Keywords: DNA/RNA transformation, Codon, Symmetric group transformation, Broken Symmetry, Non-isometric transformation.

Transcription

1 Symmetric Group Structures of Genetic Transformation Reza R. Ahangar 700 University BLVD., MS 7 Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX , e- mail: reza.ahangar@tamuk.edu ABSTRAT: Proteins are fascinating bio-molecular machines made by amino acids, generated by codon, and are connected in a linear chain. The energy created by proteins can perform operations from collecting sunlight, transporting materials, providing mechanical strength for fighting virus or bacteria. We will study symmetric transformation of codons as a function in a symmetric group transformation of a set RNA={U,,A,G} or DNA={T,,G,A}. Our goal is to translate the genetic codes into a mathematical language of composition of functions in a symmetric group of S ={,,,,}. A permutation symmetric group can be used to demonstrate natural phenomena in genetic algorithm, crossover, mutation, and natural selection. Better understanding of symmetric group transformation and an appreciation for the essential role it has played in codon formation will improve our understanding of nature s coding processes. Incorporation of algebraic structure of symmetry groups will facilitate that improvement. Keywords: DNA/RNA transformation, odon, Symmetric group transformation, Broken Symmetry, Non-isometric transformation. - Introduction and History: In 9, Schrodinger published a small book under the intriguing title What Is Life? He was describing DNA as a solid one-dimensional crystal. Now we know that DNA is not a steady state. It is a dynamic double helix library of information stored in several billion pieces linked together in 6 strings of a chromosome. For Schrodinger, writing the time evolution of DNA a building block and particle of life needs more elaboration to analyze. It may take another century to be able to analyze and understand the social structure of any living species and to write its evolution equation of motion. The discovery of bio-molecular structure of DNA by Watson and rick in 95 [() and ()] has helped to reveal the complex genetic information. This new model of double helix of DNA could explain how the information is stored in DNA in the form of sequence of nucleotides, but it could not help us to understand how it is coded, how it could be used in biological functions, or how to decipher the genetic information. Another new discovery by rick [()] et. al. in 96 was magnificent and revealed that genetic code is a triplet sequence of nucleic bases. The complete classification of the correspondence between codons and amino acids was assembled in a standard table in 966. In 957 the brilliant physicist, George Gamow [7] produced a model that he called compact triangle code for codons. Gamow proposed the transformation of an equilateral triangle in the plane and in the space to represents symmetric transformations for codons. Even the genetic table in biochemistry was compared to the periodic table of elements in hemistry at the beginning. But the table as such does not explain why there are 6 codons and only 0 amino acids. It has been discovered that this degeneracy and redundancy is associated with and is a consequence of symmetry. A mathematical structure of DNA and amino acids was presented using Geometric visualization. Mark White 007 [see ref. [8] ] testified We have merely created some linguistic and visualization tools based on fundamental codon symmetry in conjunction with the unique nature of DNA s natural two-bit set when placed within the elements of solid geometry. This exercise has generated a nifty data container with virtually no data in it, apart from the specific set of DNA nucleotides. A latest article on developing the Galois Field of five DNA base alphabet used a set {D,G,A,U,} with unspecified pairing D [Sanchez Robersy, , see [], [],[],[],[5]. Two mathematicians claimed in their article on March 99 that a genetic code, which is able to specify 0 amino acids, is derived from a simpler version which coded for only six. They came to their conclusion after examining the symmetry inherent in the redundancy of the genetic code [9]. The abstract idea of symmetric is from algebraic structures that can explain the degeneracy and redundancy of the genetic code. Symmetry can initiate discovery of an organizing principle for the way in which the genetic information is stored and regulates the process in protein synthesis. A codon is traditionally defined as an ordered set of three nucleotides selected from {U/T,, G, A}. We will try to use the mathematical approach to define codons and use them to analyze DNA or RNA transformation sequences. To do this we need the mathematical tools of permutation, symmetric mapping, and group structures. We will reproduce ayley s binary operation table for permutation group S. The symmetric group of dihedral which is denoted by D and alternating group A will be demonstrated. Out of 6 codons, there will be eight compositions of operations which will not be the product of rotations and reflections. We will present the non-isometric transformation that may produce a twist in DNA transformations. Intuitively, we may answer a persisting question what is breaking the symmetric transformation? Our conjecture will be the important rule of the evolution based on variations which we will observe under permutation group structure.. Partially ordered DNA/RNA Bases On the Bio-molecular level, we can assume that the sets of RNA or DNA are partially ordered sets. This hypothesis can be postulated either on the atomic numbers in

