Modern Homology Search
Ming Li
Canada Research Chair in Bioinformatics
Computer Science. U. Waterloo
I want to show you how one simple
theoretical idea can make big impact to
a practical field.
Outline
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Motivation, market, biology, homology search
Review dynamic programming, BLAST heuristics
Optimized spaced seeds
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Mathematical theory of spaced seeds
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The idea
How to compute them
Data models, coding regions, HMM, vector seeds
Multiple optimized spaced seeds
PatternHunter program
Why they are better
Complexity of finding optimal spaced seeds
Applications beyond bioinformatics & Open problems
1. Introduction
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Biology
Motivation and market
Definition of homology search problem
Biology
DNA
(Genotype)
Protein
Phenotype
T
A
A
T
C
G
T
A
Human: 3 billion bases, 30k genes.
E. coli: 5 million bases, 4k genes
cDNA
reverse transcription
A
G
C
G
T
C
G
T
C
G
T
A
mRNA
(A,C,G,U)
C
A
translation
transcription
Protein
(20 amino acids)
Codon: three nucleotides encode
an amino acid.
64 codons
20 amino acids, some w/more codes
A
T
A gigantic gold mine
The trend of genetic data growth
Nucleotides(billion)
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8
7
6
5
4
3
2
1
0
1980
30 billion
in year 2005
1985
1990
1995
2000
Years
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400 Eukaryote genome projects underway
GenBank doubles every 18 months
Comparative genomics  all-against-all search
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Software must be scalable to large datasets.
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Homology search
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Given two DNA sequences, find all local
similar regions, using “edit distance”
(match=1, mismatch=-1, gapopen=-5, gapext=-1).
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Example. Input:
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E. coli genome: 5 million base pairs
H. influenza genome: 1.8 million base pairs
Output: all local alignments (PH demo)
java –jar phn.jar –i ecoli.fna –j hinf.fna –o out.txt –b
Homology search vs Google search
Internet search
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Size limit: 5 billion people x homepage size
Supercomputing power used: ½ million CPU-hours/day
Query frequency: Google --- 112 million/day
Query type: exact keyword search --- easy to do
Homology search
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Size limit: 5 billion people x 3 billion basepairs +
millions of species x billion bases
10% (?) of world’s supercomputing power
Query frequency: NCBI BLAST -- 150,000/day,
15% increase/month
Query type: approximate search
Bioinformatics Companies living
on Blast
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Paracel (Celera)
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TimeLogic (recently liquidated)
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TurboGenomics (TurboWorx)
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NSF, NIH, pharmaceuticals proudly support many
supercomputing centers mainly for homology search
Hardware high cost & become obsolete in 2 years.
History
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Dynamic programming (1970-1980)
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BLAST, FASTA heuristics (1980-1990)
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Full sensitivity, but slow.
Human vs mouse genomes: 104 CPU-years
Low sensitivity, still not fast enough.
Human vs mouse genomes: 19 CPU-years
BLAST paper was referenced 100000 times
Modern technology: full sensitivity & greater
speed!
2. Old Technology
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Dynamic programming – full sensitivity
homology search
BLAST heuristics --- trading sensitivity
with time.
Dynamic Programming:
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Longest Common
Subsequence (LCS).
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V=v1v2 … vn
W=w1w2 … wm
s(i,j) = length of LCS
of V[1..i] and W[1..j]
Dynamic Programming:
s(i,j)=max
s(i-1,j)
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Sequence Alignment
(Needleman-Wunsch, 1970)
s(i,j)=max
s(i-1,j) + d(vi,-)
s(i,j-1)+d(-,wj)
s(i-1,j-1)+d(vi,wj)
0 (Smith-Waterman, 1981)
* No affine gap penalties,

