§2–1. Data availability statements

Regarding the availability statement in this paper, the data used in this manuscript were built up and prepared for the purpose of elucidating the methodology, and were not collected from actual experiments or real-world sources. They were strictly created within the text to facilitate readers’ comprehension. The data, represented in the form of equations, are exemplified within fictional scenarios and do not have a corresponding minimal data set underlying the results outside of the paper. While there are sections in which figures and methodologies are cited from other papers, this is exclusively confined to the aforementioned context (see S1 File for a proper citation).

§2–2. Preliminary explanation of categories and PNs

§2–3. Illustrations of hybrid models based on category theory and EPN

At this stage, to help the readers’ understanding, we illustrate the extended PN (EPN) using category theory [Fig 1(A)]. From the viewpoint of the standard EPN, the hybrid model that we present in this article may be regarded as one for which the standard EPN is composed using category theory as a material with some options. The main characteristics of our model may be summarized in three points: 1) places correspond to objects, 2) arcs correspond to morphisms, and 3) tokens are defined as those that can act as operators. For simplicity, throughout the article, we mainly consider a ‘state machine’ where any transition has only one input place and one output place [7−45]. However, several parts of our model are not ‘state machines’ because a description of more complex systems is needed. Although, as mentioned above, the PN is a specific form/part of a category, by hybridizing specific parts of the PN and that of the category in a heterogeneous way, we are able to give some rules to those that are lacking a corresponding standard PN: specifically, the compositionality (operationality) between 1) two morphisms, and 2) a token in a place and a morphism in the same place (to be described in detail later).

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Fig 1. Illustrations of extended Petri nets (EPNs).

(a). In the realm of formal system modeling, integrating category theory concepts within the framework of PNs presents an intriguing avenue for mathematical abstraction. The fundamental elements of category theory are objects A, B, and C and mappings f: A → B, g: B → C, and h: A → C, denoting morphisms between the objects. This abstract structure has a correspondence in PNs: Places: A (Place representing Object A). B (Place representing Object B). C (Place representing Object C). Transitions: f (Transition representing Arrow f). g (Transition representing Arrow g). h (Transition representing Arrow h). Arcs: Input arc from A to f (Signifying the starting point of Arrow f being Object A). Input arc from B to f (Denoting the endpoint of Arrow f being Object B). Input arc from B to g (Indicating the starting point of Arrow g being Object B). Input arc from C to g (Representing the endpoint of Arrow g being Object C). Input arc from A to h (Symbolizing the starting point of Arrow h being Object A). Input arc from C to h (Marking the endpoint of Arrow h being Object C). This PN representation elegantly captures the essence of category theory constructs, whereby transitions correspond to arrows and places represent distinct objects. The interconnected arcs delineate the relationships between these objects and morphisms, forming a cohesive and visually intuitive model. (b). A necessary relationship for the definition of category is ‘{potential source object p_k (place p_k)};f_(k→l) ⊆ potential target object p_l (place p_l)’. Hence, provided that ‘D_(k)j = [E₀|C₁A₂E₃T₄||E₅E₆…]’ is a token that is a potential source object in place p_k, then R_(l)j is the image of it under morphism f_(k→l) (that is, R_(l)j = {D_(k)j};f_(k→l)), ‘R_(l)j = [E₀|C₁A₂E₃U₄||E₅E₆…] ∈ place p_l. The tokens belong to place p_l; they do not belong to ‘{place p_k};f_(k→l)’ as R_(l)a = [A₀|C₁A₂E₃U₄||E₅E₆…] ∉ {p_k};f_(k→l)’, and ‘R_(l)b = [G₀|C₁A₂E₃U₄||E₅E₆…] ∉{p_k};f_(k→l)’. That is, the initial coordinate of any DNA/RNA sequence can only be E₀ because, if E₀ were only a symbol, then no partial function for the morphism can occur, and the (potential) objects may satisfy the postulates of category theory, specifically, Eq (4).

https://doi.org/10.1371/journal.pone.0302710.g001

§2-3-2. Firing rules: For a morphism (transition), there are two types of firing rules labeled I and II.

Firing rule I: Firing defined between different places (via a trans-place morphism). A morphism f_(k→l) (transition t_(k→l)) with ‘k≠l’ is defined between input and output places where an arc from the pre-place p_k to the post-place p_l is integrated with transition ‘|’ as ‘―|→’. Then, the morphism f_(k→l) ordering a token ‘a_(k)i’ (or c·a_(k)i; in the following description; this case may be omitted for simplicity without loss of generality) (a_(k)i∈A_k), denoted by ‘{a_(k)i};f_(k→l):’ or ‘f_(k→l)°a_(k)i:’ is fireable when a token a_(k)i is included in place p_k. We call f_(k→l) a ‘trans-place morphism’. Through a firing of a morphism (transition) f_(k→l) ordering ‘a_(k)i’, token a_(k)I is removed from the input place p_k, and a_i(l)i (= {a_(k)i};f_(k→l), the images of a_(k)i by f_(k→l)) (f_(k→l)∈A_l) produced in the output place p_l, determined by the ‘morphism f_(k→l): a_(k)i→a_(l)i. Examples are illustrated in (Fig 2A and 2B). The firing rule for this is stipulated as follows; subject to ‘k≠l’,

M_w+1(p_l) = M_w(p_l) + {a_(k)i};f_(k→l): p_l∈(output place of the transition) ˄ p_l∉(input place of the transition). P_k∈(input place of the transition) ˄ p_k∉(output place of the transition),

M_w+1(p_k) = M_w(p_k)–a_(k)i: p_k∈(input place of the transition) ˄ p_k∉(output place of the transition), and (3) where ‘−’ (minus) means the removal of tokens, and ‘+’ (plus) the inclusion of tokens. If the token is ‘c·a_(k)i’, only an exchange between a_(k)i and c·a_(k)i enables the formula to be true because of the previous definition for tokens.

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Fig 2. An example of a state in the standard Petri net (PN).

(a). A state that exemplifies markings is drawn. All tokens are elements (e.g., a_(k)i) of monoid or group A_k (where a_(k)i∈A_k) that are defined at place p_k (k = 1,2,…m). Different from ordinal object of category, each place does not always contain all elements of A_k. Moreover, duplication of the same elements such as ‘a_(k)i, a_(k)i, a_(k)i’ in the same place p_k is permissible. Arcs represent morphisms f_(k→l) that fire when there is at least one token ‘a_(k)i’ on which morphism ‘f_(k→l)’ orders ‘f_(k→l):{a_(k)i}:’ and is included in input place p_k. After firing, an image of a_(k)i is outputted in an input place as a token a_(l)i (= a_(k)i;f_(k→l)). We stipulate that any arc (morphism) has only one morphism (there are no multiple morphisms) that acts on one token (this is considered a ‘state machine’ in conventional PN). The index number i of a_(k)i (i = 0,1,2,…) is the only optional mark that discriminates tokens. Hence, differently indexed tokens are equivalent; that is, ‘a_(k)i = a_(k)j = a_(k)3’. (b). After a firing f_(1→2) with order a₍₁₎₂ in place p₁, token a₍₁₎₂;f_(1→2) (= a₍₂₎₂) is the output in place p₂.

https://doi.org/10.1371/journal.pone.0302710.g002

Next, a graph is produced when the compositionality between two morphisms and the existence of an identity morphism (meaning doing nothing) are omitted; therefore, if we regard the sequence of two firings ‘f_(k→l) and f_(l→m)’ under the condition for which the same token a_(k)i● as in the product of ‘a_(k)i;f_(k→l);f_(l→m) = a_(m)i’ is ordered, we may regard ‘f_(k→m)’ as the result of ‘f_(k→m) = f_(k→l);f_(l→m)’ with two firings of morphisms (transitions) f_(k→l) and f_(l→m) (here ‘;’ denotes the diagrammatic order of operations; that is, the operation occurs in the order f_(k→l), then f_(l→m), and then f_(m→n)). Furthermore, by ordering the same token a_(k)i, the associative law is confirmed as (f_(k→l);f_(l→m));f_m→n) = f_(k→l);(f_(l→m);f_m→n)) because ‘{a_(k)i}(f_(k→l);f_(l→m));f_m→n) = {a_(k)i}f_(k→l);(f_(l→m);f_m→n)) ₌ a_(n)i’ (For later discussions, we label this comment #remark 1). The index ‘i’ of a_(k)i, a_(m)i or a_(n)i… is an optional mark and only means these tokens comprise a sequence of the same/unique flow as in ‘a_(k)i→a_(m)i→a_(n)i→…’, and ‘i’ does not express a specifically ordered element within the group (or monoid) A_k (in place p_k), A_l (in place p_l) or A_m (in place p_m). Moreover, there exists an identity morphism ‘f_(k)0’ (that does nothing even to a token in place p_k) and ‘f_(l)0’ (that does nothing even to a token in place p_l) that satisfies ‘{}; f_(k→l);f_(l)0 (= f_(l)0°f_(k→l)) = f_(k→l) = {}; f_(k)0;f_(k→l) (= f_(k→l)°f_(k)0)’ for any morphism f_(k→l) (between places p_k and p_l). In category theory, provided that A_k is a domain, A_l a codomain, and f_(k→l) a morphism such as f_(k→l): A_k→A_l, the image of A_k denoted {A_k};f_(k→l) satisfies (4) We stipulate that A_k is filled with one of the elements a_(k)i (i = 0,1,2,…) (∈A_k). However, being different from A_k, place p_k is a potential source object that may be filled with any element a_(k)i (∈A_k) a multiple of times, a_(k)i, a_(k)i… or have no a_(k)j (∈A_k). Likewise, A_l is filled with one of the elements a_(l)i (i = 0,1,2,…) (∈A_l), and place p_l is a potential target object that may be filled with any element a_(l)i (A_l) many times or not at all.

Firing rule II: Firing defined within the same place (via an inner-place morphism). Suppose that an element f_(k)i is one of the morphisms (transitions) defined at place p_k (k is the place index, k = 1,2,…, n,…, and n the number of places of interest in the PN), and f_(k)i or c·f_(k)i belongs to group A_k (f_(k)i or c·f_(k)i∈group A_k). Additionally, a token a_(k)i● is one of the elements such as f_(k)i of group A_k or an accessorized element such as ‘c·f_(k)i’ (c a constant factor without any operational action). That is, ‘a_{(k)i =} f_(k)i’ or ‘a_{(k)i =} c·f_(k)i’ holds (f_(k)i or c·f_(k)i’∈group A_k). There, we stipulate that a morphism f_(k)i acts on a token a_(k)i● as (5) Similar to the ordinal binary composition within a group A_k, the binary operation-like composition is ‘{a_(k)i};f_(k)j (= f_(k)j°a_(k)i) = f_(k)i;f_(k)j (= f_(k)j°f_(k)i)’ (∈group A_k) (where the content as an operation of token ●‘a_(k)i’ is f_(k)i, which is denoted without ●) or ‘{c·a_(k)i};f_(k)j (= f_(k)j°(c·a_(k)i)) = {c·f_(k)i};f_(k)j (= f_(k)j°(c·f_(k)i)) = c·{f_(k)i;f_(k)j} (= c·(f_(k)j°f_(k)i))’ (∈group A_k) (for which the token ● is ‘c·a_(k)’), as in Eq (5). Through firing a morphism (transition) f_(k)a ordering a token a_(k)i● (= f_(k)i or c·f_(k)i), a token a_(k)i is removed from place p_k; after that, a_(k)h (= a_(k)i;f_(k)a, that is, the images of a_(k)i by f_(k)a (∈A_k)) are produced in the same place p_k, in accordance with ‘morphism f_(k)a: a_(k)i→a_(k)h (= f_(k)h or c·f_(k)h).

[Fig 3(A)]

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Fig 3. Illustrations of operations on extended Petri nets (EPNs).

