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Multiple Alignment
Anders Gorm PedersenAnders Gorm Pedersen
Molecular Evolution GroupMolecular Evolution Group
Center for Biological Sequence AnalysisCenter for Biological Sequence Analysis
[email protected]@cbs.dtu.dk
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Refresher: pairwise alignments
43.2% identity; Global alignment score: 374
10 20 30 40 50 alpha V-LSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHF-DLS-----HGSA : :.: .:. : : :::: .. : :.::: :... .: :. .: : ::: :. beta VHLTPEEKSAVTALWGKV--NVDEVGGEALGRLLVVYPWTQRFFESFGDLSTPDAVMGNP 10 20 30 40 50
60 70 80 90 100 110 alpha QVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHL .::.::::: :.....::.:.. .....::.:: ::.::: ::.::.. :. .:: :.beta KVKAHGKKVLGAFSDGLAHLDNLKGTFATLSELHCDKLHVDPENFRLLGNVLVCVLAHHF 60 70 80 90 100 110
120 130 140 alpha PAEFTPAVHASLDKFLASVSTVLTSKYR :::: :.:. .: .:.:...:. ::.beta GKEFTPPVQAAYQKVVAGVANALAHKYH 120 130 140
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Refresher: pairwise alignments
• Alignment score is Alignment score is calculated from calculated from substitution matrixsubstitution matrix
• Identities on diagonal Identities on diagonal have high scoreshave high scores
• Similar amino acids have Similar amino acids have high scoreshigh scores
• Dissimilar amino acids Dissimilar amino acids have low (negative) have low (negative) scoresscores
• Gaps penalized by gap-Gaps penalized by gap-opening + gap elongationopening + gap elongation
K L A A S V I L S D A L
K L A A - - - - S D A L
-10 + 3 x (-1)=-13
A 5R -2 7N -1 -1 7D -2 -2 2 8C -1 -4 -2 -4 13Q -1 1 0 0 -3 7E -1 0 0 2 -3 2 6G 0 -3 0 -1 -3 -2 -3 8...
A R N D C Q E G ...
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Refresher: pairwise alignments
The number of possible pairwise alignments increases explosively with the length of the sequences:
Two protein sequences of length 100 amino acids can be aligned in approximately 1060 different ways
1060 bottles of beer would fill up our entire galaxy
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Refresher: pairwise alignments
• Solution: Solution:
dynamic programmingdynamic programming
• Essentially:Essentially:
the best path through the best path through any grid point in the any grid point in the alignment matrix must alignment matrix must originate from one of originate from one of three previous pointsthree previous points
• Far fewer computationsFar fewer computations
• Best alignment Best alignment guaranteed to be foundguaranteed to be found
T C G C A
T
C
C
A
x
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Refresher: pairwise alignments
• Most used substitution matrices are themselves Most used substitution matrices are themselves derived empirically from simple multiple alignmentsderived empirically from simple multiple alignments
Multiple alignment
A/A 2.15%A/C 0.03%A/D 0.07%...
Calculatesubstitutionfrequencies
Score(A/C) = logFreq(A/C),observedFreq(A/C),expected
Convertto scores
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Multiple alignment
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Multiple alignments: what use are they?
• Starting point for studies of Starting point for studies of molecular evolutionmolecular evolution
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Multiple alignments: what use are they?
• Characterization of protein families:Characterization of protein families:– Identification of conserved (functionally important) sequence Identification of conserved (functionally important) sequence
regionsregions– Prediction of structural features (disulfide bonds, amphipathic Prediction of structural features (disulfide bonds, amphipathic
alpha-helices, surface loops, etc.)alpha-helices, surface loops, etc.)
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Scoring a multiple alignment:the “sum of pairs” score
...A...
...A...
...S...
...T...