s(i,j-1)
s(i-1,j-1)+1,vi=wj

where d, for proteins, is either
PAM or BLOSUM matrix. For
DNA, it can be: match 1,
mismatch -1, gap -3 (In fact:
open gap -5, extension -1.)
d(-,x)=d(x,-)= -a, d(x,y)=-u.
When a=0, u=infinity, it is LCS.
Misc. Concerns
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Local sequence alignment, add s[i,j]=0.
Gap penalties. For good reasons, we charge first
gap cost a, and then subsequent continuous
insertions b<a.
Space efficient sequence alignment. Hirschberg
alg. in O(n2) time, O(n) space.
Multiple alignment of k sequences: O(nk)
Blast
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Popular software, using heuristics. By Altschul, Gish, Miller,
Myers, Lipman, 1990.
E(xpected)-value: e= dmne -lS, here S is score, m is database
length and n is query length.
Meaning: e is number of hits one can “expect” to see just by
chance when searching a database of size m.
Basic Strategy: For all 3 a.a. (and closely related) triples,
remember their locations in database. Given a query
sequence S. For all triples in S, find their location in
database, then extend as long as e-value significant. Similar
strategy for DNA (7-11 nucleotides). Too slow in genome
level.
Blast Algorithm
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Find seeded (11 bases) matches
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Extent to HSP’s (High Scoring Pairs)
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Gapped Extension, dynamic programming
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Report all local alignments
BLAST Algorithm Example:
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Find seeded matches of 11 base pairs
Extend each match to right and left, until the
scores drop too much, to form an alignment
Report all local alignments
Example:
0001 1 1 011 1 11 1 1 1 11 1001101 11 10
AGCGATGTCACGCGCCCGTATTTCCGTA
G
| | | | | | x| | | | | | | |
| | | | ||
TCGGATCTCACGCGCCCGGCTTACCGTG
BLAST Dilemma:
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If you want to speed up, have to use a
longer seed. However, we now face a
dilemma:
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increasing seed size speeds up, but loses
sensitivity;
decreasing seed size gains sensitivity, but
loses speed.
How do we increase sensitivity & speed
simultaneously? For 20 years, many
tried: suffix tree, better programming ..
3. Optimized Spaced Seeds
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Why are they better
How to compute them
Data models, coding regions, HMM
PatternHunter
Multiple spaced seeds
Vector seeds
New thinking
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Three lines of existing approaches:
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Smith-Waterman exhaustive dynamic programming.
Blast family: find seed matches (11 for Blastn, 28 for
MegaBlast), then extend. Dilemma: increasing seed size
speeds up but loses sensitivity; descreasing seed size gains
sensitivity but loses speed.
Suffix tree: MUMmer, Quasar, REPuter. Only good for precise
matches, highly similar sequences.
Blast approach is the only way to deal with real and
large problems, but we must to solve Blast dilemma:
We need a way to improve sensitivity and speed
simultaneously.
Goals: Improve (a) Blast, (b) Smith-Waterman.
Optimal Spaced Seed
(Ma, Tromp, Li: Bioinformatics, 18:3, 2002, 440-445)
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Spaced Seed: nonconsecutive matches and
optimized match positions.
Represent BLAST seed by 11111111111
Spaced seed: 111*1**1*1**11*111
 1 means a required match
 * means “don’t care” position
This seemingly simple change makes a huge
difference: significantly increases hit prob. to
homologous region while reducing bad hits.
Formalization
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Given i.i.d. sequence (homology region)
with Pr(1)=p and Pr(0)=1-p for each bit:
1100111011101101011101101011111011101
111*1**1*1**11*111
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Which seed is more likely to hit this region:
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BLAST seed: 11111111111
Spaced seed: 111*1**1*1**11*111
Sensitivity: PH weight 11 seed vs Blast 11 & 10
PH 2-hit sensitivity vs Blastn 11, 12 1-hit
Expect Less, Get More
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Lemma: The expected number of hits of a weight W
length M seed model within a length L region with
similarity p is
(L-M+1)pW
Proof: The expected number of hits is the sum, over the L-M+1
possible positions of fitting the seed within the region, of the
probability of W specific matches, the latter being pW.
■
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Example: In a region of length 64 with 0.7 similarity,
PH has probability of 0.466 to hit vs Blast 0.3, 50%
increase. On the other hand, by above lemma, Blast
expects 1.07 hits, while PH 0.93, 14% less.
Why Is Spaced Seed Better?
A wrong, but intuitive, proof: seed s, interval I, similarity p
E(#hits) = Pr(s hits) E(#hits | s hits)
Thus:
Pr(s hits) = Lpw / E(#hits | s hits)
For optimized spaced seed, E(#hits | s hits)
111*1**1*1**11*111
Non overlap Prob
111*1**1*1**11*111
6
p6
111*1**1*1**11*111
6
p6
111*1**1*1**11*111
6
p6
111*1**1*1**11*111
7
p7
…..
 For spaced seed: the divisor is 1+p6+p6+p6+p7+ …
 For BLAST seed: the divisor is bigger: 1+ p + p2 + p3 + …
Finding Optimal Spaced Seeds
(Keich, Li, Ma, Tromp, Discrete Appl. Math. 138(2004), 253-263 )
Let f(i,b) be the probability that seed s hits the
length i prefix of R that ends with b.
Thus, if s matches b, then
f(i,b) = 1,
otherwise we have the recursive relationship:
f(i,b)= (1-p)f(i-1,0b') + pf(i-1,1b')
where b' is b deleting the last bit.
Then the probability of s hitting R is,
Σ|b|=M Prob(b) f(|R|,b).
Choose s with the highest probability.
Improvements
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Brejova-Brown-Vinar (HMM) and BuhlerKeich-Sun (Markov): The input
sequence can be modeled by a (hidden)
Markov process, instead of iid.
Multiple seeds
Brejova-Brown-Vinar: Vector seeds
Csuros: Variable length seeds – e.g.
shorter seeds for rare query words.
PatternHunter
(Ma, Tromp, Li: Bioinformatics, 18:3, 2002, 440-445)
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PH used optimal spaced seeds, novel
usage of data structures: red-black tree,
queues, stacks, hashtables, new gapped