(a). In the present model, we mainly consider cases where a token a_(k)i is also an operator such as f_(k)j or c·f_(k)j (f_(k)j being accessorized with constant c). Hence, through a firing of a morphism (transition) f_(k)a orders a token a_(k)i● (= f_(k)j or c·f_(k)j), denoted by f_(k)a:{a_(k)i}, f_(k)a:c{a_(k)i} may become a composition like f_(k)a;f_(k)j or f_(k)a;c·f_(k)j that also has one of each operator (∈A_k). A token a_(k)i is at once removed from place p_k; subsequently, a_(k)h (= a_(k)i;f_(k)a; that is, the image of a_(k)i by f_(k)a (∈A_k)) is produced in the same place p_k, in accordance with ‘morphism f_(k)a. a_(k)i→a_(k)h (= f_(k)h or c’·f_(k)h). Note ‘;’ means in diagrammatic order. A simple example is given in the relationship between DNA (D_(k)j) and its operator B_(k)(0→j) where ‘D_(k)j = D₀;B_(k)(0→j)’; see main text for details. Although f_(k)a and a_(k)i (= f_(k)i or c·f_(k)i) are elements belonging to the respective unique set A_k = {f_(k)1, f_(k)2,…,f_(k)i,…} or = {c·f_(k)1, c·f_(k)2,…, c·f_(k)i,…}, a_(k)i is accompanied by token ●, and f_(k)a is not. This procedure expresses the individual operation on the token. (b). Provided a unique token a_(k)i in place p_k is ordered for each operation, a binary composition (operation) between morphisms f_(k)a and f_(k)b (both∈A_k) is performable, {a_(k)i}:f_(k)a;f_(k)b = {a_(k)i}:f_(k)c when f_(k)a;f_(k)b = f_(k)c holds. In the former intermediate-state procedure, a transition (morphism) of f_(k)a and f_(k)b does not fire. In the latter intermediate-state procedure, a token a_(k)m (= a_(k)i;f_(k)c) is produced in the same place p_k when a_(k)m is the product of a_(k)i;f_(k)c. We regard the entire procedure illustrated in this figure as a one-time firing of morphism f_(k)c (= f_(k)a;f_(k)b) that occurs by ordering a_(k)i. Moreover, a sequence of these procedures (the former and latter intermediate-state procedures) defines the binary composition between an arbitrary pair of morphisms that belongs to the same group (or related concepts) A_k and place p_k.

https://doi.org/10.1371/journal.pone.0302710.g003

We call f_(k)a, f_(k)I, and f_(k)j ‘inner-place morphisms’. Although f_(k)a and a_(k)i (= f_(k)i or c·f_(k)i) are elements belonging to the respective unique sets A_k = {f_(k)1, f_(k)2,…,f_(k)i,…} and = {c·f_(k)1, c·f_(k)2,…, c·f_(k)i,…}, a_(k)i is accompanied by a token ●, and f_(k)a is not. Additionally, between morphisms f_(k)a and f_(k)b (both∈A_k), a binary composition (operation) is performable, i.e., ‘f_(k)a;f_(k)b = f_(k)c.’ An example is illustrated in a later section.

[Fig 3(B)].

In general, the operation ‘;’ is not commutative and may involve different operations depending on place p_k on which it acts. Briefly, the firing rules for f_(k)a on a_(k)i in place p_k are presented as follows:

M_w+1(p_k)=M_w(p_k) + {a_(k)i};f_(k)a − a_(k)i. Token a_(k)i is transformed into {a_(k)i};f_(k)a (= f_(k)a°a_(k)i) in the same place p_k (substitution of object) at the same instant; otherwise, (6) Furthermore, by ordering the same token a_(k)i, the associative law holds, ‘(f_(k)a;f_(k)b);f_(k)d = f_(k)a;(f_(k)b;f_(k)d)’ in place p_k if the following relationship ‘{a_(k)i};(f_(k)a;f_(k)b);f_(k)d = {a_(k)i};f_(k)a;(f_(k)b;f_(k)d) = a_(k)h (∈group A_k)’ is postulated. Together with this relation, there exists an identity morphism ‘f_(k)0’ that satisfies ‘f_(k)a;f_(k)0 (= f_(k)0°f_(k)a) = f_(k)a = f_(k)0;f_(k)a (= f_(k)a°f_(k)0)’ for any morphism f_(k)a within place p_k.

Moreover, we regard (7) as the ‘image of a_(k)i by f_(k)a’; otherwise, ‘a_(k)i;f_(k)a:’ or ‘f_(k)a°{a_(k)i}:’ means the ‘map of morphism f_(k)a on a_(k)i’. The former expresses a token, whereas the later displays the map of a morphism (operation).

Likewise, whereas ‘{a_(k)i};f_(k→l)’ (= f_(k→l)°a_(k)i) expresses a token a_(l)i, ‘{a_(k)i};f_(k→l):’ (= f_(k→l)°a_(k)i:) displays a genuine map of a morphism f_(k→l) to a_(k)i. The symbol ‘i’ of a_(k)i and a_(l)i means a relationship in which one is the image of the other in certain cases. However, this rule is optional and has no specific purpose. For example, with ‘{a_(k)i};f_{(k)a =} a_(k)h’ of Eq (7), the relationship between ‘i’ of ‘a_(k)i’ and ‘h’ of ‘a_(k)h’ is unspecified. Then, associative law-like relationships hold when combining Firing Rules I and II under the condition that a unique ‘a_(k)i’ is ordered as an initial token for the mapping of the operations; i.e., among f_(k)a, f_(k)b, f_(k→l), f_(l→m),…, (8) In fact, the meaning of the composited map f_(k)a;f_(k→l) seems unclear because the morphisms f_(k)a and f_(k→l) are defined at different locations (the former is defined in the same place, whereas the latter is defined for different places), and an integrated morphism f_(k)a;f_(k→l) seems to accomplish nothing other than a sequence of morphisms of f_(k)a and f_(k→l). Hence, Eq (8) may better be regarded only for combinations of Firing Rules I and II because they do not break the category postulates, albeit there may be no qualitative meaning.

Similarly, the following relation may arise, (9) (r, s; appropriate non-positive integers that provide adequate elements). There, the composite f_(k)a;f_(k)b may be integrated as a unique morphism f_(k)d satisfying f_(k)d = f_(k)a;f_(k)b for which the monoidal operation at place p_k is presumed. However, the combination of Firing Rules I and II f_(k)b;f_(k→m) implies only a sequence of morphisms and this may only mean that this combination does not break the category postulates. Nonetheless, the commutativity-like law might be assumed for combinations of the two firing rules in the diagrammatically ordered display of morphisms such as for Eqs (8) and (9). Naturally, the inverse morphism may be included in Eq (8) or (9) if the appropriate definition of the inverse morphism is possible.

In this EPN model, we consider instances for which the compositionality among all target objects and source objects that are necessary for the construction of the category completely satisfies Eq (4). Therefore, under this assumption, the combination of Firing Rules I and II seems to be permissible without breaking the category postulates. At this moment, we must confirm that there are two type of morphisms in our model: 1) morphisms for which the Firing Rules I and II are always assumed to act on any single token; and 2) morphisms that act on two tokens (e.g., f_Int{, }, f_spl{, }, a topic that we shall return to later) for which the group or monoid postulates are not always satisfied.

In general, suppose that C_k (k = 1,2,…,m) is an object of Category C, and object C_k is unique; then C is termed a ‘monoid’. In particular, for Firing Rule II, a pair of fireable morphisms f_(k)a and f_(k)b (a morphism is fireable if at least one token, say a_(k)i, for which index number ‘k’ is the same as that included in place p_k) is viewed as a composition of two morphisms that are monoidal operations. If any morphism has an inverse, the associated object becomes a group. In our model, the individual ‘place p_k’ appears as a substitution of the object in the category to describe an individually specific state (marking) of an EPN. For simplicity, we assume instances of places, in which inverse morphisms f_(k→l)⁻¹ exist with no regard to whether ‘k≠l’ holds.

§2–4. Annihilation/Creation morphism

To broaden the applicability of the model, we introduce an optional rule that has unique characteristics—annihilation and creation of elements; the symbol ϕ is adopted for this concept.

§2-4-2. Creation morphism (transition).

Similarly, we introduce the ‘creation morphism (transition) f_(ϕ→k)’ (also denoted by ‘{ϕ}; f_(ϕ→k): or f_(ϕ→k)°{ϕ}:’ meaning the operation f_(ϕ→k) ordering ‘ϕ’) that is fireable when ϕ is included in place p_k (in this regard, this means any f_(ϕ→k) is freely fireable at any time).

After the firing of a morphism ‘{ϕ}; f_(ϕ→k) (i.e., the image of {}; f_(ϕ→k) (= f_(ϕ→k)°{}) ordering ϕ, is denoted by {ϕ}; f_(ϕ→k) or the image of f_(ϕ→k)°{ϕ}), an implicit token ϕ (empty) is removed from place p_ϕ and an explicit a_(k)0 is produced in place p_k.

At this moment, we stipulate that a creation morphism f_(ϕ→k) is affiliated at all places with a one-tuple by an arc from a unique place p_ϕ to any arbitrary place p_k; that is, morphism f_(ϕ→k): ϕ→a_(k)0 or alternatively, {ϕ}; f_(ϕ→k) (= f_(ϕ→k)°{ϕ}) = a_(k)0.

[Fig 4(A)]

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Fig 4. Initial and terminal objects, and two types of phasor diagrams.

(a). For places p₁, p₂,..,p_k,…,p_m, there may exist ‘terminal objects’ in each place where the individual terminal object includes identity elements such as ‘a₍₁₎₀ for place p₁’ and ‘a₍₂₎₀ for place p₂’. Moreover, there exists an initial object (initial place) p_ϕ that includes an empty token, which is denoted by ϕ. If an identity token a_(k)0 is operated on by morphism f_(k→ϕ), a_(k)0 is deleted from place p_k and an empty token is generated in place p_ϕ (Note, a_(k)0 is annihilated from place p_k). Conversely, morphism f_(ϕ→k) acts on ϕ creating a_(k)0 in place p_k in accordance with {ϕ};f_(ϕ→k) = a_(k)0. There is a uniqueness between ‘f_(ϕ→k) and f_(k→ϕ)’ whereby only a unique morphism is definable between a terminal object and an initial object. In this way, the annihilation/creation morphism acts as an eraser/generator for a token. (b). In the bottom left panel (reproduced from Fig 5 of [Ref.6]), the five bases ‘E, C, A, T/U, G’ are represented on a phasor diagram (unit circle in the Gaussian plane) as five equi-spaced unit-modular phasors with inclusive angle ‘2π/5’. The general association of each base with a phasor is ‘X_m ↔ Exp(m·ω·i)’ (‘i’ denotes the ‘imaginary unit’, ω = 2π/5 (rad), and m = {0, 1, 2, 3, 4, 5}). To enhance the usability of this representation, we map the phasor diagram onto a square in the Cartesian plane (right panel) under the bijection E(1,0) ↔ E (= Exp(0)), C (1,1) ↔ C (= Exp(ω·i)), A (−1,1) ↔ A (= Exp(2·ω·i)), T/U (−1,−1) ↔ T/U (= Exp(3·ω·i)), G(1,−1) ↔ G (= Exp(4·ω·i)) between two diagrams. The position of ‘E₀’ should always be placed at the midpoint of the right side of the square.

https://doi.org/10.1371/journal.pone.0302710.g004

In fact, no explicit token would be removed in place p_ϕ because a deletion of ϕ means no change for a token, and this procedure simply means the change ‘ϕ→a_(k)0’ or the insertion of a_(k)0 in place p_k. In addition, in place p_ϕ, it is unnecessary to apply the above creation procedure because it suffices that the input and output places of a creation morphism f_(ϕ→k) are defined at the same place p_k.

By considering an operational meaning of a morphism f_(ϕ→k) to marking S₂, the following change is describable for place p_k of marking S₂, ‘f_(k→ϕ):{+a_(k)0}p_k+{−ϕ}p_ϕ’ that act on marking ‘S₂ = {ϕ}p_k+{ϕ}p_ϕ’ as (20) Here, f_(ϕ→k) means the change between the pre-/post-firing of two markings of a creation morphism.

With the use of the annihilation/creation morphism, the interchange (21) which implies reducing/increasing the number of tokens is performable almost freely.

A small novelty brought about by our idea is that an annihilation/creation morphism could be used as an eraser/generator of tokens; we demonstrate the idea below.

Firing rule: Firing defined between place p_k and place p_ϕ (via an annihilation morphism f_(k→ϕ)). Suppose that the number of an identity token a_(k)0 in place p_k (k = 1,2,..) or p_ϕ is given as N_w({a_(k)0}p_k) (w-th session, w = 1,2,…) or N_w({a_(k)0}p_ϕ), respectively.

Transition (morphism) f_(k→ϕ) is fireable ⇔ N_w({a_(k)0}p_k) ≥1;

N_w+1({a_(k)0}p_k) = N_w({a_(k)0}p_k) − 1: p_k (input place of the annihilation morphism f_(k→ϕ)) ˄ p_ϕ (output place of the annihilation morphism f_(k→ϕ)).

N_w+1({a_(k)0}p_ϕ) = N_w({a_(k)0}p_ϕ): p_ϕ (output place of the annihilation morphism f_(k→ϕ)) ˄ p_k (input place of the annihilation morphism f_(k→ϕ)), i.e., N({a_(k)0}p_ϕ) = N′({a_(k)0}p_ϕ) = 0. In addition,

N_w+1({a_(k)0}p_k) = N_w({a_(k)0}p_k), N_w+1({a_(k)0}p_ϕ) = N_w({a_(k)0}p_ϕ): otherwise.

