\[ \newcommand{\vp}{\varphi} \newcommand{\A}{\mathcal A} \newcommand{\O}{\mathcal O} \newcommand{\N}{\mathbb N} \newcommand{\ed}{\mathrm{ed}} \newcommand{\mh}{\mathrm{mh}} \newcommand{\hash}{\mathrm{hash}} \]

## Background

Quickly finding similar pieces of DNA within large datasets is at the core of computational biology. This has many applications:

Alignment: Given two pieces of related DNA, align them to find where mutations (i.e. substitutions, insertions, or deletions) occur.

Homology search: finding related genes between different species.

Assembly: DNA sequencers don’t sequence an entire genome in one piece, but produce reads for many short sequences. These reads need to be stitched together to form the full genome. There are two types of assembly:

- Short read assembly: the older of the two methods, producing sequences of hundreds of basepairs/nucleotides in length, with relatively high precision.
- Long read assembly: newer, and produces reads of length 100k - 1M,
but has a
*much*higher error rate, in particular it can have insertions or deletions (indels) up to every 6 positions.

Phylogeny reconstruction: Given a set of genes or genomes, reconstruct the evolutionary tree of how these genes/genomes evolved. This usually involves computing estimated ’evolutionary distances’ between pairs sequences and reconstructing the tree from these pairwise distances.

The classic pairwise alignment algorithm takes \(\O(nm)\) time to find the edit distance between two sequences of length \(n\) and \(m\). That is, the minimum number of insert, delete, and substitute operations to transform one into the other. This is fine for shorter sequences, but as the size of datasets has grown over time, this is now too slow for most practical applications. Furthermore, alignment methods need to consider each pair of sequences in the dataset and try to align them, which also doesn’t scale as the number of input sequences increases.

### $k$-mers

One common technique is to only look at all $k$-mers of a sequence: For each input sequence we can store all the consecutive subsequences of length \(k\) occuring in the sequence, and when such a $k$-mer occurs in two sequences, we can then check the corresponding positions in these sequences to see whether they are indeed related.

A typical choice is to set \(k\) somewhere from \(20\) to \(31\). We can do some handwavy maths to see that in ideal circumstances \(k=31\) prevents most false positives:

The probability that two uniform random $k$-mers of length \(k\) are the same is \(4^{-31} =2^{-62} < 10^{-18}\) since for each of the \(k\) positions the probability of a match is \(1/|\A| = 1/4\).

If our dataset has size 1GB, or \(10^9\) basepairs, (roughly the order of the size of a human genome) that means there are also \(10^9\) kmers, and thus \(10^{18}\) pairs of $k$-mers. Each such pair has a collision of less than \(10^{-18}\), so the expected number of collisions is less than \(1\).

For \(k=20\), the number of false positives will be larger, but since \(4^{20}\) is still a factor \(1000\) larger than \(10^9\), the fraction of $k$-mers with false positives should still be small.

In practice, this analysis doesn’t exactly work out, since nucleotide
frequencies usually are not exacly \(1/4\) and there are many *low
entropy* regions in DNA, where characters are frequently repeated and
$k$-mers are not at all uniform random. Nevertheless, this indicates
that most of the matching $k$-mers will be meaningful.

A drawback of this method is that it requires storing a lot of data: for each input sequence we need to store as many $k$-mers as the sequence has nucleotides. One solution for this is De Bruijn graphs (Wikipedia).

### Sketching

*sketching* methods are used to avoid the need for such large
datastructures. These are ways of compressing a sequence to a small hash
in (semi)linear time, so that similar sequences have similar hashes. A
datastructure can then be used to store all the hashes and quickly
retrieve similar pairs of hashes.

As for notation, we use \(\A\) to indicate the alphabet. Since we’re dealing with DNA, this will always be \(\{\texttt A, \texttt C, \texttt G, \texttt T\}\) in our case. Given two input sequences \(A\in \A^l\) and \(B\in\A^l\) of length \(l\), we denote their sketches as \(\vp (A)\) and \(\vp (B)\). The edit distance is \(\textrm{ed}(A, B)\), and ideally we would like the distance between the two sketches, \(d(\vp (A), \vp(B))\) to be a good predictor of the edit distance. The type of \(\vp (A)\) and exactly how the distance is computed depends on the sketch method being used.

