The dynamics of periodic base pairs opening in a finite stacking enthalpy DNA is investigated in this work. This is achieved by using the Joyeux-Buyukdagli DNA model, in which the polynomial approximations of the stacking interaction and Morse potential are expanded up to the fifth order nonlinear terms by using the Taylor series expansion technique. By incorporating the continuum limit approximation and the extended multiple scale asymptotic methods, higher order nonlinear Schr ödinger amplitude equations are derived. In the limit of cubic nonlinearity, the periodic base pair configurations clearly depict the open state; with linear stability analysis exposing other periodic background modes that are vital in the DNA transcription, replication, and transmission of genetic codes. The higher order modes generally display a more robust and structurally stable wave profile, which epitomizes the base pair dynamics of the DNA molecule observed from experimental investigations. Prolonged time evolution of base pairs stretching greatly modifies the higher order modes of the DNA molecule, strongly suggesting that such modes may induce abnormalities like gene mutation which is responsible for numerous diseases.
The nonlinear dynamics of deoxyribonucleic acid (DNA) remains a very fascinating and active area of research in biophysics. This is because it provides the basis for understanding intrinsic processes like transcription, replication, and transmission of genetic codes [
Concretely, bases are insoluble in water that is hydrophobic, while sugar and phosphate form bonds with water molecules that results to sugar-phosphate backbone. This backbone is generally aligned on the surface of a cylinder, while the bases are oriented toward its center. This configuration leads to a natural protection of the bases which carries most of the genetic codes, by the sugar-phosphate backbone [
Solitons are solutions of a widespread class of weakly disperse partial differential equations, and it generally originates from the balance between nonlinearity and dispersion. The soliton concept which emanates from the Fermi Pasta Ulam (FPU) paradox [
The quest to incorporate finite characteristic energies with phase transition triggered numerous research activities on DNA dynamics; tailored on capturing the appropriate phase transition predicted by statistical DNA models. This finally culminated with the brilliant works of Joyeux and Buyukdagli (BJ) model of DNA [
Unlike the aforementioned studies which deal with spatially localized excitations, the present investigation seeks to explore periodic solutions that may characterize base pairs opening. Periodic wave train solutions gives a better understanding of myriad of bio-physical activities hitherto explained only by localized solutions. For example, Vargas et al. numerically exploited localized periodic solutions in the nerve model, to rigorously explain hyper-polarization, pulse trains, and refractory periods that were experimentally observed in the nerve of locust [
In the present work, we demonstrate that the BJ model of DNA supports spatial periodic base pairs open configurations. Consequently, the organization of the paper is structured as follows. In section 2, we present the Hamiltonian of the BJ model which naturally leads us to the discrete equations of motion for the in-phase and out-of-phase motions, respectively. We expand terms of the stacking interaction and Morse potential in the out-of-phase equation of motion to the fifth order and implore the continuum limit approximation. Higher order amplitude equations are derived by using the extended multiple-scales asymptotic perturbation method. Analysis in section 3 is limited to terms up to the cubic order, in which periodic base pairs opening are captured to describe the open state configurations of the DNA. Stability analysis of these open states further reveal other localized background modes that are crucial in the transcription process. In section 4, we further explore higher order modes of the DNA open states configuration. This gives us a better opportunity to analytically and numerically investigate on the richer dynamics of DNA. Finally in section 5, we will summarize the important results obtained and articulate on some brighter perspectives.
The dynamical DNA model proposed by Joyeux and Buyukdagli is more realistic than the PBD model. This is because the JB model is based on site-specific stacking enthalpies, with the finiteness of the stacking interaction sufficiently ensuring a sharp melting transition [
H = ∑ n { 1 2 m ( x ˙ n 2 + y ˙ n 2 ) + V n ( x n , y n ) + W n ( x n , y n , x n − 1 , y n − 1 ) } , (1)
where
V n ( x n , y n ) = D 0 [ e − a ( x n − y n ) − 1 ] 2 ,
W n ( x n , y n , x n − 1 , y n − 1 ) = Δ H n C [ 2 − e − b ( x n − x n − 1 ) 2 − e − b ( y n − y n ) 2 ] + K b [ ( x n − x n − 1 ) 2 + ( y n − y n − 1 ) 2 ] .
