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Efficient Algorithms for Counting and Reporting Segregating Sites in Genomic Sequences

JOURNAL OF COMPUTATIONAL BIOLOGY

Volume14,Number7,2007

?Mary Ann Liebert,Inc.

Pp.1001–1010

DOI:10.1089/cmb.2006.0136

Ef?cient Algorithms for Counting and Reporting Segregating Sites in Genomic Sequences MANOLIS CHRISTODOULAKIS,1G.BRIAN GOLDING,2COSTAS S.ILIOPOULOS,1

YOAN JOSéPINZóN ARDILA,3and WILLIAM F.SMYTH4

ABSTRACT

The number of segregating sites provides an indicator of the degree of DNA sequence variation that is present in a sample,and has been of great interest to the biological,pharmaceutical and medical professions.In this paper,we?rst provide linear-and expected-sublinear-time algorithms for?nding all the segregating sites of a given set of DNA sequences.We also describe a data structure for tracking segregating sites in a set of sequences,such that every time the set is updated with the insertion of a new sequence or removal of an existing one, the segregating sites are updated accordingly without the need to re-scan the entire set of sequences.

Key words:segregating sites,single nucleotide polymorphisms(SNPs).

1.INTRODUCTION

T HE NUMBER OF SEGREGATING SITES is de?ned as the number of homologous DNA positions that differ in a sample of m sequences.This number,therefore,provides an indicator of the degree of DNA sequence variation that is present in a sample.

This quantity has long been of theoretical population genetic interest.It is directly modelled using what has been termed an in?nite sites model where every mutation is created at a new unique site.These models date back to1967(Karlin and McGregor,1967)and have since been extensively developed to estimate population size,mutation rate,migration rates,natural selection,and ages of polymorphisms(Innan et al., 2005;Klein et al.,1999;Fu,1996;Perlitz and Stephan,1997).

Segregating sites are also of interest to the pharmaceutical and medical professions.The number and the location of polymorphisms within humans and other groups of organisms are critical to determining the potential differences in reactions of individuals to medicines and treatments.Since polymorphisms occur every100to300bases along the three billion base human genome,they are crucial markers to map human

1002CHRISTODOULAKIS ET AL. genetic diseases.As a result,single nucleotide polymorphisms(SNPs)are searched for in a systematic manner(cf.,the SNP consortium1).

With the promise of entire genomic complements being determined within a few hours(Margulies et al., 2005)and with entire genome scans desired for SNPs(Syvanen,2005),the need to quickly determine the number of segregating sites will become ever more urgent.In future,it can be anticipated that these calculations will be required over hundreds of sequences whose length is in the tens of thousands. Segregating sites are formally de?ned for within population level phenomena but the in?nite sites model and its utility to measure genetic variation are useful in other contexts as well.The recent completion of a large number of organelle genomes chosen in a systematic fashion is just one such example.This data records the pattern of sequence variants within a hundred mitochondrial genomes(Miya et al.,2003).In this case,the data consists of approximately16,000bp from different teleostean?sh.To repeatedly query the level and pattern of variation in different subgroups requires ef?cient methodology.Another large scale biological project that will require an ef?cient algorithm is the barcode of life project(Hebert et al.,2003). This project aims to sequence the cytochrome oxidase subunit1gene from millions of animal species. Again,the variation within a species,within a genus,within different groups of genera will provide useful biological insights into the evolution of these taxa.A tool to permit the addition and removal of sequences in these queries will allow an interactive ability to perform exploratory analyses.

In this paper,we address the problem of locating the positions that constitute segregating sites in a given set S of equal-length sequences.In addition,we create a data structure that stores adequate information about the segregating sites of S,such that if one adds new sequences to S or removes sequences from S,the number of segregating sites is ef?ciently resolved accordingly.In this way,one avoids the need to recompute(from scratch)the number of segregating sites in S,every time S is updated.

In Section2,we introduce some necessary notation and state the problems of?nding segregating sites within a set of genomic sequences.Section3describes(two variants of)a very fast algorithm to locate all segregating sites positions.Section4contains an alternative algorithm to identify segregating sites with the enhanced advantage that whenever the set of sequences is updated,the segregating sites are updated accordingly.In Section5,experimental results are presented and some implementation issues are brie?y discussed.Finally,Section6contains our concluding remarks.