alignment algorithm.
Written in Java. Yet many times faster than
BLAST.
Used in Mouse/RAT Genome Consortia
(Nature, Dec. 5, 2002), as well as in
hundreds of institutions and industry.
Comparison with BLAST
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On Pentium III 700MH, 1GB
BLAST
PatternHunter
E.coli vs H.inf
716s
14s/68M
Arabidopsis 2 vs 4
-498s/280M
Human 21 vs 22
-5250s/417M
Human(3G) vs Mouse(x3=9G)* 19 years 20 days
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All with filter off and identical parameters
16M reads of Mouse genome against Human genome for MIT
Whitehead. Best BLAST program takes 19 years at the same
sensitivity
Quality Comparison:
x-axis: alignment rank
y-axis: alignment score
both axes in logarithmic scale
A. thaliana chr 2 vs 4
E. Coli vs H. influenza
Genome Alignment by PatternHunter
(4 seconds)
Prior Literature
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Over the years, it turns out, that the mathematicians
know about certain patterns appear more likely than
others: for example, in a random English text, ABC is
more likely to appear than AAA, in their study of
renewal theory.
Random or multiple spaced q-grams were used in the
following work:
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FLASH by Califano & Rigoutsos
Multiple filtration by Pevzner & Waterman
LSH of Buhler
Praparata et al
Coding Region & HMM
The Hidden Markov Model is a finite set of states, each of
which is associated with a probability distribution. Transitions
among the states are governed by a set of probabilities called
transition probabilities. In a particular state an outcome or
observation can be generated, according to the associated
probability distribution. It is only the outcome, not the state,
which is visible to an external observer and therefore states are
``hidden'' from the outside; hence the name Hidden Markov
Model. An HMM has the following components satisfying usual
probability laws:
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The number of states of the model, N.
The number of observation symbols in the alphabet, M.
A set of state transition probabilities, depending on current state.
A probability distribution of the observable symbol at each of the
states.
Modeling coding region with HMM
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PatternHunter original assumption: I is a sequence of N
independent Bernoulli variables X0, … XN-1 with P(Xi)=p.
Coding region third position usually is less conserved. Skipping it
should be a good idea (BLAT 110110110). With spaced seeds,
we can model it using HMM.
Brejova, Brown, Vinar: M(3) HMM: I is a sequence of N
independent Bernoulli random variables X0,X1, … XN-1 where
Pr(Xi=1)=pi mod 3. In particular:
p0
p1
p2
human/mouse
0.82 0.87 0.61
human/fruit fly
0.67 0.77 0.40
DP algorithm naturally extends to M(3),
Opt seed (weight 8) for coding region: 11011011000011
Picture of M(3)
Extending M(3), Brejova-Brown-Vinar proposed M(8), above, to model
dependencies among positions within codons: Σpijk = 1, each codon
has conservation pattern ijk with probability pijk.
Figures from Brejova-Brown-Vinar
Sensitivity of all seeds
Under 4 models
From Brejova-Brown-Vinar
Running PH
Download at: www.BioinformaticsSolutions.com
java –jar ph.jar –i query.fna –j subject.fna –o out.txt –b –multi 4
-Xmx512m --- for large files
-j missing: query.fna self-comparison
-db: multiple sequence input, 0,1,2,3 (no, query, subject, both)
-W: seed weight
-G: open gap penalty (default 5)
-E: gap extension (default 1)
-q: mismatch penalty (default 1)
-r: reward for match (default 1)
-model: specify model in binary
-H: hits before extension
-P: show progress
-b: Blast output format
-multi: number of seeds to be used (default 1; max 16)
Multiple Spaced Seeds: Approaching SmithWaterman Sensitivity
(Li, Ma, Kisman, Tromp, PatternHunter II, J. Bioinfo Comput. Biol. 2004)
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The biggest problem for Blast was low sensitivity (and low
speed). Massive parallel machines are built to do SmithWaterman exhaustive dynamic programming.
Spaced seeds give PH a unique opportunity of using several
optimal seeds to achieve optimal sensitivity, this was not
possible for Blast technology.
Economy --- 2 seeds (½ time slow down) achieve sensitivity of
1 seed of shorter length (4 times speed).
With multiple optimal seeds. PH II approaches Smith-Waterman
sensitivity, and 3000 times faster.
Finding optimal seeds, even one, is NP-hard.
Experiment: 29715 mouse EST, 4407 human EST. Sensitivity and
speed comparison next two slides.
Finding Multiple Seeds
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Computing the hit probability of given k seeds on a uniformly
distributed random region is NP-hard.
Finding a set of k optimal seeds to maximize the hit probability
cannot be approximated within 1-1/e unless NP=P
Given k seeds, algorithm DP can be adapted to compute their
sensitivity (probability one of the seeds has a hit).
Greedy Strategy
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(PH II)
Compute one optimal seed a. S={a}
Given S, find b maximizing hit probability of S U {b}.
Coding region, use M(3), 0.8, 0.8, 0.5 (for mod 3 positions)
We took 12 CPU-days on 3GHz PC to compute 16 weight 11
seeds. General seeds (first 4): 111010010100110111,
111100110010100001011, 11010000110001010111,
1110111010001111.
Sensitivity Comparison with Smith-Waterman (at 100%)
The thick dashed curve is the sensitivity of Blastn, seed weight 11.
From low to high, the solid curves are the sensitivity of PH II using
1, 2, 4, 8 weight 11 coding region seeds, and the thin dashed curves
are the sensitivity 1, 2, 4, 8 weight 11 general purpose seeds, respectively
Speed Comparison with Smith-Waterman
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Smith-Waterman (SSearch): 20 CPUdays.
PatternHunter II with 4 seeds: 475
CPU-seconds. 3638 times faster than
Smith-Waterman dynamic programming
at the same sensitivity.
Vector Seeds
Definition. A vector seed is an ordered pair Q=(w,T), where w is
a weight vector (w1, … wM) and T is a threshold value. An
alignment sequence x1,…,xn contains a hit of the seed Q at
position k if
i=1..M(wi∙xk+i-1) ≥ T
The number of nonzero positions in w is called the support of the
seed.
Comments. Other seeds are special cases of vector seeds.
Experiment

