Frameshift signals in genes associated with the circular code

---

Abstract

Three sets of 20 trinucleotides are preferentially associated with the reading frames and their 2 shifted frames of both eukaryotic and prokaryotic genes. These 3 sets are circular codes. They allow retrieval of any frame in genes (containing these circular code words), locally anywhere in the 3 frames and in particular without start codons in the reading frame, and automatically with the reading of a few nucleotides. The circular code in the reading frame, noted X, which can deduce the 2 other circular codes in the shifted frames by permutation, is the information used for analysing frameshift genes, i. e. genes with a change of reading frame during translation. This work studies the circular code signal around their frameshift sites. Two scoring methods are developed, a function P based on this code X and a function Q based both on this code X and the 4 trinucleotides with identical nucleotides. They detect a significant correlation between the code X and the −1 frameshift signals in both eukaryotic and prokaryotic genes, and the +1 frameshift signals in eukaryotic genes.

Keywords: frameshift gene, frameshift signal, circular code, trinucleotide, frame, statistical method

---

Introduction

Frameshift

Definition 1.

By convention, the reading frame in a gene established by a start codon ATG, GTG and TTG is the frame 0, and the shifted frames 1 and 2 are the reading frame 0 shifted by 1 and 2 nucleotides in the 5' – 3' direction, respectively.

In the reading frame, a series of 3 nucleotides (codons) is translated into a series of amino acids according to the genetic code. The correspondence between the nucleotide sequence and the protein sequence is often considered as an immutable dogma. Nevertheless, the standard rule of genetic decoding is altered in specific genes by different events that are globally termed recoding [Gesteland et al., 1992]. These events are classified into 2 main groups: those occurring at the translation termination step (stop codon readthrough) and those occurring during elongation, such as frameshifts (reviewed e. g. in Farabaugh, 1996; Namy et al., 2004; Cobucci-Ponzano et al., 2005).

Frameshift is conceptually a simple process. At a particular position in the nucleotide sequence, the ribosome shifts its reading frame into a new one. It will continue reading in a shifted frame until it encounters a stop codon. Therefore, the protein product expressed will partly be encoded in the reading frame upstream of the frameshift site and partly in the shifted frame downstream of it. Two main categories of frameshifts are distinguished. The ribosomal readthrough of a single nucleotide shifting the reading frame forward by one base (5' – 3' direction) is a +1 frameshift (Fig. 1). After the frameshift site, the new reading frame is the frame 1. The ribosomal reread of the third nucleotide of the last translated trinucleotide shifting the reading frame backward by one base (3' – 5' direction) is a −1 frameshift (Fig. 2). After the frameshift site, the new reading frame is then the frame 2. Recently, another class of frameshifts has been identified in phage Mu. The newly observed −2 frameshift has the same overall effect on the protein sequence as the +1 frameshift, except for an additional amino acid encoded at the frameshift site [Xu et al., 2004].

Natural frameshifts without particular slippery signals can occur but are highly improbable with a rate estimated at 3 × 10^-5 per codon [Parker, 1989]. Frameshifts can be much more frequent. More than 50% of the ribosomes can shift at sites of some genes. These particular shifts are called programmed frameshifts. So, translational recoding can be in competition with normal decoding. A typical eukaryotic site triggering a −1 frameshift is the slippery heptamer X,XX.Y,YY.Z, codons in the reading frame being separated by commas and codons in the shifted frame, by dots [Baranov et al., 2006]. Frameshifts generally occur in mid-passage of genes [Craigen and Caskey, 1987; Atkins et al., 1990]. They are found in eukaryotes, prokaryotes, viruses and transposons and are involved in a variety of biological processes [Namy et al., 2004].

Gene detection algorithms generally consider shifted frames to be sequencing errors or pseudogene signatures and only a few algorithms assign a frameshift as a possible regulatory process [Harrison et al., 2002]. However, frameshifts allow expression of several polypeptides from the same mRNA [Gesteland et al., 1992]. Although most of recoding events described so far have been found in small autonomous genetic elements [Bekaert et al., 2005], a few cellular genes are known to be expressed by this mode of control [Namy et al., 2004], most of them found by chance. Besides permitting the identification of new functional genes, the study of frameshifts could also be interesting for understanding the mechanisms maintaining the reading frames.

