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ISSN: 2153-0602
Journal of Data Mining in Genomics & Proteomics
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An Analysis of Palindromes and n-nary Tract Frequencies found in a Genomic Sequence

Dan Ophir*

Department of Computer Science and Mathematics, Ariel University, Israel

*Corresponding Author:
Dan Ophir
Department of Computer Science and Mathematics
Ariel University, Israel
E-mail: [email protected]

Received Date: July 24, 2013; Accepted Date: August 22, 2013; Published Date: August 29, 2013

Citation: Ophir D (2013) An Analysis of Palindromes and n-nary Tract Frequencies found in a Genomic Sequence. J Data Mining Genomics Proteomics 4:140. doi: 10.4172/2153-0602.1000140

Copyright: © 2013 Ophir D, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The motivation to investigate n-nary and palindrome tracts arose following the discovery by Chargaff and coworkers of over-representation of certain DNA binary tracts in genomes. They investigated the frequencies of various ternary tracts in diverse locations in genes of various species. The current research further examines ternary tracts and the palindromes will hereafter be called designated tracts. Does a designated tract have any extraordinary frequencies of length and location? A theoretical mathematical analysis has been performed to analyze the amount of designated tracts according to the frequencies of its single elements. The designated tracts are categorized according to those that have mixed elements from a subset of a set composing the sequence, and according to which consist of a long tract of lower n-nary order. For example, tract analysis investigates whether the special phenomena are due to the ternary tract or due to a long binary tract that is included in it. The maximal n-nary tract order of interest in the genome is of three (ternary); four is the whole gene itself. However, the higher order of n-nary tracts is of interest in other areas like “Reliability theory”. Therefore, the general formulation and treatment of designated tracts is presented here and is demonstrated for the genomic aspects, which were thoroughly investigated in the two past decades.


Palindrome; Unary tract; Binary tract; Ternary tract; n-nary tract


The intriguing question of whether ternary over-representation is due to included binary tracts is answered in the present work. As will be shown, the binary tracts are over represented to the same extent as the ternary tracts; therefore, it can be concluded that the binary tracts dominate because they have biological impacts, but the ternary tracts do not contribute to biological impacts.

The genomic sequence may be compared to the reliability sequence in the mass production of S (successes) and F (failures), which may be designated according to their severity. Thus, it is possible to receive a sequence such as SF1F2F1F2SSF1. The searching of some patterns might result in interesting answers regarding the line production behavior.

Palindromes are another investigated pattern. There were no notable special extraordinary frequencies associated with their appearances.


As previously mentioned, the current work is based on previous research [1-6]. The data used in the investigation concerned the human gene p53, as given in genebank.


A special technique has been derived that is supported by further formalism and was validated by a program [7].


Tract: A tract is a sub-sequence within a sequence of events having a mutual property.


a. A sequence of events, for example, tossing a coin. Each event can have one of two values S (the coin falling on heads) or F (the coin falling on tails) [8].

b. A genomic sub-sequence consisting of nucleotides having special properties. An event is associated with one of the nucleotides from a set of nucleotides.

c. sub-sequence of a wide spectrum of qualities of a product in a serial mass production.

Source-set: Source-Set is a set of elements that builds the sequence.

Kernel set: Kernel-Set is a set of elements. The tract is built from elements taken from the Kernel-Set. The Kernel-Set is a subset of a Source-Set.

Unary tract: A tract that is composed of one type of elements, i.e., a set of elements from which the tract elements are composed, and which contain one element only. The cardinality of the Kernel-Set of the unary tract is one.

Exact unary tract: Exact Unary Tract is a unary tract that cannot be enlarged by exactly one element (one of its neighbors’ elements) and remain a unary-tract. Namely, the exact-unary-tract cannot be contained in another unary-tract.

Binary tract: Binary Tract is a tract whose Kernel-Set has a cardinality of two.

Exact-binary-tract: Exact-Binary-Tract is a binary tract that cannot be enlarged by one neighboring element and it remains a binary tract.

