# Transition and Transversion on the Common Trinucleotide Circular Code

Authors: Emmanuel Benard; Christian J. Michel
Journal: Computational Biology Journal
 (2013)

---

## Abstract

In 1996, a trinucleotide circular code which is maximum, self-complementary, andC3, called X0, was identified statistically on a large gene population of eukaryotes and prokaryotes (Arquès and Michel (1996)). Transition and transversions I and II are classical molecular evolution processes. A comprehensive computer analysis of these three evolution processes in the code X0 shows some new results; in particular (i) transversion I on the 2nd position of any subset of trinucleotides of X0
generates trinucleotide circular codes which are always C3 and (ii) transversion II on the three positions of any subset of trinucleotides of X0
yields no trinucleotide circular codes. These new results extend our theory of circular code in genes to its evolution under transition and transversion.


---

## Body

## 1. Introduction

We continue our study of properties of trinucleotide circular codes [1–5], trinucleotide comma-free codes [1, 6], strong trinucleotide circular codes [7], and the common trinucleotide circular code X0 identified in genes [8] (see also the recent statistical analysis by [9]) which could be a translation code [10]. A trinucleotide is a word of three letters (triletter) on the genetic alphabet {A,C,G,T}. The set of 64 trinucleotides is a code (called genetic code), more precisely a uniform code but not a circular code (see Remark 2). In the past 50 years, codes, comma-free codes, and circular codes have been mathematical objects studied in theoretical biology, mainly to understand the structure and the origin of the genetic code as well as the reading frame (construction) of genes, for example, [11–13]. In order to have an intuitive meaning of these notions, codes are written on a straight line while comma-free codes and circular codes are written on a circle, but in both cases, unique decipherability is required. Circular codes only belong to some subsets of the 64 trinucleotide set while comma-free codes are even more constrained subsets of circular codes [1].Before the discovery of the genetic code, Crick et al. [11] proposed a maximum comma-free code of 20 trinucleotides for coding the 20 amino acids. This comma-free code turned out to be invalid (see, e.g., [14]). In 1996, a maximum circular code X0 of 20 trinucleotides was identified statistically on a large gene population of eukaryotes and also on a large gene population of prokaryotes [8]
(1)X0={AAC,AAT,ACC,ATC,ATT,CAG,CTC,CTG,GAA,GAC,GAG,GAT,GCC,GGC,GGT,GTA,GTC,GTT,TAC,TTC}.
This code X0 has remarkable mathematical properties as it is a C3 self-complementary maximum circular code (see the following). Since 1996, its properties have been studied in detail by different authors, for example, [9, 15–21]. Transition and transversions I and II are classical molecular evolution processes, for example, [22]. By using an algorithm based on the necklace, we perform here a comprehensive computer analysis of these three evolution processes in the code X0. Some new results are identified with the code X0 by computer analysis; in particular (i) transversion I on the 2nd position of any subset of trinucleotides of X0 generates trinucleotide circular codes which are always C3 and (ii) transversion II on the three positions of any subset of trinucleotides of X0 yields no trinucleotide circular codes.


## 2. Preliminaries

The classical notions of language theory and codes can be found in [23, 24]. Let 𝒜4={A,C,G,T} denote the genetic alphabet, lexicographically ordered by A<C<G<T. The set of words (nonempty words, resp.) on 𝒜4 is denoted by 𝒜4* (𝒜4+, resp.). The set of the 16 words of length 2 (dinucleotides or diletters) on 𝒜4 is denoted by 𝒜42={AA,AC,…,TT}. The set of the 64 words of length 3 (trinucleotides or triletters) on 𝒜4 is denoted by 𝒜43={AAA,AAC,…,TTT}.Definition 1.
A subsetX⊂𝒜4+ is a code on 𝒜4 if for each x1,…,xn,x1′,…,xm′∈X,n,m≥1, the condition x1⋯xn=x1′⋯xm′ implies n=m and xi=xi′ for i=1,…,n.Remark 2.
𝒜
4
3 is a code.Any nonempty subset of𝒜43 is a code called here trinucleotide code.Definition 3.
A trinucleotide codeX⊂𝒜43 is circular if, for each x1,…,xn,x1′,…,xm′∈X,n,m≥1,p∈𝒜4*,s∈𝒜4+, the conditions sx2⋯xnp=x1′⋯xm′ and x1=ps imply n=m, p=ε (empty word) and xi=xi′ for i=1,…,n.Notation 1.
A trinucleotide circular code is notedC.Remark 4.
𝒜
4
3 is not a trinucleotide circular code.Letl1,l2,…,ln-1,ln be letters in 𝒜4, d1,d2,…,dn-1,dn diletters in 𝒜42, and n an integer satisfying n≥2.Definition 5.
We say that the ordered sequencel1,d1,l2,d2,…,dn-1,ln,dn,ln+1 is an (n+1)LDCN (Letter Diletter Continued Necklace) for a subset X⊂𝒜43 if
(2)l1d1,l2d2,…,lndn∈X,d1l2,d2l3,…,dn-1ln,dnln+1∈X.Only a few trinucleotide codes are circular. Two propositions based on the necklace concept allow to determine if a trinucleotide code is circular or not [2, 18].Proposition 6 (see [18]).
LetX be a trinucleotide code. The following conditions are equivalent:(i)
Xis a trinucleotide circular code;(ii)
Xhas no5LDCN.Definition 7.
We say that the ordered sequencel1,d1,l2,d2,…,dn-1,ln,dn,ln+1 is an (n+1)LDCCN (Letter Diletter Continued Closed Necklace) for a subset X⊂𝒜43 if
(3)l1d1,l2d2,…,lndn∈X,d1l2,d2l3,…,dn-1ln,dnl1∈X.Proposition 8 (see [2]).
LetX be a trinucleotide code. The following conditions are equivalent:(i)
Xis a trinucleotide circular code;(ii)
Xhas nonLDCCNfor any integern∈{2,3,4,5}.Definition 9.
A trinucleotide circular codeX⊂𝒜43 is maximal if, for each x∈𝒜43, x∉X, X∪{x} is not a trinucleotide circular code.Definition 10.
A trinucleotide circular code containing exactlyk elements is called a k-trinucleotide circular code.Definition 11.
A20-trinucleotide circular code is maximum as no trinucleotide circular code can contain more than 20 words.Notation 2.
A maximum trinucleotide circular code is notedMC.Remark 12.
A20-trinucleotide circular code is both maximal and maximum.We recall two classical genetic maps: complementary and circular permutation.Definition 13.
The complementary genetic map𝒞: 𝒜4+→𝒜4+ is defined by
(4)𝒞(A)=T,𝒞(C)=G,𝒞(G)=C,𝒞(T)=A
and for all u,v∈𝒜4+ by
(5)𝒞(uv)=𝒞(v)𝒞(u).Example 14.
𝒞
(
A
C
G
)
=
C
G
T. This map 𝒞 is associated with the property of the complementary and antiparallel double helix (one DNA strand chemically oriented in a 5′-3′ direction and the other DNA strand in the opposite 3′-5′ direction).Definition 15.
The complementary map𝒞 on a trinucleotide x is naturally extended to a trinucleotide code X as follows:
(6)𝒞(X)={y∣x,y∈𝒜43,x∈X,y=𝒞(x)}.Definition 16.
The circular permutation genetic map𝒫:𝒜43→𝒜43 permutes circularly a trinucleotide l1l2l3, l1,l2,l3∈𝒜4, as follows:
(7)𝒫(l1l2l3)=l2l3l1.Example 17.
𝒫
(
A
C
G
)
=
C
G
A.Definition 18.
The circular permutation map𝒫 on a trinucleotide x is naturally extended to a trinucleotide code X as follows:
(8)𝒫(X)={y∣x,y∈𝒜43,x∈X,y=𝒫(x)}.Notation 3.
Thekth iterate of 𝒫 is denoted by 𝒫k.Remark 19.
The trinucleotide codes𝒫(X) and 𝒫2(X) are the conjugated classes of the trinucleotide code X.Definition 20.
A trinucleotide circular codeX is self-complementary if, for each x∈X, 𝒞(x)∈X.Notation 4.
A self-complementary trinucleotide circular code is notedSC.Remark 21.
Ak-trinucleotide circular code for k odd cannot be self-complementary.Definition 22.
A trinucleotide circular codeX is C3 if X, 𝒫(X), and 𝒫2(X) are trinucleotide circular codes.Notation 5.
AC3 trinucleotide circular code X is noted C3.Definition 23.
A trinucleotide circular codeX is C3 self-complementary maximum if X is maximum, X=𝒞(X) (self-complementary), and 𝒫(X) and 𝒫2(X) are trinucleotide circular codes satisfying 𝒞(𝒫(X))=𝒫2(X).Notation 6.
AC3 self-complementary maximum circular code is noted MSC3.The setX0 of 20 trinucleotides identified in the gene populations of both eukaryotes and prokaryotes is a C3 self-complementary maximum circular code MSC3 [8]; that is, X0 is maximum, X0=𝒞(X0), 𝒫(X0)=X1, and 𝒫2(X0)=X2 are trinucleotide circular codes, and 𝒞(X1)=X2.We recall three classical evolution genetic maps: transition and transversions I and II, for example, [22] and extend their definitions to the positions of a trinucleotide.Definition 24.
The transition evolution genetic map𝒯: 𝒜4+→𝒜4+ is defined by
(9)𝒯(A)=G,𝒯(C)=T,𝒯(G)=A,𝒯(T)=C.Definition 25.
The transition map𝒯 on a letter l can be applied in different positions of a trinucleotide x=l1l2l3: 𝒯i, i∈{1,2,3}, is the transition on the position i of x, 𝒯i,j, i,j∈{1,2,3} with i<j, is the transition on the two positions i and j of x, and 𝒯1,2,3 is the transition on the three positions of x.Example 26.
𝒯
1
(
A
C
G
)
=
G
C
G, 𝒯2(ACG)=ATG, 𝒯3(ACG)=ACA, 𝒯1,2(ACG)=GTG, 𝒯1,3(ACG)=GCA, 𝒯2,3(ACG)=ATA, and 𝒯1,2,3(ACG)=GTA.Definition 27.
The transition maps𝒯i,𝒯i,j,𝒯1,2,3 on a trinucleotide x are also extended to a trinucleotide code X, in a similar way to the genetic maps 𝒞 and 𝒫.Definition 28.
The transversion I evolution genetic map𝒱I: 𝒜4+→𝒜4+ is defined by
(10)𝒱I(A)=T,𝒱I(C)=G,𝒱I(G)=C,𝒱I(T)=A.Definition 29.
The transversion I map𝒱I on a letter l can also be applied in different positions of a trinucleotide x=l1l2l3: 𝒱Ii, i∈{1,2,3}, is the transversion I on the position i of x, 𝒱Ii,j, i,j∈{1,2,3} with i<j, is the transversion I on the two positions i and j of x, and 𝒱I1,2,3 is the transversion I on the three positions of x.Example 30.
𝒱
I
1
(
A
C
G
)
=
T
C
G, 𝒱I2(ACG)=AGG, 𝒱I3(ACG)=ACC, 𝒱I1,2(ACG)=TGG, 𝒱I1,3(ACG)=TCC, 𝒱I2,3(ACG)=AGC, and 𝒱I1,2,3(ACG)=TGC.Definition 31.
The transversion I maps𝒱Ii,𝒱Ii,j,𝒱I1,2,3 on a trinucleotide x are also extended to a trinucleotide code X, in a similar way to the genetic maps 𝒞 and 𝒫.Definition 32.
The transversion II evolution genetic map𝒱II: 𝒜4+→𝒜4+ is defined by
(11)𝒱II(A)=C,𝒱II(C)=A,𝒱II(G)=T,𝒱II(T)=G.Definition 33.
The transversion II map𝒱II on a letter l can also be applied in different positions of a trinucleotide x=l1l2l3: 𝒱IIi, i∈{1,2,3}, is the transversion II on the position i of x, 𝒱IIi,j, i,j∈{1,2,3} with i<j, is the transversion II on the two positions i and j of x, and 𝒱II1,2,3 is the transversion II on the three positions of x.Example 34.
𝒱
II
1
(
A
C
G
)
=
C
C
G, 𝒱II2(ACG)=AAG, 𝒱II3(ACG)=ACT, 𝒱II1,2(ACG)=CAG, 𝒱II1,3(ACG)=CCT, 𝒱II2,3(ACG)=AAT, and 𝒱II1,2,3(ACG)=CAT.Definition 35.
The transversion II maps𝒱IIi,𝒱IIi,j,𝒱II1,2,3 on a trinucleotide x are also extended to a trinucleotide code X, in a similar way to the genetic maps 𝒞 and 𝒫.Definition 36.
The evolution genetic maps inl trinucleotides of a trinucleotide circular code are defined by 𝒯(l) for transition, 𝒱I(l) for transversion I, and 𝒱II(l) for transversion II.