Firing rule: Firing defined between place p_k and place p_ϕ (via a creation morphism f_(ϕ→k)). Transition (morphism) f_(ϕ→k) is always fireable with

N_w+1({a_(k)0}p_k) = N_w({a_(k)0}p_k) + 1: p_k(output place of the creation morphism f_(ϕ→k)) ˄ p_ϕ (input place of the creation morphism f_(ϕ→k)) and

N_w+1({a_(k)0}p_ϕ) = N_w({a_(k)0}p_ϕ): p_ϕ(input place of the creation morphism f_(ϕ→k)) ˄ p_k (output place of the creation morphism f_(ϕ→k)); i.e., N({a_(k)0}p_ϕ) = N′({a_(k)0}p_ϕ) = 0. In addition

N_w+1({a_(k)0}p_k) = N_w({a_(k)0}p_k), N_w+1({a_(k)0}p_ϕ) = N_w({a_(k)0}p_ϕ): otherwise.

The following relationships are straightforward to verify:

N′({a_(k)0}p_all) = N({a_(k)0}p_all)–n_a + n_c, (N({a_(k)0}p_all): the number of the identity tokens in all places p₁, p₂, p₃,…, p_m; n_a: the number of the firing of the annihilation morphism, n_c: the number of the firing of the creation morphism) and

M′(p_all) = M(p_all)–n_a + n_c, (M(p_all): the number of tokens in all places p₁, p₂, p₃,…, p_m).

As seen above, the state of markings of an EPN may be describable in various forms as given for example in Eqs (3), (6), (11)–(14), and (18). In the present model, we take the common exhaustive approach that without loss of generality there exists a morphism f_(k→l) between any arbitrary pair of places that is potentially fireable, and for a morphism f_(k→l) that is inadequate (non-composable), it suffices that we consider the morphism (transition) f_(k→l) as being permanently non-fireable. In §3, to aid the reader’s understanding, the specific use of the annihilation and creation morphisms is explained more explicitly.

§3–1. A simplistic model to integrate/split tokens of DNA sequences

First, for integrating/splitting tokens, the following manipulation is considered provided that the two tokens

R_(k1)a● = [a_(k1)1|a_(k1)2|a_(k1)3|…|a_(k1)i|…|a_(k1)(Na−1)|a_(k1)Na|| r_{(k1)0 (Na+1)}|r_{(k1)0 (Na+2)}|…] (in place p_k1) = [{R_(k1)a}||r_{(k1)0 (Na+1)}|r_{(k1)0 (Na+2)}|…] (in place p_k1) and

R_(k2)b● = [b_(k2)1|b_(k2)2|b_(k2)3|…|b_(k2)μ|…|b_(k2)(Nb−1)|b_(k2)Nb|| r_{(k2)0 Nb+1}|r_{(k2)0 Nb+2}|…] (in place p_k2) = [{R_(k2)b}||r_{(k2)0 Nb+1}|r_{(k2)0 Nb+2}|……] (in place p_k2)

are included independently; here, N_a and N_b are the least numbers from where there are no coordinates other than the identity coordinates such as r_(k)0; the display of operations determined for each coordinate is omitted. In general, the integrated (combined) state of tokens A● and B● is denoted (22) and displayed as another token C● (‘++’ means the gluing/combining of two neighboring tokens, A and B, with the non-commutative order of these tokens noted in the map f_Int{, }. Additionally, the use of ‘||r_(k)0r_(k)0…]’ needs delicate treatment). Note that, in the above description, the display of coordinates of tokens is essentially based on our previous article [6].

Hence, the integrated state of R_(k1)a● and R_(k2)b●, denoted by R_(k3)c● is displayed as

R_(k3)c● = f_Int{R_(k1)a, R_(k2)b} = {R_(k1)a}∪{R_(k2)b}

= {R_(k1)a}++{R_(k2)b} = {R_(k3)a|R_(k3)b}● (in place p_k3)’.

[{R_(k3)a|R_(k3)b}||r_{(k3)0 Na+Nb+1}|r_{(k3)0 Na+Nb+2}|……]● (in place p_k3)

(23)

[see Fig 5(A)]. In this example, token R_(k3)c● is included in a different place p_k3.

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Fig 5. Illustrations of integration using two tokens.

(a). Provided that token ‘● R_(k1)a = [a_(k1)1|a_(k1)2|a_(k1)3||r_(k1)0r_(k1)0 …] = [{R_(k1)a}||r_(k1)0r_(k1)0 …]’ is included in place p_k1, and a token ‘●R_(k2)b = [b_(k2)1|b_(k2)2||r_(k2)0 r_(k2)0 …] = [{R_(k2)b}||r_(k2)0r_(k2)0 …]’ is also in place p_k2, by a firing of morphism f_Int{, } (= {|}; f_Int), an integrated state‘● R_(k3)c’ is produced in place p_k3; i.e., R_(k3)c = f_Int{R_(k1)a, R_(k2)b} = {R_(k1)a, R_(k2)b} = {R_(k1)a}∪{R_(k2)b} = {R_(k1)a}++{R_(k2)b} = {R_(k1)a|R_(k2)b} = [{R_(k3)a|R_(k3)b}||r_(k3)0 r_(k3)0 |……] (in place p_k3). The inverse procedure is also illustrated. A firing of f_spl{, } (= {//}) acting on token ●R_(k3)c (in place p_k3), a splitting occurs of R_(k3)c into two tokens R_(k1)a (in place p_k1) and R_(k2)b (in place p_k2), yielding ‘f_spl{R_(k1)a, R_(k2)b} = ‘{[R_(k1)a]//[R_(k2)b]} = {R_(k1)a} ((+)) {R_(k2)b} (R_(k1)a in place p_k1, R_(k2)b in place p_k2)’. (b). A simple case of assuming ‘place p_k = place p_k” is illustrated. After the firing of morphism f_Int{A, B} (= {A|B}), which performs an integration of A● and B●, the integrated state ‘●R_(k3)c’ is produced in place p_k3 as follows: R_(k)c (in place p_k) = f_Int{R_(k)a, R_(k)b} = {R_(k)a, R_(k)b} = {R_(k)a}∪{R_(k)b} = {R_(k)a}++{R_(k)b} = {R_(k)a|R_(k)b} (both, in place p_k). = [{R_(k)a|R_(k)b}||r_(k3)0 r_(k3)0 |……] = [a_(k)1|a_(k)2|a_(k)3|b_(k)1|b_(k)2||r_(k)0 r_(k)0…] (in place p_k).

https://doi.org/10.1371/journal.pone.0302710.g005

Conversely, the splitting the state is denoted by

‘f_spl{A, B} = {A//B} = {A}● (= [A||r_(k)0r_(k)0…]● (in place p_k1)) ((+)) {B}● (= [B||r_(k)0r_(k)0…] (24) In practice, for a token R_(k3)c● (= {R_(k1)a|R_(k2)b}●), the following is true;

f_spl{R_(k1)a, R_(k2)b}● = {R_(k1)a//R_(k2)b} = {R_(k1)a}● ((+)) {R_(k2)b}● (where R_(k1)a is in place p_k1, R_(k2)b is in place p_k2).

In detail,

‘R_(k1)a ● = [{R_(k1)a}||r_{(k1)0 Na+1}|r_{(k1)0 Na+2}|……] = [a_(k1)1|a_(k1)2|a_(k1)3|…|a_(k1)i|…|a_(k1)Na−1|a_(k1)Na

|| r_{(k1)0 Na+1}|r_{(k1)0 Na+2}|…] (in place p_k1)’, and (25) [see Fig 5(A)].

[Fig 5(A)] displays general cases, but here we assume conditioned cases for which places p_k1 and p_k2 are the same as place p_k (= p_k1 = p_k2 = p_k3) to simplify the examples.

For tokens R_(k)a● and R_(k)b● (both, in place p_k), the following relation can be verified; (26) (27)

= (28) [Fig 5(B)] (29) For this example, the following is also confirmed; for token R_(k)c● (= {R_(k)a|R_(k)b}●), (30) In detail, (31) and (32) We next consider rules for which the tokens are single-stranded sequences of DNA denoted by D_(k)j● and of RNA denoted by R_(k’)j● (here, the indices mean the j-th session in place p_k’). They are associated with biological products with a mature messenger RNA produced from DNA. Each place is based on the sets composed of D_(k)j, R_(k’)j, and related products. The morphism f_(k→l) has conditions pertaining to the enzyme and circumstances that act on D_(k)j and consequently change D_(k)j into another similar token D_(k)l at the same or a different place. The essential stipulations are almost equivalent to the rules of §3.

In a previous article, we suggested the DNA wallpaper pattern for which the base units are placed in a cruciform pattern (Fig 1 of [Ref. 6]; see S1 File). The pattern was introduced to develop a category-like model for genetics (in fact, molecular biology). However, in that article, the sequence of the single-strand DNA and related products were expressed marginally in a category-like formalism using a polygonal line in a plane. Because the shape of the polygonal line changed considerably, using it proved inconvenient. Therefore, we developed a more stable display.

In essence, in the canonical Watson–Crick DNA base pairing scheme, adenine (A) forms a base pair with thymine (T) and guanine (G) forms a base pair with cytosine (C) [47, 48]. Similarly, RNA is a molecule that has a much shorter chain of nucleotides but has various biological roles. The DNA sequence consisting of bases ‘A, C, T, and G’ is transcribed into RNA composed of bases ‘A, C, U, and G’; the sets only differ in that ‘U (uracil)’ replaces ‘T (thymine)’.

We introduced to these bases, a base ‘E’ to indicate an ‘empty’ base, which is treated in the same way for display purposes as an identity base. In the previous article, with five bases, we considered the set C₅ = {C, A, T, G, E}. In this example, the token D_j● is rewritten as D_(k)j●.

§3–2. A bijective expression of DNA sequences using a diagram-pole of five equi-spaced phasors and a square diagram-pole

First, we introduced a phasor diagram in which each base ‘E, C, A, T/U, G’ was assigned to one of five equi-spaced phasors located on the unit circle in the Gaussian plane, each with a ‘2π/5’ angular phase separation to the next [Fig 4(B), left panel]; reproduced from Fig 5 of [6]).

We then introduced phase angle ‘ω’ defined as the counterclockwise rotational angle ‘ω = 2π/5 (rad)’ and a binary composition of ‘ω’ denoted by ‘●’ written ‘ω₂●ω₄ = ω₁’, and an action on bases also denoted by °. A rotational group W is defined; W = {ω₀, ω₁, ω₂, ω₃, ω₄}, with the rule of ‘ω_m = ω^m’ and ‘m = {0, 1, 2, 3, 4, 5}’ (ω₀ is the identity meaning no rotation). The bijection rule; X₀ ↔ E, X₁ ↔ C, X₂ ↔ A, X₃ ↔ T, X₄ ↔ G, then allows the general form of an arbitrary base to be expressed as ‘X_m ↔ Exp(m·ω·i)’ (here, ‘i’ is the ‘imaginary unit’, m = {0, 1, 2, 3, 4}). E.g., ‘X₁·ω₃ = X₁₊₃ = X₄’ means ‘C°ω₃ = G’ (base C is transformed to G by operator ω₃). In practice, E°ω₀ = E, E°ω₁ = C, E°ω₂ = A, E°ω₃ = T, E°ω₄ = G. All compositional scenarios are based on and referenced as diagrams (Figs 2 and 3 [Ref. 6]; see S1 File).

In this paper, we introduce an additional rule; the 0-th base ‘E₀’ is always assumed explicitly in any single-strand DNA sequence or fragments such as D_(k)j = [E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…].

[Figs 6(A) and 7(A)]

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Fig 6. Illustrations of DNA sequences with/without < > on a square pole.

(a). Simple diagram of a single-strand DNA sequence ‘D_(k)j = [E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]’, and its 3-dimensional polygonal line on a square pole. We stipulate that for any D_(k)j, the 0-th coordinate E₀ and the infinite tail ‘EEE…’ are glued automatically with the actual sequence ‘C₁A₂E₃T₄E₅G₆’. (b). To highlight the actual sequence from 1st to 6-th ‘C₁A₂E₃T₄E₅G₆’ (in this case, N = 6; the appropriate range is determined by the user), and by making use of the symbol < >, planes that include ‘E’s as actual bases (i.e., C₁A₂E₃T₄E₅G₆) of <D_(k)j> are deleted completely, except those planes of the 0-th coordinate E₀ and the infinite tail ‘EEE…’; hence, <D_(k)j> = <[E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]> = [E₀|C₁|A₂|T₃|G₄||E₅E₆E₇…].

https://doi.org/10.1371/journal.pone.0302710.g006

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Fig 7. Illustrations of DNA sequences with/without < > on a pentagon pole.

