Swapan Shop Mallick

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Significance in protein analysis
Swapan Shop Mallick
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Overview

• The need for statistics
• Example BLOSUM
• What do the scores mean?
• How can you compare two scores?
• Example BLAST
• Problems with BLAST
• Review of Distributions
• Distribution of random BLAST results
• P-values and e-values
• Statistics of BLAST
• Summary and Conclusion
• Exercise

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The need for statistics

• Statistics is very important for bioinformatics.
• It is very easy to have a computer analyze the
  data and give you back a result.
• Problem is to decide whether the answer the
  computer gives you is any good at all.
• Questions
• How statistically significant is the answer?
• What is the probability that this answer could
  have been obtained by random? What does this
  depend on?

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Basics
N
n
Sample
Population

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Basics
N
Descriptive statistics
n
Sample
Population

Probability
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Example BLOSUM

• The BLOSUM matrix assigns a probability score for
  each residue pair in an alignment based on
• the frequency with which that pairing is known to
  occur within conserved blocks of related
  proteins.
• Simple since size of population size of sample
• BLOSUM matrices are constructed from observations
  which lead to observed probabilities

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BLOSUM substitution matrices

• BLOSUM matrices are used in log-odds form based
  on actually observed substitutions.
• This is because
• Ease of use Scores can be just added (the raw
  probabilities would have to be multiplied)
• Ease of interpretation
• S0 substitution is just as likely to occur as
  random
• Slt0 substitution is more likely to occur
  randomly than observed
• Sgt0 substitution is less likely to occur
  randomly than observed

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Substitution matrices
Score of amino acid a with amino acid b
Pab is the observed frequency that residues a and
b are correlated because of homology
Lambda is a scaling factor equal to 0.347, set so
that the scores can be rounded off to sensible
integers
fafb is the expected frequency of seeing residues
a and b paired together, which is just the
product of the frequency of residue a multiplied
by the frequency of residue b
Source Where did the BLOSUM62 alignment score
matrix come from? Eddy S., Nat. Biotech. 22
Aug 2004
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Substitution matrices
Lambda is a scaling factor equal to 0.347, set so
that the scores can be rounded off to sensible
integers
Pab is the observed frequency that residues a and
b are correlated because of homology
fafb is the expected frequency of seeing residues
a and b paired together, which is just the
product of the frequency of residue a multiplied
by the frequency of residue b
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i) S0 O/E ratio1 ii) Compare S5 and S10.
Ratio is based on exponential function iii)
S-10 O/E ratio 0.031 1/32. iv) Ratio of
scores S1, S2 in terms of probabilities of
observed/random
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i) S0 O/E ratio1 ii) Compare S5 and S10.
Ratio is based on exponential function iii)
S-10 O/E ratio 0.031 1/32. iv) Ratio of
scores S1, S2 in terms of probabilities of
observed/random
32.1
5.7
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i) S0 O/E ratio1 ii) Compare S5 and S10.
Ratio is based on exponential function iii)
S-10 O/E ratio 0.031 1/32. iv) Ratio of
scores S1, S2 in terms of probabilities of
observed/random
32.1
5.7
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i) S0 O/E ratio1 ii) Compare S5 and S10.
Ratio is based on exponential function iii)
S-10 O/E ratio 0.031 1/32. iv) Ratio of
scores S1, S2 in terms of probabilities of
observed/random
32.1
5.7
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Example BLAST

• Motivations
• Exact algorithms are exhaustive but
  computationally expensive.
• Exact algorithms are impractical for comparing a
  query sequence to millions of other sequences in
  a database (database scanning),
• and so, database scanning requires heuristic
  alignment algorithm (at the cost of optimality).

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Interpret BLAST results - Description
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Problems with BLAST

• Why do results change?
• How can you compare results from different BLAST
  tools which may report different types of values?
• How are results (eg evalue) affected by query
• There are _many_ values reported in the output
  what do they mean?

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Example Importance of Blast statistics

• But, first a review.

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Review

• What is a distribution?
• A plot showing the frequency of a given variable
  or observation.

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Review

• What is a distribution?
• A plot showing the frequency of a given variable
  or observation.

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Features of a Normal Distribution

• Symmetric Distribution
• Has an average or mean value at the centre
• Has a characteristic width called the standard
  deviation (S.D. s)
• Most common type of distribution known

m mean
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Standard Deviations (Z-score)
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Mean, Median Mode
Mode
Median
Mean
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Mean, Median, Mode

• In a Normal Distribution the mean, mode and
  median are all equal
• In skewed distributions they are unequal
• Mean - average value, affected by extreme values
  in the distribution
• Median - the middlemost value, usually half way
  between the mode and the mean
• Mode - most common value

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Different Distributions
Unimodal Bimodal
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Other Distributions

• Binomial Distribution
• Poisson Distribution
• Extreme Value Distribution

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Binomial Distribution
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10
10 5 1
P(x) (p q)n
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Poisson Distribution
P(x)
x
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Review

• What is a distribution?
• A plot showing the frequency of a given variable
  or observation.
• What is a null hypothesis?
• A statisticians way of characterizing chance.
  
