A model for Statistical Significance of Local Similarities in Structure' Alexander Stark, Shamil Sun

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A model for Statistical Significance of Local
Similarities in Structure. Alexander Stark,
Shamil Sunaev and Robert Russell.

• Presented by
• Viacheslav Fofanov

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Outline

• RMSD measure
• Comparison Procedure
• Model assuming independence of atoms
• Incorporating dependence of covalently linked
  atoms
• Final P-value
• Conclusions

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Sequence?Structure ?Function

• The usual methods
• Sequence comparisons
• Structure comparisons
• Alternative method
• Try to discover functionally important patterns,
  such as 3D structural patterns from known
  functional sites (e.g. active sites of a known
  protein)
• RMSD method (purely geometric)
• Evolutionary trace

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Why?

• Want to detect functionally important sites (from
  a purely geometric point of view).
• Once a site with similar geometric configuration
  has been detected want to know how likely is this
  match to happen by chance.

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RMS Deviation

• Root-mean-square deviation of the atomic position
  in a protein from the native coordinates (another
  protein). Measures the difference between the
  two sets of atoms.
• Initially used to evaluate protein structure
  prediction accuracy, so native coordinates were
  coordinates from experimentally determined
  structure.
• Here, the query is considered to have ideal
  coordinated and proteins from the database are
  compared to it.

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RMS Deviation (2)

• RMSD computed by interatomic distance method

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Querying process

• We have
• 3D coordinates (of C?) of some functionally
  important pattern/configuration (e.g. binding
  site of a protein)
• A database of protein structures, each stored as
  a set of coordinates of its residues.
• We need
• To query our pattern against those of the
  database and find close or identical matches
• Rank proteins based on how well did the query
  pattern match to a pattern on the protein.
• We get
• A protein with a possible similar function to the
  one that contained the query pattern.

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Querying process

• Brute force approach is not very feasible.
• Suppose we are searching for an 8 residue query
  pattern in a 150 residue protein.
• That leads to comparisons
• Times thousands proteins in database
• Solution Restrict comparisons to plausible
  matches, i.e. ones sufficiently close to the
  query pattern.

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Querying process

• Transformations allowed are translation and
  rotation.
• First atom can be moved into ideal position
  without any constraints
• Second atom can be placed anywhere in a shell
  defined by two spheres
• Third can lie in a ring like volume shown

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Comparison of RMSD

• Obvious problem arises in comparing RMSD scores,
  as they are highly dependent on the number of
  atoms being compared.
• RMSD of 2 Å over 150 residues does not compare
  with RMSD of 2 Å over 3 residues.
• Similar to sequence comparison, e.g. a given
  10-mer in a viral genome may be significant but
  in eukariotic genome (e.g. human) may easily
  appear by chance.

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Rationale for a statistical model of RMSD

• We cannot compare two RMSD directly, so we need
  some kind of value of how significant (unusual) a
  particular RMSD score is.
• Want to get something like a P-value, meaning
  want to know probability of finding a score equal
  or better than the one we found by chance.

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Rationale (2)

• For database searches statistical significance is
  generally assessed by an extreme value
  distribution.
• Cumulative Distribution of scores
• P(x)1 eEF(x) , where
• EF is an expectation function that predicts the
  number of matches with an equally good or better
  score found in database.
• P(x) is probability of finding a score equal or
  better than x by chance.

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Why EVD

• When querying a pattern against a much larger
  protein, there are many possible matches of
  varying RMSD.
• We are looking for the minimum of this set of
  RMSD to represent the entire protein, and hence
  Extreme Value Distribution
• What is known about EVD?

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EVD

• For any CD three possible models for asymptotic
  behavior of EVD
• Double exponent (CD decreasing quickly with good
  scores)
• Exponent of power function (CD has slowly
  decreasing tail)
• Exponent of power function w/ different sign (CD
  has finite terminal, i.e. bound).
• Correct choice of the asymptotic model is crucial
  for accurate statistics

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EVD (2)

• Authors have empirically determined the
  distribution of RMSD by searching query patterns
  against existing structural database (SCOP
  version 1.55).
• Slow increase for small RMSDs is typical for
  power but not exponential functions.

Example distribution of RMSDs for a typical
query. Plot of number of matches versus RMSD for
a search with a random pattern of C atoms (A15,
I30, D60 from PDB entry 1a6m) in a background,
non-redundant database (one member of each of the
723 folds in SCOP).
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Model assuming independence of atoms

• Assumptions
• Only one atom per residue (C?)
• Residues are independent and randomly distributed
  in space

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Model assuming independence of atoms

• Probability of residue from database to match one
  from the query
• increases with allowed volume (RM)
• proportional to the database size (D)
• proportional to residue abundance(?)

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Independence model.

• Schematic two-dimensional representation of
  allowed volumes (green) for placing atoms (grey)
  of a pattern within an RMSD limit (RM). RMSD?RM
  restricts atoms on average to an allowed volume
  described by a sphere of radius RM around the
  ideal positions (V4/3?RM3 ? RM3, top).

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Independence model.

• The first atom is not restricted, but the second
  (middle) must be placed within a shell defined by
  two spheres around the first in a volume V24/3?
  (2RM36dintra2RM) ? 8? dintra2RM ? RM (for
  RMltltdintra).

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Independence model.

• The third atom (bottom) can lie in a ring-like
  volume V3 d'intra(2RM)24 d'intraRM2 ? RM2.

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Disadvantages of independence model

• Not very realistic
• Representing a residue by only one atom is not
  sufficient
• Correct relative orientation of residues rather
  than their simple presence is crucial for
  activity
• Depends on density (rho). As density increases
  it is easier to find better matches

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Dependence model

• Possible to consider multiple atoms per residue
  when calculating RMSD
• In which case covalent bonds violate assumption
  of random and independent atom distribution
• ? the need to develop a model that accounts
  for dependence.

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Dependence model

• First residue is unconstrained
• For the second not only the position of C? has to
  match but also C?, C?, etc. Note that positions
  of C? , C? are relatively fixed.

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Modified EF

• Where S and T are the numbers of query residues
  where two and three atoms are used, respectively.

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P-value

• In square brackets are corrections depending on
  how many atoms were used per residue.

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Points of interest

• No way to tell if matched pattern is indeed on
  the surface
• The way the proteins appear in these databases
  are not random and hence certain patterns may
  appear there more often than others.
• If a query pattern appears on the surface of the
  protein at random it can still be functionally
  important.

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Points of interest

• Which three atoms in the pattern are chosen first
  might affect RMSD.
• good score may not guarantee functionality.
  e.g. little deviations in all atoms VS ideal
  matching of all but one.
• Would be interesting to know at what RMSD
  functionality is lost.

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References

• S.A. Teichmann, A.G. Murzin and C. Chothia ,
  Determination of protein function, evolution and
  interactions by structural genomics. Curr. Opin.
  Struct. Biol. 11 (2001), pp. 354363.
• F.E. Cohen and M.J. Sternberg , On the prediction
  of protein structure the significance of the
  root-mean-square deviation. J. Mol. Biol. 138
  (1980), pp. 321333.
• S. Dietmann and L. Holm , Identification of
  homology in protein structure classification.
  Nature Struct. Biol. 8 (2001), pp. 953957.
• A. Stark, S. Sunyaev, R.B. Russell, A model for
  statistical significance of local similarities in
  structure J. Mol. Biol, 326, 1307-1316, 2003.