Each base pair position is represented by n with N being the total number of the base pairs of the chain, while m is the average mass of the nucleotides. The on-site potential V n ( x n , y n ) is due to the presence of hydrogen bonds which is described by the Morse potential of depth D 0 and width a. The Morse potential opposes the breaking of the hydrogen bonds because it is an increasing function of the distance between the two bases of a pair n. The first term in the potential W n ( x n , y n , x n − 1 , y n − 1 ) describes the finite stacking interaction, while the second one models the stiffness of the sugar-phosphate backbone. Both terms are increasing functions of | x n − x n − 1 | (as well as | y n − y n − 1 | ), which implies that they oppose the de-stacking of the bases. The stacking potential which is approximated
by a Gaussian hole of depth Δ H n C , emanates from hydrophobic interactions
with the solvent and electronic interactions between successive base pairs on the same strand. The backbone stiffness is taken as a harmonic potential of constant K b , ensures that base pairs belonging to the same strand do not separate infinitely when approaching the melting temperature.
It is more convenient to introduce the coordinates u n describing the movement of a center of mass of the nucleotide pair, and v n , a stretching of the nucleotides belonging to the same pair defined as
u n = x n + y n 2 , and v n = x n − y n 2 . (2)
The in-phase motion is actually governed by u n , while v n represent the out-of-phase motion. From the Hamiltonian (1), it is possible to obtain two nonlinear discrete differential equations describing the transverse in-phase and out-of-phase dynamics of the DNA molecular chain respectively given as [
u ¨ n = 2 K b m [ ( u n + 1 − u n ) + ( u n − 1 − u n ) ] + 2 b Δ H n m C [ ( u n + 1 − u n ) e − b ( u n + 1 − u n ) 2 + ( u n − 1 − u n ) e − b ( u n − 1 − u n ) 2 ] , (3a)
v ¨ n = 2 K b m [ ( v n + 1 − v n ) + ( v n − 1 − v n ) ] + 2 b Δ H n m C [ ( v n + 1 − v n ) e − b ( v n + 1 − v n ) 2 − ( v n − 1 − v n ) e − b ( v n − 1 − v n ) 2 ] + 2 2 a D 0 m e − a 2 v n [ e − a 2 v n − 1 ] . (3b)
From a comparative analysis between Eqns (3a) and (3b), it is clear that the effects of nonlinearity are more pronounced in the out-of-phase motion Equation (3b). This is because Equation (3b) which mimics base pair stretching, incorporates hydrogen bonds interactions which is a vital component in the DNA dynamics. On the other hand the in-phase equation of motion (3a), is more associated with strong covalent bonds with little or no effects on the holistic DNA dynamics [
v ¨ n = 2 m ( K b + b Δ H n C ) ( v n + 1 − 2 v n + v n − 1 ) − 2 b 2 Δ H n m C [ ( v n + 1 − v n ) 3 − ( v n − v n − 1 ) 3 ] + b 3 Δ H n m C [ ( v n + 1 − v n ) 5 − ( v n − v n − 1 ) 5 ] − 2 a 2 D 0 m [ v n − 3 a 2 v n 2 + 7 a 2 6 v n 3 − 15 a 3 24 v n 4 + 31 a 4 120 v n 5 ] . (4)
