Genomic Sequence Analysis: A Case Study in Constrained Heaviest Segments
(Working draft)
Kun-Mao Chao
1,2,31
Graduate Institute of Biomedical Electronics and Bioinformatics
2
Department of Computer Science and Information Engineering
3
Graduate Institute of Networking and Multimedia National Taiwan University, Taipei, Taiwan 106
Email: kmchao@csie.ntu.edu.tw
Abstract
Methods for genomic sequence analysis have been studied for more than a decade. One line of investigation is to locate the biologically meaningful segments, like conserved regions or GC-rich regions in DNA sequences. A common approach is to assign a real number (also called scores) to each residue, and then look for the maximum-sum or maximum-average segment. In this chapter, we address a few interest- ing applications concerning the search for the “heaviest” segment of a numerical sequence that naturally arises in the biomolecular sequence analysis. We also introduce some fundamental algorithmic techniques for solving them.
Keywords: algorithm, maximum-average segment, maximum-sum segment, sequence analysis
1 Introduction
With the rapid expansion in genomic data, the age of large-scale biomolecular sequence analysis has arrived. An important line of research in sequence anal-
ysis is to locate biologically meaningful segments, e.g. conserved segments (Stojanovic et al., 1999), GC-rich regions (Bird, 1987; Gardiner-Garden and Frommer, 1987; Huang, 1994), non-coding RNA genes (Cs˝ur¨os, 2004), and transmembrane segments (Fariselli et al., 2003).
A common approach is to assign a value to each residue, and then look for consectutive subsequences (substring; segment) with high sum or aver- age. In order to locate these interesting segments, many combinatorial and probabilistic techniques have been proposed. Perhaps the most popular ones are window-based. That is, a window of a fixed length is moved down the sequence/alignment and the content statistics are calculated at each position that the window is moved to (Nekrutenko and Li, 2000). Since an optimal region could span several windows, the window-based approach might fail in finding the exact locations of some interesting regions.
This chapter surveys recent developments in locating constrained heaviest segments in a number sequence. We first compile a list of recent applications in this category. Then we introduce two basic algorithms for solving the maximum-sum segment problem and the maximum-average segment prob- lem. Finally, we discuss some possible extensions.
2 Heaviest Segments in Genomic Sequence Analysis
This section describes the heaviest segments and their applications in ge- nomic sequence analysis in details to put the problems and some related results in proper perspective.
2.1 GC-Rich Regions
In all organisms, the GC base composition of DNA varies between 25–75%, with the greatest variation in bacteria. Mammalian genomes typically have a GC content of 45-50%. Nekrutenko and Li (2000) showed that the extent of the compositional heterogeneity in a genomic sequence strongly correlates with its GC content. Genes are found predominantly in the GC-richest iso- chore classes. Hence, finding GC-rich regions is an important problem in gene recognition and comparative genomics.
Huang (1994) used the expression x − p · l to measure the GC richness of a region, where x is the C+G count of the region, p is a positive constant
ratio, and l is the length of the region. In other words, each of nucleotides C and G is given a reward of 1 − p, and each of nucleotides A and T is penalized by p. Similar expression was used by Sellers (1984) for recognizing patterns by mismatch density. A length cutoff L is given to avoid reporting extremely short optimal regions. Huang extended the well-known recurrence relation used by Bentley (1986) for solving the maximum sum consecutive subsequence problem, and derived a linear-time algorithm for computing the optimal segments with lengths at least L.
Here we explain briefly Huang’s idea for computing the maximum sum consecutive subsequence of length at least L. Let A = ha1, a2, . . . , ani be a DNA sequence of length n. Use w(X) to denote the score of nucleotide X, i.e. w(G)=w(C)=1 − p, and w(A)=w(T )=−p. Define S(i) to be the maximum score of regions ending at position i of A, which include the empty region. The scores S(i) can be computed by the following recurrence:
S(i) =
½ max{S(i − 1) + w(ai), 0} if i > 0,
0 if i = 0.
Now let us shift along the sequence with a window of size L. For each fixed window, we can compute its score, and then the maximum score of regions ending at the front of the window with the help of the vector S. This results in a linear-time method for computing the maximum sum consecutive subsequence of length at least L.
