Relaxing the Hypotheses of Symmetry and Time-Reversibility in Genome Evolutionary Models

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1 Relaxing the Hypotheses of Symmetry and Time-Reversibility in Genome Evolutionary Models Jacques M. Bahi, Christophe Guyeux, Antoine Perasso To cite this version: Jacques M. Bahi, Christophe Guyeux, Antoine Perasso. Relaxing the Hypotheses of Symmetry and Time-Reversibility in Genome Evolutionary Models. British Journal of Mathematics & Computer Science, 015, 5 (4), pp < < /BJMCS/015/1480>. <hal > HAL Id: hal Submitted on 7 Nov 014 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 British Journal of Mathematics & Computer Science 5(4): , 015, Article no.bjmcs ISSN: SCIENCEDOMAIN international Relaxing the Hypotheses of Symmetry and Time-Reversibility in Genome Evolutionary Models Jacques M. Bahi 1, Christophe Guyeux 1 and Antoine Perasso 1 UMR6147 Institut FEMTO-ST, Université de Franche-Comté, 5030 Besançon, France. UMR649 Chrono-environnement, Université de Franche-Comté, 5030 Besançon, France. Article Information DOI: /BJMCS/015/1480 Editor(s): (1) Carlo Bianca, Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy. () Paul Bracken, Department of Mathematics, University of Texas-Pan American Edinburg, TX 78539, USA. Reviewers: (1) Anonymous, Anhui University, China. () Anonymous, International Islamic University Malaysia, Malaysia. Complete Peer review History: Original Research Article Received: 5 September 014 Accepted: 4 October 014 Published: 14 November 014 Abstract Various genome evolutionary models have been proposed these last decades to predict the evolution of a DNA sequence over time, essentially described using a mutation matrix. By essence, all of these models relate the evolution of DNA sequences to the computation of the successive powers of the mutation matrix. To make this computation possible, hypotheses are assumed for the matrix, such as symmetry and time-reversibility, which are not compatible with mutation rates that have been recently obtained experimentally on genes ura3 and can1 of the Yeast Saccharomyces cerevisiae. In this work, authors investigate systematically the possibility to relax either the symmetry or the time-reversibility hypothesis of the mutation matrix, by investigating all the possible matrices of size ˆ and 3 ˆ 3. As an application example, the experimental study on the Yeast Saccharomyces cerevisiae has been used in order to deduce a simple mutation matrix, and to compute the future evolution of the rate purine/pyrimidine for ura3 on the one hand, and of the particular behavior of cytosines and thymines compared to purines on the other hand. *Corresponding author: antoine.perasso@univ-fcomte.fr

3 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Keywords: Genome evolutionary models; stochastic processes; discrete dynamical system; matrix calculus; nucleotides mutations. 01 Mathematics Subject Classification: 15A51; 15A16; 60G99; 9D10; 9D15; 9D0 1 Introduction Due to mutations or recombination, some variations occur in the frequency of each codon, and these codons are thus not uniformly distributed into a given genome. Since the late 60s, various genome evolutionary models have been proposed to predict the evolution of a DNA sequence as generations pass. Mathematical models allow the prediction of such an evolution, in such a way that statistical values observed in current genomes can be at least partially recovered from hypotheses on past DNA sequences. Moreover, it can be attractive to study the genetic patterns (blocs of more than one nucleotide: dinucleotides, trinucleotides...) that appear and disappear depending on mutation parameters. A first model for genomes evolution has been proposed in 1969 by Thomas Jukes and Charles Cantor [1]. This first model is very simple, as it supposes that each nucleotide has the probability m to mutate to any other nucleotide, as described in the following mutation matrix, m m m m m m m m m m m m In that matrix, the nucleotides are ordered as pa, C, G, T q, so that for instance the coefficient in row 3, column represents the probability that the nucleotide G mutates into a C during the next time interval, i.e., P pg Ñ Cq. As diagonal elements can be deduced by the fact that the sum of each row must be equal to 1, they are omitted here. This first attempt has been followed up by Motoo Kimura [], who has reasonably considered that transitions (A ÐÑ G and T ÐÑ C) should not have the same mutation rate than transversions (A ÐÑ T, A ÐÑ C, T ÐÑ G, and C ÐÑ G), this model being refined by Kimura in 1981, with three constant parameters to make a distinction between natural A ÐÑ T, C ÐÑ G and unnatural transversions, leading to: c a b c b a a b c b a c Joseph Felsenstein [3] has then supposed that the nucleotides frequency depends on the kind of nucleotide A,C,T,G. Such a supposition leads to a mutation matrix of the form: π C π G π T π A π G π T π A π C π T π A π C π G with π A, π C, π G, and π T denoting the frequency of occurance of each nucleotide, respectively. Masami Hasegawa, Hirohisa Kishino, and Taka-Aki Yano [4] have generalized the models of [] and [3], introducing in 1985 the following mutation matrix: απ C βπ G απ T απ A απ G βπ T βπ A απ C απ T. απ A βπ C απ G.. 440

