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Introduction to Evolutionary Bioinformatics
David H. Ardell,Forskarassistent
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Sequences and substitution matrices.
Alignments: basic theory and practice.
Trees: basic theory and practice.
Population sequence data: theory and practice.
Lecture Outline:
Intro. to Sequence Evolution and Substitution Matrices
Part I: Theory
Homology, paralogy and orthology
Molecular clock
Divergence, saturation and evolutionary distance
Poisson correction
PAM and other substitution matrices
Markov and other assumptions of bioinformatics
Sequence compositions
Part II: Practice
Evolving sequences on a computer
Calculating evolutionary distances
Exploring Substitution matrices
Calculating evolutionary distance with substitution matrices
HOMOLOGY: descent from a common ancestor
(Darwin, 1859)
Original definition: "the same
organ in different animals
under every variety of form
and function." (Owen, 1843).
Homology need not imply
similarity of form nor function
because of divergence.
Similarity need not imply
homology because of
convergence.
Richard Owen
(1804-1892)
Homology applied to DNA sequences:
Ancestral sequence
GCCACTTTCGCGATCA
GCCACTTTCGCGATCG
GCCACTTTCGTGATCG
GCCACTTTCGCGATCA
GCCACTTTCGCGATTA
GCCACGTTCGTGATCG
GACAGTTTCGCGATTA
GCCACGTTCGCGATCG
GCCACGTTCGCGATCG
Homologous
sequences
GGCAGTTTTGCGATTA
GGCAGTTTCGCGATTT
GGCAGTTTCGCGATTT
GCCACGTTCGCGATCG
GCCACGTTCGCGATCG
| ||
|||||||
GGCAGTCTCGCGATTT
Homologous
residues
Sequence homologs can be paralogs or orthologs.
Paralogs are members of
a “gene family.” They
arise by gene duplication.
Ex:
-hemoglobin and
-hemoglobin are
paralogs
Hardison PNAS 2001 98: 1327-1329
Paralogs arise by gene or chromosome duplications
Paralogs are members of
a “gene family.” They
arise by gene duplication.
Ex:
-hemoglobin and
-hemoglobin are
paralogs
they arose by tandem
gene duplication - a
chunks of chromosomes
duplicating locally
Hardison PNAS 2001 98: 1327-1329
Orthologs arise by speciation
(pungdjur)
Orthologs duplicate by speciation. In
practice we assume they retain the same
have function.
Ex: -hemoglobin in eutherians and
marsupials
Hardison PNAS 2001 98: 1327-1329
Evolution of the Hemoglobin Gene Family
(pungdjur)
Speciation between
marsupials and eutherians
Orthologs duplicate by speciation. In
practice we assume they are the “same”
gene in the family (have same function).
Ex: -hemoglobin in eutherians and
marsupials
Hardison PNAS 2001 98: 1327-1329
Paralogs also arise through
whole chromosome duplications
(polyploidizations).
Ancient polyploidization event
Hardison PNAS 2001 98: 1327-1329
Functional divergence can occur in orthologs
change in function
Hardison PNAS 2001 98: 1327-1329
Paralogs can be lost in some species
Hardison PNAS 2001 98: 1327-1329
Orthology is rarer than paralogy
ORTHOLOGY:
Homology by
speciation,
same function
PARALOGY:
Homology by
duplication
Hardison PNAS 2001 98: 1327-1329
Hemoglobins and other gene families evolve by
speciation, duplication, loss and divergence
duplications
losses
speciations
speciations
duplication
Hardison PNAS 2001 98: 1327-1329
The “Molecular Clock:” orthologs evolve at typical
constant rates
Emile Zuckerkandl and Linus Pauling (1965)
"Evolutionary Divergence and Convergence in Proteins,"
in Evolving Genes and Proteins, eds. V. Bryson and H. Vogel (New York: Academic Press, 1965). pp. 97-166.
• Divergence of -, -, and -Hemoglobin are about the
same regardless of which species they are in.
• Duplications preceded the divergence of mammals.
% amino acid differences
“There may thus exist a Molecular Evolutionary Clock”
Zuckerkandl & Pauling (1965)
Divergence between  and  or 
Divergence between , and 
Approx. duplication dates (mya)
from vertebrate fossil records
Different proteins “tick” at different rates
PBS Evolution Library (http://www.pbs.org/wgbh/evolution/library/)
Also, different parts of the same gene or protein evolve
at different rates
Ex: Globular proteins evolve faster at their outsides!
The molecular clock also works for DNA
Ex: influenza virus genes
Gojobori et al. 1990 PNAS 87 10015-10018
% amino acid differences
BUT: the Molecular Clock slows down after a long time
because of SATURATION (double mutations).
Approx. duplication dates (mya)
from vertebrate fossil records
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Multiple mutations
hit the same site
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Multiple mutations
hit the same site
3 mutations,
2 differences
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Multiple mutations
hit the same site
Back mutations
undo earlier
mutations
3 mutations,
2 differences
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Multiple mutations
hit the same site
Back mutations
undo earlier
mutations
3 mutations,
2 differences
4 mutations,
1 difference
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Multiple mutations
hit the same site
Back mutations
undo earlier
mutations
Parallel mutations
hide divergence
3 mutations,
2 differences
4 mutations,
1 difference
Ex: why Percent Identity (%ID) underestimates divergence
The more sequences diverge, the more substitutions we miss.
ANCESTOR
Multiple mutations
hit the same site
Back mutations
undo earlier
mutations
Parallel mutations
hide divergence
3 mutations,
2 differences
4 mutations,
1 difference
6 mutations,
1 difference
The more distantly related two sequences are, the more
we must correct for hidden mutations
Two strategies:
 Poisson correction
 Quick and dirty, can be computed by hand
 Neglects back and parallel substitutions. These are rare at low
divergence, so works better for closer-related sequences.
 Includes no information about how proteins or DNA evolve. All types of
changes are equally likely.
 Substitution matrices