2.DEFINITIONS

A string is a sequence of zero or more symbols from an alphabet?.The set of all strings over?is denoted by? .The length of a string s is denoted by j s j.The empty string,the string of length zero,is denoted by".The i th symbol of a string s is denoted by s?i .A string y is a substring of s if s D uyv, where u;v2? .We denote by s?i::j the substring of s that starts at position i and ends at position j. Consider a set of strings S D f s1;:::;s m g,where j s i j D n8i2?1::m .A position1?j?n is then called a segregating site iff there exists at least one pair of indices,.i;i0/,such that s i?j ¤s i0?j .

In Figure1,we give a simplistic example of a set S containing?ve DNA sequences,each of length14 (m D5;n D14).To?nd the segregating sites we look for those sites which are variable in the data in terms of different nucleotides;we?nd them at positions4,8,and10.For instance,site4is a segregating site since s1?4 ¤s2?4 .In contrast,positions1–3,5–7,9,and11–14are not segregating sites because all of the nucleotides are the same at each of those sites.

The problems we address in this paper are formally de?ned as follows:

Problem1.Given a set of m strings S D f s1;:::;s m g,where j s i j D n8i2?1::m ,locate all the positions1?j?n that constitute segregating sites in S.

Problem2.Given a set of m strings S D f s1;:::;s m g,where j s i j D n8i2?1::m ,whose segregating

sites have already been computed,and a set of`strings S0D f s0

1;:::;s0

`

g,where j s0

i

j D n8i2?1::` ,

ef?ciently?nd the positions1?j?n that constitute segregating sites in S[S0or S n S0.

COUNTING AND REPORTING SEGREGATING SITES1003

FIG.1.A sample and the three segregating sites.

Input:Set of strings S D f s1;:::;s m g,each of size n

Output:Positions of segregating sites in S

1:procedure SS-A LG1(S)

2:ss?1::n f0;0;:::;0g

3:for i D2to m do

4:for j D1to n do

5:if ss?j D0and s i?j ¤s1?j then

6:ss?j 1

7:return the positions of1’s in ss

1004CHRISTODOULAKIS ET AL.

FIG.2.Illustration of how a new string(s3)is processed by Algorithm1.Note that only those nucleotides marked with circles were compared.

is now being processed.Note that,for every position that has not already been marked as a segregating site,we need to compare the current nucleotide against the corresponding nucleotide in s1.For instance,at position1the symbol s3?1 is compared against s1?1 ;since they are equal,position1remains unmarked. At position6,on the contrary,s3?6 ¤s1?6 ,thus position6is marked as a segregating site(ss?6 D1). Notice,though,that when column3is processed,there is no need to check s3?3 because that position has already been marked as a segregating site.

The running time of Algorithm1is clearly?.mn/:all the sequences are scanned only once,spending a constant amount of time at each position of every sequence.The space requirements of the algorithm are limited to?.n/,the space required by vector ss.

3.2.An expected-sublinear-time algorithm

From close observations of the way Algorithm1works,we know that the larger the number of segregating sites and the earlier the sites are identi?ed as segregating,the more the symbols that do not need to be checked.For instance,in the set of sequences depicted in Figure3,approximately19%of the symbols are not checked(the symbols displayed in gray).

Although Algorithm1does not check the symbols at those positions that have been identi?ed as segregating sites,it would nonetheless check vector ss to decide whether the positions are segregating sites or not,as a result,the algorithm performs precisely.m 1/n iterations.However,for those cases when a large number of segregating sites is present(i.e.,the dense case),it seems to be bene?cial to completely skip those positions and only process non-segregating-site positions.

To this end,a simple linked list can be used to maintain the list of positions to be processed.Every time a new segregating site is discovered,the corresponding position will be removed from the list.Then for every subsequent sequence,only the positions included in this list will ever be processed.Algorithm2 describes these ideas in more detail.

FIG.3.Illustration of the fact that not all symbols in a sample need to be checked.Here8/42(19%)of the symbols are not checked—grayed symbols.

