Twist–radial normal mode analysis in double-stranded DNA chains

Physics Letters A 376 (2012) 3407–3410

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Twist–radial normal mode analysis in double-stranded DNA chains Germán Torrellas, Enrique Maciá ∗ Dpto. Física de Materiales, Fac. CC. Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 9 May 2012 Received in revised form 30 August 2012 Accepted 25 September 2012 Available online 2 October 2012 Communicated by A.R. Bishop Keywords: Double-stranded DNA Normal mode analysis Dynamical models Tight-binding approach

a b s t r a c t We study the normal modes of a duplex DNA chain at low temperatures. We consider the coupling between the hydrogen-bond radial oscillations and the twisting motion of each base pair within the Peyrard–Bishop–Dauxois model. The coupling is mediated by the stacking interaction between adjacent base pairs along the helix. We explicitly consider different mass values for different nucleotides, extending previous works. We disclose several resonance conditions of interest, determined by the fine-tuning of certain model parameters. The role of these dynamical effects on the DNA chain charge transport properties is discussed. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Studying the dynamics of the DNA double helix is a very complicated task, due to its complex structure and interactions between the nucleobases, the sugar–phosphate backbone and the environment [1]. In this work we will focus on obtaining a systematic classification of the normal modes corresponding to the system formed by three consecutive base pairs (the so-called codon unit cell). For this purpose we take into account the coupling between the H-bond radial oscillations of complementary bases and the twisting motion of each base pair as a whole through the helical sugar–phosphate backbone structure. This kind of coupling has been shown to be very important in biological processes [2] (such as denaturation and transcription), given that an untwisting of the helix can cause the local aperture of the molecule, and has not been addressed with enough detail in the recent literature [3]. Following the Peyrard–Bishop–Dauxois model [4,5], we explicitly take into account the stacking interaction along the helix, mediated by the orbital overlapping between adjacent base pairs and hydrogen-bond distortions [6]. In this way we extend results from previous works that, even considering the helical three-dimensional structure of the molecule and twisting motions, assume the radial distances to be constant [7]. To further extend previous works we also take into consideration the mass difference between the 4 different nucleobases (m G = 347.05, mC = 307.05, m A = 331.06, m T = 322.05). We can benefit, however, from the fact that the mass of each base pair as a whole is essentially the same, as M = m G + mC ∼ = m A + mT ∼ = 653.5 ± 0.5 u.m.a. 2. Model Hamiltonian Fig. 1 shows a diagram of the model. We treat each nucleotide (base + sugar + phosphate) as a point mass, each base being attached to neighboring pairs through the sugar–phosphate backbone, whose deformation is considered within the harmonic regime. We neglect environmental effects so that the position of the center of mass is constant for each base pair. Under these circumstances the radial displacements about the equilibrium position (ρn = rn − R 0 ) satisfy ρn = λn ρn , where λn = mn /mn (λGC = 1.130, λ AT = 1.028). The position of the nth nucleobase can be expressed, as xn = rn cos ϕn , yn = rn sin ϕn , and zn = c ϕn , where n labels the considered base pair along the DNA double strand, rn and ϕn are usual cylindrical coordinates, and c = h0 /θ0 , where h0  0.34 nm (B-DNA form) is the equilibrium distance between two successive base pair planes along the z axis, and θ0 is the equilibrium relative angular separation between neighboring base pairs. Thus, we can express the Euclidean distance between adjacent bases as

dn,n±1 =

*



c 2 θn2,n±1 + ( R 0 + ρn±1 )2 + ( R 0 + ρn )2 − 2( R 0 + ρn±1 )( R 0 + ρn ) cos θn,n±1

Corresponding author. E-mail addresses: [email protected] (G. Torrellas), emaciaba@fis.ucm.es (E. Maciá).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.09.036

(1)

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Fig. 1. On the left, diagram of the model showing the harmonic bonds between adjacent base pairs through the backbone and H bonds between complementary bases. On the right, detail of one of the planes containing a base pair, showing the effect the mass difference has on radial displacements, shifting the center of mass outside the axis of the helix.

