Conformational vibrations of ionic lattice in DNA

Journal of Molecular Liquids 164 (2011) 113–119

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Journal of Molecular Liquids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m o l l i q

Conformational vibrations of ionic lattice in DNA Manifestation in the low-frequency Raman spectra S.M. Perepelytsya ⁎, S.N. Volkov Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, 14-b Metrolohichna St., Kiev, 03680, Ukraine

a r t i c l e

i n f o

Article history: Received 14 January 2011 Received in revised form 15 April 2011 Accepted 15 April 2011 Available online 18 May 2011 Keywords: DNA Counterions Conformational vibrations Raman spectra

a b s t r a c t Conformational vibrations of DNA with counterions neutralizing the phosphate groups of the double helix backbone are studied within the framework of phenomenological approach developed. The counterions are considered to be localized in two possible positions: near the phosphate groups of the double helix backbone, and between the phosphate groups in DNA minor groove. For the description of DNA conformational vibrations the structure of counterions tethered to the phosphate groups of double helix backbone is represented as an ionic lattice. Using the developed approach the frequencies and Raman intensities for DNA with Na +, Cs+, and Mg2+ counterions in different positions are calculated. As a result the manifestations of influence of counterion type and their position on the low-frequency Raman spectra (b 200 cm − 1) are determined. The obtained spectrum of DNA with Na + counterions near phosphate group is characterized by intensive modes of internal conformational vibrations of the double helix (from 60 to 110 cm − 1), while modes of ion-phosphate vibrations (near 170 cm − 1) have low intensity. In case of DNA with Cs+ counterions near phosphate groups the mode of ion-phosphate vibrations (near 115 cm − 1) is the most prominent, with other modes of DNA conformational vibrations being low-intensive. The spectra of DNA with Mg2+ counterions between phosphate groups are described by the high intensity of the internal modes of the double helix (from 60 to 120 cm − 1) that was found to be essentially dependent on the minor groove width. The obtained low-frequency Raman spectra provide a way to determine the type of counterions and its position with respect to phosphate groups. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The DNA double helix is a polyelectrolyte macromolecule with a negatively charged phosphate backbone. Under the natural conditions the phosphate groups are neutralized by metal counterions that form an ion-hydrate shell around the double helix [1]. The concentration of counterions, their type and position with respect to the DNA phosphate groups determine the mechanical properties of macromolecule, such as, twisting and bending of the double helix, and interaction of DNA with biologically active compounds [2–6]. The development of methods for determination of the type of counterions and their position in DNA is of paramount importance for understanding the mechanisms of counterion influence on DNA biological functioning. For solid DNA samples the position of counterions may be determined by X-ray and NMR methods [7–13]. In the case of water solutions only the distribution of counterions in DNA ion-hydrate

⁎ Corresponding author. Tel.: + 380 445213495. E-mail address: [email protected] (S.M. Perepelytsya). 0167-7322/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2011.04.015

shell may be determined within the framework of these methods [14–17]. In this respect the methods of vibrational spectroscopy are very effective [2,18–22]. In order to determine the local contacts of counterions with the atoms of DNA structural elements the frequency modes of atom vibrations on chemical bonds are used. The influence of counterions on the conformational dynamics of the whole double helix should be the most prominent in low-frequency range of vibrational spectra (b200 cm− 1), where the vibrations of DNA structural elements (conformational vibrations) occur. The low-frequency spectra of DNA is characterized by modes of transverse vibrations of the masses of structural elements of nucleosides, phosphates, and bases [23–26]. The modes near 20 cm − 1 characterize vibrations of the double helix backbone, and the vibrations within 50–120 cm − 1 concern to the H-bond stretching in the base pairs and intranucleoside mobility. The modes of this spectra range should be interdependent with the vibrations of counterions with respect to the phosphate groups (ion-phosphate vibrations). The previous study of counterion influence on conformational dynamics of DNA [27–30] has shown that counterions affect the frequencies of internal vibrations of the double helix, but the mode of ion-phosphate vibrations have not been determined. The determination of interrelation between ion-phosphate vibrations and internal