4 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Mutation ura3 can1 T Ñ C 4 4 T Ñ A 14 9 T Ñ G 5 5 C Ñ T 16 0 C Ñ A 40 1 C Ñ G 11 9 A Ñ T 8 4 A Ñ C 6 5 A Ñ G 0 1 G Ñ T 8 0 G Ñ C 9 1 G Ñ A 6 40 Transitions Transversions Table 1: Summary of sequenced ura3 and can1 mutations [1] These efforts have been continued by Tamura, who proposed in [5, 6] a simple method to estimate the number of nucleotide substitutions per site between two DNA sequences, by extending the model of Kimura (1980). The idea is to consider a two-parameter method, for the case where a GC bias exists. Let us denote by π GC the frequency of this dinucleotide motif. Tamura supposes that π G π C π GC and π A π T 1 π GC, which leads to the following rate matrix: κp1 π GC q{ p1 π GC q{ p1 π GC q{ κπ GC { π GC { π GC { p1 π GC q{ p1 π GC q{ κp1 π GC q{ π GC{ π GC{ κπ GC{ All these models are special cases of the GTR model [7], in which the mutation matrix has the form (using obvious notations): f AC π C f AG π G f AT π T f AC π A f CG π G f CT π T f AG π A f CG π C π T f AT π A f CT π C π G Non-reversible and non-symmetric models have, for their part, been considered in practical inferences since at least a decade for phylogenetic studies, see for instance [8, 9, 10]. As they are more regarded for their interest in practical inference investigations than on the theoretical side, they will not be developed in this article. Due to mathematical complexity, matrices theoretically investigated to model evolution of DNA sequences are thus limited either by the hypotheses of symmetry and time-reversibility or by the desire to reduce the number of parameters under consideration. These hypotheses allow their authors to solve theoretically the DNA evolution problem, for instance by computing directly the successive powers of their mutation matrix. However, one can wonder whether such restrictions on the mutation rates are realistic. Focusing on this question, we used in [11] a recent research work of Lang and Murray [1], in which the per-base-pair mutation rates of the Yeast Saccharomyces cerevisiae have been experimentally measured (see Table 1), allowing us to calculate concrete mutation matrices for genes ura3 and can1. We deduced in [11] that none of the existing genomes evolution models can fit such mutation matrices, implying the fact that some hypotheses must be relaxed, even if this relaxation implies less ambitious models: current models do not match with what really occurs in concrete genomes, at least in the case of this yeast. Having these considerations in mind, the data obtained by Lang and Murray have been used in [11] in order to predict the evolution of.. 441