Pure-binary-tract: Pure-binary-tract is a tract whose elements are mixed enough, i.e. it does not contain a dominant - long enough (longer than some critical length) unary-tract.

This restriction is required in order to exclude the possibility that the properties that are assumed to belong to a binary tract are not due to some sub-unary tract that is included in the binary tract.

Exact-pure-binary-tract: Exact-Pure-Binary-Tract is a binary tract that is both: a pure binary tract and an exact binary tract.

Ternary and n-nary tracts: Similarly to the unary tract of all types, there are ternary tracts and n-nary tracts of the same types. Ternary’s kernel set possesses three elements; an n-nary tract kernel-set has n elements, i.e. n-nary tract is of order n.

According to these definitions, n-nary tract of order n-1 is also an n-nary tract. Namely, each binary tract is also a ternary tract. In the context of genomics, the highest order of n-nary tracts in the area of interest is three because the source set contains four elements and therefore there is justification for treating the quarto nary tracts in which the kernel set and the source set are equal.

Probability computations

Let p be the probability of an element to be from the kernel-set and q=1-p the probability of not being from the kernel set, i.e. the probability of belonging to the complementary set, namely to the complementary-set=source-set–kernel-set.

Random variable Xi(1)

Xi(1) is a random variable representing the length of an Exact Unary Tract starting at the i’th position of the sequence; the superscript (1) denotes the “unarity” of the tract. The general case is Xi(n) for any named “exact n-narity tract”.

equeation (1)

the exact unary tract starting at the i’ th position (i=0,…,L) is of the length k}

The random variable Xi(1) covers the whole domain as k ∞:

equeation (2)

The sum (equation 2) does not equal 1. Otherwise, each sequence should have at least one exact unary tract. This is not true; as a counterexample, see the sequence ‘FS’ (failure followed by a success), which has no exact tracts, versus the sequence ‘FF’, which has a tract (however of length 0).

Namely, the probability of the exact-unary-tract (even of length 0) converges to q. The probability of a tract not being an exact-unary-tract is 1-q=p. Such a tract should start with an element not belonging to the kernel-set and therefore its probability is p.

Exact unary tracts

The expected number of exact unary tracts of length k in a sequence of length L (assumed k<<L) can be found in [1]. Owing to the assumption about the long L compared to k, the extreme cases (i=0 and i=L – the first position is designated by 0) may be neglected and treated as the regular one in the additional formulas in this article.

The random variable Yi ={0, 1} is introduced:

equeation (3)

Using the formula for expectation, the following relations are derived (equation 5- equation 6):

equeation (4)

equeation (5)

equeation (6)

The term Exact (n,k,L,p) is a function whose name and parameters can be explained as follows: Exact – Exact tract, n- nary order: (1 stands for Unary, 2 stands for Binary, 3 stands for ternary), k-tract length, LSequence length, and p is the probability of being an element from the kernel set of the unary tract (Figure 4).

pn–The probability of an element taken from the kernel-set of the nary tract of the order n.

p1 = p (7)

p2 = 2 p1 = 2 p (8)

pn= np1 (9)

L designates the length of a sequence, i.e., the number of elements in a sequence.

It is assumed that k << L (k of an order of ten and L at least of an order of ten thousand) and therefore the events Xi(1)=k and Xj(1)=k with i+k < j are disjoint.

In (equation 5), if Pr{Xi(1)=k} is given at each i, it means that the expected number of exact unary tracts of length k at i is its probability. It follows that the sum over the all is gives the total number of expected exact unary tracts of length k in the sequence, as is supported by a MATLAB program [7].

The following equality takes place:

Exact (2,k,L,p)= Exact (1,k,L,2p) (10)

Exact binary tracts

Xi(2) is a random variable representing the length of the Exact Binary Tract starting at the i’th position of the sequence; the superscript (2) designates the “binarity” of the tract.

Pr{Xi(2)=k (11)

which represents the exact binary tract, starting at the i th position, is of the length k}.