## 3. Results

An evolution genetic map, that is,𝒯(l), 𝒱I(l), and 𝒱II(l), in l trinucleotides of the common trinucleotide circular code X0 leads to S(l)=(20l) trinucleotide codes which are potentially circular. Table 1 gives these numbers S(l).Table 1
NumberS(l)=(20l) of trinucleotide codes after an evolution genetic map (transition 𝒯(l), transversion I 𝒱I(l), and transversion II 𝒱II(l)) in l trinucleotides of the common trinucleotide circular code X0.
l
S
(
l
)
{
1,19
}
20
{
2,18
}
190
{
3,17
}
1140
{
4,16
}
4845
{
5,15
}
15504
{
6,14
}
38760
{
7,13
}
77520
{
8,12
}
125970
{
9,11
}
167960
10
184756
20
1Based on Proposition6 allowing to test if a trinucleotide code is circular or not (algorithm not detailed, see, e.g., [2]), computer analyses of a great number of trinucleotide codes allow to identify here new properties with the common trinucleotide circular code X0 observed in genes under evolution by transition and transversion.


### 3.1. Transition Map

#### 3.1.1. Transition Map𝒯i

Result 1 (Table2).
Forl=1,…,20(12)c(𝒯1(l))=c(𝒯3(l)),mc(𝒯1(l))=mc(𝒯3(l)),sc(𝒯1(l))=sc(𝒯3(l)),c3(𝒯1(l))=c3(𝒯3(l)),msc3(𝒯1(l))=msc3(𝒯3(l)).
As expected, the lists of trinucleotide circular codes C associated with c(𝒯1(l)) and c(𝒯3(l)) are different for l=1,…,13 (not shown). No trinucleotide code is circular after a certain number of transitions 𝒯i in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=14,…,20(13)c(𝒯1(l))=c(𝒯3(l))=0
and for l=10,…,20(14)c(𝒯2(l))=0.
The transition 𝒯i generates a maximum number of trinucleotide circular codes C for
(15)max{c(𝒯i(l)),i=1,2,3,l=1,…,20}=c(𝒯1(7))=c(𝒯3(7))=1436
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(16)max{msc3(𝒯i(l)),i=1,2,3,l=1,…,20}=msc3(𝒯1(6))=msc3(𝒯3(6))=20.Table 2
Transition map𝒯i(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯i(l)) of circular codes C, number mc(𝒯i(l)) of maximum circular codes MC, number sc(𝒯i(l)) of self-complementary circular codes SC, number c3(𝒯i(l)) of circular codes C3, and number msc3(𝒯i(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1
(
l
) or 𝒯3(l)
𝒯
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
12
0
0
12
0
8
0
0
8
0
2
68
6
10
68
6
28
4
8
28
4
3
240
0
0
240
0
60
0
0
56
0
4
587
15
33
586
15
92
6
18
74
6
5
1049
0
0
1040
0
108
0
0
62
0
6
1408
20
48
1372
20
96
4
8
32
4
7
1436
0
0
1352
0
60
0
0
8
0
8
1111
15
33
985
15
23
1
1
1
1
9
642
0
0
516
0
4
0
0
0
0
10
268
6
10
184
6
0
0
0
0
0
11
76
0
0
40
0
0
0
0
0
0
12
13
1
1
4
1
0
0
0
0
0
13
1
0
0
0
0
0
0
0
0
0
{
14
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


#### 3.1.2. Transition Map𝒯i,j

Result 2 (Table3).
Forl=1,…,20(17)c(𝒯1,2(l))=c(𝒯2,3(l)),mc(𝒯1,2(l))=mc(𝒯2,3(l)),sc(𝒯1,2(l))=sc(𝒯2,3(l)),c3(𝒯1,2(l))=c3(𝒯2,3(l)),msc3(𝒯1,2(l))=msc3(𝒯2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒯1,2(l)) and c(𝒯2,3(l)) are different for l=1,…,14 (not shown). No trinucleotide code is circular after a certain number of transitions 𝒯i,j in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=15,…,20(18)c(𝒯1,2(l))=c(𝒯2,3(l))=0
and for l=12,…,20(19)c(𝒯1,3(l))=0.
The transition 𝒯i,j generates a maximum number of trinucleotide circular codes C for
(20)max{c(𝒯i,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒯1,3(6))=598
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(21)max{msc3(𝒯i,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒯1,3(4))=msc3(𝒯1,3(6))=10.
The numbers c3(𝒯1,2(l))=c3(𝒯2,3(l)) of circular codes C3 have a particular growth function
(22)c3(𝒯1,2(14))=c3(𝒯2,3(14))=1,c3(𝒯1,2(l))=c3(𝒯2,3(l))=0forl=10,…,13.Table 3
Transition map𝒯i,j(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯i,j(l)) of circular codes C, number mc(𝒯i,j(l)) of maximum circular codes MC, number sc(𝒯i,j(l)) of self-complementary circular codes SC, number c3(𝒯i,j(l)) of circular codes C3, and number msc3(𝒯i,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1,2
(
l
) or 𝒯2,3(l)
𝒯
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
9
1
0
9
0
10
2
0
10
0
2
37
4
6
36
4
49
5
9
45
5
3
91
4
0
85
0
152
8
0
122
0
4
148
6
11
130
6
325
10
27
223
10
5
166
6
0
133
0
508
12
0
292
0
6
129
4
6
90
4
598
10
30
286
10
7
67
4
0
38
0
532
8
0
214
0
8
21
1
1
9
1
353
5
11
121
5
9
4
1
0
1
0
162
2
0
44
0
10
3
0
0
0
0
42
1
2
10
1
11
4
0
0
0
0
4
0
0
0
0
12
3
0
0
0
0
0
0
0
0
0
13
4
0
0
0
0
0
0
0
0
0
14
1
0
0
1
0
0
0
0
0
0
{
15
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


#### 3.1.3. Transition Map𝒯1,2,3

Result 3 (Table4).
The transition𝒯1,2,3 always generates trinucleotide circular codes. Indeed, for l=1,…,20(23)c(𝒯1,2,3(l))>0.
The lists of trinucleotide circular codes C associated with c(𝒯1,2,3(l)) and c(𝒯1,2,3(20-l)) are different for l=1,…,9 (not shown). The transition 𝒯1,2,3 generates a maximum number of trinucleotide circular codes C for
(24)max{c(𝒯1,2,3(l)),l=1,…,20}=c(𝒯1,2,3(8))=c(𝒯1,2,3(12))=72
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(25)max{msc3(𝒯1,2,3(l)),l=1,…,20}=msc3(𝒯1,2,3(4))=msc3(𝒯1,2,3(16))=6.Table 4
Transition map𝒯1,2,3(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯1,2,3(l)) of circular codes C, number mc(𝒯1,2,3(l)) of maximum circular codes MC, number sc(𝒯1,2,3(l)) of self-complementary circular codes SC, number c3(𝒯1,2,3(l)) of circular codes C3, and number msc3(𝒯1,2,3(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1,2
,
3
(
l
)
c
mc
sc
c
3
ms
c
3
{
1,19
}
8
0
0
8
0
{
2,18
}
28
4
8
28
4
{
3,17
}
56
0
0
56
0
{
4,16
}
71
7
19
70
6
{
5,15
}
64
0
0
56
0
{
6,14
}
56
8
16
28
4
{
7,13
}
64
0
0
8
0
{
8,12
}
72
8
20
1
1
{
9,11
}
64
0
0
0
0
10
56
8
16
0
0
20
1
1
1
1
1