S.M. Perepelytsya, S.N. Volkov / Journal of Molecular Liquids 164 (2011) 113–119

functioning. In the case of untwisted double helix (Fig. 1b) the counterion may be coordinated by the phosphates of the same nucleotide pair (cross-strand neutralization), and in the case of overtwisted double helix (Fig. 1c) the counterion may be coordinated by the phosphates of the neighboring nucleotide pairs (longitudinal neutralization). The number of phosphate groups that are neutralized by counterions increases as the concentration of counterions in the solution grows. Thus, under a certain concentration the regular structure of counterions and phosphate groups should be formed. In the present work such a structure is considered as an ion-phosphate lattice. According to possible cases of counterion localization with respect to the phosphate groups three types of DNA ion-phosphate lattices may be considered. The first is a lattice with counterions in single-stranded position (Fig. 1a), the second is a lattice with cross-stranded counterions (Fig. 1b), and the third is a lattice with longitudinal counterions (Fig. 1c). Herein, the conformational dynamics of these lattices is studied. To determine the frequencies and amplitudes of conformational vibrations for DNA with counterions we extend our model [31,32] developed on the basis of an approach to conformational vibrations of the double helix [23–26]. Within the framework of this approach the DNA macromolecule is represented as a double chain of phosphate group masses m0 (PO4 + C5′) and masses of nucleosides m. The nucleosides rotate as physical pendulums with respect to the phosphate groups in the plane of the nucleotide pair. The physical pendulums are characterized by reduced length l. The nucleosides of different chains are paired by H-bonds (Fig. 2). The motions of structural elements of the monomer link are considered in the plane orthogonal to the helical axis (transverse vibrations). The longitudinal vibrations of the macromolecule atomic groups have much higher frequencies [23–26] and are beyond the scope of this work. The counterions are model as masses ma tethered to the phosphate groups. In the case of single-strand neutralization one counterion neutralizes one phosphate group from the outside of the macromolecule (Fig. 2a). The counterions in cross-stranded (Fig. 2b) and longitudinal (Fig. 2c) positions are localized between phosphate groups in the minor groove of the double helix. In the case of cross-strand neutralization the counterion is tethered to the phosphate groups of the same monomer link, and in the case of longitudinal neutralization it is tethered to the phosphate groups of neighboring monomer links. The displacements of nucleosides and phosphate groups in the DNA monomer link are described by coordinates Xn1, Xn2, Yn1, and Yn2. The coordinates θn1 and θ2 describe the deviations of pendulum-nucleosides from their equilibrium position in the plane of complementary DNA pair (angle θ0). The vibrations of deoxyribose and base with respect to each other, inside the nucleoside (intranucleoside vibrations), are described

dynamics of the double helix may provide a way to determine the counterion type and their position in DNA double helix by vibrational spectra. In our previous works the approach to the description of the ionphosphate vibrations has been developed [31–34]. To obtain frequencies of vibrational modes and their Raman intensities the structure of DNA double helix with counterions is considered as an ionic lattice. The calculated frequencies of ion-phosphate vibrations for DNA with Na +, K +, Rb +, and Cs + counterions are localized within the low-frequency spectra range from 90 to 180 cm − 1 and decrease as counterion mass increases which agrees with the experimental data [27,35–40]. Thus, our study has shown that counterion type influences essentially the low-frequency spectra of DNA. At the same time, the influence of counterion position in the double helix on the conformational vibrations of DNA has not been studied yet. The goal of the present work is to study conformational vibrations of the double helix with counterions and to find features of DNA lowfrequency Raman spectra determining the position of counterions with respect to the phosphate groups. To calculate the frequencies of vibrations the model for the conformational vibrations of DNA with counterions [31,32] is extended for the case of counterions inside the double helix minor groove. The intensities of the modes of DNA conformational vibrations are calculated within the framework of semiclassical approach for DNA [33,34]. Using the calculated frequencies and intensities the low-frequency Raman spectra for DNA with Na +, Cs +, and Mg 2+ counterions are plotted. The obtained spectra show that both counterion type and counterion localization essentially influence the spectra shape. 2. Modeling DNA ion-phosphate lattice dynamics The experimental data [7–13] and molecular dynamics simulations [41–48] show that the counterions may be localized with respect to the phosphate groups in different ways. Under the natural conditions the counterions are usually localized outside the double helix, because the chemical bonds between phosphorus and free oxygen atoms in the double helix are directed outward from the double helix (Fig. 1a). In this case one counterion neutralizes one phosphate group (single-strand neutralization). Under certain conditions when the minor groove of DNA macromolecule is narrow (the case of an overtwisted or untwisted double helix) the counterions may neutralize the phosphate groups of different strands of the double helix [8,11,41–43]. In the solution the overtwisted (D-form) and untwisted (Z-form) double helixes may be realized under the influence of counterion concentration [1,2,5,6]. The helical twist of DNA may be also changed essentially during the processes of biological

a)

b) + + +

+ +

+ o

12 A

Minor groove

+ + + + + j=1

+

+ +

+

+ +n + 1

+n + n-1 + j=2

o

4.25 A

n+1

o

4.25 A

+

n+1

+

n

+

Minor groove

+

c)

