Selforganizing Map SOM in Protein Folding Based on HP Model

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Self-organizing Map (SOM) in Protein Folding
Based on HP Model

• Xiang-Sun ZHANG
• ZHANGroup_at_bioinfoamss.org
• http//zhangroup.aporc.org
• 2003.12.2
• 2 Dec. 2003 at NCSU

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Motivation

• We are all concerning what we (OR researchers and
  algorithm designers) can do in Bioinformatics?
• What is the junction of Operations research and
  Bioinfomatics?

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Abstract

• Many problems in Bioinformatics can be formulated
  as large linear/nonlinear integer programming or
  combinatorial problems which are NP-hard and
  unsolvable within existing algorithms. Then
  efficient approxi- mate methods are needed.
• As examples, a heuristic algorithm for SBH and a
  new SOM algorithm for solving the protein HP
  model are presented.
• Other related research works in our group are
  introduced.

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Problem areas in Bioinformatics

• Human Genome Project
• Large molecule data in biology, such as DNA and
  protein
• Genomics (????)
• DNA sequencing
• Gene prediction
• Sequence alignment
• Proteomics(50000 entries in google)/Protenomics
  (hundreds entries in google)(????)
• Structure prediction
• Protein alignment

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• Operations Research
• Over 8 millions entries on google

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DNA Sequencing

• ACGTGATCGATCGAGTACGAGAGTCTA
• _______________________________
• ACGTGATCGATCGAGTACGAGAGTCTA
• ACGTGATCGATCGAGTACGAGAGTCTA
• ACGTGATCGATCGAGTACGAGAGTCTA
• ACGTGATCGATCGAGTACGAGAGTCTA

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• Two pieces of a target sequence with longer
  overlap are preferably connected together, that
  needs that
• ? the average size of the pieces is as long
  
• as possible and
• ? the duplicates of the target sequence are
  
• as many as possible.

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A novel DNA sequencing technique, called
Sequencing By Hybridization (SBH), was proposed
as an alternative to the traditional sequencing
by gel electrophoresis. SBH is based on the DNA
chip (or DNA array). A DNA chip contains all
probes of length (i.e. a short k-nucleotide
fragment of DNA or called a k-tuple). Given a
probe and a target DNA, the target will bind
(hybridize) to the probe if there is a substring
of the target which fits the probe.
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DNA Sequencing

• DNA array (DNA chip)
• AAATGCG(5 3-tuples, a chip with
  3-tuples)

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SBH uses classical probing scheme, i.e., by the
hybridization of an (unknown) DNA fragment with
this chip, the unknown target DNA can be tested
and its all k-tuple compositions (called a
spectrum) determined. SBH provides information
about k-tuples presented in target DNA, but does
not provide information about positions of these
k-tuples. This results in a problem how to
reconstruct the target DNA from this data.
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• Because of the limitation of technology, k
  has not been taken as large as possible yet
  (generally less than 30---already a big chip).
  This possibly leads to the branching phenomenon
  in the sequence reconstruction and multiple
  reconstruction.
• On the other hand, there are two cases of
  errors possibly occur negative errors (i.e. some
  k-tuples in the sequence which are not
  hybridized) and positive errors (i.e. some
  hybridized probes which are not k-tuples in the
  sequence). Therefore, for larger DNA fragments,
  the problem of sequence reconstruction becomes
  rather complicated and hard to analyze.

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• In the case of error-free SBH and ideal
  spectrum (i.e. consists of n-k1 different
  k-tuples where n is the length of the DNA
  fragment), it is known that the SBH
  reconstruction problem is equivalent to finding
  an Eulerian path in a corresponding graph, and
  the algorithm can be implemented in linear time.
• An occurrence of positive and negative errors
  and repetitions of k-tuple in the DNA fragment
  will result in a computational difficulty, i.e.,
  the Problem becomes a strongly NP-hard one.

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Sequencing by Hybridization

• DNA fragment ATACGAAGA
• 
  ß
• Spectrum
• Error Positive (misread) / Negative (missing,
  repetition)

ATA TAC ACG CGA
GAA AAG AGA Ideal case
ATA TAC AGG CGA
GAA AAG AGA With errors
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• 1989,Pevzner, SBH reconstruction problem is
  equivalent to finding an Eulerian path in a
  related graph.
• 1990,Fleischner, the algorithm can be implemented
  in linear time.
• 1991,Dramanac,et al., an algorithm for SBH with
  errors under assumption that only the first or
  last nucleotide in the data can be erroneous.
• 1993,Lipshutz, use empirically derived rates of
  positive and negative errors and other
  assumptions. No convergence analysis.
• 1999,Blazewicz,et al., branch and bound method in
  the case of only positive errors.
• 2000,Blazewicz,et al., a heuristic algorithm
  producing near-optimal solutions.

