bwa.1

.TH bwa 1 "27 Feburary 2013" "bwa-0.7.0" "Bioinformatics tools"
.SH NAME
.PP
bwa - Burrows-Wheeler Alignment Tool
.SH SYNOPSIS
.PP
bwa index ref.fa
.PP
bwa mem ref.fa reads.fq > aln-se.sam
.PP
bwa mem ref.fa read1.fq read2.fq > aln-pe.sam
.PP
bwa aln ref.fa short_read.fq > aln_sa.sai
.PP
bwa samse ref.fa aln_sa.sai short_read.fq > aln-se.sam
.PP
bwa sampe ref.fa aln_sa1.sai aln_sa2.sai read1.fq read2.fq > aln-pe.sam
.PP
bwa bwasw ref.fa long_read.fq > aln.sam

.SH DESCRIPTION
.PP
BWA is a software package for mapping low-divergent sequences against a large
reference genome, such as the human genome. It consists of three algorithms:
BWA-backtrack, BWA-SW and BWA-MEM. The first algorithm is designed for Illumina
sequence reads up to 100bp, while the rest two for longer sequences ranged from
70bp to 1Mbp. BWA-MEM and BWA-SW share similar features such as long-read
support and split alignment, but BWA-MEM, which is the latest, is generally
recommended for high-quality queries as it is faster and more accurate.
BWA-MEM also has better performance than BWA-backtrack for 70-100bp Illumina
reads.

For all the algorithms, BWA first needs to construct the FM-index for
the reference genome (the
.B index
command). Alignment algorithms are invoked with different sub-commands:
.BR aln / samse / sampe
for BWA-backtrack,
.B bwasw
for BWA-SW and
.B mem
for the BWA-MEM algorithm.

.SH COMMANDS AND OPTIONS
.TP
.B index
bwa index [-p prefix] [-a algoType] <in.db.fasta>

Index database sequences in the FASTA format.

.B OPTIONS:
.RS
.TP 10
.BI -p \ STR
Prefix of the output database [same as db filename]
.TP
.BI -a \ STR
Algorithm for constructing BWT index. Available options are:
.RS
.TP
.B is
IS linear-time algorithm for constructing suffix array. It requires
5.37N memory where N is the size of the database. IS is moderately fast,
but does not work with database larger than 2GB. IS is the default
algorithm due to its simplicity. The current codes for IS algorithm are
reimplemented by Yuta Mori.
.TP
.B bwtsw
Algorithm implemented in BWT-SW. This method works with the whole human
genome.
.RE
.RE

.TP
.B mem
.B bwa mem
.RB [ -aCHMpP ]
.RB [ -t
.IR nThreads ]
.RB [ -k
.IR minSeedLen ]
.RB [ -w
.IR bandWidth ]
.RB [ -r
.IR seedSplitRatio ]
.RB [ -c
.IR maxOcc ]
.RB [ -A
.IR matchScore ]
.RB [ -B
.IR mmPenalty ]
.RB [ -O
.IR gapOpenPen ]
.RB [ -E
.IR gapExtPen ]
.RB [ -L
.IR clipPen ]
.RB [ -U
.IR unpairPen ]
.RB [ -R
.IR RGline ]
.RB [ -v
.IR verboseLevel ]
.I db.prefix
.I reads.fq
.RI [ mates.fq ]

Align 70bp-1Mbp query sequences with the BWA-MEM algorithm. Briefly, the
algorithm works by seeding alignments with maximal exact matches (MEMs) and
then extending seeds with the affine-gap Smith-Waterman algorithm (SW).

If
.I mates.fq
file is absent and option
.B -p
is not set, this command regards input reads are single-end. If
.I mates.fq
is present, this command assumes the
.IR i -th
read in
.I reads.fq
and the
.IR i -th
read in
.I mates.fq
constitute a read pair. If
.B -p
is used, the command assumes the
.RI 2 i -th
and the
.RI (2 i +1)-th
read in
.I reads.fq
constitute a read pair (such input file is said to be interleaved). In this case,
.I mates.fq
is ignored. In the paired-end mode, the
.B mem
command will infer the read orientation and the insert size distribution from a
batch of reads.