One column from alignment
AA: 4, AS: 1, AT:0AS: 1, AT: 0ST: 1
Score: 4+1+0+1+0+1 = 7
In theory, it is possible to define an alignment score for multiple alignments (there are many alternative scoring systems)
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Multiple alignment: dynamic programming is only feasible for
very small data sets • In theory, optimal multiple In theory, optimal multiple
alignment can be found by dynamic alignment can be found by dynamic programming using a matrix with programming using a matrix with more dimensions (one dimension more dimensions (one dimension per sequence)per sequence)
• BUT even with dynamic BUT even with dynamic programming finding the optimal programming finding the optimal alignment very quickly becomes alignment very quickly becomes impossible due to the astronomical impossible due to the astronomical number of computationsnumber of computations
• Full dynamic programming only Full dynamic programming only possible for up to about 4-5 protein possible for up to about 4-5 protein sequences of average length sequences of average length
• Even with heuristics, not feasible for Even with heuristics, not feasible for more than 7-8 protein sequencesmore than 7-8 protein sequences
• Never used in practiceNever used in practice
Dynamic programming matrix for 3 sequences
For 3 sequences, optimal path must comefrom one of 7 previous points
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Multiple alignment: an approximate solution
• Progressive alignment (ClustalX and other Progressive alignment (ClustalX and other programs):programs):
1.1. Perform all Perform all pairwisepairwise alignments; keep track of alignments; keep track of sequence similarities between all pairs of sequences sequence similarities between all pairs of sequences (construct “distance matrix”)(construct “distance matrix”)
2.2. Align the most similar pair of sequencesAlign the most similar pair of sequences
3.3. Progressively add sequences to the (constantly Progressively add sequences to the (constantly growing) multiple alignment in order of decreasing growing) multiple alignment in order of decreasing similaritysimilarity..
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Progressive alignment: details
1) Perform all pairwise alignments, note pairwise distances (construct “distance matrix”)
2) Construct pseudo-phylogenetic tree from pairwise distances
S1S2S3S4
6 pairwisealignments
S1 S2 S3 S4S1S2 3S3 1 3S4 3 2 3
S1 S3 S4 S2
S1 S2 S3 S4S1S2 3S3 1 3S4 3 2 3
“Guide tree”
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Progressive alignment: details
3) Use tree as guide for multiple alignment:a) Align most similar pair of sequences using dynamic programming
b) Align next most similar pair
c) Align alignments using dynamic programming - preserve gaps
S1 S3 S4 S2
S1
S3
S2
S4
S1
S3
S2
S4New gap to optimize alignmentof (S2,S4) with (S1,S3)
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Aligning profiles
S1
S3
S2
S4
+
S1
S3
S2
S4New gap to optimize alignmentof (S2,S4) with (S1,S3)
Aligning alignments: each alignment treated as a single sequence (a profile)
Full dynamic programmingon two profiles
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Scoring profile alignments
...A...
...S...
...S...
...T...
+
One column from alignment
AS: 1, AT:0
SS: 4, ST:1
Score: 1+0+4+1 = 1.54
Compare each residue in one profile to all residues in second profile. Score is average of all comparisons.
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Additional ClustalX heuristics
• Sequence weighting:Sequence weighting:– scores from similar groups of sequences are down-weightedscores from similar groups of sequences are down-weighted
• Variable substitution matrices:Variable substitution matrices:– during alignment ClustalX uses different substitution matrices during alignment ClustalX uses different substitution matrices
depending on how similar the sequences/profiles aredepending on how similar the sequences/profiles are
• Variable gap penalties:Variable gap penalties: gap penalties depend on substitution matrixgap penalties depend on substitution matrix gap penalties depend on similarity of sequencesgap penalties depend on similarity of sequences reduced gap penalties at existing gapsreduced gap penalties at existing gaps increased gap penalties CLOSE to existing gapsincreased gap penalties CLOSE to existing gaps reduced gap penalties in hydrophilic stretches (presumed reduced gap penalties in hydrophilic stretches (presumed
surface loop)surface loop) residue-specific gap penaltiesresidue-specific gap penalties and more...and more...