5 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs the rates or purines and pyrimidines in the particular case of ura3. Mathematical investigations and numerical simulations have been proposed, focusing on this particular gene and its associated matrix of size ˆ (purines vs. pyrimidines), and of size 3ˆ3 (cytosines and thymines compared to purines). Instead of focusing on two particular matrices, this extension of [11] investigates systematically all the possible mutation matrices of sizes ˆ and 3 ˆ 3. Thus, the study is finalized in this article, by investigating all the possible cases, and discussing about their mathematical and biological relevance. The remainder of this research work is organized as follows. First of all the case of mutation matrices of size ˆ is recalled in Section and applied to the ura3 gene taking into account purines and pyrimidines mutations. A simulation is then performed to compare this non reversible model to the classical symmetric Cantor model. The next sections deal with all the possible 6-parameters models of size 3 ˆ 3. In Section 3, a complete theoretical study is led encompassing all the particular situations, whereas in Section 4 an illustrative example focusing on the evolution of the purines, cytosines, and thymines triplet is given for ura3. We finally conclude this work in Section 5. General Model of Size ˆ In this section, a first general genome evolution model focusing on purines versus pyrimidines is proposed, to illustrate the method and as a pattern for further investigations. This model is applied to the case of the yeast Saccharomyces cerevisiae..1 A convergence result Let R and Y denote respectively ˆ the occurrence frequency of purines and pyrimidines in a sequence a b of nucleotides, and M the associated mutation matrix, with a P pr Ñ Rq, b P pr Ñ c d Y q, c P py Ñ Rq, and d P py Ñ Y q satisfying # a ` b 1, (.1) c ` d 1, ˆ a 1 a and thus M. c 1 c The initial probability is denoted by P 0 pr 0 Y 0q, where R 0 and Y 0 denote respectively the initial frequency of purines and pyrimidines. So the occurrence probability at generation n is P n P 0M n, where P n prpnq Y pnqq is a probability vector such that Rpnq (resp. Y pnq) is the rate of purines (resp. pyrimidines) after n generations. The following theorem states the time asymptotic behavior of the probabilit P n. We recall the following result was proved in [11]: ˆ a 1 a thm.1. Consider a DNA sequence under evolution, whose mutation matrix is M c 1 c with a P pr Ñ Rq and c P py Ñ Rq. If a 1, c 0, then the frequencies of purines and pyrimidines do not change as the generation pass. If a 0, c 1, then these frequencies oscillate at each generation between pr 0 generations) and py 0 R 0q (odd generations). Y 0q (even Else the value P n prpnq Y pnqq of purines and pyrimidines frequencies at generation n is convergent to the following limit: lim nñ8 Pn 1 ` c 1 a. c ` 1 a 44

6 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs rem.1. Note that the case a 1 c, resp. a c, translates the non symmetry property, resp. the time reversibility property.. Numerical Application For numerical application, we will consider mutations rates in the ura3 gene of the Yeast Saccharomyces cerevisiae, as obtained by Gregory I. Lang and Andrew W.Murray [1] and summed up in Table 1. They have measured phenotypic mutation rates, indicating that the per-base pair mutation rate at ura3 is m ˆ 10 7 /generation for the whole gene. For the majority of Yeasts they studied, ura3 is constituted by 804 bp: 133 cytosines, ` ` 11 thymines, 46 adenines, and 14 guanines. So R 0 «0.57, and Y 0 « Using these values in the historical model of Jukes and Cantor [1], we obtain the evolution depicted in Figure Nucleotides Evolution Rates Purine Pyrimidine Rate Time (1 unit = generations) Figure 1: Prediction of purine/pyrimidine evolution of ura3 gene in symmetric Cantor model. Theorem.1 allows us to compute the limit of the rates of purines and pyrimidines: Computation of probability a. P pr Ñ Rq p1 mq P A pnq P G pnq `P pa Ñ Gq ` P pg Ñ Aq P A pnq ` P G pnq P A pnq ` P G pnq. The use of Table 1 and the hypothesis that ˆ the base frequencies have already reached their steady states implies that a p1 mq ` m 0 46 ˆ 804 ˆ ` m ˆ. We thus obtain that 46 ` ` 11 a m 1905 « ` ` Computation of probability c. Similarly, P py Ñ Y q p1 mq ` P pc Ñ T q ` P pt Ñ P C ` P T P T Cq p1 mq`m 16 P C ` P T 46 ` 11 ˆ ` 11 `m 4 46 ` 11 ˆ m`m 133 ` P C 443

7 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Nucleotides Evolution Rates Purine Pyrimidine Rate Time (1 unit = 1 mutation) Figure : Prediction of purine/pyrimidine evolution of ura3 gene in general model of size ˆ. is: ˆ So c P py Ñ Rq m «.897 ˆ As a consequence the purine/pyrimidine mutation matrix that corresponds to the values of Table M m. (.) where m ˆ 10 7 as mentionned previously. Using the value of m for the ura3 gene leads to 1 a ˆ 10 7 and c ˆ 10 7, which can be used in Theorem.1 to conclude that the rate of pyrimidines is convergent to 49.45% whereas the rate of purines converge to 50.55%. Numerical simulations using data published in [1] are given in Figure, leading to a similar conclusion. 3 A First Genomes Evolution Model of size 3 ˆ 3 having 6 Parameters without Time-reversibility hypothesis In order to investigate the evolution of the frequencies of cytosines and thymines in the gene ura3, a model of size 3 ˆ 3 compatible with real mutation rates of the yeast Saccharomyces cerevisiae is now presented. 3.1 Formalization Let us consider a line of yeasts where a given gene is sequenced at each generation, in order to clarify explanations. The n th generation is obtained at time n, and the frequences of purines, cytosines, 444

