Beyond a Quartic Polynomial Modeling of the DNA Double-Helix Genetic Code

By combination of finite number theory and quantum information, the complete quantum information in the DNA genetic code has been made likely by Planat et al. (2020). In the present contribution a varied quartic polynomial contrasting the polynomial used by Planat et al. is proposed that considered apart from the golden mean also the fifth power of this dominant number of nature to adapt the code information. The suggested polynomial is denoted as , where − = is the golden mean. Its roots are changed to more golden mean based ones in comparison to the Planat polynomial. The new coefficients 2 4 ϕ− instead of 4 would implement the fifth power of the golden mean indirectly applying . As an outlook, it should be emphesized that the connection between genetic code and resonance code of the DNA may lead us to a full understanding of how nature stores and processes compacted information and what indeed is consciousness linking everything with each other suggestedly mediated by all-pervasive dark con-stituents of matter respectively energy. The number-the oretical approach to DNA coding leads to the question about the helical structure of the

apart from the golden mean also the fifth power of this dominant number of nature to adapt the code information. The suggested polynomial is denoted as As an outlook, it should be emphesized that the connection between genetic code and resonance code of

Introduction
In a more literary parlance, the genetic code of DNA is the very language of life, How to cite this paper: Otto lynomial containing the golden mean φ in its structure (see Appendix). Intuitively, the present author suspects that in addition to the golden mean, also the fifth power of this number should play an important role in mapping the genetic code owed to the fact that this number is observed extremely frequently in nature, covering examples from microscopic to cosmic scale [2] [3] [4] [5]. So, one may argue that this feature should also be found in some way in the genetic code. A little varied quartic polynomial can deliver such an implementation of φ 5 by replacing the starting polynomial coefficients adequately. This was described in Chapter 2. In times of a viral pandemic, it is essential to know the genetic code of DNA respectively surface proteins in order to effectively block the replication capacity of any pathogenic virus via this route. A further practical aspect of research in this direction is to pave the way for quantum computation under ambient conditions. Recently, a very creative review for understanding of what may be important when dealing with such suggestions has been given by Savelyev et al. connecting the DNA resonance code to the neural code [6]. When translating such connections in the parlance of phase transition, again one comes back to the golden ratio respectively its fifth power [4], both considered as fundamental numbers of nature.
The presented contribution is a short excerpt from a more comprehensive review coming soon about new aspects of helical structures in general [7]. Therein a didactic explanation for the characteristic diffraction pattern of a helix (doublehelix) will be presented in detail.
Finally, the algebraic-geometrical approach to the code of the DNA and secondary protein structures leads to the question what structure a fundamental particle like the electron has.
In the Appendix some supplements were made comparing the just mentioned quartic polynomials with double well potential curves respectively the Hardy probability function [2].

Varying the Quartic Polynomial f(x)
For a better understanding of the solution of polynomial equations, the recent work of Tehrani [8] is highly recommended.
The starting Planat quartic polynomial (PQP) given in [1] is Its curve is illustrated in Figure 1 and its four roots (zeros) are presented in Table 1 according to [1]. What happens, if the polynomial would be slightly varied by replacing the coefficients 4 by The coefficients now used possibly represent limits and can be recast by In this way one would implement the fifth power of the golden mean indirectly. The new polynomial is denoted as ( ) g x and is given by Its roots are changed to more golden mean based ones in comparison to the starting polynomial (see Table 1). Both quartic polynomials can be compared in

Helical Structures
Helical structures dominate or life in the form of the α-helix of proteins [9] [10] [11] or double-helices composed of double-stranded molecule chains of nucleic acids such as DNA [10]. Helical curling of cellulosic micro-fibrils leads to spring-like elastic behavior [12]. Interesting is that helical left-handedness and its opposite can be realized, when a single cellulosic fiber is supported at both ends [13]. Designers have many learnt from the multilayer tube-like structure of wood cell walls leading to superior properties of this natural material [14]. Let's stick with technical aspects such as making torque from wind. The wake behind multi-bladed rotors can be described by interlaced helical vortices [15]. But also liquid water forms helices of pentagonal or hexagonal symmetry [16]. When subtracting from the regular tetrahedron angle of 109.47˚ the angle ( ) is quite well adapted giving 104.30˚. This connection to the golden mean hierarchy is intriguing. It is reasonable to assume that water may be involved in two ways in the origin of life, as a liquid medium and as helical scaffold mediating the first assembling of a double-helix of nucleic acids. In contrast, the participation of structurally disordered clay minerals in this process is very unlikely. The fifths power of the golden mean φ 5 in turn is frequently connected to phase transformation from microscopic to cosmic scale [4] [17]. This number may be characterized as a universal one [18]. On the border between hexagonal and pentagonal respectively Fibonacci number governed symmetry our existence is located. Numerical it may be emphasized by the relation 5  Nobel laureate Linus Pauling who solved the structure of the α-helix of proteins 26 years before [9] [10] [11].
Since that time I was fascinated by the helical structure of matter, but found no time to think about it anymore considering crystal-chemical, biochemical as well as mathematical aspects.