where we have defined θn,n±1 = ±(ϕn±1 − ϕn ) as the relative angle between two neighboring base pairs. We note that in our model the distance between successive base pairs along the z direction is not assumed to be constant (as it was considered in some previous works [8–10]) but it is proportional to the twist angle. In this way, the helical structure is naturally preserved during the dynamical evolution, in agreement with dispersion relation data reported from inelastic X-ray scattering measurements [11]. For the opposite strand, dn ,n±1 can be written in an analogous way, by replacing ρn with ρn in (1). In equilibrium both distances reduce to l0 = dn,n±1 |eq. = dn ,n±1 |eq. =



h20 + 4R 20 sin2 (θ0 /2). We can write the Hamiltonian of the duplex DNA molecule as



H=

N 2 P ϕ2 ,n 1  P ρ ,n + (n) (n) 2 T 22 n=1 T 11

 + UH + US + UB

(2) (n)

(n)

(n)

where P ρ ,n , P ϕ ,n are the conjugate momenta of the nth pair’s radial and twist variables, respectively; T 11 = λn M, T 22 = T 11 (ρn2 +

ξ 2 /λn ) + 4R 0 mn ρn , and ξ =



c 2 + R 20 is related to the helical geometry of the system, so that ξ θn,n±1 measures the helix arc length in

the limit of small twist oscillations (rn = R 0 , θn,n+1 ≪ 1) [12]. The three terms of the potential describe the different interactions between the bases.

UH =

N 



D n e−

αn 2

(1+λn )ρn

−1

2

(3)

n =1

is a Morse-like potential describing the hydrogen bond between complementary bases, where the parameters D n , G–C pairs.

US =

N −1 

ks,n,n+1

n =1

8

1 + Ee

2 − b2 un++1,n − un+1,n

αn are different for A–T,

 (4)

with un±,m = (1 + λn )ρn ± (1 + λm )ρm , describes the interaction between adjacent base pairs whose relative harmonic behavior is modified by the distortion of hydrogen bonds and the overlap of the π -type orbitals through an effective constant that depends on the parameters E and b, and whose role is to inhibit configurations with large relative radial displacements between neighboring pairs. The elastic constant varies over a broad range of values (0.0017 to 0.017 eV Å−2 ) depending on the type of base considered [6], and incorporates the chirality of the molecule since k s ( AT ) = k s ( T A ), k s (GC ) = k s (C G ).

UB =

N −1 k 

2

n =1

2  (dn,n+1 − l0 )2 + dn ,n+1 − l0

(5)

describes the harmonic interaction between neighboring bases along the backbone. The spring constant k is difficult to estimate, and previous studies have reported a wide range of possible values [13–19] (0.04 to 0.5 eV Å−2 ). From the Hamiltonian we can straightforwardly obtain the equations of motion for the codon unit cell. For the sake of simplicity, we will focus on the case of a homopolymer (i.e. polyA–polyT or polyG–polyC chains) in the linear regime. In this case, we obtain for the radial and twist variables:

ρ¨n + ωρ2 ρn + Ω(ρn+1 + ρn−1 ) = γ 2 (ϕn−1 − ϕn+1 + 2θ0 ),

(6)

ϕ¨n + ωϕ2 (2ϕn − ϕn−1 − ϕn+1 ) = γ¯ 2 (ρn+1 − ρn−1 ).

(7)

For details on the parameters see Table 1.

G. Torrellas, E. Maciá / Physics Letters A 376 (2012) 3407–3410

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Table 1 Model parameters for the homopolymer. k f 0 g 0 (1+λ) M l2 λ 0 2 = ( gf00ξ )2 1+λ 2λ 2k f 0 2 2 ϕ = M ( ξ l0 )

f 0 = c 2 θ0 + R 20 sin θ0

γ2 =

γ¯ 2 =

g 0 = R 0 (1 − cos θ0 )

a

q=

α (1+λ) ω2H = D4M λ 2 ωρ = 2[aωϕ2 + (1 + q)ω2H ]

ω

2

2

λ 2 γ ξ2 k s (1+ E ) D α2

Ω = aωϕ2 − qω2H

Fig. 2. Diagram of the codon unit cell for the configuration with boundary periodic conditions. Table 2 Normal modes for each frequency for a codon with periodic boundary conditions.