2 quantum approach or by simply counting the number of hydrogen or nitrogen bonds. Nitrogen bases for Biomolecules U (uracil: N H O ), (cytosine: N H 5O), A (adenine: 5N 5H 5), and G (guanine: 5N 5H 5O) are the building blocks of set RNA= {U,, A, G}. Dupliji et al 000/005 (see []) also used the number of hydrogen bonds to study the common characteristics in 6 codons of DNA bases and arranged four nucleotides in descending order, G, U, and A. Yang 00 (see[0] and []), proposed sp N-numbers to demonstrate the RNA basis as ordered elements. We use the notation U/T, because genetic code is read from mrna, and so we will not differentiate their partial strength and order. We use this idea to introduce the RNA/DNA = {U/T,, G, A} as a partially ordered set. According to this arrangement, U / T,, G, A, we will accept the relation U / T G A and symbol "" to represent the partially ordered set. Fig..: Four DNA base with their Nitrogen and Oxygen bonds. The doublet-matrix: The matrix of the 6 possible basedoublets can be constructed by the following revised Kronecker product with concatenation (Negadi, 00, see [7]). In the first step we use partially ordered amino acids to define a vector V v where i,,,. such that i v =U/T, v =, v =G, v =A. In the second step, we assume the product of two components is defined as a binary operation, for example, (.) vi v j for i, j,,, viv. j We can use the above mentioned assumptions to define a new DNA vector product. Vector Product Operation: Given two RNA/DNA vectors V and W. Define the product (.) V v v t, W V W v v. w w w w [ v w ] i, j,,, The following is the result of the product when we apply this definition to the DNA elements; i j (.) V, V V V t U U G A G UU U A GU AU U G A UG G GG AG UA A GA AA This is consistent with the canonical matrix of doublets. From 6 possible codons one can extract 6 families each defined by the first two nucleotides. One can continue to develop the matrix representing the 6 (triplet) codons of the genetic code based on the doublet matrix of (.) to (.). There have been many attempts to explain ubiquitous existence of 6 codons and 0 (canonical) amino acids with degenerate codons. Instead of this approach, we will continue to use symmetric transformation to explain this phenomenon (see table.). In the physical interpretation of the set of n- tuplet S n, each point represents a DNA base and is a set of n points in the space with equal weights. The midpoint of each segment in DNA- space is said to be the centroid of that segment. The center of mass of a tetrahedron with equal weight on each edge will be at a point equal distance from each vertex and it will be changed when the symmetry is broken. The genetic code is an important key in the understanding of the process in the body when the DNA copy - RNA, is translated into the functional molecules, the proteins. In regions of the genome that code for protein production, each codon, such as GTA, specifies a particular amino acid - in this case histidine. Three of the 6 possible codons do not code for an amino acid but instead signify termination of the transcription process. The genetic code is an important key in the understanding of the process in the body when the DNA copy - RNA is translated into the functional molecules, the proteins. In regions of the genome that code for protein production, each codon, such as GTA, specifies a particular amino acid - in this case histidine. Three of the 6 possible codons do not code for an amino acid but instead signify termination of the transcription process. The key point to remember is that the genetic code is degenerate - different triplets may code for the same amino acid. For instance the six codons AAT, AA, GAA, GAG, GAT and GA all code for leucine. There is no great regularity to this redundancy. For example, TA is the only codon that codes for methionine. However, a definite degree of symmetry - albeit imperfect - is clearly visible in the genetic code. Often, the first two bases in a codon are enough to determine which amino acid it codes for. Let us use X for unknown code, for example GAX always codes for leucine and GX always represents arginine. In short, the code is symmetric under changes of the third base. If this symmetry were perfect, the 6 codons would break up into 6 'quadruplets' such as GA, GAG, GAA,