The BWA-MEM algorithm performs local alignment. It may produce multiple primary
alignments for different part of a query sequence. This is a crucial feature
for long sequences. However, some tools such as Picard's markDuplicates does
not work with split alignments. One may consider to use option
.B -M
to flag shorter split hits as secondary.

.B OPTIONS:
.RS
.TP 10
.BI -t \ INT
Number of threads [1]
.TP
.BI -k \ INT
Minimum seed length. Matches shorter than
.I INT
will be missed. The alignment speed is usually insensitive to this value unless
it significantly deviates 20. [19]
.TP
.BI -w \ INT
Band width. Essentially, gaps longer than
.I INT
will not be found. Note that the maximum gap length is also affected by the
scoring matrix and the hit length, not solely determined by this option. [100]
.TP
.BI -r \ FLOAT
Trigger re-seeding for a MEM longer than
.IR minSeedLen * FLOAT .
This is a key heuristic parameter for tuning the performance. Larger value
yields fewer seeds, which leads to faster alignment speed but lower accuracy. [1.5]
.TP
.BI -c \ INT
Discard a MEM if it has more than
.I INT
occurence in the genome. This is an insensitive parameter. [10000]
.TP
.B -P
In the paired-end mode, perform SW to rescue missing hits only but do not try to find
hits that fit a proper pair.
.TP
.BI -A \ INT
Matching score. [1]
.TP
.BI -B \ INT
Mismatch penalty. The sequence error rate is approximately: {.75 * exp[-log(4) * B/A]}. [4]
.TP
.BI -O \ INT
Gap open penalty. [6]
.TP
.BI -E \ INT
Gap extension penalty. A gap of length k costs O + k*E (i.e.
.B -O
is for opening a zero-length gap). [1]
.TP
.BI -L \ INT
Clipping penalty. When performing SW extension, BWA-MEM keeps track of the best
score reaching the end of query. If this score is larger than the best SW score
minus the clipping penalty, clipping will not be applied. Note that in this
case, the SAM AS tag reports the best SW score; clipping penalty is not
deducted. [5]
.TP
.BI -U \ INT
Penalty for an unpaired read pair. BWA-MEM scores an unpaired read pair as
.RI scoreRead1+scoreRead2- INT
and scores a paired as scoreRead1+scoreRead2-insertPenalty. It compares these
two scores to determine whether we should force pairing. [9]
.TP
.B -p
Assume the first input query file is interleaved paired-end FASTA/Q. See the command description for details.
.TP
.BI -R \ STR
Complete read group header line. '\\t' can be used in
.I STR
and will be converted to a TAB in the output SAM. The read group ID will be
attached to every read in the output. An example is '@RG\\tID:foo\\tSM:bar'.
[null]
.TP
.B -a
Output all found alignments for single-end or unpaired paired-end reads. These
alignments will be flagged as secondary alignments.
.TP
.B -C
Append append FASTA/Q comment to SAM output. This option can be used to
transfer read meta information (e.g. barcode) to the SAM output. Note that the
FASTA/Q comment (the string after a space in the header line) must conform the SAM
spec (e.g. BC:Z:CGTAC). Malformated comments lead to incorrect SAM output.
.TP
.B -H
Use hard clipping 'H' in the SAM output. This option may dramatically reduce
the redundancy of output when mapping long contig or BAC sequences.
.TP
.B -M
Mark shorter split hits as secondary (for Picard compatibility).
.TP
.BI -v \ INT
Control the verbose level of the output. This option has not been fully
supported throughout BWA. Ideally, a value 0 for disabling all the output to
stderr; 1 for outputting errors only; 2 for warnings and errors; 3 for
all normal messages; 4 or higher for debugging. When this option takes value
4, the output is not SAM. [3]
.RE

.TP
.B aln
bwa aln [-n maxDiff] [-o maxGapO] [-e maxGapE] [-d nDelTail] [-i
nIndelEnd] [-k maxSeedDiff] [-l seedLen] [-t nThrds] [-cRN] [-M misMsc]
[-O gapOsc] [-E gapEsc] [-q trimQual] <in.db.fasta> <in.query.fq> >
<out.sai>

Find the SA coordinates of the input reads. Maximum
.I maxSeedDiff
differences are allowed in the first
.I seedLen
subsequence and maximum
.I maxDiff
differences are allowed in the whole sequence.