(Brejova PhD. thesis, 2005)
Predicted sensitivity and false positive rate (prob. hit 0.25homology region of seed length) comparison of various types of
seeds in a Bernoulli region of 0.7 match probability, length 64.
-----------------------------------------------------------------------------------------
Name
Weight vector
T Support Sensitivity False positive rate
==================================================
BLAST
1111111111
10
10
41%
9.5x10 -7
PH-10
1110101100011011
10
10
60%
9.5x10-7
BLAST-13-15 111111111111111
VS-13-15 111111011011101111
13
13
15
15
73%
85%
9.2x10-7
9.2x10-7
VS-12-13
111101100110101111
12
13
74%
6.0x10-7
VS-11-12
111011001011010111
11
12
84%
2.2x10-6
Variable weight spaced seeds
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M. Csuros proposal: use variable
weighted spaced seeds depending on
the composition of the strings to be
hashed. Rarer strings are expected to
produce fewer false positives, use seeds
with smaller weight --- this increases
the sensitivity.
Open Question
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Can we use different patterns (of same
weight) at different positions. So this can be
called variable position-spaced seeds. At the
same weight, we do not increase expected
number of false positive hits. Can we increase
sensitivity? Find these patterns? How much
increase will there be? Practical enough?
If the above is not the case, then prove:
single optimal seed is always better than
variable positions.
Old field, new trend
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Research trend
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Dozens of papers on spaced seeds have appeared
since the original PH paper, in 3 years.
Many more have used PH in their work.
Most modern alignment programs (including
BLAST) have now adopted spaced seeds
Spaced seeds are serving thousands of users/day
PatternHunter direct users
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Pharmaceutical/biotech firms.
Mouse Genome Consortium, Nature, Dec. 5, 2002.
Hundreds of academic institutions.
4. Mathematical Theory of spaced
seeds
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Why they are better: Spaced seeds hits first
Asymptotic sensitivity: is there a “best seed” in
absolute sense?
NP hardness of computing sensitivity
Approximation algorithm for sensitivity
Why are the spaced seeds better
Theorem 1. Let I be a homologous region, homology level p. For any sequence 0≤i0 <
i1 … < in-1 ≤ |I|-|s|, let Aj be the event a spaced seed s hits I at ij, Bj be event the
consecutive seed hits I at j. (Keich, Li, Ma, Tromp, Discrete Appl. Math. 138(2004), 253-263 )
(1)
P(Uj<n Aj) ≥ P(Uj<n Bj); When ij=j, “>” holds.
Proof. Induction on n. For n=1, P(A0)=pW=P(B0). Assume the theorem holds for n≤N, we prove it
holds for N+1. Let Ek denote the event that, when putting a seed at I[0], 0th, 1st … k-1st 1s in
the seed all match & the k-th 1 mismatches. Clearly, Ek is a partition of sample space for all
k=0, .. , W, and P(Ek)=pk(1-p) for any seed. (Note, Ek for Ai and Ek for Bi have different
indices – they are really different events.) It is sufficient to show, for each k≤W:
(2)
P(Uj=0..N Aj|Ek)≥P(Uj=0..NBj|Ek)
When k=W, both sides of (2) equal to 1. For k<W since (Uj≤kBj)∩Ek=Φ and {Bk+1,Bk+2, … BN}
are mutually independent of Ek, we have
(3)
P(Uj=0..NBj|Ek)=P(Uj=k+1..NBj)
Now consider the first term in (2). For each k in {0, … W-1}, at most k+1 of the events Aj
satisfy Aj∩Ek=Φ because Aj∩Ek=Φ iff when aligned at ij seed s has a 1 bit at overlapping k-th
bit when the seed was at 0 – there are at most k+1 choices. Thus, there exist indices
0<mk+1<mk+2 … <mN ≤N such that Amj∩Ek≠Φ. Since Ek means all previous bits matched
(except for k-th), it is clear that Ek is non-negatively correlated with Uj=k+1..N Amj, thus
(4)
P(Uj=0..N Aj|Ek)≥P(Uj=k+1..N Amj|Ek)≥P(Uj=k+1..NAmj)
By the inductive hypothesis, we have
P(Uj=k+1..N Amj) ≥ P(Uj=k+1..NBj)
Combined with (3),(4), this proves (2), hence the “≥ part” of (1) of the theorem.
To prove when ij=j, P(Uj=0..n-1Aj)>P(Uj=0..n-1Bj)
--- (5)
We prove (5) by induction on n. For n=2, we have
P(Uj=0,1Aj)=2pw – p2w-Shift1>2pw-pw+1=P(Uj=0,1Bj)
(*)
where shift1=number of overlapped bits of the spaced seed s when
shifted to the right by 1 bit.
For inductive step, note that the proof of (1) shows that for all
k=0,1, … ,W, P(Uj=0..n-1Aj|Ek)≥P(Uj=0..n-1Bj|Ek). Thus to prove (5)
we only need to prove that
P(Uj=0..n-1Aj|E0)>P(Uj=0..n-1Bj|E0).
The above follows from the inductive hypothesis as follows:
P(Uj=0..n-1Aj|E0)=P(Uj=1..n-1Aj)>P(Uj=1..n-1Bj)=P(Uj=0..n-1Bj|E0).
■
Corollary
Corollary: Assume I is infinite. Let ps and pc be the first
positions a spaced seed and the consecutive seed hit
I, respectively. Then E[ps] < E[pc].
