String Metrics in Classification of Mobile Genetic Elements Discrete Mathematical Biology, Math 8803

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String Metrics in Classification of Mobile
Genetic ElementsDiscrete Mathematical Biology,
Math 8803

• Ryan Wagner
• Biology/Bioinformatics PhD student

www.yale.edu/turner/projects/ecoli.htm
www.geneticengineering.org/evolution
http//pdb.lbl.gov/microscopies
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String Metrics in Classification of Mobile
Genetic Elements

• Mathematical relevance
• Biological relevance
• Test and review of four distance methods
• What was good, bad, and ugly.

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Introduction distances on strings
Formal definition of a distance function, D
D(a, b) 0 ? a b, the identity axiom D(a, b)
D(b, a), the symmetry axiom D(a, b) D(b, c) ?
D(a, c), the triangle inequality
Tree Additivity When a tree is made from a
matrix of pairwise distance metrics, the distance
between any two leaves (sequences) equals the sum
of the edge lengths connecting them (Baake and
Heaseler, 1997).
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Introduction mathematical distance vs.
evolutionary distance

• the three metric properties comprise the basis
  for characterization
• may also be characterized by Turing Machine
  computability (Ahlbrandt et al., 2004)
• amenable to both alignment-based and
  alignment-free methods

• when obtained by common statistical correction
  techniques, fails to satisfy the triangle
  inequality
• the tree additivity property may hold where the
  triangle inequality fails
• not developed for alignment-free distances on DNA
  strings

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Introduction mobile genetic elements
Plasmid - an autonomous, self-replicating
circular piece of DNA found outside the
chromosome in bacteria. www8.nos.noaa.gov/coris_g
lossary
Bacteriophage - a virus that attacks and infects
bacterial cells.
www.ncbi.nlm.nih.gov/ICTBdb/ICTVdB
Transposon - a DNA sequence capable of moving to
new locations within the same cell
www.microbe-edu.org/etudiant
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Methods data collection and software

• DNA sequences for replication initiation (RepA)
  and division partition (ParA) in both plasmids
  and host chromosomes obtained from NCBI,
  www.ncbi.nlm.nih.gov/genomes/lproks.cgi
• DNA sequences of selected plasmids from
  gram-negative bacteria also obtained from NCBI
• Neighbor-joining trees constructed for each set
  of pairwise distances using PHYLIP,
  http//evolution.genetics.washington.edu/phylip.ht
  ml
• Custom Perl script used to generate matrices of
  pairwise distances

4 G_lovleyi 0.000000 0.864000 0.887000
0.844000 Acidovoro 0.864000 0.000000 0.664000
0.724000 Acid_JS42 0.887000 0.664000 0.000000
0.836000 Xanth_axo 0.844000 0.724000 0.836000
0.000000
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Methods edit distance
Data structure in custom script for test
input ATTGCGAGC and ATGCGACC
A T G C G A C C 0 1 2 3 4 5 6 7 8 A 1 0 1 2 3 4
5 6 7 T 2 1 0 1 2 3 4 5 6 T 3 2 1 2 3 4 5 6 7 G 4
3 2 1 2 3 4 5 6 C 5 4 3 2 1 2 3 4 5 G 6 5 4 3 2 1
2 3 4 A 7 6 5 4 3 2 1 2 3 G 8 7 6 5 4 3 2 3 4 C 9
8 7 6 5 4 3 2 3 Levenshtein distance 3, from
lower right corner, no traceback needed
Here is where horizontal gene transfer begins to
cause problems.
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Methods the problem with edit distance
Consider GTGACGTACTATTGC_ and GTGAGTACTATTGCC
1 character delete/insert Edit distance 2
Consider GTGACGTACTATTGC_ and GTACTATTGCGTGAC
5 character delete/insert Edit distance 8
Allowing block deletions, block insertions, and
block reversals confers better approximations to
the recombinant nature of DNA evolution
(Long-Hui, 2004).
However, the least-constrained application of
block edit distance has O(n3) time complexity.
Constrained block edit distance computation is
NP-hard (Lopresti and Tomkins, 1997)
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Methods Euclidean distance over dinucleotide
counts
A new paradigm complexity-based distance
metrics which do not employ alignments nor
dynamic programming
a GTGACGTACTATTGC b GTACTATTGCGTGAC
TC GA TG CA CT AG AC GT TT AA CC
GG CG AT GC TA
L2 (1/16)? ?aij ? ?bij , where ?aij
freq(ij)/(freq(i) ? freq(j)) here L2 0
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Methods compression distance by the
Burrows-Wheeler transform (scheme from Mantaci et
al., 2008)
a0 GTGACGTACTATTGC b0 GTACTATTGCGTGAC
a1 TGACGTACTATTGCG b1 TACTATTGCGTGACG
a2 GACGTACTATTGCGT b2 ACTATTGCGTGACGT
a3 ACGTACTATTGCGTG b3 CTATTGCGTGACGTA
...
...
a14 CGTGACGTACTATTG b14 CGTACTATTGCGTGA
Blue list
Red list
Merge lists
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Merged list is then sorted
ACGTACTATTGCGTG G ACGTACTATTGCGTG
G ACTATTGCGTGACGT T ACTATTGCGTGACGT
T ATTGCGTGACGTACT T ATTGCGTGACGTACT
T CGTACTATTGCGTGA A CGTGACGTACTATTG
G CTATTGCGTGACGTA A CTATTGCGTGACGTA
A GACGTACTATTGCGT T GACGTACTATTGCGT
T GCGTGACGTACTATT T GCGTGACGTACTATT
T GTACTATTGCGTGAC C GTACTATTGCGTGAC
C GTGACGTACTATTGC C GTGACGTACTATTGC
C TACTATTGCGTGACG G TACTATTGCGTGACG
G TATTGCGTGACGTAC C TATTGCGTGACGTAC
C TGACGTACTATTGCG G TGACGTACTATTGCG
G TGCGTGACGTACTAT T TGCGTGACGTACTAT
T TTGCGTGACGTACTA A TTGCGTGACGTACTA A
BRB R B R B R B R B R R B B R R B R B B R R B B R
R B
Else, sum up total unequal colors
Column of last characters is the Burrows-Wheeler
transform.
Distance 2
Note runs of nucleotides.
If color counts in each segment of runs is equal,
sum is 0.
Sequence color is then correlated to
Burrows-Wheeler column
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Methods rank distance
Related to Hamming distance, but less sensitive
to insertions/deletions (from Dinu and Sgarro,
2006)
a GTGACGTACTATTGC b GTACTATTGCGTGAC