### 3.2. Transversion I Map

#### 3.2.1. Transversion I Map𝒱Ii

Result 4 (Table5).
Forl=1,…,20(26)c(𝒱I1(l))=c(𝒱I3(l)),mc(𝒱I1(l))=mc(𝒱I3(l)),sc(𝒱I1(l))=sc(𝒱I3(l)),c3(𝒱I1(l))=c3(𝒱I3(l)),msc3(𝒱I1(l))=msc3(𝒱I3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱I1(l)) and c(𝒱I3(l)) are different for l=1,…,9 (not shown). No trinucleotide code is circular after a certain number of transversions I 𝒱Ii in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=10,…,20(27)c(𝒱I1(l))=c(𝒱I3(l))=0
and for l=18,19,20(28)c(𝒱I2(l))=0.
The transversion I 𝒱Ii generates a maximum number of trinucleotide circular codes C for
(29)max{c(𝒱Ii(l)),i=1,2,3,l=1,…,20}=c(𝒱I2(9))=24310
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(30)max{msc3(𝒱Ii(l)),i=1,2,3,l=1,…,20}=msc3(𝒱I2(8))=70.
A remarkable code property only found with transversion I 𝒱I2 is, for l=1,…,20,
(31)c(𝒱I2(l))=c3(𝒱I2(l)),
and furthermore, after a detailed computer analysis, the lists of trinucleotide circular codes C and C3 associated with c(𝒱I2(l)) and c3(𝒱I2(l)), respectively, are identical for l=1,…,17.Table 5
Transversion I map𝒱Ii(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱Ii(l)) of circular codes C, number mc(𝒱Ii(l)) of maximum circular codes MC, number sc(𝒱Ii(l)) of self-complementary circular codes SC, number c3(𝒱Ii(l)) of circular codes C3, and number msc3(𝒱Ii(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1
(
l
) or 𝒱I3(l)
𝒱
I
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
8
0
0
8
0
16
0
0
16
0
2
31
4
5
30
4
122
8
12
122
8
3
76
0
0
71
0
590
0
0
590
0
4
130
6
8
114
6
2030
28
56
2030
28
5
160
0
0
125
0
5278
0
0
5278
0
6
142
4
5
92
4
10738
56
132
10738
56
7
87
0
0
42
0
17446
0
0
17446
0
8
32
1
1
10
1
22880
70
174
22880
70
9
5
0
0
1
0
24310
0
0
24310
0
10
0
0
0
0
0
20878
56
132
20878
56
11
0
0
0
0
0
14378
0
0
14378
0
12
0
0
0
0
0
7826
28
56
7826
28
13
0
0
0
0
0
3290
0
0
3290
0
14
0
0
0
0
0
1030
8
12
1030
8
15
0
0
0
0
0
226
0
0
226
0
16
0
0
0
0
0
31
1
1
31
1
17
0
0
0
0
0
2
0
0
2
0
{
18
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


#### 3.2.2. Transversion I Map𝒱Ii,j

Result 5 (Table6).
Forl=1,…,20(32)c(𝒱I1,2(l))=c(𝒱I2,3(l)),mc(𝒱I1,2(l))=mc(𝒱I2,3(l)),sc(𝒱I1,2(l))=sc(𝒱I2,3(l)),c3(𝒱I1,2(l))=c3(𝒱I2,3(l)),msc3(𝒱I1,2(l))=msc3(𝒱I2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱I1,2(l)) and c(𝒱I2,3(l)) are different for l=1,…,12 (not shown). No trinucleotide code is circular after a certain number of transversions I 𝒱Ii,j in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=13,…,20(33)c(𝒱I1,2(l))=c(𝒱I2,3(l))=0
and for l=19,20(34)c(𝒱I1,3(l))=0.
The transversion I 𝒱Ii,j generates a maximum number of trinucleotide circular codes C for
(35)max{c(𝒱Ii,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒱I1,2(6))=c(𝒱I2,3(6))=630
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(36)max{msc3(𝒱Ii,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒱I1,2(4))=msc3(𝒱I2,3(4))=6.
The numbers sc(𝒱I1,3(l)) of self-complementary circular codes SC have a particular growth function
(37)sc(𝒱I1,3(l))=1forl=12,14,16,18,sc(𝒱I1,3(l))=0forl=8,10.
The numbers c3(𝒱I1,3(l)) of circular codes C3 have a particular growth function
(38)c3(𝒱I1,3(l))=1forl=16,18,c3(𝒱I1,3(17))=2,c3(𝒱I1,3(l))=0forl=7,…,15.Table 6
Transversion I map𝒱Ii,j(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱Ii,j(l)) of circular codes C, number mc(𝒱Ii,j(l)) of maximum circular codes MC, number sc(𝒱Ii,j(l)) of self-complementary circular codes SC, number c3(𝒱Ii,j(l)) of circular codes C3, and number msc3(𝒱Ii,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1,2
(
l
) or 𝒱I2,3(l)
𝒱
I
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
10
2
0
9
0
6
2
0
4
0
2
48
5
5
38
4
15
3
5
7
3
3
147
8
0
102
0
20
4
0
8
0
4
319
10
8
191
6
15
3
5
7
3
5
514
12
0
265
0
8
4
0
4
0
6
630
10
5
282
4
11
3
1
1
1
7
595
8
0
240
0
22
4
0
0
0
8
435
5
1
162
1
32
4
0
0
0
9
245
2
0
88
0
38
2
0
0
0
10
103
1
0
34
0
38
2
0
0
0
11
29
0
0
8
0
34
0
0
0
0
12
4
0
0
1
0
39
0
1
0
0
13
0
0
0
0
0
56
0
0
0
0
14
0
0
0
0
0
65
0
1
0
0
15
0
0
0
0
0
52
0
0
0
0
16
0
0
0
0
0
27
0
1
1
0
17
0
0
0
0
0
8
0
0
2
0
18
0
0
0
0
0
1
0
1
1
0
{
19,20
}
0
0
0
0
0
0
0
0
0
0


#### 3.2.3. Transversion I Map𝒱I1,2,3

Result 6 (Table7).
The transversion I𝒱I1,2,3 always generates trinucleotide circular codes. Indeed, for l=1,…,20(39)c(𝒱I1,2,3(l))>0.
The lists of trinucleotide circular codes C associated with c(𝒱I1,2,3(l)) and c(𝒱I1,2,3(20-l)) are different for l=1,…,9 (not shown). The transversion I 𝒱I1,2,3 generates a maximum number of trinucleotide circular codes C for
(40)max{c(𝒱I1,2,3(l)),l=1,…,20}=c(𝒱I1,2,3(10))=66
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(41)max{msc3(𝒱I1,2,3(l)),l=1,…,20}=msc3(𝒱I1,2,3(4))=msc3(𝒱I1,2,3(16))=9.Table 7
Transversion I map𝒱I1,2,3(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱I1,2,3(l)) of circular codes C, number mc(𝒱I1,2,3(l)) of maximum circular codes MC, number sc(𝒱I1,2,3(l)) of self-complementary circular codes SC, number c3(𝒱I1,2,3(l)) of circular codes C3, and number msc3(𝒱I1,2,3(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1,2
,
3
(
l
)
c
mc
sc
c
3
ms
c
3
{
1,19
}
6
0
0
6
0
{
2,18
}
17
5
7
17
5
{
3,17
}
32
0
0
32
0
{
4,16
}
47
10
17
45
9
{
5,15
}
54
0
0
46
0
{
6,14
}
51
12
19
31
7
{
7,13
}
48
0
0
12
0
{
8,12
}
53
13
11
2
2
{
9,11
}
62
0
0
0
0
10
66
14
6
0
0
20
1
1
1
1
1


### 3.3. Transversion II Map

#### 3.3.1. Transversion II Map𝒱IIi

Result 7 (Table8).
Forl=1,…,20(42)c(𝒱II1(l))=c(𝒱II3(l)),mc(𝒱II1(l))=mc(𝒱II3(l)),sc(𝒱II1(l))=sc(𝒱II3(l)),c3(𝒱II1(l))=c3(𝒱II3(l)),msc3(𝒱II1(l))=msc3(𝒱II3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱II1(l)) and c(𝒱II3(l)) are different for l=1,…,8 (not shown). No trinucleotide code is circular after a certain number of transversions II 𝒱IIi in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=9,…,20(43)c(𝒱II1(l))=c(𝒱II3(l))=0
and for l=12,…,20(44)c(𝒱II2(l))=0.
The transversion II 𝒱IIi generates a maximum number of trinucleotide circular codes C for
(45)max{c(𝒱IIi(l)),i=1,2,3,l=1,…,20}=c(𝒱II2(5))=176
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(46)max{msc3(𝒱IIi(l)),i=1,2,3,l=1,…,20}=msc3(𝒱II2(4))=6.Table 8
Transversion II map𝒱IIi(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱IIi(l)) of circular codes C, number mc(𝒱IIi(l)) of maximum circular codes MC, number sc(𝒱IIi(l)) of self-complementary circular codes SC, number c3(𝒱IIi(l)) of circular codes C3, and number msc3(𝒱IIi(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
II
1
(
l
) or 𝒱II3(l)
𝒱
II
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
6
0
0
6
0
8
0
0
8
0
2
17
3
4
17
3
30
4
8
28
4
3
30
0
0
29
0
74
0
0
58
0
4
35
3
4
31
3
132
6
18
82
6
5
28
0
0
20
0
176
0
0
76
0
6
16
1
1
7
1
170
4
8
44
4
7
6
0
0
1
0
120
0
0
12
0
8
1
0
0
0
0
61
1
1
1
1
9
0
0
0
0
0
26
0
0
0
0
10
0
0
0
0
0
10
0
0
0
0
11
0
0
0
0
0
2
0
0
0
0
{
12
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