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Other multiple alignment programs
ClustalW / ClustalX
pileup
multalign
multal
saga
hmmt
DIALIGN
SBpima
MLpima
T-Coffee
...
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Other multiple alignment programs
ClustalW / ClustalX
pileup
multalign
multal
saga
hmmt
DIALIGN
SBpima
MLpima
T-Coffee
...
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Global methods (e.g., ClustalX) get into trouble when data is
not globally related!!!
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Global methods (e.g., ClustalX) get into trouble when data is
not globally related!!!
Clustalx
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Global methods (e.g., ClustalX) get into trouble when data is
not globally related!!!
Clustalx
Possible solutions:(1) Cut out conserved regions of interest and THEN align them (2) Use method that deals with local similarity (e.g. DIALIGN)
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the Theory of Evolution
Anders Gorm PedersenAnders Gorm Pedersen
Molecular Evolution GroupMolecular Evolution Group
Center for Biological Sequence AnalysisCenter for Biological Sequence Analysis
[email protected]@cbs.dtu.dk
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Classification: Linnaeus
Carl LinnaeusCarl Linnaeus1707-17781707-1778
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Classification: Linnaeus
• Hierarchical systemHierarchical system
– KingdomKingdom(Rige)(Rige)– PhylumPhylum (Række)(Række)– ClassClass (Klasse)(Klasse)– OrderOrder (Orden)(Orden)– FamilyFamily (Familie)(Familie)– GenusGenus (Slægt)(Slægt)– SpeciesSpecies (Art)(Art)
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Classification depicted as a tree
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Classification depicted as a tree
SpeciesSpecies GenusGenus FamilyFamily OrderOrder ClassClass
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Theory of evolution
Charles DarwinCharles Darwin1809-18821809-1882
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Phylogenetic basis of systematics
• LinnaeusLinnaeus: : Ordering principle is God.Ordering principle is God.
• DarwinDarwin: : Ordering principle is shared Ordering principle is shared descent from common ancestors.descent from common ancestors.
• Today, systematics is explicitly Today, systematics is explicitly based on phylogeny.based on phylogeny.
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Darwin’s four postulates
I.I. More young are produced each generation than can More young are produced each generation than can survive to reproduce.survive to reproduce.
II.II. Individuals in a population vary in their characteristics.Individuals in a population vary in their characteristics.III.III. Some differences among individuals are based on genetic Some differences among individuals are based on genetic
differences.differences.IV.IV. Individuals with favorable characteristics have higher Individuals with favorable characteristics have higher
rates of survival and reproduction.rates of survival and reproduction.
Evolution by means of natural selectionEvolution by means of natural selection Presence of ”design-like” features in organisms:Presence of ”design-like” features in organisms:
quite often features are there “for a reason”quite often features are there “for a reason”
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Theory of evolution as the basis of biological understanding
”Nothing in biology makes sense, except in the light of evolution.
Without that light it becomes a pile of sundry facts - some of them interesting or curious but making no meaningful picture as a whole”
T. Dobzhansky
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Distance Matrix Methods
Anders Gorm PedersenAnders Gorm Pedersen
Molecular Evolution GroupMolecular Evolution Group
Center for Biological Sequence AnalysisCenter for Biological Sequence Analysis
Technical University of DenmarkTechnical University of Denmark
[email protected]@cbs.dtu.dk
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Trees: terminology
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Trees: terminology
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Trees: representations
Three different representations of the same tree
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Trees: rooted vs. unrooted
• A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.
• A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).
• In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)
Early Late
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Trees: rooted vs. unrooted
• A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.
• A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).
• In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)
Early Late
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Trees: rooted vs. unrooted
• A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.
• A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).