As noted by Huang, the lengths of the regions reported by the algorithm are usually much greater than the cutoff L. An immediate implication is that they might contain some very poor and irrelevant regions. It is there- fore natural to consider bounding the target regions with additional upper bound. Lin et al. (2002), and Fan et al. (2003) gave an algorithm that can be combined with Huang’s algorithm to yield a linear-time algorithm for computing the maximum sum consecutive subsequence of length between lower bound L and upper bound U.
Huang (1994) also proposed an interesting alternative measure for finding GC-rich regions. Namely, given a DNA sequence, one would now attempt to find segments of length at least L with the highest C+G ratio. Specifically, each of nucleotides C and G is assigned a score of 1, and each of nucleotides A and T is assigned a score of 0.
DNA sequence: ATGACTCGAGCTCGTCA Binary sequence: 00101011011011010
The maximum-average segments of the binary sequence correspond to seg- ments with the highest GC ratio in the DNA sequence.
He noted that such an optimal segment is of length at most 2L − 1.
This observation yields an O(nL)-time algorithm for computing a segment of length at least L with the highest C+G ratio, where n is the length of the input sequence. More efficient algorithms for this problem were given by Lin et al. (2002), Goldwasser et al. (2005), and Chung and Lu (2004).
2.2 CpG Islands
CpG islands are defined as regions of DNA of at least 200bp (i.e. base pairs) in length with G+C content above 50%, and a ratio of observed vs.
expected CpGs (CG di-nucleotides) at least 0.6 (Gardiner-Garden and From- mer, 1987). Most of the CpG islands are between 200 and 1400bp with a majority of them being 200–400bp.
CpG islands often occur in the 50 regions of genes (Bird, 1987). They are typically a few hundred to a few thousand bases long. Though the widely ac- cepted definition of what constitutes a CpG island was proposed by Gardiner- Garden and Frommer (1987), new definitions and methods for a CpG island are still in progress (Takai and Jones, 2002).
A Markov chain model was introduced by Durbin et al. (1998) to decide if a short DNA sequence comes from a CpG island or not. The model consists of a dinucleotide table, which, for each of the 16 different dinucleotides, gives the log likelihood ratio of the frequencies of the dinucleotide in CpG islands and in non-CpG regions. The numbers in the table range from −1.169 for the dinucleotide TA to 1.812 for the dinucleotide CG. The log-odds score of a DNA sequence is the sum of the log-odds scores of every dinucleotide in the sequence. The average score (also called normalized score) of the sequence is obtained by dividing its score by its length. It is known that CpG islands and non-CpG regions can be better discriminated by using average scores than using raw scores. The histogram of the average scores of CpG islands and non-CpG regions in (Durbin et al., 1998) shows that all non-CpG regions have average scores less than 0.1 and most of the CpG islands have average scores greater than 0.1. The average score value of 0.1 could thus be used as a cutoff in deciding if a sequence comes from a CpG island.
The above Markov chain model can be used to locate CpG islands in a long genomic sequence by computing the average score for a window of constant size around every position in the sequence and plotting the scores.
However, this approach is not very effective because CpG islands have sharp boundaries and variable lengths (Durbin et al., 1998). We consider an alter- native approach to identify CpG islands based on the Markov chain model and program MAVG.
The input genomic sequence is converted into a sequence of real num- bers using the dinucleotide table mentioned above. The average (score) of a segment of the number sequence is the sum of the numbers in the segment divided by the length of the segment. The core of a CpG island is defined as a region of length at least 250 bp with the maximum average score. The full extent of a CpG island is the longest region that does not contain any sufficiently long (i.e. 250 bp or longer) subregion with average score below the cutoff. Lin et al. (2003) implemented an algorithm for enumerating k maximum-average segments with lengths at least L, where L is given pa- rameter, as a C program called MAVG. The empirical tests suggest that the programs MAVG and NEWCPGSEEK, which is a popular existing program for finding CpG islands, are complementary in the sense that a combination of their may provide a more accurate predication of CpG islands.