3 GAT with each coding mapping for a unique amino acid. However, there are more than 6 amino acids, so sometimes the third base matters. Indeed, sometimes the second base matters as well. Either way, the perfect 'quadruplet' symmetry is broken. In this study we will not consider bio-chemical reaction in breaking the symmetries. We only try to explain geometrically that without distortion of the tetrahedron of the four letter bases, the transformation will not be possible..- Symmetric Transformation Group: A set with a binary operation "" is said to be a group if and only if under this operation i) the set is closed, ii) the set is associative, iii) has identity, iv) every element is invertible. A mapping from a partially ordered set, RNA/DNA=<U/T,, G, A> into S ={,,,} is a group, which is called a permutation group. The properties of the permutation group have been studied extensively. The set of all symmetric operations with the composition of functions "" is a non- Abelian group where the "" represents the binary operation in the group of composition functions. To investigate the codons generated by < U/T,, G, A> we will assume the following cases. - bio-molecule transformations occur in the same plane with complete symmetry of geometric shape of n- gons (square in this case for dihedral group). - transformation of the molecules that are in space have complete geometrical symmetry of a regular tetrahedron. - we will check transformation in the permutation group S that do not keep symmetry. The variations of the permutation group S over a set RNA/DNA=<U/T,, G, A> in the space is sufficiently simple to help us understand and visualize all of the possible transformations. We would like to explain and differentiate those transformations that keep the DNA bases unchanged and recognize those that lead to a new and different base. Among these, some may change content, shape, and properties. Since Darwin s evolution is based on genetic variations and natural selections, we really need to investigate what causes the variations and how much these changes will affect the internal structures of genes. Mathematical tools will help lead us to a logical conclusion. There are some simple transformations like rotation, reflection, and translation of geometric points in the space which do not cause change in the object in the space but merely change its position in the space. Studying transformations that change one molecule into another is also important. However, study of the causes of the conversion of one molecule to another molecule is beyond the scope of this article and perhaps requires quantum mechanical or thermo dynamical approaches to investigate the energy required for these transformations. To achieve our goal we need to present the following definition of isometry..- Isometric Transformation: A transformation which preserves the distance between two points is called an isometry. Assume two points in Euclidean Space E. Let M and N be the new positions of two points M, N in the space E. The transformation is said to exhibit isometry if and only if, d(m, N)=d[M,N ]. In other words, isometry transformation preserves the distances. Analytically, it can be proved that rotations and reflections preserve the distances. There are many other transformations like similarity, inversion, or conformal mapping which are not isometric. We will study some examples of isometric transformation like dihedral D and alternating group A as a subgroup of the permutation group S. The geometric views of transformation will help us to distinguish all non- isometric mapping. Study of the non-isometric permutations will help us identify genetic errors in cell transcriptions. Operator Sequence Instead of DNA Sequence: The binary operation in permutation mapping will produce a sequence of products which will be recorded in a table called ayley s table (Arthur ayley, 85). To view and interpret geometric transformation we assume that i) every motion can be described by a geometrical transformation. ii) there are some independent intrinsic and basic transformations. iii) every motion can be described by the composition of the basic transformations. Some DNA transformations can be considered planar transformation. The vertices of a square can be labeled <T/U,, G, A> or S = {,,,}. For simplicity the center of all nucleotide bases stay in the same plane and all transformations like rotation, reflections, and translations will take place in the same plane..- Geometry of the Tetrahedron Mapping: A one to one mapping from a set to itself is called permutation. It is called symmetric when the set is a subset of the positive integers S n={,,,,n}. Initially, labeling the vertex of the tetrahedron by a set of RNA/DNA={A,T/U,,G} or S ={,,,} gives us the bound for the symmetries in the group, such that: U/T,, G, A.