(a). A simple diagram for sequence ‘D_(k)j = [E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]’ and its 3-dimensional polygonal line on a pentagon pole. In addition, we stipulate that for any D_(k)j, the 0-th coordinate E₀ and the infinite tail ‘EEE…’ are glued automatically with the actual sequence ‘C₁A₂E₃T₄E₅G₆’. (b). To highlight the actual sequence from 1st to 6-th bases ‘C₁A₂E₃T₄E₅G₆’, planes that include ‘E’s as actual bases (i.e., C₁A₂E₃T₄E₅G₆) of <D_(k)j> are deleted completely, except the planes of the 0-th coordinate E₀ and infinite tail ‘EEE…’ resulting in <D_(k)j> = <[E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]> = [E₀|C₁|A₂|T₃|G₄||E₅E₆E₇…].

https://doi.org/10.1371/journal.pone.0302710.g007

The 0-th coordinate of any DNA/RNA sequence can only be E₀ if E₀ is only a symbol; hence, a partial function for a morphism (e.g., [A₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…], [G₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…] that breaks (4)) cannot occur. Therefore, the objects satisfy the postulates of category theory (the compositionality of a source object and a target object) as mentioned in (4).

[Fig 1(B)]

At this juncture, we refer the reader to our previous report in which the DNA sequences are described using an operator that is defined on a five-fold phasor diagram [Fig 4(B), left panel; see Fig 5 of [6]]. In that report, we considered a little devised Cartesian product of the group W, ‘B_(k)(j→l) = [ω_{0 0}|ω_{1 1}|ω_{0 2}|ω_{2 3}|ω_{3 4}|ω_{4 5}||ω_{0 6}|ω_{0 7}|….] (an operator/a morphism defined on DNA, RNA or related products that are expressed by the bases; each B_(k)(→) is defined in place p_k, k = 1,2,…)’. By making use of an identity DNA sequence ‘D_(k)0 (in place p_k) = [E₀|E₁|E₂|E₃|…|E_N−1|E_N||E_N+1|E_N+2…]’, any D_(k)j (in place p_k) (j; the session number, j = 1,2,3,…) is expressed as (33) For the above description, we replace the symbol ‘°’ with ‘;’ used in [6] to indicate diagrammatic ordering of the application of the operations. We stipulate that there is an infinite number of ‘E’s and the trailing sequence of ‘E’s are implicitly implied as in ‘D_(k)j = [E₀|C₁|A₂|T₃|G₄||E₅E₆E₇…]’. In addition, the general form is, for example, ‘D_(k)j = [E₀|C₁|A₂|T₃|G₄|…….. |T_N−1|A_N||E_N+1E_N+2…]’. Here, the coordinates from the 1st to N-th entry are actual base sequences that include the case in which the ‘E’s are incorporated as for example in ‘D_(k)j = [E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]’ with an infinite tail sequence of ‘E’s devised so that any D_(k)j is regarded as an element in a single set; see [Ref. 6].

§3–3. Regarding the compatibility and treatment between the explicit and implicit identity elements

In applying the creation/annihilation morphism on an explicit ‘E’ and an implicit (), the conversion ‘() →E’ (this morphism being unique) is incorporated or eliminated from any sequence as in

‘D_(k)a = [E₀|G₁|T₂|C₃|A₄||E₅E₆E₇…]

= [E₀|G₁()T₂|C₃()()A₄||E₅E₆E₇…]→[E₀|G₁|E₂|T₃|C₄|E₅|E₆|A₇||E₈E₉E₁₀…] = D_(k)b’ so long as they are recognized/tracked during the entire process. We stipulate that there are two ways of using E, specifically, the explicit base E (a sort of terminal object) and () the implicit base E (a sort of initial object). The deletion of E is performed via the inverse procedure of the above manipulation using the annihilation ‘E→()’ (this morphism being also unique). Note that the annihilation/creation of bases (coordinates) of ‘D_(k)a’, the insertion and deletion of an explicit E, and the conversion between ‘E ↔ ()’ are actual morphisms, and there exists an ordinal operator that substitutes for this change. For example, the above annihilation/creation can also be performed by an ordinal operator of the Cartesian product such that ‘B_(k)(a→b) = [ω_{0 0}|ω_{0 1}|ω_{2 2}|ω_{2 3}|ω_{4 4}|ω_{4 5}|ω_{0 6}|ω_{2 7}||ω_{0 8}|ω_{2 9}|ω_{0 10}|….]’ belongs to group W, for which the relation ‘D_(k)a°B_(k)(a→b) = D_(k)b’ can be verified. Henceforth, for this example of a hybrid model for genetic/molecular biology, all coordinates can be treated using this sort of ordinal operation as a Cartesian product such with ‘B_(k)(a→b)’ composed of multiple elements of group W.

Furthermore, X_m and X_5-m are inverse operations ‘ω_m⁻¹ = ω_5-m’. With the notation (5→3), a sequence ‘D_(k)j (5→3) = [E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]’ is transformed into its complementary base ‘D_(k)j^† (3→5) = [E₀|G₁|T₂|E₃|A₄|E₅|C₆||E₇E₈E₉…]’ simply from ‘D_(k)j^† = D₀;B_(k)(0→j)⁻¹’ in accordance with examples given in our previous report [6]. Here, because any pair of bases, such as C and G, is related by a symmetry around the arbitrary E (also for A↔T), the sequence D_(k)j^† can be obtained simply by performing a counterclockwise rotation by π (rad) around any arbitrary E of the DNA wallpaper (to be clarified later). The terms ‘(5→3)’ and ‘(3→5)’ express the prime endings <5’(five prime)→3’(three prime)> or <3’→5’> accompanying a sequence designation. The notation ‘(5→3)’ or ‘(3→5)’ is simply an additional label representing the two possible types of endings of a single-stranded DNA. Then, this pair of homologous double-stranded DNA would be described using ‘/’ meaning a pair of bases as in the following,

[D_j(5→3)/D_j^†(3→5)]

= [E₀/E₀|C₁/G₁|A₂/T₂|E₃/E₃|T₄/A₄|E₅/E₅|G₆/C₆||E₇/E₇|E₈/E₈|E₉/E₉|…].

Here, the bi-operator ‘B_j(5→3)/B_j^†(3→5)’ yields (34) where ▲ denotes the bi-operation of ‘;/;’, and its operation on the pair of identity bases sequence D₀/D₀ by an operator such as B_j is denoted ‘D₀;B_j’. In addition, the composition between B_j and B_k is rewritten as ‘B_j;B_k’ but was written as ‘B_j●B_k’ in our previous report [6]. Hence, the above bi-operator parts of Eq (34) are rewritten as (35) At this point, to enhance usability, we suggest the following optional representation in which a bijection for the vertices is considered. The vertices for bases C, A, T/U, and G on the pentagon are mapped onto the vertices of a square; the identity base E, however, is mapped onto the midpoint between the vertices of C and G [Fig 6(A)]. Although the squares in Fig 6(A) match up with the pentagons [Fig 7(A)], the positions of C, A, T/U, G on the planes are equi-spaced with an angular separation of ‘π/2’. To be specific, they have been placed at E (1,0), C (1,1), A (−1,1), T/U (−1,−1), and G (1,−1) on the plane tangent to the flow of the sequence D_(k)j. This tangent plane is not the Gaussian plane [Fig 4(B), right panel] but the simple two-dimensional Euclidean plane. With this representation, the bijection between the vertices of the two planes is written E (1,0)↔E (= Exp(0)), C (1,1)↔C (= Exp(ω·i)), A (−1,1)↔A (= Exp(2·ω·i)), T/U (−1,−1)↔T/U (= Exp(3·ω·i)), and G(1,−1)↔G (= Exp(4·ω·i)) [Figs 6(A) and 7(A)]. Here, mathematically and operationally, we assume the rotational group W over the Gaussian plane is always in the background even if we display a figure using E (1,0), C (1,1), A (−1,1), T/U (−1,−1), and G(1,−1). Furthermore, we stipulate that when a polygonal line of bases is drawn, the position of ‘E₀’ is always fixed, as in Fig 4(B). This transformation enables a simple illustration of subsequent operations.

§3–4. Notation for the identity elements that were explicitly/implicitly deleted using < > or {}

At this point, to gain more usability, we consider a further manipulation concerning the notation < >. Concerning the meaning of <D_(k)j>, we stipulate that planes that include ‘E’s other than the initial 0-th E (E₀) and the infinite tail of ‘E’s (e.g., ||E₅E₆E₇…) of D_(k)j are deleted entirely, and the remaining indexing coordinates closed up. As a result, the square pole corresponding to <D_(k)j> that does not include ‘E’s as bases other than the initial 0-th E and the infinite tail of ‘E’s is

<D_(k)j> = <[E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]> = [E₀|C₁|A₂|T₃|G₄||E₅E₆E₇…]

[Fig 6(B)]

The meaning of this rule becomes clear in a later description; however, to avoid confusion over the use of the symbol, we provide a brief description of < > and another symbol {} that was introduced in [6]. Let ‘D_(k)j = [E₀|C₁|A₂|E₃|T₄|E₅|G₆||E₇E₈E₉…]’; then, there are two optional displays:

1) in <D_(k)j>, imaginary bases ‘E’s as bases other than the initial 0-th E and the infinite tail of ‘E’s are deleted from D_(k)j; that is, <D_(k)j> = [E₀|C₁|A₂|T₃|G₄||E₅E₆E₇…]. and

2) in {D_(k)j}, bases ‘E’s remain in all sequences despite them being included explicitly in D_(k)j; {D_(k)j} = [C₁|A₂|T₄|G₅||E₇E₈E₉…] (the 0-th E, any E, and the infinite tail sequence of `E’s exist; however, the display of E in the ‘1−N’-th coordinate (in this case, N = 6) is only omitted in the display). In fact, the coordinates of D_(k)j and {D_(k)j} are equivalent. However, for <D_(k)j>, we assume that, for a sequence of single-strand DNA, there exists a bijection between the sequence of <D_(k)j> and the shape of its polygonal line along the square pole. This bijection also holds between <D_j> and the shape of its polygonal line over the DNA wallpaper (see Fig 6 of [Ref. 6]. The revised figure is drawn in [Fig 8]. Similarly, we assume that there is bijection between <D_(k)j> and the shape of its polygonal line [Fig 6(B)]. We conjecture that this bijection enables an unknown base sequence of <D_(k)j> to be identified from the shape of its polygonal line if the latter is already known.

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Fig 8. Illustration of a pair of complementary DNA sequences on the DNA wallpaper.

For the wallpaper pattern, we suppose D_(k)j (in place p_k) = [E₀|G₁|C₂|E₃|A₄|E₅|G₆|T₇|E₈|A₉|G₁₀||E₁₁E₁₂E₁₃…]’, with < > meaning the deletion of imaginary bases ‘E’ other than the initial 0-th E and the infinite tail of ‘E’s. Hence, <D_(k)j> is written <D_(k)j> = [E₀|G₁|C₂|A₃|G₄|T₅|A₆|G₇||E₈E₉E₁₀…]. A rule applies in which a polygonal line begins from an arbitrary E₀, so, in this Figure, <D_(k)j> is described as the polygonal line marked in blue. Because D_(k)j^† (the complementary sequence of D_(k)j) is given as D_(k)j^† = [E₀|C₁|G₂|E₃|T₄|E₅|C₆|A₇|E₈|T₉|C₁₀||E₁₁E₁₂E₁₃…], <D_(k)j^†> is displayed as <D_(k)j^†> = [E₀|C₁|G₂|T₃|C₄|A₅|T₆|C₇||E₈E₉E₁₀…] and is drawn as a polygonal line marked in pink. Obviously, the shape of <D_(k)j> and <D_(k)j^†> are symmetrically disposed with respect to ‘E₀’ over the wallpaper pattern. Therefore, the shape of the polygonal lines of <D_(k)j> and <D_(k)j^†> are point-symmetry-related around a common E₀’. Interestingly, we may infer there is a bijection between the shape of polygonal line <D_(k)j> and the contents of a sequence of bases ‘G₁|C₂|A₃|G₄|T₅|A₆|G₇||E₈E₉E₁₀…’. In other words, if a polygonal line is provided only with E₀’, the content of bases can be determined uniquely and automatically (The blue polygonal line instantly provides the above base sequence). Adapted from Fig 4 of [Ref. 6] with permission (S1 File).

https://doi.org/10.1371/journal.pone.0302710.g008

Naturally, to treat this issue with only a regular pentagon pole is possible [see Fig 7(A) and 7(B)]. In this article, we use the regular p-polygon pole with ‘p = 5’ (pentagon pole) where multiple phasor diagrams (planes) in the Gaussian plane are placed in parallel. By connecting the individual vertices, the DNA and other sequences can be drawn as an orbit on a regular pentagon pole [Fig 7(A) and 7(B)]. Precisely, there is a bijection between D_(k)j and the shape of its polygonal line [Fig 7(A)] and another bijection between <D_(k)j> and the shape of its polygonal line [Fig 7(B)]. In this regard, for ease of use and the advantage of this mapping, the simplification that we prefer is to focus on the reduction in the number of vertices from five to four in shape [Fig 6(A) and 6(B)] and to gain visual simplicity when presented in three dimensions.