COUNTING AND REPORTING SEGREGATING SITES1005 In the worst case,Algorithm2will yet again need O.mn/time to process a set of m sequences each of length n.The expected running time though will be sublinear with respect to the input size,since as soon as a position is identi?ed as a segregating site it will no longer be visited again.In practice(as we shall see in Section5)only when the number of segregating sites is very small,the extra time consumed in maintaining the list of positions,exceeds the time saved by the skipped positions,con?rming that this algorithm is good for samples containing more than a small number of segregating sites.

Input:Set of strings S D f s1;:::;s m g,each of size n

Output:Positions of segregating sites in S

1:procedure SS-A LG2(S)

2:ss?1::n f0;0;:::;0g

3:P D f1;2;:::;n g

4:for i D1to m do

5:for all j2P do

6:if s i?j ¤s1?j then

7:ss?j 1

8:P P nf j g

9:return the positions not in P

1006CHRISTODOULAKIS ET AL.

FIG.4.Sample S.

Clearly,the space requirements of an SSCounter is?.j?j n/.This will normally be signi?cantly less than the space required by the set S itself,since usually the number of strings m is much larger than the size of the alphabet?.

4.2.Inserting new sequences into S

Procedure I NSERT(Algorithm3)handles insertions.For each new sequence s0inserted into S,the algorithm works as follows.s0is scanned from left to right,and for every position j,the counter of the symbol occurring at s0?j is increased;in other words,if s0?j D ,then the counter c ?j is increased by 1to denote that one additional string contains the symbol at position j.Next,the value of counter c ?j is checked and if it equals1then this corresponds to a new non-zero counter and thus ac?j is increased by1.Finally,if ac?j D2,the position has just become a segregating site and has to be marked as such. Algorithm3performs a single scan through the newly inserted sequence s0.The amount of time spent at each position j depends on the type of alphabet:

For indexed alphabets,the counter c that corresponds to the current symbol ,can be identi?ed in O.1/,thus the time spent at each position j is O.1/.

For general ordered alphabets,a binary search is required to identify c ,thus the time spent at each position is O.log j?j/.

Therefore,the overall running time of Algorithm3is?.n/for indexed alphabets,and?.n log j?j/for ordered alphabets.In biological applications,of course,the DNA alphabet or the amino-acid alphabet is used,which will be indexed.

4.3.Removing sequences from S

Similarly,procedure R EMOVE(Algorithm4)handles removals of strings from S.When a sequence s0is removed from S,s0is scanned from left to right,and for every position j,the counter c ?j is decreased by one,where D s0?j .Next,we have to check whether that counter equals0.If it does,and also

FIG.5.SSCounter of S.

COUNTING AND REPORTING SEGREGATING SITES1007 Input:The SSCounter of a set of strings S,and a string s0of size n

Output:The SSCounter of S[f s0g

1:procedure I NSERT(the SSCounter of S,string s0)

2:k k C1

3:for j D1to n do

4: s0?j

5:c ?j c ?j C1

6:if c ?j D1then

7:ac?j ac?j C1

8:if ac?j D2then

9:ss?j 1

10:return the positions of1’s in ss

Algorithm4.Remove a string from an SSCounter

position j was marked as a segregating site,then one needs to re-evaluate whether position j is still a segregating site.It will still be1iff at least two symbols occur at column j;i.e.,iff ac?j >1. Likewise,as in procedure I NSERT,the time complexity of R EMOVE is also?.n/for indexed alphabets, and?.n log j?j/for ordered alphabets.

5.EXPERIMENTAL RESULTS

We implemented all of the algorithms presented in the paper using C CC and conducted a series of experiments in order to test their performance.The datasets we used consisted of both randomly generated sequences,tailored to the needs of the speci?c experiments we run,and biological sequences.2

First,we tested how the running time of the algorithms scales with respect to the number of the segregating sites present in the dataset.In Figure6,we considered a varying number of sequences(10?m?100)of length n D10,000,which contained only a small number of segregating sites,while in Figure7again a varying number of sequences of length n D10,000was considered but this time the number of segregating sites was considerably larger.

From the graph in Figure6,we can see that the running times of procedures SS-A LG1and SS-A LG2 are practically identical when the number of segregating sites is small,while Figure7shows that as the number of segregating sites increases,the running time of SS-A LG2is reduced,by a small fraction in this case.The amount of this reduction depends exclusively on the number of comparisons avoided by

1008CHRISTODOULAKIS ET AL.

