
  pairwiseAlignment(randomStrings, adapter, substitutionMatrix = submat2,
                    type = "local", gapOpening = 0, gapExtension = Inf,
                    scoreOnly = TRUE)
quantile(randomScores2, seq(0.99, 1, by = 0.001))
adapterAligns2 <-
  pairwiseAlignment(adapterStrings, adapter, substitutionMatrix = submat2,
                    type = "local", gapOpening = 0, gapExtension = Inf)
table(score(adapterAligns2) > quantile(randomScores2, 0.999), experiment[["width"]])
# Determine if the correct end was chosen
table(start(pattern(adapterAligns2)) > 37 - end(pattern(adapterAligns2)),
      experiment[["side"]])
@

Limiting consecutive matches to the ends provides better results, but it
doesn't resolve the issues related to substitutions and insertions/deletions
errors.

<<adapter5>>=
## Method 3:  Use consecutive matches on the ends with an FDR of 1e-03
submat3 <- nucleotideSubstitutionMatrix(match = 1, mismatch = -Inf, baseOnly = TRUE)
randomScores3 <-
  pairwiseAlignment(randomStrings, adapter, substitutionMatrix = submat3,
                    type = "overlap", gapOpening = 0, gapExtension = Inf,
                    scoreOnly = TRUE)
quantile(randomScores3, seq(0.99, 1, by = 0.001))
adapterAligns3 <-
  pairwiseAlignment(adapterStrings, adapter, substitutionMatrix = submat3,
                    type = "overlap", gapOpening = 0, gapExtension = Inf)
table(score(adapterAligns3) > quantile(randomScores3, 0.999), experiment[["width"]])
# Determine if the correct end was chosen
table(end(pattern(adapterAligns3)) == 36, experiment[["side"]])
@

Allowing for substitutions and insertions/deletions errors in the pairwise
sequence alignments provides much better results for finding adapter fragments.

<<adapter6>>=
## Method 4:  Allow mismatches and indels on the ends with an FDR of 1e-03
randomScores4 <-
  pairwiseAlignment(randomStrings, adapter, type = "overlap", scoreOnly = TRUE)
quantile(randomScores4, seq(0.99, 1, by = 0.001))
adapterAligns4 <-
  pairwiseAlignment(adapterStrings, adapter, type = "overlap")
table(score(adapterAligns4) > quantile(randomScores4, 0.999), experiment[["width"]])
# Determine if the correct end was chosen
table(end(pattern(adapterAligns4)) == 36, experiment[["side"]])
@

Using the results that allow for substitutions and insertions/deletions errors,
the cleaned sequence fragments can be generated as follows:

<<adapter7>>=
## Method 4 continued:  Remove adapter fragments
fragmentFound <-
  score(adapterAligns4) > quantile(randomScores4, 0.999)
fragmentFoundAt1 <-
  fragmentFound & (start(pattern(adapterAligns4)) == 1)
fragmentFoundAt36 <-
  fragmentFound & (end(pattern(adapterAligns4)) == 36)
cleanedStrings <- as.character(adapterStrings)
cleanedStrings[fragmentFoundAt1] <-
  as.character(narrow(adapterStrings[fragmentFoundAt1], end = 36,
         width = 36 - end(pattern(adapterAligns4[fragmentFoundAt1]))))
cleanedStrings[fragmentFoundAt36] <-
  as.character(narrow(adapterStrings[fragmentFoundAt36], start = 1,
         width = start(pattern(adapterAligns4[fragmentFoundAt36])) - 1))
cleanedStrings <- DNAStringSet(cleanedStrings)
cleanedStrings
@

\subsection{Exercise 6}
\begin{enumerate}
\item Rerun the simulation time using the \Rfunction{simulateReads} function
with a \Rfunarg{substitutionRate} of 0.005 and \Rfunarg{gapRate} of 0.0005.
How do the different pairwise sequence alignment methods compare?
\item (Advanced) Modify the \Rfunction{simulateReads} function to accept
different equal length adapters on either side (left \& right) of the reads.
How would the methods for trimming the reads change?
\end{enumerate}

[Answers provided in section \ref{sec:Answers6}.]


