TREES. ChimpHumanGorilla HumanChimpGorilla = ChimpGorillaHuman == GorillaChimp Trees

Chimp HumanGorillaHuman ChimpGorilla

Chimp GorillaHuman

Human GorillaChimp

Same thing…

s4 s5s1 s3s2s4 s5s1 s3s2

Terminology

A branch =An edge

External node - leaf

Human ChimpChicken Gorilla

The root

Internal nodes

אלו מהמשפטים הבאים נכון, בהתייחס לעץ הנ"ל?

א. האדם והגורילה יותר קרובים זה לזה מהשימפנזה והגורילה.ב. האדם קרוב לתרנגולת ולברווז באותה מידה.

ג. התרנגולת יותר קרובה לגורילה מהאדם.ד. א'+ב'.ה. א'+ג'.ו. ב'+ג'.

ז. א'+ב'+ג'.ח. אף תשובה אינה נכונה.

תרגיל

The maximum parsimony principle.Tree building

Genes: 0 = absence, 1 = presence

speciesg1g2g3g4g5g6

s1100110

s2001000

s3110000

s4110111

s5001110

Tree building

s1 s4 s3 s2 s5

Evaluate this tree…

Tree building

s1 s4 s3 s2 s5

Gene number 1

1 1 1 0 0

Tree building

s1 s4 s3 s2 s5

Gene number 1, Option number 1.

1 1 1 0 0

Tree building

s1 s4 s3 s2 s5

Number of changes for gene 1 (character 1) = 1

1 1 1 0 0

Tree building

s1 s4 s3 s2 s5

0 1 1 0 0

Tree building

s1 s4 s3 s2 s5

0 1 1 0 0

Tree building

s1 s4 s3 s2 s5

0 1 1 0 0

Tree building

s1 s4 s3 s2 s5

0 0 0 1 1

Tree building

s1 s4 s3 s2 s5

0 0 0 1 1

Tree building

s1 s4 s3 s2 s5

1 1 0 0 1

Tree building

s1 s4 s3 s2 s5

1 1 0 0 1

Tree building

Gene number 5 is the same as Gene number 4

Tree building

s1 s4 s3 s2 s5

Gene number 6, 1 option only:

0 1 0 0 0

Tree building

Sum of changes

Sum of changes for this tree topology = 9

Can we do better ???

Tree building

s1 s4 s3 s2 s5

The MP (most parsimonious) tree:

Sum of changes for this tree topology = 8

Tree building

How to efficiently compute the MP score of a tree

The Fitch algorithm (1971):

A GC CA

Human ChimpChicken GorillaDuck

{A,C,G}

Postorder tree scan. In each node, if the intersection between the leaves is empty: we apply a union operator. Otherwise, an intersection.

Number of changes

Total number of changes = number of union operators => 3 in this case.

{A,C,G}

GACA GGGACAAG GCGAGAAA

Find minimum number of changes.

תרגיל

Chimpanzee

HumanGorilla

Gorilla

Position 3 A A T

Chimp HumanGorilla

Position 1 A A A

Position 4 A A APosition 5 T C G

Position 2 A C C

Gorilla

Position 3 A A T

Chimp HumanGorilla

Position 1 A A A

Position 4 A A APosition 5 T C G

Position 2 A C C

Gorilla

Position 3 A A T

Gorilla HumanChimp

Position 1 A A A

Position 4 A A APosition 5 C T G

Position 2 C A C

Gorilla

Gorilla HumanChimp

Chimp HumanGorillaChimp HumanGorilla

These 3 trees will ALWAYS get the same score

The unrooted tree represents a set of rooted trees

A general observation: the position of the root does not affect the MP score.

D E C A BBC

A B C E D A B C E D

s1 s4 s3 s2 s5

1 1 1 0 0

Intuition as to why rooting does not change the score.

The change will always be on the same branch, no matter where the root is positioned…

Which is not a rooted version of this tree?

C E D A BBC

A B D E C A B C D E

תרגיל T3

Gorilla gorilla

(Gorilla)

Homo sapiens (human)

Pan troglodytes (Chimpanzee)

Gallus gallus (chicken)

Evaluate all 3 possible UNROOTED trees:

Chicken

Gorilla

Chicken

Gorilla

MP tree

Rooting based on a priori knowledge:

Chicken

Gorilla

Ingroup / Outgroup:

INGROUPOUTGROUP

Subtrees

A subtree

Monophyletic groups

The Gorilla+Human+Chimp are monophyletic.A clade is a monophyletic group.

