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## Unary Operator and RNA/DNA Strands An RNA or DNA strand could be interpreted

as being generated by the unary operator.

Since RNA and DNA strands are linear structures, where each element(base) has two neighbors except for the first and last that have one neighbor each, it is possible to define a unary operator that would enable conversion of each base into another base(or into itself), and, in this way, generation of the entire strand.

We will begin with a simple case, to define  unary operator for two elements that would enable generating binary RNA or DNA strands.

Let Q2 be a set of two elements: Q2 {a,b}. Within this set we will define  unary operator u2 that transforms each of the element into itself or another element of the same set (Fig.1). We can see that u2 is expressed through four different operations. The first operation 1 transforms a into a and b into a. Operation 2 transforms a into a and b into b, while 3 transforms a into b and b into a. Finally, operation 4 transforms a into b and b into b. We will also define a “strand” S as a series of elements (values), including repetition, listed in a linear order and having the first element. A strand can be generated by implementing one or more operations of the operator u2. Those strands generated by a single operation we will call the “basic strands”. With four u2 operations we can generate 8 basic strands (Fig.2). In order to generate complex strands we should apply different operations, each operation on a particular position(Fig.3). If we identify elements a and b with bases T and A (a=T, b=A), these two arbitrary strands would look like this in both alphabet and visual representation where T= white, A=gray(Fig.4). This algorithm could help describing already existing DNA/RNA strands thus reducing them to a smaller number of data, and on the other hand, “explaining” a strand’s structure . Instead of listing all the bases of a particular strand, we now need to know only the first base, and the triggering positions of each unary operation. Since Tetrahymena thermophila strain CU427/CU428 macronuclear isoform gene has quite a few binary stretches here are some of them interpreted through a series of unary operations. Among those TA stretches there are four cases of highly organized states(marked with *), but no case of high entropy.

However, a case of high entropy was noticed in a AC stretch presented at the end, but there are not unary operations applied here.        Unary operators could be defined on the sets of three and four elements as presented below and their detailed applications on DNA are possible to find at www.visualdna.org

Let Q3 be a set of three elements a,b and c: Q3 {a,b,c}. Within this three-element set we can define unary operator u3 which transforms each element of this set into itself or another element of the same set. This unary operator (u3) is defined by 27 operations . With these 2 operations we could generate 81 basic strands out of which 33 are different, three of them trivial(no change).

These 33 strands we will consider to be basic for the three-element set Q3,and we will assign one number from 1 to 33 for each basic strand. Thus we will consolidate all the operations that generate identical strands with given first element, into one operation that generates that particular strand.

We will now define a set of four elements Q4 with elements a, b, c and d: Q4 {a,b,c,d}. These four elements are selected to correspond to the visual representation of the four bases of DNA (C,A,G,T). Within this set we will define unary operator u4 represented by 256 operations. With these 256 operations we could generate 1024 basic strands out of which only 196 are different, four of them being trivial. We could consolidate all operations that generate identical strands with given first element (starting point), and assign to each of them a number from 1 to 196 . These basic four-element strands could be then represented as 196 corresponding D-icons.

And in case of four-element set, we could generate complex strands with successive application of two or more different unary operations.

Gregor Mobius

New York 2001-2017