R. El Kinani, H. Kaidi, M. Benhamou., Investigation of DNA denaturation from generalized Morse potential Materials and Devices, Vol 3 #2,0207 (2018) – DOI: 10.23647/ca.md20180207 Article type: A-Regular research paper Investigation of DNA denaturation from generalized Morse potential R. El Kinani (1), H. Kaidi (2), M. Benhamou (3) (1) Physics Department, Faculty of Sciences P.O. Box 11201, Moulay Ismail University, Meknes, Morocco, radouane_kin@yahoo.fr (2) CRMEF, P.O. Box 255, Meknes, Morocco, hamidkaidi@gmail.com (3) Physics Department, Faculty of Sciences P.O. Box 11201, Moulay Ismail University, Meknes, Morocco, benhamou.mabrouk@gmail.com Correspondingauthor: radouane_kin@yahoo.fr RECEIVED: 18 january 2018 / RECEIVED IN FINAL FORM: 30 june 2018 / ACCEPTED: 02 july 2018 Abstract: In this paper, we present a non-linear model for the study of DNA denaturation transition. To this end, we assume that the double-strands DNA interact via a realistic generalized Morse potential that reproduces well the features of the real interaction. Using the Transfer Matrix Method, based on the resolution of a Schrödinger equation, we first determine exactly their solution, which are found to be bound states. Second, from an exact expression of the ground state, we compute the denaturation temperature and the free energy density, in terms of the parameters of the potential. Then, we calculate the contact probability, which is the probability to find the double-strands at a (finite) distance apart, from which we determine the behaviour of the mean-distance between DNA-strands. The main conclusion is that, the present analytical study reveals that the generalized Morse potential is a good candidate for the study of DNA denaturation. Keywords : DNA, INTERACTIONS, GENERALIZED MORSE POTENTIAL, DENATURATION TRANSITION, FREE ENERGY. Introduction Deoxyribonucleic acid, more commonly known as DNA, is a complex molecule that contains all of the information necessary to build and maintain an organism. All living things have DNA within their cells. In fact, nearly every cell in a multicellular organism possesses the full set of DNA required for that organism. However, the dynamics of DNA transcription is among the most fascinating problems of modern biophysics, because it is at the basis of life. It is also a very difficult problem due to the complex role played by RNA polymerases in the process. It is now well established (Freifelder, 1987) that the local denaturation of DNA is involved, so that it is interesting to investigate the thermal denaturation of the double-helix as a preliminary step to the understanding of the transcription (Fig. 1). DNA denaturation is a process that refers to the melting of the double-stranded DNA into two single strands. In fact, such a phenomenon originates from the breaking of hydrogen bonds between the bases in the duplex. Thermodynamically speaking, the major contribution to DNA helix stability is the stacking of the bases on top of one another. Hence, in order to denature DNA, the main obstacle to overcome is the stacking energies providing cohesion between adjacent base pairs. Moreover, studies of molecular dynamics and structure of nucleic acids are of general importance for the understanding of their functions. As such they have been subject of several theoretical and experimental investigations. Because of the importance of the functions played by DNA denaturation in replication, transcription and recombination processes, many theoretical and experimental techniques have been developed to study melting or strand separation of DNA. http://www.synonymo.fr/synonyme/investigation