Paraphyletic = Non-monophyletic groups

Whale ChimpDrosophila Zebrafish

The Zebrafish+Whale are paraphyletic

Chicken

Gorilla

Chicken + Rat seems to be monophyletic but they are not, since the root of the tree is between Chicken and the rest.

Human and Gorilla are not monophyletic no matter where the root is…

When an unrooted tree is given, you cannot know which groups are monophyletic. You can only say which are not.

HOW MANY TREES

How many rooted trees

a ba b c b a c c a b

N=3, TR(3) = 3

b c da c b da d b ca a c db c a db

TR = “TREE ROOTED”

N=2, TR(2) = 1

d a cb a b dc b a dc d a bc a b cd

b a cd c a bd b c da c b da d b ca

N=4, TR(4) = 15

How many rooted trees

TR = “TREE ROOTED”

2 branches. 3 possible places to add “c”

b a cdd b ca

4 branches. 5 possible places to add “d”

6 branches. 7 possible places to add “e”

The number of branches is increased by 2 each time. The number of branches is an arithmetic series.0,2,4,6,8,…. A(n) = A(1)+(n-1)d. A(1) = 0; d=2. => A(n) = (n-1)*2 = 2n-2

How many rooted treesTR = “TREE ROOTED”

The number of branches is increased by 2 each time. The number of branches is an arithmetic series.0,2,4,6,8,…. A(n) = A(1)+(n-1)d. A(1) = 0; d=2. => A(n) = (n-1)*2 = 2n-2

2 branches. 3 possible places to add “c”c c

Each time we can add a new branch in Br(n)+1 places. [Br(n)=number of branches]

TR(n+1) = TR(n)*(BR(n)+1)=TR(n)*(2n-1)TR(5) = TR(4)*7=TR(3)*5*7=TR(2)*3*5*7=1*3*5*7…TR(n) = 1*3*5*7*…..*(2n-3)

[Tr(n)=number of trees with n sequences]

n!=1*2*3*4*5*6…..*n = n factorial.

TR(n) = 1*3*5*7*…..*(2n-3) =

2*4*6*8*….*(2n-4) =

1*2*3*4*5*6*7*…*(2n-3)

(2*1)*(2*2)*(2*3)*(2*4)*….*(2*(n-2)) =

1*2*3*4*5*6*7*…*(2n-3)

(2(n-2))*(1*2*3*4*….(n-2)) =

(2n-3)!

(2(n-2))*(n-2)!

(2n-3)! =

TR(n) = 1*3*5*7*…..*(2n-3) =

(2(N-2))*(n-2)!

(2n-3)! =

=(2n-3)!!

HEURISTIC SEARCH

There are many trees..,

We cannot go over all the trees. We will try to find a way to find the best tree.These are approximate solutions…

Finding the maximum is the same thing as finding the minimum

Say we have a computer procedure that given a function, it finds its minimum, andwe want to find the maximum of a function f(x). We can just find the minimum of -f(x) and this is minus the maximum of f(x).

Example.

f(0) = 3; f(1) = 7; f(2) = -5; f(3) = 0; max f(x) = 7. argmax f(x) = 1;-f(0)=-3; -f(1) = -7; -f(2) = 5; -f(3) =0; min(-f(x)) = -7. argmax –(f(x) = 1;

Score = 1700

Score = 1825

Score = 1710

Score = 1410

Score = 1695

Score = 1825

Score = 1828

Score = 1910

Score = 1800

Max score = 2900

Score = 2100

Problem number 1: local maximum

Score = 3100

Score = 2900

Local max

Global max

This algorithm is “greedy” – it seizes the first improvement encountered.

One way to avoid local maxima is to start from many random starting points

Several options to define a neighbor.

Option 1Option 2

Nearest-neighbor interchange

Each internal branchdefines two neighbors

How many neighbors do we check each time?

For unrooted trees of n taxa, we have 2n-3 branches. However, only internal branches are interesting, thus we have n-3. Each defines two neighbors, thus the total number of neighbors in each NNI cycle is 2n-6.

Internal branches

External branches

NNI is possible only in internal branches

I am greedy

(1)Most greedy: Start searching your neighbors. If you find something better – move there, and start the search again.

(2)Just greedy: Check ALL your neighbors. Move to the one that is the highest.

(3)Smart greedy: Try all NNI of trees that are tied for the best score.

Greedy variants

There are many other variants of the greedy search

that would not be discussed in this course.

Parsimony has many shortcomings. To name a few:

(1) All changes are counted the same, which is not true for biological systems (Leu->Ile is much more likely than Leu->His).