As we mentioned in our previous publications (reviewed in Michel, 2007), some particular sets of trinucleotides, called circular codes, may have a role in detecting and maintaining the reading frames in genes. Therefore, the circular code property may be altered with the -1 and +1 frameshifts. A possible relation between circular codes and frame alteration is investigated here by developing a new approach based on 2 score functions with the 3 circular codes X and its 2 permuted sets X₁ and X₂ of 20 trinucleotides preferentially associated with the reading frame and their 2 shifted frames 1 and 2 of both eukaryotic and prokaryotic genes. Therefore, the main code X with its properties allows retrieval of any frame in genes (containing these circular code words), locally anywhere in the 3 frames and in particular without start codons in the reading frame, and automatically with the reading of 13 nucleotides in each frame. The frameshift positions are statistically significantly identified for the −1 and +1 frameshifts of eukaryotic populations, and for the −1 frameshifts of prokaryotic population. This approach does not use any recoding model but only a very short sliding window of only 4 trinucleotides associated with the theoretical property of reading with the circular code.

After a brief presentation of the definition and the main properties of circular codes, the 2 score functions developed for analysing the circular code signal around the frameshift sites are detailed.

Common circular code

Identification

In 1996, a simple occurrence study of the 64 trinucleotides = {AAA,...,TTT} in the 3 frames of genes has shown that the trinucleotides are not uniformly distributed in these 3 frames. By excluding the 4 trinucleotides with identical nucleotides _id = {AAA,CCC,GGG,TTT} and by assigning each trinucleotide to a preferential frame (frame of its highest frequency), the same 3 subsets X₀, X₁ and X₂ of 20 trinucleotides are identified in the frames 0, 1 and 2 respectively of 2 large and different gene populations of eukaryotes (26,757 sequences, 11,397,678 trinucleotides) and prokaryotes (13,686 sequences, 4,708,758 trinucleotides) [Arquès and Michel, 1996]. These 3 trinucleotide subsets present several strong biomathematical properties, in particular the property of circular code. The subset X₀ = {AAC,AAT,ACC,ATC,ATT,CAG,CTC,CTG,GAA,GAC,GAG,GAT,GCC,GGC,GGT,GTA,GTC,GTT,TAC,TTC} of 20 trinucleotides in frame 0 is a common circular code as it is observed in the reading frames of both eukaryotic and prokaryotic genes and furthermore, a C³ code as its 2 permuted sets X₁ and X₂ are also circular codes (Tab. 1). X₀ is simply noted C³ code X. Due to the law of large numbers, these 3 trinucleotide subsets are (obviously) retrieved in these 2 gene populations with the actual statistical studies (results not shown).

We recall the main properties of the common C³ code X which will be involved in this paper. A recent review of circular codes in genes details the research context, the history and their different properties [Michel, 2007].

Notation 1.

being a finite alphabet, * denotes the words over of finite length including the empty word ε of length 0 and ⁺, the words over of finite length greater or equal to 1. Let w₁w₂ be the concatenation of the 2 words w₁ and w₂.

Definition 2.

The (left circular) permutation of a trinucleotide w₀ = l₀l_ll₂, w₀ , is the permuted trinucleotide (w₀) = w₁ = l₁l₂l₀, e. g. (AAC) = ACA, and ((w₀)) = (w₁) = w₂ = l₂l₀l₁, e. g. ((AAC)) = CAA. This definition is naturally extended to the trinucleotide set permutation: the permutation of a set of trinucleotides is the permuted trinucleotide set obtained by the permutation of all its trinucleotides.

Definition 3.

The complementarity of a trinucleotide w₀ = l₀l₁l₂, w₀ , is the complementary trinucleotide (w₀) = (l₂)(l₁)(l₀) with (A) = T, (C) = G, (G) = C, (T) = A, e. g. (AAC) = GTT. This definition is naturally extended to the trinucleotide set complementarity.

Definition 4.

A subset X of is a circular code if n, m ≥ 1 and x₁, x₂, ..., x_n, y₁, y₂, ..., y_m X and r *, s , the equalities sx₂ ... x_nr = y₁y₂ ...y_m and x₁ = rs imply n = m, r = ε and x_i = y_i, 1 ≤ i ≤ n.

A circular code allows the reading frames in genes to be retrieved. It is a set of words over an alphabet such that any word written on a circle (the next letter after the last letter of the word being the first letter) has a unique decomposition into words of the circular code. As an example, let the set Y be composed of the 6 following words: Y = {AAT,ATG,CCT,CTA,GCC,GGC} and the word w, be a series of the 9 following letters: w = ATGGCCCTA. The word w, written on a circle, can be factorized into words of Y according to 2 different ways: ATG,GCC,CTA and AAT,GGC,CCT, the commas showing the way of decomposition (Fig. 3). Therefore, Y is not a circular code. In contrast, if the set Z obtained by replacing the word GGC of Y by GTC is considered, i. e. Z = {ATT,ATG,CCT,CTA,GCC,GTC}, then there never exists an ambiguous word with Z, such as w for Y, and then Z is a circular code.