\([l]\) indicates the set \(\{1, \dots, n\}\). We write \(X^{(l)}\) for the set of subsets of \(X\) of size \(l\). Given \(L \subseteq [l]\) and \(A\in \A^l\) we denote by \(A_L\) the subsequence of characters of \(A\) at positions in \(L\): \(A_L := (A_{L_i})_{i\in [|L|]}\).

The classic sketching algorithm is MinHash, explained below. Another approach is Tensor Sketch, which is explained briefly here.

### MinHash

As an example, MinHash (Wikipedia) is a well known sketch method that sketches a sequence into a list of integers of some length \(h\). Formally, we take \(h\) independent hash functions \(H_i : \A^k \to \N\) (\(1\leq i\leq h\)) that hash $k$-mers to integers. The $i$th component of the MinHash sketch will be the minimal value of \(H_i\) over all $k$-mers in the input sequence \(A\):

\begin{align} \vp_\mh &: \A^l \to \N^h\\ (\vp_\mh(A))_i &:= \min_{j\in [l-k+1]} H_i(A[j: j+k]) \end{align}

The distance between two sequences can then be estimated as the fraction of the \(h\) positions where the two sequences differ.:

\begin{align} d_\mh &: \N^h \times \N^h \to \mathbb R \\ (\vp_1, \vp_2) &\mapsto \frac 1h \big|\{i \in [h] : \vp_{1, i} \neq \vp_{2, i}\}\big|. \end{align}

> TODO: some small example of MinHash.

### Terminology

- k-mer
- A (sub)string of \(k\) consecutive characters.
- Spaced seed
- A
*pattern*of length \(l\) and weight \(w\): given a substring of length \(l\), the \(w\) positions indicate the subset of positions we consider. A pattern can be given as a binary string containing \(w\) ones, or as a subset of size \(w\) of \(\{1, \dots, l\}\).In the literature, a spaced seed usually includes the first and last character (those at positions \(1\) and \(l\)). We do not assume this.

- Spaced k-mer
- The \(w\) characters induced by a spaced seed of weight \(w\).
- Matching positions
- We say that two positions
*match*(assuming a spaced seed) if the spaced k-mers induced by the spaced seed starting at these positions are the same. - Window
- The \(l\) characters surrounding a spaced k-mer, corresponding to the
chosen spaced seed. The window may extend outside the range of the leftmost and
right most character in the spaced k-mer.
TODO: Maybe we actually should include both ends, as that should reduce vulnerability to accidentally having undetected indels at the ends, which is bad for measuring the hamming distance.

#### Example

The string `ACTGC`

contains:

- the $4$-mers
`ACTG`

and`CTGC`

; - for the spaced seed
`1101`

, the spaced k-mers`ACG`

and`CTC`

. - the window
`ACTG`

and`CTGC`

. These are the same as the $4$-mers, but we will use this term explicitly for the characters surrounding a spaced k-mer.

## Introduction

While playing around with some datasets to test homology search methods, we realised that indels between related sequences are quite rare in practice (less than once every \(100\) positions). Thus, a sketching method for Hamming distance (which only allows substitutions, not indels) should be able to find most homologous sequences, while being a much simpler problem to solve.

The remainder of this post presents an efficient sketch algorithm for Hamming distance.

Note that this is joint work between Amir Joudaki and myself.

TODO: Write (a separate post) on literature search. In particular, spaced $k$-mers is a very similar technique, but we need to search a bit more to see whether it has been used in a way similar to this algorithm.

## Spaced $k$-mer Seeded Distance^{1}

**Problem:** Given one or more DNA sequences, find all homologous regions.

Instead of looking at entire sequences, we will only consider
subsequences of length exactly \(w\). We’ll call these subsequences
*windows*. Typically these windows will have length \(64\) or \(128\).

So, take for example these two windows

```
GGCGGGGATTTACGCGGATTGCATGTGGTATCCACCGGGTAGCGGTGCTAGGGAACATCGGTGC
GGCAGGGATTTATGGGGATTGCATGTGGTTACCACCGGGTAGCGGAGCTAGGGATCGTCGGTGC
* * * ** * * *
```

Because of the mutations, these sequences wouldn’t be matched when only looking at matching $k$-mers or their MinHash, since they don’t share a $k$-mer of length \(20\).

