CS 581 Paper Presentation Muhammad Samir Khan Recovering the - - PowerPoint PPT Presentation

CS 581 Paper Presentation

Muhammad Samir Khan

Recovering the Treelike Trend of Evolution Despite Extensive Lateral Genetic Transfer: A Probabilistic Analysis by Sebastien Roch and Sagi Snir

Overview

• Introduction (what is LGT?)
• Notation
• Model
• Bounded-rates Model
• Yule Process
• Quartet Based Approach
• Bounded Rates Model
• Yule Process
• Preferential LGT
• Further Results

What is LGT?

• Non-vertical transfer of genes
• Overall evolution is tree-like
• Particularly common in

bacteria

• Primary Reason for the spread
• f antibiotic resistance 1
• 1. https://en.wikipedia.org/wiki/Horizontal_gene_transfer
• 2. http://www.nature.com/nrmicro/journal/v3/n9/images/nrmicro1253-f1.gif

Species Phylogeny

• 𝑈

𝑡 = (𝑊 𝑡, 𝐹𝑡, 𝑀𝑡: 𝑠, 𝜐)

• 𝑊

𝑡

vertices

• 𝐹𝑡

edges

• 𝑀𝑡

leaves

• 𝑠

root

• 𝜐(𝑓)

interspeciation times

• Number of leaves 𝑜 = 𝑜+ + 𝑜−
• 𝑜+ > 0 extant species
• 𝑜− ≥ 0 extinct species

𝑠 extinct extant 𝜐(𝑓)

Extant Phylogeny

• Denoted 𝑈

𝑡 + = (𝑊 𝑡+, 𝐹𝑡 +, 𝑀𝑡 +: 𝑠+, 𝜐+)

• Restrict to extant leaves 𝑈

𝑡|𝑀𝑡 +

• Suppress vertices of degree 2 (add up

the branch lengths)

• Root at the most recent common

ancestor of 𝑀𝑡

+

• 𝑈

𝑡 + is ultrametric

• Want to recover the extant

phylogeny

𝑠 time

Gene Trees

• 𝑈

𝑕 = (𝑊 𝑕, 𝐹𝑕, 𝑀𝑕: 𝜕𝑕) for a gene 𝑕 is an unrooted tree

• 𝑊

𝑕

vertices

• 𝐹𝑕

edges

• 𝑀𝑕

leaves subset of 𝑀𝑡

• 𝜕𝑕(𝑓)

branch lengths (expected number of substitutions)

• Each vertex of degree 2 or 3
• 𝒰

𝑕 = 𝒰[𝑈 𝑕] is the topology of 𝑈 𝑕 with degree 2 vertices suppressed

• Not ultrametric

LGT Transfer – Subtree Prune and Regraft

• LGT Transfer takes place on

locations along the edges

• Recipient location: pruning
• Donor location: regrafting
• A new node at donor location
• 1. Roch, S., & Snir, S. (2013). Recovering the treelike trend of evolution despite extensive lateral genetic transfer: a probabilistic analysis. Journal of Computational

Biology, 20(2), 93-112.

Contemporaneous Locations

• Two locations 𝑦, 𝑧 are contemporaneous if their 𝜐-distance to the root

is identical: 𝜐 𝑦, 𝑠 = 𝜐(𝑧, 𝑠)

• For 𝑆 > 0, 𝐷𝑦

(𝑆) is the set of locations contemporaneous to 𝑦 and

with MRCA at 𝜐-distance at most 𝑆 from 𝑦: 𝐷𝑦

(𝑆) =

𝑧 ∶ 𝜐 𝑦, 𝑠 = 𝜐 𝑧, 𝑠 , 𝜐 𝑦, 𝑧 ≤ 2𝑆

Random LGT

• Species phylogeny fixed 𝑈

𝑡 = 𝑊 𝑡, 𝐹𝑡, 𝑀𝑡: 𝑠, 𝜐

• 0 < 𝑆 ≤ ∞ (possibly depending on 𝑜)
• Each edge has a rate of LGT 𝜇 𝑓 : 0 < 𝜇 𝑓 < +∞
• Λ 𝑓 = 𝜇 𝑓 𝜐 𝑓
• Λ𝑢𝑝𝑢 = σ𝑓∈𝐹𝑡 Λ 𝑓
• Λ = σ𝑓∈𝐹(𝑈

