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response to reviewers' comments, round 2
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Heng Li
+16
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@@ -61,13 +61,6 @@
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Volume = {32},
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Year = {2016}}
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@misc{Suzuki:2016,
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title = {Fast and accurate alignment tool for PacBio and Nanopore long reads},
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author = {Hajime Suzuki},
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journal = {Unpublished},
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howpublished = {\href{https://github.com/ocxtal/minialign}{https://github.com/ocxtal/minialign}},
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year = {2016}}
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@misc{Ruan:2016,
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title = {Ultra-fast de novo assembler using long noisy reads},
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author = {Jue Ruan},
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@@ -172,14 +165,6 @@
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Volume = {29},
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Year = {2011}}
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@article {Suzuki130633,
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author = {Suzuki, Hajime and Kasahara, Masahiro},
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title = {Acceleration Of Nucleotide Semi-Global Alignment With Adaptive Banded Dynamic Programming},
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year = {2017},
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note = {doi:10.1101/130633},
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publisher = {Cold Spring Harbor Labs Journals},
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journal = {bioRxiv}}
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@article{Gotoh:1982aa,
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Author = {Gotoh, O},
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Journal = {J Mol Biol},
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@@ -337,3 +322,19 @@
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Title = {{MUMmer4}: A fast and versatile genome alignment system},
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Volume = {14},
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Year = {2018}}
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@article{Li:2009ys,
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Author = {Li, Heng and others},
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Journal = {Bioinformatics},
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Pages = {2078-9},
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Title = {The {Sequence Alignment/Map format and SAMtools}},
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Volume = {25},
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Year = {2009}}
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@article{Suzuki:2018aa,
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Author = {Suzuki, Hajime and Kasahara, Masahiro},
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Journal = {BMC Bioinformatics},
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Pages = {45},
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Title = {Introducing difference recurrence relations for faster semi-global alignment of long sequences},
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Volume = {19},
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Year = {2018}}
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+18
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@@ -19,7 +19,7 @@
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\begin{document}
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\firstpage{1}
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\title[Aligning nucleotide sequences with minimap2]{Minimap2: versatile pairwise alignment for nucleotide sequences}
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\title[Aligning nucleotide sequences with minimap2]{Minimap2: pairwise alignment for nucleotide sequences}
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\author[Li]{Heng Li}
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\address{Broad Institute, 415 Main Street, Cambridge, MA 02142, USA}
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@@ -64,7 +64,7 @@ the thought that 10kb long sequences should be easier to map than 100bp reads
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because we can more effectively skip repetitive regions, which are often the
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bottleneck of short-read alignment. We confirmed our speculation by achieving
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approximate mapping 50 times faster than BWA-MEM~\citep{Li:2016aa}.
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\citet{Suzuki130633} extended our work with a fast and novel algorithm on
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\citet{Suzuki:2018aa} extended our work with a fast and novel algorithm on
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generating base-level alignment, which in turn inspired us to develop minimap2
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with added functionality.
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@@ -88,7 +88,9 @@ the versatility of minimap2.
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Minimap2 follows a typical seed-chain-align procedure as is used by most
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full-genome aligners. It collects minimizers~\citep{Roberts:2004fv} of the
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reference sequences and indexes them in a hash table. Then for each query
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reference sequences and indexes them in a hash table, with the key being the
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hash of a minimizer and the value being a list of locations of the minimizer
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copies. Then for each query
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sequence, minimap2 takes query minimizers as \emph{seeds}, finds exact matches
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(i.e. \emph{anchors}) to the reference, and identifies sets of colinear anchors as
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\emph{chains}. If base-level alignment is requested, minimap2 applies dynamic
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@@ -118,9 +120,12 @@ distance between two anchors is too large); otherwise
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\begin{equation}\label{eq:chain-gap}
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\beta(j,i)=\gamma_c\big((y_i-y_j)-(x_i-x_j)\big)
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\end{equation}
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In implementation, a gap of length $l\not=0$ costs
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In implementation, a gap of length $l$ costs
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\[
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\gamma_c(l)=0.01\cdot \bar{w}\cdot|l|+0.5\log_2|l|
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\gamma_c(l)=\left\{\begin{array}{ll}
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0.01\cdot \bar{w}\cdot|l|+0.5\log_2|l| & (l\not=0) \\
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0 & (l=0)
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\end{array}\right.
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\]
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where $\bar{w}$ is the average seed length. For $N$ anchors, directly computing all $f(\cdot)$ with
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Eq.~(\ref{eq:chain}) takes $O(N^2)$ time. Although theoretically faster
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@@ -164,7 +169,7 @@ empirical formula:
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\[
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{\rm mapQ}=40\cdot (1-f_2/f_1)\cdot\min\{1,m/10\}\cdot\log f_1
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\]
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where $m$ is the number of anchors on the primary chain, $f_1$ is the chaining
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where $\log$ denotes natural logarithm, $m$ is the number of anchors on the primary chain, $f_1$ is the chaining
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score, and $f_2\le f_1$ is the score of the best chain that is secondary to the
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primary chain. Intuitively, a chain is assigned to a higher mapping quality if
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it is long and its best secondary chain is weak.
