Phylogenetic inference

1
Phylogenetic inference
Minicourse Molecular Phylogenetics an update of
new methodological developments during the 48th
Annual Meeting of the Sociedade Brasileira de
Genetica, Aguas de Lindoia, Sao Paulo, Brasil
(Sept 2002)

• Many methods available, using different
  techniques, many software packages
• For molecular data, the trend is towards using
  methods based on explicit models based on
  realistic assumptions
• New improved methods and tests appear in the
  literature constantly

2
Phylogenetic inference

• This minicurso will review some of the widely
  used (traditional) approaches and introduce two
  recent developments
• Bayesian inference
• Genetic algorithms (MetaGA)
• Review
• algorithmic vs. optimality criteria approaches
  (parsimony, distance methods and ML)
• Tree searching (heuristic search)
• Models (using Modeltest to choose one)

3
Classification of phylogenetic methods
4
Distance and discrete data
5
Algorithms versus optimality criteria

• Phylogenetic inference is an estimation procedure
  (best estimate)
• Only have information about the contemporary
  molecules (and organisms)
• How do we choose a tree from the set of all
  possible trees?
• Two basic approaches
• Algorithmic just follow a sequence of steps
• Optimality criterion how to compare trees

6
Algorithmic methods

• Combine tree inference and the definition of a
  preferred tree into a single statement
• Include UPGMA and all forms of pair-group cluster
  analysis, and neighbor joining
• Computationally fast because they go straight to
  the final solution
• The task of finding an optimal tree can not be
  separated from that of evaluating a specific tree

7
Optimality criteria

• Two logical steps
• Define an optimality criterion (objective
  function for evaluating trees)
• Find the tree(s) with the best value for the
  objective function (may use algorithms)
• Evolutionary assumptions made in the first step
  are decoupled from the computations involved in
  the second step
• Price for logical clarity is that these methods
  can be very slow

8
Use of algorithms

• Different use in the two approaches
• In purely algorithmic methods, the algorithm
  defines the tree selection criterion and is
  fundamental
• In criterion-based methods, algorithms are merely
  tools used in evaluating and searching for
  optimal trees
• Reliability of the tree?

9
Optimality criteria

• Parsimony select the tree that minimizes the
  total tree length (number of steps or character
  transformations required to explain a given set
  of data)
• Some methods are based on models of evolutionary
  change assumptions are made explicit.
• Is parsimonys model-free nature an advantage
  or a disadvantage?
• Parsimony does make assumptions (consistency)

10
Optimality methods (cont.)

• Maximum likelihood evaluates the probability
  that a proposed model of evolution and the
  hypothesized history could give rise to the
  observed data (attempts to estimate the actual
  amount of change)
• Usually more consistent estimates with lower
  variance than other methods robust to violations
  of assumptions

11
Optimality criteria (cont.)

• Pairwise distance methods also minimize the
  effect of multiple hits when using appropriate
  model to estimate the true evolutionary distance
  between two sequences (less desirable than full
  ML)
• Additive and ultrametric distances can be fitted
  to a tree such that all pairwise distances are
  equal to the sum of the branches along the path
  connecting them in the tree

12

• Observed distances are obtained directly from the
  sequences themselves and patristic distances
  from a tree
• For additive and ultrametric distances, the
  observed and tree distances match exactly

Additive
Ultrametric
For real data this is rarely the case, indicating
that observed distances cannot be completely
accurately represented by a tree.
13
Classification of phylogenetic methods
14
UPGMA an algorithmic method

• Cluster analysis Unweighted pair group method
  using arithmetic averages (Sneath and Sokal 1973)
• Assumes ultrametricity