The discrete coupled nonlinear Equation (4) is non integrable, but an approximation can be implored which preserves the nonlinearity of the system and reduce Equation (4) to an integrable form of a partial differential equation in order to obtain analytic solutions. Let us assume that the base pairs stretching v n ( t ) changes only slightly from one site to the next such that v n ( t ) = v ( z = n r , t ) , where z is a dimensionless variable that measures the position along the DNA strand. r is a measure of the equilibrium distance between two successive neighboring nucleotides in the same strand, with numerical value of 3.4Å in a real DNA molecule [
v n ± 1 ( t ) ≈ v ± r ∂ v ∂ z + r 2 2 ∂ 2 v ∂ z 2 + ⋯ . (5)
The continuum limit approximation (5), transforms Equation (4) to
∂ 2 v ∂ t 2 = [ 2 r 2 m ( K b + b Δ H n C ) + 6 r 4 b 2 Δ H n m C ( ∂ v ∂ z ) 2 + ⋯ ] ∂ 2 v ∂ z 2 − 2 a 2 D 0 m [ v − 3 a 2 v 2 + 7 a 2 6 v 3 − 5 a 3 8 v 4 + 31 a 4 120 v 5 ] , (6)
with all the key features involved in the DNA dynamics maintained, and terms of order ○ ( r 5 ) or higher are neglected. It is more convenient to transform Equation (6) into a an appealing form by considering the dimensionless variables [
V = a v , t = m a 2 D 0 τ , K 0 = ( 2 r 2 a 2 D 0 ) ( K b + b Δ H n C ) , K 1 = 6 r 4 b 2 Δ H n a 4 C D 0 , ω 2 = 2 , α = 3 / 2 , β = − 7 / 6 , γ = 5 / 8 , σ = − 31 / 120 . (7)
Equation (6) is now written in the dimensionless form
∂ 2 V ∂ τ 2 = [ K 0 + K 1 ( ∂ V ∂ z ) 2 + ⋯ ] ∂ 2 V ∂ z 2 − ω 2 [ V − α V 2 − β V 3 − γ V 4 − σ V 5 ] , (8)
in which in the new dimensionless time unit, we have that τ = 1.00 , corresponds to t = 0.198 ps .
To find the solution of Equation (8), we must first obtain a useful and manageable equation by assuming a more appropriate ansatz; which we postulate that the asymptotic series is more generalized while preserving the essential features of the DNA system. This can be done using reductive perturbative analysis, which is more robust in that it can work for a wide range of problems. In fact, we explore the extended multiple-scales asymptotic approach to reduce Equation (8) to higher order nonlinear Schrödinger amplitude equations. The main idea behind the asymptotic approach is to introduce fast and slow time and spatial variables into Equation (8), by exploiting the perturbation parameter ε ≪ 1 . The hierarchies of new independent variables replacing z and τ are
( t 0 = τ , t 2 = ε 2 τ , t 4 = ε 4 τ , ⋯ z 1 = ε z , z 3 = ε 3 z , ⋯
so that the τ - and z-derivatives are replaced by
∂ ∂ τ = ∂ ∂ t 0 + ε 2 ∂ ∂ t 2 + ε 4 ∂ ∂ t 4 + ⋯ ,
∂ ∂ z = ε ∂ ∂ z 1 + ε 3 ∂ ∂ z 3 + ⋯ . (9)
According to the extended reductive perturbation method, we consider the following ansatz for the solution of V ( z , τ ) as [
V ( z , τ ) = ε ( ψ e i ω t 0 + ψ * e − i ω t 0 ) + ε 2 ( G 2 + H 2 e 2 i ω t 0 + H 2 * e − 2 i ω t 0 ) + ε 3 ( F 3 e i ω t 0 + F 3 * e − i ω t 0 + J 3 e 3 i ω t 0 + J 3 * e − 3 i ω t 0 ) + ε 4 ( G 4 + H 4 e 2 i ω t 0 + H 4 * e − 2 i ω t 0 + K 4 e 4 i ω t 0 + K 4 * e − 4 i ω t 0 ) , (10)
where the amplitudes ψ , G 2 , H 2 , F 3 , J 3 , G 4 , H 4 , K 4 and their complex conjugates ψ * , H 2 * , F 3 * , J 3 * , H 4 * , K 4 * are functions of ( z 1 , t 2 , z 3 , t 4 ) , while ω stands for the angular frequency. Upon substitution of the ansatz (10) into the DNA Equation (8) yields a series of inhomogeneous equations at different orders of [ ε , e i ω t 0 ] .