• In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)
Early Late
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Trees: rooted vs. unrooted
• In unrooted trees there is no directionality: we do not know In unrooted trees there is no directionality: we do not know if a node is earlier or later than another nodeif a node is earlier or later than another node
• Distance along branches directly represents node distanceDistance along branches directly represents node distance
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Trees: rooted vs. unrooted
• In unrooted trees there is no directionality: we do not know In unrooted trees there is no directionality: we do not know if a node is earlier or later than another nodeif a node is earlier or later than another node
• Distance along branches directly represents node distanceDistance along branches directly represents node distance
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Reconstructing a tree using non-contemporaneous data
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Reconstructing a tree using present-day data
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Data: molecular phylogeny
• DNA sequencesDNA sequences– genomic DNAgenomic DNA– mitochondrial DNAmitochondrial DNA– chloroplast DNAchloroplast DNA
• Protein sequencesProtein sequences
• Restriction site polymorphismsRestriction site polymorphisms
• DNA/DNA hybridizationDNA/DNA hybridization
• Immunological cross-reactionImmunological cross-reaction
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Morphology vs. molecular data
African white-backed vulture(old world vulture)
Andean condor (new world vulture)
New and old world vultures seem to be closely related based on morphology.
Molecular data indicates that old world vultures are related to birds of prey (falcons, hawks, etc.) while new world vultures are more closely related to storks
Similar features presumably the result of convergent evolution
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Molecular data: single-celled organisms
Molecular data useful for analyzing single-celled organisms (which have only few prominent morphological features).
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Distance Matrix Methods
1.1. Construct multiple Construct multiple alignment of sequencesalignment of sequences
2.2. Construct table listing all Construct table listing all pairwise differences pairwise differences (distance matrix)(distance matrix)
3.3. Construct tree from Construct tree from pairwise distancespairwise distances
Gorilla : ACGTCGTAHuman : ACGTTCCTChimpanzee: ACGTTTCG
GoGo HuHu ChCh
GoGo -- 44 44
HuHu -- 22
ChCh --
Go
Hu
Ch
2
11
1
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Finding Optimal Branch Lengths
SS11 SS22 SS33 SS44
SS11 -- DD1212 DD1313 DD1414
SS22 -- DD2323 DD2424
SS33 -- DD3434
SS44 --Observed distance
S1
S3
S2
S4
a
b
c
d e
Distance along tree
D12 d12 = a + b + cD13 d13 = a + dD14 d14 = a + b + eD23 d23 = d + b + cD24 d24 = c + eD34 d34 = d + b + e
Goal:
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Optimal Branch Lengths: Least Squares
• Fit between given tree Fit between given tree and observed distances and observed distances can be expressed as “sum can be expressed as “sum of squared differences”:of squared differences”:
Q = Q = (D(Dijij - d - dijij))22
• Find branch lengths that Find branch lengths that minimize Q - this is the minimize Q - this is the optimal set of branch optimal set of branch lengths for this tree.lengths for this tree.
S1
S3
S2
S4
a
b
c
d e
Distance along tree
D12 d12 = a + b + cD13 d13 = a + dD14 d14 = a + b + eD23 d23 = d + b + cD24 d24 = c + eD34 d34 = d + b + e
Goal:
j>i
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Least Squares Optimality Criterion
• Search through all (or many) tree topologiesSearch through all (or many) tree topologies
• For each investigated tree, find best branch For each investigated tree, find best branch lengths using least squares criterionlengths using least squares criterion
• Among all investigated trees, the best tree is the Among all investigated trees, the best tree is the one with the smallest sum of squared errors. one with the smallest sum of squared errors.
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Exhaustive search impossible for large data sets
No. No. taxataxa
No. treesNo. trees
33 11
44 33
55 1515
66 105105
77 945945
88 10,39510,395
99 135,135135,135
1010 2,027,0252,027,025
1111 34,459,42534,459,425
1212 654,729,075654,729,075
1313 13,749,310,57513,749,310,575
1414 316,234,143,225316,234,143,225
1515 7,905,853,580,6257,905,853,580,625
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Heuristic search
1.1. Construct initial tree; determine sum of squaresConstruct initial tree; determine sum of squares
2.2. Construct set of “neighboring trees” by making small Construct set of “neighboring trees” by making small rearrangements of initial tree; determine sum of squares for rearrangements of initial tree; determine sum of squares for each neighboreach neighbor
3.3. If any of the neighboring trees are better than the initial tree, If any of the neighboring trees are better than the initial tree, then select it/them and use as starting point for new round of then select it/them and use as starting point for new round of rearrangements. (Possibly several neighbors are equally good)rearrangements. (Possibly several neighbors are equally good)
4.4. Repeat steps 2+3 until you have found a tree that is better Repeat steps 2+3 until you have found a tree that is better than all of its neighbors.than all of its neighbors.