ω=0

 0 = (3ξ 2 M )−1/2 (0 0 0 1 1 1) N

ω = ω1

 1 = (3λ M )−1/2 (1 1 1 0 0 0) N

ω = ω2,±

2  ± = [2M (λC ± N + 3ξ 2 )]−1/2 (−C ± 0 C ± 1 −2 1)

(a)

2  ± = [ 2 M (λC ± N + 3ξ 2 )]−1/2 (− 3

(b)

C± 3

2C ± 3

− C3± −1 0 1)

3. Normal modes To obtain the normal modes and frequencies for the codon system in the homopolymer case, one can take into account different configurations (e.g., an isolated codon or a codon attached to fixed base pairs on each end). Considering a codon with periodic boundary conditions (Fig. 2), and making use of the Hessian matrices we obtain the normal frequencies

ω0 = 0,

(8)

ω12 = 2Ω + ωρ2 , ω22,± =

1 2

ωρ2 − Ω + 3ωϕ2 ±



(9)

2 ωρ2 − Ω − 3ωϕ2 + 12a˜ ωϕ4 ,

(10)

g ξ

where a˜ = λ1 [ 20f (1 + λ)]2 , and whose corresponding normal modes and their schematic representations are shown in Table 2 and Fig. 3, 0 

where we have defined C ± =

ω22,± 2 2ωϕ

ξ2

a˜ λ

.

4. Results From the analysis of the dynamical equations of motion the codon unit cell arises as a clearly important dynamic entity, in the sense that it is possible to extend the study of a single base pair to the codon, the codon of codons, and so on, maintaining the same equation structure, which shows a nested self-similarity. The presence of the normal frequency ω0 = 0 is related to the mode in which the codon rotates as a whole moving in the direction along the helix axis. The normal mode related to ω12 = 2Ω + ωρ2 is a “breathing” mode, in the

+ , N  + modes are of particular interest since sense that the whole molecule expands or contracts collectively in the radial direction. The N they present very noticeable deformations, bringing adjacent base pairs together, which would enhance orbital overlapping, significantly improving the charge transfer efficiency. As expected, the normal frequencies increase by increasing the value of the backbone elastic constant k, and also in the case of the G–C homopolymer, due to its stronger triple H bond. The normal frequencies corresponding to the configurations of the connected codon (either fixed or with periodic boundary conditions) are greater than those of the isolated codon, and their corresponding modes also exhibit more pronounced deformations. The parameter Ω = aωϕ2 − qω2H may vanish, in principle, within the range of values considered for k and k s . In this situation the radial equation of motion reduces to that of a harmonic oscillator for the radial variables. Values for k obtained in recent works by various methods [13–19], suggest that this resonance could in fact be reached, and even fine-tuned taking into account the mass difference among nucleotides. This difference, in principle fixed for each type of base pair, could be modified by environmental changes leading to the adhesion of small molecules to the sugar–phosphate backbone or due to epigenetic processes such as methylation (the addition of a methyl group (–CH3 ) to one of the bases). Whether this resonance is favorable or adverse to the biological functionality of DNA, or may endow it with specific chemical sensitivity properties will be addressed (a)

(b)

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Fig. 3. Mixed normal modes as a composition of twist (T) and radial (R) pure ones.

in a forthcoming work, along with the study of the system in the nonlinear regime, where coupling effects become significant and local openings of the molecule are more likely to happen [2]. A preliminary study of the charge transfer through the molecule making use of a 1D tight-binding Hamiltonian shows that the transfer integral between adjacent base pairs can be written as



2 2 tn,n±1 = t 0 1 − χ θn2,n±1 − χ R − 0 (ρn − ρn±1 )



(11)

where t 0 is the transfer integral of the 2D conformation and χ ∼ = 2.92 is a parameter that depends on geometry and neighboring p z orbital overlapping. The structure of this transfer integral suggests that both radial and twist relative displacements play a similar role. However, the twist term, being close to unity, almost cancels out in the transfer integral, and the radial term rules over. When relative displacements are large enough, the transfer integral becomes negative, hence favoring delocalization and charge transfer efficiency. (a)  (b) + , N + normal modes dynamics Broadly speaking, these results agree with our previous qualitative estimations on the basis of the N (which exhibit large twist and radial relative displacements). A more in-depth study on these issues, of prime importance for possible biotechnological applications of DNA dynamics is currently in progress and will be reported elsewhere [20]. Acknowledgements This work is supported by the Consejería de Educación de la Comunidad de Madrid and the Universidad Complutense de Madrid through project No. CCG10-UCM/MAT-4628. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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