Genome evolution

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Genome evolution

• Lecture 5 Inference through sampling. Basic
  phylogenetics

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Reminder Inference
We assume the model (structure, parameters) is
given, and denote it by q
Alignment
Evolutionary rates
Tree
Ancestral Inference on a phylogenetic tree
The Total probability of the data s
Learning a model
This is also called the likelihood L(q).
Computing Pr(s) is the inference problem
Easy!
Given the total probability it is easy to compute
Exponential?
Total probability of the data
Posterior of hi given the data
Marginalization over hi
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Two examples
What makes these examples difficult?
We want to perform inference in an extended tree
model q expressing context effects
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We want to perform inference on the tree
structure itself! Each structure impose a
probability on the observed data, so we can
perform inference on the space of all possible
tree structures, or tree structures branch
lengths
q
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More terminology make sure you know how to
define these
InferenceParameter learningLikelihoodTotal
probability/Marginal probabilityExact
inference/Approximate inference
Sampling is the a natural way to do approximate
inference
Marginal Probability (integration over A sample)
Marginal Probability (integration over all space)
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Sampling from a BN
How to sample from the CPD?
Naively If we could draw h,s according to the
distribution Pr(h,s) then Pr(s)
(samples with s)/( samples)
Forward sampling use a topological order on the
network. Select a node whose parents are already
determined sample from its conditional
distribution (all parents already
determined!) Claim Forward sampling is correct
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Focus on the observations
What is the sampling error?
Two tasks P(s) and P(f(h)s), how to approach
each/both?
Naïve sampling is terribly inefficient, why?
This can be done, but we no longer sample from
P(h,s), and not from P(hs) (why?)
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Likelihood weighting
Likelihood weighting weight 1 use a
topological order on the network. Select a node
whose parents are already determined if the
variable was not observed sample from its
conditional distribution else weight
P(xipaxi), and fix the observation Store the
sample x and its weight wx Pr(hs) (total
weights of samples with h) / (total weights)
Weight
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Importance sampling
Our estimator from M samples is
But it can be difficult or inefficient to sample
from P. Assume we sample instead from Q, then
Prove it!
Unnormalized Importance sampling
Claim
To minimize the variance, use a Q distribution is
proportional to the target function
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Correctness of likelihood weighting Importance
sampling
For the likelihood waiting algorithm, our
proposal distribution Q is defined by fixing the
evidence at the nodes in a set E and ignoring the
CPDs of variable with evidence. We sample from
Q just like forward sampling from a Bayesian
network that eliminated all edges going into
evidence nodes!
Unnormalized Importance sampling with the
likelihood weighting proposal distribution Q and
any function on the hidden variables
Proposition the likelihood weighting algorithm
is correct (in the sense that it define an
estimator with the correct expected value)
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Normalized Importance sampling
When sampling from P(hs) we dont know P, so
cannot compute wP/Q We do know
P(h,s)P(hs)P(s)P(hs)aP(h)
So we will use sampling to estimate both terms
Using the likelihood weighting Q, we can compute
posterior probabilities in one pass (no need to
sample P(s) and P(h,s) separately)
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Limitations of forward sampling
observed
Likelihood weighting is effective here
unobserved
But not here
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Symmetric and reversible Markov processes
Definition we call a Markov process symmetric if
its rate matrix is symmetric
What would a symmetric process converge to?
Definition A reversible Markov process is one
for which
Time t ? s
i
j
j
i
qji
Claim A Markov process is reversible iff
such that
pi
pj
If this holds, we say the process is in detailed
balance and the p are its stationary distribution.
qij
Proof Bayes law and the definition of
reversibility
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Reversibility
qji
Claim A Markov process is reversible iff
such that
pi
pj
If this holds, we say the process is in detailed
balance.
qij
Proof Bayes law and the definition of
reversibility
Claim A Markov process is reversible iff we can
write
where S is a symmetric matrix.
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Markov Chain Monte Carlo (MCMC)
We dont know how to sample from P(h)P(hs) (or
any complex distribution for that matter) The
idea think of P(hs) as the stationary
distribution of a Reversible Markov chain
Find a process with transition probabilities for
which
Process must be irreducible (you can reach from
anywhere to anywhere with pgt0)
Then sample a trajectory
Theorem (C a counter)