Proof. E[ps]=Σk=0..∞ kP(ps=k)
=Σk=0..∞ k[P(ps>k-1)-P(ps>k)]
=Σk=0..∞P(ps>k)
=Σk=0..∞(1-P(Uj=0..kAj))
<Σk=0..∞(1-P(Uj=0..kBj))
=E[pc].
■
Asymptotic sensitivity
(Buhler-Keich-Sun, JCSS 2004, Li-Ma-Zhang, SODA’2006)
Theorem. Given a spaced seed s, the asymptotic hit
probability on a homologous region R can be computed, up
to a constant factor, in time exponentially proportional to
|s|, but independent of |R|.
Proof. Let d be one plus the length of the maximum run of 0's in s. Let M'=|s|+d.
Consider an infinite iid sequence R=R[1]R[2] ... , Prob(R[i]=1)=p.
Let T be set of all length |s| strings that do not dominate s (not matched by s).
Let Z[i,j] denote the event that s does not hit at any of the positions R[i], R[i+1],
… , R[j]. For any ti  T, let
xi(n) = P[Z[1,n] | R[n..n+|s|-1] = ti ] .
That is, xi(n) is the no-hit probability in the first n positions when they end with ti.
We now establish a relationship between xi(n) and xi(n-M’) . Let
Ci,j = P[Z[n-M'+1,n] | ti at n, tj at n-M'] x P[tj at n-M' | ti at n].
Thus Ci,j is the probability of generating tj at position n-M' (from ti at n) times the
probability not having a hit in a region of length |s|+M' beginning with tj and
ending with ti. Especially Ci,j ≠ 0 for any nontrivial seed of length greater than 0
(because of M’ with d spacing).
Proof Continued
Then,
xi(n) = j=1K P[Z[1,n-M'] | tj at n-M']
x P[tj at n-M' | ti at n] x P[Z[n-M'+1,n] | ti at n, tj at n-M']
= j=1K P[Z[1,n-M'] | tj at n-M'] x Ci,j
= j=1K Ci,j xj(n-M’)
Define the transition matrix C=(Ci,j). It is a K x K matrix. Let x(n) = (x1(n),
x2(n), … , xK(n)). Then, assuming that M' divides n,
x(n) = x(1) ◦ (CT)n/M’.
Because Ci,j > 0 for all i,j, C is a positive matrix. The row sum of the i-th
row is the probability that a length-(|s|+M') region ending with ti does
not have a hit. The row sum is lower bounded by (1-p)d (1-pW ), where
(1-p)d is the probability of generating d 0's and 1-pW is the probability of
generating a string in T. It is known that the largest eigenvalue of C is
positive and unique, and is lower bounded by the smallest row sum and
upper bounded the largest row sum of C. Let λ1>0 be the largest
eigenvalue, and λ2, 0 < ||λ2|| < λ1 be the second largest. There is a
unique eigenvector corresponding to λ1.
x(n) / ||x(n)|| = x(1) ◦ (CT )n/M’ / ||x(1) ◦ (CT)n/M’||
converges to the eigenvector corresponding to λ1. As λ1n tends to zero,
we can use standard techniques to normalize xn as
y(n) = x(n) / ||x(n)||2
and then x(n+M’) = C y(n) .
Proof Continued
…
Then (by power method) the Rayleigh quotient of
λ = (y(n))T C y(n) / (y(n))T y(n) = (y(n))T x(n+M’)
converges to λ1. The convergence speed depends on the ratio λ1/ |λ2|, it is
known that we can upper bound the second largest eigenvalue λ2 in a
positive matrix by:
λ2 ≤ λ1 (K-1)/(K+1)
where K= maxi,j,k,l  (Ci,j Ck,l / Ci,l Ck,j). For any i,j, we have
pa (1-p)b (1-p)d ≤ Ci,j ≤ pa (1-p)b
where a is the number of 1's in tj, b the number of 0's in tj. K = O(1/(1-p)d ).
As (1 - 1/K)K goes to 1/e, the convergence can be achieved in O(K) = O(1/(1p)d ) steps. The time complexity is therefore upper bounded by an
exponential function in |s|.
QED
Theorem 2. It is NP hard to find the optimal seed
(Li, Ma, Kisman, Tromp, J. Bioinfo Comput. Biol. 2004)
Proof. Reduction from 3-SAT. Given a 3-SAT instance C1 … Cn, on Boolean variables x1, … ,
xm, where Ci= l1 + l2 + l3, each l is some xi or its complement. We reduce this to seed
selection problem. The required seed has weight W=m+2 and length L=2m+2, and is
intended to be of the form 1(01+10)m1, a straightforward variable assignment encoding.
The i-th pair of bits after the initial 1 is called the code for variable xi. A 01 code
corresponds to setting xi true, while a 10 means false. To prohibit code 11, we introduce,
for every 1 ≤ i ≤ m, a region
Ai=1 (11)i-101(11)m-i 10m+1 1(11)i-110(11)m-i 1.
Since the seed has m 0s, it cannot bridge the middle 0m+1 part, and is thus forced to hit at
the start or at the end. This obviously rules out code 11 for variable xi. Code 00
automatically becomes ruled out as well, since the seed must distribute m 1s among all m
codes. Finally, for each clause Ci, we introduce
Ri =1(11)a-1 c1(11)m-a 1 000 1(11)b-1c2(11)m-b 1 000 1(11)c-1c3(11)m-c 1.
The total length of Ri is (2m+2)+3+(2m+2)+3+(2m+2)=6m+12; it consists of three parts
each corresponding to a literal in the constraint, with ci being the code for making that
literal true.
Thus a seed hits Ri iff the assignment encoded by the seed satisfies Ci .
■
The complexity of computing spaced
seed sensitivity
Theorem. It is also NP hard to compute
sensitivity of a given seed. It remains to
be NPC even in a uniform region, at any
homology level p.
Proof.
Very complicated. Omitted.
PTAS for computing sensitivity of
a seed