Grouping terms to orders [ ε 2 , e 0 i ω t 0 ] , [ ε 2 , e 2 i ω t 0 ] , and [ ε 3 , e i ω t 0 ] , respectively yields the equations
0 = − ω 2 G 2 + ω 2 α | ψ | 2 , (11a)
− 4 ω 2 H 2 = − ω 2 H 2 + ω 2 α ψ 2 , (11b)
2 i ω ∂ ψ ∂ t 2 = K 0 ∂ 2 ψ ∂ z 1 2 + 2 α ω 2 [ ψ G 2 + ψ * H 2 + ψ G 2 * ] + 3 β ω 2 | ψ | 2 ψ . (11c)
Simplification of Equation (11a) gives G 2 = α | ψ | 2 , while that of Equation (11b) result in H 2 = − α ψ 2 / 3 , and Equation (11c) is eventually reduced to
i ∂ ψ ∂ t 2 − P ∂ 2 ψ ∂ z 1 2 + Q | ψ | 2 ψ = 0 , (12)
where
( P = r 2 a 2 ω D 0 [ K b + b Δ H n C ] , Q = − ω 6 [ 10 α 2 + 9 β ] .
The dispersion coefficient P, and nonlinear coefficient Q, are inextricably linked to the intrinsic constants r , a , ω , D 0 , K b , b , Δ H n , C , α , β , which are vital in the DNA dynamics; because it determines the type of wave profile in the system. The NLS Equation (12) is a prominent equation used to model a plethora of weakly nonlinear quasi-harmonic wave packets, like the propagation of electromagnetic waves in optical fibers [
This work is quite unique because it considers higher order modes of the DNA dynamics by incorporating the terms F 3 , J 3 , G 4 , H 4 , K 4 , which highly depend on the type of solution emanating from Equation (12). Consequently, terms to orders [ ε 3 , e 3 i ω t 0 ] , [ ε 4 , e 0 i ω t 0 ] , and [ ε 4 , e 2 i ω t 0 ] , respectively gives
− 9 ω 2 J 3 = − ω 2 J 3 + 2 α ω 2 ψ H 2 + β ω 2 ψ 3 , (13)
0 = K 0 ( ∂ 2 G 2 ∂ z 1 2 + ∂ 2 G 2 * ∂ z 1 2 ) − ω 2 ( G 4 + G 4 * ) + 6 γ ω 2 | ψ | 4 + α ω 2 [ 2 ψ F 3 * + 2 ψ * F 3 + ( G 2 + G 2 * ) 2 + 2 | H 2 | 2 ] + 3 β ω 2 [ ψ 2 H 2 * + ψ * 2 H 2 + 2 | ψ | 2 ( G 2 + G 2 * ) ] , (14)
− 4 ω 2 H 4 + 4 i ω ∂ H 2 ∂ t 2 = K 0 ∂ 2 H 2 ∂ z 1 2 − ω 2 H 4 + 4 γ ω 2 | ψ | 2 ψ 2 + 3 β ω 2 [ ψ 2 ( G 2 + G 2 * ) + 2 | ψ | 2 H 2 ] + α ω 2 [ 2 ψ F 3 + 2 ψ * J 3 + 2 G 2 H 2 + 2 G 2 * H 2 ] . (15)
Finally, grouping terms to orders [ ε 4 , e 4 i ω t 0 ] and [ ε 5 , e i ω t 0 ] , respectively generates the equations
− 16 ω 2 K 4 = − ω 2 K 4 + α ω 2 H 2 2 + 2 α ω 2 ψ J 3 + γ ω 2 ψ 4 + 3 β ω 2 ψ 2 H 2 , (16)
and
∂ 2 ψ ∂ t 2 2 + 2 i ω ∂ ψ ∂ t 4 + 2 i ω ∂ F 3 ∂ t 2 = K 0 ∂ 2 F 3 ∂ z 1 2 + 2 K 0 ∂ 2 ψ ∂ z 1 ∂ z 3 + 10 σ ω 2 | ψ | 4 ψ + 4 γ ω 2 [ ψ 3 H 2 * + 3 | ψ | 2 ψ ( G 2 + G 2 * ) + 3 | ψ | 2 ψ * H 2 ] + 3 β ω 2 [ ψ ( G 2 + G 2 * ) 2 + 2 ψ * H 2 ( G 2 + G 2 * ) + 2 | H 2 | 2 ψ + ψ 2 * J 3 + ψ 2 F 3 * + 2 | ψ | 2 F 3 ] + 2 α ω 2 [ ψ ( G 4 + G 4 * ) + ψ * H 4 + F 3 * H 2 + F 3 ( G 2 + G 2 * ) + H 2 * J 3 ] . (17)
Equations (12) to (17) will be used to comprehensively study periodic base pair opening in a DNA double strand, in order to give satisfactory explanations to the transcription and replication processes from a biophysical perspective. However the analysis in section III will be limited just to Equation (12), in which we focus only on the periodic DNA dynamics in the cubic limit [