FIG.6.Running time for1000segregating sites.

FIG.7.Running time for9000segregating sites.

COUNTING AND REPORTING SEGREGATING SITES1009

FIG.8.Running time for updating an SSCounter.

T ABLE1.P ERFORMANCE OF THE A LGORITHMS FOR B IOLOGICAL S EQUENCES

Set of

sequences n m Seg.sites SS-Alg1SS-Alg2Insert

SS-A LG2,which itself depends on both the number of segregating sites and how early these segregating sites are identi?ed;the earlier the better.Finally,we observe that the running time of I NSERT is practically independent of the number of segregating sites.This is true,given that this algorithm performs almost the same amount of work for each position of any input sequence,regardless of whether it is a segregating site or not.

Secondly,we compared the running time of procedure I NSERT against that of R EMOVE.An existing SSCounter,ssc,was passed to the two procedures,and then the same sets of sequences were inserted to/removed from ssc,using I NSERT and R EMOVE,respectively.As can be seen in Figure8,the two algorithms perform identically in terms of running time for the values chosen in this experiment. Finally,we run our procedures on sets of biological sequences.The?rst set consisted of106sequences of intron4from the licorne gene of Drosophila simulans and Drosophila melanogaster strains taken from Haddrill et al.(2005).The Anoplura data set are sequences of cytochrome oxidase subunit1,downloaded from GenBank.3The third set contained various Drosophila complete mitochondrial genomes,also down-loaded from GenBank.And the last data set is54newly sequenced complete mtDNA genomes from teleost ?sh from Miya et al.(2003).The results are shown in Table1,where n is the common length of all the sequences in each set,and m is the number of sequences in the set.

1010CHRISTODOULAKIS ET AL.

6.DISCUSSION

In this paper,we presented algorithms for?nding all the segregating sites of a given set of strings S.The ?rst two algorithms are very fast and are useful when S is a new,not already processed,set of sequences. Both algorithms operate in O.mn/time,where m is the number of input sequences and n is their length, however the second algorithm’s expected running time will be on average less than O.mn/.

We also presented two more algorithms,which not only can compute the segregating sites of a new set of strings S,but also can update the number and positions of the segregating sites of S,whenever the latter is updated with an insertion of a new string or the removal of an existing one.

ACKNOWLEDGMENTS

We thank J.Holub and G.Valiente for their useful comments and suggestions.

REFERENCES

Fu,Y.X.1996.Estimating the age of the common ancestor of a DNA sample using the number of segregating sites. Genetics144,829–838.

Haddrill,P.R.,Thornton,K.R.,Charlesworth,B.,et al.2005.Multilocus patterns of nucleotide variability and the demographic and selection history of Drosophila melanogaster populations.Genome Res.15,790–799. Hebert,P.D.,Cywinska,A.,Ball,S.L.,et al.2003.Biological identi?cations through DNA barcodes.Proc.Biol.Sci. 270,313–321.

Innan,H.,Zhang,K.,Marjoram,P.,et al.2005.Statistical tests of the coalescent model based on the haplotype frequency distribution and the number of segregating sites.Genetics169,1763–1777.

Karlin,S.,and McGregor,J.L.1967.The number of mutant forms maintained in a population.Proc.Fifth Berkeley Symp.Math.Statist.Prob.4,415–438.

Klein,E.K.,Austerlitz,F.,and Laredo,C.1999.Some statistical improvements for estimating population size and mutation rate from segregating sites in DNA sequences.Theor.Popul.Biol.55,235–247.

Margulies,M.,Egholm,M.,Altman,W.E.,et al.2005.Genome sequencing in microfabricated high-density picolitre reactors.Nature437,376–380.

Miya,M.,Takeshima,H.,Endo,H.,et al.2003.Major patterns of higher teleostean phylogenies:a new perspective based on100complete mitochondrial DNA sequences.Mol.Phylogenet.Evol.26,121–138.

Perlitz,M.,and Stephan,W.1997.The mean and variance of the number of segregating sites since the last hitchhiking event.J.Math.Biol.36,1–23.

Syvanen,A.C.2005.Toward genome-wide SNP genotyping.Nat.Genet.37s,S5–S10.

