The activation of natural killer (NK) cells, involved in innate immune response, is controlled by the ligands of the Toll-like receptors (TLRs) (see Figure 7.1 and Bulet et al., 1999; Elkon et al., 2007; Miyake et al., 2000). The gene GATA-3 is activating the gene BCL10, which is crucial for NFκB activation by T- and B-cell receptors (Zhou et al., 2004), and the protein ICAM1 is a type of intercellular adhesion molecule continuously present in low concentrations in the membranes of leucocytes involved in the blood adaptive immune response. The network controlling TLR and ICAM1 expression contains a couple of circuits, one positive five-circuit tangent to a negative five circuits, giving only one attractor (see Figure 7.1 and Table 7.1, red circle), which corresponds to the activation of the gene TLR.

Table 7.1

Total number of attractors in parallel dynamics (with T = 0), with 2 tangent circuits, left-circuit being negative of length l and right-circuit positive of length r (cf. Demongeot et al., 2012, and Annex A6). .

2.2 The links with the microRNAs

Most of the genes introduced here have links with microRNAs exerting a negative control on them and then, susceptible to deciding if the unique physiologic attractor will occur, by cancelling their target gene activity. Here are two examples of such microRNAs, negatively regulated by the circular RNA ciRs7 (Hansen et al., 2011, 2013):

1. for the subsequence pUNO-hRP105 of the TLR 2 gene (4937 bp) (http://mirdb.org/miRDB/; http://mirnamap.mbc.nctu.edu.tw/; Miyake et al., 2000), close to the reference sequence AL (cf. Annex A6 and Demongeot, 1978; Demongeot and Besson, 1983, 1996; Demongeot et al., 2003, 2009a, 2009b; Demongeot and Moreira, 2007), the hybridization is made by the microRNA miR 200a:

5’-CCAUUCAAGAUGAAUGGUACUG-3’ AL 14 anti-matches

5’-UCAUUGUUAUGCUACAGGUAUU-3’ ciRs7 14 anti-matches

3’-UGUAGCAAUGGUCUGUCACAAU-5’ hsa miR 200a 12 anti-matches

5’-UUGUGCUCAUUGAGAUGAAUGG-3’ pUNO-hRP105 mRNA starting in position 531

5’-UACUGCCAUUCAAGAUGAAUGG-3’ AL 15 matches

5’-CUGCCAUUCCUGAAGAAUAGCA-3’ ciRs7 17 matches

5’-AGGGAGCUACAAUUCAAGAUGA-3’ ciRs7 17 matches (significance of 2x17 matches: 2.5‰)

2. For the GATA 3 gene, hybridization is made by miR 200c (http://mirdb.org/miRDB/; http://mirnamap.mbc.nctu.edu.tw/):

5’-GCCAUUCAAGAUGA–AUGGUACU-3’ AL 13 anti-matches

5’-ACCAUCAUUAUCCCUAUUUUACA-3’ ciRs7 15 anti-matches

3'-AGGUAGUAAUGGGC–CGUCAUAA-5' has miR 200c 15 anti-matches

5’-UCUGCAUUUUUGCAGGAGCAGUA-3' GATA 3 mRNA starting in position 57

The gene expressing TLR contains the AGAUGAAUGG subsequence, belonging both to the D-loop of many tRNAs, to the reference sequence AL (Demongeot, 1978; Demongeot and Besson, 1983, 1996; Demongeot et al., 2003, 2009a, 2009b; Demongeot and Moreira, 2007) and to the circular RNA ciRs7, which signifies its affiliation to an ancestral genome, confirming the old origin of the innate immunologic system (Bulet et al., 1999; Elkon et al., 2007; Miyake et al., 2000) (see Annex A5 for the significance of the matches). In case of parallel updating (with T = 0), the network controlling the TLR production has 4 (resp. 1) attractor, if miR200c is (resp. not) expressed (see Tables 7.1 and 7.6 left bottom, red circles).

2.3 The adaptive immunetworks

The adaptive immunetworks are essentially made of three couples of tangent circuits (Figure 7.2) concerning the key genes GATA 3, transcriptional activator binding to DNA sites with the consensus sequence [AT]GATA[AG]), which controls negatively T cell receptors β (TCRβ), PU.1 (PUrine-rich box-1 gene), controlling negatively the recombination-activating gene (RAG) responsible of the V(D)J rearrangements giving birth to the TCRα receptors, and Zap70 (Zeta-chain-associated protein kinase 70 gene), controlling negatively TCRβ synthesis (Demongeot et al., 2012; Georgescu et al., 2008). These circuits are inserted into a global immunetwork (Figure 7.3), whose attractors are those of the three couples of circuits, the rest of the network being reducible to up- and down-trees connected to the circuits.

