Statistical properties of nucleotide clusters in DNA sequences

Statistical properties of nucleotide clusters in DNA sequences 1 first-author †E-mail:Jh_Chengjun@163.com †E-mail:Jh_Chengjun@163.com Jh_Chengjun@163.com 2 Department of Physics, Jinhua University, Jinhua 321017, China Department of Physics, Jinhua University, Jinhua 321017, China Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Physics, Zhejiang University, Hangzhou 310027, China Using the complete genome of Plasmodium falciparum 3D7 which has 14 chromosomes as an example, we have examined the distribution functions for the amount of C or G and A or T consecutively and non-overlapping blocks of m bases in this system. The function P ( S ) about the number of the consecutive C-G or A-T content cluster conforms to the relation P ( S )∝ e − αs ; values of the scaling exponent α CG are much larger than α AT ; and α AT of 14 chromosomes are hardly changed, whereas α CG of 14 chromosomes have a number of fluctuations. We found maximum value of A-T cluster size is much larger than C-G, which implies the existence of large A-T cluster. Our study of the width function ξ ( m ) of cluster C-G content showed that follows good power law ξ ( m )∝ m − γ . The average γ ̄ for 14 chromosomes is 0.931. These investigations provide some insight into the nucleotide clusters of DNA sequences, and help us understand other properties of DNA sequences. DNA sequence Plasmodium falciparum 3D7 Nucleotide clusters Power law INTRODUCTION 1 The genetic information on organisms is stored in DNA sequences, represented by a string of just four letters, each corresponding to a definite type of nucleotides: adenine A (purines), guanine G (purines), cytosine C (pyrimidines), and thymine T (pyrimidines) (Albert et al., 1994 1994 ). These letters can form different combinations. Each DNA sequence is packed in a chromosome, varying in length from 10 5 base pairs (bp) in yeast to 10 9 bp in human. During the past few years, many biologists, chemists and physicists have researched intensively the intriguing statistical behaviour of DNA sequences. The most common research fields include mutual information functions (Herzel et al., 1998 1998 ; Herzel and Grobe, 1997 1997 ; Li, 1992 1992 ), the power spectra (Li, 1997 1997 ; Voss, 1992 1992 ; Viswanathan et al., 1998 1998 ; Li and Kaneko, 1992 1992 ), auto-correlation analysis (Herzel et al., 1998 1998 ; Azbel, 1995 1995 ; De Sousa Vieira, 1999 1999 ), detrended fluctuation analysis (Sun et al., 2004 2004 ; Peng et al., 1994 1994 ; Buldyrev et al., 1995 1995 ). However, some statistical aspects of the DNA sequences are still obscure. The clustering of similar nucleotides can be clarified by studying the properties of the cluster size distribution on various real DNA sequences, ranging from viral to higher eukaryotic sequences. The term “cluster of similar (homologous) nucleotides” means a string in a sequence containing only one type of nucleotides, e.g., only A’s or only C’s, etc. The term “cluster” was defined by Provata and Almirantis ( 1997 1997 ; 2002 2002 ). They did not differentiate between adenine and guanine, both considered by them as Pu; and between cytosine and thymine, both considered by them as Py. They defined the Pu-cluster as an ensemble of consecutive Pus bound by at least one Py respectively on the left and on the right, and to be equivalent to Py-clusters. Here the Pu-cluster and Py-cluster are both called “nucleotide cluster”. Therefore, an entire DNA sequence may be considered as a collection of nucleotide clusters. Some statistical properties of nucleotide clusters are linked to a higher level of organization. The statistical dynamics of clustering in the genome structure were also investigated (Provata and Almirantis, 2002 2002 ). In fact, we found that Pu and Py clusters appear alternately in those clusters. If we can know the statistical properties and size distributions of sequential Pu-clusters and Py-clusters, we will have more detailed understanding of the DNA sequences. In this work, we investigated the size distribution of purine and pyrimidine clusters in the complete nucleotide sequences of Plasmodium falciparum 3D7 which has 14 chromosomes. We found that the distribution values for every chromosome are similar and obtained some information on the statistical properties of nucleotide clusters in DNA sequences. METHOD OF CALCULATION 1 Here we modify the definition of the Py-cluster and