Alignment-free sequence comparison

Analysis of Biological Sequences 140.638

Why not just align the sequences?

• Alignment scoring can be arbitrary• current alignment algorithms are not scalable: tedious and slow to do

sequence alignment on a large scale (especially short read sequencing)• sequences may not align to each other well enough to give recognizable

distances (gaps etc)• alignment algorithms assume generally collinear sequences

Why not just align the sequences?

• below “twilight zone” of 60-65% identity (nucleotide) or 20-35% identity (protein), alignments are not accurate

• memory and time consuming (prohibitive for multiple genomes)• algorithms make implicit assumptions about evolutionary trajectories of

sequences being compared

resolution-free sequence comparison methods

• word counting/composition comparison• Universal sequence maps (CGR)• Kolmogorov complexity• Complete composition vectors

word-based distance

word-based distances

• word size 2-6 works well for protein comparisons• 8-10nt words useful for DNA or RNA• long words (~25nt) can distinguish very closely related bacterial species in

metagenomics applications

word-based approach

word-based distances

• determine relative frequency of each word:

Oij = # times word Oj appears in sequence i

fij = frequency of word Oj in sequence i

word-based distances

comparing two sequences x and y:

used this method to compare mitochondrial DNA from primate species

improved word-based methods

a single change in a word creates a new word -- biologically realistic?

instead use word neighborhoods e.g.CATTATT, CATTATA, CATTAAT...

N2 similarity score

N2 similarity score

• defines a (potentially weighted) set of words that are the “neighborhood” of any word

• compute word neighborhood counts• correct for inter-variable frequency (e.g. observations of CAAAA and AAAAA

are strongly correlated)• correct for word covariance• normalize so that all word frequencies sum to one

N2 similarity score: distinguishing enhancers

D2 score

if XW is the count of word W in the sequence X,

D2 is Poisson distributed

improvements to D2 score

D2S is normally distributed, D2* is the sum of independent normally distributed variables

metagenomics with 5-tuples

k-tuple scores and metagenomics

metagenomics with D2*

clustering of gut bacteria from foregut fermenters, hindgut fermenters, and carnivores

more metagenomics

speeding up kmer distances

CAFE workflow

source sequences can be whole genomes, contigs, or short reads

CAFE results

resolution-free sequence comparison methods

word counting/composition comparisonComplete composition vectorsUniversal sequence maps (CGR)Kolmogorov complexity

Universal sequence maps

Chaos theory / Chaos game representationIterative functions to represent biological sequencesCan be generalized to any order alphabet (thus “universal”)

Chaos game representation

Plot sequence in a square with vertices labeled A,C,T,G1st nt is plotted halfway between the center of the square and the vertex labeled with that ntSubsequent nts are plotted halfway between previous dot and the vertex labeled with the new nt-> 2D plot representing 1° DNA sequence for ANY lengthpatterns are usually fractal

Chaos game representation

ACG C

A T

G

Chaos game representation

Chaos game representation

Sierpiński Sieve

Chaos game representation

Genomic signature: dinucleotide & trinucleotide relative abundance profiles distinguish between organisms and sequence segments, and can be used in phylogenetic analysis

We see less variation of CGR along genomes than between genomes -> related to genomic signature?

Chaos game representation

What determines the pattern in a CGR?Short nucleotide frequencies don’t solely determine the patternFor a DNA sequence, one can construct a simulated sequence with the same length and nucleotide compositionIF CGRs are the same, then nucleotide and dinucleotide frequencies are all that’s important

making CGR computable

• Hatje & Kollmar divide the CGR grid to outline short oligos & then get frequencies of those oligos

• Almeida et al proved that the length of the common prefix between two CGR is the dissimilarity distance

3D Chaos game representation (HPV)

computing feature vectors from 3D CGR (HPV)

Kolmogorov complexity

K(x) is the shortest binary program that can compute the string x on a binary computerK(x|y) is the shortest binary program that can compute the string x, given the information from y

NID(x,y) = max[K(x|y), K(y|x)]/max[K(y), K(x)] is the normalized information distance (0 ≤ NID ≤ 1)Can be shown that NID(x,y) can express all other distances between x and y!. . . Not computable though :(

Kolmogorov complexity

NID(x,y) = max[K(x|y), K(y|x)]/max[K(y), K(x)]

NCD = normalized compression distance, approximation to NID

NCD(x,y) = (min[C(xy),C(yx)] - min[C(x),C(y)])/max[C(x),C(y)]

Where C(x) is the compressibility of the string x, C(xy) is the compressibility of the string x concatenated to y etc. Can just use gzip!