+

j=1

+

n n-1

+ + j=2

Minor groove

114

+

n-1

+ +

j=2

j=1

Fig. 1. Schemes of counterion localization with respect to the phosphate groups in DNA ion-phosphate lattice. (a) Single-stranded position of counterions. Width of the minor grove is shown. (b) Cross-stranded position of counterions. Ion-phosphate distance is shown. (c) Longitudinal position of counterions. Ion-phosphate distance is shown. φ is angle determining direction of ion-phosphate bond in the minor groove.

S.M. Perepelytsya, S.N. Volkov / Journal of Molecular Liquids 164 (2011) 113–119

a)

b)

(c) m

n

n1 n1

Xn1

n2

n n1

n1

X n2

ma

Yn2

Yn1

X n1

m0

0

m0 n1

n2

n1

n2

n2

m

n2

ma

n1

Xn2 n

Xn1

n2 n2

m ma

X n2 n

0

m0 Yn2

Yn2

Yn1

X

Y

n

0

Yn1

X Z

115

X

Y

Z

Z

Y

Fig. 2. Monomer link of the model of conformational vibrations for DNA with counterions in single-stranded (a), cross-stranded (b), and longitudinal (c) positions. XYZ is a reference frame connected with the monomer link; l is reduced length of pendulum-nucleoside; θ0 is equilibrium angle; m, m0, and ma are masses of nucleosides, phosphate groups, and counterions, respectively; Xnj, Ynj, θnj, ρnj, and ξnj are vibrational coordinates of the model (see text). The arrows indicate positive directions of displacements.

by changes of pendulum lengths ρn1 and ρn2. The vibrations of counterions in single-stranded positions are described by coordinates ξn1 and ξn2. For the description of vibrations of a counterion between phosphate groups the coordinate ξn is used. The vibrational coordinates of the model and the positive directions of displacements are shown in Fig. 2. The heterogeneity of the DNA nucleosides is not taken into account for being essential only for the modes of H-bond stretching in base pairs [23–26]. Therefore, the average values for nucleotide masses m, reduced length l, and equilibrium angle θ0 are considered. 3. Vibrational modes of DNA ion-phosphate lattice Within the framework of the introduced model of DNA ionphosphate lattice the energy of vibrations of structural elements of the double helix may be written as follows:

E = ∑ ðKn + Un Þ;

ð1Þ

n

where Kn and Un are kinetic and potential energies of vibrations of masses in the monomer link n of DNA ion-phosphate lattice. The kinetic energy of the monomer link may be written as: Kn = K0n + Kcn, where K0n and Kcn are kinetic energies of vibrations of nucleosides and counterions, respectively. The potential energy of displacements of the masses in the monomer link may be written as: Un = U0n + Ucn + Gn, n − 1, where U0n and Ucn are potential energies of vibrations of nucleosides and counterions, respectively; Gn, n − 1 is the interaction energy of structural elements in the monomer link along the double helix. The energies K0n and U0n have been determined earlier [24–26] as follows:

K0n =

h    1 2 2 2 2 ∑ M X˙ nj + Y˙ nj + m ρ˙ nj + l θ˙ nj 2 j + 2lsθ˙ njY˙ nj + 2b ρ˙ njY˙ nj −2a X˙ nj ρ˙ nj + 2lbX˙ nj θ˙ nj

i ;

ð2Þ

and in the present work it is determined the same as in Ref. [24–26]: δn ≈ ls(θn1 + θn1) + Yn1 + Yn1 + b(ρn1 + ρn1). Kinetic and potential energies of counterion vibrations in case of single-strand neutralization of the DNA phosphate groups may be written as follows:

Kcn =

  ma γ 2 2 2 ∑ ξ˙ nj + Y˙ nj ; Ucn = ∑ ξnj ; 2 j 2 j

where γ is force constant of counterion vibrations. In the case of counterions in the DNA minor groove (cross-strand and longitudinal neutralizations) the energy of counterion vibrations is as follows: Kcn =

ma ˙ 2 ξ ; 2 n

Ucn =

Gn;n−1 =

ð3Þ

where ls = la; a = sinθ0; b = cosθ0; the index j = 1, 2 enumerates the chain of the double helix; the force constants α, σ, and β describe Hbond stretching in base pairs, intranucleoside mobility, and rotation of nucleosides with respect to the backbone chain in base-pair plane, respectively; δn describes stretching of H-bonds in the base pairs,

