Scaling behavior of nucleotide cluster in DNA sequences

Scaling behavior of nucleotide cluster in DNA sequences 1 first-author †E-mail:jh_chengjun@163.com †E-mail:jh_chengjun@163.com jh_chengjun@163.com 1 2 Department of Physics, Jinhua College of Profession and Technology, Jinhua 321017, China Department of Physics, Jinhua College of Profession and Technology, Jinhua 321017, China Department of Physics, Wenzhou University, Wenzhou 325027, China Department of Physics, Wenzhou University, Wenzhou 325027, China In this paper we study the scaling behavior of nucleotide cluster in 11 chromosomes of Encephalitozoon cuniculi Genome. The statistical distribution of nucleotide clusters for 11 chromosomes is characterized by the scaling behavior of P ( S )∝e − αS , where S represents nucleotide cluster size. The cluster-size distribution P ( S 1 + S 2 ) with the total size of sequential C-G cluster and A-T cluster S 1 + S 2 were also studied. P ( S 1 + S 2 ) follows exponential decay. There does not exist the case of large C-G cluster following large A-T cluster or large A-T cluster following large C-G cluster. We also discuss the relatively random walk length function L ( n ) and the local compositional complexity of nucleotide sequences based on a new model. These investigations may provide some insight into nucleotide cluster of DNA sequence. Nucleotide cluster DNA sequences Scaling behavior INTRODUCTION 1 The sequence dependency of DNA sequences is important for protein-DNA recognition (Gromiha, 2005 2005 ; Gromiha et al., 2004 2004 ; Olson et al., 2004 2004 ). Statistical analysis of DNA chain bending profiles for complete genome sequences revealed that long-range correlations in the 10~200 bp range are the signatures of the nucleosomal in structure and that over larger distances ( > 200 bp) are likely to play a role in the hierarchical packaging of DNA. To which extent sequence-dependent DNA mechanical properties help to regulate the structure and dynamics of chromatin is an issue of fundamental importance. We recently explored long-range correlations in the base composition of DNA and adopted a new method to study the scaling behaviors of C-G clusters in different organism chromosomes (Cheng and Zhang, 2005a 2005a ; 2005b 2005b ). The term cluster was first defined by Provata and Almirantis ( 1997 1997 ; 2002 2002 ). They did not differentiate between adenine and guanine, both considered as Pu, cytosine and thymine, both considered as Py, and they defined the Pu-cluster as an ensemble of consecutive Pu bound by at least one Py on the left and right respectively, equivalent to Py-clusters. Here the Pu-cluster and Py-cluster are both called “nucleotide cluster”, which may be considered to constitute an entire DNA sequence. Some statistical properties of nucleotide clusters are linked to a higher level of organization, and statistical dynamics of clustering in the genome structure were investigated (Provata and Almirantis, 2002 2002 ; Zhang and Jiang, 2004 2004 ; Sun et al., 2004 2004 ; Chen et al., 2004 2004 ; Gromiha et al., 1997 1997 ; Hogan and Austin, 1987 1987 ). Here we change the definition of the cluster (Chen et al., 2004 2004 ). It is well known that the nucleotide A is paired with T, and C is paired with G. The CG content of DNA sequences was used to gain understanding of several properties of DNA, such as curvature, bending, etc. (Harrington and Winicov, 1994 1994 ; Sugiarto et al., 2006 2006 ; Vaillant et al., 2003 2003 ; Arnéodo, 1998 1998 ). In particular, in possible relation to the isochore structure of the human genome, it had been clearly shown that the long-range correlation properties of human DNA sequences are dependent upon their CG content (Vaillant et al., 2003 2003 ; Arnéodo, 1998 1998 ). If we can know the statistical properties and size distributions of sequential CG and AT clusters, we can then understand the DNA sequences in more detail. In this work, we investigate the size distribution of C-G clusters and A-T clusters in the complete nucleotide sequences for Encephalitozoon cuniculi Genome which has 11 chromosomes. We picked this organism as example since the size of the genome is manageable and evidence for a power law is particularly impressive in this species. Our aim is to investigate the scaling behaviors of nucleotide clusters in DNA sequences. METHOD OF CALCULATION 1 In this paper we change the definition of the Py- and Pu-cluster proposed by Provata and Almirantis ( 1997 1997 ; 2002 2002 ), and adopt hydrogen bond energy rule (Poland, 2004 2004 ; Azbel et al., 1982 1982 ; Azbel, 1973 1973 ). We refer 1’s to strongly bonded pairs (C or G) and 0’s to weakly bonded pairs (A or T) and define the average number n ̄ of C-G clusters in each block (Cheng and Zhang, 2005a 2005a ; 2005b 2005b ; Zhang and Chen, 2005 2005 ) as follows: $$\bar n = {{\sum {S_i } } \mathord{\left/{\vphantom {{\sum {S_i } } {\sum i }}} \right.\kern-\nulldelimiterspace} {\sum i }} = {{\sum\limits_{i = 1}^N {S_i } } \mathord{\left/{\vphantom {{\sum\limits_{i = 1}^N {S_i } } N}} \right.