§3–5. On the mapping from DNAs to RNAs and the conversion of their dimensions using a unit of Descartes’ geometry

Next, apart from the above stipulation, and to view the mapping between D_(k)j (DNA) into R_(k’)j^† (RNA), we consider the following cyclic group Γ = {σ, n}, where element n is the identity operation that does not change any base; e.g., A;n = A, T;n = T and E;n = E. Conversely, element σ produces the change {E, C, A, T, G}↔{E, C, A, U, G}, and ‘σ² = n (no operation)’ holds. In practice, σ yields the following changes, (36)

Hence, the simple conversion between DNA and RNA is written (37)

Because the group Γ = {σ, n} is {Z₂,; or ×}, D_(k)j and R_(k’)j^† may be expressible as a Cartesian product {Z₅^×∞×Z₂,; or ×} where ‘; ‘ (= ×) denotes the operation between ‘D_(k)j’s or ‘R_(k’)j^†’s.

If the primary DNA sequence is written ‘D_(k)j = [E₀|G₁|E₂|T₃|A₄|C₅||E₆|E₇|…]’, the corresponding RNA sequence R_(k’)j^† is written [E₀|G₁|E₂|T₃|A₄|C₅||E₆|E₇|…];σ (or [E₀|G₁|E₂|T₃|A₄|C₅||E₆|E₇|…]×σ) (σ acting on any coordinate independently) = [E₀;σ|G₁;σ|E₂;σ|T₃;σ|A₄;σ|C₅;σ||E₆;σ|E₇;σ|…] = [E₀|G₁|E₂|U₃|A₄|C₅||E₆|E₇|…]. In other words, any DNA (RNA) sequence is an element of the Cartesian product Z₅^×∞×Z₂ (or D_(k)j×Γ), such as ‘R_(k’)j^† = [E₀|G₁|E₂|U₃|A₄|C₅||E₆|E₇|…];n (or [E₀|G₁|E₂|T₃|A₄|C₅||E₆|E₇|…]×σ)’. In this regard, the expression ‘D_(k)j’ is also regarded as an abbreviation of ‘D_(k)j;n (∈Z₅^×∞×Z₂)’. In practice, this is always possible if we are given ‘D_(k)j = [E₀|G₁|E₂|T₃|A₄|C₅||E₆|E₇|…]’ as ‘D_(k)j;n;n;n;…;n (∈Z₅^×∞×Z₂×Z₂×Z₂×…×Z₂)’. If we regard ‘D_(k)j;n (∈Z₅^×∞×Z₂)’ as the product after ‘m−1’ shrinkages ‘Z₂×…×Z₂ (m-tuple) = Z₂ (one-tuple)’, then that is considered possible. In other words, the essential form of the token D_(k)j● is expressed as Z₅^×∞×Z₂; the optional expression (38) is permissible. If we shrink the parts of ‘…×Z₂×Z₂×Z₂×…×Z₂’, only the appropriate power of the binary operation n;n (= n) is necessary and sufficient to meet the conditions stipulated. Conversely, if a specific form of ‘Z₅^×∞×Z₂×…×Z₂ (r-tuple)’ is required from ‘Z₅^×∞×Z₂×…×Z₂ (s-tuple)’ (for non-negative integers r,s; r≥s), ‘r − s’ applications of the operation ‘n;n = n’ completes this transformation between the two tuples. In this way, there is freedom to choose the number of applications ‘×n’ in accordance with the operations of the group Γ = {σ, n}. For D_(k)j ‘(∈Z₅^×∞×Z₂×…×Z₂ (r-tuple))’ and D_(k)l ‘(∈Z₅^×∞×Z₂×…×Z₂ (s-tuple))’, with r≠s, the corresponding tokens D_(k)j and D_(k)l do not seem to belong to the same place p_k. A difference clearly exists, and usually we only assume the essential case, (39) We introduce the rule that the 3-dimensional polygonal line of ‘D_(k)j = D_(k)j;n’ or ‘R_(k’)j^† = D_(k)j;σ’ (∈Z₅^×∞×Z₂)’ is expressed as ‘[D_(k)j] = D_(k)j;n’ and ‘[R_(k’)j^†] = D_(k)j;σ = R_(k’)j^†;n’. In this way, for these extreme circumstances, we find an arbitrary/appropriate number ‘;n (∈Z₂)’ at the end of any D_(k)j or R_(k’)j^† if needed. This rule is an application of the use of the ‘unit 1’ in Descartes’ geometry [49]. Therefore, we can regard D_(k)j (∈Z₅^×∞) as (40) If D_(k)j also expresses the state D_(k)j where all ‘n;n;n;…n’ are deleted from ‘D_(k)j;n;n;n;…;n’, then, the following relations hold; D_(k)j = D_(k)j;n;n;n;…;n or D_(k)j;n = D_(k)j, D_(k)j;σ = R_(k’)j^†. Similarly, for R_(k’)j^†, (41) Moreover, the binary compositions between two DNAs such as ‘D_j;D_k’, ‘D_j;D_j^†’ or ‘D_j;D_j⁻¹’ are regarded as being nonsensical as an operation because D_j and D_j⁻¹ are not operators. However, note that the primary single-strand DNA sequence ‘D_(k)j’ and the complementary sequence D_(k)j^† have a symmetry with respect to the horizontal plane that includes all Es [see Fig 9(A) and 9(B)].

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Fig 9. Illustrations of the complementary region of a pair of DNA sequences on two types of poles.

(a). Given the two polygonal lines <D_(k)j> = […|C_m|A_m+1|T_m+2|G_m+3|…] (marked in blue) and <D_(k)l> = <D_(k)j^†> = […|G_m|T_m+1|A_m+2|C_m+3|…] (marked in pink), they are symmetry-related in the ‘x-z’ plane that includes the line ‘…E_mE_m+1E_m+2E_m+3…’. This symmetry helps to identify a pair of two complementary segments of single-strand DNA (RNA) sequences. Note that the shape of their polygonal lines underscores a bijection between <D_(k)j> and <D_(k)j^†> (= <D_(k)l>). However, in general, even if the shapes of <D_(k)a> and <D_(k)b> are complementary-related, they may not always be bijective or complementary. (b). Given the two polygonal lines <D_(k)j> = […|C_m|A_m+1|T_m+2|G_m+3|…] (marked in blue) and <D_(k)l> = <D_(k)j^†> = […|G_m|T_m+1|A_m+2|C_m+3|…] (marked in pink), they are clearly symmetry-related in the ‘x-z’ plane that includes the line ‘…E_mE_m+1E_m+2E_m+3…’.

https://doi.org/10.1371/journal.pone.0302710.g009

§3–6. Method for the detection of the complementary region of a pair of DNA(RNA) sequences using two polygonal lines

For more generality, we introduce another rule: if we interpret any single-strand DNA fragment such as ‘A₃T₄’ in ‘D_{(k)r =} [E₀|G₁|E₂|A₃|T₄|C₅||E₆|E₇|E₈|…]’, this part is also recognized as ‘D_(k)s = [E₀|A₁|T₂||E₃|E₄|E₅|…]’. With this understanding, any DNA sequence ‘D_(k)r’ or its partial sequence ‘D_(k)s’ may be related by bijection with the shape of its polygonal line for either the DNA wallpaper, the square pole or the pentagon pole.

For the DNA wallpaper, the bijection is repeated, and the shape of the polygonal line of <D_(k)j> may be determined uniquely over the DNA wallpaper [Fig 8]; adapted from Fig 4 of [Ref. 6]), where naturally, any polygonal line of the <D_(k)j>s never crosses over the ‘E’s. If we distinguish <D_(k)j> from <D_(k)j>_θ that has been rotated through θ (rad) around E₀, the polygonal line of <D_(k)j^†> (the complementary sequence of <D_(k)j>) may be obtained automatically by the counterclockwise rotation of π (rad) around E₀ of the DNA wallpaper [Fig 8]. In the three cases—DNA wallpaper, square pole, and pentagon pole—the polygonal line of <D_(k)j> and the actual single-strand DNA sequence such as ‘CATG…’ are in correspondence, and there is a bijection between the two polygonal lines [Figs 8 and 9(A), 9(B)].

Therefore, the specific sequence of bases in any arbitrary D_(k)j or a pair of DNA sequences have complementary sequences that may be identified graphically. They can be converted into complementary sequences of DNA that can be performed easily from the shape of the polygonal line. For the complementary region of D_(k)j and D_(k)l (single-strand DNA sequences in place p_k), the composition of the operator parts of ‘D_(k)j = […|C_m|A_m+1|T_m+2|G_m+3|…] = […|E°ω_{1 m}|E°ω_{2 m+1}|E°ω_{3 m+2}|E°ω_{4 m+3}|…]’ and ‘D_(k)l = […|G_m|T_m+1|A_m+2|C_m+3|…] = […|E°ω_{4 m}|E°ω_{3 m+1}|E°ω_{2 m+2}|E°ω_{1 m+3}|…]’ (i.e., B_(k)j●B_(k)l = […|ω_{1 m}|ω_{2 m+1}|ω_{3 m+2}|ω_{4 m+3}|…]●[…|ω_{4 m}|ω_{3 m+1}|ω_{2 m+2}|ω_{1 m+3}|…]) is considered to satisfy the following relation, (42)

meaning that B_(k)j and B_(k)l are ‘inverses’ in accordance with their shapes [Fig 9(A) and 9(B)]. This relationship may aid in assessing whether arbitrary regions of two single-strand DNA sequences are complementary, although careful treatment is necessary regarding the direction of <5’(five prime)→3’(three prime)> because there may be instances for which D_(k)j and an inverse sequence of D_(k)l are related, as in Eq (42), but have been omitted. Clearly, this similarity relationship and usage may be used on the pentagon pole representation [Fig 9(B)].

§3–7. Supplemental explanations concerning the explicit/implicit identity sequences in category theory

In the present article, we introduce a rule by which an explicit identity sequence ‘D₀ = D_(k)0 (in place p_k) = [E₀|E₁|E₂|E₃|…|E_N−1|E_N||E_N+1E_N+2…]’ may be converted into an implicit identity sequence ‘D_ϕ (in place p_ϕ) = [() () () …() () ||() () …] = ()’ [Fig 10(A) and 10(B)], and vice versa using the annihilation/creation morphism (as D₀ ↔ D_ϕ), and ‘<D₀> = <D_(k)0> = <[E₀|E₁|E₂|E₃|…|E_N−1|E_N||E_N+1E_N+2…]> = D_(k)0’ (in place p_k). Because <D₀> may be obtained by the map f_(→ϕ) on D₀ as ‘D₀;f_(→ϕ) = D_ϕ’, then from a certain viewpoint, for any ‘D₀’, the map <D₀> is a result of the map of an n-tuple of annihilation morphisms f_(→ϕ);f_(→ϕ);…f_{(→ϕ) =} f_(→ϕ)^×n (n; natural number or ∞). However, we must stipulate that f_(→ϕ) changes all ‘E₀E₀…E₀’ in D₀ to () () …(); henceforth, we write f_(→ϕ)^×n only as f_(→ϕ). Significantly, similar to ‘S_! ≠ S₀’ given in Eq (16) and (17) in §2-2-3, because D_(k)0 (in place p_k) may be regarded as a terminal object that is defined on category A_k (at place p_k obtained via viewing group (or monoid) A_k as category), and D_ϕ (at place p_ϕ) may be seen as the initial object of category A_k, there is no zero object ‘D₀ = D_ϕ’ in category A_k (at place p_k).

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Fig 10. Illustrations of the relationships between explicit and implicit identity DNA sequences on two types of poles.