8 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs and thymines at time n are respectively denoted by P R pnq, P C pnq, and P T pnq. Let a be the probability that a purine is changed into a cytosine between two generations, that is: a P pr Ñ Cq. Similarly, denote by b, c, d, e, f the respective probabilities: P pr Ñ T q, P pc Ñ Rq, P pc Ñ T q, P pt Ñ Rq, and P pt Ñ Cq. Contrary to existing approaches, P pr Ñ Cq is not supposed to be equal to P pc Ñ Rq, and the same statement holds for the other probabilities. For the sake of simplicity, we will suppose in all that follows that a, b, c, d, e, f are not time dependent. Let 1 a b a b M c 1 c d d e f 1 e f be the mutation matrix associated to the probabilities mentioned above, and P n the vector of occurrence, at time n, of each of the three kind of nucleotides. In other words, P n pp R pnq P C pnq P T pnqq. Under that hypothesis, P n is a probability P N, P Rpnq, P Cpnq, P T pnq P r0, 1s, P R pnq ` P C pnq ` P T pnq 1, Let P 0 pp Rp0q P Cp0q P T p0qq P r0, 1s 3 be the initial probability vector. We have obviously: P Rpn ` 1q P RpnqP pr Ñ Rq ` P CpnqP pc Ñ Rq ` P T pnqp pt Ñ Rq, with similar equalities for P C pn ` 1q and P T pn ` 1q so that P n P n 1 M P 0 M n. (3.1) In all that follows we wonder if, given the parameters a, b, c, d, e, f as in [1], one can determine the frequency of occurrence of any of the three kind of nucleotides when n is sufficiently large, in other words if the limit of P n is accessible by computations. 3. Resolution This section, that is a preliminary of the convergence study, is devoted to the determination of the powers of matrix M in the general case and some particular situations 3..1 Determination of M n in the general case The characteristic polynomial of M is equal to χ M pxq x 3 ` ps 3qx ` pp s ` 3qx 1 ` s p px 1q `x ` ps qx ` p1 s ` pq, where s a ` b ` c ` d ` e ` f, p ad ` ae ` af ` bc ` bd ` bf ` ce ` cf ` de, detpmq 1 s ` p. The discriminant of the polynomial of degree in the factorization of χ M is equal to ps q 4p1 s pq s 4p. Let x 1 and x the two roots (potentially complex or equal) of χ M, given by x 1 s ` as 4p and x s ` ` as 4p. (3.) 445

9 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Let n P N, n ě. As χ M is a polynomial of degree 3, a division algorithm of X n by χ M pxq leads to the existence and uniqueness of two polynomials Q n and R n, such that X n Q npxqχ pxq ` R npxq, (3.3) where the degree of R n is lower than or equal to the degree of χ M, i.e., R n pxq a n X ` b n X ` c n with a n, b n, c n P R for every n P N. By evaluating (3.3) in the three roots of χ M, we find the system & 1 a n ` b n ` c n % x n 1 a n x 1 ` b n x 1 ` c n a n x ` b n x ` c n This system is equivalent to & c n ` b n ` a n 1 b npx 1 1q ` a npx 1 1q x n 1 1 % b npx 1q ` a npx 1q x n 1 & % x n If we suppose that x 1 1, x 1, and x 1 x, then standard algebraic computations give ȷ 1 x n a n 1 x x 1 x 1 xn 1 1, x 1 1 b n x1 ` 1 x 1 x x n 1 x 1 ` x ` 1 x x 1 x n 1 1 x 1 1, c n 1 a n b n. Using for i 1, and n P N the following notation, and since x x 1?, the system above can be rewritten as a n X i pnq xn i 1 x i 1, (3.4) Xpnq X1pnq?, & b n px ` 1qX1pnq px1 ` 1qXpnq?, (3.5) % c n 1 ` x1x pnq x X 1 pnq?. By evaluating (3.3) in M and due to the theorem of Cayley-Hamilton, we finally have for every integer n ě 1, M n a nm ` b nm ` c ni 3, (3.6) where I 3 is the identity matrix of size 3, a n, b n, and c n are given by (3.5), and M is given by a ` ab ` ac a a ab ac ab ` ad b `b ` be b ` 1 ad ` a ` bf be bf ` b M ac bc c ac ` c ` cd c bc cd d cd ` c ` de `d ` df d ` 1 de df ` d. ae be ` cf ae cf df be ` df ` e ` ef e ef ` e ef f ` f e ` f f ` 1 446