We now consider the profile of the solution dictated by the derived amplitude Equation (12). Based on experimental parameters [
In order to look for stationary solutions to Equation (12), we assume an ansatz of the form
ψ ( z 1 , t 2 ) = a ( z 1 ) e i b t 2 , (18)
where b is the modulation frequency of the envelope and a ( z 1 ) is a real constant that represent the amplitude of the field envelope. Upon substituting Equation (18) into Equation (12) gives
− b a − P ∂ 2 a ∂ z 1 2 + Q a 3 = 0 , (19)
which can be conveniently transformed to a first-order integral equation
( d a d z 1 ) 2 = − b P a 2 + Q 2 P a 4 + C . (20)
It is important to note that the integration constant C is a key parameter that determines the nature of solution of the amplitude a ( z 1 ) . For a localized profile solution in which a ( z 1 ) vanishes as z 1 → ± ∞ , C naturally turns to zero. Consequently the solution to Equation (20) is given by
a ( z 1 ) = 2 b Q sech [ − b P z 1 ] . (21)
It is more appropriate to set b = ( 2 u e u c − u e 2 ) / 4 P , where u e and u c are real numbers which respectively represent envelope and carrier wave velocities measured in units of Å per dimensionless time τ [
ψ ( z 1 , t 2 ) = 2 u e u c − u e 2 2 P Q sech [ u e 2 − 2 u e u c 4 P 2 z 1 ] e i [ u e 2 − 2 u e u c 4 P ] t 2 . (22)
When the integration constant C is nonzero, the single pulse solution (22) becomes very unstable and difficult to be sustained in the system. Consequently, Equation (12) now admits periodic solution of the form [
ψ ( z 1 , t 2 ) = a 0 d n [ L z 1 , k ] e i ζ t 2 , (23)
where dn is a Jacobi elliptic function of modulus k ( 0 ≤ k ≤ 1 ) ,
a 0 = u e 1 − 2 η − 2 ( 2 − k 2 ) P Q , L = a 0 − Q 2 P , ζ = − P L 2 , (24)
and η = u c / u e , with 0 ≤ η < 0.5 . The solution of Equation (8) in the cubic limit (i.e. γ = σ = F 3 = J 3 = G 4 = H 4 = K 4 = 0 ), now gives
V ( z , τ ) = 2 a 0 ε d n [ L ε z , k ] cos ( Ω τ ) + 2 a 0 ε 2 d n 2 [ L ε z , k ] { α 2 − α 3 cos ( 2 Ω τ ) } + ○ ( ε 3 ) , (25)
where
a 0 ε = ε u e 1 − 2 η − 2 ( 2 − k 2 ) P Q , L e = ε a 0 ε − Q 2 P , ζ ε = − P L ε 2 , Ω = ω + ε 2 ζ ε , t 0 = τ . (26)
The constants a 0 ε , 1 / L e , and Ω , are respectively the amplitude, width, and angular frequency of the soliton solution that mimics the DNA open state configuration in the cubic limit. They are all dependent on the perturbation parameter ε , elliptic modulus k, and the experimental parameters of the DNA system as depicted in the contour plot in
henceforth adopt ε = 1.00 , k > 0.8 , without loss of generality. The choice of the values of ε and k, coupled with the experimental parameters given in
Concretely at the initial stage for k = 0.88 , and k = 0.98 , as in
in the DNA double strand chain [
We further plot the long-time evolution of base pair stretching, at different times as shown in
