Let’s say you get a complex sample:

• Environmental studies
• Human microbiota
• Archeology

You extract total DNA, you sequence all.

You got millions of reads

What is on these reads?

Now you ask:

• What are the organisms on this sample?
• How many of them?
• What can they do?

Most of the reads do not align to any reference genome
(most of Prokaryotes have not been isolated)

• similarity based: i.e. alignment
  • highly specific
  • low sensitivity
• composition based: \(k\)-mer frequency
  • low specificity
  • high sensitivity

Given a DNA read \(r\) we want to find which species \(S\) is the most probable origin of the read

We want to find an \(S\) that maximizes \(\Pr(\text{species is }S\vert \text{we saw }r)\)

Thus, the question is how to evaluate this probability. Some approaches:

• Naïve Bayes Classification (Rosen 2008)
• RaiPhy (Nalbantoglu 2011)

Given a sequence \(s\) of length \(L\), we characterize it by a vector counting all subwords of size \(k\)

Given a DNA “word” \(w\in {\cal A}^k\) of length \(k\), we define

• Absolute frequency \(n_{(k)}(w;s)=\sum_j[s_j\ldots s_{j+k-1}=w]\)
• Relative frequency \(f_{(k)}(w;s)=n_{(k)}(w;s)/\sum_{v\in\cal A} n_{(k)}(v;s)\)

Remember that \(\mathcal A = \{A,C,G,T\}\) and that \([Q]=1\) iff \(Q\) is true.

The classification must be invariant to reverse complement

The representation should not change if we use either strand

• \(D_k=4^{k/2} + 4[k\text{ even}]\) distinct values
• \(D_k-1\) linearly independent values
• So for \(k=1\) we have only one value: GC content

We built a platform that allow us to test these hypothesis:

• Phylogenetically close species have similar distribution of \(k\)-mer frequencies

• Phylogenetically distant species have very different \(k\)-mer distributions

We used the values published in TimeTree.org

• Hedges, S. Blair, Joel Dudley, and Sudhir Kumar. “TimeTree: A Public Knowledge-Base of Divergence Times among Organisms.” Bioinformatics 22, no. 23 (2006): 2971–72. doi:10.1093/bioinformatics/btl505.
• Hedges, S. Blair, Julie Marin, Michael Suleski, Madeline Paymer, and Sudhir Kumar. “Tree of Life Reveals Clock-like Speciation and Diversification.” Molecular Biology and Evolution 32, no. 4 (2015): 835–45. doi:10.1093/molbev/msv037.
  • 2274 Studies, 50K Species

Every species is represented by the frequency of each \(k\)-mer
(i.e. empirical probability distributions)

We can compare two probability distributions using the Total Variation distance: \[\mathrm{dist_{TV}}(p,q)=\frac{1}{2}\Vert p-q\Vert_1 = \frac{1}{2}\sum_i\vert p_i-q_i\vert\] (it happens to be half the Manhattan distance)

Since all probability distributions follow \(\sum_i\vert p_i\vert=\Vert p\Vert_1=1\), it is easy to see that the Manhattan distance is a good one.