10 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Determination of M n in particular situations Formulations of (3.5) only hold for x 1 x, x 1 1, and x 1. We now investigate these latter cases. Preliminaries Let us firstly remark that, as the mutation matrix M is stochastic, we have necessarily 0 ď a`b ď 1, 0 ď c`d ď 1, and 0 ď e`f ď 1. These inequalities imply that s P r0, 3s. Consequently from the definition of p one can check that p ad ` ape ` fq ` bpc ` dq ` bf ` cpe ` fq ` de ď ad ` a ` b ` bf ` c ` de ď s, as each parameter is in r0, 1s. To sum up, 0 ď p ď s ď 3. (3.7) Suppose now that ě 0. Then (3.) and (3.7) imply that?? s ` s ` ` x 1 P r ; 1s, x P 1 ; 5 ȷ (3.8) Note that, as we deal with a stochastic process, the module of the eigenvalues of M are smaller than 1, so x 1 ď 1 and x ď 1. Suppose that x 1 1 Then s a s 4p ðñ s p 0. So a b c d e f 0, and the mutation matrix is equal to the identity of size 3. Conversely, if a b c d e f 0, then x 1 1. In that situation, the system does not evolve. Suppose that x 1 (and x 1 1) Then s a s 4p ðñ p 0. In that situation, x 1 1 s and 1 is root of multiplicity of χ, whereas x 1 1 s is its third root. As the case x 1 1 has already been regarded, we can consider that s 0. Using (3.3), These facts lead to the following system: & % 1 a n ` b n ` c n, n a n ` b n, p1 sq n p1 sq a n ` p1 sqb n ` c n. Standard computations then give the following formula: & % 1 ` sn ` p1 sqn a n, s b n p3 sq ` ps sqn ` ps 3qp1 sq n, s c n ps 1qps 1q sps 1q n ps 3s ` 1qp1 sq n. s (3.9) Case x 1 x 1 ( 0) Then (3.8) implies that x 1 1 s{ P 1, 1. From a differentiation of (3.3) one deduces that x 1 satisfies the following system for every n P N, & % 1 a n ` b n ` c n x n 1 a nx 1 ` b nx 1 ` c n nx n 1 1 a nx 1 ` b n 447

11 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Standard algebraic computations give, since x 1 1, xn 1 1 a n n x & 1 1 X1pnq x 1 1 % b n X 1 pnq a n px 1 ` 1q c n 1 a n b n (3.10) where X 1 pnq is defined in (3.4). 3.3 Convergence study Convergence study in the general case We suppose in this section that x 1 x, x 1 1, and x 1. So formulations of (3.5) hold for a n, b n, and c n. We split the study convergence in several sub-cases, that are the objects of Theorems thm 3.1. Suppose that x 1 ă 1 and x ă 1. Then the frequencies P R pnq, P C pnq, and P T pnq of occurrence at time n of purines, cytosines, and thymines in the considered gene, converge to the following values: ce ` cf ` de P Rpnq ÝÑ p bf ` df ae ` af ` bf P C pnq ÝÑ p bf ` df ad ` bc ` bd P T pnq ÝÑ p bf ` df Proof. If x 1 ă 1 and x ă 1 then X ipnq ÝÑ 1 for i 1, and so 1 x i a n ÝÑ? 1 ˆ x 1 x 1 Denote by a 8 this limit. We have and finally a 8 x x 1? p1 x qp1 x 1 q 1 p1 x qp1 x 1 1q s `? s? a 8 Similarly, b n X 1 pnq a n px 1 ` 1q satisfies The following computations 1 1 x 1 4 s 1 p. b n ÝÑ 1 x1 ` 1. 1 x 1 p? ps q? s ` s s?, p? x 1 ` 1 s ` 4, p p, 448