In the case of parallel updating (with T = 0; cf. Annex A1), Table 7.7 shows that a negative six-circuit tangent to a negative two-circuit has only one attractor (Table 7.7, blue circle at the bottom right), less than an isolated negative six-circuit that has six attractors (Table 7.2 blue circle), showing that for controlling RAG, Bcl1, and NK cells (i.e., in the case of both adaptive and innate mechanisms), if Notch2/CSL is silent, PU.1 is on and hence can activate both RAG and Bcl1, as well as promote NK cells. On the contrary, if Notch2/CSL inhibits PU.1, the immunologic system is paralyzed. In the same way, we can show that GATA 3 and ZAP70 networks each have three attractors (Table 7.7, orange circles on the right).

Table 7.7

Left: Total number of attractors of period p for positive (top) and negative (bottom) circuits of order n. Right: Total number of attractors in case of tangent circuits, where (a) the left circuit is negative and the right circuit positive and (b) both side circuits are negative with parallel updating and T = 0

(after Demongeot et al., 2012]). .

Table 7.2

Total Number of Attractors of Period p in Parallel Updating for a Unique Isolated Negative Circuit of Size n

(After Demongeot et al., 2012). .

The mathematical object modeling a real genetic regulatory network is called a genetic threshold Boolean regulatory network (denoted in the following as getBren). A getBren N can be considered as a set of random automata, defined by the following criteria (Robert, 1980):

1. Any random automaton i of the getBren N owns at time t a state x_i(t) valued in {0,1}, 0 (resp. 1) meaning that gene i is inactivated or in silence (Respectively activated or in expression). The global state of the getBren at time t, called configuration in the sequel, is then defined by x(t) = (x_i(t))_{i ∈ {1,n}} ∈ Ω = {0,1}ⁿ

2. A getBren N of size n is a triplet (W, Θ, P), where

• W is a matrix of order n, where the coefficient w_ij ∈ IR represents the interaction weight gene j has on gene i. Sign(W) = (α_ij = sign(w_ij)) is the adjacency (or incidence) matrix of a graph G, called the interaction graph.

• Θ is a threshold of dimension n, its component θ_i being the activation threshold attributed to automaton i.

• M: P(Ω) → [0,1]^m × m, where P(Ω) is the set of all subsets of Ω and m = 2ⁿ is a Markov transition matrix, built from local probability transitions P_i giving the new state of the gene i at time t + 1 according to W, Θ, and configuration x(t) of N at time t such that

$\forall g\in \{0,1\},\beta \in \Omega ,{P_{i,g}}^{\beta }\left(\left\{x_i\left(t+1\right)=g|x\left(t\right)=\beta \right\}\right)=exp\left[g\left({{\Sigma }_j}_{\in Ni}{w_i}_j{\beta }_j-{\theta }_i\right)/T\right]/Z_i\text{,}$

where Z_i = [1 + exp[(Σ_j∈ Niw_ijβ_j - θ_i)/T], N_i is the neighborhood of the gene i in the getBren N; i.e., the set of genes j (including possibly i) such that w_ij ≠ 0, and P_i,g^β is the probability for the gene i of passing to the state g at time t + 1, from the configuration β at time t on N_i. M denotes the transition matrix built from the P_i,g^β’s. M depends on the update mode chosen for changing the states of the getBren automata. In this chapter, we use the parallel or synchronous mode of updating.

For the extreme values of the randomness parameter T, we have the following:

1. If T = 0, getBren becomes a deterministic threshold automata network and the transition can be written as

$x_i\left(t+1\right)=h\left({{\mathrm{\Sigma }}_j}_{\in Ni}{w}_{ij}{x}_j\left(t\right)-{\theta }_i\right)\text{,}$

where h is the Heaviside function: h(y) = 1, if y > 0;

$\mathrm{h}\left(\mathrm{y}\right)=0,\mathrm{if}\mathrm{y}<0\text{,}$

except for the case Σ_j∈ Niw_ijx_j(t) − θ_I = 0, for which, if necessary, 1 and 0 are both chosen with probability ½. In this chapter, we chose T = 0.

2. When T tends to infinity, then P_i,g^β = ½ and each line M_i of M becomes the uniform distribution on the basin of the final class of the Markov matrix M to which i belongs (corresponding to an attraction basin when T = 0).