15
UPGMA example

• Given a matrix of pairwise distances, find the
  clusters (taxa) i and j such that dij is the Min
  value in the table
• Define the depth of the branching between i and j
  (lij) to be dij/2
• If i and j were the last two clusters, the tree
  is complete. Otherwise, create a new cluster
  called u
• Define a distance from u to each other cluster
  (k, with k ? i or j) to be the average of the
  distances dki and dkj
• Go back to step 1 with one less cluster clusters
  i and j have been eliminated, and cluster u has
  been added

16
Distance Matrices and phenogram
17
Classification of phylogenetic methods
18
Parsimony methods

• The most widely-used method, familiar notion in
  science (simplicity)
• Shared attributes among taxa are inherited from
  common ancestors
• When character conflicts occur, ad hoc hypotheses
  cannot be avoided if you want to explain all the
  data, and assumptions of homoplasy must be
  invoked

19
Parsimony

• From all sets of possible trees, find all trees ?
  such that L(?) is minimal
• B is the number of branches
• N is the number of characters
• k and k are the two nodes incident to each
  branch k
• xkj and xkj represent either elements of the
  input matrix or optimal-character assignments
  made to internal nodes
• Diff(y,z) is a function specifying the cost of
  transformation from y to z along any branch--for
  unrooted trees diff(y,z)diff(z,y) Diff may be
  defined by cost matrix
• The coefficient w is the weight assigned to each
  character (a priori or a posteriori)

20
Other parsimony variants

• Dollo parsimony every derived character must be
  uniquely derived (originate only once in the
  tree)
• Homoplasy only reversals are allowed (no
  parallelism or convergence)
• In practice, Dollo parsimony does not require
  inclusion of hypothetical ancestors just
  character polarity (unrooted Dollo)
• Convenient for restriction-site characters
  (easier to loose that to gain a site)

21
Dollo parsimony and RFLP data
Relaxed Dollo criterion, may be appplied using
generalized parsimony
22
Generalized Parsimony

• All parsimony variants can be subsumed into a
  generalized method that assigns a cost for each
  possible transformation
• Costs are represented in a m-by-m cost matrix S,
  where each element Sij represents the increase in
  tree length due to a transformation from state i
  to j
• The cost of each transformation (weight) can be
  determined a priori (e.g. for RFLPs or for
  transition/transversion changes) or a posteriori
  (using the same data, e.g. successive
  approximations method)

23
Generalized Parsimony Cost matrices
24
Protein parsimony

• A 20x20 matrix specifies the cost for each
  possible transformation
• The matrix may be based on the genetic code
  (PROTPARS matrix) and/or the biochemical
  properties of the amino acids themselves (Dayhoff
  matrices)

25
Difference in perspective MP and ML

• Parsimony seeks solutions that minimize the
  amount of change required to explain the data
  (underestimates superimposed changes)
• ML attempts to estimate the actual amount of
  change (by specifying the evolutionary model that
  will account for the data with the highest
  likelihood)
• Methods that incorporate models of evolutionary
  change can make more efficient use of the data

26
Classification of phylogenetic methods
27
Distance methods

• Experimentally derived distances are assumed to
  be estimates of true distances
• We want to fit them to a mathematical model
  (additive tree) and find the optimal value for
  the adjustable parameters
• Branching pattern
• Branch lenghts
• Some methods Fitch Margoliash, minimum evolution
  (ME)

28
Distance Methods

• Alternative approach to ML for minimizing the
  impact of the underestimation problem if
  corrected distances are used
• Corrected distances are assumed to be estimates
  of the true evolutionary distance (between a pair
  of sequences)
• Distance methods are less desirable
  approximations to a full ML approach, but much
  faster
• But some drawbacks of character data-to-distance
  transformations are information loss and
  difficulty for combination of two or more data
  sets

29
The problem...

• We have uncertain data (distance estimates) that
  we want to fit to a particular mathematical model
  (an additive tree) and find optimal values for
  the adjustable parameters
• The branching pattern
• The branch lengths