Pu-cluster proposed by Provata and Almirantis ( 1997 1997 ; 2002 2002 ), using hydrogen bond energy rule (Azbel, 1973 1973 ; Azbel et al., 1982 1982 ). We regard 1’s as strongly bonded pairs (C or G) and 0’s as weakly bonded pairs (A or T). We define the function of the size S , the number N ( S ) of continuous C-G cluster and the average number n ̄ of C-G units in each block (Chen and Zhang, 2005 2005 ; Sun et al., 2004 2004 ) as follows: $$\bar n = \frac{{n_{{\text{C - G}}} }}{{N_{{\text{C - G}}} }}$$ where n C–G represents the number of 1’s (C or G) per block and N C–G is the total number of C-G units (consecutive C and G are treated as one unit; Fig. 1 1 for illustration) in the block. The division of the genome into consecutive non-overlapping blocks containing m bases is illustrated in Fig. 1 1 . We selected the snippet with 45 nucleotide letters as the example to show this change. The upper row illustrates the assignment of bases to blocks in the case of m =10. In the lower row, we indicate how we assign 0’s to A or T bases and 1’s to C or G bases and count the number of 1’s (C-G units) in each box (Sun et al., 2004 2004 ). We calculate n ̄ using Eq.( 1 1 ). If the value of m is small to a certain extent, the average number n ̄ can show all the characteristics of the whole sequence per block including the C-G content and A-T content. The parameter number n is unilateral as it neglects the array sequence and covers the feature of different sequences with the same value m , while n ̄ can include more information on the sequence perfectly. Then we collect the statistics of the boxes in the complete nucleotide sequence which contain n C-G units or n A-T units, thus obtaining the distribution function for the C-G content or the A-T content in m-blocks. The function P ( S ) which represents the fraction of clusters corresponding to a certain cluster size S is called the “cluster-size distribution”. To simplify the parameter N ( S ), we rewrite the function as P ( S )= N ( S )/ N ( S =1) where N ( S =1) is the number of units on cluster size S =1, which is apparently the largest in one DNA sequence. So the values of this parameter are always smaller than 1. On the other hand, we can get the distribution function for a block of m bases if we assume independent units. In terms of the definition of the average number n ̄, we obtain the following relations giving the first two moments of the distribution $$\left\{ \begin{gathered}\mu _2 (m) = \left\langle {\bar n^2 } \right\rangle \\\mu _1 (m) = \left\langle {\bar n} \right\rangle \\\end{gathered} \right.$$ We take as a standard measure of the breadth of the distribution the root-mean-square width (Poland, 2004 2004 ) $$\sigma _m = \sqrt {\mu _2 (m) - \mu _1 (m)^2 }$$ This is a useful method for defining a parameter in terms of the differences between the average of the square and the square of the average. We are interested in how the actual C-G cluster distributions obtained from 14 chromosomes of Plasmodium falciparum 3D7 genome deviate from the dependence on m given in Eq.( 4 4 ). To this end, we define the following function to investigate the relations between the width of the empirical distribution σ m and the m -dependence. $$\xi (m) = \sigma _m /\sqrt m$$ From Eq.( 5 5 ), we can note that if the distribution of the C-G content in m -blocks is random, ξ ( m ) should be equal to a constant value or close to a constant value as the length of the random sequence is increased while the slope of the line fitting Eq.( 5 5 ) should equal or approach 1 too. To test this conclusion we computed the random sequences 1000000 bp and get the slope 0.999 and the correlation coefficient 0.999 that we presupposed (Chen and Zhang, 2005 2005 ). The species we chose as example was the Plasmodium falciparum 3D7, which has 14 chromosomes with length of 643292 bp to 13515526 bp. The complete genome sequences data we use 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. We chose this organism as an example since the size of the genome is manageable and the evidence for power law is particularly impressive in this species. RESULTS AND DISCUSSIONS 1 In terms of Eq.