We define (Demongeot et al., 2003) the energy U and frustration F of a getBren N by

$\forall x\in \Omega ,U\left(x\right)={{\mathrm{\Sigma }}_i}_{,j\in \left\{1,n\right\}}{\alpha }_{ij}{x}_i{x}_j=Q_{+}\left(N\right)-F\left(x\right)\text{,}$

where Q₊(N) is the number of positive edges in the interaction graph G of the network N and F(x) the global frustration of x; i.e., the number of pairs (i,j) where the values of x_i and x_j are contradictory with the sign α_ij of the interaction between genes i and j: F(x) = Σ_i,j∈{1,n}F_ij(x), where F_ij is the local frustration of the pair (i,j) defined by

$\begin{array}{l}{F}_{ij}\left(x\right)=1,\mathrm{if}{\alpha }_{ij}=1,x_j=1\mathrm{and}{x}_i=0,\mathrm{or}{x}_j=0\mathrm{and}{x}_i=1,\mathrm{and}\mathrm{if}{\alpha }_{ij}=-1,x_j=1\mathrm{and}{x}_i=1,\\ \mathrm{or}{x}_j=0\mathrm{and}{x}_i=0,\\ F_{ij}\left(x\right)=0,\mathrm{elsewhere}\text{.}\end{array}$

Eventually, we define the random global dynamic frustration D by

$D\left(x\left(t\right)\right)={{\Sigma }_i}_{,j\in \left\{1,n\right\}}{D}_{ij}\left(x\left(t\right)\right)\text{,}$

where D_ij is the local dynamic frustration of the pair (i,j) defined by

$\begin{array}{l}{D}_{ij}\left(x\left(t\right)\right)=1,\mathrm{if}{\alpha }_{ij}=1,x_i\left(t\right)\ne h\left({{\mathrm{\Sigma }}_j}_{\in Ni}{w}_{ij}{x}_j\left(t\right)-{\theta }_i\right)\mathrm{or}{\alpha }_{ij}=-1,x_i\left(t\right)=h\left({{\mathrm{\Sigma }}_j}_{\in Ni}{w}_{ij}{x}_j\left(t\right)-{\theta }_i\right),\\ D_{ij}\left(x\left(t\right)\right)=0,\mathrm{elsewhere}\text{.}\end{array}$

A2 First propositions

Based on these definitions, we can prove the following propositions [cf. Demongeot and Waku (2012) and Demongeot et al. (2013b) for complete results]:

Proposition 1 is used to estimate the evolution of the robustness of a network because from Demongeot et al. (2014a, 2014b), it results that the quantity E = E_μ − E_attractor (= log2ⁿ − E_attractor, if μ is uniform), called evolutionary entropy, serves as a robustness parameter, being related to the capacity that a getBren has to return to μ, the equilibrium measure, after endogenous or exogenous perturbation. E_attractor can be evaluated by the quantity

${\mathrm{E}}_{\mathrm{attractor}}=-{\mathrm{\Sigma }}_{\mathrm{k}=1,\mathrm{m}\le 2}\mathrm{n}\mathrm{\mu }\left({\mathrm{C}}_{\mathrm{k}}\right)\mathrm{log\mu }\left({\mathrm{C}}_{\mathrm{k}}\right)\text{,}$

where m is the number of attractors and C_k = B(A_k)∪A_k is the union of the attractor A_k and of its attraction basin B(A_k). A systematic calculation of E_attractor allows quantifying the complexification of a network ensuring a dedicated regulatory function in different species. For example, the increase of the inhibitory sources in up-trees converging on a conserved subgraph of a genetic network causes a decrease of its attractor number by cutting some inhibited circuits, hence a decrease of E_attractor and an increase of the evolutionary entropy E, showing that the robustness of a network is positively correlated with its connectivity (i.e., the ratio between the numbers of interactions and genes in the network). Propositions 2 and 3 give examples of extreme cases where the networks are either discrete potential (or gradient) systems, generalizing previous works on continuous dynamics in which authors attempt to explicit Waddington and Thom chreode’s energy functions, conserved or dissipated. In Demongeot et al. (2012), a method was proposed for calculating the number of attractors in the case of circuits with Boolean transitions reduced to identity or negation. These results about attractors counting constitute a partial response to the discrete version of the 16th Hilbert’s problem and can be approached by using Hamiltonian energy levels. For example, for a positive circuit of order 8, it is easy to prove that, in case of parallel updating, we have only even values for the global frustration D (they are odd for a negative circuit), corresponding to different values of the period of the attractors (Table 7.6).