30
An additive distance measure defines a tree...
For any 2 sequences, the value in the distance
matrix should correspond to the sum of the branch
lengths along the path between the 2 sequences on
the tree.
31
When distances are not ultrametric but only
metric they can still be represented by a tree
An additive tree
Additive trees also represent additive distances
exactly...
32

• While this tree is additive, it is not
  ultrametric
• Notice that sequences b and c are the most
  similar (3), but ARE NOT the most closely related
• Similarity and and evolutionary relationship will
  only coincide exactly if the distances are
  ultrametric

33
Additive-tree methods

• Due to the finite amount of available data,
  stochastic variation will cause deviations of the
  estimated evolutionary distances from perfect
  tree additivity...
• even when evolution proceeds according to the
  model used for distance correction (JC, K2P,
  HKY85, etc)
• Many methods exist that derive a tree (w/ branch
  lengths) from a distance matrix to come closest
  to being additive
• The discrepancy (distortion) between observed
  and tree distances can be used as an indicator
  (optimality criterion) of how well observed
  distances fit a tree like representation (but
  confusion with algorithm)

34
Fitch-Margoliash and related methods

• E definition of disagreement between data and
  tree
• Alpha and weights must be defined
• If alpha1 then this is an absolute difference
  criterion
• If alpha2 thenthis is a least squares criterion
• Weighting schemes (w) more commonly used are 1,
  1/dij, 1/dij2, and 1/variance(dij)

35
Minimum Evolution Method

• Use unweighted least squares criterion (w1,
  alpha2) to fit branch lengths, but a different
  criterion to evaluate and compare trees

Optimality criterion sum of the absolute values
of the BL that minimize the squared deviations
between observed and path-length distances
2T-3 is the number of independent branches in an
unrooted tree
36
Computation and tree-searchproblems

• Sometimes negative branch lengths will be defined
  to optimize the fit (E in the equation) some
  solutions
• Outright rejection of the tree with negative
  branch lenghts (too drastic)
• Constrain the optimization process to disallow
  negative branch lengths (set them to zero)
• Least-square and minimum-abs-deviation methods
  assume that each pairwise distance is independent
  (not generally true because of common
  evolutionary history of the molecules)
• Also remember loss of information when
  summarizing discrete data as a distance matrix

37
Classification of phylogenetic methods
38
Maximum Likelihood methods

• Evaluates a hypothesis about evolutionary history
  in terms of the probability that a proposed model
  of the evolutionary process and the hypothesized
  history would give rise to the observed data
• L Pr (DH)
• A history with a higher probability of giving
  rise to the current state of affairs is preferred
• Cavalli-Sforza and Edwards (1967) and Felsenstein
  (1981, 1993) and others.

39
ML Objective

• Data are observed sequences (DNA or prot)
• Unknowns are branching order (topology) and
  branch lengths of a tree
• A concrete model of evolution that transforms one
  sequence into another needs to be specified
  (fully defined or with uncertain parameters that
  need to be estimated from the data)
• L Pr (DH)
• Trees with higher likelihoods are preferred

40
Calculating L for a tree

• Aligned sequences for 4 taxa
• We want to evaluate the tree shown
• What is the prob that this tree generated the
  data?

41
Calculating L for a tree

• Root the tree at any internal node (models are
  time-reversible)
• Assumption of independence allows to calculate L
  for each site separately
• Then combine the likelihoods into a total value
  at the end

42
Calculating L for a tree

• To calculate L for some site j, we must consider
  all possible scenarios by which the tip sequences
  could have evolved
• Specifically, the root (6) may have had A, C, T,
  or G
• For each of these possibilities, the other
  internal node (5) also might have possessed any
  of the 4 nucleotides

43
Calculating L for a tree

• Thus, there are 4x416 possibilities to consider

44
Calculating L for a tree

• Calculate the probability of each and sum them to
  obtain the total probability for site j
• Assume that the changes along each branch are
  independent (Markov model)
• Thus, the Pr of any single scenario is equal to
  the product of the Pr of the changes required by
  that scenario

