Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Pre Darwin phylogenetic trees

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Origin sole figure

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Cytochrome C Gene Tree (Fitch, 1967)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

• Processes of speciation

• Evolution of traits

• Biogeography

• Epidemiology

• Co-Evolution (host/parasite)

• Domestication

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Selecting a “Duck”

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Molecular Clock (early ’60s)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Models of Sequence Evolution

JC69 model (Jukes and Cantor, 1969)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Kingman Coalescent (1982)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Wright-Fisher Population (1931)

• The individuals were randomly sampled

from a population of size N.

• The parent of any individual is chosen

uniformly at random from all potential

parents

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Coalescent

The larger the population, the longer (on average)you have to travel back in time for the commonancestor.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Coalescent for multiple individuals

The waiting time for the first common ancestor oftwo individuals out of m (going backwards in time)

is exponential with a rate of (m2)/Ne

.Ne is the Wright-Fisher effective population size.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

From Models to Inference

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Bayes’ Theorem (a Reminder)

P(A ∧ B) = P(A)P(B |A)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Bayes’ Theorem (2)

P(B)P(A|B) = P(A ∧ B) = P(A)P(B |A)

P(A|B) = P(B |A)P(A)P(B)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Bayesian Inference

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Models (so far)

Substitution model: A stochastic process for the evolution(change) of genetic data (sequences) overtime.

Clock model: How substitution rates change over time.

Coalescent model: A stochastic process for the ancestralrelationship between a group ofhomologous sequences from severalindividuals.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Models (Math Notation)

Coalescent model: f (T |Ne)

Substitution model: f (G |T )

Where G is the gene (sequence data) and T is theancestral relationships (tree).

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Biological Species Concept

The conventional definition of “a species” amongstevolutionary biologists is “a group of organisms whosemembers interbreed among themselves, but are separatedfrom other groups by genetically-based barriers to geneflow.”

Jerry Coyne “Why Evolution is True” blog.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Species “tree”

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Gene(s) tree

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tree Ancestral Reconstruction

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Multiple Individuals from each Species

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Multispecies coalescent – Kingman Coalescent per SpeciesTree Branch

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Multiple Independent Loci

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Multispecies Posterior

P(S |D) =

∫g

P(S , g |D)

∝∫

g

P(D|S , g)P(S , g)

=

∫g

P(D|g)P(S , g)

=

∫g

P(D|g)P(g |S)P(S)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

A Complex Posterior

P(S |D) =

∫g

f (S , g |D)

=

∫g

f (D|S , g)f (S , g)f (D)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Problem 1: f(D)

The prior probability of obtaining data D.

P(S |D) =

∫g

f (D|S , g)f (S , g)

f(D)

We don’t know the value of f (D).

f (D) =

∫g ,S

f (D|S , g)f (S , g)

However, it is a constant.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Problem 2: The Whole Damn Thing

The posterior is a distribution defined by a complexmultidimensional integral.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Enter MCMC

Markov Chain Monte Carlo (MCMC) is a class ofmethods for stochasticly sampling from probabilitydistributions based on constructing a Markov chain.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Very short history of MCMC

1953: Metropolis algorithm published in Journal ofChemical Physics (Metropolis et al.)

1970: Hastings algorithms in Biometrika (Hastrings)

1974: Gibbs sampler and Hammersley-Clifford theorempaper by Besag

1980s: Image analysis and spatial statistics enjoyed MCMCalgorithms, not popular with others due to the lackof computing power

1995: Reversible jump algorithm in Biometrika (Green)

groundtruth.info/AstroStat/slog/2008/mcmc-historyo

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (2)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (3)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (4)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (5)

If we propose to go from B or A to either A or B with equalprobability, then

2

1

A B

Flow from A to B is 2/3 · 1/4 = 1/6, and from B to B is1/3 · 1/2 = 1/6, 1/3 in total.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (6)

Hastings Ratio (x to y) =p(y → x)f (y)

p(x → y)f (x)

So far we had p(y → x) = p(x → y), that is theprobability going from x to y was equal to goingback.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (7)

If at A we always propose to go to B but from B we go to A or Bwith equal propability, that is,

Table: p(x → y)

A B

A 0 1B 1/2

1/2

Then

HR(A→ B) =1/2 · 11 · 2 = 1/4,

AndHR(B → A) = 4.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

MCMC in a nutshell (8)

2

1

A B

Flow from B to A is 1/3 · 1/2 and from A to A is 2/3 · 3/4 = 1/2,2/3 in total.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Tree(s) Visualisation

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Traditional Tree Visualization

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Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Traditional Tree Visualization (2)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tree with Population Sizes

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tree with Gene Trees

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tree (Densitree)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The “Star Tree”

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tree (Densitree)

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Taxa Order Matters (When Drawing Multiple Trees)

0 1 2 3 4 5 6

73%

0 1 5 6 2 3 4

17%

2 3 4 5 6 0 1

10%

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Some Orders are Better than Others

0 1 2 3 4 5 6

73%

0 1 5 6 2 3 4

17%

2 3 4 5 6 0 1

10%

2 3 4 0 1 5 6

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Disadvantages:

• Population size changes inone branch have a visualeffect on other branches.