4 When vertices of a tetrahedron change, their image in the second set will also change. Mathematically we define a one-to-one function from the domain of objects RNA/DNA elements to a set of four integers {,,,}. Let us demonstrate the mapping by U / T G A. This kind of mapping is called permutation. There are choices for the position of vertex. For the second position there are choices for vertex. Thus there are! = * **= symmetric permutations in S (see [],[6], and [0]). entire object, but may be expressed as a composition of other single transformation portions of the object. The following is an example which demonstrates one sample of non-isometric transformation U / T G A ( ) ( G) We wanted to show the geometric meaning of this change without considering other scientific restrictions. In this permutation two amino acid bases U/T and A will not be changed, but we will interchange and A. Geometrically we can reflect the segment A symmetrically without rotating or reflecting the object. Tetrahedron (b) demonstrates axes of revolution which are midpoint connectors of opposite edges: MN, PQ, and RS. There are four symmetric axes for rotations in regular tetrahedron (a) and three in (b). All symmetric axes pass through a fixed point center O. (a) (a) (b) (b) Fig..: Tetrahedron TGA in (b) ( or UGA) in (a) demonstrates symmetric axis of rotations passing through each vertex and the centroid of the opposite face. Non-Isometric Transformation in DNA/RNA Permutation: In the previous sections, two subgroups of transformation are discussed in both A and D. In both cases the new position of the entire object will be determined after a sequence of rotations or reflections. The following transformations are not a single rotation or reflection of the Fig..: an we transform tetrahedron (a) to (b) without geometrical distortion? This transformation is not possible without with a simple rotation and reflection of the object twisting part of the object..- Dihedral Group D : We denote D n (n>= ) dihedral group of symmetries for each regular n-sided polygon. In each case there are n- rotations (including the identity ) and n- flips so that the order of D n is n. This argument demonstrates that D and S have the same order of six. Thus D represents all rotations and reflections for equilateral triangles and D also is a group of all rotations and reflections in a square. Assume symmetry group of the square (Fig..) and that r represents a 90 degree rotation counterclockwise and s a reflection across a horizontal axis. We can observe that r =s =e where e=() is denoted for identity transformation.

5 All other transformations in this group can be interpreted by these two elements r and s. Thus, D is a group that is generated by a pair of transformations on a mapping such that, r=()=(uga) U / T G A Fig..: The Square UGA is a regular polygon with symmetric axes: PQ, NM, A, and GU. In the following matrix, the first row represents the RNA/DNA bases with their original order in the second row. The third row demonstrates the position after transformation. We call it identity transformation if it comes back to the same order. Identity Transformation U / T () G A e In a regular tetrahedron, U the center of triangle GA is the image of the point U. We will present another example to explain the rotation of 0 about UU (or equivalently TT in DNA) will produce the following transformation. U / T G A () ( GA) Symbolically, a simple transformation will be (GA). In this permutation model, U/T will map to itself, to G, G to A, and A will map to. All of the elements in D, A, and S can be described by graphical and vector approaches..- : Non-isometric Transformation: ayley s Binary Multiplication Table for Alternating Symmetric Group A :The order of the alternating group A, according to the formula A n =n!/ will be. For simplicity, we will call each element a letter that represents the associated transformation. In the ayley s table (), the set of binary operations can be observed A ={e, a, b, c, g, h, i, j, r, s, t, u}. It can be verified that {e, a, b, c} is a subgroup of the alternating group. In addition, all of the elements in A can be generated by the elements of this subgroup. That is: e=(), a=()(), b=()(), ab=()(), g=(), ag=(), bg=(), abg=(), g =(), bg =(), abg =(). Space of Symmetric Transformation in DNA/RNA: Let us call the midpoint connectors of all opposite edges in tetrahedron by MN, PQ, and RS. Due to the symmetry properties of a regular tetrahedron all of them are concurrent at a point O. This is the centroid of the tetrahedron ATG. Each face from the set {AT, AG, AGT, GT} represents a triplet codon with a centroid {A,G,,T }. Thus, in addition to MN, PQ, and RS, segments AA,, GG, and TT are symmetric axes of rotations or reflections of the tetrahedron in D space. We are planning that the geometric approach used in this article can explain the puzzle of degenerate codons. Further research will be required to explain the link of this approach of using the geometric interpretation for symmetric transformation in DNA with other advanced level genetic errors during transcription, apoptosis, and mutation. There will be significant differences between our approach and the traditional approach. Our approach is based on the geometry of permutation group structure which has been a trend of research during the past few decades. - We will look at the sequence of the operation functions rather than DNA sequence. - We can observe all operation functions listed in Table (). - There are isometric transformations in the Alternating Symmetric group A and 8 isometric transformations in dihedral group D with four elements in common. - The total number of permutations of four letter alphabets in RNA/DNA={U/T,,G,A} is equal to!= and the total number of isometric transformations is 6. - There will be a difference of (-6=8) eight nonisometric transformations. These non-isometric transformations will cause the shape of the codons deformation of DNA. - ayley s table (Table ()) demonstrates only functional operators as a result of operations in the symmetric group. - The puzzle that that are degenerate and redundant partially explained. Actually, 0 of them are explained very well to reduce 6 positions in ayley s table to 0 amino acids..- oncluding Discussion: - We have studied the dynamic functional operations instead of static positions of single DNA bases. - The understanding of searching a DNA sequence should be changed into the effect of dynamic functional operators to analyze a genetic code. - Nature may dictate these operators in their physical, chemical, or biological operations.

