What is a tree sequence?#
A succinct tree sequence, or “tree sequence” for short, represents the ancestral relationships between a set of DNA sequences. Tree sequences are based on fundamental biological principles of inheritance, DNA duplication, and recombination; they can be created by evolutionary simulation or by inferring genealogies from empirical DNA data.
Tree sequences provide an efficient way of storing genetic variation data, and can power analyses of millions of whole genomes. Plots (a) and (b) summarize results presented further down this tutorial.
How to think about tree sequences#
A sequence of trees…#
As the name suggests, the simplest way to think about a tree sequence is that it describes a sequence of correlated “local trees” — i.e. genetic trees located at different points along a chromosome. Here’s a tiny example based on ten genomes, \(\mathrm{a}\) to \(\mathrm{j}\), spanning a short 1000 letter chromosome.
Show code cell source
import string
import tskit
mutated_ts = tskit.load("data/whatis_example.trees")
ts = mutated_ts.delete_sites(list(range(mutated_ts.num_sites)))
# Extra code to label and order the tips alphabetically rather than numerically
labels = {i: string.ascii_lowercase[i] for i in range(ts.num_nodes)}
genome_order = [n for n in ts.first().nodes(order="minlex_postorder") if ts.node(n).is_sample()]
labels.update({n: labels[i] for i, n in enumerate(genome_order)})
style1 = (
".node:not(.sample) > .sym, .node:not(.sample) > .lab {visibility: hidden;}"
".mut {font-size: 12px} .y-axis .tick .lab {font-size: 85%}")
sz = (800, 250) # size of the plot, slightly larger than the default
ticks = [0, 5000, 10000, 15000, 20000]
ts.draw_svg(
size=sz, node_labels=labels, style=style1, y_label="Time ago",
y_axis=True, y_ticks=ticks)
The tickmarks on the X axis and background shading indicate the genomic positions covered by the trees. For the first short portion of the chromosome, from the start until position 189, the relationships between the ten genomes are shown by the first tree. The second tree shows the relationships between positions 189 and 546, and the third from position 546 to the end. We can say that the first tree spans 189 base pairs, the second 357, and the third 454.
… created by a genealogical network#
In fact, succinct tree sequences don’t store each tree separately, but instead are based on an interconnected genetic genealogy, in which genetic recombination has led to different regions of the chromosome having different histories. Another way of thinking about the tree sequence above is that it describes the full genetic family “tree” (strictly, “network”) of our 10 genomes.
An efficient encoding of DNA data#
A tree sequence can be used to describe patterns of genetic variation by combining the trees with a knowledge of where mutations occur on their branches. Here’s how that might look in our simple example:
Show code cell source
mut_labels = {} # An array of labels for the mutations, listing position & allele change
l = "{:g} ({}→{})"
for mut in mutated_ts.mutations(): # This entire loop is just to make pretty labels
site = mutated_ts.site(mut.site)
older_mut = mut.parent >= 0 # is there an older mutation at the same position?
prev = mutated_ts.mutation(mut.parent).derived_state if older_mut else site.ancestral_state
mut_labels[mut.id] = l.format(site.position, prev, mut.derived_state)
mutated_ts.draw_svg(
size=sz, style=style1, node_labels=labels, mutation_labels=mut_labels)
There are now twelve single nucleotide mutations in the tree sequence. They are shown on the branches of the trees, and the positions of the twelve variable sites associated with the mutations are shown along the X axis.