ð5Þ

h  2  2 1 ∑ f Y −Yn−1; j + fX Xnj −Xn−1; j 2 nj Y nj  2  2 i ; + fθ θnj −θn−1; j + fρ ρnj −ρn−1; j

ð6Þ

where fY, fX, fθ, and fρ are force constants describing changes of interaction energy due to transverse displacements of the masses of nucleosides, phosphate groups, and counterions along coordinates Ynj, Xnj, θnj, and ρnj, respectively. The interaction of counterions with the charges of the ion-phosphate lattice are included in the constant γ, which expression is determined in the following section. Taking into consideration the fact that a DNA double helix consists of two chains kinetic and potential energies of the system may be split into two parts: Kn = Kn+ + Kn− and Un = Un+ + Un−. To obtain Kn+, Un+ and Kn−, Un− the variables qn+ and qn− should be used: þ

h i 1 2 1 2 2 αδn + ∑ σ ρnj + βθnj ; 2 2 j

h i2 γ j ∑ Ynj + ð−1Þ ξn : 2 j

The energy of interaction along the double helix may be determined analogically to [23,24]:

qn = qn1 + qn2 ; U0n =

ð4Þ

−

qn = qn1 −qn2 ;

ð7Þ

where qn1 = {Xn1, Yn1, θn1, ρn1, ξn1}, and qn2 = {Xn2, Yn2, θn2, ρn2, ξn2}. In case of counterions in cross-stranded and longitudinal positions, there is only one coordinate of counterion vibrations ξn− ≡ ξn. To write the equations of motion let us use a limited long range approximation that is enough for the interpretation of experimental spectra. Within the framework of this approximation the sum in Eq. (1) turns into integrals: ∑n → 1h ∫dz, and the differences of coordinates in Eq. (6) turn into derivatives: ðqn −qn−1 Þ→h ∂q = hq′ . ∂z

116

S.M. Perepelytsya, S.N. Volkov / Journal of Molecular Liquids 164 (2011) 113–119

In the variables of Eq. (7) the system of equations of motions splits into “+” and “−” subsystems of coupling equations that in the limited long range approximation is as follows: d ∂K0F d ∂KcF ∂U0F ∂UcF + − F− F dt ∂q˙ F dt ∂q˙ F ∂q ∂q

  ′ d ∂G qj F + = 0: dz ∂q′F

ð8Þ

Here, q + and q − are some coordinates {X +, Y +, θ +, ρ +, ξ +} and {X −, Y −, θ −, ρ −, ξ −, ξ}, respectively [see definition (Eq. 7)]. The first subsystem of the equation of motions refers to q + coordinates describing symmetric vibrations of the masses in the monomer link. The second one refers to q − coordinates describing antisymmetric vibrations of the masses in the monomer link. To solve the equations of motions we use the substitution: qF = q˜ F expðiωt−kzÞ, where q˜ F and ω are amplitudes and frequencies of vibrations, and k is wave number. To compare the calculated frequencies of vibrations with experimental spectra the modes of optic type should be determined under k → 0, since only the limited long range vibrations of optic type interact with the electro-magnetic fields and manifest themselves in vibrational spectra [49]. Taking this into consideration the last term in the Eq. (8) may be neglected, since it is proportional to k 2. As a result from the first subsystem of equations of motion (Eq. 8) the following equation for frequencies is obtained:    α + p γ ma −ω2  0 a 0 M     0        b α M −ω2  0 m       a M 2   l α0 m −ω   2  −pb ω

0

  mb M 2 α0 −ω M m

  mls M 2 α0 −ω m M

−1

ma M

mlb − M

Mb 2 α0 + σ 0 −ω m

Ml b α0 s m

aω −

2

2

bω l

2

α0

0

2

Mls b m

α0

Ma 2 + β0 −ω m

0

0

 ma 2  ω  M     0      = 0: 0       0    2  pa γ0 −ω

−pb

ð9Þ From the second subsystem of equations of motion (Eq. 8) it follows that:    pa γ0 ma −ω2  M     0    −bω2    a 2  − ω  l    −p γ −p ω2 a 0

b

0 −1

mb M ma M

−

mls M mlb − M −

aω2

σ 0 −ω2

0

b 2 − ω l

0

β0 −ω2

0

0

0

−

  ma  2 pb ω + 2pa γ0  M     0    = 0: 0     0    2  γ ðp + 1Þ−ω 0

a

the Raman spectra at the right angle geometry is determined as follows:

