Clustering, Phylogenetic Trees,
and Inferences about Evolution
BMMB597E
Protein Evolution
1
Given a set of organisms:
• Can we measure similarities, and cluster the
organisms into subsets?
• Can we form hierarchical clusterings (that is,
clusters of clusters of clusters …) that
correspond to an evolutionary tree?
• Can we calibrate rates of divergence, and
thereby date branching events during life
history?
2
Note that
• We can observe similarities among organisms, or
species, both among extant organisms; and, with
greater difficulty, extinct ones
• It is rare that we can observe evolutionary
relationships directly. Generally evolutionary
relationship (homology) is an inference from
similarities that we can observe
• Some dating can be calibrated from geology.
However, much dating depends on models and
assumptions, and is therefore questionable
3
What is a cluster?
• Given a set of objects (species, people, literary
texts, protein structures, minerals …)
• A cluster is a subset of these objects such that the
similarity among the objects in the subset is
generally higher than the similarity among the
objects in the full set
• Clustering depends on property chosen to
measure similarity
• For instance, focussing on wings would cluster
bats with birds; not separate mammals and birds
4
Linnaeus’ Systema Naturae
• Linnaeus (1707-1778) developed a taxonomic
system for species
• Based on clustering together species with
similarities
• Main clusters called Kingdoms
– Animal, vegetable, mineral
• Within each main cluster are subclusters
• Hierarchical clustering:
clusters of clusters of clusters …
5
http://www.mun.ca/biology/scarr/139416_Natural_classification.jpg
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Linnaean hierarchy
•
•
•
•
•
Kingdom
Class
Order
Genus
Species
• Linnaeus introduced binomial nomenclature:
genus/species: For example Homo sapiens, Bos taurus
– Higher levels are implied; that is: humans and cows are
mammals
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Linnaean hierarchy
•
•
•
•
•
Kingdom
Class
Order
Genus
Species
Titian: Rape of Europa
• Linnaeus introduced binomial nomenclature:
genus/species: For example Homo sapiens, Bos taurus
– Higher levels implied; that is humans and cows are
mammals
8
Taxonomy now has more levels of clustering
•
•
•
•
•
•
•
Kingdom
Phylum
Class
Order
Family
Genus
Species
There are many
intermediate levels
also: superfamily,
subfamily
Below species:
variety, strain
9
Objective and subjective
aspects of clustering
• We have already mentioned the problem of which
characters to choose on which to base
measurements of similarity.
• Even if people agree on the degrees of similarity
among element of a set of objects, they may
disagree on how finely to cluster them
• People are called “lumpers” or “splitters”
– To a music major, all chemistry courses one cluster
– To a chemistry major: important distinction between
physical, analytical, inorganic, organic, biochemical
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Linnaeus and evolution
• When Linnaeus created his taxonomy, it was
based solely on his perceived similarities among
species
• It turned out that the hierarchy largely reflects
evolutionary relationships
• All the creatures within the same genus or family
should be more closely related to each other than
they are to creatures in different genuses or
families.
• Usually true, although Linnaean hierarchy does
not always correspond to modern taxonomy
11
Linnaeus v. Huxley
• Linnaeus divided the animal kingdom into six
classes: mammals, birds, amphibia (including
reptiles), fishes, insects and worms.
• Linnaeus therefore considered crocodiles more
closely related to salamanders than to birds.
• Thomas Huxley, in the 19th century, grouped
reptiles and birds together. This is now believed
to be correct .