\kern-\nulldelimiterspace} N}$$ , here N refers to the number of CG clusters, and S i the size of CG clusters of i th cluster in the consecutive, non-overlapping block, where the size of non-over-lapping block is m . If the value of m is small to a certain extent, the average number n ̄ can show all characteristics of a whole sequence per block including the C-G content and A-T content. For example, there are two blocks with the sequence: 0101001111 and 1010101011 respectively. We can obtain the average size of CG cluster n ̄( m )=6/3 and n ̄( m )=6/5 respectively, and get to know that the distribution of CG cluster is different in those two sequences although the number of strong pairs is the same. Therefore, the total number of CG cluster n in the consecutive, non-overlapping block is unilateral because it neglects the array sequence and covers the feature of different sequences with the same number of CG cluster n , and n ̄ can include more information of the sequence perfectly (Zhang and Jiang, 2004 2004 ; Poland, 2004 2004 ). First, we collect statistics for boxes in the complete nucleotide sequence containing n C-G units, thus giving the distribution function for the C-G content in m -blocks. The function P ( S ) which represents the fraction of clusters corresponding to a certain cluster size S is called cluster-size distribution. We define the cluster-size distribution of CG (or AT) clusters P ( S ) as $$P(S) = \frac{{N(S)}}{{N(S = 1)}}$$ , where N ( S =1) is the number of clusters with cluster size S =1, and it is apparently the largest one in the DNA sequence. Therefore, the cluster-size distribution of CG (or AT) clusters P ( S ) is always smaller than 1.0. Second, we can gain some insight into the correlations in DNA sequences by interpreting the sequence as a random walk (Mandelbrot, 1982 1982 ; Feder, 1989 1989 ), and make the following assignments to each base in the sequences: α i =−1, if C or G; α i =+1, if A or T. The two equations can be regarded as a random walk where α i =+1 indicates a step to the right and α i =−1 indicates a step to the left. The distance of the walk from the origin after n steps ( n bases) is then defined as $$L(n)' = \sum\limits_{i = 1}^n {\alpha _i }$$ . On the average, this function will have the value < L ( n )′ > = n Δ f , where Δ f = f at − f cg . It is convenient to define the length of the walk compared to the average walk given above, i.e., $$L(n) = \sum\limits_{i = 1}^n {\alpha _i } - n\Delta f$$ . The average value of this function L ( n ) is zero, i.e., < L ( n ) > =0. Third, we study the local compositional complexity of nucleotide sequences. Given a window of length L , we define the local complexity state by the vector for the nucleotide sequence as n =[ n 1 , n 2 , n 3 , n 4 ], such that n 1 ≥ n 2 ≥ n 3 ≥ n 4 are all nonnegative integers that satisfy n 1 + n 2 + n 3 + n 4 = L . The number n 1 represents the number of occurrences of the most frequent nucleotide α 1 , n 2 the number of the second most frequent nucleotide α 2 and so on. Here α i ∈{A, C, G, T}. The multinomial coefficient is given by $$\Omega = \frac{{L!}}{{n_1 ! n_2 ! n_3 ! n_4 !}}$$ , and it is the total number of distinguishable arrangements of n 1 outcomes of α 1 , n 2 outcomes of α 2 , and so forth. Over a sequence window length L , we adopt the measure of complexity (Salamon and Konopka, 1992 1992 ; Salamon et al., 1993 1993 ; Wootton and Federhen, 1993 1993 ), K =(ln Ω )/ L . At last we introduce a new way to study the local compositional complexity of nucleotide sequences. Given a window of length L , we define the local complexity state by the vector for the nucleotide sequence as n =[ n 1 , n 2 , n 3 , …, n 16 ], $$\sum\limits_{i = 1}^{16} {n_i } = L - 1$$ . The number n 1 represents the number of occurrences of the most frequent nucleotide α 1 β 1 , n 2 the number of the second most frequent nucleotide α 2 β 2 and so on, where α i β i ∈{AA, AC, AG, AT, …, TG, TT}. The multinomial coefficient is given by $$\Omega ' = {{(L - 1)!} \mathord{\left/{\vphantom {{(L - 1)!} {\prod\limits_{i = 1}^{16} {n_i !} }}} \right.\kern-\nulldelimiterspace} {\prod\limits_{i = 1}^{16} {n_i !} }}$$ , and it is the total number of distinguishable arrangements of n 1 outcomes of α 1 β 1 , n 2 outcomes of α 2 β 2 , and so forth, over a sequence window length L . Then we adopt the measure of complexity, K ′=(ln Ω )/( L −1). The species we have chosen to be treated as examples are Encephalitozoon cuniculi Genome, which has 11 chromosomes with the range of lengths from 194 439 bp to 267 509 bp. The complete genome sequences data used in this paper were all taken from the www at the National Center for Biotechnology Information (USA) ( http://www.ncbi.nih.gov/genbank/genomes/ http://www.ncbi.nih.gov/genbank/genomes/ ) in GenBank format. RESULTS AND DISCUSSION 1 Cluster-size distribution P ( S ) 2 The cluster-size distribution P ( S ) of A-T cluster for Encephalitozoon cuniculi Genome serving cluster size S is shown in Fig. 1a 1a . In the upper right of Fig. 1a 1a , the straight line represents an exact power law with