Js ≈

3κJ0 ðν0 −νs Þ4

s ∑ Aj j

s θ˜ j +

ρ ˜ sj l

!#2

" +

s ∑ Bj j

s θ˜ j +

ρ ˜ sj l

!#2 ) ;ð11Þ

h  i bjyy −bjxx sin2θ0 + 2ð−1Þj bjxy cos2θ0 ; h  i Bsj = bjyy −bjxx ð−1Þj cos2θ0 −2bjxy sin2θ0 ;

Asj =

where κ = 13 ⋅ 2 8π 5/(9c 4); J0 and ν0 are intensity and frequency of incident light; νs is frequency of a molecular normal vibration that has + + usual interdependence with cyclic frequency 2πνs = ωIon , ωH+, ωHS , − ωB+, ωIon , ωS−, ωB−; index s enumerates the mode of normal vibrations; c is velocity of light; h is the Plank constant; kB is the Boltzmann constant; and bjxx, bjxy, and bjyy are components of nucleoside polarizability tensor. To estimate the frequencies and intensities of the modes of DNA conformational vibrations by the formulae in Eqs. (9)–(11) force constants (α, β, σ, and γ), structural parameters of the model (l and θ0), and components of nucleoside polarizability tensor (bjxx, bjxy, and bjyy) are necessary. 4. Parameters of the model The parameters of the model α, β, σ, l, and θ0 have been determined in [23–26] on the basis of experimental data and conformational analysis of DNA nucleosides. In the present work we use them for the case of B-DNA (Table 1). The equilibrium angle (θ0) is considered the same as in Refs. [23–26] only in the case of counterions in singlestranded and longitudinal positions. In the case of counterions in crossstranded position it is considered equal to θ0 = 55° that corresponds to the distance between phosphate groups of the monomer link 8.5 Å. The X-ray data [11] show that such distance is the most appropriate for localization of a counterion in the DNA minor groove. The necessary polarizabilities of nucleosides were estimated by us earlier [33] within the framework of additive scheme of bond polarizabilities. The temperature is 300°K. The constant of ion-phosphate vibrations (γ) for the harmonic approximation of interaction energy between counterion and phosphate group is determined by expending the interaction potential in the Taylor series to the second order: V ≈ γ(r − r0) 2/2. For such approximation the potential describin well energy of ionic crystals is used [49]:

ð10Þ In Eqs. (9) and (10) the designations are introduced: α0 = 2α/M; σ0 = σ/m; β0 = β/ml 2; γ0 = γ/ma. In the case of single-stranded localization of counterions pb = 1, pa = 0, and in the case of longitudinal and cross-stranded localization of counterions pb = 0, pa = 1. The equation for frequencies (Eq. 9) describes the modes of symmetric vibrations of the atomic groups in the monomer link: the + modes of H-bond stretching in the base pairs (ωH+ and ωHS ), the mode of pendulum-nucleoside vibrations (ωB+), and the mode of ion+ phosphate vibrations (ωIon ). The equation of motions (Eq. 10) describes the modes of antisymmetric vibrations: the mode intranucleoside vibrations (ωS−), the mode of pendulum-nucleoside vibra− tions (ωB−), and the mode of ion-phosphate vibrations (ωIon ). The Raman intensities for the modes of DNA conformational vibrations are calculated within the framework of phenomenological approach developed in our previous works [33,34]. It is based on the valence-optic theory [50] and the four-mass model with counterions [31,32]. According to this approach the intensity of some mode of DNA conformational vibrations that is observed in the Stokes part of

hνs BT

−k

1−e

("

V =−

  2 Mα Ze r + B exp − ; g 4πεε0 r

ð12Þ

Table 1 Parameters of the model. Positions of counterions are denoted as follows: S-s—singlestranded, C-s—cross-stranded, and L—longitudinal. Parameter

Reference

α (kcal/molÅ2) β (kcal/mol) σ (kcal/molÅ2) l (Å) θ0 (rad) r0 (Å) ε g (Å) Mα γ (kcal/molÅ2)

[25] [25] [25] [25] [25] [11,13] [32] [49]