𝑡|𝑀𝑡 +) Λ 𝑓

• Taxon sampling probability 𝑞 ∶ 0 < 𝑞 ≤ 1

Random LGT

• LGT locations:
• Start from root (chronologically)
• Along each edge 𝑓 ∈ 𝐹𝑡, select a recipient location according to a continuous-

time Poisson process with rate 𝜇 𝑓

• If 𝑦 is selected as a recipient location, donor location is selected uniformly at

random from 𝐷𝑦

𝑆

• Keep each extant leaf independently with probability 𝑞, to get 𝑀𝑕
• Gene tree 𝑈

𝑕 is obtained by keeping the subtree restricted to 𝑀𝑕

Bounded Rates Model

• Constants:
• 𝜍𝜇 ∶ 0 < 𝜍𝜇 < 1
• 𝜍𝜐 ∶ 0 < 𝜍𝜐 < 1
• ҧ

𝜐 ∶ 0 < ҧ 𝜐 < +∞

• ҧ

𝜇 possibly depending on 𝑜+ : 0 < ҧ 𝜇 < +∞

• Used to control the amount of LGT
• Under the bounded rates model:

𝜍𝜇 ҧ 𝜇 ≤ 𝜇 𝑓 ≤ ҧ 𝜇 ∀𝑓 ∈ 𝐹𝑡 𝜍𝜐 ҧ 𝜐 ≤ 𝜐+ 𝑓+ ≤ ҧ 𝜐 ∀𝑓+ ∈ 𝐹𝑡

+

Yule Process

• Branching process that starts with two species
• Each species generates a new offspring at rate 𝜉 ∶ 0 < 𝜉 < +∞
• No extinct species
• Stop when number of species = 𝑜 + 1 (ignore the last species)
• 𝜍𝜇 ҧ

𝜇 ≤ 𝜇 𝑓 ≤ ҧ 𝜇 for every edge 𝑓 ∈ 𝐹𝑡

• 𝜍𝜇 constant: 0 < 𝜍𝜇 < 1
• ҧ

𝜇 possibly depending on 𝑜: 0 < ҧ 𝜇 < +∞

Quartet Based Approach

• Input: Gene trees 𝑈

𝑕1, … , 𝑈 𝑕𝑂

Output: Estimated extant species phylogeny ෠ 𝑈

• Let 𝑌 = 𝑏, 𝑐, 𝑑, 𝑒 be a four-tuple of extant species
• Three possible quartets
• 𝑟1 = 𝑏𝑐|𝑑𝑒
• 𝑟2 = 𝑏𝑑|𝑐𝑒
• 𝑟3 = 𝑏𝑒|𝑐𝑑
• Frequency of quartet:

𝑔

𝑌 𝑟𝑗 = 𝑕𝑘∶𝑌⊆𝑀𝑕𝑘,𝒰

𝑕𝑘|𝑌=𝑟𝑗

𝑕𝑘∶𝑌⊆𝑀𝑕𝑘

Quartet Based Approach

• 1. Roch, S., & Snir, S. (2013). Recovering the treelike trend of evolution despite extensive lateral genetic transfer: a probabilistic analysis. Journal of Computational

Biology, 20(2), 93-112.

Bounded Rates Model

• 1. Roch, S., & Snir, S. (2013). Recovering the treelike trend of evolution despite extensive lateral genetic transfer: a probabilistic analysis. Journal of Computational

Biology, 20(2), 93-112.

Yules Process

• 1. Roch, S., & Snir, S. (2013). Recovering the treelike trend of evolution despite extensive lateral genetic transfer: a probabilistic analysis. Journal of Computational

Biology, 20(2), 93-112.

Preferential LGT

• 1. Roch, S., & Snir, S. (2013). Recovering the treelike trend of evolution despite extensive lateral genetic transfer: a probabilistic analysis. Journal of Computational

Biology, 20(2), 93-112.

Further Results

• Highways of LGT
• The same model as before with additional “highways”
• Highways are pairs of edges where LGT occurs deterministically
• Highways can be different for different genes
• Same result holds under the bounded rates model
• Assuming no extinctions
• Frequency of genes affected by highways is low
• Distance Based Approach under the GTR model
• Compute the distance matrix by using the median of distances
• Use any statistically consistent distance based method

Questions?