Address reprint requests to:

Dr.Manolis Christodoulakis

King’s College London

Department of Computer Science

Strand

London WC2R2LS,UK

E-mail:manolis@https://www.doczj.com/doc/0512403870.html,

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Unit5 Reading An Exciting Job 说课稿 Liu Baowei Part 1 My understanding of this lesson The analysis of the teaching material:This lesson is a reading passage. It plays a very important part in the English teaching of this unit. It tells us the writer’s exciting job as a volcanologist. From studying the passage, students can know the basic knowledge of volcano, and enjoy the occupation as a volcanologist. So here are my teaching goals: volcanologist 1. Ability goal: Enable the students to learn about the powerful natural force-volcano and the work as a volcanologist. 2. Learning ability goal: Help the students learn how to analyze the way the writer describes his exciting job. 3. Emotional goal: Make the Students love the nature and love their jobs. Learn how to express fear and anxiety Teaching important points: sentence structures 1. I was about to go back to sleep when suddenly my bedroom became as bright as day. 2. Having studied volcanoes now for more than twenty years, I am still amazed at their beauty as well as their potential to cause great damage. Teaching difficult points: 1. Use your own words to retell the text. 2. Discuss the natural disasters and their love to future jobs. Something about the S tudents: 1. The Students have known something about volcano but they don’t know the detailed information. 2. They are lack of vocabulary. 3. They don’t often use English to express themselves and communicate with others.

an exciting job 翻译

我的工作是世界上最伟大的工作。我跑的地方是稀罕奇特的地方,我见到的是世界各地有趣味的人们,有时在室外工作,有时在办公室里,有时工作中要用科学仪器,有时要会见当地百姓和旅游人士。但是我从不感到厌烦。虽然我的工作偶尔也有危险,但是我并不在乎,因为危险能激励我,使我感到有活力。然而,最重要的是,通过我的工作能保护人们免遭世界最大的自然威力之一,也就是火山的威胁。 我是一名火山学家,在夏威夷火山观测站(HVO)工作。我的主要任务是收集有关基拉韦厄火山的信息,这是夏威夷最活跃的火山之一。收集和评估了这些信息之后,我就帮助其他科学家一起预测下次火山熔岩将往何处流,流速是多少。我们的工作拯救了许多人的生命,因为熔岩要流经之地,老百姓都可以得到离开家园的通知。遗憾的是,我们不可能把他们的家搬离岩浆流过的地方,因此,许多房屋被熔岩淹没,或者焚烧殆尽。当滚烫沸腾的岩石从火山喷发出来并撞回地面时,它所造成的损失比想象的要小些,这是因为在岩石下落的基拉韦厄火山顶附近无人居住。而顺着山坡下流的火山熔岩造成的损失却大得多,这是因为火山岩浆所流经的地方,一切东西都被掩埋在熔岩下面了。然而火山喷发本身的确是很壮观的,我永远也忘不了我第一次看见火山喷发时的情景。那是在我到达夏威夷后的第二个星期。那天辛辛苦苦地干了一整天,我很早就上床睡觉。我在熟睡中突然感到床铺在摇晃,接着我听到一阵奇怪的声音,就好像一列火车从我的窗外行驶一样。因为我在夏威夷曾经经历过多次地震,所以对这种声音我并不在意。我刚要再睡,突然我的卧室亮如白昼。我赶紧跑出房间,来到后花园,在那儿我能远远地看见基拉韦厄火山。在山坡上,火山爆发了,红色发烫的岩浆像喷泉一样,朝天上喷射达几百米高。真是绝妙的奇景! 就在这次火山喷发的第二天,我有幸做了一次近距离的观察。我和另外两位科学被送到山顶,在离火山爆发期间形成的火山口最靠近的地方才下车。早先从观测站出发时,就带了一些特制的安全服,于是我们穿上安全服再走近火山口。我们三个人看上去就像宇航员一样,我们都穿着白色的防护服遮住全身,戴上了头盔和特别的手套,还穿了一双大靴子。穿着这些衣服走起路来实在不容易,但我们还是缓缓往火山口的边缘走去,并且向下看到了红红的沸腾的中心。另外,两人攀下火山口,去收集供日后研究用的岩浆,我是第一次经历这样的事,所以留在山顶上观察他们

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