The trees inform us that, for example, the final mutation (at position 987) is inherited by genomes \(\mathrm{h}\) to \(\mathrm{j}\). These genomes must have an T at that position, compared to the original value of G. In other words, once we know the ancestry, placing a relatively small number of mutations is enough to explain all the observed genetic variation. Here’s the resulting “variant matrix”:
Show code cell source
haplotypes = [" ".join(h) for h in mutated_ts.haplotypes()]
print("Position: " + "".join(f"{s.position:^4g}" for s in mutated_ts.sites()))
print("\n".join(sorted(
[f"Genome {labels[i]}: {h}" for i, h in zip(mutated_ts.samples(), haplotypes)])))
Position: 6 51 108 417 496 569 617 731 822 863 907 987
Genome a: A C A T T C A A G C C G
Genome b: A C A A T C A A C C C G
Genome c: A C A A T G A A C C C G
Genome d: A C A A T C A A C C C G
Genome e: A C A A T C A A C C C G
Genome f: A C A A T C A A C C C G
Genome g: A C C T C C A A C A C G
Genome h: C G A T T C T T C A G T
Genome i: C G A T T C T T C A G T
Genome j: C G A T T C T T C A G T
This approach scales effectively to millions of genomes, and to chromosomes of hundreds of megabases in length. The ability to deal with huge datasets comes down to one key feature of genomic data: adjacent trees along a chromosome are highly correlated, that is, they share structure. In our example this becomes evident if we highlight the branches (“edges” in tree sequence terminology) that remain unchanged between the first and the second tree.
Show code cell source
# Highlight certain edges in certain trees. Other visualization possibilities in tutorials/viz.html
kept_edges = [e for e in ts.edges() if e.left==0 and e.right>ts.breakpoints(True)[1]]
style3 = (
",".join(f"#svg1 .tree:not(.t2) .node.a{e.parent}.n{e.child} > .edge" for e in kept_edges)
+ "{stroke:#00DD00; stroke-width: 2px}"
+ style1)
ts.draw_svg(
size=(500, 250), x_lim=(0, 500), root_svg_attributes={'id':'svg1'}, y_ticks=ticks,
node_labels=labels, style=style3)
A branch can be shared by many adjacent trees, but is stored as a single edge in the tree sequence. For large datasets this is a great saving, because typically each tree-change affects only a few branches at a time, regardless of the tree size.
Below is an extension of the plot at the top of this page, showing predicted file sizes when storing not just millions, but billions of human-like genomes: enough to encompass every human on the planet. This demonstrates that the tree sequence encoding leads to savings of many orders of magnitude, even when compared against compressed versions of the standard VCF storage format (original published data here). It’s also worth noting that the efficiency extends to processing time too: tree sequences are often several orders of magnitude faster to process than other storage formats.
A record of genetic ancestry#
Often, we’re not interested so much in the DNA sequence data as the genetic ancestry itself (discussed e.g. here). In other words, the main consideration is the actual trees in a tree sequence, rather than the distributions of mutations placed upon them — indeed in genetic simulations, it may not be necessary to incorporate neutral mutations at all. The trees reflect, for example, the origin and age of alleles under selection, the spatial structure of populations, and the effects of hybridization and admixture in the past.
The tree sequence in this tutorial was actually generated using a model of population splits and expansions as shown in the following schematic, plotted using the DemesDraw package. Our 10 genomes were sampled from modern day populations A (a constant-size population) and B (a recently expanding one), where limited migration is occuring from A to B.