As in the unary tracts, also in the binary tracts, k<<L is assumed. This assumption enables treating the extreme cases in which i=0 or i=L as regular ones without loss of much accuracy.

equeation (12)

Exact n-nary tracts

Xi(n) is a random variable representing the length of the Exact Binary Tract starting at the i’th position of the sequence.

Equation (12) Pr{Xi(n)=k | , representing the exact n-nary tract, starting at the i’th position, is of length k}.

equeation (13)

Pure -exact n-nary tracts

Zi(n) is a random variable representing the length of the Pure Exact n-nary Tract at the i’th position of the sequence.

Pr{Zi(n)=k (14)

representing the pure exact n-nary tract starting, at the i’th (i=0,…,L) position, is of the length k}.

equeation (15)

The developed formula (equation 15) is an approximated formula. The closed form of the definition 6.1 for small k values is under development [9]. The idea is to eliminate from the ‘average’ pure–exact tracts those exact n-nary tracts that are inside the original tract, but are of lower order and above some threshold length (l min). The expression is multiplied by n, because there are n possible combinations of subtracts.

The general form ϕ(α ,β ,γ ,δ) represents the following terms:

Φ-the expected number of tract functions of PE or E

α-the tract order

β-tract length

γ-sequence length

δ-probability of one element in the tract.

n-ary algorithm

The GUI (Figure 1) of the n-ary algorithm (Figure 2) is demonstrated. The GUI requires the nucleotides to be found in an n-ary tract. Its output is displayed in the GUI showing the found tracts, indicating the position of the nucleotides in their tract.


Figure 1: GUI of the n-ary algorithm (its Pseudo code, see Figure 2).


Figure 2: Pseudo-code (for its GUI, see Figure 1) of an algorithm for finding tracts and numeration of the corresponding nucleotides in the tracts.

Figure 2 shows the pseudo-code of an algorithm for finding tracts and numerating their corresponding nucleotides.

The complexity of an algorithm is O(n), where n is the length of the sequence in which the tracts are searched. The specific nucleotides to be checked in the n-ary tracts are treated as ASCII characters. The values of the elements of an array named codes (i) are 0, 1, corresponding to the element number; however, if its position in the array equals the value of one of the specific nucleotides, then the value of array element is 1; otherwise it is 0. Using the codes (i) array significantly simplifies the algorithm.


A palindrome is a symmetric tract. In order to find the distribution of palindromes in the p53 sequence, a more rigorous palindrome identification approach is required.

Definitions: There are two distinguishable types of palindromes, depending on the parity of the number of their elements: even or odd.



An Exact-palindrome is a symmetric tract that cannot be extended any more. The following tract is an Exact-even-palindrome:


And it has a length of 2n if the following condition holds:


The following tract is an Exact-odd-palindrome:


And has a length of 2n+1 if the following condition holds:


Exact-pure-palindrome is an exact-palindrome that does not include any sub-palindrome.

For example, ACGCA is an exact-pure-palindrome,

where as AGCGTGCGA is not an Exact-pure-palindrome, because it contains the GCG palindrome.

An algorithm for finding the palindromes

The palindrome algorithm given in Figure 3 above is of O(n) complexity.


Figure 3: An algorithm for constructing an exact-palindrome.


Figure 4: The graph that describes the convergence of the formula 5.4.4 (for p=0.5) for the values received numerically from pseudo-random sequences.

Random sequence: A random sequence is generated for comparison with the p53 sequence to search for extraordinary properties of the gene sequence. For a comparison, the term uniformity is defined and used.

Uniformity is an index to measure the randomness of the distribution of the nucleotides in the sequence. Nucleotide uniformity of the nucleotide at the i’th position in the tract receives the following values: 1 if two of its neighbors are different and differ from the nucleotide at the i’th be consistent with I’th position; 0.5 if two of its neighbors of the i’th nucleotide differ from the i’th nucleotide but they are equal themselves; 0.25 if one of the neighbors of the i’th nucleotide and only one is the same as the i’th nucleotide; 0 for the other cases.