Trivial alg. Sample R poly times, compute the
frequency s hits. Use that to approximate the
real probability.
When p is large, then using Chernoff bound,
we have a PTAS.
But when p is very small, or seed weight is
very large, hitting probability is actually very
small, then we do not have a PTAS.
Efficient approximation of the hit
probability of a seed
Theorem. Let the hit probability of s be x.
For any ε>0, let N= 6|R|2log |R| / ε2.
Then with high probability, in
polynomial time and N samples, we can
output a value y such that |y-x| ≤ εx.
Proof.
We give a transformation to transform the low
probability event ``s hits R'' dependent on p and weight of s
to several events with probability sum ≥ 1, independent of p
and seed weight W. Let sr be the reverse string of s.
P[s hits R] = P[sr hits R]
= i=0|R|-|s| P[sr hits at i, but misses at 0, … , i-1]
= i=0|R|-|s| P[sr hits at i] x P[sr misses 0, … , i-1 | sr hits at i]
= i=0|R|-|s| pW x P[s misses 1… i | s hits 0]
(1)
Because P[s hits R] ≥ pW, (1) induces that
i=0|R|-|s| P[s misses 1 … i | s hits 0] ≥ 1
Now, we can do efficient sampling, and using Chernoff bounds,
the rest of the proof is standard.
QED
Expected hit distance
Corollary. E(second hit position | s hits at 0) = p-W.
Proof.
i=0|R|-|s| P[s misses 1… i | s hits 0]
= i=0|R|-|s| j=i+1|R|-|s|+1 P[s second hits j | s hits 0]
= j=0|R|-|s|+1 i=0j-1 P[s second hits j | s hits 0]
= i=0|R|-|s|+1 (j-1)P[s second hits j | s hits 0]
Combining with (1), we have:
P[s hits R]xpW=i=0|R|-|s|+1 (j-1)P[s second hits j|s hits 0]
Letting |R|  ∞, the left hand is p-W, the right is
E(second hit position | s hits at 0).
QED
Chernoff bound
Consider a sequence of n independent
tosses of a coin with heads prob. p. The
expected number of heads is m=np.
What is the probability that the number
of heads deviates a lot from this? Let X
be the number of successes, then:
Pr[|X-m|≥εm] ≤ 2e-εεm/3
Complexity of finding the optimal
spaced seeds
Theorem 1 [Li-Ma]. Given a seed and it is NP-hard to find its sensitivity, even in a
uniform region.
Theorem 2 [Li-Ma]. The sensitivity of a given seed can be efficiently approximated
by a PTAS.
Theorem 3. Given a set of seeds, choose k best can be approximated with ratio 11/e.
Theorem 4 [Buhler-Keich-Sun, Li-Ma] The asymptotic hit probability is computable
in exponential time in seed length, independent of homologous region length.
Theorem 5 [L. Zhang] If the length of a spaced seed is not too long, then it strictly
outperforms consecutive seed, in asymptotic hit probability.
5. Applications beyond bioinformatics
Stock Market Prediction
Zou-Deng-Li: Detecting Market Trends by Ignoring It, Some Days, March, 2005