• Index each base and correlate it to its position
  in the sequence relative to the other sequence
• e.g. count the first occurrence of G in a and b,
  compute the difference in their positions,
• count the second occurrence of G in a and b,
  compute the difference in their positions,

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Methods rank distance
a GTGACGTACTATTGC b GTACTATTGCGTGAC
position difference 0
position difference 6
position difference 5
Sum rank counts for G
Repeat procedure for T, A, and C
Sum rank counts for all four bases and normalize
by arithmetic mean of sequence length
Distance 0.01667, c.f. normed edit distance
0.5333
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Results of attempt to cluster by mobile element
type
Sequences of different taxonomic groups paired
closely - diagnostic of mobile genetic elements
Multiple sequence alignment-based NJ tree -
customary bioinformatics.
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Results of attempt to cluster by mobile element
type
Edit distance tree gives same topology
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Results of attempt to cluster by mobile element
type
Dinucleotide counts over Euclidean distance and
Rank distance successfully group two plasmids
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Results of attempt to cluster by mobile element
type
Burrows-Wheeler compression pairwise distances do
not give a clear clustering.
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Why did the BWT distances not perform well?
RepA-ParA sequence data were too short for useful
shared repeat regions to appear.
Remedy Run complete plasmid sequences through
BWT distance script
Insurmountable problem the BWT distance script,
as given, could not compute distances on whole
plasmids.
Diagnosis time-complexity of BWT is O(nlog(n)),
but space complexity is O(n2)
Mantaci et al. also found their BWT distance does
not satisfy the triangle inequality (2008)
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Can dinucleotide counts or rank distance be made
to perform better in separating mobile elements?
Li et al (2004) used trinucleotide counts
combined with higher-order nucleotide word counts
to accurately infer an evolutionary tree of
mammalian mitochondrial DNA.
Such simple methods cannot hope to approximate
Kolomogorov complexity distance.
Recall that Kolmogorov complexity is related to
the length of the Turing Machine needed to
transform sequence a into sequence b (Li et al.,
2004).
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Open issues

• So far, only dinucleotide counts have been
  developed for clustering of mobile elements
  (Blaisdell and Karlin, 1996)
• BWT distance and Rank distance were developed to
  cluster mammalian mitochondrial DNA (Mantaci et
  al.,2008 Dina and Sgarro, 2006).
• Rank distance not shown to satisfy triangle
  inequality
• Can it be proven whether or not a pairwise
  distance satisfying the triangle inequality
  yields an additive tree.

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References

• Ahlbrandt, C., Benson, G., and Casey, W. (2004)
  Minimal entropy probability between genome
  families. Journal of Mathematical Biology.
  48563-590.
• Baake, E. (1998) What can and cannot be inferred
  from pairwise sequence comparisons?
  Mathematical Biosciences. 1541-21
• Blaisdell, B.E., Campbell, A.M., and Karlin, S.
  (1996) Similarities and dissimilarities of phage
  genomes. Proc. Natl. Acad. Sci. 935854-5859.
• Dinu, L.P. and Sgarro, A. (2006) A
  low-complexity distance for DNA strings.
  Fundamenta Informaticae. 76361-372.
• Li. M, Chen, X., Li, X., Ma, B., and Vianyi, P.
  (2004) The similarity metric. IEEE
  Transactions on Information Theory XX(Y)
• Long-Hui, W., Juan, L., Zhou, H-B., and Feng,
  Shi. (2004) "A new distances metric and its
  application in phylogenetic tree construction."
  Proceedings of the 2004 IEEE Symposium on
  Computational Intelligence in Bioinformatics and
  Computational Biology.
• Lopresti, D. and Tomkins, A. (1997) "Block edit
  models for approximate string matching."
  Theoretical Computer Science. 181159-179
• Mantaci, S., Restivo, A., and Sciortino. (2008)
  Distance measure for biological sequences Some
  recent approaches. International Journal of
  Approximate Reasoning. 47109-124.