String Metrics in Classification of Mobile Genetic Elements

Discrete Mathematical Biology, Math 8803

www.yale.edu/turner/projects/ecoli.htm www.geneticengineering.org/evolution

Ryan WagnerRyan Wagner

Biology/Bioinformatics PhD studentBiology/Bioinformatics PhD student

http://pdb.lbl.gov/microscopies

String Metrics in Classification of Mobile Genetic Elements

1. Mathematical relevance

2. Biological relevance

3. Test and review of four distance methods

4. What was “good, bad, and ugly”.

Introduction: distances on strings

D(a, b) = 0 a = b, the identity axiomD(a, b) = D(b, a), the symmetry axiomD(a, b) + D(b, c) D(a, c), the triangle inequality

Formal definition of a distance function, D

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).

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

Introduction: mobile genetic elements

Plasmid - an autonomous, self-replicating circularpiece of DNA found outside the chromosome

in bacteria. www8.nos.noaa.gov/coris_glossary

Bacteriophage - a virus that attacks and infects bacterial cells.www.ncbi.nlm.nih.gov/ICTBdb/ICTVdB

www.microbe-edu.org/etudiant

Transposon - a DNA sequence capable of moving to new locations within the same cell

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.html

• Custom Perl script used to generate matrices of pairwise distances:

4G_lovleyi 0.000000 0.864000 0.887000 0.844000Acidovoro 0.864000 0.000000 0.664000 0.724000Acid_JS42 0.887000 0.664000 0.000000 0.836000Xanth_axo 0.844000 0.724000 0.836000 0.000000

Methods: edit distance

Here is where horizontal gene transfer begins to cause problems.

Data structure in custom script for test input:ATTGCGAGC and ATGCGACC

A T G C G A C C0 1 2 3 4 5 6 7 8

A 1 0 1 2 3 4 5 6 7T 2 1 0 1 2 3 4 5 6T 3 2 1 2 3 4 5 6 7G 4 3 2 1 2 3 4 5 6C 5 4 3 2 1 2 3 4 5G 6 5 4 3 2 1 2 3 4A 7 6 5 4 3 2 1 2 3G 8 7 6 5 4 3 2 3 4C 9 8 7 6 5 4 3 2 3

Levenshtein distance = 3, from lower right corner, no traceback needed

Methods: the problem with edit distance

Consider GTGACGTACTATTGC_ and

GTACTATTGCGTGAC

Consider GTGACGTACTATTGC_ and

GTGAGTACTATTGCC1 character delete/insert

Edit distance = 2

5 character delete/insertEdit 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)

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

Computation of counts vectors for a and b

dinucleotide a b L2

TC + GA 1 1 0

TG + CA 2 2 0

CT + AG 1 1 0

AC + GT 4 4 0

TT + AA 1 1 0

CC + GG 0 0 0

CG 1 1 0

AT 1 1 0

GC 1 1 0

TA 2 2 0

L2 = (1/16)[∑ | a*ij b*ij |],

where a*ij = freq(ij)/(freq(i) freq(j))

here L2 = 0

TC + GA

TG + CA

CT + AG

AC + GT

TT + AA

CC + GG

CG

AT

GC

TA

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

Merge lists“Blue” list “Red” list

Merged list is then sorted:

ACGTACTATTGCGTG GACGTACTATTGCGTG GACTATTGCGTGACGT TACTATTGCGTGACGT TATTGCGTGACGTACT TATTGCGTGACGTACT TCGTACTATTGCGTGA ACGTGACGTACTATTG GCTATTGCGTGACGTA ACTATTGCGTGACGTA AGACGTACTATTGCGT TGACGTACTATTGCGT TGCGTGACGTACTATT TGCGTGACGTACTATT TGTACTATTGCGTGAC CGTACTATTGCGTGAC CGTGACGTACTATTGC CGTGACGTACTATTGC CTACTATTGCGTGACG GTACTATTGCGTGACG GTATTGCGTGACGTAC CTATTGCGTGACGTAC CTGACGTACTATTGCG GTGACGTACTATTGCG GTGCGTGACGTACTAT TTGCGTGACGTACTAT TTTGCGTGACGTACTA ATTGCGTGACGTACTA A

BRBRBRBRBRBRRBBRRBRBBRRBBRRB

Column of last characters is the Burrows-Wheeler transform.

Note “runs” of nucleotides.

Sequence “color” is then correlated to Burrows-Wheeler column

If color countsin each segment of runs is equal, sum is 0.

Else, sum up total unequal colors

Distance = 2

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, …

Methods: “rank” distance

a = GTGACGTACTATTGC b = GTACTATTGCGTGAC

positiondifference= 0

positiondifference= 6

positiondifference= 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

Results of attempt to cluster by mobile element type

Multiple sequence alignment-based NJ tree - customary bioinformatics.

Sequences of different taxonomic groups paired closely - diagnostic of mobile genetic elements

Results of attempt to cluster by mobile element type

Edit distance tree gives same topology

Results of attempt to cluster by mobile element type

Dinucleotide counts over Euclidean distance and Rank distance successfully group two plasmids

Results of attempt to cluster by mobile element type

Burrows-Wheeler compression pairwise distances do not give a clear clustering.

Why did the BWT distances not perform well?

Insurmountable problem: the BWT distance script, as given, could not compute distances on whole plasmids.

Diagnosis: time-complexity of BWT is O(n·log(n)), but space complexity is O(n2)

RepA-ParA sequence data were too short for useful shared repeat regions to appear.

Remedy: Run complete plasmid sequences through BWT distance script

Mantaci et al. also found their BWT distance does not satisfy the triangle inequality (2008)

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).

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.

ReferencesAhlbrandt, C., Benson, G., and Casey, W. (2004) “Minimal entropy probability between

genome families.” Journal of Mathematical Biology. 48:563-590.

Baake, E. (1998) “What can and cannot be inferred from pairwise sequence comparisons?” Mathematical Biosciences. 154:1-21

Blaisdell, B.E., Campbell, A.M., and Karlin, S. (1996) “Similarities and dissimilarities of phage genomes.” Proc. Natl. Acad. Sci. 93:5854-5859.

Dinu, L.P. and Sgarro, A. (2006) “A low-complexity distance for DNA strings.” Fundamenta Informaticae. 76:361-372.

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. 181:159-179

Mantaci, S., Restivo, A., and Sciortino. (2008) “Distance measure for biological sequences: Some recent approaches.” International Journal of Approximate Reasoning. 47:109-124.