.B OPTIONS:
.RS
.TP 10
.BI -n \ NUM
Maximum edit distance if the value is INT, or the fraction of missing
alignments given 2% uniform base error rate if FLOAT. In the latter
case, the maximum edit distance is automatically chosen for different
read lengths. [0.04]
.TP
.BI -o \ INT
Maximum number of gap opens [1]
.TP
.BI -e \ INT
Maximum number of gap extensions, -1 for k-difference mode (disallowing
long gaps) [-1]
.TP
.BI -d \ INT
Disallow a long deletion within INT bp towards the 3'-end [16]
.TP
.BI -i \ INT
Disallow an indel within INT bp towards the ends [5]
.TP
.BI -l \ INT
Take the first INT subsequence as seed. If INT is larger than the query
sequence, seeding will be disabled. For long reads, this option is
typically ranged from 25 to 35 for `-k 2'. [inf]
.TP
.BI -k \ INT
Maximum edit distance in the seed [2]
.TP
.BI -t \ INT
Number of threads (multi-threading mode) [1]
.TP
.BI -M \ INT
Mismatch penalty. BWA will not search for suboptimal hits with a score
lower than (bestScore-misMsc). [3]
.TP
.BI -O \ INT
Gap open penalty [11]
.TP
.BI -E \ INT
Gap extension penalty [4]
.TP
.BI -R \ INT
Proceed with suboptimal alignments if there are no more than INT equally
best hits. This option only affects paired-end mapping. Increasing this
threshold helps to improve the pairing accuracy at the cost of speed,
especially for short reads (~32bp).
.TP
.B -c
Reverse query but not complement it, which is required for alignment in
the color space. (Disabled since 0.6.x)
.TP
.B -N
Disable iterative search. All hits with no more than
.I maxDiff
differences will be found. This mode is much slower than the default.
.TP
.BI -q \ INT
Parameter for read trimming. BWA trims a read down to
argmax_x{\\sum_{i=x+1}^l(INT-q_i)} if q_l<INT where l is the original
read length. [0]
.TP
.B -I
The input is in the Illumina 1.3+ read format (quality equals ASCII-64).
.TP
.BI -B \ INT
Length of barcode starting from the 5'-end. When
.I INT
is positive, the barcode of each read will be trimmed before mapping and will
be written at the
.B BC
SAM tag. For paired-end reads, the barcode from both ends are concatenated. [0]
.TP
.B -b
Specify the input read sequence file is the BAM format. For paired-end
data, two ends in a pair must be grouped together and options
.B -1
or
.B -2
are usually applied to specify which end should be mapped. Typical
command lines for mapping pair-end data in the BAM format are:

    bwa aln ref.fa -b1 reads.bam > 1.sai
    bwa aln ref.fa -b2 reads.bam > 2.sai
    bwa sampe ref.fa 1.sai 2.sai reads.bam reads.bam > aln.sam
.TP
.B -0
When
.B -b
is specified, only use single-end reads in mapping.
.TP
.B -1
When
.B -b
is specified, only use the first read in a read pair in mapping (skip
single-end reads and the second reads).
.TP
.B -2
When
.B -b
is specified, only use the second read in a read pair in mapping.
.B
.RE

.TP
.B samse
bwa samse [-n maxOcc] <in.db.fasta> <in.sai> <in.fq> > <out.sam>

Generate alignments in the SAM format given single-end reads. Repetitive
hits will be randomly chosen.

.B OPTIONS:
.RS
.TP 10
.BI -n \ INT
Maximum number of alignments to output in the XA tag for reads paired
properly. If a read has more than INT hits, the XA tag will not be
written. [3]
.TP
.BI -r \ STR
Specify the read group in a format like `@RG\\tID:foo\\tSM:bar'. [null]
.RE

.TP
.B sampe
bwa sampe [-a maxInsSize] [-o maxOcc] [-n maxHitPaired] [-N maxHitDis]
[-P] <in.db.fasta> <in1.sai> <in2.sai> <in1.fq> <in2.fq> > <out.sam>

Generate alignments in the SAM format given paired-end reads. Repetitive
read pairs will be placed randomly.