A real gold mine: 4.6 billion dollars are
traded at NYSE daily.
Buy low, sell high.



Best thing: God tells us market trend
Next best thing: A good market indicator.
Essentially, a “buy” indicator must be:


Sensitive when the market rises
Insensitive otherwise.
My goal


Provide a sensitive, but not aggressive
market trend indicator.
Learning/inference methods or
complete and comprehensive systems
are beyond this study. They can be
used in conjunction with our proposal.
Background

Hundreds of market indicators are used (esp.
in automated systems). Typical:






Common sense: if the past k days are going up,
then the market is moving up.
Moving average over the last k days. When the
average curve and the price curve intersect,
buy/sell.
Special patterns: a wedge, triangle, etc.
Volume
Hundreds used in automated trading systems.
Note: any method will make money in the
right market, lose money in the wrong market
Problem Formalization



The market movement is modeled as a 0-1 sequence, one bit per day,
with 0 meaning market going down, and 1 up.
S(n,p) is an n day iid sequence where each bit has probability p being
1 and 1-p being 0. If p>0.5, it is an up market
Ik=1k is an indicator that the past k days are 1’s.



Iij is an indicator that there are i 1’s in last j days.



I811 has high sensitivity 0.96 in S(30,0.7)
But it is too aggressive at 0.139 false positive rate in S(100, 0.3).
Spaced seeds 1111*1*1111 and 11*11111*11 combine to



I8 has sensitivity 0.397 in S(30,0.7), too conservative
I8 has false positive rate 0.0043 in S(100, 0.3). Good
have sensitivity 0.49 in S(30,0.7)
False positive rate 0.0032 in S(100, 0.3).
Consider a betting game: A player bets a number k. He wins k dollars
for a correct prediction and o.w. loses k dollars. We say an indicator A
is better than B, A>B, if A always wins more and loses less.
Sleeping on Tuesdays and Fridays

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Spaced seeds are beautiful indicators: they are
sensitive when we need them to be sensitive and
not sensitive when we do not want them to be.
11*11*1*111
always beats I811
if it bets 4 dollars
for each dollar
I811 bets. It is >I8
too.
11111111
11*11*1*111
8-out-of-11
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Two spaced seeds
1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1111*1*1111; 11*11111*11
1.E-06
11111111
1.E-07
Observe two spaced
Seeds curve vs I8, the
spaced seeds are
always more sensitive
in p>0.5 region, and
less sensitive when p<0.5
9-out-of-11
1.E-08
0.1
0.2
0.3
0.4
0.5
1
0.9
0.8
0.7
0.6
0.5
0.4
1111*1*1111; 11*11111*11
0.3
11111111
0.2
9-out-of-11
0.1
0
0.5
0.6
0.7
0.8
0.9
1
Two experiments

