Introduction to Evolutionary Bioinformatics David H. Ardell,Forskarassistent.

Introduction to Evolutionary Bioinformatics

David H. Ardell,Forskarassistent

Outline, v. 4

Wed. Jan. 26 Sequences and substitution matrices.

Thurs. Jan. 27 Alignments: basic theory and practice.

Fri. Jan. 28 Trees: basic theory and practice.

Mon. Jan. 31 Population sequence data: theory and practice.

Lecture Outline: Intro. to Sequence Evolution and Substitution Matrices

Part I: TheoryHomology, 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: PracticeEvolving sequences on a computer

Calculating evolutionary distances

Exploring Substitution matrices

Calculating evolutionary distance with substitution matrices

Richard Owen (1804-1892)

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.

Homology applied to DNA sequences:

GCCACTTTCGCGATCA

GCCACTTTCGCGATCA

GCCACTTTCGCGATTA

GACAGTTTCGCGATTA

GGCAGTTTTGCGATTA

GGCAGTTTCGCGATTT

GCCACTTTCGCGATCG

GCCACTTTCGTGATCG

GCCACGTTCGTGATCG

GCCACGTTCGCGATCG

GCCACGTTCGCGATCG

GCCACGTTCGCGATCG| || |||||||GGCAGTCTCGCGATTT

Ancestral sequence

Homologous sequences

GCCACGTTCGCGATCG GGCAGTTTCGCGATTT

Homologous residues

Hardison PNAS 2001 98: 1327-1329

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

they arose by tandem gene duplication - a chunks of chromosomes duplicating locally

Paralogs are members of a “gene family.” They arise by gene duplication. Ex: -hemoglobin and-hemoglobin are paralogs

Hardison PNAS 2001 98: 1327-1329

Orthologs arise by speciation

Orthologs duplicate by speciation. In practice we assume they retain the same have function.Ex: -hemoglobin in eutherians and marsupials

(pungdjur)

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

Ancient polyploidization event

Paralogs also arise through whole chromosome duplications (polyploidizations).

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 byspeciation, 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

speciations

speciations

duplication

losses

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.

The “Molecular Clock:” orthologs evolve at typical constant rates

“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

% a

min

o a

cid d

iffere

nce

s

PBS Evolution Library (http://www.pbs.org/wgbh/evolution/library/)

Different proteins “tick” at different rates

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 DNAEx: influenza virus genes

Gojobori et al. 1990 PNAS 87 10015-10018

BUT: the Molecular Clock slows down after a long time because of SATURATION (double mutations).

Approx. duplication dates (mya) from vertebrate fossil records

% a

min

o a

cid d

iffere

nce

s

Ex: why Percent Identity (%ID) underestimates divergence

The more sequences diverge, the more substitutions we miss.

ANCESTOR



Multiple mutations hit the same site

ANCESTOR




ANCESTOR

3 mutations,2 differences




Back mutations undo earlier mutations

ANCESTOR






ANCESTOR

4 mutations,1 difference






Parallel mutations hide divergence

ANCESTOR



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:


1. Want to estimate avg. evolutionary distance = r t (# mutations per sequence length in sites) from %ID = 100 x (p/N).




2. Assume mutations occur independently in space and time.





3. 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 ).





3. 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 ).

4. Therefore, if we see (p/N) sites not mutated and assume no back- or parallel mutations, we can estimate distance = – ln (p/N).




2. Assume mutations occur independently in space and time.3. 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 ).

4. Therefore, if we see (p/N) sites not mutated and assume no back- or parallel mutations, we can estimate distance = – ln (p/N).

5. 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.


Poisson-Corrected Evolutionary Distance vs. %ID

%ID

Su

bst

itu

tion

s p

er

site

61%ID = 0.5

37%ID = 1.0

Poisson-Corrected Evolutionary Distance vs. %ID

%ID

Su

bst

itu

tion

s p

er

site

61%ID = 0.5

37%ID = 1.0

Something wrong here though:Real proteins don’t evolve less than about 5% ID, and they do it much slower than this.

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




t = 2




t = 2

t = 3




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 2Species 1

Ancestor

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.





2. Memorylessness: Probability of mutation/substitution at

a site depends only on its present state, not on its

history.







history.

3. Stationarity: Sequence composition is the same or will

become the same as in the alignments that were used

to make the matrix.







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.

A G

TC

Markov models of DNA evolution: The Jukes-Cantor model

A G

TC


“Pools”

A G

TC


“Flows out”

A G

TC


“Flows in”

A G

TC


Because of symmetry, sequences evolve uniform base composition (25%A, 25%G, 25%C, 25%T).

A G

TC

€

€

Markov models of DNA evolution: The Kimura 2-parameter model

€

€

A G

TC

€

€


€

€

Transitions

Transversions

A G

TC

€

€


€

€

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 W Y V

JTT MDM-15 Score Matrix, 85% expected ID between proteins


JTT MDM-120 Score Matrix, 36% expected ID


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 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 refinementsEx: Jones, Taylor, Thornton 1992 (JTT) Gonnet, Benner and Cohen

Some matrices are for specialized kinds or parts of proteins. Ex: JTT transmembrane protein matrixGoldstein secondary structure matrices

Some matrices have different assumptionsEx: 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 1 Seq 2

Seq 2Seq 1

Root

Introduction to Evolutionary Bioinformatics David H. Ardell,Forskarassistent.

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Transcript of Introduction to Evolutionary Bioinformatics David H. Ardell,Forskarassistent.