(a). By making use of the annihilation morphism f_(→ϕ), an explicit identity sequence D₀ (a line that passes through the vertex of ‘E₀s on the pentagon pole) is converted into D_ϕ (the implicit identity sequence or nothing) in accordance with ‘D₀;f_(→ϕ) = D_ϕ’. This procedure is also written as ‘<D₀> = <[E₀|E₁|E₂|E₃|…|E_N−1|E_N||E_N+1E_N+2…]> = D_ϕ’. In contrast, f_(ϕ→) yields an explicit identity sequence D₀ from D_ϕ in accordance with ‘D_ϕ;f_{(ϕ→) =} D₀’. (b). With no regard to the shape of the polygonal line, the same conversion is considered performable as ‘D₀;f_(→ϕ) = D_ϕ’ and ‘D_ϕ;f_{(ϕ→) =} D₀’.

https://doi.org/10.1371/journal.pone.0302710.g010

Similarly, in §2-2-2, a unique place p_ϕ for D_ϕ (= () () …(); implicit identity sequence of single-strand DNA) need not always be expressed explicitly in the above conversion procedure because it suffices to consider it somewhere in the entire EPN for an arbitrary annihilation/creation morphism like f_(k→ϕ) or f_(ϕ→k). If it is imposed, R_ϕ (= () () …(); implicit identity sequence of a single-strand RNA) is equivalent to ‘D_ϕ = () () …()’.

In place p_k, a morphism f_(k)a (an operator, for example, such as ‘f_(k)a = B_(k)a = [ω_{2 1}|ω_{3 2}|ω_{0 3}|ω_{4 4}||ω_{0 5}|ω_{0 6}|….], B_(k)b, B_(k)c,…; ∈group W’) may act on any token D_(k)j● and produce an image. If the image of f_(k)a to D_(k)j is denoted by ‘D_(k)j;f_(k)a’ or ‘f_(k)a°D_(k)j’, the operation itself is denoted ‘D_(k)j;f_(k)a;’ or ‘f_(k)a°D_(k)j:’. In fact, D_(k)j is a similar product of B_(k)a (a; non-negative integer) because D_(k)j is determined as D_(k)j = D_(k)0;B_(k)(0→j); thereby, a simple operation may be performable among any arbitrary D_(k)j and morphism f_(k)a similar to the example in §3.

§3–8. An explicit/implicit identity protein and its treatment

Furthermore, an interpretation of the codon (triplet of bases) from the polygonal line comprising bases given in Fig 6(A) and 6(B), and Fig 9(A) and 9(B) is possible in accordance with the conventional codon table [see Fig 11(A)]. The combination of three vertices (to each, a base ‘C, A, T, G or E’ is allocated) may be translated into an amino-acid sequence (protein: a sequence of amino acids) in parallel via morphism f_(4→6) or f_(5→7). If limited to the procedure of a morphism f_(4→6), f_(6→4), f_(5→7) or f_(7→5), the mapping from a sequence of bases to that of an amino-acid sequence (protein) is performable. There, we believe that the inverse procedure is automatically performable using electronic tools such as a scanner for sequences of molecules. We stipulate that by recognizing the shape of the polygonal line of the amino-acid sequence (protein), any sequence of triplets of lines of codons may be interpreted from the corresponding amino-acid sequences (proteins) such as ‘AUG→(M)’, ‘CUG→(L)’, and ‘AGU→(S)’; that is, Rs_(4)j^† (in place p₄, j a non-negative session number) = [E₀|A₁U₂G₃C₄U₅G₆A₇G₈U₉||E₁₀E₁₁E₁₂…] → ‘Pr_(6)j = (M)(L)(S)ΔΔΔ…’(in place p₆) → ‘<Pr_(7)j> = (M)(L)(S)’ (in place p₇), where Δ means an explicit identity (empty) amino acid (protein), and a conversion ‘Δ↔() (implicit Δ, nothing) may be considered. In more detail, we may instigate a rule by which a triplet ‘EEE’ is mapped to an explicit identity amino acid (protein) ‘Δ’; for example, with ‘Rs₍₄₎₁^† = [E₀|A₁U₂G₃E₄C₅U₆G₇A₈G₉U₁₀||E₁₁E₁₂E₁₃E₁₄…]’ (in place p₄) and the appropriate use (firing) of an operator f_(4)a (that is, an operator like B_(j→k) in place p₄), we produce the following changes (43)

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Fig 11. Codon table and illustration of the canonical central dogma using an EPN.

(a). The conventional codon table is displayed. A codon (triplet of bases) is translated into an amino acid that is incorporated in the growing polypeptide chain. The number of triplet combinations is 64. The stop codons are UAA, UAG, and UGA. (b). The canonical central dogma from the double-strand DNA to the protein Q (Glutamine)… is displayed in simplified form using the EPN. First, a token ● is included in place p₁. Subsequently, morphisms f_(1→2), f_(2→3), f_(3→4),, …,f_(6→7) are fired in order. There are two pathways via ‘place p₄→p₆→p₇’ or ‘place p₄→p₅→p₇’. Although multiple tokens are drawn, only one token is included in one of the places in p₁, p₂,…, p₇. Multiplying a marking x by an incidence matrix produces one move of a token. However, the incidence matrices of the EPN are slightly different depending on which pathway is selected from the above three.

https://doi.org/10.1371/journal.pone.0302710.g011

Additionally, the morphism f_{total-splicing} (= f_t-sp = f_(4→6)) leads to further changes;

{Rs_(4)1a^†}; f_t-sp = Pr₍₆₎₁ = Δ₀(M₁)Δ₂(L₃)(S₄)Δ₅Δ₆Δ₇…’ (in place p₆) (we stipulate that an initial Δ₀ and an infinite trail such as ‘Δ₅Δ₆Δ_7…’ are glued automatically) [Fig 11(B)].

Then, Pr₍₆₎₁ (in place p₆) is translated into <Pr₍₇₎₁> = Δ₀(M₁)(L₂)(S₃)Δ₄Δ₅Δ₆… (in place p₇) via morphism f_post-prot. (= f_po-pr = f_(6→7)). In another example, with ‘Rs₍₄₎₂^† = [E₀|E₁A₂U₃G₄E₅C₆U₇G₈A₉G₁₀U₁₁||E₁₂E₁₃E₁₄E₁₅…]’, a specific morphism f_(4→5) (that is, an operator like B_(j→k) defined from place p₄ in place p₅) produces the result, Rs₍₄₎₂^† (in place p₄)→‘<Rs₍₅₎₂^†> (in place p₅) = [E₀|() A₁U₂G₃() C₄U₅G₆A₇G₈U₉||E₁₀E₁₁E₁₂E₁₃…]

= [E₀|(A₁U₂G₃)(C₄U₅G₆)(A₇G₈U₉)||E₁₀E₁₁E₁₂E₁₃…]’; once firing of the specific morphism f_total-prot. (= f_t-pr = f_(5→7)) produces the next change <Rs₍₅₎₂^†> (in place p₅)→‘{<Rs₍₅₎₂^†>};f_(5→7) = <Pr₍₇₎₂> = [Δ₀|(M₁)(L₂)(S₃)||Δ₄Δ₅Δ₆…] in place p₇ (via morphism f_(5→7))’. The above manipulations of ‘E’s are to adjust their number so that all bases form complete triplets. Note that, for the firing rules, we stipulate that when a morphism f_(4→6) (= f_t-sp) is assumed, we disregard ‘pathway place p₄→p₅→p₆’ because f_(5→6) is an inconvenience and nonsensical if f_(4→6) and f_(5→6) are always fireable/non-fireable contemporaneously. Because sequences in place p₅ such as <Rs₍₅₎₂^†> = [E₀|(A₁U₂G₃)(C₄U₅G₆)(A₇G₈U₉)||E₁₀E₁₁E₁₂E₁₃…] do not include ‘E’s as coordinates, other than an initial E₀ and an infinite tail of ‘E₀’s at the end of the sequence, only products that can go through morphism f_(5→7) (when f_(5→7) is fireable) such as ‘<Rs₍₅₎₂^†>;f_(5→7) = <Pr₍₇₎₂> = [Δ₀|(M₁)(L₂)(S₃)||Δ₄Δ₅Δ₆…]’ in place p₇ can also be outputs in place p₆ such as a sequence like Pr₍₆₎₂ = [Δ₀|(M₁)(L₂)(S₃)||Δ₄Δ₅Δ₆…] via morphism f_(5→6) (when f_(5→6) is fireable). Therefore, there are cases where morphism f_(5→6) is fireable. However, morphism f_(5→7) is sufficient, and morphism f_(5→6) is not always necessary. For simplicity of the model, we omit morphism f_(5→6). In this manner, any amino-acid sequence (protein) in place p₆ may be expressed as a sequence of polygonal lines appended by an infinite explicit trailing ΔΔΔ…. Similarly, any amino-acid sequence (protein) in place p₇ may be expressed as ϕ, a sequence of polygonal lines for which there is no infinite explicit trailing ΔΔΔ…. At this stage, an identity sequence of amino acids (proteins) is also definable where the infinite number of Δ is ordered as ‘Pr_(k)0 = ΔΔΔΔΔΔΔ…’ (in this case, k = 6, because place p₆ is designed for an amino-acid sequence (protein) that may include Δ’s). We call Pr_(k)0 an explicit identity protein sequence. Moreover, <Pr_(k)0> is defined as the state where the ‘Δ’s of the explicit identity protein are deleted via the annihilation morphism for Pr_(k)0; ‘{Pr_(k)0};f_(k→ϕ) = <Pr_(k)0> = [Δ₀|Δ₁Δ₂Δ₃…]≠[() () () …] = ϕϕϕ…ϕ = ϕ = Pr_ϕ’ (in this instance, k = 6). Note the relation ‘<Pr_(k)0> = Pr_(k)0 (in place p_k) = [Δ₀|Δ₁Δ₂Δ₃…]’. We refer to it as an implicit identity protein sequence. Naturally, an inverse procedure is considered via the creation morphism, ‘{ϕ};f_(ϕ→k) = {Pr_ϕ};f_{(ϕ→k) =} {<Pr_(k)0>};f_{(ϕ→k) =} Pr_(k)0 = ΔΔΔ…Δ’ (with k = 6). By this rule, any sequence of arbitrary amino-acid sequences (protein) Pr_(k)j (j; the session number, non-negative number j = 0, 1,2,…) can be defined so that Pr_(k)j belongs to set P (the set of amino-acid-sequences (protein)), although no operational meaning is assumed between the ‘Pr_(k)j’s at this moment. The inverse translation (firing) for the token from place p₆ to place p₄ is considered to be always possible; however, that from place p₇ to place p₆ or from place p₅ to place p₄ seems to be impossible because in the latter cases, the bijection is lost. Furthermore, using a notation similar to that in (#remark 1), for morphisms f_t-sp (= f_(4→6)) and f_po-pr (= f_(6→7)), the associative law holds if an adequate adjustment is made to the index numbers; for morphism f_t-sp (= f_(4→6)), the index numbers of the amino-acid sequences (proteins) Pr_(k)j that are transformed from Rs_(k)j^† do not change until all manipulations to the sequence of operations is completed; in detail, the associative law within Eq (43) is obvious because this procedure in place p₄ obeys the group operation.

§3–9. Regarding the symbols [[]], (()) and << >> that treat the explicit/implicit deletion of identity elements

Then, for the ‘part of the operation f_t-sp (= f_(4→6))’ and, with, say, token ‘Rs₍₄₎₂^† (expressed in Eqs (44) and (45))’, the intron parts are changed into appropriate ‘E’s in accordance with the former part of the splicing procedure in place p₄.

If the deletion of explicit ‘E’s from the r-th to the s-th coordinate (s,r: non-negative integers with s≤r; s,r = 0,1,2,…) of token Rs₍₄₎₂^† is denoted ‘Rs₍₄₎₂^†;< >_s,r’, the following manipulation is confirmed; (44) (45) (the place is not determined because this is an intermediate state of places p₄ and p₅), where the index numbers 10–12 remain until the sequence of operations is completed; this is displayed using double round brackets (()) as below. In this way, Rs₍₄₎₂^†;< >_4,6;< >_10,12;< >_16,18 (essentially, the operations are performed in diagrammatic order, from left to right on Rs₍₄₎₂^†),

= [E₀|A₁U₂G₃()_4-6C₇U₈G₉()_10-12C₁₃G₁₄U₁₅()_16-18A₁₉C₂₀G₂₁||E₂₂E₂₃E₂₄E₂₅…]

(the place is not determined for the reason previously stated),

= [E₀|A₁U₂G₃C₄U₅G₆C₇G₈U₉A₁₀C₁₁G₁₂||E₁₃E₁₄E₁₅E₁₆…]

(the place is not determined), using double round brackets (()),

= ((Rs₍₄₎₂^†;< >_4,6;< >_10,12;< >_16,18)) (the place is not determined).