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The Metropolis(-Hastings) Algorithm
Why reversible? Because detailed balance makes it
easy to define the stationary distribution in
terms of the transitions So how can we find
appropriate transition probabilities?
We want Define a proposal distribution And
acceptance probability
F
x
y
What is the big deal? we reduce the problem to
computing ratios between P(x) and P(y)
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Acceptance ratio for a BN
To sample from
We will only have to compute
For example, if the proposal distribution changes
only one variable hi what would be the ratio?
We affected only the CPDs of hi and its
children Definition the minimal Markov blanket
of a node in BN include its children, Parents and
Childrens parents. To compute the ratio, we
care only about the values of hi and its Markov
Blanket
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Gibbs sampling
A very similar (in fact, special case of the
metropolis algorithm) Start from any state h do
Chose a variable Hi Form ht1 by sampling a
new hi from
This is a reversible process with our target
stationary distribution
Gibbs sampling is easy to implement for BNs
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Sampling in practice
We sample while fixing the evidence. Starting
from anywere but waiting some time before
starting to collect data
How much time until convergence to P? (Burn-in
time)
Mixing
Sample
Burn in
Consecutive samples are still correlated! Should
we sample only every n-steps?
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Inferring/learning phylogenetic trees
Distance based methods computing pairwise
distances and building a tree based only on those
(how would you implement this?) More elaborated
methods use a scoring scheme that take the whole
tree into account, using Parsimony vs
likelihood Likelihood methods universal rate
matrices (BLOSSOM62, PAM) Searching for the
optimal tree (min parsimony or max likelihood) is
NP hard Many search heuristics were developed,
finding a high quality solution and repeating
computation using partial dataset to test for the
robustness of particular features (Bootstrap)
Bayesian inference methods assuming some prior
on trees (e.g. uniform) and trying to sample
trees from the probability space P(qD). Using
MCMC, we only need a proposal distribution that
span all possible trees and a way to compute the
likelihood ratio between two trees (polynomial
for the simple tree model) From sampling we can
extract any desirable parameter on the tree
(number of time X,Y and Z are in the same clade)
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Curated set of universal proteins
Eliminating Lateral transfer
Multiple alignment and removal of bad domains
Maximum likelihood inference, with 4 classes of
rate and a fixed matrix
Bootstrap Validation
Ciccarelli et al 2005
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How much DNA? (Following M. Lynch)
Viral particles in the oceans 1030 times 104
bases 1034
Global number of prokaryotic cells 1030 times
3x106 bases 1036
107 eukaryotic species (1.6 million were
characterized) One may assume that they each
occupy the same biomass, For human, 6x109
(population) times 6x109 (genome) times 1013
(cells) 1032 Assuming average eukaryotic
genome size is 1 of the human, we have 1037 bases
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?
?
RNA Based Genomes
Ribosome Proteins Genetic Code
DNA Based Genomes
Membranes
Diversity!
Think ecology..
3.4 3.8 BYA fossils?? 3.2 BYA good
fossils 3 BYA metanogenesis 2.8 BYA
photosynthesis .. .. 1.7-1.5 BYA
eukaryotes .. 0.55 BYA camberian explosion
0.44 BYA jawed vertebrates 0.4 land
plants 0.14 flowering plants 0.10 - mammals
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PROKARYOTES EUKARYOTES
(Also present in the Planktomycetes) Presence of a nuclear membrane
(also in b-protebacteria) Organelles derived from endosymbionts
Tubulin-related protein, no microtubules Cytoskeleton and vesicle transport
- Trans-splicing
Rare almost never in coding Introns in protein coding genes, spliceosome
Short UTRs Expansion of untranslated regions of transcripts
Ribosome binds directly to a Shine-Delgrano sequence Translation initiation by scanning for start
Nonsense mediated decay pathway is absent mRNA surveillance
Single linear chromosomes in a few eubacteria Multiple linear chromosomes, telomeres
Absent Mitosis, Meiosis
- Gene number expansion
Some exceptions, but cells are small Expansion of cell size
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Eukaryotes
Uniknots
Biknots
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Eukaryotes
Uniknots one flagela at some developmental
stage Fungi Animals Animal parasites Amoebas
Biknots ancestrally two flagellas Green
plants Red algea Ciliates, plasmoudium Brown
algea More amobea
Strange biology!
A big bang phylogeny speciations across a short
time span? Ambiguity and not much hope for
really resolving it
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Vertebrates
Fossil based, large scale phylogeny
Sequenced Genomes phylogeny
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Primates
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1.2
0.8
3
1.5
0.5
0.5
Gorilla
Chimp
Gibbon
Human
Baboon
Macaque
Marmoset
Orangutan
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Flies
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Yeasts