http://www.synonymo.fr/synonyme/investigation R. El Kinani, H. Kaidi, M. Benhamou., Investigation of DNA denaturation from generalized Morse potential Materials and Devices, Vol 3 #2,0207 (2018) – DOI: 10.23647/ca.md20180207 Figure 1: Double-helix structure of DNA shown in a full atomic representation bursts (left) and representation compact (right). The physics of DNA thermal denaturation (or melting), that is the separation of the two strands upon heating, is a subject which has a long history [1]-[3], because it can be viewed as a preliminary for the understanding transcription. The separation of the two complementary strands starts locally by the formation of small denatured regions, very similar to the transcription bubble. Another important motion of the DNA molecule is its "breathing" or fluctuational opening. In these very large fluctuations, the base-pairs are temporarily broken, and the two bases are exposed for chemical reaction, for a very short time (10−7s). These fluctuational openings can be considered as intrinsic precursors for the denaturation, and they could play a role in carcinogenesis by external molecules [4]. The wide interest in this problem is also motivated by the fact that UV absorption spectroscopy experiments [5]-[6] indicate that the whole denaturation process looks like a chain of sharp first-order-like phase transitions, in the sense that large portions of inhomogeneous DNA sequences separate over narrow temperature intervals. The molecular deformation involved in melting or in fluctuational openings, is so large that they cannot be described by linear approximations Therefore, biomolecular dynamics is a fascinating topic for nonlinear science, because it is related to basic phenomena of life and one knows that it has to be fundamentally nonlinear. We point out that the first model for the study of DNA denaturation is Ising-like, which represents a base-pair by a variable taking only two values, 0 and 1, i.e., closed and open. The denaturation transition is then analyzed by treating the Statistical Mechanics of this one-dimensional Ising-spin chain. The best known model in this category is that introduced by Poland and Scheraga [7]. In our case, we have adopted some model introduced by Dauxois, Peyrard and Bishop [8], with a change in the level of the interaction potential between the bases of the double-helices. The model by Dauxois, Peyrard and Bishop considers a simplified geometry for DNA chain, where one neglects the asymmetry of the molecule and represents each strand by a set of point-like masses corresponding to the nucleotides. We emphasize that, in their model, Peyrard and Bishop [8] adopted the standard Morse potential (MP) [9] that depends on three kinds of parameters. Such a potential reproduces well the main characteristics of the real potential, namely repulsive at short-distance (steric interaction) and attractive (van der Waals interaction), at high-distance. We discuss here some aspects of DNA dynamics and we show that simple nonlinear models can provide a good description of the large amplitude distortions of the molecule, which are observed experimentally. Our basic approach can be viewed as an extension of Ising models, which have been widely used to study the Statistical Mechanics of the melting, in which we treat completely the dynamics of the bases. In this work, we re-examine the study of the denaturation of DNA, using an extended Peyrard and Bishop model [8], where the standard MP is replaced by a more realistic one we introduce for the first time, termed q-MP, with a real parameter, q (see below). Such a new potential reduces to the standard MP for q = 0, and to