2.3 Annotating Multiple Sequence Alignments
Conserved regions in biomolecular sequences are strong candidates for func- tional elements. The most popular methods to compute conserved regions all start with a given multiple sequence alignment (Stojanovic et al., 1999;
Stojanovic and Dewar, 2004). Stojanovic et al. (1999) gave several meth- ods for finding highly conserved regions within previously computed multiple alignments. Three of the methods are based on assigning a numerical score to each column of a multiple alignment and then looking for runs of columns with high cumulative scores. Since the assigned scores may be all positive (e.g. in the information content case), each examined column could increase the cumulative score. It follows that the entire alignment could be reported erroneously as a conserved region. Therefore, it is imperative that each col- umn score is adjusted by subtracting a positive anchor value. Determining such an anchor value appropriately for each dataset could make the use of a program based on the above approach very complicated. An alternative solution to the above problem is to look for runs of sufficiently many columns in the multiple alignment with the maximum average (or normalized) score instead. This can be efficiently computed by the algorithm for the length- constrained maximum average consecutive subsequence problem.
2.4 Post-Processing Sequence Alignments
A new popular approach to gene prediction in the human genome is based on comparative analysis of human and mouse DNA. The rationale behind this approach is that similarity between corresponding human and mouse exons is 85% on average, while similarity between introns is 35% on average (Ar- slan, E˘gecio˘glu, and Pevzner, 2001). Though the Smith-Waterman (Smith and Waterman, 1981) local alignment approach has been very successful in revealing highly conserved regions by discarding poorly conserved surround- ing regions, a potential drawback of the method is that it may lead to the inclusion of arbitrarily poor internal regions (called the mosaic effect).
In an attempt to fix the mosaic effect problem, Zhang et al. (1999) sug- gested to first run Smith-Waterman type of alignment algorithms and then post-process the computed alignments. They developed an elegant linear- time algorithm that decomposes a long alignment into sub-alignments to avoid the mosaic effect. The method for computing the length-constrained maximum average consecutive subsequences can be used to locate within an alignment the region that is sufficiently long and has the maximum degree of normalized similarity.
2.5 All Maximal Scoring Segments
Ruzzo and Tompa (1999) gave a linear-time algorithm for finding all maximal- sum segments. The problem arises in biological sequence analysis, where the maximal-sum segments correspond to regions of unusual composition in a nucleic acid or protein sequence. See Karlin and Brendel (1992) and Karlin and Altschul (1993) for some potential applications.
The idea of this linear-time algorithm is to maintain a list of candidates while processing the sequence. For each candidate segment (i, j), there are two crucial values: (1) the prefix sum of the first i − 1 positions, i.e. P (i − 1), which is referred to as the left prefix sum, and (2) the prefix sum of the first j positions, i.e. P (j), which is referred to as the right prefix sum. Whenever we scan in a new positive value, we search the candidate segments from right to left for the leftmost segment to merge with. The segments to be merged with must have a lower left prefix sum, and a lower right prefix sum.
2.6 Maximum-Scoring Segment Sets
Cs˝ur¨os (2004) considers the problem of finding maximum-scoring segment sets. Given a sequence of length n and an integer k ≤ n, we wish to find a non-intersecting set of k segments with maximum total score. A k-cover C = {S1, S2, ..., Sk} is a nonintersecting set of segments. The score of a k-cover C is the sum of its elements’ scores. Cs˝ur¨os (2004) proposed two algorithms which are based on the incremental nature of maximal covers. He first showed that a maximal (k + 1)-cover can be obtained from any maximal k-cover, either (1) by adding a new segment to it, or (2) by removing the middle of segment in it. This immediately yields an algorithm that finds a k-cover with maximum score for k ≤ K in O(nK) time where K is an upper bound on the cover size. On the other hand, he also showed that a maximal (k − 1)-cover can be created from any maximal k-cover by merging two segments, or by removing one. This yields an algorithm for finding a maximal k-cover in O(n log n) time. An improved algorithm was given by Bengtsson and Chen (2006).
Cs˝ur¨os (2004) tested his program on M. jannaschii (1.66 Mbp, GenBank accession NC 000909.1). By a maximum likelihood estimation of segments, he employed the scores -0.66 if the corresponding nucleotide was A (adenine), T (thymine), or W (weak), and the scores 0.72 for G (guanine), C(cytosine), or S (strong). The smallest maximal cover that includes all tRNAs has size k = 38. That cover also includes all rRNAs, RNase P RNA, and SRP 7S genes. The maximal 46-cover contains all RNA genes of Klein et al. (2002), including Mj6a, not discovered by either the HMM or the sliding windows of Schattner (2002). The 46-cover intersects only one more protein-coding gene than the 38-cover.
3 Two Basic Algorithms
This section introduces some algorithmic techniques for solving the maximum- sum segment problem and the maximum-average problem.