A3 Tangent and intersecting circuits

The study of tangent and intersecting circuits in strong connected components of a genetic network is possible when interactions are either identity or negation (Demongeot et al., 2012; http://dev.biologists.org/content/131/12/2911/ F8.large.jpg), with a mixing rule monotonic when different arcs come on the same node. For example, such circuits with one or two genes in common are shown in Figure 7.10.

By looking on the networks of Figure 7.10, we see that each is made of four main paths of opposite senses: two are up (A and C), and two down (B and D), with the respective lengths of ℓ_A, ℓ_B, ℓ_C, and ℓ_D, and parities of s_A, s_B, s_C, and s_D, equal to 1 (resp. − 1) if they have an even (resp. odd) number of negative arcs, with s_A = Π_a∈A s_a,, where the sign s_a of the arc a of A is equal to − 1 if a is negative (inhibition) and 1 if a is positive (activation).

For example, in Figure 7.10, the path A of N₁ is such as ℓ_A = 3 and s_A = − 1. The main paths have in N₁ two common nodes, 10 and 13, and in N₂, only one common node 23. Finally, with N₁, having four paths and two common nodes, there are four combinations giving four possible circuits (A,B), (B,C), (A,D), and (D,C), with a circuit like (A,C) having the parity s_A,C = s_A s_C. In the following discussion, to facilitate the reasoning, we will suppose that the state 0 of a gene is replaced by the state − 1.

Let us denote as ((J,L),(L,M)) the general couple of circuits inside the set of the six possible couples created from these four circuits. If N((J,K),(L,M)) denotes the number of possible attractors of ((J,K),(L,M)), then the attractor number of N₁ is the minimum of the values of N((J,K),(L,M)) for the six couples of circuits. We conjectured (Demongeot and Moreira, 2007) that this number was less than the attractor number of N₂, the attractors of N₁ being those that allow the two common nodes to have the same state for each couple of circuits.

The minimum can be made more precise: suppose that two circuits (K,L) and (M,) have two common nodes with the same state. Any attractor of these intersected circuits has the same configurations as one of those of the tangent circuits that we can build by decoupling one on their two common nodes. Indeed, if not, there is at least one node different from the common nodes having an asymptotic sequence of states different for N₁ and N₂. Starting from this node and following the path going from this node to the common node of N₂, we would find for this node a sequence of states different that those observed in N₂ attractors, which is impossible. This reduces considerably the possible attractors for intersecting circuits, since they must always have the same state on their two intersected nodes. Then, counting attractors corresponds to a known combinatorial problem generalizing the necklace problem (cf. Demongeot et al., 2012; and Table 7.7). Then two new questions remain open:

Q1: Is a given attractor of the network N₂ respecting the constraint ${x_{20}}_{{}_0}\left(t\right)=x_{2_0}\left(t\right))$, identical to an attractor of N₂?

Q2: If the answer to Q1 is yes, what are the constraints of its period?

The following propositions partially address Q1 and Q2.

A4 State-dependent updating schedule

A last important feature of the getBren dynamics is the existence of genes influencing directly the opening of the DNA inside the chromatine, hence allowing or disallowing the gene expression. If these genes are controlled by microRNA, it is necessary to generalize the getBren structure by considering that the possibility to update a block of genes at iteration t depends on the state of r clock genes (i.e., involved in the chromatine updating clock) k₁,…k_r (like histone acetyltransferase, endonucleases, exonucleases, helicase, replicase, and polymerases) depending on s microRNAs, l₁,…,l_s. Then the transition for a gene i, such as i, does not belong to {k₁,…k_r}, could be written as

$\forall g\in \{0,1\},\beta \in {\left\{0,,,1\right\}}^n,i\mathrm{f}\forall j=1,\dots ,r,{x_k}_j\left(t\right)=1\text{,}$

then

(i) P_i,g^β({x_i(t + 1) = gx(t) = β}) = exp[g(Σ_j∈ Niw_ijβ_j-θ_i)/T]/[1 + exp[(Σ_j∈ Niw_ijβ_j-θ_i)/T], if microRNAs l₁,…,l_s are dominant;

(ii) P_i,g^β({x_i(t + 1) = β_ix(t) = β}) = 1, if not.