• There are, however, much more serious problems
in the relation between taxonomy and evolution
12
Similarities are not relationships
• Forming hierarchical clusters on the basis of
similarities do not necessarily imply biological
relationships
• Choice of characters with which to measure
similarities often ambiguous
• Classical methods: palaeontology, comparative
anatomy, embryology, in hands of experts, did
extremely well
• Molecular methods (especially DNA and protein
sequences) perhaps more reliable
13
Early attempts to use molecular
properties in taxonomy
• Especially important for prokaryotes, where
standard properties such as skeletal anatomy
not possible
• Nature of biochemicals – chemotaxonomy
• Immunological cross-reactivity
• Electrophoretic ‘fingerprinting’ – spread
proteome out on a gel
• Hybridization of DNAs
14
Genotype and phenotype
• Evolutionary relationships fundamentally
based on genotype
• Palaeontology, comparative anatomy,
embryology attempt to reason from
phenotype to genotype
• So sequence-based methods more direct
• However, sequences don’t always give
unambiguous answer
15
The species as the ‘atom’ of taxonomy
• Taxonomy has been fundamentally the
classification of species.
• Remember that before Darwin, it was believed
that species were immutable
• We are still interested in evolutionary trees of
species
• But it has become clear that it is more difficult
to define the concept of species
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Difficulty in defining species
• At base of hierarchy is idea of species.
• It is species that Linnaeus and subsequent taxonomists
are trying to cluster.
– Note: subspecies, varieties, strains …
• Difficult to define species
• Mutual fertility within group, and infertility outside
group, is a major conceptual ingredient
• But even for mammals this doesn’t
quite work: there are mutually fertile
species (tigers, lions) that do not mate
in the wild
A “tiglon”
17
Whole concept of hierarchy in
question
• Horizontal gene transfer: incorporation of
genetic material from another organism that is
not a parent
• Example: plasmid exchange among bacteria to
distribute antibiotic resistance
• Known mechanisms:
– Transformation (Avery, McLeod and McCarty, 1944:
proof DNA genetic material)
– Transduction: virus carries DNA from one organism to
another (bacteria, human retroviruses)
– Bacterial conjugation: DNA transfer by cell-cell contact
18
From: “Studies on the Chemical Nature of the
Substance Inducing Transformation of
Pneumococcal Types: Induction of
Transformation by a Desoxyribonucleic Acid
Fraction Isolated from Pneumococcus Type
III.” J. Exp. Med., 79: 137-158. January 1944.
Bacterial conjugation. (Image by C; C. Brinton, Jr.,
http://biosciences-people.bham.ac.uk/About/
staff_profiles_research.asp?ID=205)
Bacteriophage infecting E. coli
http://www.washington.edu/alumni/partnership
s/biology/200710/kerr.html
19
Bacterial transformation
http://slic2.wsu.edu
Bacterial conjugation
knowledgerush.com
Bacteriophage infecting cell
biology.about.com
20
Horizontal gene transfer makes
nonsense of the “tree of life”
Picture by W.F. Doolittle
21
To summarize
• We want to construct a “tree of life” stating the
genealogy of all organisms (or at least all species)
• Classical methods based on phenotype not bad
• Molecular data – especially DNA sequences – are
based directly on genotype
• Work as well as anything could
• Still problems with horizontal gene transfer
• These problems are worse in prokaryotes, worst in
earliest life forms (see Doolittle’s picture).22
22
Remaining general problems
• Once you choose measures of similarity you can derive
a hierarchy
• Still left with how you define clusters (lumper-splitter
problem)
• Whether this represents evolutionary relationships is a
question – in view of HGT
• Differential rates of change can complicate picture
• Can we calibrate molecular similarities to date events
in life history?
23
How do we represent hierarchies
• Idea of graph: nodes and edges
• A tree is a special kind of graph, in which there
is only one path from any node to any other
node
This graph is a tree
This graph is not a tree
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25
http://www.genealogyintime.com/NewsStories/2009/April/inbreeding_of_spanish_royalty_page2.html
Family `tree’ NOT a tree!