The construction frame of a word generated by any concatenation of words of a circular code can be retrieved after the reading, anywhere in the generated word, of a certain number of nucleotides depending on the code. This series of nucleotides is called the window of the circular code. Then, the minimal window length is the size of the longest ambiguous word which can be read in at least 2 frames, more one letter.

Main properties

We point out only the circular code properties related to the aim of the paper.

Property 1:

Maximality: X₀ is a maximal circular code (20 trinucleotides) as it is not contained in a larger circular code, i. e. in a circular code with more words.

A circular code has several constraints:

1. The 4 trinucleotides _id must be excluded from such a code. Indeed, the concatenation of AAA with itself, for example, does not allow the reading (original) frame to be retrieved as there are 3 possible decompositions: ...AAA,AAA,AAA,..., ...A,AAA,AAA,AA... and ...AA,AAA,AAA,A...
2. The permuted trinucleotides by the permutation of a trinucleotide, for example (AAC) = ACA, must also be excluded from such a code. Indeed, the concatenation of AAC with itself, for example, also does not allow the reading frame to be retrieved as there are 2 possible decompositions: ...AAC,AAC,AAC,... and ...A,ACA,ACA,AC...

Therefore, by excluding the 4 trinucleotides _id and by gathering the 60 remaining trinucleotides in 20 classes of 3 trinucleotides such that, in each class, 3 trinucleotides are deduced from each other by circular permutations, e. g. AAC, (AAC) = ACA and ((AAC)) = CAA, a circular code has only one trinucleotide per class and therefore contains at most 20 trinucleotides. Then, any 20-long circular code is maximal and there are 3²⁰ 3.5 billions of potential codes of length 20.

Property 2:

Permutation: (X₀) = X₁ and ((X₀)) = X₂ (X₀ generates X₁ by one permutation and X₂ by another permutation).

Property 3:

Complementarity: (X₀) = X₀ (X₀ is self-complementary) and, (X₁) = X₂ and (X₂) = X₁ (X₁ and X₂ are complementary to each other).

Property 4:

C³ code: X₁ and X₂ obtained by permutation of X₀ (property 2) are also maximal circular codes (property 1). Therefore, if X₀, X₁ and X₂ are circular codes, then X₀, X₁ and X₂ are C³ codes. As the code X₀ is coding for the reading frame (frame 0) in genes, i. e. the most important frame, and as it is self-complementary (property 3), it is considered as the main C³ code and noted X simply.

Remark 1.

A circular code Y₀ does not necessarily imply that Y₁ and Y₂ obtained by permutation of Y₀, are also circular codes.

Property 5:

Rarity: the occurrence probability of the complementary C³ code X is equal to 216/3²⁰ 6 × 10^-8.

This probability is equal to the computed number of complementary C³ codes (216) divided by the number of potential codes (3²⁰ = 3,486,784,401, see property 1).

Property 6:

Largest window length: the lengths of the minimal windows of X₀, X₁ and X₂ for retrieving the frames 0, 1 and 2 respectively, are all equal to 13 nucleotides and represent the largest window length.

Property 7:

Common: the C³ code X is identified in both eukaryotic and prokaryotic genes.

Property 8:

The common C³ code X (properties 4 and 7) with its particular structure (property 5) allows retrieval of any frame in genes (containing these circular code words), locally anywhere in the 3 frames and in particular without start codons in the reading frame, and automatically with the same window length of 13 nucleotides in each frame (property 6).

---

Method

Definition of a score function P based on the common C³ code X

In order to identify genes with frameshift sites, we develop an algorithm based on the common C³ code X and its property 8.

