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**Prokaryotic Gene Structure**

Prokaryotic genes have a simple one-dimensional structure 5’→ 5’→ ←5’ ←5’ ATG AAA ATG GCA . . . GCA TTG CTA TAG Start codon Stop codon Note that the ATG codon encodes both start and methionine

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**Prokaryotic Gene Structure**

Prokaryotic gene prediction begins with ORF finding Possible start ATG AAA GCA Alternate start . . . GCA TTG CTA TAG Stop codon Because of the possibility of alternate start sites, it’s not unusual for several ORFs to share a common stop codon An ORF finder needs to be able to find overlapping ORFs, whether they end with the same stop codon, or overlap in a different frame

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**Prokaryotic Gene Structure**

Prokaryotic gene prediction begins with ORF finding 'ATG(...)*?(TAA|TAG|TGA)' A regular expression crafted to find ORFs must also exhibit “non-greedy” behaviour Note that many bacteria also employ rarer alternate start codons, most commonly GTG and TTG. But we’ll pretend this doesn’t happen!

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**Higher Order Markov Chains**

We don’t need to always just consider the most recent state An nth order Markov process is a stochastic process where the probabilities associated with an event depend on the previous n events in the state path 𝑷 𝒙 𝒊 𝒙 𝒊−𝟏 , 𝒙 𝒊−𝟐 ,…, 𝒙 𝟏 =𝑷( 𝒙 𝒊 | 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏 ) So far all the Markov models we have seen so far have been of order 1 In the case of a first order process this statement reduces to our statement of the Markov property

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**𝑷( 𝒙 𝒊 | 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏 ) = 𝑷 𝒙 𝒊 , 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏+𝟏 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏**

Higher Order Markov Chains Higher order models have an equivalent first order model An nth order Markov chain over alphabet A is equivalent to a first order Markov chain over the alphabet An of n-tuples… 𝑷( 𝒙 𝒊 | 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏 ) = 𝑷 𝒙 𝒊 , 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏+𝟏 𝒙 𝒊−𝟏 ,…, 𝒙 𝒊−𝒏 Practically, this says we can implement a higher order model just by expanding the alphabet size of a first order model This follows trivially from P(X,Y|Y) = P(X|Y)

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**Higher Order Markov Chains**

Consider this first order Markov process A = {A, B} A S e B Here the alphabet A (our set of states) consists of just A and B How would we convert this to a second order Markov process?

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**Higher Order Markov Chains Now reconfigured as a second order model**

AA AB BB BA A = {AA, AB, BA, BB} Note how we have disallowed certain transitions (i.e. set their probability to zero). Start and End omitted for clarity

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**1 2 3 Inhomogeneous Markov Chains ATG GTC AAA GCA**

A Markov model of genes should model codon statistics ATG GTC AAA GCA In true coding genes, each of the three positions within a codon will be statistically distinct This can be accomplished in a few different ways…

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**1 2 3 Inhomogeneous Markov Chains**

A Markov model of genes should model codon statistics ATG GTC AAA GCA 𝒂 𝒙 𝟏 𝒙 𝟐 𝟏 𝒂 𝒙 𝟐 𝒙 𝟑 𝟐 𝒂 𝒙 𝟑 𝒙 𝟒 𝟑 𝒂 𝒙 𝟒 𝒙 𝟓 𝟏 𝒂 𝒙 𝟓 𝒙 𝟔 𝟐 𝒂 𝒙 𝟔 𝒙 𝟕 𝟑 One idea is to intersperse three different Markov chains in alternating fashion This can also be recast as an HMM with additional states in obvious way

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**Histograms with matplot lib**

We’ll use this to look at log-odds per NT distributions import numpy as np import pylab as P . . . # probs here should be your list of probabilities # 50 here corresponds to the number of desired n, bins, patches = P.hist(probs, 50, normed=1, histtype='stepfilled') P.setp(patches, 'facecolor', 'g', 'alpha', 0.75) P.show() More histogram examples may be found at: matplotlib.org/examples/pylab_examples/histogram_demo_extended.html

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Objective The student will be able to: recognize and use the commutative and associative properties and the properties of equality.

Objective The student will be able to: recognize and use the commutative and associative properties and the properties of equality.

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