How to turn a set of pairwise
similarities into a tree
• UPGMA method (Unweighted Pair Group Method
with Arithmetic Mean)
• Start by taking each item as a separate subset
• Take most closely related pair, form a new node that
is their parent
• The original pair becomes a two-element subset
associated with the higher node
• Then take next most closely-related pair of subsets
• Similarity/difference between two subsets is the
average of the similarities/differences between all
pairs of elements from the subsets
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Number of sequence differences
between cytochromes c
Data from W.M. Fitch & E. Margoliash, Science 155, 279-284 (1967)
27
UPGMA tree from cytochrome c sequences
First two steps: BF closest, join them
Then AD next closest, join them
http://www.nmsr.org/upgma.htm
28
UPGMA tree from cytochrome c
sequences, subsequent steps
http://www.nmsr.org/upgma.htm
29
UPGMA method is suitable for deriving
hierarchy from any set of objects given
a measure of similarity
• Problems arise when trying to infer
evolutionary relationships, and dating
divergences
• Consider items sold in different sections of a
department store:
– Reasonable that men’s and women’s shoes have a
common ancestor
– Not reasonable that shoes and furniture have a
common ancestor
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Can we trust UPGMA tree as reflecting
evolution?
• If choice of different similarity measures gives
inconsistent results, then there is a problem
• Typical similarity measures for molecular biology
are sequence similarities, either nucleic acid or
protein
• Many different measures suggested, basic idea is
that the more substitutions in optimal alignment,
the more distant the sequences
• Quantitatively, correct for back mutation in highly
diverged sequences
31
UPGMA assumes constant divergence
rates
• Suppose there are species that are indeed
closely related, but suppose that the
cytochrome c of one of them is changing
much faster than the other
• Then that pair will appear very dissimilar and
be separated in the phylogenetic tree
• This is an error if we wish to assume that the
similarities in the cytochromes c indicate the
closeness of the evolutionary relationships
32
Defenses against non-uniform rates of change
• Sometimes unusually large rate of change the result
of selective pressure
• Choose third-base changes as non selective?
• Detection of non-uniform rates of change: choose
‘outgroup’
• For instance, if we are dealing with sequences from
primates, choose another mammal: cow
• Similarity of cow sequence to all primate sequences
should be approximately equal
• If not, some primate species is changing faster than
others
33
Unrooted and rooted trees
A rooted tree contains one more bit of information: what node in the graph
corresponds tao the last common ancestor
Inclusion of an outgroup can allow ‘rooting’ of the tree
34
Cladistic methods
• Explicitly assume evolutionary relationship
and evolutionary model
• Deal specifically with sequences
• Start from multiple sequence alignment
• Two classical methods:
– Maximum parsimony
– Maximum likelihood
35
Maximum parsimony
• Find tree that postulates fewest mutations
• Given sequences: ATGC, ATGG, TCCA, TTCA
(These appear on bottom line of trees)
● Tree on left postulates four mutations
● Tree on right postulates eight mutations
(T→A at position 1 occurs twice)
● Ancestral sequence at each node shown
36
How could you test a method for
determining a phylogenetic tree?
• Use real data: sequence samples of virus
taken from same patient at different times
• Use simulated data: set up a model of
sequence change, write computer program to
implement it, producing a known tree with
known final generation of sequences – see
whether methods correctly reproduce the tree
37
Time calibration of phylogenetic trees:
the ‘Molecular Clock’
• ‘Molecular clock’ hypothesis (Pauling &
Zuckerkandl, 1962): suggested that the rate of
evolutionary change in the amino acid sequence
of each protein family was approximately
constant over time, independent of lineage
• E. Margoliash (1963): ‘It appears that the number
of residue differences between cytochrome C of
any two species is mostly conditioned by the time
elapsed since the lines of evolution leading to
these two species originally diverged.’
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Problems with molecular clock (F. Ayala)
• Different generation times – should the ‘clock’
run at a constant rate per year or per generation?