• Fails when trying to showthe whole posterior a laDensiTree.

• No obvious way to extendfor trees with constantpopulation size per branch.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

The Imperial AT-AT Tree

Provide some space between branches.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

A double Act

Target species tree (blue) and ?BEAST posterior summary(orange).

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tree with Constant Population Sizes

To extend to constant branches, we need a rule to place thebottom of the branch on top of the descendant branches. We usethe proportion rule.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Position,Position,Position

A species tree specifies heights and widths. The challenge is topick good X-axis positions.The star tree builds the tree from root towards the tips. Buildingfrom the tips towards the root is simpler when drawing speciestrees. When building from the tips the descendants X-positionsdetermine the parent position.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Position,Position,Position

However, there are many ways to place nodes. Here are four ofthem:

Descendants Mean Halfway between direct descendants.

Tips Mean Average of all tips in the sub-tree.

Middle Halfway between rightmost tip of left sub-treeand leftmost tip of right sub-tree.

Balanced by Population At point minimizing the difference betweenbranch bottom and top centers.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Node Positioning

The methods are similar for balanced trees. The difference is in thehandling of unbalanced trees.

D-MeanT-MeanMiddleBalanced

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Node Positioning

D-MeanBalanced

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Species Tress Posterior

0 20 40 60 80

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Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Gene Trees Within Species Trees: Preliminary

Next we would like to draw the gene tree within the species tree.

Hurdle 1: Obtain a suitable gene tree.

The gene tree has to be compatible with the species tree. This isnot a problem when drawing a specific MCMC state, but is aproblem when using summary trees.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Tips Positioning

Hurdle 2: Branches with non-constant width complicatespositioning of tips.

−1 0 1 2 3 4 5 6

0.0

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Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Tips Positioning (automatic)

The placing insures that extrema points are at least ε apart(horizontally).

−2 0 2 4 6 8 100.0

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Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Tips Positioning (automatic)

Even with Python, it is far from trivial to implement. Remember,we want a tight fit, and placing should work for all modes ofinternal positioning.Basically, we build the tree from bottom to top up by joiningclades. the X position of extrema points for the clade is a linearfunction of the spacing between the sub-trees (where the spacinginside the two sub trees are fixed). So each extrema points sets alower limit on the spacing, and the largest is taken as the finalseparation.The best way to pick ε for a tree still needs to be worked out.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Drawing the Gene Tree

Hurdle 3: A suitable policy for drawing the gene tree.

We reuse the ideas of the Star Tree. Given the position of aninternal node, the left/right branches is drawn as a straight linestowards the “middle” of the left/right sub-trees. But we still needto handle the species transitions.

From the bottom up, we(linearly) map the lineagesleaving the branch to the topof the branch. The top ofthe clade is then put in themiddle of the mapped taxa.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Visual Clutter

0.0

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200.0

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Contained Tree

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Gene Tree Inside Species Tree: As-Is

0 2 4 6 8 10

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1.5

2.0

2.5

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ID2 ID3 ID4 ID0 ID1 ID5 ID6

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Reducing Visual Clutter (1)

a b x ab x

The size (number of tips) of the sub-tree (b,x) is 2, but the span(number of tips between leftmost and rightmost) is 3.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Reducing Visual Clutter (2)

For every sequential arrangement of the gene tree taxa we can geta rough measure of the amount of crossings,∑

n∈Internal Nodesspan(n)− size(n)

size(n) is the number of taxa in the sub tree. span(n) is thenumber of taxa in the group bounded by the leftmost andrightmost tips of the sub tree. The difference is the excess taxa,the number of potential lineages that may need to cross out of theclade.Note that valid arrangements depend on the orientation of thespecies tree, so optimization should be over both species orderingand gene tips arrangements compatible with that order. Since thenumber is typically large, we resort to multiple tries of hillclimbing. Number of tries might be fixed or bounded by time.

Introduction The Coalescent Bayesian Inference MCMC Visualisation Tree Shape Taxa Position

Unresolved Conflicts

Optimized tree on the right.