To work around this, our sketch method looks at a subsample of
\(l\approx 20\) of the \(w\) positions, and matches two sequences when
they match in all these \(l\) positions. Thus, we pick a random subset
\(L \in [w]^{(l)}\) of \(\{1, \dots w\}\) of size \(l\) and hash the window \(W\) to
\(W_{L} := (W_{L_i})_{i\in [l]}\). For the particular pair of sequences
above, the algorithm could have been looking at the positions in
\(L = \{10, 25, 27, 39, 52, 55, 56\}\), marked `|`

(with \(l=7\)).

```
GCTTCAACCCGCACTGTCTCACGATTGTACAGCAAAGTACGTGTATTTGGGCCTATTTCCAGCT
CTTTTAACCCGCAATATATCACGATTGTACCGCATAGGACGTGTATTTCGGCTTATTGCAAGGT
** * | * * * | | * * *| * |* || * * *
```

The subsampled $k$-mer is `CTGACAT`

for both sequences, thus hashing
them into the same *bucket* and making them candidates for being a
matching pair.

Since we may be unlucky and pick one of the positions with a mutation in \(L\), we repeat the entire algorithm \(r\) times, which will be chosen such that we discover most matching windows (below some Hamming distance away from each other) with high probability. Windows that match but have indels will likely be missed by this method, as their Hamming distance is typically large.

Thus, a naive python implementation of the algorithm using typical parameter values proceeds as follows:

```
r = 20 # The number spaced seed and filter pairs to average over.
l = 128 # The length of the spaced seeds.
w = 20 # The weight of the spaced seeds.
f = 3 # The size of the filter
# Input: a pair of sequences.
# Output: a list of pairs of matching positions.
def get_matching_positions(A, B):
# A list of pairs of matching pairs of positions.
matching_positions = []
for _ in range(r): # Repeat r times.
# Pick a random spaced seed of length l and weight w.
# Note that the code uses 0-based indices while the analysis is 1-based.
W = random.sample(range(l), k=w)
# Pick a random filter.
F = random.choose('ACTG', k=f)
# Hash all windows to their spaced kmers.
# Hash table mapping spaced kmers the windows where they occur.
spaced_kmer_occurences = defaultdict(list)
# Loop over the two sequences
for s in [A, B]:
# Loop over start positions
for j in range(len(s) - l + 1):
window = s[j : j+l]
spaced_kmer = [window[Li] for Li in L]
# Does the spaced kmer pass the filter?
if spaced_kmer[0:f] == F:
# A window is identified by its sequence and start position..
spaced_kmer_occurences[spaced_kmer].append((s, j))
# All pairs of windows that hash to the same bucket are candidate matches.
for spaced_kmer in spaced_kmers:
# Loop over all unordered pairs of windows inducing this spaced k-mer.
for p1, p2 in itertools.combinations(spaced_kmer_occurences[spaced_kmer], 2):
# Add the pair into the list of candidate matches
matching_positions.append((p1, p2))
return matching_positions
```

### Improving performance

**Memory usage**

The major bottleneck of the algoritm is memory usage: for each window it needs to store its hash and the identifier of the window. In practice, this mean that the algorithm will use at least eight times as much memory as the total size of the input dataset. While this is feasible for small datasets, it becomes a problem when running on more than a few gigabytes of data (on my 64GB RAM laptop, anyway).

One way of reducing the memory usage is to simply not consider all windows, but only a subset of them.

Given similar sequences, we don’t need to know this for every pair of corresponding positions – it is sufficient to know the similarity once every, say, \(d\approx 64\) positions, since each match typicalle has length at least \(64\) anyway.

We do this as follows. First fix the filter size \(f=3\) and \(f\) random characters \(F \in \A^f\). Now only consider windows for which the first \(f\) characters of their hash are exactly equal to \(F\). For uniform random input sequences, this keeps \(4^{-f} = 1/64\) of the sequences.

Thus, our example window hash from earlier with has `CTGACAT`

would only
be processed when `F = CTG`

.