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SBH Reconstruction Problem

• Design efficient heuristic algorithms
• Ji-Hong Zhang, Ling-Yun Wu and Xiang-Sun Zhang. A
  new approach to the reconstruction of DNA
  sequencing by hybridization. Bioinformatics, vol
  19(1), pages 14-21, 2003.
• Xiang-Sun Zhang, Ji-Hong Zhang and Ling-Yun Wu.
  Combinatorial optimization problems in the
  positional DNA sequencing by hybridization and
  its algorithms. System Sciences and Mathematics,
  vol 3, 2002. (in Chinese)
• Ling-Yun Wu, Ji-Hong Zhang and Xiang-Sun Zhang.
  Application of neural networks in the
  reconstruction of DNA sequencing by
  hybridization. In Proceedings of the 4th ISORA,
  2002.

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Basic Observation

• The spectrum corresponds to a graph each k-tuple
  to a vertex and two connected k-tuples to an
  edge. The structure of the graph is represented
  by
• the adjacency matrix
• A reconstruction of the spectrum is a path in
  the graph. Information about all
• paths are implied in the power of the
  adjacency matrix

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• Some criteria, using information in the power of
  adjacency matrix, which can determine the most
  possible k-tuples at both ends and in the middle
  of all possible reconstructions of the target DNA
  in a polynomial time
• are given.
• A novel means which can transform the negative
  errors into the positive errors is proposed. It
  enables us to handle both types of errors easily.

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Protein Structure Prediction

• Predict protein 3D structure from (amino acid)
  sequence
• Sequence secondary structure 3D structure
  function

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Proteins Secondary Structure

• a-helix (30-35)a-??
• b-sheet / b-strand (20-25)b-??
• Coil (40-50) ?????
• Loop ?
• b-turn b-??

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3D Structure of Protein
Turn or coil
Alpha-helix
Beta-sheet
Loop and Turn
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Protein 3D Structure Detection

• X-ray diffraction
• X-?????
• Expensive
• Slow

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Protein Structure Prediction

• Prediction is possible because
• Sequence information uniquely determines 3D
  structure
• Sequence similarity (gt50) tends to imply
  structural similarity
• Prediction is necessary because
• DNA sequence data protein sequence data
  structure data

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Three Methods of Protein StructurePrediction

• Goal
• Find best fit of sequence to 3D structure
• Comparative (homology) modeling (?????)
• Construct 3D model from alignment to protein
  sequences with known structure
• Threading (fold recognition) (?????)
• Pick best fit to sequences of known 2D / 3D
  structures (folds)
• Ab initio / de novo methods (?????)
• Attempt to calculate 3D structure from scratch
• Molecular dynamics
• Energy minimization
• Lattice models

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Lattice Models

• Suppose that each amino acid occupies one point
  in a space lattice

• It is called an Exact Model

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HP Model (Simple Model)

• Twenty amino acids can be divided into two
  classes Hydrophobic/Non-polar (H)
  (??) Hydrophilic/Polar (P) (??)
• The contacts between H points are favorable

hydrophobic amino acid hydrophilic
amino acid Covalent bond H-H contact

• Goal maximize the number of H-H contacts

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Basic Ideas

• Each acid (neuron) in the primary sequence
  occupies one lattice point (city).
• The distance between two cities mapped by two
  neighboring neurons is forced to be 1 as a
  covalent bond length between the amino acids in a
  protein molecule.
• Move the neurons to have more H-H contacts, I.e.,
  emphasis on forming hydrophobic core.

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Main Observation

• A Traveling Salesman Problem with an energy
  function concerning the H-H contacts that would
  be maximized.

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Mathematical Model (in square lattice)

• Let the both of sequence and lattice size be ,
  let
• for the i-th acid taking the j-th lattice point
  or not. Let
• be the neighboring set of point j.
  