\section{Application:  Quality Assurance in Sequencing Experiments}

Due to its flexibility, the \Rfunction{pairwiseAlignment} function is able to
diagnose sequence matching-related issues that arise when
\Rfunction{matchPDict} and its related functions don't find a match. This
section contains an example involving a short read Solexa sequencing experiment
of bacteriophage $\phi$ X174 DNA produced by New England BioLabs (NEB). This
experiment contains slightly less than 5000 unique short reads in
\Robject{srPhiX174}, with quality measures in \Robject{quPhiX174}, and frequency
for those short reads in \Robject{wtPhiX174}.

In order to demonstrate how to find sequence differences in the target, these
short reads will be compared against the bacteriophage $\phi$ X174 genome
NC\_001422 from the GenBank database.

<<genome1>>=
data(phiX174Phage)
genBankPhage <- phiX174Phage[[1]]
nchar(genBankPhage)

data(srPhiX174)
srPhiX174
quPhiX174
summary(wtPhiX174)

fullShortReads <- rep(srPhiX174, wtPhiX174)
srPDict <- PDict(fullShortReads)
table(countPDict(srPDict, genBankPhage))
@

For these short reads, the \Rfunction{pairwiseAlignment} function finds that
the small number of perfect matches is due to two locations on the bacteriophage
$\phi$X174 genome.

Unlike the \Rfunction{countPDict} function, the \Rfunction{pairwiseAlignment}
function works off of the original strings, rather than \Rfunction{PDict}
processed strings, and to be computationally efficient it is recommended that
the unique sequences are supplied to the \Rfunction{pairwiseAlignment}
function, and the frequencies of those sequences are supplied to the
\Rfunarg{weight} argument of functions like \Rfunction{summary},
\Rfunction{mismatchSummary}, and \Rfunction{coverage}. For the purposes of this
exercise, a substring of the GenBank bacteriophage $\phi$ X174 genome is
supplied to the \Rfunarg{subject} argument of the \Rfunction{pairwiseAlignment}
function to reduce the computation time.

<<genome2>>=
genBankSubstring <- substring(genBankPhage, 2793-34, 2811+34)

genBankAlign <-
  pairwiseAlignment(srPhiX174, genBankSubstring,
                    patternQuality = SolexaQuality(quPhiX174),
                    subjectQuality = SolexaQuality(99L),
                    type = "global-local")
summary(genBankAlign, weight = wtPhiX174)

revisedPhage <-
  replaceLetterAt(genBankPhage, c(2793, 2811), "TT")
table(countPDict(srPDict, revisedPhage))
@

The following plot shows the coverage of the aligned short reads along the
substring of the bacteriophage $\phi$ X174 genome. Applying the
\Rfunction{slice} function to the coverage shows the entire substring is
covered by aligned short reads.

<<genome3, fig=TRUE>>=
genBankCoverage <- coverage(genBankAlign, weight = wtPhiX174)
plot((2793-34):(2811+34), as.integer(genBankCoverage), xlab = "Position", ylab = "Coverage",
     type = "l")
nchar(genBankSubstring)
slice(genBankCoverage, lower = 1)
@

\subsection{Exercise 7}
\begin{enumerate}
\item Rerun the global-local alignment of the short reads against the entire
genome. (This may take a few minutes.)
\item Plot the coverage of these alignments and use the \Rfunction{slice}
function to find the ranges of alignment. Are there any alignments outside of
the substring region that was used above?
\item Use the \Rfunction{reverseComplement} function on the bacteriophage
$\phi$ X174 genome. Do any short reads have a higher alignment score on this
new sequence than on the original sequence?
\end{enumerate}

[Answers provided in section \ref{sec:Answers7}.]


\section{Computation Profiling}
The \Rfunction{pairwiseAlignment} function uses a dynamic programming algorithm
based on the Needleman-Wunsch and Smith-Waterman algorithms for global and
local pairwise sequence alignments respectively. The algorithm consumes memory
and computation time proportional to the product of the length of the two
strings being aligned.