.B OPTIONS:
.RS
.TP 8
.BI -a \ INT
Maximum insert size for a read pair to be considered being mapped
properly. Since 0.4.5, this option is only used when there are not
enough good alignment to infer the distribution of insert sizes. [500]
.TP
.BI -o \ INT
Maximum occurrences of a read for pairing. A read with more occurrneces
will be treated as a single-end read. Reducing this parameter helps
faster pairing. [100000]
.TP
.B -P
Load the entire FM-index into memory to reduce disk operations
(base-space reads only). With this option, at least 1.25N bytes of
memory are required, where N is the length of the genome.
.TP
.BI -n \ INT
Maximum number of alignments to output in the XA tag for reads paired
properly. If a read has more than INT hits, the XA tag will not be
written. [3]
.TP
.BI -N \ INT
Maximum number of alignments to output in the XA tag for disconcordant
read pairs (excluding singletons). If a read has more than INT hits, the
XA tag will not be written. [10]
.TP
.BI -r \ STR
Specify the read group in a format like `@RG\\tID:foo\\tSM:bar'. [null]
.RE

.TP
.B bwasw
bwa bwasw [-a matchScore] [-b mmPen] [-q gapOpenPen] [-r gapExtPen] [-t
nThreads] [-w bandWidth] [-T thres] [-s hspIntv] [-z zBest] [-N
nHspRev] [-c thresCoef] <in.db.fasta> <in.fq> [mate.fq]

Align query sequences in the
.I in.fq
file. When
.I mate.fq
is present, perform paired-end alignment. The paired-end mode only works
for reads Illumina short-insert libraries. In the paired-end mode, BWA-SW
may still output split alignments but they are all marked as not properly
paired; the mate positions will not be written if the mate has multiple
local hits.

.B OPTIONS:
.RS
.TP 10
.BI -a \ INT
Score of a match [1]
.TP
.BI -b \ INT
Mismatch penalty [3]
.TP
.BI -q \ INT
Gap open penalty [5]
.TP
.BI -r \ INT
Gap extension penalty. The penalty for a contiguous gap of size k is
q+k*r. [2]
.TP
.BI -t \ INT
Number of threads in the multi-threading mode [1]
.TP
.BI -w \ INT
Band width in the banded alignment [33]
.TP
.BI -T \ INT
Minimum score threshold divided by a [37]
.TP
.BI -c \ FLOAT
Coefficient for threshold adjustment according to query length. Given an
l-long query, the threshold for a hit to be retained is
a*max{T,c*log(l)}. [5.5]
.TP
.BI -z \ INT
Z-best heuristics. Higher -z increases accuracy at the cost of speed. [1]
.TP
.BI -s \ INT
Maximum SA interval size for initiating a seed. Higher -s increases
accuracy at the cost of speed. [3]
.TP
.BI -N \ INT
Minimum number of seeds supporting the resultant alignment to skip
reverse alignment. [5]
.RE

.SH SAM ALIGNMENT FORMAT
.PP
The output of the
.B `aln'
command is binary and designed for BWA use only. BWA outputs the final
alignment in the SAM (Sequence Alignment/Map) format. Each line consists
of:

.TS
center box;
cb | cb | cb
n | l | l .
Col	Field	Description
_
1	QNAME	Query (pair) NAME
2	FLAG	bitwise FLAG
3	RNAME	Reference sequence NAME
4	POS	1-based leftmost POSition/coordinate of clipped sequence
5	MAPQ	MAPping Quality (Phred-scaled)
6	CIAGR	extended CIGAR string
7	MRNM	Mate Reference sequence NaMe (`=' if same as RNAME)
8	MPOS	1-based Mate POSistion
9	ISIZE	Inferred insert SIZE
10	SEQ	query SEQuence on the same strand as the reference
11	QUAL	query QUALity (ASCII-33 gives the Phred base quality)
12	OPT	variable OPTional fields in the format TAG:VTYPE:VALUE
.TE

.PP
Each bit in the FLAG field is defined as:

.TS
center box;
cb | cb | cb
c | l | l .
Chr	Flag	Description
_
p	0x0001	the read is paired in sequencing
P	0x0002	the read is mapped in a proper pair
u	0x0004	the query sequence itself is unmapped
U	0x0008	the mate is unmapped
r	0x0010	strand of the query (1 for reverse)
R	0x0020	strand of the mate
1	0x0040	the read is the first read in a pair
2	0x0080	the read is the second read in a pair
s	0x0100	the alignment is not primary
f	0x0200	QC failure
d	0x0400	optical or PCR duplicate
.TE

.PP
The Please check <http://samtools.sourceforge.net> for the format
specification and the tools for post-processing the alignment.