3.1 The maximum-sum segment problem
Given a sequence of real numbers A = ha1, a2, . . . , ani, the maximum-sum segment problem is to find a consecutive subsequence (i.e., a substring or segment) in A with the maximum sum. For each position i, we can compute
9 –3 1 7 –15 2 3 –4 2 –7 6 –2 8 4 -9 S(i) 9 6 7 14 –1 2 5 1 3 –4 6 4 12 16 7
Figure 1: A = h 9, -3, 1, 7, -15, 2, 3, -4, 2, 7, 6, -2, 8, 4, -9 i. The maximum-sum segment of A is h 6, -2, 8, 4 i, whose sum is 16.
the maximum-sum segment ending at that position in O(i) time. Therefore, a naive algorithm runs in Pn
i=1O(i) = O(n2) time.
Now let us describe a more efficient dynamic-programming algorithm for this problem (Bentley, 1986; Cormen et al., 1999). Define S(i) to be the maximum sum of segments ending at position i of A. The value S(i) can be computed by the following recurrence:
S(i) =
½ ai+ max{S(i − 1), 0} if i > 1,
a1 if i = 1.
If S(i − 1) < 0, concatenating ai with its previous elements will give smaller sum than ai itself. In this case, the maximum-sum segment ending at position i is ai itself.
By a tabular computation, each S(i) can be computed in constant time from i = 1 to i = n, therefore in total O(n) time. During the computation, we also need to record the largest entry computed so far in order to report where the maximum-sum segment ends. We also record the traceback information for each position i so that we can trace back from the end position of the maximum-sum segment to its start position. If S(i − 1) > 0, we need to concatenate with previous elements for a larger sum, therefore the traceback symbol for position i is “←.” Otherwise, “↑” is recorded. The traceback information can be used to quickly construct the maximum-sum segment by following the arrows until a “↑” is reached. Figure 1 illustrates the process.
Let prefix sum P (i) = Pi
j=1aj be the sum of the first i elements. It can be easily seen that Pj
k=iak = P (j) − P (i − 1). Therefore, if we wish to compute for a given position the maximum-sum segment ending at it, we could just look for a minimum prefix sum ahead of this position. This yields another linear-time algorithm for the maximum-sum segment problem.
3.2 The maximum-average segment problem
Given a sequence of real numbers, A = ha1, a2, . . . , ani, the maximum-average segment problem is to find, for each position i, a consecutive subsequence of A starting at that position such that the average value of the numbers in the subsequence is maximized. Lin et al. (2002) gave a very interesting linear-time algorithm for solving this problem.
By using a technique of partitioning each suffix of A into chain of right- skew substrings, which will be defined later, we can answer for each position i, a consecutive subsequence of A starting at that position such that the average value of the numbers in the subsequence is maximized. We show that this right-skew decomposition can be done in O(n) time by scanning all suffices from the shortest to the longest. The algorithmic details of the techniques used are somewhat involved. For a reader interested in them, we continue with a more detailed overview of the right-skew decomposition.
Let w(A) = Pn
i=1ai be the sum of elements of A. Furthermore, let d(A) = |A| = n, be the length of the sequence A. The average of A is defined as µ(A) = w(A)/d(A). The definition below is the key to the construction.
Definition 1 A sequence A = ha1, a2, . . . , ani is right-skew if and only if the average of any prefix ha1, a2, . . . , aii is always less than or equal to the average of the remaining suffix subsequence hai+1, ai+2, . . . , ani. A partition A = A1A2· · · Akis decreasingly right-skew if each segment Ai of the partition is right-skew and µ(Ai) > µ(Aj) for any i < j .
The following are some useful properties of right-skew segments and their averages.
Lemma 1 (Combination) Let A, B be two sequences with µ(A) < µ(B).
Then µ(A) < µ(AB) < µ(B).
Proof. Let λ = d(A)/d(AB). We have µ(AB) = λµ(A) + (1 − λ)µ(B). The
result is true because 0 < λ < 1. ¤
Lemma 2 Let A, B be two right-skew sequences with µ(A) ≤ µ(B). Then the sequence AB is also right-skew.
Proof. Consider a prefix P of AB. Clearly, µ(P ) ≤ µ(B) if P = A. If P is a proper prefix of A, i.e. A = P Y for some nonempty sequence Y , then
Report-DRS-Part(i, p[·])
Input: i denoting the suffix sequence hai, ai+1, . . . , ani; p[·]: right-skew pointers of A.