<<profiling1, fig=TRUE>>=
N <- as.integer(seq(500, 5000, by = 500))
timings <- rep(0, length(N))
names(timings) <- as.character(N)
for (i in seq_len(length(N))) {
  string1 <- DNAString(paste(sample(DNA_ALPHABET[1:4], N[i], replace = TRUE), collapse = ""))
  string2 <- DNAString(paste(sample(DNA_ALPHABET[1:4], N[i], replace = TRUE), collapse = ""))
  timings[i] <- system.time(pairwiseAlignment(string1, string2, type = "global"))[["user.self"]]
}
timings
coef(summary(lm(timings ~ poly(N, 2))))
plot(N, timings, xlab = "String Size, Both Strings", ylab = "Timing (sec.)", type = "l",
     main = "Global Pairwise Sequence Alignment Timings")
@

When a problem only requires the pairwise sequence alignment score, setting the
\Rfunarg{scoreOnly} argument to \Robject{TRUE} will more than halve the
computation time.

<<profiling2>>=
scoreOnlyTimings <- rep(0, length(N))
names(scoreOnlyTimings) <- as.character(N)
for (i in seq_len(length(N))) {
  string1 <- DNAString(paste(sample(DNA_ALPHABET[1:4], N[i], replace = TRUE), collapse = ""))
  string2 <- DNAString(paste(sample(DNA_ALPHABET[1:4], N[i], replace = TRUE), collapse = ""))
  scoreOnlyTimings[i] <- system.time(pairwiseAlignment(string1, string2, type = "global", scoreOnly = TRUE))[["user.self"]]
}
scoreOnlyTimings
round((timings - scoreOnlyTimings) / timings, 2)
@

\subsection{Exercise 8}
\begin{enumerate}
\item Rerun the first set of profiling code, but this time fix the number of
characters in \Robject{string1} to 35 and have the number of characters in
\Robject{string2} range from 5000, 50000, by increments of 5000. What is the
computational order of this simulation exercise?
\item Rerun the second set of profiling code using the simulations from the
previous exercise with \Rfunarg{scoreOnly} argument set to \Robject{TRUE}. Is
is still twice as fast? 
\end{enumerate}

[Answers provided in section \ref{sec:Answers8}.]


\section{Computing alignment consensus matrices}

The \Rfunction{consensusMatrix} function is provided for computing a consensus matrix
for a set of equal-length strings assumed to be aligned. To illustrate, the
following application assumes the ORF data to be aligned for the first 10
positions (patently false):
<<doal>>=
file <- system.file("extdata", "someORF.fa", package="Biostrings")
orf <- readDNAStringSet(file)
orf
orf10 <- DNAStringSet(orf, end=10)
consensusMatrix(orf10, as.prob=TRUE, baseOnly=TRUE)
@

The information content as defined by Hertz and Stormo 1995 is computed as
follows:
<<infco>>=
informationContent <- function(Lmers) {
 zlog <- function(x) ifelse(x==0,0,log(x))
 co <- consensusMatrix(Lmers, as.prob=TRUE)
 lets <- rownames(co)
 fr <- alphabetFrequency(Lmers, collapse=TRUE)[lets]
 fr <- fr / sum(fr)
 sum(co*zlog(co/fr), na.rm=TRUE)
}
informationContent(orf10)
@


\section{Exercise Answers}

\subsection{Exercise 1}
\label{sec:Answers1}
\begin{enumerate}
\item Using \Rfunction{pairwiseAlignment}, fit the global, local, and overlap
pairwise sequence alignment of the strings \Robject{"syzygy"} and
\Robject{"zyzzyx"} using the default settings.
<<ans1a>>=
pairwiseAlignment("zyzzyx", "syzygy")
pairwiseAlignment("zyzzyx", "syzygy", type = "local")
pairwiseAlignment("zyzzyx", "syzygy", type = "overlap")
@
\item Do any of the alignments change if the \Rfunarg{gapExtension} argument
is set to \Robject{-Inf}? \textit{Yes, the overlap pairwise sequence alignment
changes.}
<<ans1b>>=
pairwiseAlignment("zyzzyx", "syzygy", type = "overlap", gapExtension = Inf)
@
\end{enumerate}

\subsection{Exercise 2}
\label{sec:Answers2}
\begin{enumerate}
\item What is the primary benefit of formal summary classes like
\Rclass{PairwiseAlignmentsSingleSubjectSummary} and \Rclass{summary.lm}
to end-users?
\textit{These classes allow the end-user to extract the summary output for
further operations.}
<<ans2a>>=
ex2 <- summary(pairwiseAlignment("zyzzyx", "syzygy"))
nmatch(ex2) / nmismatch(ex2)
@
\end{enumerate}