The case (i) implies that ∀ j = 1,…, s, x_lj(t-1) = 0. To make the transition rule more precise, we can, for the sake of simplicity, decide that the indices k₁,…k_r of the r clock genes are 1,…, r and then we have the three possible following behaviors:

• If y(t) = Π_{i = 1,…,r}x_i(t) = 1, then rule (ii) is available.

• If y(t) = 0 and Σ_{s = t,…,t-c}y(s) > 0, then x(t + 1) = x(t-s*), where s* is the last time before t, where y(s*) = 1.

• If y(t) = 0 and Σ_{s = t,…,t-c}y(s) = 0, then x(t + 1) = 0 (by exhaustion of the pool of genes still in expression).

The dynamical system remains autonomous with respect to the time t [i.e., it depends on t only through the set of state variables {x(t-c),…, x(t-1)}], but a theoretical study of its attractors (as in Demongeot et al., 2012), with a state-dependent updating schedule is difficult to perform and will be investigated further in the future.

A5 The circular Hamming distance

The most usual way to compare vectors with values in a finite alphabet is through the Hamming distance. Given two vectors x, y ∈ Aⁿ, the Hamming distance between them is

${\mathrm{d}}_{\mathrm{H}}\left(\mathrm{x},\mathrm{y}\right)=\#\left\{\mathrm{i}\in \left\{0,...,\mathrm{n}-1\right\}:{\mathrm{x}}_{\mathrm{i}}\ne {\mathrm{y}}_{\mathrm{i}}\right\}\text{.}$

In other words, it is the number of positions in which the values of the vectors differ. The function d_H is a metric: it is nonnegative and symmetric, it satisfies the triangle inequality, and a null distance implies identity of the vectors. It is also easy to see that

$\forall \mathrm{i}\in \left\{0,...,\mathrm{n}-1\right\},{\mathrm{d}}_{\mathrm{H}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{d}}_{\mathrm{H}}\left({\mathrm{\sigma }}^{\mathrm{i}}\left(\mathrm{x}\right),{\mathrm{\sigma }}^{\mathrm{i}}\left(\mathrm{y}\right)\right),\mathrm{and}\mathrm{hence}{\mathrm{d}}_{\mathrm{H}}\left(\mathrm{x},{\mathrm{\sigma }}^{\mathrm{i}}\left(\mathrm{y}\right)\right)={\mathrm{d}}_{\mathrm{H}}\left({\mathrm{\sigma }}^{-\mathrm{i}}\left(\mathrm{x}\right),\mathrm{y}\right)\text{.}$

Using this last property, we define the circular Hamming distance between two rings [x] and [y] as

$\begin{array}{l}{{\mathrm{d}}^{\mathrm{c}}}_{\mathrm{H}}\left(\left[\mathrm{x}\right],\left[\mathrm{y}\right]\right)=min{\mathrm{d}}_{\mathrm{H}}\left(\mathrm{x},{\mathrm{\sigma }}^{\mathrm{k}}\left(\mathrm{y}\right)\right)\\ 0\le \mathrm{k}\le \mathrm{n}-1\end{array}$

In general, the minimum between two metrics is not necessarily a metric, but here it holds.

The cumulative distribution function F_n,k of the circular Hamming proximity p^c_H = n - d^c_H, defined by k permutations of a ring of length n can be calculated from the cumulative function G_n,p of the binomial law B(n,p) of order n, by the formula:

$\forall \mathrm{i}\in \mathrm{N},{\mathrm{F}}_{\mathrm{n},\mathrm{k}}\left(\mathrm{i}\right)=\mathrm{P}\left({{\{\mathrm{p}}^{\mathrm{c}}}_{\mathrm{H}}\le \mathrm{i}\}\right)=\mathrm{P}\left(\left\{{sup}_{\mathrm{j}=1,\mathrm{k}}{\mathrm{p}}_{\mathrm{H},\mathrm{j}}\le \mathrm{i}\right\}\right)=\mathrm{P}\left({\cap }_{\mathrm{j}=1,\mathrm{k}}\left\{{\mathrm{p}}_{\mathrm{H},\mathrm{j}}\le \mathrm{i}\right\}\right)={\mathrm{G}}_{\mathrm{n},\mathrm{p}}{\left(\mathrm{i}\right)}^{\mathrm{k}}$

For example, if n = 22, k = 22, and p = 4/16 = 1/4 (i.e., the case of the circular Hamming distance between a small RNA of length 22 and the sequence AL):