12 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs finally yield So and to sum up, the distribution limit is given by b 8 s p. c n ÝÑ 1 a 8 b 8 p s ` 1, p a 8 1 p & % b 8 s p c 8 p s ` 1 p (3.11) Using the latter values in (3.6), we can determine the limit of M n, which is a 8 M ` b 8 M ` c 8 I 3. All computations done, we find the following limit for M n, ce ` cf ` de ae ` af ` bf ad ` bc ` bd 1 ce ` cf ` de ae ` af ` bf ad ` bc ` bd. p bf ` df ce ` cf ` de ae ` af ` bf ad ` bc ` bd Using (3.1), we can thus finally determine the limit of P n P 0M n pp Rp0q P Cp0q P T p0qqm n. thm 3.. Suppose that x 1 1, x 1 1, and x 1. Then the evolutionary model is not convergent. More precisely, we have: P R pnq pa ` ab ` ac a ` b ` be b ` 1qP R p0q ` p a ab ac ad ` a ` bfqp C p0q ` p ab ` ad b be bf ` bqp T p0q, P Rpn ` 1q p1 a bqp Rp0q ` ap Cp0q ` bp T p0q, P C pnq p ac bc c cd ` c ` deqp R p0q ` pac ` c ` cd c ` d ` df d ` 1qP C p0q ` pbc cd d de df ` dqp T p0q, P Cpn ` 1q cp Rp0q ` p1 c dqp Cp0q ` dp T p0q, P T pnq p ae be ` cf e ef ` eqp R p0q ` pae cf df ef f ` fqp C p0q ` pbe ` df ` e ` ef e ` f f ` 1qP T p0q, P T pn ` 1q ep R p0q ` fp C p0q ` p1 e fqp T p0q, Proof. Suppose that x 1 1 and x 1. Then x 1, x P R, and so x 1 1 or x 1 1. The first case has yet been regarded. If x 1 1, then s `? (due to (3.)). So s 4?, and so s 4p 4 4s ` s. Consequently, p s 1. But x 1 x 1 s ` p, so x 1 x 0, which leads to x 0. Using (3.5), we can thus conclude that a n 1 p 1qn 1 1 ` p 1qn. So a n 1 and a n`1 0. Similarly, b n 0 and b n`1 1, and finally c n P N. These values for a n, b n, and c n lead to the following values for M n : " M n M M n`1 M. 449

13 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs rem 3.1. The case x 1 1 and x 1 necessarilly implies that x 1, which is in contradiction with the assumptions made in preamble of Section thm 3.3. If x 1 x, but x 1, x P CzR, then pp R pnq P C pnq P T pnqq pp R p0q P C p0q P T p0qq ˆ pa n M ` b n M ` c n I 3 q, where sin ` nθ sin pn 1qθ a n sin `, θ sinpθq sin ` nθ sin pn qθ cos ` θ b n sinpθq sin `, θ sin ` nθ sin pn 3qθ c n 1 sinpθq sin `. θ with e iθ x 1. Proof. Suppose that x 1 x, but x 1, x P CzR. Then x 1 and x are complex and conjugate, of the form x 1 e iθ, x e iθ, with θ ı 0rπs. So x 1 x? e iθ e iθ i sinpθq, and a n X pnq X 1 pnq? X 1pnq X pnq i sinpθq i sinpθq a n e inθ 1 e iθ 1 einθ 1 e iθ 1 Finally, Similarly, e in θ e i θ pn 1qθ e i e in θ e in θ θ ein e i θ e in θ e in θ e i θ e i θ e i θ e i θ i sin ` nθ i sin ` e i pn 1qθ i sin ` nθ θ i sin ` θ sin ` nθ sin ` pn 1qθ e i e i pn 1qθ. θ a n sin ` nθ sin pn 1qθ sin ` θ sinpθq.? bn px ` 1qX 1 pnq px 1 ` 1qX pnq i sinpθqb n `e iθ pn 1qθ sin ` 1 ` nθ e i sin ` `e iθ ` 1 e i pn 1qθ sin ` nθ θ sin ` θ sin ` nθ ı sin ` pn 3qθ pn 1qθ e i ` e i e i pn 3qθ e i pn 1qθ θ b n sin ` nθ sinpθqsin ` θ sin pn 3qθ ` sin pn 1qθ. 450