Na+

Cs+

Mg2+

Mg2+

S-s

S-s

C-s

L

80 40 43 4.8 28 2.88 2.3 0.328 1.07 44

80 40 43 4.8 28 3.18 2.6 0.319 1.07 35

80 40 43 4.8 55 4.25 2.3 0.328 1.50 62

80 40 43 4.8 28 4.25 2.3 0.328 1.50 18

S.M. Perepelytsya, S.N. Volkov / Journal of Molecular Liquids 164 (2011) 113–119

117

where Mα is the Madelung constant of ion-phosphate lattice; Z is valence of counterion; e is electron charge; ε is the dielectric constant of DNA ion-hydrate shell; ε0 is the vacuum dielectric constant; g is constants describing repulsion between counterion and DNA phosphate group; B is determined using the condition for equilibrium position: ð∂V =∂r Þr0 = 0; r is distance between counterion and phosphate group; and r0 is the equilibrium ion-phosphate distance. As a result the constant of ion-phosphate vibrations for counterion in single-stranded and cross-stranded positions is obtained as follows:

chain d0 = 6.7 Å and the minor groove width L = 8.5 Å are considered the same as in D-form of DNA [1]. According to these models the Madelung constant may be written as follows:

  Mα Ze2 r0 γ= −2 : 4πεε0 r03 g

The form of the function Mα(n) depends on counterion localization. In the case of counterions in single-stranded position and counterions in the DNA minor groove (cross-stranded and longitudinal positions) it may be expressed as:

ð13Þ

To obtain the constant of ion-phosphate vibrations for counterions in longitudinal position the constant γ determined by the formula in Eq. (13) should be multiplied by cos2φ, where φ is the angle determining the direction of the ion-phosphate bond in the minor groove (Fig. 1c). To estimate constant γ the parameters r0, g, Mα, and ε should be determined. The equilibrium distance for monovalent counterions in single-stranded position is determined in the same way as in our previous work [32] using the X-ray data for nucleic acids with alkali metal counterions [13]. In the case of counterions in longitudinal and cross-stranded positions r0 is considered as half of the distance between oxygen atoms of the phosphate groups in the narrow minor groove. According to the X-ray data [11] it should be about r0 = 4.25 Å. The repulsion constant g is considered the same as in case of ionic crystals of alkali metal chlorides (Table 1). The dielectric constant of DNA core is known to be lower than in the solution [51–54]. To estimate values of ε for DNA with counterions in single-stranded position a spatial approach has been developed by us earlier [32]. This approach is based on the theory of ionic crystals [49] and experimental data for wet films of DNA with different counterions [55–57]. Herein, we use ε values the same as in Ref. [32] for all considered cases of counterion localization (Table 1). To determine the Madelung constant Mα the following models of the DNA ion-phosphate lattices are used. In the case of counterions in single-stranded position the ion-phosphate lattice is presented as an unlimited double chain of phosphate groups with tethered counterions outside the chain (Fig. 3a). The distance between phosphate groups along the chain (d0 = 7 Å) and the minor groove width (L = 12 Å) are considered the same as in B-form of DNA. In case of counterions in the DNA minor groove (longitudinal and cross-stranded position of counterions) the ion-phosphate lattice is presented as an unlimited double chain of phosphate groups with counterions between phosphate groups (Fig. 3b). The distance between phosphate groups along the

a)

+

j=1

d1 -1 r1 -1

-

+

d11 r11

r0

-

+

-

Mα = ∑ Mα ðnÞ:

ð14Þ

n

! r0 Zr0 − ; rnj dnj

2

Mα1 ðnÞ = ∑

j=1

2

Mα2 ðnÞ = ∑

j=1

∞

Mα ≈ Mα ð0Þ + Mα ð1Þ + ∫2 Mα ðnÞdn:

j=2

b) j=1

r2 -1

-d +

d0

2 -1

n = -1

r21

-d

20

+

n=0

d21

-

ð16Þ

For functions Mα(n), defined in Eq. (15), the integral may be determined analytically. The result shows that in the case of ionphosphate lattice with counterions in single-stranded position this integral converges only for Z = 1, while in the case of counterions in cross-stranded and longitudinal positions it converges only for Z = 2. Thus, the analysis of the convergence of the sum (Eq. 14) shows that the Madelung constants of considered ion-phosphate lattices have a physical sense only in the case of electrically neutral systems. The respective values of the Madelung constants calculated by the formulae in Eqs. (14) and (15) for electrically neutral systems are shown in Table 1. The obtained values of Mα in the case of counterions in cross-stranded and longitudinal positions are higher than in singlestranded position. Taking into account the condition of convergence of the sum (Eq. 14) the case of single-strand neutralization is considered for monovalent counterions Na + and Cs+, and the cases of cross-strand and longitudinal neutralizations are considered for bivalent counterions Mg2+. Due to high energy of hydration of Mg2+ we presume that six water molecules of hydration shell are attached to the ion, therefore, the compound [Mg