A major benefit of “tree sequence thinking” is the close relationship between the tree sequence and the underlying biological processes that produced the genetic sequences in the first place, such as those pictured in the demography above. For example, each branch point (or “internal node”) in one of our trees can be imagined as a genome which existed at a specific time in the past, and which is a “most recent common ancestor” (MRCA) of the descendant genomes at that position on the chromosome. We can mark these extra “ancestral genomes” on our tree diagrams, distinguishing them from the sampled genomes (\(\mathrm{a}\) to \(\mathrm{j}\)) by using circular symbols. We can even colour the nodes by the population that we know (or infer) them to belong to at the time:
Show code cell source
colours = {0: "#1f77b4", 1: "#ff7f0e", 2: "#2ca02c"}
style2 = ".y-axis .tick .lab {font-size: 85%}"
style2 += "#svg2 .node > .sym {visibility: visible;}" # force-show all nodes: not normally needed
style2 += "".join([f".p{n.population} > .sym {{fill: {colours[n.population]}}}" for n in ts.nodes()])
mutated_ts.draw_svg(
size=sz, root_svg_attributes={'id':'svg2'}, y_label="Time ago (generations)",
y_axis=True, y_ticks=ticks, node_labels=labels, mutation_labels={}, style=style2)
The diagram shows that most of the ancestral genomes \(\mathrm{k}\) to \(\mathrm{u}\) lived much longer ago than the population split, 1000 generations back, and resided in the ancestral (blue) population. The tree sequence also allows us to easily deduce these MRCA genomes, simply by looking at which mutations they have inherited:
Show code cell source
import numpy as np
tables = mutated_ts.dump_tables()
# Flip sample and nonsample flags, making the haplotypes() method print out nonsample nodes
s_flags = tables.nodes.flags[ts.samples()[0]]
no_flags = s_flags-s_flags
tables.nodes.flags = np.where(tables.nodes.flags & tskit.NODE_IS_SAMPLE, no_flags, s_flags)
ts_flipped = tables.tree_sequence()
haplotypes = [" ".join(h) for h in ts_flipped.haplotypes(missing_data_character=" ")]
print(" " * ts_flipped.num_sites, " " * (ts_flipped.num_sites-4), "")
print(
"||ANCESTRAL GENOMES|| Position:",
"".join(f"{s.position:^4g}" for s in ts_flipped.sites()))
print(
"\n".join(reversed(sorted([
f"Genome {labels[i]} ({ts.node(i).time:7.1f} {ts_flipped.time_units} ago): {h}"
for i, h in zip(ts_flipped.samples(), haplotypes)]))))
||ANCESTRAL GENOMES|| Position: 6 51 108 417 496 569 617 731 822 863 907 987
Genome u (19697.9 generations ago): A C A T T C T T C A G G
Genome t (12059.9 generations ago): A C A T T
Genome s (10172.9 generations ago): T T C A A C A C G
Genome r ( 9770.8 generations ago): A C A
Genome q ( 3795.0 generations ago): C A A C C C G
Genome p ( 3215.9 generations ago): A C A A T C A A C C C G
Genome o ( 3200.2 generations ago): C G A T T C T T C A G T
Genome n ( 1428.2 generations ago): C G A T T C T T C A G T
Genome m ( 1249.5 generations ago): A C A A T C A A C C C G
Genome l ( 449.6 generations ago): A C A A T C A A C C C G
Genome k ( 377.0 generations ago): A C A A T C A A C C C G
You can see that some ancestors are missing genomic regions, because those parts of their genome have not been inherited by any of the sampled genomes. In other words, that ancestral node is not present in the corresponding local tree.
A framework for efficient computation#
Using tree structures is a common way to implement efficient computer algorithms, and many phylogenetic methods use the structure provided by the evolutionary tree to implement efficient dynamic programming algorithms. The tree sequence structure allows these approaches to be extended to the particular form of phylogenetic network defined by multiple correlated trees along a genome.
For example, statistical measures of genetic variation can be thought of as a calculation combining the local trees with the mutations on each branch (or, often preferably, the length of the branches: see this summary). Because a tree sequence is built on a set of small branch changes along the chromosome, statistical calculations can often be updated incrementally as we move along the genome, without having to perform the calculation de novo on each tree. Using tree sequences can result in speed-ups of many orders of magnitude when perfoming calculations on large datasets, as in this example of calculating Tajima’s D (from here):
The tskit library has extensive support for these sorts of population genetic calculations. It provides efficient methods for traversing through large trees and tree sequences, as well as providing other phylogenetically relevant methods such as parsimonious placement of mutations, and the counting of topologies embedded within larger trees.
If you are new to tree sequences, and want to start finding out about tskit, you might now want to continue to the next tutorial: Terminology & concepts.
Further reading#
Jump straight in: if you already have a tree sequence you wish to deal with, the Getting started with tskit tutorial show you how to do a number of common tasks.
How is a tree sequence stored: details in the Tables and editing tutorial
The offical tskit documentation