Output: The decreasingly right-skew partition of the suffix.
1 while i ≤ n do B Reports hai, . . . , aji as a right-skew segment.
2 Output (i, p[i]); i ← p[i] + 1
Figure 2: Reporting the decreasingly right-skew partition of a suffix sequence.
DRS-Point(A)
Input: A sequence A = ha1, a2, . . . , ani.
Output: n right-skew pointers of A, encoded by array p[·].
1 for i ← n downto 1 do
2 p[i] ← i; w[i] ← w(ai); d[i] ← d(ai); B Each haii alone is right-skew.
3 while (p[i] < n) and (w[i]/d[i] ≤ w[p[i] + 1]/d[p[i] + 1]) do 4 w[i] ← w[i] + w[p[i] + 1]
5 d[i] ← d[i] + d[p[i] + 1]
6 p[i] ← p[p[i] + 1]
Figure 3: Setting up the right-skew pointers in O(n) time.
we have µ(P ) ≤ µ(A) ≤ µ(Y ) by Lemma 1. Hence, µ(P ) ≤ µ(Y B) since µ(P ) ≤ µ(B).
On the other hand, if P contains a proper prefix of B, i.e. B = CD and P = AC for some nonempty sequences C and D, then µ(C) ≤ µ(B) ≤ µ(D).
Hence, µ(P ) = µ(AC) ≤ µ(D) since µ(A) ≤ µ(B) ≤ µ(D). ¤ Each suffix of A, hai, . . . , ani, defines a decreasingly right-skew partition, denoted as A(i)1 A(i)2 · · · A(i)k , for some k ≥ 1. Suppose that A(i)1 = hai, . . . , ap[i]i, where p[i] is called the right-skew pointer of index i. Note that the right-skew pointers of A implicitly encode the decreasingly right-skew partitions for each suffix hai, . . . , ani of A. Given the right-skew pointers, one can easily report the decreasingly right-skew partitions of a suffix as illustrated in Figure 2.
Interestingly, we can compute all right-skew pointers in linear time.
Lemma 3 The algorithm DRS-Point given in Figure 3 computes all right- skew pointers for a length n sequence in O(n) time.
Proof. Consider the algorithm DRS-Point shown in Figure 3. The working
pointer i scans the elements of A from right to left. By Lemma 2, two increasingly right-skew segments can be grouped into one right-skew segment and hence, the pair (i, p[i]) always represents a segment of A that is right-skew throughout the entire algorithm. The correctness of the algorithm follows from the fact that the right-skew pointers found by the algorithm encode a partition of each suffix of A with strictly decreasing averages.
We can analyze the time complexity of the algorithm by an amortized argument. We conclude that the amortized cost of each iteration of the for-loop is just a constant.
In short, we deposit a credit whenever a correct value of the right-skew pointer p[·] is found. Later on, when the algorithm needs to advance the p[·] pointer in the while-loop, the skipping cost can be charged to the pre- deposited credits. Since exactly n credits are deposited in the entire process, the while-loop spends at most overall O(n) time. ¤ It should be noted that (i, p[i]) is the maximum-average segment of A starting at position i. Readers are encouraged to define the “left-skew de- composition” and locate the maximum-average segments ending at each po- sition.
4 Discussion
This chapter briefly surveys a few fundamental problems concerning the search for the “heaviest” segment of a numerical sequence that naturally arises in the biomolecular sequence analysis. Two fundamental algorithms are presented to help the readers understand the basic techniques used in this line of investigation.
This has been an exciting new research theme in recent years. Other related interesting topics include the problems of locating the longest and shortest heaviest segments (Allison, 2003; Wang and Xu, 2003; Chen and Chao, 2005), the maximum segments with a constrained length (Fariselli et al., 2003), the range maximum-sum segment query (Chen and Chao, 2004), the k maximum-sum segments (Bae and Takaoka, 2004; Bengtsson and Chen, 2004; Cheng et al., 2005), and the disjoint segments with maximum sum of densities (Bergkvist and Damaschke, 2005; Chen et al., 2005; Liu and Chao, 2006).
Acknowledgements
Kun-Mao Chao was supported in part by NSC grants 94-2213-E-002-018 and 95-2221-E-002-126-MY3 from the National Science Council, Taiwan.
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