\subsection{Exercise 3}
\label{sec:Answers3}
For the overlap pairwise sequence alignment of the strings \Robject{"syzygy"}
and \Robject{"zyzzyx"} with the \Rfunction{pairwiseAlignment} default settings,
perform the following operations:
<<ans3>>=
ex3 <- pairwiseAlignment("zyzzyx", "syzygy", type = "overlap")
@
\begin{enumerate}
\item Use \Rfunction{nmatch} and \Rfunction{nmismath} to extract the number
of matches and mismatches respectively.
<<ans3a>>=
nmatch(ex3)
nmismatch(ex3)
@
\item Use the \Rfunction{compareStrings} function to get the symbolic
representation of the alignment.
<<ans3b>>=
compareStrings(ex3)
@
\item Use the \Rfunction{as.character} function to the get the character string
versions of the alignments.
<<ans3c>>=
as.character(ex3)
@
\item Use the \Rfunction{pattern} function to extract the aligned pattern and
apply the \Rfunction{mismatch} function to it to find the locations of the
mismatches.
<<ans3d>>=
mismatch(pattern(ex3))
@
\item Use the \Rfunction{subject} function to extract the aligned subject and
apply the \Rfunction{aligned} function to it to get the aligned strings.
<<ans3e>>=
aligned(subject(ex3))
@
\end{enumerate}

\subsection{Exercise 4}
\label{sec:Answers4}
\begin{enumerate}
\item Use the \Rfunction{pairwiseAlignment} function to find the
Levenshtein edit distance between \Robject{"syzygy"} and \Robject{"zyzzyx"}.
<<ans4a>>=
submat <- matrix(-1, nrow = 26, ncol = 26, dimnames = list(letters, letters))
diag(submat) <- 0
- pairwiseAlignment("zyzzyx", "syzygy", substitutionMatrix = submat,
                    gapOpening = 0, gapExtension = 1, scoreOnly = TRUE)
@
\item Use the \Rfunction{stringDist} function to find the Levenshtein edit
distance for the vector
\Robject{c("zyzzyx", "syzygy", "succeed", "precede", "supersede")}.
<<ans4b>>=
stringDist(c("zyzzyx", "syzygy", "succeed", "precede", "supersede"))
@
\end{enumerate}

\subsection{Exercise 5}
\label{sec:Answers5}
\begin{enumerate}
\item Repeat the alignment exercise above using \Robject{BLOSUM62}, a gap
opening penalty of 12, and a gap extension penalty of 4.
<<ans5a>>=
data(BLOSUM62)
pairwiseAlignment(AAString("PAWHEAE"), AAString("HEAGAWGHEE"), substitutionMatrix = BLOSUM62,
                  gapOpening = 12, gapExtension = 4)
@
\item Explore to find out what caused the alignment to change. \textit{The sift
in gap penalties favored infrequent long gaps to frequent short ones.}
\end{enumerate}


\subsection{Exercise 6}
\label{sec:Answers6}
\begin{enumerate}
\item Rerun the simulation time using the \Rfunction{simulateReads} function
with a \Rfunarg{substitutionRate} of 0.005 and \Rfunarg{gapRate} of 0.0005.
How do the different pairwise sequence alignment methods compare? \textit{The
different methods are much more comprobable when the error rates are lower.}
<<ans6a>>=
adapter <- DNAString("GATCGGAAGAGCTCGTATGCCGTCTTCTGCTTGAAA")
set.seed(123)
N <- 1000
experiment <-
  list(side = rbinom(N, 1, 0.5), width = sample(0:36, N, replace = TRUE))
table(experiment[["side"]], experiment[["width"]])
ex6Strings <-
  simulateReads(N, adapter, experiment, substitutionRate = 0.005, gapRate = 0.0005)
ex6Strings <- DNAStringSet(ex6Strings)
ex6Strings

## Method 1:  Use edit distance with an FDR of 1e-03
submat1 <- nucleotideSubstitutionMatrix(match = 0, mismatch = -1, baseOnly = TRUE)
quantile(randomScores1, seq(0.99, 1, by = 0.001))
ex6Aligns1 <-
  pairwiseAlignment(ex6Strings, adapter, substitutionMatrix = submat1,
                    gapOpening = 0, gapExtension = 1)
table(score(ex6Aligns1) > quantile(randomScores1, 0.999), experiment[["width"]])