Complex to compute
Accounts for back and parallel substitutions,more accurate
A complete model of evolution about how sequences evolve
Can be used for making alignments, database searches and trees
The Poisson Correction
Imagine mutations “raining down” on sequences:
The Poisson Correction
Imagine mutations “raining down” on sequences:
The Poisson Correction
Imagine mutations “raining down” on sequences:
The Poisson Correction
Imagine mutations “raining down” on sequences:
The Poisson Correction
Imagine mutations “raining down” on sequences:
1.
Want to estimate avg. evolutionary distance  = r t (# mutations
per sequence length in sites) from %ID = 100 x (p/N).
The Poisson Correction
Imagine mutations “raining down” on sequences:
1.
2.
Want to estimate avg. evolutionary distance  = r t (# mutations
per sequence length in sites) from %ID = 100 x (p/N).
Assume mutations occur independently in space and time.
The Poisson Correction
Imagine mutations “raining down” on sequences:
1.
2.
3.
Want to estimate avg. evolutionary distance  = r t (# mutations
per sequence length in sites) from %ID = 100 x (p/N).
Assume mutations occur independently in space and time.
Normalize sequence to length 1. Then each site has probability
/N of mutating at distance . The average fraction of sites not
mutated at this distance is then: (1 - /N)N ≈ e– ( as N   ).
The Poisson Correction
Imagine mutations “raining down” on sequences:
1.
2.
3.
4.
Want to estimate avg. evolutionary distance  = r t (# mutations
per sequence length in sites) from %ID = 100 x (p/N).
Assume mutations occur independently in space and time.
Normalize sequence to length 1. Then each site has probability
/N of mutating at distance . The average fraction of sites not
mutated at this distance is then: (1 - /N)N ≈ e– ( as N   ).
Therefore, if we see (p/N) sites not mutated and assume no backor parallel mutations, we can estimate distance  = – ln (p/N).
The Poisson Correction
Imagine mutations “raining down” on sequences:
1.
2.
3.
4.
5.
Want to estimate avg. evolutionary distance  = r t (# mutations
per sequence length in sites) from %ID = 100 x (p/N).
Assume mutations occur independently in space and time.
Normalize sequence to length 1. Then each site has probability
/N of mutating at distance . The average fraction of sites not
mutated at this distance is then: (1 - /N)N ≈ e– ( as N   ).
Therefore, if we see (p/N) sites not mutated and assume no backor parallel mutations, we can estimate distance  = – ln (p/N).
Ex: %ID of 37.8 ≈ 100 x e–1 implies  = -ln( 1/e) = 1. About as
many mutations as the length of the sequence have occurred.
Substitutions per site
Poisson-Corrected Evolutionary Distance vs. %ID
37%ID = 1.0
61%ID = 0.5
%ID
Substitutions per site
Poisson-Corrected Evolutionary Distance vs. %ID
Something wrong here though:
Real proteins don’t evolve less
than about 5% ID, and they do it
much slower than this.
37%ID = 1.0
61%ID = 0.5
%ID
For most bioinformatics work we need something more
sophisticated… substitution matrices.
 The Poisson correction…
 … neglects back and parallel substitutions: %ID goes falsely to
zero at large evolutionary divergences.
 … uses information only from sites that are identical. Throws out
information from the mutated sites.
 … includes no information about which kinds of changes are more
likely to occur than other kinds of changes (Ex: hydrophobic