6 Further studies are needed to find the outcomes in generating or producing the natural process. Future Research: The importance of this work is not that it solves the question of evolution, life, birth, death, mutation, and structure of the genetic code, but that it opens up a new line of direction to the genetic transformation. However Further investigation needed to determine the nature of eight non-isometric symmetric group structures. We still need to demonstrate the cause of four more degeneracies in the 6 codons of ayley s table. References:. Akinori Sarai and Yoshinori Takeda, Lamda Repressor recognizes the approximately -fold symmetric half operator sequences asymmetrically, proc. Natl. Acad. Sci. USA, Vol. 86, pp , September 989, Biochemistry.. rick F.H.., Barnett L., Brenner S., Watston in R.J., Nature 9, 7 (96).. Dean, R. A., lassical Abstract Algebra, Harper and Row, Publishers, New York, (990).. Duplij Diana and Duplij Steven, DNA sequence representation by trainders and determinative degree of nucleotides, Journal of Zhejiang University Science (JZUS), 005 6B(8): Beland, P., T. F. Allen, The Origin and Evolution of the Genetic ode, Journal of Theoretical Biology, (99). 6. Gilbert, J., Gilbert, L., Elements of Modern Algebra, Thompson Brooks/ole, (005). 7. Gamow, G., Possible relation between deoxyribonucleic acid and Protein Structure., Nature, vol. 7, pp.8, (95). 8. Luscombe N.M., Austin S.E., Berman H. M., Thorton J.M. An overview of the Structures of protein-dna complexes*, Published: 9 June 000, Genome Biology 000, (): reviews The electronic version of this article is the complete one and can be found online at GenomeBiology.com (Print ISSN ; Online ISSN 65-69). 9. Ian Stewrt, Science Magazine, Broken symmetry in the genetic code? 05 March 99 by IAN STEWART. Hornos and Hornos, Physical Review Letters, vol 7, p Moore, J. T., Elements of Abstract Algebra, The Macmillan ompany, New York, (96).. Sanchez Robersy, and Grau Ricardo, Vector Space of Extended Base-triplets over the Galois Field of Five DNA Bases Alphabet, International Journal of Biological and Medical Sciences : 008, Bioinformatic Group, Santo Domingo, Vila lara, uba, p Sanchez Robersy, Morgado Eberto, and Grau Ricardo, A Genetic ode Boolean Structure: I. The Meaning of the Boolean Dections. Math Biol doi: 0.06/j. bulm.00.05(00).. Sanchez, R. E. Morgado, R. Grau, Abelian Finite Group of DNA Genomic Sequences, Quantitative Biology, (005).. Sanchez R., Morgado E., and Grau R., Gene Algebra from a Genetic ode Algebraic Structure, Journal of Mathematical Biology, (005), Vila lara, uba.. 5. Sanchez Robersy, Morgado Eberto, and Grau Ricardo, The Genetic ode Boolean Lattice: Match ommun. Math omput. hem. 5, 9-6 (00). 6. Shcherbak, V. I. A new manifestation of the decimal system in the genetic code, In: Proceedings of the th 7. International onference Origin of Life, Book of Abstracts, San-Diego, July 6, USA Negadi Tidjani: Symmetric Groups for the Rumer- Konopel chenko-shcherbak Bisections of the Genetic ode and Applications, Internet Electronic Journal, Molecular Design. 