Let = {AAA,...,TTT} be the set of 64 trinucleotides over the alphabet {A,C,G,T}. Let t be a trinucleotide. Let F be a frameshift gene population with n(F) sequences s. Each sequence s has a frameshift site in the nucleotide position i = 0. Let t₀ be the first downstream trinucleotide after the frameshift site. Precisely, in the +1 frameshift, the 1st nucleotide after the shifted nucleotide is the 1st nucleotide of t₀ (Fig. 4). In the −1 frameshift, the shifted nucleotide is the 1st nucleotide of t₀ (Fig. 5). By convention, all trinucleotides before the frameshift site have a negative position i < 0, all trinucleotides after the frameshift site have a non-negative position i ≥ 0 and the method reading frame is ...,t₋₁, t₀, t₁,.... The sequence s is considered as a series of trinucleotides t_i. Let w_i = t_i0t_i1t_i2t_i3 be a window of length |w| = 4 trinucleotides in the sequence s, where t_i0 is the ith trinucleotide in s and, t_ij, the jth trinucleotide in w_i. This sliding window length is the length of the minimal windows of the 3 circular codes X₀, X₁ and X₂ for retrieving the frames 0, 1 and 2 respectively in genes (property 6). Let X_f, f {0,1,2}, be the 3 codes X₀, X₁ and X₂ in the 3 frames f.

In a given window w_i, the function δ_f(t_ij) indicates if the trinucleotide t_ij belongs or not to the code X_f

with f {0,1,2} and j {0,1,2,3}. Each sequence s is associated with a frame {0, 1, 2}. In s₀ (s₁ and s₂ resp.), t₀ is the first downstream trinucleotide (more one nucleotide and 2 nucleotides resp.) after the frameshift site. Then, the score P(X_f, i, ) of the code X_f in a window w_i of a given frame of a sequence s is

The score P(i, ) of the C³ code X in a window w_i of a given frame of a sequence s measuring the frame retrieval intensity, is defined as

Then, the score P(i, s) of the C³ code X in a window w_i in the average frame of a sequence s is

Finally, the score P(i, F) of the C³ code X in a window w_i in the average frame of a gene population F is

Proposition 1.

If the words of the 3 codes X₀, X₁ and X₂ are uniformly distributed in the windows w_i in each frame of the sequences of a population F, then P(X_f , i, ) = P(X_{f '} , i, ) with f ' f leading to P(i, F) = 0. Zero is the minimum value for P(i, F).

Proposition 2.

If the words of only one code X_f occur in the windows w_i in each frame of the sequences of a population F, then P(X_f , i, ) = 1, P(X_{f '} , i, ) = 0 with f ' f leading to P(i, F) = 1. One is the maximum value for P(i, F).

Proposition 3.

0 ≤ P(i, F) ≤ 1 (consequence of propositions 1 and 2).

Definition of a score function Q based on both the common C³ code X and the trinucleotide set _id

The score function P based only on the common C³ code X allows detection of frameshift signals in genes. It does not consider the 4 trinucleotides with identical nucleotides _id = {AAA,CCC,GGG,TTT} which are excluded in a circular code (property 1). In contrast to the circular code, the 4 trinucleotides _id cannot identify frames in genes. This information is now considered in the score function Q, obviously in a contrary way to the circular code information. The function Q based on both the common C³ code X and the trinucleotides _id is developed in a way similar to the function P for detecting frameshift signals in genes.

In a given window w_i, the function δ_id(t_ij) indicates if the trinucleotide t_ij belongs or not to the set _id

with j {0,1,2,3}. Then, the score Q(_id, i, ) of the set _id in a window w_i of a given frame of a sequence s is

Thus, the score Q(i, ) of the C³ code X in a window w_i of a given frame of a sequence s by withdrawing the score Q(_id, i, ) associated with the set _id which does not allow retrieval of the frame, is

Then, the score Q(i, s) of the C³ code X in a window w_i in the average frame of a sequence s is

Finally, the score Q(i, F) of the C³ code X in a window w_i in the average frame of a gene population F is

Proposition 4.

If only one trinucleotide of _id occur in the sequences of a population F, then Q(_id, i, ) = 1 leading to Q(i, F) = −1. Minus one is the minimum value for Q(i, F).

Proposition 5.

If the words of only one code X_f occur in the windows w_i in each frame of the sequences of a population F, then P(i, ) = 1 and Q(_id, i, ) = 0 leading to Q(i, F) =1 . One is the maximum value for Q(i, F).

Proposition 6.

−1 ≤ Q(i, F) ≤ 1 (consequence of propositions 4 and 5).

Data acquisition

The frameshift genes used in this study are extracted from the RECODE database [Baranov et al., 2001]. The RECODE database is a compilation of programmed translational recoding events. It deals with programmed ribosomal frameshifts, codon redefinition and translational bypass occurring in a variety of organisms. Each entry includes the gene, its encoded protein for both normal and alternate decoding, the type of the recoding event involved and the trans-factors and cis-elements that influence recoding.