In the preceding subsection, we obtained localized periodic wave trains that mimic base pairs stretching in the finite stacking enthalpy DNA molecular chain. To discuss the stability of these periodic breathing modes, one must superimpose a small perturbation on this solution and analyze the evolution of the
perturbation. Note that stability analysis is an important issue related to the study of nonlinear dynamical systems because it provides an effective way of testing the robustness of the soliton trains against small perturbation in the amplitude. Stability analysis is applied in a diverse manner, based on the complexity of the physical system under review and the type of solution involved.
In order to investigate the linear stability of the spatial periodic soliton mode, we consider small perturbations a p ( z 1 ) to the amplitude of the DNA excitation mode denoted by a 0 ( z 1 ) , so that
ψ ( z 1 , t 2 ) = [ a 0 ( z 1 ) + ε a p ( z 1 ) ] e i b t 2 , (27)
where ε ≪ 1 . After the nonlinear interactions, the resultant internal modes of vibration carrying the genetic codes is obtained by substituting Equation (27) into Equation (12) and considering terms to the various orders of ε :
Order ε 0 ,
P d 2 a 0 d z 1 2 − Q a 0 3 + b a 0 = 0. (28)
Order ε 1 ,
P d 2 a p d z 1 2 − 3 Q a 0 2 a p + b a p = 0. (29)
We have already obtained periodic solution of Equation (28) as
a 0 ( z 1 ) = 2 b Q ( 2 − k 2 ) d n [ − b P ( 2 − k 2 ) z 1 , k ] , (30)
which can be substituted into Equation (29) and simplified to obtain [
d 2 a p d ξ 2 + [ E ( k ) − 6 k 2 s n 2 ( ξ , k ) ] a p = 0, (31)
where λ = − b P ( 2 − k 2 ) , ξ = λ z 1 , and E ( k ) = 6 P λ 2 + b P λ 2 . Equation (31) is known as the Lamé equation of the second order with five distinct localized solutions [
a p 1 ( ξ ) = A 1 ( k ) c n ( ξ , k ) d n ( ξ , k ) , b 1 = ( 5 − k 2 ) | P | λ 2 , (32a)
a p 2 ( ξ ) = A 2 ( k ) c n ( ξ , k ) s n ( ξ , k ) , b 2 = ( 2 − k 2 ) | P | λ 2 , (32b)
a p 3 ( ξ ) = A 3 ( k ) s n ( ξ , k ) d n ( ξ , k ) , b 3 = ( 5 − 4 k 2 ) | P | λ 2 , (32c)
a p 4 , p 5 ( ξ ) = A 4 , 5 ( k ) [ s n 2 ( ξ , k ) − 1 + k 2 3 k 2 ∓ 1 − k 2 ( 1 − k 2 ) 3 k 2 ] , b 4 , 5 = [ 2 − k 2 ± 1 − k 2 ( 1 − k 2 ) 2 ] | P | λ 2 . (32d)
The amplitudes A i ( k ) for i = 1 , 2 , 3 , 4 , 5 can be obtained by using appropriate orthogonality and normalization relations [
The five bound states given in
In order to give a comprehensive account of the base pairs stretching that mimics the DNA open state, we now incorporate all the terms in the ansatz (10). However we set F 3 = 0 , without of loss of generality, since we are dealing with periodic modes. The term F 3 generally introduces secularity which makes the solution cumbersome, and blurs the basic physics that characterize periodic open state dynamics of finite stacking enthalpy DNA. It is important to note that in the investigation of stationary breather modes of generalized nonlinear Klein-Gordon lattices, F 3 ≠ 0 [