x: AAAAAAAy: ACGAATAxy: AAAAAAAACGAATAyx: ACGAATAAAAAAAA

>seq1TAGAAATAAATGGAAAGTCAGTAAATGTGTGGCCTGTTAAAATTCTTGGAGAATATACATCACCACTTTCCTCCAAAAATGGGAATAGAATTAGTTCGAATAATTTAGAGAAAAGCACCAACAAACAAATCCACTCAGAATTCTCCATTTCTAGATTGCCCAGAACTAGGCCACGGCAGCTGGGTTCTGAGCAAGACAGTGAGGTTTTCCCTTCCGACCAGGGTGTCAAGAAGAATTGTAAGCAGATTGAATCTGCTAAATTATTACCTGATACACCCGTTCAATTCATACCTCCAAATACATTGAACCTTCGTAGCTTTACCAAGATCATAAAGAGACTGGCTGAACTGCATCCAGAAGTCAGCAGAGACCATATTATAAATGCACTTCAGGAAGTGAGAATAAGACATAAAGGTTTTCTGAATGGCTTATCTATTACTACTATTGTGGAGATGACTTCATCTCTTCTGAAAAACTCTGCTTCCAGTTAGGAATTCAAAAAACAATAAAGAGAACTTCCTTGGAAAGTGTGTTTCCTCCTTCAGAGAATGTTCTACAGCACTTAGGAAAAAGTAGTAATAACAAGATGATGTAATTAAATAGGCTCTATAAATGGGCTAAGCTGTTAAAATATTCTACTTTATATCCCTCCTTTAAAATCTAGCAACAGTTGTCTATACAATATTAAGATCTTCTCTATATATTTAAAGTTAAAATATAATTTTTAATAAGTTTTTAAATTTTTTTATTTCAATTTTGTTACTTAGAACATTAAGATGCATATTTGTGATCTAAAGAAATTGTCTTGTCCATTTTAAAAACCTTTATTAAGTCACTTTTAAAATGTATTGACCAAGAAGGAGGTTTGTTGTTACATCAATGTTTGTGAAATGATTTCCATACATAAAAAATGTAATTTACCTGAACTTTGTCTTAAGACTCTTACATTGGATTATAGGATAACAGATAAATAAACTGTATAGATACATTCAGTATCATACAACATTTTGGAATGTGTATGCTTTCAGGCTTCCAAGATAATTAAATTACTAGAAATAAATGGAAAGTCAGTAAATGTGTGGCCTGTTAAAATTCTTGGAGAATATACATCACCACTTTCCTCCAAAAATGGGAATAGAATTAGTTCGAATAATTTAGAGAAAAGCACCAACAAACAAATCCACTCAGAATTCTCCATTTCTAGATTGCCCAGAACTAGGCCACGGCAGCTGGGTTCTGAGCAAGACAGTGAGGTTTTCCCTTCCGACCAGGGTGTCAAGAAGAATTGTAAGCAGATTGAATCTGCTAAATTATTACCTGATACACCCGTTCAATTCATACCTCCAAATACATTGAACCTTCGTAGCTTTACCAAGATCATAAAGAGACTGGCTGAACTGCATCCAGAAGTCAGCAGAGACCATATTATAAATGCACTTCAGGAAGTGAGAATAAGACATAAAGGTTTTCTGAATGGCTTATCTATTACTACTATTGTGGAGATGACTTCATCTCTTCTGAAAAACTCTGCTTCCAGTTAGGAATTCAAAAAACAATAAAGAGAACTTCCTTGGAAAGTGTGTTTCCTCCTTCAGAGAATGTTCTACAGCACTTAGGAAAAAGTAGTAATAACAAGATGATGTAATTAAATAGGCTCTATAAATGGGCTAAGCTGTTAAAATATTCTACTTTATATCCCTCCTTTAAAATCTAGCAACAGTTGTCTATACAATATTAAGATCTTCTCTATATATTTAAAGTTAAAATATAATTTTTAATAAGTTTTTAAATTTTTTTATTTCAATTTTGTTACTTAGAACATTAAGATGCATATTTGTGATCTAAAGAAATTGTCTTGTCCATTTTAAAAACCTTTATTAAGTCACTTTTAAAATGTATTGACCAAGAAGGAGGTTTGTTGTTACATCAATGTTTGTGAAATGATTTCCATACATAAAAAATGTAATTTACCTGAACTTTGTCTTAAGACTCTTACATTGGATTATAGGATAACAGATAAATAAACTGTATAGATACATTCAGTATCATACAACATTTTGGAATGTGTATGCTTTCAGGCTTCCAAGATAATTAAATTAC