45
Calculating L for a tree

• Because the Pr of any single observation is an
  extremely small number, we evaluate the log of
  the likelihood instead
• Probabilities are accumulated as the sum of logs
  of the single-site likelihoods

46
Difference in perspective MP and ML

• Parsimony seeks solutions that minimize the
  amount of change required to explain the data
  (underestimates superimposed changes)
• ML attempts to estimate the actual amount of
  change (by specifying the evolutionary model that
  will account for the data with the highest
  likelihood)
• Methods that incorporate models of evolutionary
  change can make more efficient use of the data

47
Difference in perspective ML vs MP
48
Parsimony and likelihood
ML and MP scores for all 15 unrooted trees for
mtDNA sequence data
49
MP and Inconsistency
50
Long Branch Attraction

• The Felsenstein Zone
• What are the assumptions for MP?
• How can we tell if theres LBA in our data?

51
Searching for optimal trees

• Methods with explicit optimality criteria
• Parsimony
• Maximum likelihood
• Additive-tree distance
• Separate the problem of
• evaluating the tree
• finding the optimal tree(s)
• Can we evaluate all possible trees for a
  particular problem?

52
Searching for optimal trees

• For small to moderate data sets, with as many as
  8-20 taxa, we can use exact methods
• Exact methods guarantee the discovery of all
  optimal trees
• Exact methods include
• Exhaustive search
• Branch-and-bound search

53
How many trees?
54
And for more than 10 taxa?
55
Exhaustive search enumerate al possible trees
56
Branch-and-bound Does not require exhaustive
search and yet provides an exact solution. 1.
Traverse a search tree in a depth-first
sequence 2. Select upper bound (L) on optimal
value of chosen criterion. 3. Move along path to
tips and evaluate trees. If tree is L then
dispense the rest of that path.
57
Approximate methods

• For larger data sets computing time becomes
  prohibitive and we only explore some subset of
  all possible trees (hoping that the optimal trees
  will be found in the subset explored)
• Heuristic approaches sacrifice the guarantee of
  optimality in favor of reduced computer time
• Use hill climbing methods. Initial tree starts
  the process, then we seek to improve its score
• When we can find no way to further improve the
  score, we stop.We dont know if we reached a
  local or a global optimum

58
Initial trees

• May be obtained by stepwise addition, the most
  commonly used method
• Similar to exhaustive search but evaluate trees
  at every step, each time you add a new taxon and
  only follow the path derived from the optimal
  tree
• Which taxa do you choose first? Which do you
  connect next?
• These are greedy algorithms

59
Stepwise addition
60

• Initial trees also may be obtained by star
  decomposition, another greedy algorithm

61
Branch swapping

• To improve the initial estimate we can perform
  sets of predefined rearrangements on the tree
• Any of these rearrangements amounts to a stab in
  the dark
• Globally optimal trees may be several
  rearrangements away from the starting tree
• If a better tree is found, a new round of
  rearrangements is then performed in the new tree
• Several branch-swapping algorithms are available

62
Branch swapping by tree bisection and
reconnection (TBR) 1. Tree is bisected along a
branch, yielding two disjunct subtrees 2. The
subtrees are reconnected by joining a pair of
branches, one from each subtree 3. All possible
bisections and pairwise reconnections are
evaluated
63
Branch swapping by subtree prunning and
regrafting 1. A subtree is pruned from the tree
(e.g. A,B) 2. The subtree is then regrafted to a
different location on the tree 3. All possible
subtree removals and reattachment points are
evaluated
64
Branch swapping by nearest-neighbor interchanges
(NNI) 1. Each interior branch of the tree
defines a local region of four subtrees
2. Interchanging a subtree on one side of the
branch with one from the other constitutes an
NNI 3. Two such rearrangements are possible for
each interior branch (all interior branches are
swapped)
65
Landscapes and the problem of islands of trees