Simplification of Equations (13) to (19), naturally lead us to the following
important quantities:
J 3 = 1 24 [ 2 α 2 − 3 β ] ψ 3 , (33a)
G 4 = K 0 α ω 2 [ ∂ 2 | ψ | 2 ∂ z 1 2 ] + [ 3 γ + 5 α β + 19 9 α 3 ] | ψ | 4 , (33b)
H 4 = 2 K 0 α 9 ω 2 [ ∂ ψ 2 ∂ z 1 − ψ ∂ 2 ψ ∂ z 1 2 ] − [ 4 3 γ + 31 12 α β + 59 54 α 3 ] | ψ | 2 ψ 2 , (33c)
K 4 = [ 1 12 α β − 1 15 γ − 1 54 α 3 ] ψ 4 . (33d)
Consequently, Equation (17) can now be re-written as
2 i ω ∂ ψ ∂ t 4 + ∂ 2 ψ ∂ t 2 2 = 2 K 0 ∂ 2 ψ ∂ z 1 ∂ z 3 + 4 K 0 α 2 ψ ∂ 2 | ψ | 2 ∂ z 1 2 + 2 K 0 α ω 2 [ ψ * ∂ ψ 2 ∂ z 1 − | ψ | 2 ∂ 2 ψ ∂ z 1 2 ] + ω 2 [ 84 3 α γ + 635 36 α 2 β − 1005 162 α 4 − 3 8 β 2 + 10 σ ] ψ | ψ | 4 . (34)
Based on the values obtained for J 3 , G 4 , H 4 , K 4 , we can conveniently modify solution (23) to now read [
ψ ( z 1 , t 2 , z 3 , t 4 ) = a 0 d n [ L z 1 + L 3 z 3 , k ] e i ( ζ t 2 + ζ 4 t 4 ) , (35)
where L 3 and ζ 4 are real constants. To determine the values of L 3 and ζ 4 , we substitute the modified solution (35) into Equation (34) and group coefficients of terms of the orders d n 0 [ ] , d n 2 [ ] and so on. After some rigorous calculations and simplifications, this leads us to
L 3 = L a 0 2 α [ 4 α ( 3 − 2 k 2 ) + 1 ω ( k 2 − 1 ) ] , (36a)
ζ 4 = K 0 α L 2 a 0 2 ( k 2 − 1 ) 2 [ 1 − 4 α ω 2 ( 4 − k 2 ) ] ω 3 − ζ 2 ω . (36b)
The most appropriate form of the analytic solution of the higher order stretching of nucleotide base pairs is now obtained by substituting the solution (35), into the ansatz (10) to have
V ( z , τ ) = 2 a ′ 0 ε d n [ L ′ ε z , k ] cos ( Ω ′ τ ) + a ′ 0 ε 2 d n 2 [ L ′ ε z , k ] { 3 2 − cos ( 2 Ω ′ τ ) } + 2 3 a ′ 0 ε 3 d n 3 [ L ′ ε z , k ] cos ( 3 Ω ′ τ ) − a ′ 0 ε 4 d n 4 [ L ′ ε z , k ] { 1 2 + 23 50 cos ( 4 Ω ′ τ ) } + 33 50 ε a ′ 0 ε 3 d n 3 [ L ′ ε z , k ] cos ( [ 3 Ω ′ − 2 ] τ ) + 3 ε 2 K 0 L 2 a ′ 0 ε 2 d n [ L ′ ε z , k ] { d n 3 [ L ′ ε z , k ] + ( k 2 − 1 ) d n ( L ′ ε z , k ) − 2 L k 2 s n [ L ′ ε z , k ] c n [ L ′ ε z , k ] } cos ( 2 Ω ′ τ ) + 3 2 ε 2 K 0 L 2 a ′ 0 ε 2 d n [ L ′ ε z , k ] { k 2 − 1 + 2 ( 1 − k 2 ) d n 2 [ L ′ ε z , k ] − 2 d n 4 [ L ′ ε z , k ] } . (37)
The constants in solution (37) are
a ′ 0 ε = ε u e 1 − 2 η − 2 ( 2 − k 2 ) P Q , L ′ ε = ε a ′ 0 ε − Q 2 P { 1 + ε 2 a ′ 0 ε 2 ( 25.9 − 16.9 k 2 ) } ,
Ω ′ = 2 + 0.325 ε 4 Q a ′ 0 ε 2 + 0.53 ε 4 K 0 L 2 a 0 2 ( k 2 − 1 ) 2 ( 12 k 2 − 47 ) , (38)
and the values of L and a 0 are given in Equation (24).
The profiles of the higher order modes are given in
[
The long time evolution of the gradual unzipping of the DNA molecule is captured in
[