• Generally, a mathematical model of randomness
  with respect to a particular set of observations.
• The purpose of most statistical tests is to
  determine whether the observed data can be
  explained by the null hypothesis.

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Review

• What is a distribution?
• A plot showing the frequency of a given variable
  or observation.
• What is a null hypothesis?
• A statisticians way of characterizing chance.
  
• Generally, a mathematical model of randomness
  with respect to a particular set of observations.
• The purpose of most statistical tests is to
  determine whether the observed data can be
  explained by the null hypothesis.

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Review

• Examples of null hypotheses
• Sequence comparison using shuffled sequences.
• A normal distribution of log ratios from a
  microarray experiment.
• LOD scores from genetic linkage analysis when the
  relevant loci are randomly sprinkled throughout
  the genome.

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Empirical score distribution

• The picture shows a distribution of scores from a
  real database search using BLAST.
• This distribution contains scores from
  non-homologous and homologous pairs.

High scores from homology.
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Empirical null score distribution

• This distribution is similar to the previous one,
  but generated using a randomized sequence
  database.

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Review

• What is a p-value?

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Review

• What is a p-value?
• The probability of observing an effect as strong
  or stronger than you observed, given the null
  hypothesis. I.e., How likely is this effect to
  occur by chance?
• Pr(x gt Snull)

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Review

• What is the name of the distribution created by
  sequence similarity scores, and what does it
  look like?
• Extreme value distribution, or Gumbel
  distribution.
• It looks similar to a normal distribution, but it
  has a larger tail on the right.

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Review

• What is the name of the distribution created by
  sequence similarity scores, and what does it
  look like?
• Extreme value distribution, or Gumbel
  distribution.
• It looks similar to a normal distribution, but it
  has a larger tail on the right.

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Statistics

• BLAST (and also local i.e. Smith-Waterman and
  BLAT scores) between random, unrelated sequences
  follow the Gumbel Extreme Value Distribution
  (EVD)
• Pr(sgtS) 1-exp(-Kmn e-lS)
• This is the probability of randomly encountering
  a score greater than S.
• S alignment score
• m,n query sequence lengths, and length of
  database resp.
• K, l parameters depending on scoring scheme and
  sequence composition
• Bit score S lS log(K)
  log(2)

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BLAST output revisited
S S E
n m
? K
From Expasy BLAST
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Review

• EVD for random blast
• Upper tail behaviour Pr( s gt S ) Kmn e-lS
  

This is the EXPECT value Evalue
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How to Calculate E-values

• Think of the databank as one very long random
  sequence, length G
• Alignments with sgtS occur randomly across the
  genome, with a Poisson distribution
• Pr (highest-scoring alignment sgtS) KmGe-lS
• Pr( no alignment sgtS ) 1 - KmGe-lS
• Expected number m of alignments with sgtS given by
• 1-e-m 1 - KmGe-lS (Poisson property)
• m -log(KmG) lS
• Threshold S log(KmG) m /l

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Summary

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)

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Summary

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)
• Bit score
• S lS log(K) log(2)

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Summary
Score and bit score grow linearly with the length
of the alignment

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)
• Bit score
• S lS log(K) log(2)

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Summary
Score and bit score grow linearly with the length
of the alignment

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)
• Bit score
• S lS log(K) log(2)
• E-value of bit score
• E mn2-S

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Summary
Score and bit score grow linearly with the length
of the alignment
E-Value shrinks really fast as bit score grows

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)
• Bit score
• S lS log(K) log(2)
• E-value of bit score
• E mn2-S

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Summary
Score and bit score grow linearly with the length
of the alignment
E-Value shrinks really fast as bit score grows

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)
• Bit score
• S lS log(K) log(2)
• E-value of bit score
• E mn2-S

E-Value grows linearly with the product of target
and query sizes.
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Summary
Score and bit score grow linearly with the length
of the alignment
E-Value shrinks really fast as bit score grows

• Want to be able to compare scores in sequences of
  different compositions or different scoring
  schemes
• Score S sum(match) sum(gap costs)
• Bit score
• S lS log(K) log(2)
• E-value of bit score
• E mn2-S

E-Value grows linearly with the product of target
and query sizes.
Doubling target set size and doubling query
length have the same effect on e-value
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Conclusion

• You should now be able to compare BLAST results
  from different databases, converting values if
  they are reported differently (which happens
  frequently)
• You should now know why BLAST results might
  change from one day to the next, even on the same
  server
• You should understand also the dependance of
  query length on E-value.
• Statistical rankings are reported for (almost)
  every database search tool. When making
  comparisons between databases, between sequences
  it is useful to know how the statistics are
  derived to know if comparisons are meaningful.

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• THE END

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• Supplemental
• Section

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• Look through Patterns in sequences (Searching
  for information within sequences) - Some common
  problems and their solutions
• http//lepo.it.da.ut.ee./mremm/kurs/pattern.htm
• What is the structure of my sequence?
• http//speedy.embl-heidelberg.de/gtsp/flowchart2.h
  tml (clickable!)