5.5. This tree is a “local optimum” (not necessarily a global This tree is a “local optimum” (not necessarily a global optimum!) optimum!)
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Clustering Algorithms
• Starting point: Distance matrixStarting point: Distance matrix
• Cluster least different pair of sequences:Cluster least different pair of sequences:– Tree: pair connected to common ancestral node, compute branch Tree: pair connected to common ancestral node, compute branch
lengths from ancestral node to both descendantslengths from ancestral node to both descendants
– Distance matrix: combine two entries into one. Compute new Distance matrix: combine two entries into one. Compute new
distance matrix, by finding distance from new node to all other distance matrix, by finding distance from new node to all other nodesnodes
• Repeat until all nodes are linkedRepeat until all nodes are linked
• Results in only one tree, there is no measure of tree-Results in only one tree, there is no measure of tree-goodness.goodness.
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Neighbor Joining Algorithm
• For each tip compute For each tip compute uuii = = jj DDijij/(n-2)/(n-2)
(this is essentially the average distance to all other tips, except the (this is essentially the average distance to all other tips, except the denominator is n-2 instead of n)denominator is n-2 instead of n)
• Find the pair of tips, i and j, where Find the pair of tips, i and j, where DDijij-u-uii-u-ujj is smallest is smallest
• Connect the tips i and j, forming a new ancestral node. The branch Connect the tips i and j, forming a new ancestral node. The branch lengths from the ancestral node to i and j are:lengths from the ancestral node to i and j are:
vvii = 0.5 D = 0.5 Dijij + 0.5 (u + 0.5 (uii-u-ujj))
vvjj = 0.5 D = 0.5 Dijij + 0.5 (u + 0.5 (ujj-u-uii))
• Update the distance matrix: Compute distance between new node and Update the distance matrix: Compute distance between new node and each remaining tip as follows:each remaining tip as follows:
DDij,kij,k = (D = (Dikik+D+Djkjk-D-Dijij)/2)/2
• Replace tips i and j by the new node which is now treated as a tipReplace tips i and j by the new node which is now treated as a tip
• Repeat until only two nodes remain.Repeat until only two nodes remain.
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Superimposed Substitutions
• Actual number of Actual number of
evolutionary events:evolutionary events: 55
• Observed number of Observed number of
differences:differences: 22
• Distance is (almost) always Distance is (almost) always underestimatedunderestimated
ACGGTGC
C T
GCGGTGA
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Model-based correction for superimposed substitutions
• Goal: try to infer the real number of evolutionary Goal: try to infer the real number of evolutionary events (the real distance) based onevents (the real distance) based on
1.1. Observed data (sequence alignment)Observed data (sequence alignment)
2.2. A model of how evolution occursA model of how evolution occurs
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Jukes and Cantor Model
• Four nucleotides assumed to Four nucleotides assumed to be equally frequent (f=0.25)be equally frequent (f=0.25)
• All 12 substitution rates All 12 substitution rates assumed to be equalassumed to be equal
• Under this model the Under this model the corrected distance is:corrected distance is:
DDJCJC = -0.75 x ln(1-1.33 x = -0.75 x ln(1-1.33 x DDOBSOBS))
• For instance:For instance:
DDOBSOBS=0.43 => =0.43 => DDJCJC=0.64=0.64
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A -3
C -3
G -3
T -3
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Other models of evolution