that of Dang-Fan potential [10], for q = −1. In this sense, the q-MP is more general. In addition, the introduction of this new potential is that, the experimental adjustment of q-parameter may model various physical situations. The remaining of presentation proceeds as follows. In Section 2, we describe the used model for DNA denaturation. A study of the denaturation transition is presented in Section 3. Finally, some concluding remarks are drawn in the last section. A simple model for DNA denaturation: The main characteristics of our model are as follows: (i) One takes into account the transverse motions. The displacements from equilibrium of the n-th nucleotide are denoted as wn, for one chain, and vn, for the other. The longitudinal displacements are not considered, because their typical amplitudes are significantly smaller than the amplitudes of the smaller ones [11]. (ii) Two neighboring nucleotides of the same strands are connected by an harmonic potential, in order to keep the model as simple as possible. On the other hand, the bonds connecting the two bases belonging to different strands, are extremely stretched, when the double-helix opens locally, so that their nonlinearity must not be ignored. We use a generalized Morse potential to represent the transverse interaction of the bases in a pair. It describes not only the hydrogen bonds, but the repulsive interactions of the phosphate groups, partly screened by the surrounding solvent as well. The Hamiltonian of the model is then the following 𝐻 = ∑ { 1 2 𝑚(�̇�𝑛 2 − �̇�𝑛 2) 𝑛 + 1 2 𝐾[(𝑤𝑛 − 𝑤𝑛−1) 2 + (𝑣𝑛 − 𝑣𝑛−1) 2] + 𝑉𝑞 (𝑤𝑛 − 𝑣𝑛 )} , (1) where 𝑚 is the reduced mass of the bases. The contribution, 𝑉𝑞 (𝑤𝑛 − 𝑣𝑛 ), represents the interaction potential (q-MP) of the bases in a pair, whose form is the following http://www.synonymo.fr/synonyme/investigation R. El Kinani, H. Kaidi, M. Benhamou., Investigation of DNA denaturation from generalized Morse potential Materials and Devices, Vol 3 #2,0207 (2018) – DOI: 10.23647/ca.md20180207 𝑉𝑞 (𝑤𝑛 − 𝑣𝑛 ) = 𝐷 ((1 − 𝑏 𝑒𝛼(𝑤𝑛−𝑣𝑛)/√2 + 𝑞 ) 2 − 1) . (1a) We assume that the real parameter 𝑞 is such that 𝑞 > 0 or −1 ≤ 𝑞 < 0. In fact, the latter acts as an important deformation parameter. There, 𝑏 is some parameter that is related to the minimum of the interaction potential, 𝐷 is the potential depth and 𝛼 > 0 defines the potential-range. Notice that, the proposed potential is a four parameter exponential-type one, which is already pointed out in Ref. [12], and it may reduce to the most well-known exponential-type molecule potentials by choosing appropriate parameters (𝐷, 𝑏, 𝛼, 𝑞). The values 𝑞 = 0 and 𝑞 = −1 describe the Morse and Deng-Fan potentials, respectively. Hence, the potential 𝑉𝑞 (𝑤𝑛 − 𝑣𝑛 ) is more general and may be a good candidate for the study of a large class of interacting molecular systems. In Fig. 2, we report the 𝑞-generalized MP upon distance, for various values of the parameter 𝑞 keeping fixed the other ones, (𝐷, 𝑏, 𝛼). In particular, this figure shows that, the potential minimum is shifted towards its smaller values, as the parameter 𝑞 is increased. Figure 2: Reduced Morse potential, 𝐕𝐪 (𝐰𝐧 − 𝐯𝐧), versus distance, for different values of 𝐪. Now returning to equation (1) when it is expressed in terms of the variables 𝑥𝑛 = (𝑤𝑛 + 𝑣𝑛 ) √2⁄ and 𝑦𝑛 = (𝑤𝑛 − 𝑣𝑛 ) √2⁄ , yielding 𝐻 = ∑ * 1 2 𝑚�̇�𝑛 2 + 𝐾 2 (𝑥𝑛 − 𝑥𝑛−1) 2 𝑛 + ( 1 2 𝑚�̇�𝑛 2 + 𝐾 2 (𝑦𝑛 − 𝑦𝑛−1) 2) + 𝐷 ((1 − 𝑏 𝑒𝛼𝑦𝑛 + 𝑞 ) 2 − 1)+ . (2) The part of the Hamiltonian that depends on the variables 𝑥𝑛 ′𝑠, describes the displacement of the centre of mass of each base pair. It is well behaved, so that we can ignore this term in the Statistical Mechanics of the model. Therefore, we henceforth consider only the part of 𝐻 which depends only on 𝑦𝑛, because it possesses the complete information on the associative dynamics of the DNA, 𝐻′ = ∑ * 1 2 𝑚�̇�𝑛 2 + 𝐾 2 (𝑦𝑛 − 𝑦𝑛−1) 2 + 𝑉𝑞 (𝑦𝑛 )+ .