>seq2ATTTATAGAGAAGCCAGTGTTAAGCCGTACTTAAGGTTCACATTTGTAATGAAATAGGTAACTGGGCCTCCACAAGTTCCATGGGAATCGCAGACTAACCATTTGGTTTTCCTCTGCCTCATTTTCTCCTCCTCCTCCTGCTCCTCCTCTTCCTCCTCCCCTCTCTTTAGCATCCTCCTCCTCCTTCTTCTTCTACATCCTCCTTTTCCTCTTCCTCCTCCATCTTCTCCTCTCCTTCTCCTCTTCCTCCCCTTCTTCATCTATTCATTCTTCCTTGAGCCTCCTGGCCCACTAGGGCCCTTCTATCTTGCATCACCTCTGCCCTCTCAAGGCATGCAATATCCTGTATCTCATTCTTCCTTTAGTTCAGCTGCCTTCTCTTCACATGGTGGTCTATCTTGGGCTGTCTGCTCAGACCACATCTCACCCAATTTCCTTGCTACATTCCCAGTGGACAAGCCCGGTGATTCACTCTTGATCTTTGGACAATATTCAGAATGAAGCAGGAAGAAAGCAAGCGGTAGTCTTTTGTGAGTACCTAAGTCTTCATTTTTCTTCAGGTCCTTTCTTATTGCCTTTAAGAGGAACATAATTCTTCATCAGCTATCATAGCCTCAGAGCAAGCCTTGTCACTTGGAGCTGTATCTTCAGGTTTCACCTTTTCCTTTGTAGGCATGAAGGTCCTCTCCAAGAACTCAGCAAAGCTGACTGGACCCAGGCATTTCTTTCTGTTCTCCTGGAAGTCTGCAGGAAGACAGCTCCTGGGCCTTTTCTTCCTCCAGCCAACCCAGTCTCCTTCACCCAAGGTGACCCATGGCGTGCGGGGAGAAGGGGGGCTCTATCTGAGTGGGCTTTTTCCTGAGTCCAAACCAGATGCTTCCTTCTCCATACGATTGTCAGCTGGCTTCACTTTTCATATTATTTTAAGCTTTAATTATTTTTCTCTCCTTGCAGAGCAACAATTGTGGTAATAAAACCAGATACCAACTCTTATCTCAGGTTAGTAATAAAGTTGTTGCCTACTATCTAGAAATGTACCTGCCTTTTCTTTTTTCTTTCCTTTTCTCTTTCCTTTCCTTTCCTTTCCTTTCCTTTTCGTTTCTTTTCTTTTCTGTAAAATGTGGCAATTTACAGGTTGGGATGTATCACCGTTGGTGGAGTGTTTACCTAGCTAGTATATACAAAGCCCTTGTTTAAATTCCTAGCACTGGGTAGGTATGGTGACTCGTGTCTGTAATCTCAGAACTCTACAGGTAGACATGTGGGAAGCAGAAATTCATCCTCAGCATACAGTGAGTTTCAAGTTGGCCTGACCCAGAAGAACTCAGGGGAAAAAAGCTGATGTCTTTTCTCTCTCTCTCTCTCTCTTTCTCTCTCTCTCTCTCTCTCTCTCTCTTTCTCTCTTTCATAATTCTTTTGGTAGAGAGAAGGAAAGAGATGAACATGTATTAAGTTCCCTGGTATCTACCAAATTTGTGTATTACTTGTCGGTTAATATTATAACAAACATTAAATTGTATTCAGAACCATATTTTGATTATTATCTTTGTGTGCTTTGGATCTCACGACAGTAATAGTTACCTGAGGTGCTTAACTACCGTTTCTGTGACAGTAAATTATTTAAGTTTACTCTCTCCCTCTACAGCCCAACAGTGTGTAGTTTGTATGGTTCATTTGTTGTTGGCTTGTTGTTATTGATGTTGTTTGTGTTGCTGATGCTAGAGTCTGGGGCCTTGGACATATTCGGCAGGCAAATGCTCCACCACTGAGCCTCCAGCCACTTTGCTGGAGGTTTTTGTAGCTGTAGATTGTAATGAAGAAGTTTTTCATCTTTTATATTTGAAAAAGATACCACGGCACGATACACAGCTACAACCAATGCACTAAGATAAATAACCAACCCAACAGAGTGACATTATGATGCAGTAGTTGTAAGAATCAATTTAAAAGATATATCACTTCATCCTTGGGTTTGCCTATGTTCTCATCTGTGAGATTTAAAATCTTTTGAAACATTGAATGAAGCCTCTCATCTATCATCAACTGCCATTAAATATCACATATTCACAGCTGGAGAAATGGACCAGCCGACATCCGGAA

sequence comparison by gzip

bytes filename 2130 seq1 2130 seq2 4260 seq1.seq1 4260 seq2.seq2 4260 seq1.seq2 4260 seq2.seq1

sequence comparison by gzip

bytes filename 2130 seq1 2130 seq2 4260 seq1.seq1 4260 seq2.seq2 4260 seq1.seq2 4260 seq2.seq1NCD(x,y) = (min[C(xy),C(yx)] - min[C(x),C(y)])/max[C(x),C(y)]= (min(1104/4260, 1088/4260) - min(766/4260, 443/4260))/max(766/4260, 443/4260)= (1088/4260 - 443/4260) / (766/4260) = 0.84

bytes filename422 seq1.gz735 seq2.gz443 seq1.seq1.gz766 seq2.seq2.gz1104 seq1.seq2.gz1088 seq2.seq1.gz