A Number-Theoretical Golden Helix Gimmick
Now we turn to the α-helix peculiarities of proteins where 18 subunits perform 5 turns to reach an identical position when projected down the fiber axis [9] [10]. This means one needs 3.6 subunits per turn. This number is nearby another golden mean derived number (see Equation (2) In case of the α-helix, when tentatively using 3.61803… subunits per turn instead of 3.6, one would end with 5 18 ϕ + subunits after 5 turns ( ) ( ) and again the fifth power of the golden mean would be involved.
What marginal change would such approach of helix breathing produce in the Fourier transform possibly overseen as yet by the interpretation of the diffraction patterns showing quite broad smeared out diffraction spots? An elaborated study of structural variations of B-DNA by interpreting features of its diffraction pattern was recently given in [20].
The value of 2 3 1 ϕ ϕ − + = + is also related to an icosahedron as its circumradius when this polyhedron with an edge length of a = 2 is centered at the Journal of Applied Mathematics and Physics origin [5] [21]. It may be considered as a very important distance frequently found in the following reflections.
Turning to five-fold symmetrical nucleic acid structures discussed in [1], the regular pentagon with side length a has an area of A 5 calculated by [21] Another example should be considered with respect to the seven-fold symmetry of complexes like the Lsm1-7 one recently analyzed by Planat et al. [ The relative area difference between heptagon and pentagon can be approximated by Another example from number theory where the golden mean in form of the square root term 3 ϕ + is involved but not yet expected is the series of heptagonal numbers generated when heptagons with ascending size are combined [24]. Such figurative heptagonal numbers are generated by the formula [24] indicating a parity sequence of odd, odd, even, even … and successive Lucas numbers 7 and 18 [25] respectively Fibonacci numbers 34 and 55 [26].
The sum of reciprocals of this number series shows a result in full glory, if we recast for the first time the already known expression [24] using the golden mean as Alternatively, it applies because and further ( ) 2 1 − represents the silver mean here approximated by an expression for the golden mean.
It indicates that even in the case of seven-fold symmetry a sophisticated numerical relationship to φ, the omnipresent golden code number of life, is given and should be considered in future work on the genetic code and resonance behavior of life. The relation between 3 ϕ + respectively 5 ϕ with the circle number π is also remarkable (see also Appendix)

Simple Didactic Approach for the Fourier Transform of a Helix
The Fourier transform of a helical structure resulting in the integral representation of Bessel functions of different order was first introduced in 1952 by Cochran [9] and can be studied in detail in the monograph "Direct Analysis of Diffraction by Matter" by Hosemann and Bagchi [27]. Another recent contribution by Latychevskaia and Fink [28] is recommended as well as that given by Strutz in German [29].
Our simplified but didactic approach is as follows. If one looks through the projection of a helical structure perpendicular to the filament direction, one will see the mirror image on the back at the helix diameter distance to the front image, somewhat offset in filament direction depending on the helix winding parameters. One can perform the Fourier summation to obtain diffraction intensities exemplified by the projected hexagonal nets of a special Fibonacci mi-  Figure 3 respectively Figure 4 as a quantum computer bio-template.
Information may be preserved in tubulin molecules by structural adaption and sequence fluctuations disrupting the helical coherence.
When removing from the Fibonacci net some atomic stripes one can simulate a double helix. Detailed intensity calculation of such a construct will be given in a forthcoming contribution [7] showing the characteristic X shaped cross of diffraction intensity distribution of a double helix similar to the electron diffraction pattern of B-DNA fibers [36] respectively to the historical X-ray pattern (photo 51) of B-DNA taken by the felt Nobel laureate of hearts Rosalind Franklin [37].
When one works with the projected curved net instead of a flat net, the reflections smear out as result of the introduced coordinate statistics. Figure 3 may

Beyond the Icosahedron Equation
If the genetic information is ultimately coiled and compacted into an icosahedral shape as found in viruses, then another function may come into play, represented by the icosahedron equation mapping for instance the positions of the face centers of an icosahedron with unit in-radius projected onto a complex plane where z is the coordinates [ Based on the preliminary work by Gordon [42], Klein showed the connection between the regular icosahedron, one of the five Platonic bodies, and the solution of the quintic polynomial. Instead of following Klein The four root of this polynomial have been calculated giving The reader is frequently confronted with the Fibonacci number 13, which ob-Journal of Applied Mathematics and Physics viously plays an important role besides ϕ and 5 ϕ when assessing bio-coding and related storage and processing of information.