Tract-uniformity is the average uniformity of all its internal nucleotides.


Analysis of the frequencies of the exact tracts

The question is as follows: do the n-nary exact tracts in the genomes have exceptional frequency? For the binary (n=2) tracts the answer [10,11] is positive, namely, the frequencies of binary tracts are overrepresented in the genomes.

In the following section the analysis of exact tracts will be described, performed on the human oncostatic gene, p53 (entry HSP53G, accession number X54156, 20303 bases) and on a randomly generated sequence of bases.

Unary exact tracts: As mentioned, relation 5.4.4 is an approximate one. Since that formula is the basis of deriving the other formulas, verification was made in order to check how close this formula describes the actual situation.

This verification is shown (Table 1 with specific columns: A, B, C, …) by checking the E(1,k,L,p) (p was taken 0.25, 0.5, 0.75– corresponding to the probability of the nucleotide appearing in the unary, binary, and the ternary tracts) convergence for growing L (the length of the sequence); (Column B) min_tract size is the limit for minimal tracts allowed.

Lengths Data p=0.25 p=0.5 p=0.75
Expected Lengths Result Expected Lengths Result Expected Lengths Result
sequence length min-tract length Random Sequence Formula Relative Difference (%) Random Sequence Formula Relative Difference (%) Random Sequence Formula Relative Difference (%)
10 1 1 1.33 24.8120301 1 1.3 23.0769231 3 3.4033 11.85026298
20 1 1.25 1.33 6.01503759 1.4 2 30 5.333 3.9964 33.44510059
50 1 1.33 1.33 0 2.09 2 4.5 3.0909 4 22.7275
20 5 0 5.333 100 0 5.999 100 10 7.838 27.58356724
50 5 0 5.333 100 0 6 100 9 7.9999 12.50140627
100 5 0 5.3333 100 5.5 6 8.33333333 8.1667 8 2.08375
1000 5 7 5.3333 31.2508203 6.31 6 5.16666667 9.1765 8 14.70625
10000 5 5.28 5.3333 0.99938125 6.13 6 2.16666667 8.0543 8 0.67875
20000 5 5.3 5.3333 0.6243789 6.015 6 0.25 8.0666 8 0.8325
40000 5 5.249 5.3333 1.58063488 6.0235 6 0.39166667 7.8913 8 1.35875
100000 5 5.117 5.3333 4.05565035 5.9705 6 0.49166667 8.0096 8 0.12

Table 1: Comparison of the expected tract lengths under various conditions.

(Columns C, F, I) values received their sequence generated randomly; (Columns D, G, J) values of the expected number of exact unary tracts were obtained by formula 5.4.4.

It was deduced from Table 1 that the error received using the formula compared to the values received using a random sequence was for greater L (20, 000 <L values, i.e. the sequence-length) of about 0.03%.

Another approach for treating this issue (statistics of unary exact tracts) is demonstrated in Gera and Ophir [9].

Binary Exact tracts: The exceptions for the frequencies of binary exact tracts were introduced before [10,11]. In the current paper the results of over representing the binary exact tracts in the p53 gene are validated and compared to the derived general formula dealing with n-nary tracts of any order.

Comparing the two tables, Table 2 and 3, we can see that the binary exact tracts found in the p53 gene do not always decrease as do their counterparts (the distribution of similar tracts in the randomized sequence). The results received from the randomized sequence show that the exact tracts are shorter in general than in the p53 gene sequence; however, the frequencies of the nucleotides in the gene are almost the same as the theoretical one (Table 2).

Nucleotides Frequencies in  p53
Nucleotide Frequency Theoretical Probability Difference
A 0.2584 0.25 0.0084
C 0.2448 0.25 -0.0052
G 0.2515 0.25 0.0015
T 0.2453 0.25 -0.0047
    σ 0.006375474

Table 2: Frequencies of the nucleotides in the given gene p53.