( 2 2 ), we discuss the cluster-size distribution P ( S ) of the Plasmodium falciparum 3D7 genome. The A-T clusters distribution of chromosome XII is shown legibly in Fig. 2 2 . In this single-logarithm, we can link these squares to a straight line with slope of −0.087. The correlation coefficient is 0.972. We found that maximum value of A-T Cluster size is 100. We can note from the graph that the result complies with the power law: $$P(S) \propto e^{ - \alpha S}$$ where α is the absolute value of the line slope. The other 13 chromosomes were also similar to chromosome XII in the statistical property of the A-T cluster. The results are shown in Table 1 1 . The values of α AT range from 0.085 to 0.090 and the average is 0.087. Apparently, α AT of 14 chromosomes are hardly changed. The values of the correlation coefficient are 0.960 to 0.974. The maximum values of A-T Cluster size are 98 to 100. This means that there exists large A-T cluster in the DNA sequences. Likewise, the C-G clusters distribution of chromosome XII is shown in Fig. 3 3 legibly. The straight line represents a power law with exponent α CG =1.114 and correlation coefficient 0.982. The other 13 chromosomes are also similar to chromosome XII in the statistical property of the C-G cluster. The results are also shown in Table 1 1 . The values of α CG are 0.877 to 1.287 and the average is 1.103. The α CG values of 14 chromosomes have a number of fluctuations. The values of correlation coefficient are 0.961 to 0.996. The maximum values of C-G Cluster size are 10 to 15, which means that large C-G cluster does not exist in sequences different from those of the A-T cluster. 2 No. 2 Sequence 5 P ( S )∝e − αs 2 ξ ( m )∝ m − γ 2 Maximum value of cluster Length α AT Correlation coefficient α CG Correlation coefficient γ Correlation coefficient A-T cluster C-G cluster 1 Chromosome I 643292 0.085 0.960 1.011 0.968 0.918 0.999 99 10 2 Chromosome II 947102 0.087 0.969 1.158 0.984 0.920 0.998 100 11 3 Chromosome III 1060087 0.087 0.968 1.121 0.985 0.938 0.998 100 12 4 Chromosome IV 1204112 0.087 0.968 0.974 0.978 0.856 0.998 99 13 5 Chromosome V 1343552 0.087 0.969 1.296 0.996 0.948 0.998 100 14 6 Chromosome VI 1378756 0.090 0.969 1.094 0.989 0.927 0.999 98 15 7 Chromosome VII 13515526 0.087 0.969 1.045 0.973 0.917 0.998 100 12 8 Chromosome VIII 1325595 0.085 0.967 1.234 0.994 0.907 0.999 100 10 9 Chromosome IX 1541723 0.087 0.972 1.119 0.989 0.951 0.998 100 12 10 Chromosome X 1694445 0.090 0.974 1.287 0.995 0.948 0.998 99 10 11 Chromosome XI 2035250 0.090 0.974 1.002 0.967 0.949 0.998 100 14 12 Chromosome XII 2271477 0.087 0.972 1.114 0.982 0.945 0.998 100 12 13 Chromosome XIII 2732359 0.090 0.974 0.877 0.961 0.932 0.999 100 15 14 Chromosome XIV 3291006 0.087 0.974 1.094 0.964 0.976 0.999 100 14 2 Average 0.087 1.103 0.931 99.6 12.4 The width function ξ ( m ) is defined in Eq.( 5 5 ). Fig. 4 4 shows the results of ξ ( m ) based Eq.( 5 5 ) with m =400 on chromosome XII. The straight line represents an exact power law with exponent equal to 0.945 correlation coefficient of 0.998. We can get the direct conclusion when the results are expressed in natural logarithms for both coordinates. ξ ( m ) in Fig. 4 4 strongly suggests a power-law behavior. $$\xi (m) \propto m^{ - \gamma }$$ Here γ is the absolute value of the slope of the fit line in Fig. 4 4 . This linear fit agrees nicely with the results obtained by Poland ( 2004 2004 ). All of the other 13 chromosomes exhibit power-law behavior for the width of the C-G distributions similar to that shown in Fig. 4 4 . The values of γ are 0.856 to 0.976 and the average is 0.931. The γ values of 14 chromosomes have a number of fluctuations. The correlation coefficient values are from 0.998 to 0.999. At last, the values of the scaling exponents for 14 chromosomes of Plasmodium falciparum 3D7 are shown in Fig. 5 5 . The solid dots, squares and pentagons refer to the α AT , α CG and γ respectively. From the above discussion, we conclude that large C-G and A-T clusters are formed as a result of the correlations in the unabridged DNA sequences. In this process the cluster-size distribution P ( S ) follows exponential decay law and the widths of the C-G cluster distribution ξ ( m ) follow a power law. In addition to the behaviour of the genome for the Plasmodium falciparum 3D described here, we studied the genomes of yeast, nematode, erenothecium gossypii and human (Cheng and Zhang, 2005 2005 ). The question of course arises as to whether the occurrence of a power law with a characteristic exponent γ describing the width of the C-G distributions and α AT or α CG describing the cluster-size distributions is a general feature of the DNA of all organisms. 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