In the above example, by performing the additional operation < >_13,∞ to Rs₍₄₎₂^†, ((Rs₍₄₎₂^†;< >_4,6;< >_10,12;< >_16,18));< >_13,∞

= [E₀|A₁U₂G₃C₄U₅G₆C₇G₈U₉A₁₀C₁₁G₁₂] = <Rs₍₄₎₂^†> (in place p₅).

Hence, the associative law for Rs₍₄₎₂^†, (< >_4,6;< >_10,12);< >_16,18 = < >_4,6;(< >_10,12;< >_16,18) is evidently obeyed. In an ordinal <Rs₍₄₎₂^†>, s = 1, r = ∞ is imposed, and <Rs₍₄₎₂^†> = <Rs₍₄₎₂^†>_1,∞ = [E₀|A₁U₂G₃C₄U₅G₆C₇G₈U₉A₁₀C₁₁G₁₂] (in place p₅).

For the ‘part of the operation f_t-sp (= f_(4→6)) and f_post-prot. (= f_po-pr = f_(6→7)).’; we see f_(4→6));f_(6→7) and f_(4→5));f_(5→7) commute because all tokens of place p₅ are products of Rs₍₄₎₂^†;< >_r,s;…) to a sequence of amino acids (protein) (in place p₆) from the r-th to the s-th coordinate (s,r: non-negative integers with s≤r; s,r = 0,1,2,…) of, say, token Rs₍₄₎₂^†, denoted ‘<Rs₍₄₎₂^†>;[] _s,r’, the following manipulation is verified for ‘Rs₍₄₎₂^† = [E₀|A₁U₂G₃()_4-6C₇U₈G₉()_10-12C₁₃G₁₄U₁₅()_16-18A₁₉C₂₀G₂₁] (in place p₅)’, (46) The combination of amino acids (protein) and bases for which the part underlined L_7–9 means the amino-acid sequence (protein) of molecule ‘L (leucine)’ and the place to which <Rs₍₄₎₂^†;[] _7,9> belongs is not determined because this is an intermediate state for places p₅ and p₆ for which the index numbers 7–9 remain until the sequence of operations is completed and displayed using double round brackets (()).

Additionally, for the ‘entire operation f_(4→6);f_(6→7)) or f_(4→5);f_(5→7)’, we shall see whether the associative law holds. If the conversion procedure from a sequence of bases including explicit ‘E’s (place p₄) to a sequence of amino acids (protein) including explicit identity (empty) amino acids (proteins) ‘Δ’s (in place p₆) from the r-th to the s-th coordinate (s,r: non-negative integers with s≤r; s,r = 0,1,2,…) of, say, token Rs₍₄₎₂^† denoted ‘Rs₍₄₎₂^†;[[]] _s,r’, the following manipulation is verified for ‘Rs₍₄₎₂^†, = [E₀|A₁U₂G₃E₄E₅ E₆C₇U₈G₉E₁₀E₁₁E₁₂C₁₃G₁₄U₁₅E₁₆E₁₇E₁₈A₁₉C₂₀G₂₁||E₂₂E₂₃E₂₄E₂₅…] (in place p₄)’, (47) The combination of amino acids and bases that is underlined L_7–9 denotes the amino-acid molecule ‘L (leucine)’, and the place to which Rs₍₄₎₂^†;[[]] _7,9 belongs is not determined because this is an intermediate state of places p₄ and p₆ for which the index numbers 7–9 remain until the sequence of operations is completed and is displayed using double round brackets (()). Therefore, because ([[]] _7,9;[[]] _4,6);[[]] _19,21 = [[]] _7,9;([[]] _4,6;[[]] _19,21), the associative law for Rs₍₄₎₂^† holds.

In this way, ‘Rs₍₄₎₂^†;[[]] _7,9;[[]] _4,6;[[]] _19,21’ is calculated to be (48) where the underlined Δ_4–6, L_7–9, and T_19–21 refer to the amino-acid molecule (T: tyrosine), (the place is not determined for the reason previously stated).

In the present state, the morphism f_t-sp (= f_(4→6)) is not fired completely for similar reasons stated for Eq (47). Moreover, if all triplets (codons) are converted into any appropriate amino-acid sequence (protein), the product may be included in place p₆ as a token. That is, an additional operation may change any coordinate into a suitable amino-acid sequence (protein); for example,

‘((Rs₍₄₎₂^†;[[]] _7,9;[[]] _4,6;[[]] _19,21)); [[]] _1,3;[[]] _7,9;[[]] _10,18;[[]] _22,∞,

= [E₀|A₁U₂G₃(Δ_4–6)(L_7-9)(E₁₀E₁₁E₁₂)C₁₃G₁₄U₁₅(E₁₆E₁₇E₁₈)(T_19-21)

||E₂₂E₂₃E₂₄E₂₅…];[[]] _1,3;[[]] _7,9;[[]] _10,18;[[]] _22,∞,

= [Δ₀|M₁Δ₂L₃Δ₄R₅Δ₆T₇||Δ₈Δ₉Δ₁₀…] = Pr₍₆₎₂ (in place p₆)’, where the parts to the left of || belong to an amino-acid sequence (protein), and that on the right of || is an explicit identity base of ‘E’s (This sequence denotes a combination of amino acids (protein) and bases. Additionally, Δ₀ is required to be placed automatically). In detail, there are conversions: ‘A₁U₂G₃→M₁ (methionine)’, ‘E₄E₅E₆→Δ₂ (an explicit identity/empty an amino acid (protein))’, ‘C₇U₈G₉→L₃ (leucine)’, ‘E₁₀E₁₁E₁₂→Δ₄’, ‘C₁₃G₁₄U₁₅→R₅ (arginie)’, ‘E₁₆E₁₇E₁₈→Δ₆’, ‘A₁₉C₂₀G₂₀→T₇ (tyrosine)’, ‘E₂₂E₂₃E₂₄E₂₅…→Δ₈Δ₉Δ₁₀… (infinite identities)’ (in place p₆).

Here, the associative law for Rs₍₄₎₂^†, ([[]] _7,9;[[]] _4,6);[[]] _19,21 = [[]] _7,9;([[]] _4,6;[[]] _19,21), holds. In this regard, the practical utility of intermediate states such as those in Eqs (47) and (48) is unclear and less valuable because the associative law does not hold between morphism f_(4→6), f_(4→5), and f_(5→6).

Similarly, for morphism f_po-pr (= f_(6→7)); if the deletion of explicit ‘Δ’s from the r-th to the s-th coordinate (s,r: non-negative integers with s≤r; s,r = 0,1,2,…) of, say, token ‘Pr₍₆₎₂ is denoted ‘Pr₍₆₎₂;<< >>_s,r’, the following manipulation is verified for ‘Pr₍₆₎₅ = [Δ₀|M₁Δ₂Δ₃L₄R₅Δ₆Δ₇Δ₈T₉R₁₀Δ₁₁Δ₁₂L₁₃M₁₄||Δ₁₅Δ₁₆Δ₁₆…] = (in place p₆)’,

Pr₍₆₎₂^†;<< >>_2,3 = [Δ₀|M₁()_2-3 L₄R₅Δ₆Δ₇Δ₈T₉R₁₀Δ₁₁Δ₁₂L₁₃M₁₄||Δ₁₅Δ₁₆Δ₁₇…].

In this way, Pr₍₆₎₂^†;<< >>_2,3;<< >>_6,8;<< >>_11,12 = [Δ₀|M₁()_2-3L₄R₅()_6-8 T₉R₁₀Δ₁₁Δ₁₂L₁₃M₁₄||Δ₁₅Δ₁₆Δ₁₇…]. Then,

((Pr₍₆₎₂^†;<< >>_2,3;<< >>_6,8<< >>_11,12));<< >>_15,∞

= [Δ₀|M₁()_2-3L₄R₅()_6-8T₉R₁₀()_11-12 L₁₃M₁₄||()₁₅()₁₆()₁₇…] (49) The associative law ‘(<< >>_2,3;<< >>_6,8);<< >>_11,12 = << >>_2,3;(<< >>_6,8<< >>_11,12)’ holds for morphism f_po-pr (= f_(6→7)), although the proof is omitted.

In general, the associative law is necessary for composition in category theory and seems to be satisfied for morphisms f_t-sp (= f_(4→6)), f_po-pr (= f_(6→7)), and f_(4→7)). The identity morphisms f_(t-sp)0 (= f_(4→6)0), f_(po-pr)0 (= f_(6→7)0), and f_(4→7)0 (no operation) are also considered.

In summary, if a double-strand DNA of arbitrary length is given and is expressed as D_(k)j (primary DNA sequence) or [D_(k)j/D_(k)j^†] (double-stranded DNA sequences), then the following occurs: i) the firing of a morphism from f_(1→2) and f_(2→3) can be performed automatically; ii) then, in place p₃, the appropriate order that determines what parts are introns and exons is necessary so that an adequate region of exons can be translated into amino acids (protein); iii) if that is conducted, then, for a sequence of bases, an appropriate adjustment for the composition of the complete sequence of triplets is necessary in place p₄, and iv) the pathway either via f_(4→5)—f_(5→7) or via f_(4→6)—f_(6→7) can be performed automatically. As a result, an arbitrary D_(k)j or [D_(k)j/D_(k)j^†] can be translated into an amino-acid sequence (protein). For step iv), a visual graphic approach is possible using electronic tools that read/translate them.

§3–10. On a potential method to describe the protein sequences using the rotation group

In general, a sequence of amino acids contains biological information concerning the synthesis of various proteins. Conventionally, 20 amino acids such as ‘I (isoleucine), L (leucine), F (phenylalanine), G (glycine), S (serine), and R (arginine)’ exist in the human body and they are used as codings for protein synthesis. Here, we believe there are more than 20 different amino acids plus a ‘start amino acid’ ┴ (methionine (M) often plays this role), a ‘stop amino acid’ ┬, and a ‘do-nothing amino acid (an identity amino acid/protein) Δ’. They match up with the 23-fold phasor diagram on the Gaussian plane and the 23 polygon poles like the pentagon poles of Figs 4(B), 7(A), 7(B) and 9(B). Here, the ‘start amino acid’ ┴ acts as an order to start reading the amino-acid code; the ‘stop amino acid’ ┬ acts as an order to stop reading, and the ‘do-nothing amino acid (an identity amino acid/protein) Δ does nothing. The rotational group of order 23 {α₀, α₁, α₂,…α₂₁, α₂₂} is defined as ‘α_i;α_{j =} α_i+j (mod 23)’ with ‘α₁ = 2π/23 (rad)’. Similar to instances of a bijection between <D_(k)j> and the shape of the polygonal line of a single-strand DNA sequence, we infer that there is a bijection between <Pr_(7)j> (from Pr_(7)j, all ‘Δ’s other than the initial Δ₀ (0-th Δ) and infinite trail ‘ΔΔΔ…’ are deleted) and the shape of the polygonal line of the amino-acid sequence. The detection of an amino-acid sequence from only the shape of its polygonal line is possible, and vice versa. However, we believe that the development of this idea from the viewpoint of interpretation needs more analysis concerning the higher usable forms, e.g., ┴ and ┬, which may better be placed symmetrically to the line that is drawn from protein Δ to the center of the 23 polygons. To some degree, this proposition is unrealistic because we were not able to provide a meaningful order for the 23 proteins for which an inverse of α_i⁻¹ that satisfies ‘α_i⁻¹;α₁ = α₁;α_i⁻¹ ₌ α₀’ could have more meanings than appears in the definition of D_j or R_j; the shape of the polygonal line for the inverse D_(k)j^† is immediately obtained from the polygonal line of the symmetrical position of ‘D_(k)j’ around an arbitrary E₀. For this reason, although it seems possible, the composition of the cyclic group using amino acids has to be curtailed at this stage.

Fortunately, for the transformation/exchange of sequences of amino acids (proteins), there is a considerable number of models that treat the manner of exchange using for example the rotational group based on the regular icosahedron, the 20 vertices of which are associated with the 20 amino acids. Therefore, the transformation between the amino-acid sequences may be suitably treated in this existing formalism.

As a result, the canonical central dogma may be describable in an EPN [Fig 11(B)] where individual tokens express one of the products including DNA, RNA, pre-mRNA, mRNA, and related biological products.

§3–11. A treatment of the central dogma using matrices (Naganuma’s theory of operation matrix)

To overview the rules developed for the present EPN model, we refer to our previous report. The following morphisms are assumed to express the behaviors of DNA/RNA sequences with some adjustments; in this regard, for the present description, ‘°’ means the left translation rule, ‘;’ operations are executed in diagrammatic order for all processes, and {}; f (= f°{}) morphism f acts on element in the argument of object {}.