Binary Exact Tracts
Exact Tract Expected Length Results
Sequence Origin
Symbol Name p53 Gene Random Diff
AC Keto 6.6049 5.9385 0.6664
AG Pyramidin 6.8898 6.0997 0.7901
AT Strong 7.6508 6.009 1.6418
CG Weak 6.0468 5.8696 0.1772
CT Purines 6.5942 6.0031 0.5911
GT Imino 6.4223 6.0228 0.3995
      σ 0.50418

Table 3: Comparing the Exact Tract Frequencies in the p53 gene to a randomly generated sequence (σ is 0.50418).

Ternary exact tracts: Figures 5 and 6 describe graphs of the number of tracts for each length. They compare the difference between the exact ternary tracts’ distributions of a randomly generated sequence and a p53 gene sequence of the same length. The ternary exact tracts in the randomly generated sequence are uniformly generated, similarly to a graph of an exponential function, where as they appear in the p53 gene, which is not a monotonically descending function, and consequently, there is a peak with the value 12 (Figures 7 and 8).


Figure 5: Exponentially distributed lengths of the binary (AC–Keto) Exact tracts of a Random sequence.


Figure 6: Distribution of the binary (AC–Keto) Exact tracts of p53.


Figure 7: V-Ternary Exact Tracts’ Distribution in a Random Sequence.


Figure 8: V-Ternary Exact Tracts’ Distribution in a p53 Gene DNA Sequence.

Table 4 presents several indications of the frequencies of ternary tracts in the chosen gene in comparison to the frequencies of ternary tracts in a randomly generated sequence.

Ternary Exact Tracts
  Min. Tract Length - 5 Min. Tract Length - 10 Min. Tract Length - 15
 Tract Designation Expected Length (theoretical 8) Results Expected Length (theoretical 13) Results Expected Length (theoretical 18) Results
Sequence Origin Sequence Origin Sequence Origin
Symbol Name p53 Gene Random Diff p53 Gene Random Diff p53 Gene Random Diff
ACG V 8.5262 7.8877 0.6385 14.318 13.244 1.074 19.6635 18.5441 1.1194
ACT H 9.0367 7.9486 1.0881 16.327 13.0667 3.2603 22.744 18.724 4.02
AGT C 8.9774 8.074 0.9034 15.9188 13.0207 2.8981 22.7025 17.6264 5.0761
CGT B 8.2962 8.0421 0.2541 14.1567 12.9412 1.2155 19.444 18.333 1.111
      σ 0.36186   σ 1.12808   σ 2.02831

Table 4: Comparison of a ternary exact tract frequency in a gene and in a random sequence.

The ternary tracts are over represented and especially long tracts appear more frequently in the gene than in the random sequence. In the random sequence, long ternaries (more than 20 nucleotides) are not seen in the random sequence at all.

Each ternary tract is composed of binary subtracts. Each binary tract is (by definition) a special case of a ternary tract. The decomposition of the ternary tract into its components, the binary tracts, is shown (Table 5), where the ternary tracts used for the demonstration are of the V type, longer than 20 nucleotides, which appear in the p53 gene. The longest one, whose length is 40 nucleotides, starts at nucleotide 11806 in the gene. The longest sub-tract in each tract is denoted by a pale (red) color and an italic font.

The question being investigated here is whether the properties of ternary tracts are due to their binary sub-tracts or are independent of the binary sub-tracts included in them.

In order to answer this question, the following examination was performed. The pure exact tract was defined and a formula (5.7.2) was derived to approximate the frequencies of the pure exact tracts. In addition, an algorithm was developed to filter only those ternary exact tracts that are pure, i.e., they do not include binary sub-tracts longer than some specified length.

A question that arises here is: are the ternary tracts that appear in the gene dominant? As mentioned before, it is obvious that each ternary tract is composed of binary subtracts, so maybe from a biological point of view, the binary subtracts included in the ternary tracts are responsible for the biological phenomena in the unwinding process [12] and the ternary tracts that compose the binary one have no special purpose.