Self-Duality as a Suggested Property for Image Storage, Memory and Consciousness
Self-duality is a very important concept of mathematics as well as physics and should be applied to biophysics respectively biochemistry in order to understand the very stable information storage within the human brain. Self-dual objects include polyhedra, graphs, configurations and codes. The skeleton of a self-dual polyhedron is a self-dual graph. For instance, self-dual solitons as self-reinforcing and shape-maintaining wave packets are expected to be very stable. Among the self-dual polyhedra are the regular tetrahedron respectively pyramids in general.
Also the Kummer quartic surface with 16 nodes and 16 singular tangent conics, believed by Planat et al. [22] [23] to explain in part the genetic code of secondary protein structures besides DNA, is self-dual [43]. Journal of Applied Mathematics and Physics The information storage respectively processing certainly doesn't work via simple bits or qubits but via complex pictures generated by superposition of stable (soliton-like) wave packets, burned into a multiplex microtubule array or a comparable matrix. This happens against the background of the omnipresent dark energy with all the information collected up to then. Each speedy moving body or particle is surrounded by dark matter respectively energy [17], and when seeking for an assumed resonance field that links DNA or tubulin resonances with the work of the brain according to [6], the first candidate to fulfill all conditions is simply dark energy.

What Is the Code for an Assumed Electron Helix?
Importantly, also elementary particles such as the electron are thought to exhibit a helical structure [44] [45] indicating that nature used this principle of curling again and again [46]. Remarkably, the arithmetic mean between the proposed maximal superluminal electron velocity and the minimal one yields c v ϕ = . The application of IRT transformations [17] to the problem would certainly present the electron helix approach correctly and in full glory. We will work on this. The frequency of electron's Zitterbewegung is found to equal that of the composing photons [44]. Besides the results given in [47] [48] [49] and contrasting the QED coding that uses an unbelievable cascade of Feynman diagrams [50] [51] [52] one should try to encode and understand a numerical golden mean relation for the electron as a quite stable helically wound and less spherically than icosahedron-shaped compacted entity where The experimental value of e g ∆ was determined slightly larger [51] [52], but a new relativistic correction was proposed shifting this value towards the considered smaller one [48].
If we proceed with this helix quantum-vortex approach we may at least understand electron's pairing in superconductors, too [53]. Historically, already in 1978 L. S. Levitt had raised the question whether the photon is a double helix [54]. A photon with high energy decomposes by interaction with matter to an electron and a positron, both helically twisted. Recently, Williamson suggested a Journal of Applied Mathematics and Physics possible structure of the spherical electron (positron) by an elaborated new linear theory of light and matter [55], but used once again the questionable relativistic where the Doppler factor is given as However, if one could ultimately prove the double helix structure of the electron, it would seem natural to think further of its possible fine structure making one electron not equal to another one when interacting with the environment, despite their same mass and bulk properties. It would revolutionize our ideas when it comes to storing and processing exorbitantly large amounts of information.

Helical Helix Construct
Single respectively double-strand helices may not only be found in nature as rotational blurred linearly aligned fibers. The fiber axis of a helix or double-helix may itself be proposed in turn helically wound to strengthen its spring-like elastic behavior suggested for application in building constructions. Such construct would not represent the densest packing, but an extremely flexible one. I am not sure whether nature used this construct, for instance as DNA partial packing in viruses or in the structure of the electron, but designers could certainly gain profit in its application. Also energy harvesting may be possible when such construct was set in vibration by wind or water.

Conclusion
The and further ( ) The product of these fundamental numbers is found as volume quotient of the in-sphere of the Great Pyramid to the pyramid volume itself [5] [21].

Solution of a Special Quadric Polynomial Equation
The approach ( ) , which can also be tackled by an infinite continued fraction representation [58].

About Double Well Potentials
Because the polynomial f(x) (Equation (1)) looks like an asymmetric double well potential curve, let us linger a little bit on such linear superposition of "classical" states as a concept important in quantum information theory [60].
On can suggest from the expression for the baryonic matter energy density [17] a quartic polynomial of a symmetric double-well form ( Figure A1 which is depicted in Figure A3 in comparison with Hardy's function. Figure A3. Polynomial approximation (red) of Hardy's function (blue) according to Equation (54).