#### 3.3.2. Transversion II Map𝒱IIi,j

Result 8 (Table9).
Forl=1,…,20(47)c(𝒱II1,2(l))=c(𝒱II2,3(l)),mc(𝒱II1,2(l))=mc(𝒱II2,3(l)),sc(𝒱II1,2(l))=sc(𝒱II2,3(l)),c3(𝒱II1,2(l))=c3(𝒱II2,3(l)),msc3(𝒱II1,2(l))=msc3(𝒱II2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱II1,2(l)) and c(𝒱II2,3(l)) are different for l=1,…,5,13,14 (not shown). The distribution of trinucleotide codes which are not circular under transversions II 𝒱IIi,j in the trinucleotides of the common trinucleotide circular code X0 is very unusual. Indeed, for l=6,…,12,15,…,20(48)c(𝒱II1,2(l))=c(𝒱II2,3(l))=0
and for l=15,…,20(49)c(𝒱II1,3(l))=0.
The transversion II 𝒱IIi,j generates a maximum number of trinucleotide circular codes C for
(50)max{c(𝒱IIi,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒱II1,3(6))=662
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(51)max{msc3(𝒱IIi,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒱II1,3(4))=6.
The numbers c(𝒱II1,2(l))=c(𝒱II2,3(l)) of circular codes C have a particular growth function
(52)c(𝒱II1,2(l))=c(𝒱II2,3(l))=1forl=13,14,c(𝒱II1,2(l))=c(𝒱II2,3(l))=0forl=6,…,12.Table 9
Transversion II map𝒱IIi,j(l) in l trinucleotide of the common trinucleotide circular code X0. Number c(𝒱IIi,j(l)) of circular codes C, number mc(𝒱IIi,j(l)) of maximum circular codes MC, number sc(𝒱IIi,j(l)) of self-complementary circular codes SC, number c3(𝒱IIi,j(l)) of circular codes C3, and number msc3(𝒱IIi,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
II
1,2
(
l
) or 𝒱II2,3(l)
𝒱
II
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
5
1
0
5
0
8
0
0
8
0
2
11
2
3
10
2
40
4
8
30
4
3
13
2
0
11
0
132
0
0
72
0
4
8
1
1
7
1
312
6
22
117
6
5
2
1
0
2
0
532
0
0
130
0
6
0
0
0
0
0
662
4
24
92
4
7
0
0
0
0
0
592
0
0
34
0
8
0
0
0
0
0
389
1
9
4
1
9
0
0
0
0
0
188
0
0
0
0
10
0
0
0
0
0
87
0
1
2
0
11
0
0
0
0
0
54
0
0
6
0
12
0
0
0
0
0
34
0
2
7
0
13
1
0
0
0
0
14
0
0
4
0
14
1
0
0
0
0
3
0
1
3
0
{
15
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


#### 3.3.3. Transversion II Map𝒱II1,2,3

Proposition 37.
Forl=1,…,19(53)c(𝒱II1,2,3(l))=0
and obviously, by letter invariance, c(𝒱II1,2,3(20))=1 as in Tables 4 and 7.Proof.
The common trinucleotide circular codeX0 can be partitioned according to the maps 𝒱II1,2,3, 𝒫, and 𝒫2 as shown in Table 10.
Let a partitionPi={x,x′}, i∈{1,…,10}, composed of two trinucleotides x,x′∈X0. For l=1, any transversion II of a trinucleotide x∈Pi generates a trinucleotide y which is a permuted trinucleotide of the other trinucleotide x′∈Pi. So, any transversion II of a trinucleotide x∈X0 leads to a trinucleotide code which is not circular. For 2≤l≤19, the proof needs a computer analysis of the necklace for the nontrivial cases when two transversions II occur with two trinucleotides in the same partitions.Table 10
Partition ofX0
X
0
𝒱
II
1,2
,
3
(
X
0
)
=
X
2
𝒫
(
X
2
)
=
X
0
′
𝒱
II
1,2
,
3
(
X
0
′
)
=
X
1
𝒫
2
(
X
1
)
=
X
0
P
1
=
{
A
A
C
,
A
C
C
}
A
C
C
C
A
A
A
A
C
C
C
A
A
C
C
P
2
=
{
A
A
T
,
G
C
C
}
G
C
C
T
A
A
A
A
T
C
C
G
G
C
C
P
3
=
{
A
T
C
,
G
A
C
}
A
T
C
C
G
A
G
A
C
T
C
A
A
T
C
P
4
=
{
A
T
T
,
G
G
C
}
A
T
T
C
G
G
G
G
C
T
T
A
A
T
T
P
5
=
{
C
A
G
,
T
A
C
}
T
A
C
G
C
A
C
A
G
A
C
T
T
A
C
P
6
=
{
C
T
C
,
G
A
A
}
C
T
C
A
G
A
G
A
A
T
C
C
C
T
C
P
7
=
{
C
T
G
,
G
T
A
}
C
T
G
A
G
T
G
T
A
T
G
C
C
T
G
P
8
=
{
G
A
G
,
T
T
C
}
T
T
C
G
G
A
G
A
G
T
C
T
T
T
C
P
9
=
{
G
A
T
,
G
T
C
}
G
T
C
T
G
A
G
A
T
T
C
G
G
T
C
P
10
=
{
G
G
T
,
G
T
T
}
G
T
T
T
G
G
G
G
T
T
T
G
G
T
TRemark 38.
Very surprisingly, for the three maps of transition, transversions I and II,𝒯i(l), 𝒱Ii(l), and 𝒱IIi(l), i∈{1,2,3}, 𝒯i,j(l), 𝒱Ii,j(l), and 𝒱IIi,j(l), i,j∈{1,2,3} with i<j and 𝒯1,2,3(l) (not for 𝒱I1,2,3(l) and 𝒱II1,2,3(l)), the numbers msc3 of self-complementary maximum circular codes MSC3 for the first even values of l follow a series of binomial coefficients. For 𝒯i(l), 𝒱Ii(l), and 𝒱IIi(l), i∈{1,2,3}, 𝒯1,2(l), 𝒯2,3(l), and 𝒱IIi,j(l), i,j∈{1,2,3} with i<j, the numbers mc of maximum circular codes MC for the first even values of l follow a series of binomial coefficients. For 𝒯1,2,3(l), the numbers c3 of circular codes C3 for the values l and (20-l) with l=1,…,8 follow a series of binomial coefficients. These binomial properties with some numbers of circular codes for the three maps of transition, transversions I and II have no combinatorial explanation so far.


## 3.1. Transition Map

### 3.1.1. Transition Map𝒯i

Result 1 (Table2).
Forl=1,…,20(12)c(𝒯1(l))=c(𝒯3(l)),mc(𝒯1(l))=mc(𝒯3(l)),sc(𝒯1(l))=sc(𝒯3(l)),c3(𝒯1(l))=c3(𝒯3(l)),msc3(𝒯1(l))=msc3(𝒯3(l)).
As expected, the lists of trinucleotide circular codes C associated with c(𝒯1(l)) and c(𝒯3(l)) are different for l=1,…,13 (not shown). No trinucleotide code is circular after a certain number of transitions 𝒯i in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=14,…,20(13)c(𝒯1(l))=c(𝒯3(l))=0
and for l=10,…,20(14)c(𝒯2(l))=0.
The transition 𝒯i generates a maximum number of trinucleotide circular codes C for
(15)max{c(𝒯i(l)),i=1,2,3,l=1,…,20}=c(𝒯1(7))=c(𝒯3(7))=1436
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(16)max{msc3(𝒯i(l)),i=1,2,3,l=1,…,20}=msc3(𝒯1(6))=msc3(𝒯3(6))=20.Table 2
Transition map𝒯i(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯i(l)) of circular codes C, number mc(𝒯i(l)) of maximum circular codes MC, number sc(𝒯i(l)) of self-complementary circular codes SC, number c3(𝒯i(l)) of circular codes C3, and number msc3(𝒯i(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1
(
l
) or 𝒯3(l)
𝒯
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
12
0
0
12
0
8
0
0
8
0
2
68
6
10
68
6
28
4
8
28
4
3
240
0
0
240
0
60
0
0
56
0
4
587
15
33
586
15
92
6
18
74
6
5
1049
0
0
1040
0
108
0
0
62
0
6
1408
20
48
1372
20
96
4
8
32
4
7
1436
0
0
1352
0
60
0
0
8
0
8
1111
15
33
985
15
23
1
1
1
1
9
642
0
0
516
0
4
0
0
0
0
10
268
6
10
184
6
0
0
0
0
0
11
76
0
0
40
0
0
0
0
0
0
12
13
1
1
4
1
0
0
0
0
0
13
1
0
0
0
0
0
0
0
0
0
{
14
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


### 3.1.2. Transition Map𝒯i,j

Result 2 (Table3).
Forl=1,…,20(17)c(𝒯1,2(l))=c(𝒯2,3(l)),mc(𝒯1,2(l))=mc(𝒯2,3(l)),sc(𝒯1,2(l))=sc(𝒯2,3(l)),c3(𝒯1,2(l))=c3(𝒯2,3(l)),msc3(𝒯1,2(l))=msc3(𝒯2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒯1,2(l)) and c(𝒯2,3(l)) are different for l=1,…,14 (not shown). No trinucleotide code is circular after a certain number of transitions 𝒯i,j in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=15,…,20(18)c(𝒯1,2(l))=c(𝒯2,3(l))=0
and for l=12,…,20(19)c(𝒯1,3(l))=0.
The transition 𝒯i,j generates a maximum number of trinucleotide circular codes C for
(20)max{c(𝒯i,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒯1,3(6))=598
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(21)max{msc3(𝒯i,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒯1,3(4))=msc3(𝒯1,3(6))=10.
The numbers c3(𝒯1,2(l))=c3(𝒯2,3(l)) of circular codes C3 have a particular growth function
(22)c3(𝒯1,2(14))=c3(𝒯2,3(14))=1,c3(𝒯1,2(l))=c3(𝒯2,3(l))=0forl=10,…,13.Table 3
Transition map𝒯i,j(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯i,j(l)) of circular codes C, number mc(𝒯i,j(l)) of maximum circular codes MC, number sc(𝒯i,j(l)) of self-complementary circular codes SC, number c3(𝒯i,j(l)) of circular codes C3, and number msc3(𝒯i,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1,2
(
l
) or 𝒯2,3(l)
𝒯
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
9
1
0
9
0
10
2
0
10
0
2
37
4
6
36
4
49
5
9
45
5
3
91
4
0
85
0
152
8
0
122
0
4
148
6
11
130
6
325
10
27
223
10
5
166
6
0
133
0
508
12
0
292
0
6
129
4
6
90
4
598
10
30
286
10
7
67
4
0
38
0
532
8
0
214
0
8
21
1
1
9
1
353
5
11
121
5
9
4
1
0
1
0
162
2
0
44
0
10
3
0
0
0
0
42
1
2
10
1
11
4
0
0
0
0
4
0
0
0
0
12
3
0
0
0
0
0
0
0
0
0
13
4
0
0
0
0
0
0
0
0
0
14
1
0
0
1
0
0
0
0
0
0
{
15
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