the exponents as 0.647 and the correlation coefficient as 0.991 for Chromosome I sequence. Similar results are also obtained for other chromosomes from Fig. 1a 1a . In the meantime, the cluster-size distribution P ( S ) of C-G cluster for Encephalitozoon cuniculi Genome serving cluster size S is given in Fig. 1b 1b . In the upper right of Fig. 1b 1b , the straight line represents an exact power law with the exponents as 0.812 and the correlation coefficient as 0.988 for Chromosome I sequence. Similar results are obtained from Fig. 1 1 . Apparently, we can find that the results conform to the power law from Fig. 1 1 : $$P(S) \propto {\text{e}}^{ - \alpha S}$$ . It can be found that the value of the scaling exponent α for C-G and A-T clusters ranges from 0.771 to 0.887 ( α C-G ) and 0.470 to 0.662 ( α A-T ) respectively, and that the average of the scaling exponent α ̄ C-G for CG cluster (0.825) is greater than that for AT cluster (0.582). The correlation coefficient for those scaling exponents ranges from 0.949 to 0.999 with most of them being double numbers 9 after the radix point. It is nearly equal to 1, and this fit is perfectly good. In Fig. 1 1 , we find that the maximum value of A-T and C-G cluster size is different for these chromosomes. The maximum value of A-T cluster is 27 (Chromosome XI). However, the maximum value of C-G cluster of is only 16 (Chromosome V). Cluster-size distribution P ( S 1 + S 2 ) with the total size of sequential C-G and A-T S 1 + S 2 2 In DNA sequence, the C-G and A-T clusters appear alternately. Here we discuss the cluster-size distribution P ( S 1 + S 2 ) with the total size of sequential C-G and A-T S 1 + S 2 . In order to explain our calculation method in more details, we express the sequence 0001011111101010110111111100… as B 3 A 1 B 1 A 6 B 1 A 1 B 1 A 1 B 1 A 2 B 1 A 7 B 2 … Here A i represents C-G clusters of size i and B j represents A-T clusters of size j . For example, B 3 represents A-T clusters of size 3. Here S 1 , S 2 stand for the size of sequential C-G cluster and A-T cluster respectively. The cluster-size distribution P ( S 1 + S 2 ) versus S 1 + S 2 for Encephalitozoon cuniculi Genome is shown in Fig. 2 2 . In the upper right of Fig. 2 2 , the straight line represents an exact power law with the exponents as 0.566 and the correlation coefficient as 0.996 for Chromosome I. Similar results are obtained for other chromosomes in Fig. 2 2 . Apparently, we can find that the results conform to the power law in Fig. 2 2 : $$P(S_1 + S_2 ) \propto {\text{e}}^{ - \alpha _{S_1 + S_2 } S}$$ . The scaling exponent α S 1 + S 2 ranges from 0.494 to 0.596, and the correlation coefficient for those scaling exponents ranges from 0.971 to 0.996 with the average 0.989. In Fig. 2 2 , we find that the maximum value of S 1 + S 2 is different for these chromosomes. The maximum value of S 1 + S 2 for these chromosomes is only 28 (Chromosome XI). This means that there does not exist the case of large C-G cluster following large A-T cluster or large A-T cluster following large C-G cluster. At last, the values of the scaling exponents for 11 chromosomes of Encephalitozoon cuniculi Genome are all shown in Fig. 3 3 . The solid squares, dots, and triangles refer to the α CG , α CAT , and α S 1 + S 2 , respectively. The dashed straight lines give the average of them respectively. Fractional Brownian walk 2 In Fig. 4 4 , the upper solid curve gives the random walk function L ( n )′ defined in Eq.( 4 4 ) for Chromosome I of the Encephalitozoon cuniculi Genome. The upper dashed straight line gives the locus of the average length of the walk as a function of the number of steps given in Eq.( 5 5 ). The lower solid curves give the random walk as a function of the average length of the walk as defined in Eq.( 7 7 ) and for a random sequence on the same scale respectively. The average of this function is zero as indicated by the lower solid straight line. In the meantime, we study the relative random walk length function L ( n ) defined in Eq.( 7 7 ) for the other 10 chromosomes of Encephalitozoon cuniculi Genome. We find that the small amplitude curves give L ( n ) a random sequence for comparison in each case. Apparently, L ( n ) is larger for a given actual sequence as compared to a corresponding random sequence, which means the information for the DNA of chromosomes is greatly different in a corresponding random sequence. This investigation can provide some insight into nucleotide clusters properties, and help us understand the long-range correlation and other properties of DNA sequences. DNA sequence complexity analysis 2 Fig. 5 5 is about complexity versus bp position ( L =70 bp). The upper solid curve gives the local compositional complexity of nucleotide sequences K ′ defined in Eq.( 11 11 ) and the lower solid curve gives K difined in Eq.( 9 9 ) for Chromosome I of Encephalitozoon cuniculi Genome. We find that the value of K ′ is almost twice that of K . The changes of K ′ versus bp position are extremely similar to that of K . 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