𝑛 (3) Figure 3: Schematic representation of model. In the above expression, m is the mass of strands and K > 0 accounts for their elastic constant. Study of DNA denaturation and its Statistical Mechanics: Since we are interested in the thermal denaturation of DNA, the natural approach is to make use of Statistical Mechanics. Due to the one-dimensional character of the system, and since the interactions are restricted to nearest neighbor interactions, it can be treated exactly, including fully the nonlinearities, with the transfer operator method [13]. In the canonical ensemble, the partition function of the model is related to the Hamiltonian, 𝐻𝑦, by 𝑍 = ∫ ∏ 𝑑𝑦𝑛 𝑑𝑝𝑛 𝑒 −𝛽𝐻𝑦 𝛿(𝑝1 − 𝑝𝑁 ) 𝑁 𝑛=1 +∞ −∞ 𝛿(𝑦1 − 𝑦𝑁 ) (4) with the notation 𝛽 = 1 𝑘𝐵 𝑇⁄ , and the Dirac-function, 𝛿, ensures the periodic boundary conditions. The integrals over variables 𝑝𝑛 are decoupled Gaussian integrals, which can be easily calculated to give 𝑍 = (2𝜋𝑚𝑘𝐵 ) 𝑁/2𝑍𝑦, and due to the one- dimensional character of the model, the configurationally part, 𝑍𝑦, is a product of terms that couple two consecutive variables 𝑦𝑛 𝑍𝑦 = ∫ ∏ 𝑑𝑦𝑛 𝑒 −𝛽𝑓(𝑦𝑛,𝑦𝑛−1) 𝑁 𝑛=1 +∞ −∞ 𝛿(𝑦1 − 𝑦𝑁 ) , (5) with the notation 𝑓(𝑦𝑛 , 𝑦𝑛−1) = 𝐾 2 (𝑦𝑛 − 𝑦𝑛−1) 2 + 𝑉𝑞 (𝑦𝑛 ) . (5a) Therefore, calculation of 𝑍𝑦 can be performed by the transfer integral (TI) method [14]-[16]. Let us define TI operator 𝑦𝑛−1 → 𝑦𝑛 and its eigenfunctions, 𝜑𝑖, by ∫ 𝑑𝑦𝑛−1𝑒 −𝛽𝑓(𝑦𝑛,𝑦𝑛−1)𝜑𝑖(𝑦𝑛−1) = 𝑒 −𝛽𝜖𝑖 𝜑𝑖 (𝑦𝑛 ) . (5b) http://www.synonymo.fr/synonyme/investigation R. El Kinani, H. Kaidi, M. Benhamou., Investigation of DNA denaturation from generalized Morse potential Materials and Devices, Vol 3 #2,0207 (2018) – DOI: 10.23647/ca.md20180207 The 𝛿-function in 𝑍𝑦 expression can be expanded in the basis of the eigenfunctions of TI operator as 𝛿(𝑦𝑁 − 𝑦1) = ∑ 𝜑𝑖 ∗(𝑦𝑛 )𝜑1(𝑦1) . (5c) Performing successively integrals over variables (𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑁 ), one gets 𝑍𝑦 = ∑ 𝑒 −𝛽𝜖𝑖 ∫ 𝑑𝑦𝑛 |𝜑𝑖 (𝑦𝑛 )| 2 = ∑ 𝑒−𝛽𝑁𝜖𝑖 . 𝑛𝑛 (5d) We used the fact that the functions 𝜑𝑖’s are normalized to unity. In the thermodynamic limit (𝑁 → ∞), the above sum is dominated by the term having the smallest value of 𝜖𝑖 we denote as 𝜖1. Therefore, according to Ref. [3], we arrive at an equation which is formally identical to the Schrödinger equation for a particle experienced a 𝑞-generalized MP, − (𝑘𝐵 𝑇) 2 2𝐾 𝑑2 𝑑𝑦2 𝜑𝑖 (𝑦) + 𝑉𝑞 (𝑦)𝜑𝑖 (𝑦) = 𝐸𝑖 𝜑𝑖 (𝑦) . (7) This Schrödinger equation has been exactly solved [10], and in particular, we find that the associated ground state and the ground state energy are exactly given by 𝜑0(𝑦) = 𝑁0𝑒 𝑄1𝑦 ( 𝑒𝛼𝑦 𝑒𝛼𝑦 + 𝑞 ) 𝑄2 𝛼𝑞 , (8) 𝐸0 = − (𝑘𝐵 𝑇) 2 2𝐾 𝑄1 2 . (9) with a known normalization constant, 𝑁0, and 𝑄1 = 2𝐾𝐷𝑏 (𝑘𝐵𝑇) 2 ( 𝑏−2𝑞 𝑞 ) 1 2𝑄2 × 𝑄2 2𝑞 , (10) 𝑄2 = 1 2 [−𝛼𝑞 + √𝛼2𝑞2 + 8𝐾 (𝑘𝐵 𝑇) 2 𝐷𝑏2] . (11) We find that the free energy (per site) is given by 𝑓 = −𝑘𝐵 𝑇 ln 𝑍 = 𝜖0 − 𝑘𝐵 𝑇 2 ln(2𝜋𝑚𝑘𝐵 𝑇) . (12) On the other hand, the computed ground state, 𝜑0(𝑦), gives the contact probability as follows 𝑃(𝑦) = 𝐶0𝑒 2𝑄1𝑦 ( 𝑒𝛼𝑦 𝑒𝛼𝑦+𝑞 ) 2𝑄2 𝛼𝑞⁄ , (13) with the known normalization constant, 𝐶0. Now, notice that the system undergoes a denaturation transition, if the ground state energy, 𝐸0, vanishes; this condition then defines the denaturation temperature given by, 𝑇𝑑 = 2√2𝐾𝐷 𝛼𝑘𝐵 × √𝑏 √𝑏 − 2𝑞 (14) that naturally depends on the parameters of the problem, which are (𝐾, 𝐷, 𝛼, 𝑞). Before discussing the above expression of the denaturation temperature, we emphasize that the above