### 3.1.3. Transition Map𝒯1,2,3

Result 3 (Table4).
The transition𝒯1,2,3 always generates trinucleotide circular codes. Indeed, for l=1,…,20(23)c(𝒯1,2,3(l))>0.
The lists of trinucleotide circular codes C associated with c(𝒯1,2,3(l)) and c(𝒯1,2,3(20-l)) are different for l=1,…,9 (not shown). The transition 𝒯1,2,3 generates a maximum number of trinucleotide circular codes C for
(24)max{c(𝒯1,2,3(l)),l=1,…,20}=c(𝒯1,2,3(8))=c(𝒯1,2,3(12))=72
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(25)max{msc3(𝒯1,2,3(l)),l=1,…,20}=msc3(𝒯1,2,3(4))=msc3(𝒯1,2,3(16))=6.Table 4
Transition map𝒯1,2,3(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯1,2,3(l)) of circular codes C, number mc(𝒯1,2,3(l)) of maximum circular codes MC, number sc(𝒯1,2,3(l)) of self-complementary circular codes SC, number c3(𝒯1,2,3(l)) of circular codes C3, and number msc3(𝒯1,2,3(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1,2
,
3
(
l
)
c
mc
sc
c
3
ms
c
3
{
1,19
}
8
0
0
8
0
{
2,18
}
28
4
8
28
4
{
3,17
}
56
0
0
56
0
{
4,16
}
71
7
19
70
6
{
5,15
}
64
0
0
56
0
{
6,14
}
56
8
16
28
4
{
7,13
}
64
0
0
8
0
{
8,12
}
72
8
20
1
1
{
9,11
}
64
0
0
0
0
10
56
8
16
0
0
20
1
1
1
1
1


## 3.1.1. Transition Map𝒯i

Result 1 (Table2).
Forl=1,…,20(12)c(𝒯1(l))=c(𝒯3(l)),mc(𝒯1(l))=mc(𝒯3(l)),sc(𝒯1(l))=sc(𝒯3(l)),c3(𝒯1(l))=c3(𝒯3(l)),msc3(𝒯1(l))=msc3(𝒯3(l)).
As expected, the lists of trinucleotide circular codes C associated with c(𝒯1(l)) and c(𝒯3(l)) are different for l=1,…,13 (not shown). No trinucleotide code is circular after a certain number of transitions 𝒯i in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=14,…,20(13)c(𝒯1(l))=c(𝒯3(l))=0
and for l=10,…,20(14)c(𝒯2(l))=0.
The transition 𝒯i generates a maximum number of trinucleotide circular codes C for
(15)max{c(𝒯i(l)),i=1,2,3,l=1,…,20}=c(𝒯1(7))=c(𝒯3(7))=1436
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(16)max{msc3(𝒯i(l)),i=1,2,3,l=1,…,20}=msc3(𝒯1(6))=msc3(𝒯3(6))=20.Table 2
Transition map𝒯i(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯i(l)) of circular codes C, number mc(𝒯i(l)) of maximum circular codes MC, number sc(𝒯i(l)) of self-complementary circular codes SC, number c3(𝒯i(l)) of circular codes C3, and number msc3(𝒯i(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1
(
l
) or 𝒯3(l)
𝒯
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
12
0
0
12
0
8
0
0
8
0
2
68
6
10
68
6
28
4
8
28
4
3
240
0
0
240
0
60
0
0
56
0
4
587
15
33
586
15
92
6
18
74
6
5
1049
0
0
1040
0
108
0
0
62
0
6
1408
20
48
1372
20
96
4
8
32
4
7
1436
0
0
1352
0
60
0
0
8
0
8
1111
15
33
985
15
23
1
1
1
1
9
642
0
0
516
0
4
0
0
0
0
10
268
6
10
184
6
0
0
0
0
0
11
76
0
0
40
0
0
0
0
0
0
12
13
1
1
4
1
0
0
0
0
0
13
1
0
0
0
0
0
0
0
0
0
{
14
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


## 3.1.2. Transition Map𝒯i,j

Result 2 (Table3).
Forl=1,…,20(17)c(𝒯1,2(l))=c(𝒯2,3(l)),mc(𝒯1,2(l))=mc(𝒯2,3(l)),sc(𝒯1,2(l))=sc(𝒯2,3(l)),c3(𝒯1,2(l))=c3(𝒯2,3(l)),msc3(𝒯1,2(l))=msc3(𝒯2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒯1,2(l)) and c(𝒯2,3(l)) are different for l=1,…,14 (not shown). No trinucleotide code is circular after a certain number of transitions 𝒯i,j in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=15,…,20(18)c(𝒯1,2(l))=c(𝒯2,3(l))=0
and for l=12,…,20(19)c(𝒯1,3(l))=0.
The transition 𝒯i,j generates a maximum number of trinucleotide circular codes C for
(20)max{c(𝒯i,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒯1,3(6))=598
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(21)max{msc3(𝒯i,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒯1,3(4))=msc3(𝒯1,3(6))=10.
The numbers c3(𝒯1,2(l))=c3(𝒯2,3(l)) of circular codes C3 have a particular growth function
(22)c3(𝒯1,2(14))=c3(𝒯2,3(14))=1,c3(𝒯1,2(l))=c3(𝒯2,3(l))=0forl=10,…,13.Table 3
Transition map𝒯i,j(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯i,j(l)) of circular codes C, number mc(𝒯i,j(l)) of maximum circular codes MC, number sc(𝒯i,j(l)) of self-complementary circular codes SC, number c3(𝒯i,j(l)) of circular codes C3, and number msc3(𝒯i,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1,2
(
l
) or 𝒯2,3(l)
𝒯
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
9
1
0
9
0
10
2
0
10
0
2
37
4
6
36
4
49
5
9
45
5
3
91
4
0
85
0
152
8
0
122
0
4
148
6
11
130
6
325
10
27
223
10
5
166
6
0
133
0
508
12
0
292
0
6
129
4
6
90
4
598
10
30
286
10
7
67
4
0
38
0
532
8
0
214
0
8
21
1
1
9
1
353
5
11
121
5
9
4
1
0
1
0
162
2
0
44
0
10
3
0
0
0
0
42
1
2
10
1
11
4
0
0
0
0
4
0
0
0
0
12
3
0
0
0
0
0
0
0
0
0
13
4
0
0
0
0
0
0
0
0
0
14
1
0
0
1
0
0
0
0
0
0
{
15
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


## 3.1.3. Transition Map𝒯1,2,3

Result 3 (Table4).
The transition𝒯1,2,3 always generates trinucleotide circular codes. Indeed, for l=1,…,20(23)c(𝒯1,2,3(l))>0.
The lists of trinucleotide circular codes C associated with c(𝒯1,2,3(l)) and c(𝒯1,2,3(20-l)) are different for l=1,…,9 (not shown). The transition 𝒯1,2,3 generates a maximum number of trinucleotide circular codes C for
(24)max{c(𝒯1,2,3(l)),l=1,…,20}=c(𝒯1,2,3(8))=c(𝒯1,2,3(12))=72
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(25)max{msc3(𝒯1,2,3(l)),l=1,…,20}=msc3(𝒯1,2,3(4))=msc3(𝒯1,2,3(16))=6.Table 4
Transition map𝒯1,2,3(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒯1,2,3(l)) of circular codes C, number mc(𝒯1,2,3(l)) of maximum circular codes MC, number sc(𝒯1,2,3(l)) of self-complementary circular codes SC, number c3(𝒯1,2,3(l)) of circular codes C3, and number msc3(𝒯1,2,3(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒯
1,2
,
3
(
l
)
c
mc
sc
c
3
ms
c
3
{
1,19
}
8
0
0
8
0
{
2,18
}
28
4
8
28
4
{
3,17
}
56
0
0
56
0
{
4,16
}
71
7
19
70
6
{
5,15
}
64
0
0
56
0
{
6,14
}
56
8
16
28
4
{
7,13
}
64
0
0
8
0
{
8,12
}
72
8
20
1
1
{
9,11
}
64
0
0
0
0
10
56
8
16
0
0
20
1
1
1
1
1


## 3.2. Transversion I Map

### 3.2.1. Transversion I Map𝒱Ii

Result 4 (Table5).
Forl=1,…,20(26)c(𝒱I1(l))=c(𝒱I3(l)),mc(𝒱I1(l))=mc(𝒱I3(l)),sc(𝒱I1(l))=sc(𝒱I3(l)),c3(𝒱I1(l))=c3(𝒱I3(l)),msc3(𝒱I1(l))=msc3(𝒱I3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱I1(l)) and c(𝒱I3(l)) are different for l=1,…,9 (not shown). No trinucleotide code is circular after a certain number of transversions I 𝒱Ii in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=10,…,20(27)c(𝒱I1(l))=c(𝒱I3(l))=0
and for l=18,19,20(28)c(𝒱I2(l))=0.
The transversion I 𝒱Ii generates a maximum number of trinucleotide circular codes C for
(29)max{c(𝒱Ii(l)),i=1,2,3,l=1,…,20}=c(𝒱I2(9))=24310
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(30)max{msc3(𝒱Ii(l)),i=1,2,3,l=1,…,20}=msc3(𝒱I2(8))=70.
A remarkable code property only found with transversion I 𝒱I2 is, for l=1,…,20,
(31)c(𝒱I2(l))=c3(𝒱I2(l)),
and furthermore, after a detailed computer analysis, the lists of trinucleotide circular codes C and C3 associated with c(𝒱I2(l)) and c3(𝒱I2(l)), respectively, are identical for l=1,…,17.Table 5
Transversion I map𝒱Ii(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱Ii(l)) of circular codes C, number mc(𝒱Ii(l)) of maximum circular codes MC, number sc(𝒱Ii(l)) of self-complementary circular codes SC, number c3(𝒱Ii(l)) of circular codes C3, and number msc3(𝒱Ii(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1
(
l
) or 𝒱I3(l)
𝒱
I
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
8
0
0
8
0
16
0
0
16
0
2
31
4
5
30
4
122
8
12
122
8
3
76
0
0
71
0
590
0
0
590
0
4
130
6
8
114
6
2030
28
56
2030
28
5
160
0
0
125
0
5278
0
0
5278
0
6
142
4
5
92
4
10738
56
132
10738
56
7
87
0
0
42
0
17446
0
0
17446
0
8
32
1
1
10
1
22880
70
174
22880
70
9
5
0
0
1
0
24310
0
0
24310
0
10
0
0
0
0
0
20878
56
132
20878
56
11
0
0
0
0
0
14378
0
0
14378
0
12
0
0
0
0
0
7826
28
56
7826
28
13
0
0
0
0
0
3290
0
0
3290
0
14
0
0
0
0
0
1030
8
12
1030
8
15
0
0
0
0
0
226
0
0
226
0
16
0
0
0
0
0
31
1
1
31
1
17
0
0
0
0
0
2
0
0
2
0
{
18
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


