Stochastic Models and Statistic Tools Used in Phylogenetic Reconstruction

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Stochastic Models and Statistic Tools Used in
Phylogenetic Reconstruction

• Zhen Zhu

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What is Phylogentics?

• The study of evolutionary relatedness among
  various groups of organisms.
• Phylogenetics treats a species as a group of
  lineage-connected individuals over time.
• In short words, phylogentic reconstruction is to
  construct a tree of life.

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Animals
Plants
Fungi
Protists
Eukaryotes
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Some Terminology

• Character (variables, e.g. the color of the egg)
• States (domain of the variable, e.g, ATGC for DNA
  nucleotides)
• Taxa (the species being studied)
• Marker (the character used for a particular
  study, mostly DNA sequences, could be combo)

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Why to use stochastic models in phylogenetic
reconstruction

• Why?
• We dont know a lot of things in the tree, have
  to guess.
• Even for the leaf species that exists today, it
  is hard to get all the samples we want.
• We know certain things about DNA mutation, its
  easy to use those knowledge with the model.
• Stochastic models also allow us to predict the
  unknowns.

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How to use stochastic models in phylogenetic
reconstruction

• Known, unknown and assumption
• Known sample data set D, and perhaps, some
  probability gathered from other sources.
• Unknown almost everything!
• A lot of methods of building the model are
  suggested based on different assumptions.
• The mostly used model (a tree!)
• Normally, one study only focus on a relatively
  small sub-tree of the whole tree of life. T is
  used to denote the tree we are studying. V is the
  set of all vertices in the tree. Each edge e can
  be represented as a pair of vertices (v,w) where
  v is closer to the root ?.

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Detail the Model

• Once we have this rough idea that the model will
  look like a tree, well have to think about how
  to construct it.
• There are a lot of methods to construct the
  model. Basically, by different method, the
  vertices and edges store different information.
  There will always be a probability matrix
  associated with each vertex. How matrices change
  differs according to the assumption.
• If we know the state of a root and all the
  transition matrices, we can uniquely define a
  tree (usually the models are identifiable). We
  could have a lot of estimation tree, but we want
  the best that can fits the sample data we have.

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Example tree model
A
A
T
G
C
.


A
C
C
A
..
..
..
..

• In this model, the marker is only one
  nucleotide, it has 4 states A,T,G,C. Each level
  represents a time unit increment, the time unit
  is small enough that within the unit, no mutation
  will happen. After this unit of time, the
  character will either remains the same or change
  to another state with certain probability. We
  could assign different evolution rate to each
  edge. And we will have a transition associated
  with each vertex.

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Ways to reconstruct the tree

• Once we have the estimated tree, we need to find
  a true evolutionary tree out of those
  candidates using the sample data we have.
• There are so many different ways, totally depend
  on what data is available to you!
• Because its hard to get the samples millions of
  years ago, so usually the samples will be
  associated with the leaf nodes and leave the
  internal vertices unlabeled.

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Top-down or Bottom-up

• Top-down
• You know the probability from some other sources.
  So you can construct the tree using simulation
  such as MCMC (Markov Chain Monte Carlo).
• You have several hypothesis to test, and you just
  throw them to the simulator, and gather the data
  from the output, and have a scheme to determine
  whether the hypothesis is valid or not according
  to your sample data.
• The result is not one single best tree, but a
  distribution of the possible best results. (the
  probability of each hypothesis being the true
  tree)

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• Bottom-up
• A lot of math involved. GTR (Generalized Time
  Reversible) Model. Tree is a little bit
  different. (the tree is predetermined in shape so
  that number of leaves is the same as number of
  samples).
• U means unlabelled, we dont really care about
  what is exactly in those nodes. Transition
  matrices here are associated with the edge rather
  than the vertex.

?
U
U
U
U
U
U
1
2
3
4
5
6
7
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• Suppose the root has a probability pi of being in
  state i (is fixed).
• Given an edge e(v,w), we have a transition matrix
  M(e) associated with it, where an element in the
  matrix M(e)rs means the conditional probability
  that w being in state s given that v being in
  state r. (Thats why we dont really care about
  what is in the unlabelled vertices.)
• Suppose that the probability in the transition
  matrices only depend on its ancestor.
• We are able to get the probability of the leaves
  from
• Suppose the determinant of all the transition
  matrix is not 0, 1 or -1 and pi will not be 0.
• Its proved that under such assumption that if we
  know that which leaf is in what state, we can
  reconstruct the tree (except the root).

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Other statistical methods used in phylogenetic
reconstruction

• Sampling (extremely important but problematic)
• How to fit in the sample data (maximum
  likelihood)
• Measure the result with confidence
• In short, we are just guessing with the hope to
  have good luck.

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• Reference
• An overview of Phylogeny Reconstruction
  (C.R.Linder T.Warnow)
• Recovering a tree from the leaf colurations it
  generates under a Markov model (Mike Steel)
• A Probability Model for Inferring Evolutionary
  Trees (James S. Farris)
• Wikipedia and other internet sources

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Thank You For Not Sleeping