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@@ -253,7 +258,7 @@ performance of minimap2. Traditional SSE implementations~\citep{Farrar:2007hs}
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based on Eq.~(\ref{eq:ae86}) can achieve 16-way parallelization for short
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sequences, but only 4-way parallelization when the peak alignment score reaches
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32767. Long sequence alignment may exceed this threshold. Inspired by
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\citet{Wu:1996aa} and the following work, \citet{Suzuki130633} proposed a
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\citet{Wu:1996aa} and the following work, \citet{Suzuki:2018aa} proposed a
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difference-based formulation that lifted this limitation.
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In case of 2-piece gap cost, define
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\[
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@@ -320,7 +325,7 @@ y_{rt}&=&\max\{0,y_{r-1,t}+u_{r-1,t}-z_{rt}+q\}-q-e\\
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\end{equation*}
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In this formulation, cells with the same diagonal index $r$ are independent of
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each other. This allows us to fully vectorize the computation of all cells on
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the same anti-diagonal in one inner loop. It also simplifies banded alignment,
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the same anti-diagonal in one inner loop. It also simplifies banded alignment (500bp band width by default),
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which would be difficult with striped vectorization~\citep{Farrar:2007hs}.
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On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, the initial
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@@ -355,7 +360,7 @@ times as fast as Parasail's 4-way vectorization~\citep{Daily:2016aa}. Without
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banding, our implementation is slower than Edlib~\citep{Sosic:2017aa}, but with
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a 1000bp band, it is considerably faster. When performing global alignment
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between anchors, we expect the alignment to stay close to the diagonal of the
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DP matrix. Banding is applicable most of time.
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DP matrix. Banding is applicable most of the time.
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\subsubsection{The Z-drop heuristic}
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@@ -478,7 +483,7 @@ both C and Python. It is distributed under the MIT license, free to both
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commercial and academic uses. Minimap2 uses the same base algorithm for all
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applications, but it has to apply different sets of parameters depending on
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input data types. Similar to BWA-MEM, minimap2 introduces `presets' that
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modify multiple parameters with a simple invokation. Detailed settings
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modify multiple parameters with a simple invocation. Detailed settings
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and command-line options can be found in the minimap2 manpage. In addition to
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the applications evaluated in the following sections, minimap2 also retains
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minimap's functionality to find overlaps between long reads and to search
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@@ -570,7 +575,7 @@ Peak RAM (GByte) & 8.9 & 14.5 & 3.2 & 29.2\vspace{1em}\\
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\% approx. introns & 91.8\% & 96.9\% & 92.5\% & 82.4\% \\
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\botrule
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\end{tabular}
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}{Mouse reads (AC:SRR5286960; R9.4 chemistry) were mapped to the primary assembly of mouse
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}{Mouse cDNA reads (AC:SRR5286960; R9.4 chemistry) were mapped to the primary assembly of mouse
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genome GRCm38 with the following tools and command options: minimap2 (`-ax
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splice'); GMAP (`-n 0 --min-intronlength 30 --cross-species'); SpAln (`-Q7 -LS
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-S3'); STARlong (according to
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@@ -579,7 +584,7 @@ compared to the EnsEMBL gene annotation, release 89. A predicted intron
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is \emph{novel} if it has no overlaps with any annotated introns. An intron
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is \emph{exact} if it is identical to an annotated intron. An intron is
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\emph{approximate} if both its 5'- and 3'-end are within 10bp around the ends
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of an annotated intron.}
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of an annotated intron. Chimeric alignments are defined in the SAM spec~\citep{Li:2009ys}.}
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\end{table}
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We next aligned real mouse reads~\citep{Byrne:2017aa} with GMAP~(v2017-06-20;
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@@ -687,7 +692,7 @@ involving $>$100kb introns, which was impractically slow ten years ago. The
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minimap2 chaining algorithm is fast and highly accurate by itself. In fact,
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chaining alone is more accurate than all the other long-read mappers in
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Fig.~\ref{fig:eval}a (data not shown). This accuracy helps to reduce downstream
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base-level alignment of candidate chains, which is still times slower than
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base-level alignment of candidate chains, which is still several times slower than
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chaining even with the Suzuki-Kasahara improvement. In addition, taking a
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general form, minimap2 chaining can be adapted to non-typical data types such as
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spliced reads and multiple reads per fragment. This gives us the opportunity to
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