(Appendix A in S1 Appendix)

Conventionally, for the standard PN, an incidence matrix is often used to express the flow of the markings. Although not an incidence matrix, the matrix represents the flow of processes associated with the canonical central dogma. The entries of this “incidence-matrix-like” matrix A are operators with tokens for the respective markings. Multiplications involving A are interpreted as a flow of process information. Using the list of morphisms (Appendix A in S1 Appendix) and a flowchart [Fig 11(B)], examples of the operators and the entries in A (and A’) are shown in Appendix B in S1 Appendix. This application is referred to as the “theory of the operation matrix” proposed by Naganuma where the ‘operation matrix’ contains operators as entries in a square matrix and mathematically expresses general physical aspects of various models, with a special focus on the theoretical/mathematical limitations of simple systems [50]. In our model, as a specific example of that theory, the operators that act on ‘D_(k)j’s and ‘R_(k)j’s are included in the entries of matrix A, and as a result, time flows associated with the canonical central dogma are expressed discretely as multiplications by matrix A provided all morphisms (transitions) for the flow [Fig 11(B)] are fireable. Details of the calculation are described in Appendix B in S1 Appendix.

§3–12. A treatment of the somatic recombination (VDJ recombination) for antibodies using the concept of colimit

We now turn our attention to a more advanced use of our concepts, specifically, the expression of the limit or colimit (‘tensor product’ by ignoring the arrows and terminal object) that may apply to somatic recombination (VDJ recombination) for antibodies [51]. That is, the V segment is the sequence V = [V₁|V₂|V₃|…|V_w1||E|E|E|…]; in each part of ‘V₁,V₂,V₃,…,V_i,…,V_w1’, except for the infinite tail of ‘E’s (||EEE…), each V_i (i = 1,2,…,w₁) is also a sequence of bases that are parts of set V as follows. Suppose ‘N_v’ is the number equal to or greater than the maximum number of recognized non-E bases among (50) then, for example,

V_i = [C₁|A₂|…|G_(Nv−1)|T_Nv||E_(Nv+1)|E_(Nv+2)|E_(Nv+3)|…] = [C₁|A₂|G₃|T₄||E₅|E₆|E₇|…], where the number of recognized bases is ‘N_v = 4’. That is, in a certain V, with ‘N_v = 4’, the number of recognized non-E bases is smaller than N_v, the coordinates being filled with explicit ‘E’s as in V₃ = [T₁|G₂|T₃||E₄E₅…] so that N_v (in this case ‘N_v = 3’) can be chosen equivalent among V₁, V₂,…,V_w1 (∈object V) with ‘V_N ≥ V₁, V₂,…,V_w1’.

For a more practical example, we put ‘V₃ = [A₁|C₂||E₃E₄…]’; we find that the sequence of bases V₁, V₂, and V₃ belong to a unique object, (51) where the minimum N_v is ‘N_v = 4’ implying there are no ‘non-E’s on the right side of the N_v-th component in Eq (51). Note that in this case, N_v ≥4 suffices in satisfying condition (51).

Similarly, the D segment is the sequence D = [D₁|D₂|D₃|…|D_w2||E|E|E|…], a direct combination of ‘D₁,D₂,D₃,…,D_i,…,D_w2’ except for the infinite tail of ‘E’s (||EEE…); e.g., D₁ (i = 1,2,…,w₂) is also a sequence of bases that are parts of object D. Let ‘N_D’ be the number equal to or greater than the maximum number of recognized non-E bases among D₁, D₂,…, D_w2 (∈object D). For example,

D₂ = [G₁|…|C_ND||E_(ND+1)|E_(ND+2)|E_(ND+3)|…] = [G₁|A₂|C₃||E₄|E₅|E₆|…],

where the number of recognized bases is ‘N_D = 3’. Similarly, the J segment is the sequence J = [J₁|J₂|J₃|…|J_w3||E|E|E|…], a direct combination of ‘J₁,J₂,J₃,…,J_i,…,J_w3’ except for the infinite tail of ‘E’s (||EEE…); e.g., J₁ (i = 1,2,…,w₃) is also a sequence of bases of object J. Let ‘N_J’ be the number that is equal to or greater than the maximum number of recognized non-E bases among J₁, J₂,…, J_w3 (∈object J). Hence, as an example, we have

J₂ = [A₁|C2|…|GNJ||E(NJ+1)|E(NJ+2)|E(NJ+3)|…] = [A1|C2|T3|G₄|A₅||E₆|E₇|E₈|…],

where the number of recognized bases is ‘N_J = 5’. N_D and N_J must be counted in a similar manner as N_V in object V.

Therefore, because the following limit or colimit Π is considerable as in the other place p_k” (k”; an appropriate number), V⊗D⊗J = [V|D|J||EE…] in place p_k” (the product ⊗ means that for suitable numbers N_v, N_D, and N_J the sequences are glued together). Thus, (52) (∈ object Π; Π is the object of V⊗D⊗J with suitable N_v, N_D, and N_J). In the above description, ‘V_a⊗D_b⊗J_c’ (a, b, c are appropriate non-negative integers) is for example an element (a Cartesian product) of the tensor V⊗D⊗J. Here, we introduce the rule that a sequence of infinite tails of ‘E’s (||EEE…) be written automatically for any product in this model and disregarded when we discuss attributes such as the colimit (or tensor product), in practical theorems in category theory. Note that in category theory, the tensor product is viewed as a sort of colimit of objects V, D, and J where the forgetful functor acts on V⊗D⊗J so that the initial and terminal objects are ignored. Hence, we have the following notation; for appropriate numbers N_v, N_D, and N_J, (53) (∈ object Π; (co)limit V⊗D⊗J with an infinite tail of ‘E’s (||EEE…). Adjustments to the index number are necessary after trimming; however, we disregard strict counting of the index number provided that the respective bases are recognized rigorously.

Suppose that one of the elements of colimit (or tensor) Π is a Cartesian product as for {V₃, D₁, J₂} denoted by [V₃⊗D₁⊗J₂||EE…] in place p_k”, where an infinite tail ‘||EEE…’ accompanies automatically the element. Being an arbitrary product, it does not break the definition of a tensor product provided in that the number of recognized bases are accurate; in the above case, ‘N_V = 4’, ‘N_D = 3’, and ‘N_J = 5’. As an example, given ‘V_{3 =} [G₁|C₂|T₃|A₄|C₅||E₆|E₇|E₈|…], D₁ = [T₁|G₂|C₃||E₄|E₅|E₆|…], and J_{2 =} [C₁|T₂|A₃|G₄||E₅|E₆|E₇|…]’, their Cartesian product is written {V₃, D₁, J₂} = [V₃⊗D₁⊗J₂||EE…] and (54)

Here, the infinite tail of ‘E’s (||EEE…) is automatically placed outside of the scope of the (co)limit (or Cartesian product). The projection from object V×D×J to that of V×D, D×J, V, D, and J are describable in accordance with the conventional definition of a category.

[Fig 12(A)]

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Fig 12. Illustrations of a procedure for the recombination of an antibody using an EPN and conventional category theory.

(a). For reference, a procedure for the recombination of an antibody using compartments from segments V, D, and J is drawn as an EPN. In the present article, we mainly assume a ‘state machine’ where any transition has only a single input place and a single output place. However, this EPN does not comply with this rule. Using the map f_Int{, } for compartments of the V, D, and J segments, the recombination of an antibody may be treated automatically if transition f_Int{, } is free firing. Here, V_i×D_j×J_k (or V_i⊗D_j⊗J_k) is assumed to be a simple combination of V, D, and J. For the unique product V_i×D_j×J_k, there exist multiple pathways that commute in the diagram although actually, there exists a majority or minority between them owing to some biological mechanism. (b). In the diagram, V×D×J is composed as a limit of V, D, J from which the projection to V×D, D×J, V, D, and J are describable in accordance with conventional definitions of category theory. For example, π^1,2_V×D×J (projection from object V×D×J yields object V×D, etc). In this regard, we disregard any infinite tail of ‘E’s (||EEE…) of each DNA sequences. In the dual diagram, V×D×J is composed as a colimit of V, D, J where the direction of all projections is inverted. In the diagrams, the symbol for morphism (transition) ‘―|→’ is drawn as an ordinal arrow ‘→’ so as to be interpreted in the context of conventional category theory. For the composition of this EPN model, the morphism with transition | is also possible.

https://doi.org/10.1371/journal.pone.0302710.g012

These manipulations are interpreted as the result of the following operation via the morphism for the combination morphism f_Int{, } = {} ∪{} = {|}. Specifically, for the state ‘{D_j}(+){J_k} where (+) means the independent existence (or insertion) in the same place (in place p_k1), (55) Similarly, for the state ‘{V_i}(+){D_jJ_k} where (+) means the independent existence (or inclusion) in the same place,

f_Int{V_i, D_jJ_k} = {V_i}∪{D_jJ_k}: {[V_i||EE…]}∪{[D_jJ_k||EE…]}

= [{V_i|D_jJ_k}||EE…] = [{V_iD_jJ_k}||EE…] in place p_k3 holds.

By iterating the map of this morphisms f_Int{, }, the regions V, D, and J can be integrated into a unique sequence giving (56) [Fig 12(B)]

Note that because we regard object T as an object of tensor product V⊗D⊗J, simple sequences such as Eq (56) are not an element of object T but an element of object D (a simple base sequence with an infinite tail of ‘E’s). There may occur an instance for which an inverse of morphism f_spl{, } for the splitting procedure is considered, The splitting morphism f_spl{, } = {} ((+)) {} = {} (+) {} can be defined in accordance with Eq (29) with recognized tokens all included in the same place p_k (hence, in this example, we can simply replace ‘((+))’ by ‘(+)’); e.g., for token V_iD_jJ_k (in place p_k3), f_spl{V_iD_j, J_k} = {V_iD_j//J_k} = [{V_iD_j}||EE…] (+) [{J_k}||EE…] = [V_iD_j||EE…] (+) [J_k||EE…] (in place p_k). Because ‘f_spl{D_j,J_k} = {D_j//J_k} = [{D_j}||EE..] (+) [{J_k}||EE…] = [D_j||EE..] (+) [J_k||EE…] (in place p_k1)’, the following holds, (57)

Note that the associative law holds because the result is equivalent, (58)

In general, for morphisms f_Int{, } and f_spl{, }, the associative law holds, (59) (60) where j, k, and l are any appropriate non-negative integers. A category relation involving them is displayed in Fig 12(B).

Here, relationships between the morphisms are drawn although the infinite tails of ‘E’s are omitted in the display.

§3–13. Concerning an operation that perform indels of DNA sequences and its notation

For problems with mismatches in the index numbers (#remark 2) for the respective places p_k (k = 1,2,…), almost all coordinates of D_(k)j or R_(k’)j and Rs_(k)j seem to be simple; there may often occur instances for which the products of operations such as D_(k)l, R_(k’)l, and Rs_(k”)m, D_(k)m, R_(k’)m, and Rs_(k”)m may be regarded as being included in the same place p_k in extreme cases where all tokens are indexed with the same natural number ‘k’ as (k). Hence, the problem (#remark 2) does not matter so much. This circumstance seems to contribute to simplifications as exemplified in the following scenario.

With the developed notation for the recombination of DNA sequences, the following rule may be established; let D_(k)a and D_(k)b (in place p_k) be a certain finite DNA (or RNA; in this regard the place for DNA and that for RNA may be distinct with no specific use) sequence such as ‘D_(k)a = [A₁|T₂|C₃|G₄]’ and ‘D_(k)b = [G₁|C₂|T₃|A₄] (in place p_k)’, and put ‘D_(k)j = [G|C|T|A|C{D_(k)a}{D_(k)b}C|T|A|G||E|E|E|…] in place p_k1’. In fact, the specific content is ‘D_(k)j = [G₁|C₂|T₃|A₄|C₅{A₆|T₇|C₈|G₉}{G₁₀|C₁₁|T₁₂|A₁₃}C₁₄|T₁₅|A₁₆|G₁₇||E₁₈|E₁₉|E₂₀|…] (in place p_k1)’. The morphism f_excl {//} is applied to D_(k)j,

f_excl{}: [G|C|T|A|C//{D_(k)a}//{D_(k)b}C|T|A|G||E|E|E|…]

= [G₁|C₂|T₃|A₄|C₅//{A₆|T₇|C₈|G₉}//{D_(k)b}C₁₄|T₁₅|A₁₆|G₁₇||E₁₈|E₁₉|E₂₀|…].