00,,7-70, 9. Watson J.D. and rick F.H.., Nature 7, 7 (95) and Nature 7, 96 (95). 0. White, M. The G-Ball, a New Icon for odon Symmetry and the Genetic ode, Qualitative Biology, (007).. White M., The G-Ball, a New Icon for odon Symmetry and the Genetic ode, by Mark White, MD, opyright Rafiki, Inc Yang hi Ming, 00, The Naturally Designed Spherical Symmetry in the Genetic ode, arxiv.org/ftp/q- bio/papers/0090 Sept. Yang, D. J., Ying, Z. J., Detection of Permutation Symmetry in Pattern Sets, Discrete Dynamics in Nature and Society, (006). U A G U Phe Ser Tyr ys U Phe Ser Tyr ys Leu Ser STOP STOP A Leu Ser STOP Trp G Leu Pro His Arg U Leu Pro His Arg Leu Pro Gln Arg A Leu Pro Gln Arg G A Ile Thr Asn Ser U Ile Thr Asn Ser Ile Thr Lys Arg A Met Thr Lys Arg G G Val Ala Asp Gly U Val Ala Asp Gly Val Ala Glu Gly A Val Ala Glu Gly G Table (): The codon triplet table- nobelprize.org/ educational/ medicine/ gene-code/ how.html

7 ayley's table for DNA Transformation in S Binary omposition " " RNA Group e a b c g h i j r=g s=i t=j u=h N O P R S T U V W X Y Z e=() (UGA) e r=g s=i t=j u=j a b c g h i j N O P R S T U V W X Y Z a=()()=s (U)(GA) a s=i r=g u=j t=h e c b h g j i O N R P U V S T Z Y X W b=()()=r (UG)(A) b t=h u=h r=g s=i c e a i j g h R P O N T S V U Y Z W X c=()()=r s (UG)(G) c u=j t=j s=i r=g b a e j i h g P R N O V U T S X W Z Y g=() (UG) r=g t=j u=j s=i e g i j h b c a W X Y Z N O P R S T U V h=() (UGA) s=i u=j t=h r=g a h j i g c b e X W Z Y P R N O V U T S i=() (AG) t=h r=g s=i u=h b i g h j e a c Z Y X W O N R P U V S T j=() (UA) u=j s=i r=g t=j c j h g i a e b Y Z W X R P O N T S V U g =()=r (UG) r=g u=j s=i t=j g e c a b j h i S T U V W X Y Z N O P R i =()=s (GA) s=i t=h r=g u=j a b e c h i g j T S V U Y Y W X R P O N h =()=t (UAG) t=h s=i u=h r=g b a c e i h j g V U T S X W Z Y P R N O j =()=u (UA) u=j r=g t=j s=i c e b a j g i h U V S T Z Y X W O N R P e r=g t=j u=j s=i N=() (U)(GA) N O P R S T U V W X Y Z a b c g i j h a t=h r=g s=i u=h O=() (GA) O N R P U V S T Z Y X W e c b i g h j c s=i u=j t=h r=g 5 P=() (UGA) P R N O V U T S X W Z Y b a e h j i g b u=j s=i r=g t=j 6 R() (UAG) R P O N T S V U Y Z W X c e a j h g i r=g u=j s=i t=j e g 7 S=() (G) S T U V W X Y Z N O P R a b c j h i s=i t=h r=g u=j b 8 T=() (UAG) T S V U Y Y W X R P O N c e a h i g j u=j r=g t=j s=i a 9 U=() (UAG) U V S T Z Y X W O N R P e c b j g i h t=h s=i u=h r=g c 0 V=() (UA) V U T S X W Z Y P R N O b a e i h j g r=g u=j s=i t=j e W=()=rs (UG) W X Y Z N O P R S T U V g i j h a b c t=h s=i u=h r=g c X=()=r (UGA) X W Z Y P R N O V U T S h j i g b a e Y=()=r s=i t=h r=g u=j b s (A) Y Y W X R P O N T S V U j h g i c e a Z=()=r u=j r=g t=j s=i a (UAG) Z Y X W O N R P U V S T i g h j e c b Table (): ayley s Table for all periodic permutation in S demonstrates codons transformation group and subgroups.