• Let and the
  coordinates of point j be
• 
  

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Complexity

• NP-hard problem even in the case of two
  dimensional HP model
• P.Crescenzi, et al.
• On the complexity of protein folding,
• Journal of Computational Biology, 5(3)
• 423-, 1998
• Many local solutions
• GA MC SA ----- time consuming

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SOM Approach

• Existing algorithm
• Motivated by Self-Organizing-Map for TSP
• Incorporation of HP Information
• Compact lattice
• (the sequence
• exactly fills the
• lattice)
• A 36-long sequence
• In a 6x6 lattice

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New SOM Approach

• Motivation
• Consider a bigger lattice than
• the sequence to have more
• flexible shapes than the only
• rectangular shape
• Equivalent to a PCTSP
• (Price Collecting Traveling
• Salesman Problem) a man
• travels only a part of the city
• set with some expectation.
• Difficulties caused
• Number of cities gt number of neurons

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PCTSP

• A traveling salesman who gets a prize in
  every
• city k that he visits and pays a penalty for
  
• every city that he fails to visit, and who
  travels
• between cities i and j at cost , wants to
  minimize
• the sum of his travel cost and net penalties,
  while
• including in his tour enough cities to collect a
• prescribed amount of prize money.

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The New SOM model is corresponding to the integer
programming

• where mgtn and the total variables are (n1)m.

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New SOM Approach

• Innovate Points
• Heuristic initialization to imitate a protein
• Learning sample set partition strategy
• Learning sample set reduction strategy
• Local search procedure to overcome the
  multi-mapping phenomena

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Numerical Results

• Constructed HP sequences
• (Length of 17)

• HP benchmark (up to 36 amino acids)

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SOM Approach for 2D HP-Model

• Xiang-Sun Zhang, Yong Wang, Zhong-Wei Zhan,
  Ling-Yun Wu, Luonan Chen. A New SOM Approach for
  2D HP-Model of Proteins' Structure Prediction.
  Submitted to RECOMB04.
• Yong Wang, Zhong-Wei Zhan, Ling-Yun Wu, Xiang-Sun
  Zhang. Improved Self-Organizing Map Algorithm for
  Protein Folding and its Realization. Submitted
  to  J. of Systems Science and Mathematical
  Sciences. (in Chinese)

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Main Inprovements

• Find the global maximum H-H contacts
  configurations in all the tests
• Find more optimal conformations
• Fast -- running time is linear with the sequence
  length

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Unique Optimal Folding Problem

• What proteins in the two dimensional HP model
  have unique optimal (minimum energy) folding?
  (Brian Hayes, 1998)
• Oswin Aichholzer proved that in square lattice
• There are closed chains of monomers with this
  property for all even lengths.
• There are open monomer chains with this property
  for all lengths divisible by four.

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Square Lattice and Triangular Lattice
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Our Results

• For any n 18k (k is a positive integer), there
  exists an n-node (open or closed) chain with at
  least optimal foldings all with isomorphic
  contact graphs of size n/2.
• On 2D triangular lattice, for any integer ngt 19,
  there exist both closed and open chains of n
  nodes with unique optimal folding.

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Proteins With Unique Optimal Foldings

• Zhen-Ping Li, Xiang-Sun Zhang, Luo-Nan Chen,
  Protein with Unique Optimal Foldings on a
  Triangular Lattice in the HP Model, Submitted to
  Journal of Computational Biology.

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Examples of Optimal Foldings
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3D Protein Structure Alignment

• Motivation
• Group proteins by structural similarity
• Determine impact of individual residues on
  protein structure
• Identify distant homologues of protein families
• Predict function of proteins with low sequence
  similarity
• Identify new folds / targets for x-ray
  crystallography

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3D Protein Structure Alignment

• Correspondence between atoms
• Pairwise sequence alignment
• Locations of atoms
• Protein Data Bank (in PDB file)
• Bond angles / lengths
• X,Y,Z atom coordinates
• Evaluation metric
• 6 degrees of freedom
• 3 degrees of translation (A)
• 3 degrees of rotation (R)
• Root Mean Square Deviation (RMSD)
• n number of atoms
• di distance between corresponding atoms i

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Structure Alignment Problem
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Match two rigid bodies by rotating and removing
them in the 3D space
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Structure Alignment Problem

• A nonlinear integer programming problem

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Structure Alignment Problem

• Luo-Nan Chen, Tian-Shou Zhou, Yun Tang, Xiang-Sun
  Zhang. Structure of Alignment of Protein by Mean
  Field Annealing. Submitted to ICSB2003.

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On-going Research

• Protein structure prediction
• Algorithms for HP model
• Threading methods
• Protein structure alignment
• Novel model for structure alignment
• SBH reconstruction
• Algorithms for new pattern SBH methods
• SNP(Single Nucleotide Polymorphism) and Haplotype
  analysis

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Summary

• Problems in Bioinformatics are simple in
  description but complicated in solving
• Many problems in Proteomics are in deterministic
  nature
• Combinatorial
• Continuous model
• while many problems in Genomics are in
• stochastic nature
• Model a problem accurately but solves it
• approximately