14 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs and finally, b n As c n 1 a n b n, we have: sin ` nθ sin pn qθ cos ` θ sinpθq sin `. θ c n 1 sin ` nθ sin pn 3qθ sinpθq sin ` θ Convergence study in particular situations The case where x 1 1 has already been discussed, it implies that a b c d e f 0, and so the system does not evolve. The other particular situations are invastigated in the two following theorems. thm 3.4. Suppose that x 1 and x 1 1 (or equivalently p 0). Then the system is well formulated if and only if M ` sps qm ps 1q I 3 0. In that situation, we have: either s Ps0, r, and so pp Rpnq P Cpnq P T pnqq ÝÑ pp Rp0q P Cp0q P T p0qq ˆ 1 s r M ` sp3 sqm ` ps 1qps 1qI 3s. or s, and so pp Rpnq P Cpnq P T pnqq ÝÑ pp Rp0q P Cp0q P T p0qq whereas pp Rpn ` 1q P Cpn ` 1q P T pn ` 1qq pp Rp0q P Cp0q P T p0qq ˆ p M ` 4M ` I 3q. Proof. Using (3.9), we can deduce that M n is equal to: a n M ` b n M ` c n I 3 n s rm ` sps qm ps 1q I 3 s Several cases can be deduced from this equality. ` 1 s p1 sqn rm ` sps 3qM ps 3s ` 1qI 3 s ` 1 s r M ` sp3 sqm ` ps 1qps 1qI 3s. If s Ps0, r, then M n is bounded if and only if M `sps qm ps 1q I 3 0. In that condition, M n ÝÑ 1 s r M ` sp3 sqm ` ps 1qps 1qI 3 s. If s, then another time M n is bounded if and only if M ` sps qm ps 1q I 3 0. In that condition, M n ÝÑ I 3, whereas M n`1 ÝÑ M ` 4M ` I 3. Finally, if s ą, then as s a ` b ` c ` d ` e ` f and a, b, c, d, e, f P r0, 1s, we have necessarily at least three coefficients in a, b, c, d, e, f that are non zero. So at least one product in abc, abd, abe, abf, acd, ace, acf, ade, adf, aef, bcd, bce, bcf, bde, bdf, bef, cde, cdf, cef, def is strictly positive. This is impossible, as p ad ` ae ` af ` bc ` bd ` bf ` ce ` cf ` de is equal to 0. thm 3.5. Suppose that x 1 x 1 (or equivalently s 4p). Then the probabilities P R pnq, P C pnq, and P T pnq of occurrence at time n of a purine, cytosine, and thymine on the considered nucleotide, converge to the following values: P R pnq ÝÑ 4 pce ` cf ` deq, s 451

15 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs P C pnq ÝÑ 4 pae ` af ` bfq, s P T pnq ÝÑ 4 pad ` bc ` bdq. s Proof. In that case 0, meaning that (3.10) holds. Since x 1 P 1, 1, one gets the following limits, lim X1pnq 1, nñ8 1 x 1 lim nñ8 xn 1 0, lim nxn 1 1 0, nñ8 and finally pa n, b n, c n q converges to pa 8, b 8, c 8 q with 1 a 8 p1 x 1 q 4 s & b 8 x 1 p1 x 1 q 4 s s % c 8 x1 1 ˆ p1 x ` 1 1 1q s Using these values in (3.6), we can determine the limit of M n, which is a 8M ` b 8M ` c 8I 3, where I 3 is the identity matrix of size 3. All computations done, we find M n ÝÑ 4 M 11 M 1 M 13 M s 1 M M 3 M 31 M 3 M 33 with M 11 s 4 p ` ce ` cf ` de, M 1 ae ` af ` bf, M 13 ad ` bc ` bd, M 1 ce ` cf ` de, M s p ` ae ` af ` bf, M3 ad ` bd ` bc, M31 ce ` de ` cf, M3 ae ` af ` bf, and 4 M 33 s 4 p ` ad ` bc ` bd. However, since x 1 x, we have s 4p 0 and so M n ÝÑ 4 s ce ` cf ` de ae ` af ` bf ad ` bc ` bd ce ` cf ` de ae ` af ` bf ad ` bc ` bd ce ` cf ` de ae ` af ` bf ad ` bc ` bd, 4 Application in Concrete Genomes Prediction We consider another time the numerical values for mutations published in [1]. Gene ura3 of the Yeast Saccharomyces cerevisiae has a mutation rate of 3.80ˆ10 10 /bp/generation [1]. As this gene is constituted by 804 nucleotides, we can deduce that its global mutation rate per generation is equal to m 3.80 ˆ ˆ ˆ Let us compute the values of a, b, c, d, e, and f. The first line of the mutation matrix is constituted by 1 a b P pr Ñ Rq, a P pr Ñ T q, and b P pr Ñ Cq. P pr Ñ Rq takes into account the fact that a purine can either be preserved (no mutation, probability 1 m), or mutate into another purine (A Ñ G, G Ñ A). As the generations pass, authors of [1] have counted 0 mutations of kind A Ñ G, and 6 mutations of kind G Ñ A. Similarly, there were 8 mutations G Ñ T and 8: A Ñ T, so 36: R Ñ T. Finally, 6: A Ñ C and 9: G Ñ C lead to 15: R Ñ C mutations. The total of mutations to consider when evaluating the first line 45