amino acids, transition bias in DNA mutation).
 …provides only a “back-of-the envelope” model of evolution.
 Substitution matrices…
 …give a complete accounting of all possible mutational paths is
made.
 …use information from all sites, changed or unchanged.
 …provide a superior model of sequence evolution.
 …can be used to make alignments, search databases (GenBank)
for homologs, and make phylogenetic trees.
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
t=2
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
t=2
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
t=2
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
t=2
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
t=2
t=3
Q: What is a “substitution?”
A: A substitution is the fixation of a mutation in a population. It
has been “accepted” by natural selection.
Population of 5
individuals at
generation t = 0
t = 2: 2 mutations occur
t=3
t = 4: 1 substitution occurs
HINT: Sequence
differences between
species are often
assumed to be
substitutions (fixed
differences).
Species 1
Ancestor
Species 2
Margaret Oakley Dayhoff (1925-1983)
Inventor of PAM Amino Acid
Substitution Matrices
Basic ideas:
1. Collect a big dataset of closely related proteins.
2. Count up amino acid changes and the total composition of
amino acids in the dataset.
3. Calculate from this the transition probabilities for any amino
acid to change into any other amino acid after 1% sequence
divergence.
4. This defines the PAM1 matrix (“Point Accepted Mutation,” where
“accepted” means “by natural selection”).
5. Assume that the transition probabilities after N% sequence
divergence is given by “powering up” the PAM1 matrix.
Ex: PAM250 = PAM1250
Q: What does PAM250 – 250% change to a protein –
mean?
A: just over 18%ID
Assumptions of PAM Substitution Matrices
1. Site-Independence: Probability of mutation/substitution
at a site is independent of which amino acids/bases
occupy all other sites in any protein in the organism.
Assumptions of PAM Substitution Matrices
1. Site-Independence: Probability of mutation/substitution
at a site is independent of which amino acids/bases
occupy all other sites in any protein in the organism.
2. Memorylessness: Probability of mutation/substitution at
a site depends only on its present state, not on its
history.
Assumptions of PAM Substitution Matrices
1. Site-Independence: Probability of mutation/substitution
at a site is independent of which amino acids/bases
occupy all other sites in any protein in the organism.
2. Memorylessness: Probability of mutation/substitution at
a site depends only on its present state, not on its
history.
3. Stationarity: Sequence composition is the same or will
become the same as in the alignments that were used
to make the matrix.
Assumptions of PAM Substitution Matrices
1. Site-Independence: Probability of mutation/substitution
at a site is independent of which amino acids/bases
occupy all other sites in any protein in the organism.
2. Memorylessness: Probability of mutation/substitution at
a site depends only on its present state, not on its
history.
3. Stationarity: Sequence composition is the same or will
become the same as in the alignments that were used
to make the matrix.
4. Markov Assumption: The probabilities of change remain
the same throughout history.
Markov models of DNA evolution:
The Jukes-Cantor model