We considered some other ways in which sampling windows could be done, but these don’t get the same coverage when considering a fixed fraction of windows.

- Take one window every \(d\) positions.
- Find a subset of positions \(S\subseteq \N\), such that taking all windows starting at positions in \(S\) in both sequence \(A\) and \(B\) guarantees a matching starting position once every \(d\) positions.
- Take each window independently with probability \(1/d\).

**Speed**

To further improve the speed of the algorithm, we can parallelize the loop over all windows. One issue is that hashtables typically do not support multithreaded write operations. We can work around this by splitting the hashtable into disjoint parts. Consider the next \(s=2\) characters of the hash (after the initial \(f\) which are already fixed), and create a total of \(4^s\) hashtables. The \(s\) characters determine in which of the hashtables the current window should be stored.

Continuing the example with hash `CTGACAT`

, the fourth and fifth
basepairs, `AC`

, will be used to select which of the \(16\) hashtables
will be used, and the remainder of the hash, `AT`

, will be used as a key
in this hashtable.

### Analysis

This analysis assumes that the input sequences are uniform random sequences over \(\A = \{\texttt A, \texttt C, \texttt G, \texttt T\}\).

We will compute two numbers:

- False positives: Given two unrelated sequences, what is the probability that we consider them as candidate matches.
- Recall: Given two related windows where a fraction \(p\) of the nucleotides is substituted, what is the probability that we return this pair of windows as a candidate match.

**False positives**

This is similar to the analysis we did for $k$-mers. The probability that two random windows match in all \(l=20\) positions is \(4^{-20} \approx 10^{-12}\). When the total size of the data is 1GB (\(10^9\) windows), we have a total of \(10^{18}\) pairs of sequences, and we can expect \(10^-9\cdot 10^{18} = 10^6\) of these to be false positives. This is sufficiently low to iterate over them and discard them during further analysis.

**Recall**

Suppose that between two matching windows \(A\) and \(B\) each character is substituted with probability \(p\), where typically \(p\) is less than \(0.1\), i.e. at most \(10\%\) of the characters has changed. The probability that the \(l=20\) character hashes of these windows are equal is

\[ \mathbb P(\hash(A)=\hash(B)) = (1-p)^l \geq (1-0.1)^20 \approx 0.12 \]

If we repeat the algorithm \(r=20\) times with different random hash functions, the probability of a match is boosted to

\[ \mathbb P(\exists i\in [r] : \hash_i(A)=\hash_i(B)) = 1-(1-(1-p)^l)^r \geq 1-0.88^r \approx 0.92. \]

Thus, we are able to recover \(92\%\) of all matching windows with an edit distance of \(10\%\). By running the algorithm with more repeats, even more of these high distance pairs can be found.

For windows with a distance of only \(5\%\), doing \(20\) repeats already covers more than \(99.8\%\) of the pairs.

**TODO: Separation**

Some result on the probability of reporting \(f(A_1, A_2) < f(B_1, B_2)\) given the actual distances \(d(A_1, A_2)\) and \(d(B_1, B_2)\).

**TODO: Compare sensitivity**

Other papers introduce a sensitivity. We could compare with this.

### Pruning false positive candidate matches

When \(l\) is chosen too low and the dataset is sufficiently large, the algorithm will produce false positives: windows that match in the sampled \(l\) positions, but are otherwise unrelated. This may seem like a probem, but in practice these pairs are easily identified and discarded because there is a dichotomy (large gap) between the expected Hamming distance between related sequences and the expected Hamming distance between unrelated sequences.

TODO: A plot here would be nice.

In particular the expected relative hamming distance between two random
sequences matching in \(l\) positions will be \(\frac34(w-l)/w\), which
for \(w=64\) and \(l=20\) comes out as \(0.51\). For truely related
sequences on the other hand, a relative distance of \(0.1\) is already
somewhat large, and distances of \(0.2\) are quite rare^{2}.

To discard false positive pairs of matching windows, we can simply compute the Hamming distance between the two windows, and if it is larger than \(0.3 \cdot w\), we discard this candidate match.

## Phylogeny reconstruction

Given this algorithm, we can attempt to solve the problem of phylogeny reconstruction.