• Population size – genetic drift stronger in small
populations, more of evolution is neutral in small
populations
• General species-specific differences
• Functional change in protein studied – stick to
non-coding (=??? non-functional??) DNA or silent
mutations
• Differential selective pressure
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Calibration of molecular clock
• Use dates of species divergence available from
classical palaeontology
• Dating by geological methods
• If there are enough calibration points, then can
interpolate
• Some well known exceptions to constant rate of
sequence divergence have arisen
• For instance, the clock runs about 5 times as fast
in rodents as in humans (generation time?)
40
Bayes’ theorem:
P(A|B) = P(B|A) P(A) / P(B)
• The theorem is telling us how to calculate P(A|B) = the conditional
probability of A, given the observation B, on which A may depend.
• For example: if A = rain and B can = cloudy or sunny, then
– P(rain|cloudy) is the probability that it will rain on a cloudy day
– P(rain|sunny) is the probability that it will rain on a sunny day
(small but not zero)
– P(rain|cloudy) is likely to be greater than P(rain|sunny)
• P(A) = the prior probability of A. (Without checking the sky today, what is
the probability that it will rain = the number of rainy days per year/365)
• P(B) = the prior probability of B (number of cloudy days per year/365 or
number of sunny days per year /365)
• P(B|A) = the conditional probability of B, given A. In our example,
P(cloudy|rain) = the probability that it is cloudy, if we know it is raining 41
P(A|B) = P(B|A) P(A) / P(B)
• Suppose that in State College there are 66 rainy days per
year and 299 (= 365 – 66) dry days
• There are 100 cloudy days per year and 265 clear days
• 95% of rainy days are cloudy; 5% of rainy days clear
• We want to guess whether it will rain today
• If we don’t look at the sky we can only estimate: 66/365 =
18% chance of rain
• If we observe that it is cloudy, probability of rain is:
P(rain|cloudy) = P(cloudy|rain)×P(rain)/P(cloudy)
=
0.95
× 0.18 / (100/365)
= 62.4% chance of rain
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P(A|B) = P(B|A) P(A) / P(B)
• If we don’t look at the sky we can only estimate: 66/365 =
18% chance of rain
• If we observe that it is cloudy, probability of rain is:
P(rain|cloudy) = P(cloudy|rain)×P(rain)/P(cloudy)
=
0.95
× 0.18 / (100/365)
= 62.4% chance of rain
• Observation of a contingent quantity (cloudy sky) allows us
to correct our a priori probability, 18%, to 62.4%
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What if we don’t know the numbers?
• In the simple example, we had a completely
parameterized model and tried to predict an outcome
• Alternatively we don’t have the statistics – they are
‘unknown parameters’ – and we observe the sky and
the weather over many consecutive days. These are
our data.
• For any value of the parameters, we can calculate the
probability of observing the data.
• Those values of parameters that give the highest
probability to the data actually observed are our
estimate of their values
44
Bayesian inference of phylogenetic trees
• Observable: multiple sequence alignment
• What phylogenetic tree best accounts for this
alignment
• Trees depend on model of evolutionary change; a
general model being specified by values of
parameters such as mutation rate
• For any model, can compute the probability of
different observed sequence alignments
• The likelihood of certain parameter values is the
computed probability of observing the actual
data, if the parameters have those values
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More detailed description of parameters
• Tree topology and branch lengths
• nucleotide or amino acid frequencies
• Substitution model parameters
– transition/transversion ratio
– substitution matrix such as BLOSUM62
• Ancestral sequences
• We want to use the observed data to
determine the parameters
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Power of Bayesian methods
• Allow for more complex models of evolutionary
process
• Avoid assumption of constancy of molecular clock
along different branches
• Allow determination of branching times and rates
of evolution along different branches
• Calculations can be done with a Markov-ChainMonte-Carlo (MCMC) approach; this is an
efficient way of optimising search in parameter
space
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Suggested reading
Holder, M. & Lewis, P.O. (2003).
Phylogeny estimation: traditional and Bayesian
approaches.
Nature Reviews Genetics 4, 275--284.
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