(2) Cannot take biological context into account (secondary structures, dependencies among sites, evolutionary distances between the analyzed organisms, etc).

(3) Statistical basis questionable.

Alternative:

MAXIMUM-LIKELIHOOD METHOD.

Maximum likelihood uses a probabilistic model of evolution

Each amino acid has a certain probability to change and this probability depends on the evolutionary distances.

Evolutionary distances are inferred from the entire set of sequences.

Evolutionary distances

Positions can be conserved because of two reasons. Either because of functional constraints, or because of short evolutionary time.

5 replacements in 10 positions between 2 chimps, is considered very variable. 5 replacements between human, and cucumber, is not considered that variable…

Maximum likelihood takes this information into account.

Maximum Parsimony

Maximum Likelihood

All changes counted the same

Different probabilities to the different types

of substitutions

Statistically questionable

Statistically robust

Ignores biological context

Accounts for biological context

)]()()()(

)()()([

)]()()()(

)()()([

)]()()()(

)()()([

tPtPtPtP

tPtPXP

tPtPtPtP

tPtPXP

tPtPtPtP

tPtPXPDataP

FZEZZYCY

X Y ZYXGX

AZTZZYCY

X Y ZYXLX

AZMZZYCY

X Y ZYXKX

The likelihood computations

We can infer the phylogenetic tree using maximum likelihood. This is more accurate than maximum parsimony.

Maximum likelihood tree reconstruction

This is incredibly difficult (and challenging) from the computational point of view, but efficient algorithms to find approximate solutions were developed.

HIV evolution – an example of using phylogeny tools

The virus = HIV

The disease = AIDS (Aquired Immunodeficiency Syndrome)

First recognized clinically in 1981

By 1992, it had become the major cause of death in individuals 25-44 years of age in the States.

Human Immunodeficiency Virus (HIV)

Till Dec 2007: 25 million people died of AIDS (20 million in 2002)

People living with HIV/AIDS in 2007 33.2 million

Africa has 12 million AIDS orphans (2007). 1 out of 3 children in some areas lost at least one of his/her parents

HIV Statistics

HIV is a lentivirus

Species = HIVGenus = LentivirusesFamily = Retroviridae

Lentiviruses have long incubation time, and are thus called “slow viruses”.

In 1986, a distinct type of HIV prevalent in certain regions of West Africa was discovered and was termed HIV type 2.

Individuals infected with type 2 also had AIDS, but had longer incubation time and lower morbidity (# of cases/population size).

HIV-1 and HIV-2

HIV subtypes

published by the International AIDS Vaccine Initiative

Five lines of evidence have been used to substantiate zoonotic transmission of primate lentivirus:

1. Similarities in viral genome organization;2. Phylogenetic relatedness;3. Prevalence in the natural host;4. Geographic coincidence;5. Plausible routes of transmission.

For HIV-2, a virus (SIVsm) that is genomically indistinguishable and closely related phylogenetically was found in substantial numbers of wild-living sooty mangabeys whose natural habitat coincides with the epicenter of the HIV-2 epidemic

מנגבי, קוף ארוך זנב מסוג סרקוסבוס מצוי באזורי היערות של אפריקה

Close contact between sooty mangabeys and humans is common because these monkey are hunted for food and kept as pets.

No fewer than six independent transmissions of SIVsm to humans have been proposed.

The origin of HIV-1 is much less certain.

HIV and SIV tree based on maximum parsimony

This tree can be explained by co-evolution of virus and host.

Virus A

Primate B

Primate C

Primate A

Virus C

Virus B

Host-pathogen co-evolution in other SIV

There are at least two different HIV-1 clades, and two different SIVcpz clades

Phylogenetic tree

2006. Nature

“We tested 378 chimpanzees and 213 gorilla fecal samples from remote forest regions in Cameroon for HIV-1 cross-reactive antibodies”

“Surprisingly, 6 of 213 fecal samples from wild-living gorillas also gave a positive HIV-1 signal”

The origin of HIV-O

Bayesian analysis

HIV-1 O is a sister clade of SIV from Gorilla!

It seems that chimpanzee transmitted SIV to gorilla and gorilla to human type O, or

Chimpanzee transmitted to both gorilla and to human type O

Note: gorilla and chimps rarely interact + gorilla are herbivores

The origin of HIV-O

Thanksתודה

Thank You…

TREES. ChimpHumanGorilla HumanChimpGorilla = ChimpGorillaHuman == GorillaChimp Trees

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