BWA generates the following optional fields. Tags starting with `X' are
specific to BWA.

.TS
center box;
cb | cb
cB | l .
Tag	Meaning
_
NM	Edit distance
MD	Mismatching positions/bases
AS	Alignment score
BC	Barcode sequence
_
X0	Number of best hits
X1	Number of suboptimal hits found by BWA
XN	Number of ambiguous bases in the referenece
XM	Number of mismatches in the alignment
XO	Number of gap opens
XG	Number of gap extentions
XT	Type: Unique/Repeat/N/Mate-sw
XA	Alternative hits; format: (chr,pos,CIGAR,NM;)*
_
XS	Suboptimal alignment score
XF	Support from forward/reverse alignment
XE	Number of supporting seeds
.TE

.PP
Note that XO and XG are generated by BWT search while the CIGAR string
by Smith-Waterman alignment. These two tags may be inconsistent with the
CIGAR string. This is not a bug.

.SH NOTES ON SHORT-READ ALIGNMENT
.SS Alignment Accuracy
.PP
When seeding is disabled, BWA guarantees to find an alignment
containing maximum
.I maxDiff
differences including
.I maxGapO
gap opens which do not occur within
.I nIndelEnd
bp towards either end of the query. Longer gaps may be found if
.I maxGapE
is positive, but it is not guaranteed to find all hits. When seeding is
enabled, BWA further requires that the first
.I seedLen
subsequence contains no more than
.I maxSeedDiff
differences.
.PP
When gapped alignment is disabled, BWA is expected to generate the same
alignment as Eland version 1, the Illumina alignment program. However, as BWA
change `N' in the database sequence to random nucleotides, hits to these
random sequences will also be counted. As a consequence, BWA may mark a
unique hit as a repeat, if the random sequences happen to be identical
to the sequences which should be unqiue in the database.
.PP
By default, if the best hit is not highly repetitive (controlled by -R), BWA
also finds all hits contains one more mismatch; otherwise, BWA finds all
equally best hits only. Base quality is NOT considered in evaluating
hits. In the paired-end mode, BWA pairs all hits it found. It further
performs Smith-Waterman alignment for unmapped reads to rescue reads with a
high erro rate, and for high-quality anomalous pairs to fix potential alignment
errors.

.SS Estimating Insert Size Distribution
.PP
BWA estimates the insert size distribution per 256*1024 read pairs. It
first collects pairs of reads with both ends mapped with a single-end
quality 20 or higher and then calculates median (Q2), lower and higher
quartile (Q1 and Q3). It estimates the mean and the variance of the
insert size distribution from pairs whose insert sizes are within
interval [Q1-2(Q3-Q1), Q3+2(Q3-Q1)]. The maximum distance x for a pair
considered to be properly paired (SAM flag 0x2) is calculated by solving
equation Phi((x-mu)/sigma)=x/L*p0, where mu is the mean, sigma is the
standard error of the insert size distribution, L is the length of the
genome, p0 is prior of anomalous pair and Phi() is the standard
cumulative distribution function. For mapping Illumina short-insert
reads to the human genome, x is about 6-7 sigma away from the
mean. Quartiles, mean, variance and x will be printed to the standard
error output.

.SS Memory Requirement
.PP
With bwtsw algorithm, 5GB memory is required for indexing the complete
human genome sequences. For short reads, the
.B aln
command uses ~3.2GB memory and the
.B sampe
command uses ~5.4GB.

.SS Speed
.PP
Indexing the human genome sequences takes 3 hours with bwtsw
algorithm. Indexing smaller genomes with IS algorithms is
faster, but requires more memory.
.PP
The speed of alignment is largely determined by the error rate of the query
sequences (r). Firstly, BWA runs much faster for near perfect hits than
for hits with many differences, and it stops searching for a hit with
l+2 differences if a l-difference hit is found. This means BWA will be
very slow if r is high because in this case BWA has to visit hits with
many differences and looking for these hits is expensive. Secondly, the
alignment algorithm behind makes the speed sensitive to [k log(N)/m],
where k is the maximum allowed differences, N the size of database and m
the length of a query. In practice, we choose k w.r.t. r and therefore r
is the leading factor. I would not recommend to use BWA on data with
r>0.02.
.PP
Pairing is slower for shorter reads. This is mainly because shorter
reads have more spurious hits and converting SA coordinates to
chromosomal coordinates are very costly.