The length of the binary subtracts included in the ternary tracts was investigated (Table 5). The chosen represented tract is the one whose number in the ordered list of V tracts and whose length is greater than 4 is 1196. The longest subtract in this tract is the M-sub-tract (Table 6) having a length of 12 nucleotides, which is 32.5% of the whole tract.

No.   Begin   End  Length  Type  Tract
8 7031 7051 21 intron CAGGAGGCAGAGGCAGGAGAA
9 13913 13933 21 intron CAAAAAAAAAAAAAAAAGGCC
10 2003 2024 22 intron CCCGGAGAAAAAAAAAAAAGAA
11 12636 12657 22 intron CAAAGAGGCCAAGGCAGGCAGA
12 14591 14612 22 intron AAGCAAGCAGGACAAGAAGCGG
13 14707 14728 22 exon CCCCAGCCAAAGAAGAAACCAC
14 18529 18550 22 Intron CAGGGAAAAGGGGCACAGACCC
18 12882 12904 23 Intron CAAAAAAAAAAAAAAAGAAAAGC
19 16914 16936 23 Intron GCAGGGAGCCAAGACGGCGCCAC

Table 5: The ternary tracts of the V type, their location, and their decomposition.

No.   Begin   End  Length  Type  Code Binary subtract
1 1 7 7 AG R GGAAGGG
2 5 8 4 CG  S GGGC
3 5 7 3 G H GGG
4 9 11 3 AG  R AGG
5 10 13 4 CG S GGCC
6 12 21 10 AC M CCACCACCCC
7 18 22 5 CG S CCCCG
8 18 21 4 C D CCCC
10 24 27 4 C D CCCC
11 30 33 4 C D CCCC
12 35 40 6 CG S GCCCCC
13 36 40 5 C D CCCCC

Table 6: The decomposition of the longest V-ternary tract into its exact binary subtracts. The tract is shown at the top of the table.

Analysis of the frequencies of palindromes

The expected palindrome length of the p53 gene is compared with the random sequence of nucleotides (Table 7).

p53 palindromes’ quantities
Type: Missing A
Length Number Number
p53 random
1 3576 3666
2 1329 1258
3 1602 1573
4 410 426
5 384 371
6 74 103
7 77 75
8 12 13
9 15 19
10 6 4
11 3 5
12 0 1
13 0 1
14 0 0
Expected length 2.124332 2.128809

Table 7: Comparison of the number of palindrome tracks (type A) in the p53 gene and in a random sequence of the same length.

The comparison demonstrates that the difference between the quantities of palindromes of the two sequences of the same length, the p53 gene, and the random sequence are of the same order of magnitude. The classification of the palindromes to types is performed according to the types of the nucleotides composing the palindromes. The type case in which A (adenine) is missing means that the palindrome consists only of the nucleotides {C, G, T}.



The MATFREQ [7] (Figure 9) program for tract frequency analysis was developed especially for parsing the genes, searching for the tracts, investigating them, generating pseudo-random sequences, and comparing the results with the given gene sequence and with the developed formula. The results shown here are produced by MATFREQ.


Figure 9: The GUI of MATFREQ.

Palindrome program

The GUI of the programs is presented in Figures 10a and 10b.


Figure 10: A program counting the number of palindromes of different types and of various lengths. (a) Input, (b) Output.


The binary subtracts dominate the ternary tracts (Table 5), i.e. their length is more than 50% of the ternary tracts in which they reside (ternary tracts longer than 20 bases). The average uniformity of ternary tracts longer than 20 for the p53 gene is 0.24334, whereas the average uniformity of such tracts in a pseudo-randomly generated tract is 0.3924, namely, the uniformity in the p53 gene is 62.5% more than in the random sequence. This confirms the assumption that the ternary tracts’ proprieties are received from the binary subtracts included in them. This means that general ternary tracts do not play any significant role in the gene. However, special ternary tracts, including a binary subtract sufficiently long, play a significant role.

Analysis and comparison of the palindrome track frequencies of various types in the p53 gene and in a random sequence resulted in not finding any significant differences.


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