16 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs Nucleotides Evolution Rates Purine Cytosine Thymine Rate Time (1 unit = 1 mutation) Figure 3: Prediction of evolution concerning the purine, thymine, and cytosine rates in ura3. Non-symmetric Model of size 3 ˆ 3. is so equal to 77. All these considerations lead to the fact that 1 a b p1 mq ` m 6 77, a 36m 77, and b 15m 19m. A similar reasoning leads to c 77 3, d 4m 3, e 51m 16m, and f In that situation, s a ` b ` c ` d ` e ` f 05m ˆ So s 4p 8541m 1 m ˆ c to Theorem 3.1: P R pnq ÝÑ P Cpnq ÝÑ P T pnq ÝÑ ce ` cf ` de p bf ` df ae ` af ` bf p bf ` df ad ` bc ` bd p bf ` df «8.134 ˆ 10 7, and p 07488m « ą 0, x 1 1 m ˆ c `, and x As x 1 « P r0, 1s and x « P r0, 1s, we have, due Using the data of [1], we find that P R p0q «0.57, PCp0q «0.165, and P T p0q «0.63. So P Rpnq ÝÑ 0.549, P C pnq ÝÑ 0.9, and P T pnq ÝÑ Simulations corresponding to this example are given in Fig Conclusion In this document, a formulation of the non symmetric discrete model of size ˆ has been proposed, which studies a DNA evolution taking into account purines and pyrimidines mutation rates. A simulation 453

17 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs has been performed, to compare the proposal to the well known Jukes and Cantor model. Then all non-symmetrical models of size 3x3 that have 6 parameters have been studied theoretically. They have been tested with numerical simulations, to make a distinction between cytosines and thymines in the former proposal. These two models still remain generic, and can be adapted to a large panel of applications, replacing either the couple (purines, pyrimidines) or the tuple (purines, cytosines, thymines) by any categories of interest. Remark that the ura3 gene is not the unique example of a DNA sequence of interest such that none of the existing nucleotides evolution models cannot be applied due to a complex mutation matrix. For instance, a second gene called can1 has been studied too by the authors of [1]. Similarly to gene ura3, usual models cannot be used to predict the evolution of can1, whereas a study following a same canvas than what has been proposed in this research work can be realized. In future work, biological consequences of the results produces by these models will be systematically investigated. Then, the most general non symmetric model of size 4 will be regarded in some particular cases taken from biological case studies, and the possibility of mutations non uniformly distributed will then be regarded. Finally, this 4 ˆ 4 general case will be investigated using Perron- Frobenius based approaches instead of using methods directly inspired by linear algebra, in order to obtain the most global results on mutation matrices. Competing Interests The authors declare that no competing interests exist. References [1] Jukes TH, Cantor CR. Evolution of protein molecules. Academy Press; [] Kimura M. A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. Journal of Molecular Evolution. 1980;16: /BF [3] Felsenstein J. A view of population genetics. Science. 1980;08(4449):153. [4] Hasegawa M, Kishino H, Yano T. Dating of the human-ape splitting by a molecular clock of mitochondrial dna. J Mol Evol. 1985;(): [5] Tamura K. Estimation of the number of nucleotide substitutions when there are strong transitiontransversion and g+c-content biases. Molecular Biology and Evolution. 199;9(4): [6] Tamura K, Nei M. Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees. Molecular Biology and Evolution. 1993;10(3): [7] Yang Z. Estimating the pattern of nucleotide substitution. Journal of Molecular Evolution. 1994;10: [8] Klosterman, Peter S, Uzilov, Andrew V, Bendaa, Yuri R, Bradley, Robert K, Chao, Sharon and Kosiol, Carolin and Goldman, Nick and Holmes, Ian. XRate: a fast prototyping, training and annotation tool for phylo-grammars. BMC Bioinformatics. 006;7:48. [9] Boussau, Gouy. Efficient likelihood computations with nonreversible models of evolution. Systematic Biology. 006;55(5): [10] Yap B, Speed T. Rooting a phylogenetic tree with nonreversible substitution models. BMC Evolutionary Biology. 005;5(). 454

18 Bahi et al.; BJMCS, 5(4), , 015; Article no.bjmcs [11] Bahi JM, Guyeux C, Perasso A. Predicting the evolution of gene ura3 in the yeast saccharomyces cerevisiae. Procedia Computer Science. 01;11:4-16. [1] Lang GI, Murray AW. Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae. Genetics. 008;178(1): c 015 Bahi et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) 455