computed contact probability, relation (13), may be used to determine the mean- distance, ξ = 〈y〉, between adjacent strands. We find that, in the vicinity of the denaturation temperature, Td, this length scales as ξ = 〈y〉 = ∫ yP(y)dy ∞ 0 ~(Td − T) −1 . (15) The divergence of this length is a signature of a denaturation transition. Now, come back to the main result dealt with the denaturation temperature expression, relation (14). Firstly, such a characteristic temperature depends naturally of the parameters (D, b, α, q) of the q-MP and the elastic constant, K, between neighboring strands. Secondly, this temperature is well defined, because of the conditions: −1 ≤ q < 0. Thirdly, the denaturation temperature increases, as it should be, with the elastic constant, K. This means that the denaturation transition occurs at high temperature for higher elastic strands. Fourthly, the denaturation temperature increases with increasing q-parameter. Fifthly, in the q = 0 limit, we find that result from the standard MP [8], that is 𝑇𝑑,𝑞=0 = 2√2𝐾𝐷 𝛼𝑘𝐵 , (16) Finally, the value q = −1 defines the denaturation temperature from Deng-Fan potential [10], 𝑇𝑑,𝑞=−1 = 2√2𝐾𝐷 𝛼𝑘𝐵 × √𝑏 √𝑏 + 2 . (17) Conclusion The aim of this work is an analytical study of DNA denaturation from a 𝑞-defomed Morse potential, using the Schrödinger equation method. Finally, the main conclusion is that, our analytical studies reveal that the 𝑞-deformed Morse potential is a good candidate for the description of DNA denaturation and its statistical properties. Acknowledgements: We are much indebted to Professor T. Bickel for fruitful correspondences and helpful discussions, and for our referee, for its useful suggestions and pertinent remarks. http://www.synonymo.fr/synonyme/investigation R. El Kinani, H. Kaidi, M. Benhamou., Investigation of DNA denaturation from generalized Morse potential Materials and Devices, Vol 3 #2,0207 (2018) – DOI: 10.23647/ca.md20180207 REFERENCES 1. D.Poland, H.A.Scheraga, Theory of helix-coil transitions in biopolymers, Academic Press, New York (1970) 2. R.M.Wartell, A.S.Benight,Thermal denaturation of DNA molecules: a comparison of theory with experiment, Physics Reports, vol.126, p 67 (1985) 3. M.Peyrard, Nonlinear dynamics and statistical physics of DNA, Nonlinearity, vol.17, p R1 (2004) 4. O.Gotoh, Prediction of melting profiles and local helix stability for sequenced DNA, Advances in biophysics, vol.16, p 1 (1983) 5. R.Thomas, Structure secondaire et denaturation des acides nucleiques, Bulletin de la Société de Chimie Biologique, vol.35, p 609 (1953) 6. A.Segerer, T.A.Langworthy, K.O.Stetter, Thermoplasma acidophilum and thermoplasma volcanium sp. nov. from solfatara fields, Systematic and Applied Microbiology, vol.10, p 161 (1988) 7. 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D.J.Sears, R.A.M.Ferrell, Statistical mechanics of one-dimensional Ginzburg-Landau fields, Physical Review B, vol.6, p 3409 (1972) 15. J.A.Krumhansl, J.R.Schrieffer, Dynamics and statistical mechanics of a one-dimensional model Hamiltonian for structural phase transitions, Physical Review B, vol.11, p 3535 (1975) 16. A.R.Bishop, J.A.Krumhansl, S.E.Trullinger, Solitons in condensed matter: a paradigm, Physica D: Nonlinear Phenomena, vol.1, p 1 (1980) http://www.synonymo.fr/synonyme/investigation R. El Kinani, H. Kaidi, M. Benhamou., Investigation of DNA denaturation from generalized Morse potential Materials and Devices, Vol 3 #2,0207 (2018) – DOI: 10.23647/ca.md20180207 Important:Articles are published under the responsability of authors, in particular concerning the respect of copyrights. 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