### 3.2.2. Transversion I Map𝒱Ii,j

Result 5 (Table6).
Forl=1,…,20(32)c(𝒱I1,2(l))=c(𝒱I2,3(l)),mc(𝒱I1,2(l))=mc(𝒱I2,3(l)),sc(𝒱I1,2(l))=sc(𝒱I2,3(l)),c3(𝒱I1,2(l))=c3(𝒱I2,3(l)),msc3(𝒱I1,2(l))=msc3(𝒱I2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱I1,2(l)) and c(𝒱I2,3(l)) are different for l=1,…,12 (not shown). No trinucleotide code is circular after a certain number of transversions I 𝒱Ii,j in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=13,…,20(33)c(𝒱I1,2(l))=c(𝒱I2,3(l))=0
and for l=19,20(34)c(𝒱I1,3(l))=0.
The transversion I 𝒱Ii,j generates a maximum number of trinucleotide circular codes C for
(35)max{c(𝒱Ii,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒱I1,2(6))=c(𝒱I2,3(6))=630
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(36)max{msc3(𝒱Ii,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒱I1,2(4))=msc3(𝒱I2,3(4))=6.
The numbers sc(𝒱I1,3(l)) of self-complementary circular codes SC have a particular growth function
(37)sc(𝒱I1,3(l))=1forl=12,14,16,18,sc(𝒱I1,3(l))=0forl=8,10.
The numbers c3(𝒱I1,3(l)) of circular codes C3 have a particular growth function
(38)c3(𝒱I1,3(l))=1forl=16,18,c3(𝒱I1,3(17))=2,c3(𝒱I1,3(l))=0forl=7,…,15.Table 6
Transversion I map𝒱Ii,j(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱Ii,j(l)) of circular codes C, number mc(𝒱Ii,j(l)) of maximum circular codes MC, number sc(𝒱Ii,j(l)) of self-complementary circular codes SC, number c3(𝒱Ii,j(l)) of circular codes C3, and number msc3(𝒱Ii,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1,2
(
l
) or 𝒱I2,3(l)
𝒱
I
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
10
2
0
9
0
6
2
0
4
0
2
48
5
5
38
4
15
3
5
7
3
3
147
8
0
102
0
20
4
0
8
0
4
319
10
8
191
6
15
3
5
7
3
5
514
12
0
265
0
8
4
0
4
0
6
630
10
5
282
4
11
3
1
1
1
7
595
8
0
240
0
22
4
0
0
0
8
435
5
1
162
1
32
4
0
0
0
9
245
2
0
88
0
38
2
0
0
0
10
103
1
0
34
0
38
2
0
0
0
11
29
0
0
8
0
34
0
0
0
0
12
4
0
0
1
0
39
0
1
0
0
13
0
0
0
0
0
56
0
0
0
0
14
0
0
0
0
0
65
0
1
0
0
15
0
0
0
0
0
52
0
0
0
0
16
0
0
0
0
0
27
0
1
1
0
17
0
0
0
0
0
8
0
0
2
0
18
0
0
0
0
0
1
0
1
1
0
{
19,20
}
0
0
0
0
0
0
0
0
0
0


### 3.2.3. Transversion I Map𝒱I1,2,3

Result 6 (Table7).
The transversion I𝒱I1,2,3 always generates trinucleotide circular codes. Indeed, for l=1,…,20(39)c(𝒱I1,2,3(l))>0.
The lists of trinucleotide circular codes C associated with c(𝒱I1,2,3(l)) and c(𝒱I1,2,3(20-l)) are different for l=1,…,9 (not shown). The transversion I 𝒱I1,2,3 generates a maximum number of trinucleotide circular codes C for
(40)max{c(𝒱I1,2,3(l)),l=1,…,20}=c(𝒱I1,2,3(10))=66
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(41)max{msc3(𝒱I1,2,3(l)),l=1,…,20}=msc3(𝒱I1,2,3(4))=msc3(𝒱I1,2,3(16))=9.Table 7
Transversion I map𝒱I1,2,3(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱I1,2,3(l)) of circular codes C, number mc(𝒱I1,2,3(l)) of maximum circular codes MC, number sc(𝒱I1,2,3(l)) of self-complementary circular codes SC, number c3(𝒱I1,2,3(l)) of circular codes C3, and number msc3(𝒱I1,2,3(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1,2
,
3
(
l
)
c
mc
sc
c
3
ms
c
3
{
1,19
}
6
0
0
6
0
{
2,18
}
17
5
7
17
5
{
3,17
}
32
0
0
32
0
{
4,16
}
47
10
17
45
9
{
5,15
}
54
0
0
46
0
{
6,14
}
51
12
19
31
7
{
7,13
}
48
0
0
12
0
{
8,12
}
53
13
11
2
2
{
9,11
}
62
0
0
0
0
10
66
14
6
0
0
20
1
1
1
1
1


## 3.2.1. Transversion I Map𝒱Ii

Result 4 (Table5).
Forl=1,…,20(26)c(𝒱I1(l))=c(𝒱I3(l)),mc(𝒱I1(l))=mc(𝒱I3(l)),sc(𝒱I1(l))=sc(𝒱I3(l)),c3(𝒱I1(l))=c3(𝒱I3(l)),msc3(𝒱I1(l))=msc3(𝒱I3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱I1(l)) and c(𝒱I3(l)) are different for l=1,…,9 (not shown). No trinucleotide code is circular after a certain number of transversions I 𝒱Ii in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=10,…,20(27)c(𝒱I1(l))=c(𝒱I3(l))=0
and for l=18,19,20(28)c(𝒱I2(l))=0.
The transversion I 𝒱Ii generates a maximum number of trinucleotide circular codes C for
(29)max{c(𝒱Ii(l)),i=1,2,3,l=1,…,20}=c(𝒱I2(9))=24310
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(30)max{msc3(𝒱Ii(l)),i=1,2,3,l=1,…,20}=msc3(𝒱I2(8))=70.
A remarkable code property only found with transversion I 𝒱I2 is, for l=1,…,20,
(31)c(𝒱I2(l))=c3(𝒱I2(l)),
and furthermore, after a detailed computer analysis, the lists of trinucleotide circular codes C and C3 associated with c(𝒱I2(l)) and c3(𝒱I2(l)), respectively, are identical for l=1,…,17.Table 5
Transversion I map𝒱Ii(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱Ii(l)) of circular codes C, number mc(𝒱Ii(l)) of maximum circular codes MC, number sc(𝒱Ii(l)) of self-complementary circular codes SC, number c3(𝒱Ii(l)) of circular codes C3, and number msc3(𝒱Ii(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1
(
l
) or 𝒱I3(l)
𝒱
I
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
8
0
0
8
0
16
0
0
16
0
2
31
4
5
30
4
122
8
12
122
8
3
76
0
0
71
0
590
0
0
590
0
4
130
6
8
114
6
2030
28
56
2030
28
5
160
0
0
125
0
5278
0
0
5278
0
6
142
4
5
92
4
10738
56
132
10738
56
7
87
0
0
42
0
17446
0
0
17446
0
8
32
1
1
10
1
22880
70
174
22880
70
9
5
0
0
1
0
24310
0
0
24310
0
10
0
0
0
0
0
20878
56
132
20878
56
11
0
0
0
0
0
14378
0
0
14378
0
12
0
0
0
0
0
7826
28
56
7826
28
13
0
0
0
0
0
3290
0
0
3290
0
14
0
0
0
0
0
1030
8
12
1030
8
15
0
0
0
0
0
226
0
0
226
0
16
0
0
0
0
0
31
1
1
31
1
17
0
0
0
0
0
2
0
0
2
0
{
18
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


## 3.2.2. Transversion I Map𝒱Ii,j

Result 5 (Table6).
Forl=1,…,20(32)c(𝒱I1,2(l))=c(𝒱I2,3(l)),mc(𝒱I1,2(l))=mc(𝒱I2,3(l)),sc(𝒱I1,2(l))=sc(𝒱I2,3(l)),c3(𝒱I1,2(l))=c3(𝒱I2,3(l)),msc3(𝒱I1,2(l))=msc3(𝒱I2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱I1,2(l)) and c(𝒱I2,3(l)) are different for l=1,…,12 (not shown). No trinucleotide code is circular after a certain number of transversions I 𝒱Ii,j in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=13,…,20(33)c(𝒱I1,2(l))=c(𝒱I2,3(l))=0
and for l=19,20(34)c(𝒱I1,3(l))=0.
The transversion I 𝒱Ii,j generates a maximum number of trinucleotide circular codes C for
(35)max{c(𝒱Ii,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒱I1,2(6))=c(𝒱I2,3(6))=630
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(36)max{msc3(𝒱Ii,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒱I1,2(4))=msc3(𝒱I2,3(4))=6.
The numbers sc(𝒱I1,3(l)) of self-complementary circular codes SC have a particular growth function
(37)sc(𝒱I1,3(l))=1forl=12,14,16,18,sc(𝒱I1,3(l))=0forl=8,10.
The numbers c3(𝒱I1,3(l)) of circular codes C3 have a particular growth function
(38)c3(𝒱I1,3(l))=1forl=16,18,c3(𝒱I1,3(17))=2,c3(𝒱I1,3(l))=0forl=7,…,15.Table 6
Transversion I map𝒱Ii,j(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱Ii,j(l)) of circular codes C, number mc(𝒱Ii,j(l)) of maximum circular codes MC, number sc(𝒱Ii,j(l)) of self-complementary circular codes SC, number c3(𝒱Ii,j(l)) of circular codes C3, and number msc3(𝒱Ii,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1,2
(
l
) or 𝒱I2,3(l)
𝒱
I
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
10
2
0
9
0
6
2
0
4
0
2
48
5
5
38
4
15
3
5
7
3
3
147
8
0
102
0
20
4
0
8
0
4
319
10
8
191
6
15
3
5
7
3
5
514
12
0
265
0
8
4
0
4
0
6
630
10
5
282
4
11
3
1
1
1
7
595
8
0
240
0
22
4
0
0
0
8
435
5
1
162
1
32
4
0
0
0
9
245
2
0
88
0
38
2
0
0
0
10
103
1
0
34
0
38
2
0
0
0
11
29
0
0
8
0
34
0
0
0
0
12
4
0
0
1
0
39
0
1
0
0
13
0
0
0
0
0
56
0
0
0
0
14
0
0
0
0
0
65
0
1
0
0
15
0
0
0
0
0
52
0
0
0
0
16
0
0
0
0
0
27
0
1
1
0
17
0
0
0
0
0
8
0
0
2
0
18
0
0
0
0
0
1
0
1
1
0
{
19,20
}
0
0
0
0
0
0
0
0
0
0