## Method 2:  Use consecutive matches anywhere in string with an FDR of 1e-03
submat2 <- nucleotideSubstitutionMatrix(match = 1, mismatch = -Inf, baseOnly = TRUE)
quantile(randomScores2, seq(0.99, 1, by = 0.001))
ex6Aligns2 <-
  pairwiseAlignment(ex6Strings, adapter, substitutionMatrix = submat2,
                    type = "local", gapOpening = 0, gapExtension = Inf)
table(score(ex6Aligns2) > quantile(randomScores2, 0.999), experiment[["width"]])
# Determine if the correct end was chosen
table(start(pattern(ex6Aligns2)) > 37 - end(pattern(ex6Aligns2)),
      experiment[["side"]])

## Method 3:  Use consecutive matches on the ends with an FDR of 1e-03
submat3 <- nucleotideSubstitutionMatrix(match = 1, mismatch = -Inf, baseOnly = TRUE)
ex6Aligns3 <-
  pairwiseAlignment(ex6Strings, adapter, substitutionMatrix = submat3,
                    type = "overlap", gapOpening = 0, gapExtension = Inf)
table(score(ex6Aligns3) > quantile(randomScores3, 0.999), experiment[["width"]])
# Determine if the correct end was chosen
table(end(pattern(ex6Aligns3)) == 36, experiment[["side"]])

## Method 4:  Allow mismatches and indels on the ends with an FDR of 1e-03
quantile(randomScores4, seq(0.99, 1, by = 0.001))
ex6Aligns4 <- pairwiseAlignment(ex6Strings, adapter, type = "overlap")
table(score(ex6Aligns4) > quantile(randomScores4, 0.999), experiment[["width"]])
# Determine if the correct end was chosen
table(end(pattern(ex6Aligns4)) == 36, experiment[["side"]])
@
\item (Advanced) Modify the \Rfunction{simulateReads} function to accept
different equal length adapters on either side (left \& right) of the reads.
How would the methods for trimming the reads change?
<<ans6b>>=
simulateReads <-
function(N, left, right = left, experiment, substitutionRate = 0.01, gapRate = 0.001) {
  leftChars <- strsplit(as.character(left), "")[[1]]
  rightChars <- strsplit(as.character(right), "")[[1]]
  if (length(leftChars) != length(rightChars))
    stop("left and right adapters must have the same number of characters")
  nChars <- length(leftChars)
  sapply(seq_len(N), function(i) {
    width <- experiment[["width"]][i]
    side <- experiment[["side"]][i]
    randomLetters <-
      function(n) sample(DNA_ALPHABET[1:4], n, replace = TRUE) 
    randomLettersWithEmpty <-
      function(n)
      sample(c("", DNA_ALPHABET[1:4]), n, replace = TRUE,
             prob = c(1 - gapRate, rep(gapRate/4, 4)))
    if (side) {
      value <-
        paste(ifelse(rbinom(nChars,1,substitutionRate), randomLetters(nChars), rightChars),
              randomLettersWithEmpty(nChars),
              sep = "", collapse = "")
      value <-
        paste(c(randomLetters(36 - width), substring(value, 1, width)),
                sep = "", collapse = "")
    } else {
      value <-
        paste(ifelse(rbinom(nChars,1,substitutionRate), randomLetters(nChars), leftChars),
              randomLettersWithEmpty(nChars),
              sep = "", collapse = "")
      value <-
        paste(c(substring(value, 37 - width, 36), randomLetters(36 - width)),
                sep = "", collapse = "")
    }
    value
  })
}

leftAdapter <- adapter
rightAdapter <- reverseComplement(adapter)
ex6LeftRightStrings <- simulateReads(N, leftAdapter, rightAdapter, experiment)
ex6LeftAligns4 <- 
  pairwiseAlignment(ex6LeftRightStrings, leftAdapter, type = "overlap")
ex6RightAligns4 <- 
  pairwiseAlignment(ex6LeftRightStrings, rightAdapter, type = "overlap")
scoreCutoff <- quantile(randomScores4, 0.999)
leftAligned <-
  start(pattern(ex6LeftAligns4)) == 1 & score(ex6LeftAligns4) > pmax(scoreCutoff, score(ex6RightAligns4))
rightAligned <-
  end(pattern(ex6RightAligns4)) == 36 & score(ex6RightAligns4) > pmax(scoreCutoff, score(ex6LeftAligns4))
table(leftAligned, rightAligned)
table(leftAligned | rightAligned, experiment[["width"]])
@
\end{enumerate}