The nucleotide is an elementary unit which consists of sugar, phosphate, and base. The segment of a DNA which is responsible for biosynthesis of a single polypeptide chain is called a gene, which averagely contains about 900 to 1500 nucleotide base pairs. During the transcription and replication processes, these genes are effectively transferred to single stranded shorter RNA. In fact, bases freely interact with enzymes during the local opening of DNA chain and therefore play a key regulatory role in the transcription process. This study seeks to expose the various forms of periodic base pairs stretching of DNA, in order to elucidate on intrinsic processes like transmission, transcription, and replication of genetic codes. Concretely, the DNA remains the basic organ of a cell which stores all vital information that ensures the effective growth and reproduction of all living organisms. Some distorted profile of base pairs stretching identified in this study may make the system so vulnerable to external attack, and hinders the smooth flow of genetic codes.
We effectively derived higher order nonlinear Schrödinger amplitude equations from the BJ model of DNA, by using the extended multiple scale asymptotic methods. Periodic solutions of these amplitude equations were used to mimic the open state configurations of a DNA strand under appropriate physiological conditions. The BJ model can also provide valuable information relating to the thermodynamic properties of DNA, by showing that the finite enthalpy may be responsible for the DNA denaturation. The stability analysis shows the existence of other background modes that may provide a possible physical mechanism for the effect of finite enthalpy stacking on DNA dynamics to be investigated. The stacking interaction of the BJ model equally provides both linear and nonlinear coupling parameters in the system which can independently control the dynamics and stability of the periodic solutions. We believe that this work opens up new vision on the concept of nonlinear periodic waves in DNA, and can also be exported in the study of many other physical systems.
In perspective, it will be very interesting to investigate on the impact of the helicoidal interactions on the periodic stretching of base pairs. The helicoidal term can best be appreciated because of the twisted nature of a real DNA molecule. Secondly, the interactions between the oscillating nucleotides and the aqueous environment naturally induce frictional forces, that must not be neglected in order to holistically describe the DNA dynamics. Lastly, quantum theory needs to be fully incorporated in order to comprehensively describe the DNA dynamics at the level much smaller than the nucleotides [
N. Oma Nfor appreciates the enriching discussions with S. Bruno Yamgoué, on the dynamics of DNA molecule.
The author declares no conflicts of interest regarding the publication of this paper.
Nfor, N.O. (2021) Higher Order Periodic Base Pairs Opening in a Finite Stacking Enthalpy DNA Model. Journal of Modern Physics, 12, 1843-1865. https://doi.org/10.4236/jmp.2021.1213107