-

r1 -1

L 2+ d-1 r20

ð15Þ

Let us study conditions of convergence of the sum (Eq. 14). The sum in Eq. (14) may be estimated with the sufficient accuracy by integrating with respect to n if r0/rnj b b 1 and r0/dnj b b 1. Such condition is true for the terms beginning with n = 2. Hence, an estimative formula for the Madelung constant may be determined as:

r0

L

! r0 Zr − 0: rnj dn

j=2

r2 -1

-

n = -1

2+

r0 d0

-

n=0

r11

-

d1 2+ r21

-

n=1

+

n=1

Fig. 3. Models of the DNA ion-phosphate lattices for calculation of the Madelung constant. a) The case of counterions in single-stranded position. b) The case of counterions between phosphate groups of the double helix (longitudinal and cross-stranded positions).

118

S.M. Perepelytsya, S.N. Volkov / Journal of Molecular Liquids 164 (2011) 113–119

Table 2 Frequencies of conformational vibrations of DNA with Na+, Cs+, Mg2+ (cm−1). Positions of counterions are denoted as follows: S-s—single-stranded, C-s—crossstranded, and L—longitudinal. Ion +

Na Cs+ Mg2+ Mg2+

Position

+ ωIon

− ωIon

+ ωH

ωS−

+ ωHS

ωB+

ωB−

S-s S-s C-s L

167 116 – –

167 102 136 95

111 89 110 114

79 55 79 58

57 41 90 71

15 12 57 26

16 13 15 16

(H2O)6] 2+ is taken. The Na+ and Cs + ions have much lower energy of hydration, hence, they are modeled without water molecules. Using the determined parameters the constants of ion-phosphate vibrations are calculated (Table 1). The obtained results show that in the case of single-strand neutralization the constant γ is the highest for Na+ counterions. In the case of Mg 2+ counterions in the DNA minor groove it is the highest for the counterion in cross-stranded position. It should be noted, that the constant γ depends on hydration rate of Mg 2+ counterion. 5. The low-frequency Raman spectra of DNA with counterions Using the determined parameters the frequencies of DNA conformational vibrations are calculated by the formulae in Eqs. (9)–(10). The results (Table 2) show that in the case of light Na + counterions the frequencies of internal conformational vibrations ωH+, + ωHS , ωB+, ωS−, and ωB− are close to the values that have been obtained within the framework of the four-mass model without counterions [25]. That is because light counterions do not influence the internal dynamics of the double helix. The vibrations of light counterions + − concern the modes of ion-phosphate vibrations (ωIon and ωIon ) that are localized essentially higher (near 170 cm − 1) than the frequencies of internal vibrations of the double helix. Due to influence of counterion mass the frequencies of ion-phosphate vibrations decrease to 115 cm − 1 in the case of heavy Cs + counterions. These frequencies are close to the frequencies of internal vibrations of the double helix which only slightly decrease as counterion mass increases.

The calculated frequencies of vibrations of DNA with Mg 2+ counterions between phosphate groups differ from the case of DNA with counterions in single-stranded position (Table 2). The most essential difference is obtained in the case of the modes of backbone + vibrations ωB+ and H-bond stretching ωHS . The frequencies of these modes increase several times. Such frequency changes are caused by an additional link between phosphate groups of different strands of the double helix that appears due to the counterions in the DNA minor groove. The modes of internal vibrations ωH+, ωS−, and ωB− of Mg-DNA do not differ essentially from the case Na-DNA. For these frequencies the influence of counterion localization is not essential, insofar as during the conformational vibrations the distance between counterion and phosphate group does not change. The frequency of ion-phosphate − vibrations ωIon for counterions in cross-stranded position is higher than for counterions in longitudinal position. That is because the constant of ion-phosphate vibrations is larger in the case of crossstranded position. Using the determined frequencies of vibrations the Raman intensities are calculated by the formula in Eq. (11). The necessary amplitudes of vibrations are estimated analogically to our previous works [24,32]. The results show that the modes of backbone vibrations (ωB+ and ωB−) within a frequency range lower than 50 cm− 1 have very high intensity for all counterion types that agree with the experimental data [35,38]. As it follows from the formulae in Eq. (11) the intensity increases mostly due to a temperature factor that is higher about one order for the modes ωB+ and ωB− comparing to the other modes of conformational vibrations [33,34]. To analyze the frequency range from 50 to 200 cm− 1 the lowfrequency Raman spectra are plotted (Fig. 4). In the obtained spectrum + of Na-DNA, there are intensive modes of H-bond stretching (ωHS and ωH+), as well as the mode of intranucleoside vibrations (ωS−). The intensities of these modes are essential due to large amplitudes of vibrations θ˜ and ρ ˜ that determine the intensity of DNA low-frequency spectra (Eq. 11). Due to low mass of sodium counterions the amplitudes of vibrations θ˜ and ρ ˜ are not essential in the case of the modes of ion+ − phosphate vibrations (ωIon and ωIon ). Therefore, these modes have low intensity in the obtained spectra of Na-DNA (Fig. 4a).