## 3.2.3. Transversion I Map𝒱I1,2,3

Result 6 (Table7).
The transversion I𝒱I1,2,3 always generates trinucleotide circular codes. Indeed, for l=1,…,20(39)c(𝒱I1,2,3(l))>0.
The lists of trinucleotide circular codes C associated with c(𝒱I1,2,3(l)) and c(𝒱I1,2,3(20-l)) are different for l=1,…,9 (not shown). The transversion I 𝒱I1,2,3 generates a maximum number of trinucleotide circular codes C for
(40)max{c(𝒱I1,2,3(l)),l=1,…,20}=c(𝒱I1,2,3(10))=66
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(41)max{msc3(𝒱I1,2,3(l)),l=1,…,20}=msc3(𝒱I1,2,3(4))=msc3(𝒱I1,2,3(16))=9.Table 7
Transversion I map𝒱I1,2,3(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱I1,2,3(l)) of circular codes C, number mc(𝒱I1,2,3(l)) of maximum circular codes MC, number sc(𝒱I1,2,3(l)) of self-complementary circular codes SC, number c3(𝒱I1,2,3(l)) of circular codes C3, and number msc3(𝒱I1,2,3(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
I
1,2
,
3
(
l
)
c
mc
sc
c
3
ms
c
3
{
1,19
}
6
0
0
6
0
{
2,18
}
17
5
7
17
5
{
3,17
}
32
0
0
32
0
{
4,16
}
47
10
17
45
9
{
5,15
}
54
0
0
46
0
{
6,14
}
51
12
19
31
7
{
7,13
}
48
0
0
12
0
{
8,12
}
53
13
11
2
2
{
9,11
}
62
0
0
0
0
10
66
14
6
0
0
20
1
1
1
1
1


## 3.3. Transversion II Map

### 3.3.1. Transversion II Map𝒱IIi

Result 7 (Table8).
Forl=1,…,20(42)c(𝒱II1(l))=c(𝒱II3(l)),mc(𝒱II1(l))=mc(𝒱II3(l)),sc(𝒱II1(l))=sc(𝒱II3(l)),c3(𝒱II1(l))=c3(𝒱II3(l)),msc3(𝒱II1(l))=msc3(𝒱II3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱II1(l)) and c(𝒱II3(l)) are different for l=1,…,8 (not shown). No trinucleotide code is circular after a certain number of transversions II 𝒱IIi in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=9,…,20(43)c(𝒱II1(l))=c(𝒱II3(l))=0
and for l=12,…,20(44)c(𝒱II2(l))=0.
The transversion II 𝒱IIi generates a maximum number of trinucleotide circular codes C for
(45)max{c(𝒱IIi(l)),i=1,2,3,l=1,…,20}=c(𝒱II2(5))=176
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(46)max{msc3(𝒱IIi(l)),i=1,2,3,l=1,…,20}=msc3(𝒱II2(4))=6.Table 8
Transversion II map𝒱IIi(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱IIi(l)) of circular codes C, number mc(𝒱IIi(l)) of maximum circular codes MC, number sc(𝒱IIi(l)) of self-complementary circular codes SC, number c3(𝒱IIi(l)) of circular codes C3, and number msc3(𝒱IIi(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
II
1
(
l
) or 𝒱II3(l)
𝒱
II
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
6
0
0
6
0
8
0
0
8
0
2
17
3
4
17
3
30
4
8
28
4
3
30
0
0
29
0
74
0
0
58
0
4
35
3
4
31
3
132
6
18
82
6
5
28
0
0
20
0
176
0
0
76
0
6
16
1
1
7
1
170
4
8
44
4
7
6
0
0
1
0
120
0
0
12
0
8
1
0
0
0
0
61
1
1
1
1
9
0
0
0
0
0
26
0
0
0
0
10
0
0
0
0
0
10
0
0
0
0
11
0
0
0
0
0
2
0
0
0
0
{
12
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


### 3.3.2. Transversion II Map𝒱IIi,j

Result 8 (Table9).
Forl=1,…,20(47)c(𝒱II1,2(l))=c(𝒱II2,3(l)),mc(𝒱II1,2(l))=mc(𝒱II2,3(l)),sc(𝒱II1,2(l))=sc(𝒱II2,3(l)),c3(𝒱II1,2(l))=c3(𝒱II2,3(l)),msc3(𝒱II1,2(l))=msc3(𝒱II2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱II1,2(l)) and c(𝒱II2,3(l)) are different for l=1,…,5,13,14 (not shown). The distribution of trinucleotide codes which are not circular under transversions II 𝒱IIi,j in the trinucleotides of the common trinucleotide circular code X0 is very unusual. Indeed, for l=6,…,12,15,…,20(48)c(𝒱II1,2(l))=c(𝒱II2,3(l))=0
and for l=15,…,20(49)c(𝒱II1,3(l))=0.
The transversion II 𝒱IIi,j generates a maximum number of trinucleotide circular codes C for
(50)max{c(𝒱IIi,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒱II1,3(6))=662
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(51)max{msc3(𝒱IIi,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒱II1,3(4))=6.
The numbers c(𝒱II1,2(l))=c(𝒱II2,3(l)) of circular codes C have a particular growth function
(52)c(𝒱II1,2(l))=c(𝒱II2,3(l))=1forl=13,14,c(𝒱II1,2(l))=c(𝒱II2,3(l))=0forl=6,…,12.Table 9
Transversion II map𝒱IIi,j(l) in l trinucleotide of the common trinucleotide circular code X0. Number c(𝒱IIi,j(l)) of circular codes C, number mc(𝒱IIi,j(l)) of maximum circular codes MC, number sc(𝒱IIi,j(l)) of self-complementary circular codes SC, number c3(𝒱IIi,j(l)) of circular codes C3, and number msc3(𝒱IIi,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
II
1,2
(
l
) or 𝒱II2,3(l)
𝒱
II
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
5
1
0
5
0
8
0
0
8
0
2
11
2
3
10
2
40
4
8
30
4
3
13
2
0
11
0
132
0
0
72
0
4
8
1
1
7
1
312
6
22
117
6
5
2
1
0
2
0
532
0
0
130
0
6
0
0
0
0
0
662
4
24
92
4
7
0
0
0
0
0
592
0
0
34
0
8
0
0
0
0
0
389
1
9
4
1
9
0
0
0
0
0
188
0
0
0
0
10
0
0
0
0
0
87
0
1
2
0
11
0
0
0
0
0
54
0
0
6
0
12
0
0
0
0
0
34
0
2
7
0
13
1
0
0
0
0
14
0
0
4
0
14
1
0
0
0
0
3
0
1
3
0
{
15
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


### 3.3.3. Transversion II Map𝒱II1,2,3

Proposition 37.
Forl=1,…,19(53)c(𝒱II1,2,3(l))=0
and obviously, by letter invariance, c(𝒱II1,2,3(20))=1 as in Tables 4 and 7.Proof.
The common trinucleotide circular codeX0 can be partitioned according to the maps 𝒱II1,2,3, 𝒫, and 𝒫2 as shown in Table 10.
Let a partitionPi={x,x′}, i∈{1,…,10}, composed of two trinucleotides x,x′∈X0. For l=1, any transversion II of a trinucleotide x∈Pi generates a trinucleotide y which is a permuted trinucleotide of the other trinucleotide x′∈Pi. So, any transversion II of a trinucleotide x∈X0 leads to a trinucleotide code which is not circular. For 2≤l≤19, the proof needs a computer analysis of the necklace for the nontrivial cases when two transversions II occur with two trinucleotides in the same partitions.Table 10
Partition ofX0
X
0
𝒱
II
1,2
,
3
(
X
0
)
=
X
2
𝒫
(
X
2
)
=
X
0
′
𝒱
II
1,2
,
3
(
X
0
′
)
=
X
1
𝒫
2
(
X
1
)
=
X
0
P
1
=
{
A
A
C
,
A
C
C
}
A
C
C
C
A
A
A
A
C
C
C
A
A
C
C
P
2
=
{
A
A
T
,
G
C
C
}
G
C
C
T
A
A
A
A
T
C
C
G
G
C
C
P
3
=
{
A
T
C
,
G
A
C
}
A
T
C
C
G
A
G
A
C
T
C
A
A
T
C
P
4
=
{
A
T
T
,
G
G
C
}
A
T
T
C
G
G
G
G
C
T
T
A
A
T
T
P
5
=
{
C
A
G
,
T
A
C
}
T
A
C
G
C
A
C
A
G
A
C
T
T
A
C
P
6
=
{
C
T
C
,
G
A
A
}
C
T
C
A
G
A
G
A
A
T
C
C
C
T
C
P
7
=
{
C
T
G
,
G
T
A
}
C
T
G
A
G
T
G
T
A
T
G
C
C
T
G
P
8
=
{
G
A
G
,
T
T
C
}
T
T
C
G
G
A
G
A
G
T
C
T
T
T
C
P
9
=
{
G
A
T
,
G
T
C
}
G
T
C
T
G
A
G
A
T
T
C
G
G
T
C
P
10
=
{
G
G
T
,
G
T
T
}
G
T
T
T
G
G
G
G
T
T
T
G
G
T
TRemark 38.
Very surprisingly, for the three maps of transition, transversions I and II,𝒯i(l), 𝒱Ii(l), and 𝒱IIi(l), i∈{1,2,3}, 𝒯i,j(l), 𝒱Ii,j(l), and 𝒱IIi,j(l), i,j∈{1,2,3} with i<j and 𝒯1,2,3(l) (not for 𝒱I1,2,3(l) and 𝒱II1,2,3(l)), the numbers msc3 of self-complementary maximum circular codes MSC3 for the first even values of l follow a series of binomial coefficients. For 𝒯i(l), 𝒱Ii(l), and 𝒱IIi(l), i∈{1,2,3}, 𝒯1,2(l), 𝒯2,3(l), and 𝒱IIi,j(l), i,j∈{1,2,3} with i<j, the numbers mc of maximum circular codes MC for the first even values of l follow a series of binomial coefficients. For 𝒯1,2,3(l), the numbers c3 of circular codes C3 for the values l and (20-l) with l=1,…,8 follow a series of binomial coefficients. These binomial properties with some numbers of circular codes for the three maps of transition, transversions I and II have no combinatorial explanation so far.