\subsection{Exercise 7}
\label{sec:Answers7}
\begin{enumerate}
\item Rerun the global-local alignment of the short reads against the entire
genome. (This may take a few minutes.)
<<ans7a>>=
genBankFullAlign <-
  pairwiseAlignment(srPhiX174, genBankPhage,
                    patternQuality = SolexaQuality(quPhiX174),
                    subjectQuality = SolexaQuality(99L),
                    type = "global-local")
summary(genBankFullAlign, weight = wtPhiX174)
@
\item Plot the coverage of these alignments and use the \Rfunction{slice}
function to find the ranges of alignment. Are there any alignments outside of
the substring region that was used above? \textit{Yes, there are some
alignments outside of the specified substring region.}
<<ans7b>>=
genBankFullCoverage <- coverage(genBankFullAlign, weight = wtPhiX174)
plot(as.integer(genBankFullCoverage), xlab = "Position", ylab = "Coverage", type = "l")
slice(genBankFullCoverage, lower = 1)
@
\item Use the \Rfunction{reverseComplement} function on the bacteriophage
$\phi$ X174 genome. Do any short reads have a higher alignment score on this
new sequence than on the original sequence? \textit{Yes, there are some strings
with a higher score on the new sequence.}
<<ans7c>>=
genBankFullAlignRevComp <-
  pairwiseAlignment(srPhiX174, reverseComplement(genBankPhage),
                    patternQuality = SolexaQuality(quPhiX174),
                    subjectQuality = SolexaQuality(99L),
                    type = "global-local")
table(score(genBankFullAlignRevComp) > score(genBankFullAlign))
@
\end{enumerate}

\subsection{Exercise 8}
\label{sec:Answers8}
\begin{enumerate}
\item Rerun the first set of profiling code, but this time fix the number of
characters in \Robject{string1} to 35 and have the number of characters in
\Robject{string2} range from 5000, 50000, by increments of 5000. What is the
computational order of this simulation exercise? \textit{As expected, the
growth in time is now linear.}
<<ans8a, fig=TRUE>>=
N <- as.integer(seq(5000, 50000, by = 5000))
newTimings <- rep(0, length(N))
names(newTimings) <- as.character(N)
for (i in seq_len(length(N))) {
  string1 <- DNAString(paste(sample(DNA_ALPHABET[1:4], 35, replace = TRUE), collapse = ""))
  string2 <- DNAString(paste(sample(DNA_ALPHABET[1:4], N[i], replace = TRUE), collapse = ""))
  newTimings[i] <- system.time(pairwiseAlignment(string1, string2, type = "global"))[["user.self"]]
}
newTimings
coef(summary(lm(newTimings ~ poly(N, 2))))
plot(N, newTimings, xlab = "Larger String Size", ylab = "Timing (sec.)",
     type = "l", main = "Global Pairwise Sequence Alignment Timings")
@
\item Rerun the second set of profiling code using the simulations from the
previous exercise with \Rfunarg{scoreOnly} argument set to \Robject{TRUE}. Is
is still twice as fast? \textit{Yes, it is still over twice as fast.}
<<ans8b>>=
newScoreOnlyTimings <- rep(0, length(N))
names(newScoreOnlyTimings) <- as.character(N)
for (i in seq_len(length(N))) {
  string1 <- DNAString(paste(sample(DNA_ALPHABET[1:4], 35, replace = TRUE), collapse = ""))
  string2 <- DNAString(paste(sample(DNA_ALPHABET[1:4], N[i], replace = TRUE), collapse = ""))
  newScoreOnlyTimings[i] <- system.time(pairwiseAlignment(string1, string2, type = "global", scoreOnly = TRUE))[["user.self"]]
}
newScoreOnlyTimings
round((newTimings - newScoreOnlyTimings) / newTimings, 2)
@
\end{enumerate}


\section{Session Information}
All of the output in this vignette was produced under the following
conditions:

<<sessinfo>>=
sessionInfo()
@


\begin{thebibliography}{}

\bibitem{Durbin:1998}
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\end{document}