We performed two trading experiments


One artificial
One on real data (S&P 500, Nasdaq
indices)
Experiment 1: Artificial data





This simple HMM
generates a very
artificial simple model
5000 days (bits)
Indicators: I7, I711, 5
spaced seeds.
Trading strategy: if
there is a hit, buy, and
sell 5 days later.
Reward is: #(1)-#(0) in
that 5 days times the
betting ratio
4/5
EVEN:
P(1)=0.5
1/50
1/60
1/10
UP
P(1)=0.7
1/50
1/10
DOWN
P(1)=0.3
1/60
29/30
24/25
Results of Experiment 1.
R
I7=1111111
$30
I711
$15
5 Spaced seeds $25
#Hits Final MTM #Bankrupcies
12
47
26
$679
$916
$984
16
14
13
Experiment 2.



Historical data of S&P 500, from Oct 20, 1982 to Feb.
14, 2005 and NASDAQ, from Jan 2, ’85 to Jan 3,
2005 were downloaded from Yahoo.com.
Each strategy starts with $10,000 USD. If an
indicator matches, use all the money to buy/sell.
While in no ways this experiment says anything
affirmatively, it does suggest that spaced patterns
can serve a promising basis for a useful indicator,
together with other parameters such as trade
volume, natural events, politics, psychology.
Open Questions
I have presented a simple idea of optimized spaced seeds and its
applications in homology search and time series prediction.
Open questions & current research:

Complexity of finding (near) optimal seed, in a uniform region.
Note that this is not an NP-hard question.

Tighter bounds on why spaced seeds are better.

Applications to other areas. Apparently, the same idea works for
any time series.

Model financial data

Can we use varying patterns instead of one seed? When
patterns vary at the same weight, we still have same number of
expected hits. However, can this increase sensitivity? One seed
is just a special case of this.
Acknowledgement





PH is joint work with B. Ma and J. Tromp
PH II is joint work with Ma, Kisman, and Tromp
Some joint theoretical work with Ma, Keich,
Tromp, Xu, Brown, Zhang.
Financial market prediction: J. Zou, X. Deng
Financial support: Bioinformatics Solutions Inc,
NSERC, Killam Fellowship, CRC chair program,
City University of Hong Kong.
Here is an example of a BLASTP match (E-value 0) between gene 0189 in C. pneumoniae and
gene 131 in C. trachomatis.
Query: CPn0189
Score (bits)
Aligned with CT131 hypothetical protein
Query: 1
0.0
MKRRSWLKILGICLGSSIVLGFLIFLPQLLSTESRKYLVFSL I HKESGLSCSAEELKISW 60
MKR
Sbjct: 1
1240
E-value
W KI G L
+L
L LP+ S+ES KYL
S+++KE+GL
E+L +SW
MKRSPWYKIFGYYLLVGVPLALLALLPKFFSSESGKYLFLSVLNKETGLQF EIEQLHLSW 60
Query: 61 FGRQTARKIKLTG-EAKDEVFSAEKFELDGSLLRLL I YKKPKGITLSGWSLKINEPASID 119
FG QTA+KI++ G ++ E+F+AEK + GSL RLL+Y+ PK + TL+GWSL+I+E S++
Sbjct: 61
FGSQTAKKIRIRGIDSDSEIFAAEKI IVKGSLPRLLL YRFPKALTLTGWSLQIDESLSMN 120
Etc.
Note: Because of powerpoint character alignment problems, I inserted some white space to make it
look more aligned.
Summary and Open Questions






Simple ideas: 0 created SW, 1 created Blast. 1 & 0
created PatternHunter.
Good spaced seeds help to improve sensitivity and
reduce irrelevant hit.
Multiple spaced seeds allow us to improve sensitivity to
approach Smith-Waterman, not possible with BLAST.
Computing spaced seeds by DP, while it is NP hard.
Proper data models help to improve design of spaced
seeds (HMM)
Open Problems: (a) A mathematical theory of spaced
seeds. I.e. quantitative statements for Theorem 1.
(b) Applications to other fields.
Smaller (more solvable) questions:
(perhaps good for course projects)




Multiple protein seeds for SmithWaterman sensitivity?
Applications in other fields, like internet
document search?
Prove Theorem 1 for other models – for
example, M(3) model 0.8,0.8,0.5
together with specialized seeds.
More extensive study of Problem 3.