A Black Hole-Aided Deep-Helix Channel Model for DNA

In this article, we present a black-hole -aided deep -helix (bh-dh) channel model to enhance information bound and mitigate a multiple-helix directional issue in Deoxyribonucleic acid (DNA) communications. The recent observations 1 , 2 , 3 , 4 of DNA do not match with Shannon 5 , 6 , 7 bound due to their multiple-helix 8 , 9 directional issue. Hence, we propose a bh-dh channel model in this paper. The proposed bh-dh channel model follows a similar fashion of DNA and enriches the earlier DNA observations 1 , 2 , 3 , 4 as well as achieving a composite like information bound. To do successfully the proposed bh-dh channel model, we ﬁrst deﬁne a black-hole -aided Bernoulli -process and then consider a symmetric bh-dh channel model. After that, the geometric and graphical insight shows the resemblance of the proposed bh-dh channel model in DNA and Galaxy layout. In our exploration, the proposed bh-dh symmetric channel geometrically sketches a deep-pair-ellipse when a deep-pair information bit or digit is distributed in the proposed channel. Furthermore, the

However, DNA is a fundamental appearance of life, which assists to trace out the hereditary information within chromosomes.
It is mainly an information stream that relies on the nitrogenous bases 1,2,17,18 : adenine (A), cytosine (C), guanine (G), and thymine (T ). The quantity of each nucleobase is the key factor to trace out the hereditary information within chromosomes.
The main idea of DNA was first introduced in 1869 by Friedrich Miescher. Almost 7 to 8 decades later, many scientists were investigated the physical structure of DNA molecules. For example, in 1950, Erwin Chargaff's team investigated the composition of human DNA and proposed Chargaff's rules of DNA base pairing 17 . Zamenhof et al showed that different living organisms contain different quantities of each DNA nucleobase 18 . In 1953, Linus Pauling and Robert Corey studied an unsatisfactory structure of nucleic acid in which their model consisted of three intertwined chains with the phosphates near the fiber axis and the bases on the outside 2,3 . Around the same time, the X-ray fiber diffraction pattern was obtained by Rosalind Franklin 19 , and finally, on April 25, 1953, James Watson and Francis Crick suggested a satisfactory right-handed double 1helix structure of DNA. After Watson and Crick model, several researchers have significantly studied the physical structure of DNA, for example, a twisted 9 -circular structure and a four 4 -stranded structure. It is observed that, most of the relevant above literature's investigated the DNA model utilizing the chemical and biological method. To the best of our knowledge, the concept of the proposed black-hole-aided deep-helix (bh-dh) channel model for DNA communications has not yet been studied in the literature.
In this paper, we propose a bh-dh channel model to expand information bound and help conquer a multiple-helix directional issue for DNA communications. The geometrical form of the proposed bh-dh channel is a deep-pair ellipse, which graphically illustrates as a beautiful circulant ring. It is observed that the proposed bh-dh channel model follows a similar fashion of DNA and enriches the Watson-Crick 1 DNA model as well as achieving a composite like information bound. In general, a stochastic process is a Bernoulli process. Hence, we first define a black hole-aided Bernoulli (bh-B) process. Based on the bh-B process, we then consider a symmetric bh-dh channel model and measure the entropy and capacity of the channel. Through computer simulations, we finally verify the effectiveness of the proposed bh-dh symmetric channel with the conventional binary 5,6,7 symmetric channel, in terms of Shannon entropy and capacity bound.
Proposed bh-dh channel model: In this Section, we will design the proposed bh-dh channel considering a black hole-aided technique. To do successfully design the proposed bh-dh channel, we call Proposition 1 as below: Proposition 1: A black or white hole is a constant blind source, which produces a central-hole to help expand a circular symmetric channel space.
Proof of Proposition 1: To proof Proposition 1, we first define a black-hole-aided Bernoulli (bh-B) process and then consider a bh-dh symmetric channel model to analyze the geometric and graphical illustration as bellow: Definition of a bh-B process: Consider η be a constant blind or black-hole source and x ∈ X be a binary or a discrete random source, which construct a composite input source x d . The composited input source x d satisfy η ≤ x d ≤ 1 + η with 0 ≤ x ≤ 1. Now, using 20,21 , we define a bh-B multiplicative form as follows.
where, p d denotes the composited probability mass function satisfy η ≤ p d ≤ 1 + η with 0 ≤ p ≤ 1 and p denotes the probability mass function.