**Problem**: Reconstruct the phylogeny (evolutionary tree) of a given set
of genes/genomes.

**Input**: A set of (possibly unassembled) genes or genomes.

**Output**: Pairwise distances between all sequences, from which the
phylogeny can be constructed.

We are only computing pairwise distances instead of the actual tree since there are well established algorithms to compute a phylogeny from these distances: UPGMA and neighbor-joining.

The returned distances are typically some measure of evolutionary distance. In our approach, we estimate the distance between sequence \(A\) and \(B\) as the average hamming distance between matching windows between sequence \(A\) and \(B\).

Python pseudocode for this would be

```
# Input: a pair of sequences.
# Output: the average distance
def distance(A, B):
matching_positions = get_matching_positions([A, B])
total_distance = 0
count = 0
for pa, pb in matching_positions:
window_a = A[pa:pa+l]
window_b = B[pb:pb+l]
total_distance += hamming_distance(pa, pb)
count += 1
return total_distance / count
```

In practice, we implemented this in a slightly different way: We observed that for many buckets in the hash table, there are many windows from a single sequence. This is to be expected because many genes and other parts of DNA can be repeated. For example the E.coli dataset gives the following bucket:

```
Seq Pos
...
B4Sb227 3707750 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCGGAGACTCTGCGTGTCTGGGTACGCCAGCATGAGCGGGATACCGGAGGC
B4Sb227 3748505 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCGGAGACTCTGCGTGTCTGGGTACGCCAGCATGAGCGGGATACCGGAGGC
B4Sb227 3866449 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCGGAGACTCTGCGTGTCTGGGTACGCCAGCATGAGCGGGATACCGGAGGC
B4Sb227 4203113 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCGGAGACTCTGCGTGTCTGGGTACGCCAGCATGAGCGGGATACCGGAGGC
B4Sb227 4444086 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCGGAGACTCTGCGTGTCTGGGTACGCCAGCATGAGCGGGATACCGGAGGC
EDL933 2137122 TGGTTCTGGAAAGTCAGGATGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACGCCGGAGACTCTGCGTGTCTGGGTTCGCCAGCATGAGCGGGATACCGGGGGC
EDL933 2171829 TGGTTCTGGAAAGTCAGGATGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACGCCGGAGACTCTGCGTGTCTGGGTTCGCCAGCATGAGCGGGATACCGGGGGC
EDL933 2524436 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAATGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCAGAGACTCTGCGTGTGTGGGTTCGTCAGCATGAGCGGGATACCGGGGGC
EDL933 2756369 TGGTTCTGGAAAGTCAGGATGAATATGACTCACAGTGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACGCCGGAGACTCTGCGTGTCTGGGTTCGCCAGCATGAGCGGGATACCGGGGGC
EDL933 2813013 TGGTTCTGGAAAGTCAGGGCGAATATGACTCACAATGGGCGGCAATTTGTTCCATTGCCCCAAAGATTGGCTGTACACCAGAGACTCTGCGTGTGTGGGTTCGTCAGCATGAGCGGGATACCGGGAGT
...
```

To prevent the average distance to be too much skewed to one particular repeated window, we only pick one random representative in these cases and ignore all repeats of the window within a sequence.

### Running the algorithm

We tested this phylogeny reconstruction algorithm on a few datasets from the Alignment Free project (afproject.org).

The algorithm performs very well on the genome-based phylogeny tasks. In particular, on the unassembled E.coli task with a coverage of \(5\), our algorithm returns a phylogeny with an Robinson-Foulds distance of \(2\) to the ground truth, while the current best (select coverage 5 in the dropdown) has a distance of \(6\) .

For assembled genomes, the algorithm consistently ranks in the top \(10\) of currently tested methods.

On the other hand, the algorithm completely fails on some other tasks and may need more tuning, or may just not work well at all in specific circumstances.

In general, my feeling is that it works very well to find matches between long sequences, but currently isn’t suitable for estimating distances between sequences only a few hundreds basepairs in length.

TODO: Plot RF distance as function of \(l\) for one/a few datasets.

## TODO Assembly

We’ll also test the algorithm for both long and short read assembly.