a)

b)

c)

d)

Fig. 4. The low-frequency Raman spectra of DNA with counterions in different positions. (a) Na+ counterions in single-stranded position. (b) Cs+ counterions in single-stranded position. (c) Mg2 + counterions in cross-stranded position. (d) Mg2 + counterions in longitudinal position. The halfwidth of spectra lines is 5 cm− 1.

S.M. Perepelytsya, S.N. Volkov / Journal of Molecular Liquids 164 (2011) 113–119

The spectrum of DNA with Cs + counterions (Fig. 4b) is characterized by an intensive band near 115 cm− 1 consisting of modes of ion+ − phosphate vibrations (ωIon and ωIon ). An increase in intensity of these modes is induced by influence of Cs+ counterions on internal dynamics of the double helix: amplitudes of internal vibrations of the double helix ˜ ) grow as counterion mass increases. In contrast, the internal (θ˜ and ρ + modes (ωHS , ωS−, and ωH+) of Cs-DNA are less intensive due to the changes of character of vibrations. As a result, the spectrum of Cs-DNA at frequencies higher than 50 cm− 1 is characterized by a single band near 115 cm− 1. This band is much higher than the respective modes of NaDNA. Such a difference between Na- and Cs-DNA spectra has been observed in the experimental Raman spectra of DNA water solutions with Na +, Cs + counterions [40] that confirms the existence of the ionphosphate vibrations in DNA low-frequency spectra. The vibrations of DNA with Mg 2 + counterions in cross-stranded position are characterized by large amplitudes of internal vibrations in case of the modes ωH+, ωS−, and ωB+. Therefore, these modes have high intensity in the obtained spectrum (Fig. 4c). In the case of counterions in longitudinal position of counterions the intensity of the mode of intranucleoside vibrations (ωS−) decreases, while the intensity of mode + of H-bond stretching (ωHS ) increases due to changes of amplitudes of vibrations. The vibrations of counterions in the DNA minor groove do not enlarge the amplitudes of internal vibrations of the double helix, − hence, the mode of ion-phosphate vibrations (ωIon ) has a low intensity. The obtained low-frequency Raman spectra essentially depend on counterion type and localization with respect to the phosphate groups. Such difference originated from different characters of conformational vibrations of DNA with different counterions. 6. Conclusions The manifestations of counterion influence on conformational vibrations of the double helix in the low-frequency Raman spectra of DNA are determined. Using the developed phenomenological approach for conformational vibrations of DNA with counterions the frequencies and the Raman intensities of transverse vibrational modes are calculated for DNA with Na +, Cs +, and Mg 2 + counterions in different positions. According to the calculated frequencies and intensities of vibrations the low-frequency Raman spectra are plotted. The obtained spectra strongly depend on counterion type and position with respect to the phosphate groups of the double helix. In the case of DNA with Cs + counterions within a frequency range higher than 50 cm − 1 the most intensive is the band near 115 cm − 1 consisting of modes of ion-phosphate vibrations. The spectrum of Na-DNA at this frequency range is characterized by intensive modes of internal vibrations of the double helix, while the modes of sodium-phosphate vibrations have a low intensity. The spectra of DNA with Mg 2 + counterions in the double helix minor groove have characteristic bands within the frequency range from 50 to 120 cm − 1 that are featured by modes of internal vibrations of the double helix. The frequencies and intensities of these modes essentially depend on the minor groove width. The determined frequency shifts and changes of intensities induced by counterions may be used for the interpretation of the experimental low-frequency Raman spectra of DNA and determination of counterion type and position with respect to the phosphate groups of the double helix. References [1] W. Saenger, Principles of Nucleic Acid Structure, Springer, New York, 1984. [2] V.I. Ivanov, L.E. Minchenkova, A.K. Schyolkina, A.I. Poletayev, Biopolymers 12 (1973) 89.

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