## 3.3.1. Transversion II Map𝒱IIi

Result 7 (Table8).
Forl=1,…,20(42)c(𝒱II1(l))=c(𝒱II3(l)),mc(𝒱II1(l))=mc(𝒱II3(l)),sc(𝒱II1(l))=sc(𝒱II3(l)),c3(𝒱II1(l))=c3(𝒱II3(l)),msc3(𝒱II1(l))=msc3(𝒱II3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱II1(l)) and c(𝒱II3(l)) are different for l=1,…,8 (not shown). No trinucleotide code is circular after a certain number of transversions II 𝒱IIi in the trinucleotides of the common trinucleotide circular code X0. Precisely, for l=9,…,20(43)c(𝒱II1(l))=c(𝒱II3(l))=0
and for l=12,…,20(44)c(𝒱II2(l))=0.
The transversion II 𝒱IIi generates a maximum number of trinucleotide circular codes C for
(45)max{c(𝒱IIi(l)),i=1,2,3,l=1,…,20}=c(𝒱II2(5))=176
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(46)max{msc3(𝒱IIi(l)),i=1,2,3,l=1,…,20}=msc3(𝒱II2(4))=6.Table 8
Transversion II map𝒱IIi(l) in l trinucleotides of the common trinucleotide circular code X0. Number c(𝒱IIi(l)) of circular codes C, number mc(𝒱IIi(l)) of maximum circular codes MC, number sc(𝒱IIi(l)) of self-complementary circular codes SC, number c3(𝒱IIi(l)) of circular codes C3, and number msc3(𝒱IIi(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
II
1
(
l
) or 𝒱II3(l)
𝒱
II
2
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
6
0
0
6
0
8
0
0
8
0
2
17
3
4
17
3
30
4
8
28
4
3
30
0
0
29
0
74
0
0
58
0
4
35
3
4
31
3
132
6
18
82
6
5
28
0
0
20
0
176
0
0
76
0
6
16
1
1
7
1
170
4
8
44
4
7
6
0
0
1
0
120
0
0
12
0
8
1
0
0
0
0
61
1
1
1
1
9
0
0
0
0
0
26
0
0
0
0
10
0
0
0
0
0
10
0
0
0
0
11
0
0
0
0
0
2
0
0
0
0
{
12
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


## 3.3.2. Transversion II Map𝒱IIi,j

Result 8 (Table9).
Forl=1,…,20(47)c(𝒱II1,2(l))=c(𝒱II2,3(l)),mc(𝒱II1,2(l))=mc(𝒱II2,3(l)),sc(𝒱II1,2(l))=sc(𝒱II2,3(l)),c3(𝒱II1,2(l))=c3(𝒱II2,3(l)),msc3(𝒱II1,2(l))=msc3(𝒱II2,3(l)).
The lists of trinucleotide circular codes C associated with c(𝒱II1,2(l)) and c(𝒱II2,3(l)) are different for l=1,…,5,13,14 (not shown). The distribution of trinucleotide codes which are not circular under transversions II 𝒱IIi,j in the trinucleotides of the common trinucleotide circular code X0 is very unusual. Indeed, for l=6,…,12,15,…,20(48)c(𝒱II1,2(l))=c(𝒱II2,3(l))=0
and for l=15,…,20(49)c(𝒱II1,3(l))=0.
The transversion II 𝒱IIi,j generates a maximum number of trinucleotide circular codes C for
(50)max{c(𝒱IIi,j(l)),i,j=1,2,3,i<j,l=1,…,20}=c(𝒱II1,3(6))=662
and a maximum number of C3 self-complementary maximum circular codes MSC3 for
(51)max{msc3(𝒱IIi,j(l)),i,j=1,2,3,i<j,l=1,…,20}=msc3(𝒱II1,3(4))=6.
The numbers c(𝒱II1,2(l))=c(𝒱II2,3(l)) of circular codes C have a particular growth function
(52)c(𝒱II1,2(l))=c(𝒱II2,3(l))=1forl=13,14,c(𝒱II1,2(l))=c(𝒱II2,3(l))=0forl=6,…,12.Table 9
Transversion II map𝒱IIi,j(l) in l trinucleotide of the common trinucleotide circular code X0. Number c(𝒱IIi,j(l)) of circular codes C, number mc(𝒱IIi,j(l)) of maximum circular codes MC, number sc(𝒱IIi,j(l)) of self-complementary circular codes SC, number c3(𝒱IIi,j(l)) of circular codes C3, and number msc3(𝒱IIi,j(l)) of C3 self-complementary maximum circular codes MSC3.
l
𝒱
II
1,2
(
l
) or 𝒱II2,3(l)
𝒱
II
1,3
(
l
)
c
mc
sc
c
3
ms
c
3
c
mc
sc
c
3
ms
c
3
1
5
1
0
5
0
8
0
0
8
0
2
11
2
3
10
2
40
4
8
30
4
3
13
2
0
11
0
132
0
0
72
0
4
8
1
1
7
1
312
6
22
117
6
5
2
1
0
2
0
532
0
0
130
0
6
0
0
0
0
0
662
4
24
92
4
7
0
0
0
0
0
592
0
0
34
0
8
0
0
0
0
0
389
1
9
4
1
9
0
0
0
0
0
188
0
0
0
0
10
0
0
0
0
0
87
0
1
2
0
11
0
0
0
0
0
54
0
0
6
0
12
0
0
0
0
0
34
0
2
7
0
13
1
0
0
0
0
14
0
0
4
0
14
1
0
0
0
0
3
0
1
3
0
{
15
,
…
,
20
}
0
0
0
0
0
0
0
0
0
0


## 3.3.3. Transversion II Map𝒱II1,2,3

Proposition 37.
Forl=1,…,19(53)c(𝒱II1,2,3(l))=0
and obviously, by letter invariance, c(𝒱II1,2,3(20))=1 as in Tables 4 and 7.Proof.
The common trinucleotide circular codeX0 can be partitioned according to the maps 𝒱II1,2,3, 𝒫, and 𝒫2 as shown in Table 10.
Let a partitionPi={x,x′}, i∈{1,…,10}, composed of two trinucleotides x,x′∈X0. For l=1, any transversion II of a trinucleotide x∈Pi generates a trinucleotide y which is a permuted trinucleotide of the other trinucleotide x′∈Pi. So, any transversion II of a trinucleotide x∈X0 leads to a trinucleotide code which is not circular. For 2≤l≤19, the proof needs a computer analysis of the necklace for the nontrivial cases when two transversions II occur with two trinucleotides in the same partitions.Table 10
Partition ofX0
X
0
𝒱
II
1,2
,
3
(
X
0
)
=
X
2
𝒫
(
X
2
)
=
X
0
′
𝒱
II
1,2
,
3
(
X
0
′
)
=
X
1
𝒫
2
(
X
1
)
=
X
0
P
1
=
{
A
A
C
,
A
C
C
}
A
C
C
C
A
A
A
A
C
C
C
A
A
C
C
P
2
=
{
A
A
T
,
G
C
C
}
G
C
C
T
A
A
A
A
T
C
C
G
G
C
C
P
3
=
{
A
T
C
,
G
A
C
}
A
T
C
C
G
A
G
A
C
T
C
A
A
T
C
P
4
=
{
A
T
T
,
G
G
C
}
A
T
T
C
G
G
G
G
C
T
T
A
A
T
T
P
5
=
{
C
A
G
,
T
A
C
}
T
A
C
G
C
A
C
A
G
A
C
T
T
A
C
P
6
=
{
C
T
C
,
G
A
A
}
C
T
C
A
G
A
G
A
A
T
C
C
C
T
C
P
7
=
{
C
T
G
,
G
T
A
}
C
T
G
A
G
T
G
T
A
T
G
C
C
T
G
P
8
=
{
G
A
G
,
T
T
C
}
T
T
C
G
G
A
G
A
G
T
C
T
T
T
C
P
9
=
{
G
A
T
,
G
T
C
}
G
T
C
T
G
A
G
A
T
T
C
G
G
T
C
P
10
=
{
G
G
T
,
G
T
T
}
G
T
T
T
G
G
G
G
T
T
T
G
G
T
TRemark 38.
Very surprisingly, for the three maps of transition, transversions I and II,𝒯i(l), 𝒱Ii(l), and 𝒱IIi(l), i∈{1,2,3}, 𝒯i,j(l), 𝒱Ii,j(l), and 𝒱IIi,j(l), i,j∈{1,2,3} with i<j and 𝒯1,2,3(l) (not for 𝒱I1,2,3(l) and 𝒱II1,2,3(l)), the numbers msc3 of self-complementary maximum circular codes MSC3 for the first even values of l follow a series of binomial coefficients. For 𝒯i(l), 𝒱Ii(l), and 𝒱IIi(l), i∈{1,2,3}, 𝒯1,2(l), 𝒯2,3(l), and 𝒱IIi,j(l), i,j∈{1,2,3} with i<j, the numbers mc of maximum circular codes MC for the first even values of l follow a series of binomial coefficients. For 𝒯1,2,3(l), the numbers c3 of circular codes C3 for the values l and (20-l) with l=1,…,8 follow a series of binomial coefficients. These binomial properties with some numbers of circular codes for the three maps of transition, transversions I and II have no combinatorial explanation so far.


## 4. Conclusion

A comprehensive computer analysis of transition and transversions I and II in theC3 self-complementary maximum circular code X0 shows some new results; in particular (i) transversion I 𝒱I2(l) on the 2nd position of any subset of trinucleotides of X0 generates trinucleotide circular codes which are always C3 and (ii) transversion II 𝒱II1,2,3 on the three positions of any subset of trinucleotides of X0 yields no trinucleotide circular codes. In addition to the classical self-complementary (Definition 20) partition of X0 known since 1996, a new partition of X0 based on the transversion II map 𝒱II1,2,3 (Definition 33) and the circular permutation maps 𝒫 and 𝒫2 (Definition 18) is also identified here. These results here extend our theory of circular code in genes to its evolution under transition and transversion.



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*Source: 795418-2013-05-29.xml*