G


    


C
T

A
Markov models of DNA evolution:
The Jukes-Cantor model
A
 
C






“Pools”
G
  
T
Markov models of DNA evolution:
The Jukes-Cantor model

G


    


C
T

A
“Flows out”
Markov models of DNA evolution:
The Jukes-Cantor model

G


    


C
T

A
“Flows in”
Markov models of DNA evolution:
The Jukes-Cantor model

G


    


C
T

A
Because of symmetry, sequences evolve uniform base
composition (25%A, 25%G, 25%C, 25%T).
Markov models of DNA evolution:
The Kimura 2-parameter model
A
 

 C







G
  
T
Markov models of DNA evolution:
The Kimura 2-parameter model
A
 

 C







G
  
T
Transitions
Transversions
Markov models of DNA evolution:
The Kimura 2-parameter model
A
 

 C







G
  
T
Markov models of DNA evolution:
The Kimura 2-parameter model
A
 

 C







G
  
T
Jones, Taylor, Thornton (1992) “JTT” MDM-1 score matrix
A R N D C Q E G H I L K M F P S T WYV
JTT MDM-15 Score Matrix, 85% expected ID between proteins
A R N D C Q E G H I L K M F P S T WYV
JTT MDM-120 Score Matrix, 36% expected ID
A R N D C Q E G H I L K M F P S T WYV
Score Matrices vs. Substitution Matrices
To make evolutionary matrices, calculate avg. composition Ma = p(a)
and transition probabilities Mab = p(b|a*)p(a*|a) that an amino
acid/base mutates to b and substitutes in the population.
Score Matrices vs. Substitution Matrices
To make evolutionary matrices, calculate avg. composition Ma = p(a)
and transition probabilities Mab = p(b|a*)p(a*|a) that an amino
acid/base mutates to b and substitutes in the population.
Substitution matrices are made only from the transition probabilities
Mab. Because Mab  Mba, they are not symmetric about the
diagonal.
Score Matrices vs. Substitution Matrices
To make evolutionary matrices, calculate avg. composition Ma = p(a)
and transition probabilities Mab = p(b|a*)p(a*|a) that an amino
acid/base mutates to b and substitutes in the population.
Substitution matrices are made only from the transition probabilities
Mab. Because Mab  Mba, they are not symmetric about the
diagonal.
Score Matrices (or “MDMs”) are made from both Mab and Ma. They
give the log-odds of two residues in a sequence being biologically
homologous relative to chance.
Score Matrices vs. Substitution Matrices
To make evolutionary matrices, calculate avg. composition Ma = p(a)
and transition probabilities Mab = p(b|a*)p(a*|a) that an amino
acid/base mutates to b and substitutes in the population.
Substitution matrices are made only from the transition probabilities
Mab. Because Mab  Mba, they are not symmetric about the
diagonal.
Score Matrices (or “MDMs”) are made from both Mab and Ma. They
give the log-odds of two residues in a sequence being biologically
homologous relative to chance.
Score matrices are symmetrical:
Sab = log (Mab / Mb) = log (Mba / Ma) = Sba.
Score Matrices vs. Substitution Matrices
To make evolutionary matrices, calculate avg. composition Ma = p(a)
and transition probabilities Mab = p(b|a*)p(a*|a) that an amino
acid/base mutates to b and substitutes in the population.
Substitution matrices are made only from the transition probabilities
Mab. Because Mab  Mba, they are not symmetric about the
diagonal.
Score Matrices (or “MDMs”) are made from both Mab and Ma. They
give the log-odds of two residues in a sequence being biologically
homologous relative to chance.
Score matrices are symmetrical:
Sab = log (Mab / Mb) = log (Mba / Ma) = Sba.
Score matrices are used for many bioinformatic applications we will
soon cover such as alignment and database searching.
Q: Score matrices are “log-odds” matrices.
What are log-odds?
Odds are ratios of probabilities. Usually written like “4:1”
(said like “4 to 1”) they tell you the relative chance of
two events.
Score Matrices are made from the odds-ratio
p(AB):p(A)p(B) that two amino acids or bases A and B
are likely to be found in homologous positions in a
sequence p(AB), relative to the chance of picking the
pair at random p(A)p(B)
Log-odds L are made by taking the log of the odds-ratio:
log p(AB):p(A)p(B) = log p(AB) – log p(A) – log p(B)
they are more convenient to compute with and
understand: if L > 0, A and B more likely to occur by
evolution than by chance and vice versa.
Other Amino Acid Substitution/Score Matrices
Some matrices are updates of the original Dayhoff method with
more data or some technical refinements
Ex: Jones, Taylor, Thornton 1992 (JTT)
Gonnet, Benner and Cohen
Some matrices are for specialized kinds or parts of proteins.
Ex: JTT transmembrane protein matrix
Goldstein secondary structure matrices
Some matrices have different assumptions
Ex: BLOSSUM: removes Markov assumption. They make a
series of matrices from alignments at different %IDs.
OBS: BLOSSUMs are labeled by expected %ID, so while
PAM250 > PAM100,
BLOSSUM30 > BLOSSUM62 !!
One last point: evolutionary distance between two
sequences:
Root
Seq 1
Seq 2
One last point: evolutionary distance between two
sequences:
Root
Seq 2
Seq 1
Root
Seq 1
Seq 2