A symmetric bh-dh channel model: If we set x ∈ {0, 1} in (1), then using 6,7 and (1), we can model a symmetric bh-dh channel with transition matrix P d , which is given by It is noted that the proposed channel P d leads two different circulant matrices when two different bits or digits are applied in (2). We also noted that if p d satisfies 0 ≤ p d ≤ 1 in (2) I, we can summarize the following points as: (i) The proposed P d matrix pursues a right-handed DNA channel when η satisfies 0 < η ≤ 1 with T G > AC.
(ii) The proposed P d matrix becomes a left-handed DNA channel when η satisfies −1 ≤ η < 0 with T G < AC and (iii) Otherwise P d matrix be a P s when η = 0 with T G = AC and P s denotes the central hole-free 5,6,7 symmetric circulant channel.
(b) Graphical illustration of the proposed bh-dh symmetric channel: According to the geometrical construction of the bh-dh symmetric channel, we analyze the graphical illustration in this subsection. The graphical illustration of the proposed bh-dh symmetric channel model is depicted in Fig. 1. In Fig. 1, we observe that the area of both inner and outer ellipses depend Example 3 at p ∈ {0, 1} and η wl = 0 when T G = AC where constant η generates a center-hole in Fig. 2. In Fig. 2, we also observe that the proposed bh-dh symmetric channel model conducts a deep-pair ellipse when a deep-pair information bit or digit is applied in (2). For example, when we set η = 0 and a 1 × 198 binary-chain in (2), the proposed bh-dh symmetric channel model turns back into a binary symmetric channel and draws a central hole-free circle in Fig. 2a. In contrast, when we set η = 0.35 and a 1 × 198 binary-chain in (2), the proposed bh-dh symmetric channel model illustrates a beautiful-circulant-ring with central-hole as shown in Fig. 2b.
The most important graphical view describes in Fig. 3 and Fig. 4. Fig. 3 shows the graphical inspection of the proposed bh-dh channel model, which follows the physical structure of DNA. An X-ray pattern of the double 1 -helix DNA model is investigated in Fig. 3a. A polar diagram depicts the proposed bh-dh channel model using by a 1 × 81 binary-chain in Fig. 3b.
According to the definition of a bh-B process, geometrical and graphical illustration, we can see that (2) is a bh-dh symmetric channel, which has expanded a circular symmetric space by producing a central hole. Thus, Proposition 1 has been proven.
Information entropy and capacity bound of the proposed bh-dh channel model: Particularly, the information capacity is the maximum amount of information that can pass through a channel without error. In contrast, information entropy is a function of a transitional error probability that is usually measured uncertainty in Shannons 5 (bits) or natural units (nats) or decimal digits (dits). Since the Shannon bound does not match with the DNA information bound, we investigate the entropy and capacity bound of the proposed bh-dh channel model in this Section.
Consider random input and output variables X and Y for a communication channel where x ∈ X and y ∈ Y. Hence, the well-known capacity formula 5,6,7 is given by where p(x) denotes the input distribution function, C s satisfy 0 ≤ C s ≤ 1 with 0 ≤ x ≤ 1 and 0 ≤ p ≤ 1, and I(X; Y ) Circle when p d = 0 or 1.
Outer ellipse when p d > 1 or p d < 0. denotes the mutual information 6 , that is given by where H(X) and H(Y ) are the marginal entropies, H(X|Y ) and H(Y |X) are the conditional entropies, the p is the error probability satisfy x∈{0,1} p(x) = 1, and the Shannon entropy H s (p) is given by 6,7 the following equation where H s (p) satisfy 0 ≤ H s (p) ≤ 1. Similarly, based on (1), (3) and (4), we can write the following capacity formula for the proposed bh-dh channel model as for right-handed DNA case when 0 < η ≤ 1 with T G > AC, where H d (Y d ) = 1 when p d = 1/2 is applied in (6), H d (p d ) denotes the entropy of the proposed bd-dh channel model.
Similarly in (5), the H d (p d ) is given by     TABLE II and TABLE III. In TABLE II, when we set p = 0 for both η = 1 and η = −1 in (6) and (7), the measured entropy H d (p d ) is given by 0 bit and -2 bit and the measured capacity C d is given by 1 bit and 3 bits, respectively. In contrast,   when we set p = 1 for both η = 1 and η = −1, the measured entropy H d (p d ) is given by -2 bit and 0 bit and the measured capacity C d is 3 bits and 1 bit, respectively.