.SH CHANGES IN BWA-0.6
.PP
Since version 0.6, BWA has been able to work with a reference genome longer than 4GB.
This feature makes it possible to integrate the forward and reverse complemented
genome in one FM-index, which speeds up both BWA-short and BWA-SW. As a tradeoff,
BWA uses more memory because it has to keep all positions and ranks in 64-bit
integers, twice larger than 32-bit integers used in the previous versions.

The latest BWA-SW also works for paired-end reads longer than 100bp. In
comparison to BWA-short, BWA-SW tends to be more accurate for highly unique
reads and more robust to relative long INDELs and structural variants.
Nonetheless, BWA-short usually has higher power to distinguish the optimal hit
from many suboptimal hits. The choice of the mapping algorithm may depend on
the application.

.SH SEE ALSO
BWA website <http://bio-bwa.sourceforge.net>, Samtools website
<http://samtools.sourceforge.net>

.SH AUTHOR
Heng Li at the Sanger Institute wrote the key source codes and
integrated the following codes for BWT construction: bwtsw
<http://i.cs.hku.hk/~ckwong3/bwtsw/>, implemented by Chi-Kwong Wong at
the University of Hong Kong and IS
<http://yuta.256.googlepages.com/sais> originally proposed by Nong Ge
<http://www.cs.sysu.edu.cn/nong/> at the Sun Yat-Sen University and
implemented by Yuta Mori.

.SH LICENSE AND CITATION
.PP
The full BWA package is distributed under GPLv3 as it uses source codes
from BWT-SW which is covered by GPL. Sorting, hash table, BWT and IS
libraries are distributed under the MIT license.
.PP
If you use the BWA-backtrack algorithm, please cite the following
paper:
.PP
Li H. and Durbin R. (2009) Fast and accurate short read alignment with
Burrows-Wheeler transform. Bioinformatics, 25, 1754-1760. [PMID: 19451168]
.PP
If you use the BWA-SW algorithm, please cite:
.PP
Li H. and Durbin R. (2010) Fast and accurate long-read alignment with
Burrows-Wheeler transform. Bioinformatics, 26, 589-595. [PMID: 20080505]
.PP
If you use the fastmap component of BWA, please cite:
.PP
Li H. (2012) Exploring single-sample SNP and INDEL calling with whole-genome de
novo assembly. Bioinformatics, 28, 1838-1844. [PMID: 22569178]
.PP
The BWA-MEM algorithm has not been published yet.

.SH HISTORY
BWA is largely influenced by BWT-SW. It uses source codes from BWT-SW
and mimics its binary file formats; BWA-SW resembles BWT-SW in several
ways. The initial idea about BWT-based alignment also came from the
group who developed BWT-SW. At the same time, BWA is different enough
from BWT-SW. The short-read alignment algorithm bears no similarity to
Smith-Waterman algorithm any more. While BWA-SW learns from BWT-SW, it
introduces heuristics that can hardly be applied to the original
algorithm. In all, BWA does not guarantee to find all local hits as what
BWT-SW is designed to do, but it is much faster than BWT-SW on both
short and long query sequences.

I started to write the first piece of codes on 24 May 2008 and got the
initial stable version on 02 June 2008. During this period, I was
acquainted that Professor Tak-Wah Lam, the first author of BWT-SW paper,
was collaborating with Beijing Genomics Institute on SOAP2, the successor
to SOAP (Short Oligonucleotide Analysis Package). SOAP2 has come out in
November 2008. According to the SourceForge download page, the third
BWT-based short read aligner, bowtie, was first released in August
2008. At the time of writing this manual, at least three more BWT-based
short-read aligners are being implemented.

The BWA-SW algorithm is a new component of BWA. It was conceived in
November 2008 and implemented ten months later.

The BWA-MEM algorithm is based on an algorithm finding super-maximal exact
matches (SMEMs), which was first published with the fermi assembler paper
in 2012. I first implemented the basic SMEM algorithm in the
.B fastmap
command for an experiment and then extended the basic algorithm and added the
extension part in Feburary 2013 to make BWA-MEM a fully featured mapper.