Our study concerns the −1 and +1 frameshifts of eukaryotic and prokaryotic genes as the common C³ code X has been identified only in these genes and not, for example, in viral genes [Arquès and Michel, 1996]. Therefore, 4 gene populations F are extracted according to the frameshift type and the organism kingdom: −1 frameshifts of eukaryotes EUK⁻¹ and prokaryotes PRO⁻¹, and +1 frameshifts of eukaryotes EUK⁺¹ and prokaryotes PRO⁺¹. Tab. 2 shows the size of the studied frameshift gene populations.

The score functions P and Q are applied on these 4 frameshift gene populations. In each population F, frameshift genes from different species and producing proteins with different functionalities are found. Therefore, the gene size and the frameshift site vary from one gene to another. This data variation implies that these 2 functions will be computed in the nucleotide interval around the frameshift site i = 0 where there is a sufficient number of sequences (not detailed), explaining thus the variations of abscisses i in the figures below according to the population F.

---

Results

Results for the −1 frameshifts

Figs. 6 and 7 show the results using the function P with the frameshift gene populations F = EUK⁻¹ and F = PRO⁻¹, respectively. The abscissa represents the position i in the population F by varying i in the range {−400,600} for EUK⁻¹ and {−100,200} for PRO⁻¹, and the ordinate, the value of the function P.

Both figures show a significant lowest value of the function P around the frameshift site i = 0, precisely P(−3, EUK⁻¹) = 0.34 (Fig. 6) and P(−2, PRO⁻¹) = 0.27 (Fig. 7). As expected by the circular code property, a frameshift region is associated with a mixing of frames, i. e. a random mixing of codes X₀, X₁ and X₂ which is revealed by a low value of the function P (proposition 1).

Figs. 8 and 9 show the results using the function Q with the populations F = EUK⁻¹ and F = PRO⁻¹, respectively. There is a very significant lowest value of the function Q around the frameshift site i = 0 in both figures, precisely Q(−3, EUK⁻¹) = 0.07 (Fig. 8) and Q(−2, PRO⁻¹) = −0.05 (Fig. 9). A better result is observed with the function Q compared to the function P as the −1 frameshift signals contain also trinucleotides _id, such as the slippery heptamer XXXYYYZ [Baranov et al., 2006], which increase the loss of the circular code property.

Results for the +1 frameshifts

Fig. 10 shows the result for the frameshift gene population F = EUK⁺¹ using the function P and by varying i in the range {−50,150}. Surprisingly, a lowest value is observed around the frameshift site i = 0 with P(1, EUK⁺¹) = 0.38. Furthermore, the function Q does not show a lowest value around the frameshift site in EUK⁺¹. Therefore, the +1 frameshift eukaryotic signals are associated only with the loss of the circular code property.

No interesting result has been found until now in the frameshift gene population F = PRO⁺¹ using P or Q (see "Discussion").

---

Discussion

Recoding is widely distributed between organisms. It is thus likely that numerous novel recoded cellular genes remain to be discovered. However, the prediction of frameshift sites from genomic databases is currently a difficult task. Our new method shows a strong correlation between the circular code signal and the frameshift sites in genes. It can be performed without a prior knowledge of the mechanism involved in recoding. Indeed, it is based on the common C³ code X found in eukaryotic and prokaryotic genes, i. e. a structural feature of genes, which allows retrieval of any frame in genes (containing these circular code words), locally anywhere in the 3 frames and in particular without start codons in the reading frame, and automatically with the same window length of 13 nucleotides in each frame.

This method has been applied on 4 gene populations: −1 and +1 frameshifts of eukaryotic and prokaryotic genes. Frameshift sites in these populations are associated with low values for the functions P and Q. The function P is directly based on the property of the C³ code X while the function Q is based on both this property and the 4 trinucleotides with identical nucleotides which are considered in a contrary way to the circular code information. The absence of detection of frameshift signals with the +1 frameshift prokaryotic genes (PRO⁺¹) could be due to the special nature of these genes or to an insufficiency of the method.

Except for this population PRO⁺¹, the developed method is generic since it is applicable on both +1 and -1 frameshift genes even if the frameshift motifs are completely different in these various genes. It is also very efficient since frameshift signals are revealed with a window of very small size (4 trinucleotides) associated with the theoretical property of reading with the circular code. This new approach based on the circular code information is a contribution to the existing methods. We are currently investigating this approach for predicting frameshift sites at the gene level by stressing that the results have been obtained with gene populations of small sizes, e. g. 15 genes with PRO⁻¹ (Tab. 2). Indeed, this method can obviously be improved by varying the size of the sliding window, by using circular codes specific to the genomes [Frey and Michel, 2006] and by adding recoding information.