$\begin{array}{l}{\mathrm{F}}_{22,22}\left(12\right)={\mathrm{G}}_{22,1/4}{\left(12\right)}^{22}\approx {\left(0.9993\right)}^{22}\approx 0.985\\ {\mathrm{F}}_{22,22}\left(13\right)={\mathrm{G}}_{22,1/4}{\left(13\right)}^{22}\approx {\left(0.99998\right)}^{22}\approx 0.9996\end{array}$

Note the following: the probability that the number of matches of a ring of length 22 with a linear sequence of length 22 is k or more is equal to P_k = 1 − F_22,22(k − 1) = 1 − G_22,1/4(k-1)²²; i.e., the probability that the circular Hamming distance is strictly less than k. For k = 14, P₁₃ = 1 − G_22,1/4(12)^22:; is about 1.5%. In the same way, the probability P₁₄ = 1-G_22,1/4(13)²² is about 0.4‰, and P₁₇ = 1-G_22,1/4(16)²² is about 8.8 10^- 6. If the linear sequence is of length 129126 (like ciRs7), then the probability to observe at least 17 matches once is about 5%.

In addition, if n = 22 and p = 6/16 = 3/8 = 0.375 (i.e., the case of Hamming proximity with the constraint to match a substring of length 5 like AUGGU or UGGUA and authorize A–U, C–G and U–G coupling, between a small RNA of length 22 containing the substring and the sequence AL):

${\mathrm{F}}_{22}\left(13\right)={\mathrm{G}}_{22,3/8}\left(13\right)\approx 0.9885$

Hence, the probability P₁₄ = 1-F₂₂(13) that Hamming proximity is 14 or more, is about 1.5‰.

A6 The ArchetypaL sequence AL

In Demongeot and Moreira (2007), a sequence of bases called AL (for ArchetypaL) is described as follows: 5’-UGCCAUUCAAGAUGAAUGGUAC-3’ corresponding to a putative circular RNA with a possible hairpin form (cf. Figure 7.6, and Demongeot and Moreira, 2007; Meyer and Nelson, 2011; Turk-Mcleod et al., 2012; de Vladar, 2012; Yarus, 1988, 2010, 2013). AL can serve as a primitive ribosome in the sense that its circular form can bind any amino acid [with the weak electromagnetic or van der Waals interactions described in the Direct RNA Templating (DRT) hypothesis on the origin of the genetic code, which is still under debate (Demongeot and Moreira, 2007; Meyer and Nelson, 2011; Turk-Mcleod et al., 2012; de Vladar, 2012; Yarus, 1988, 2010, 2013) to one of the triplets of its synonymy class in the genetic code, allowing the formation of small peptides (Yarus, 2013).

The sequence AL share many subsequences, like quintuplets, with small RNAs coming from Rfam (Griffiths-Jones et al., 2005), a database containing information about noncoding RNA families and structured RNA molecules, like transfer RNA (tRNA).

If we compare AL to the sequence of the circular RNA ciRs7, we find qualitative similarities, with 17/22 quintuplets passing the 5% upper threshold of significance (the 5%-threshold number of occurrences of a quintuplet in ciRs being equal to 129126/1024 + 1.6 x 11 = 144) of an unrandom frequency of common triplets (Figure 7.11). The occurrence numbers of the 22 successive quintuplets of AL inside the sequence ciRs7 of length 129126 have the following values, with the local maxima in red:

uucaa (Tψ-loop) 250 ucaag 154 caaga 146 aagau 163 agaug 163 gauga 122 augaa 211 ugaau 238 (articulation loop) gaaug 152 aaugg 145 auggu 156 (D-loop) uggua 120 gguac 62 guacu 90 uacug 143 acugc 129 cugcc 160 (anticodon-loop) ugcca 155 gccau 121 ccauu 198 cauuc 155 auuca 206.

The similarity between the function of circular RNAs, tRNAs and the sequence AL could come from the fact that they are all concerned by the protein synthesis, directly for the tRNAs and its ancestor AL, and indirectly, by inhibiting translational inhibitors as the microRNAs. This functional proximity could explain the frequent presence as relics of subsequences of AL inside the ciRs and tRNAs.

In the case of the tRNAs, the similitude concerns the conserved (inside and between species) bases of their loops (cf. Figure 7.12 and Alexander et al., 2010; Brown et al., 1986; Demongeot and Moreira, 2007; Shigi et al., 2002; Ueda et al., 1992; Yu et al., 2011), as well as of some particular tRNAs, such as where the ordered sequences of the loops is identical to AL except for two bases (Demongeot and Moreira, 2007).