Stretching and relaxation dynamics in double stranded DNA

arXiv:nlin/0306019v1 [nlin.PS] 12 Jun 2003 Stretching and relaxation dynamics in double stranded DNA D. Hennig1∗and J.F.R. Archilla2 Freie Universit¨at Berlin, Fachbereich Physik Institut f¨ur Theoretische Physik Arnimallee 14, 14195 Berlin, Germany 2Group of Nonlinear Physics Departamento de F´ isica Aplicada I ETSI Inform´atica, University of Sevilla, Avda Reina Mercedes, s/n. 41012 - Sevilla, Spain January 6, 2014 PACS numbers: 87.-15.v, 63.20.Kr, 63.20.Ry Abstract We study numerically the mechanical stability and elasticity properties of duplex DNA molecules within the frame of a network model incorporating microscopic degrees of freedom related with the arrangement of the base pairs. We pay special attention to the opening-closing dynamics of double-stranded DNA molecules which are forced into non-equilibrium conformations. Mechanical stress imposed at one terminal end of the DNA molecule brings it into a partially opened configuration. We examine the subsequent relaxation dynamics connected with energy exchange processes between the various degrees of freedom and structural rearrangements leading to complete recombination to the double-stranded conformation. The similarities and differences between the relaxation dynamics for a planar ladder-like DNA molecule and a twisted one are discussed in detail. In this way we show that the attainment of a quasi-equilibrium regime proceeds faster in the case of the twisted DNA form than for its thus less flexible ladder counterpart. Furthermore we find that the velocity of the complete recombination of the DNA molecule is lower than the velocity imposed by the forcing unit which is in compliance with the experimental observations for the opening-closing cycle of DNA molecules. ∗Corresponding author: Email:[email protected] 1 1 Introduction Nowadays powerful micromanipulation techniques allow for the selective excitation of DNA in single molecule experiments under controlled conditions (see, e.g., [1],[2],[3]). Particularly mechanical properties of DNA molecules under tension and the inducement of structural transitions with external stress have received a lot of attention [4]-[9]. In this context several force measurements were performed on single DNA molecules to examine their elastic response to applied stretching [6] and twisting forces, respectively [4],[7],[9]. It has been demonstrated that with the application of stronger forces to the molecule in stretching experiments, even highly cooperative conformational transitions were discovered, for which, e.g., B-DNA is converted into a new overstretched conformation called S-DNA [6]. Furthermore, with sufficiently high tension the mechanical unzipping of single DNA molecules is achievable [8]. The opening process of DNA has been the subject of a number of studies due to its immense relevance for the transcription process. Simple models taking into account the elasticity properties of DNA, such as the wormlike and rodlike chain model, respectively have been invoked to describe these unzipping experiments [10]. The statistical mechanics of the torque induced denaturation of DNA has been studied in [11]. Recently, a semi-microscopic model of the binding between the two strands has been used to explain the effects of applied tension to DNA unzipping [12]. In order to determine the geometry and deformability of DNA at a local level, i.e. in terms of base pairs, energy functions based on discrete network models have been derived in [14]-[17]. In an early network model for the base pair opening in DNA involving microscopic degrees of freedom, Peyrard and Bishop (PB) proposed a planar ladder-like oscillator model of DNA assigning each base pair a vertical inter-strand vibrational degree of freedom which simulates the stretchings and compressings of the corresponding H-bridges [18]. The binding forces of the hydrogen bridges are described by a Morse potential. The bases itself are treated as point masses. Horizontally, the bases on the same strand are coupled via the stacking interaction described by harmonic potentials. The PB model has been extensively studied and localized oscillating solutions (breathers) have been found reflecting successfully some typical features of the DNA opening dynamics such as the magnitude of the amplitudes and the time scale of the breathing of the ’bubble’ occurring prior to thermal denaturation [18]. On the other hand, as the coiled form of double-stranded DNA is concerned, it has been found that the bubble formation is strongly correlated with twist deformations and local openings are always connected with a local untwist of the double helix [19],[20]. In order to account for the helicoidal structure of DNA the PB model has been significantly extended by Barbi, Cocco and Peyrard (BCP) [19]. In their model of the twisted form of DNA (the WatsonCrick double helix), two degrees of freedom per base pair are introduced. There is a radial variable measuring the distance between two H-bridged bases along a line that connects them in the base plane being perpendicular to the helix axis. Further, the twist angle between this connecting line and a reference direction determines the orientation of the H-bridge. In this model a further degree of 2 freedom has been included, namely the axial distance variation [12]. Interactions between the bases are described then by appropriate potential terms in the energy of the system. The process of DNA unzipping by applied force has been studied in great detail [11],[12],[13]. The merit of these works is that many of the theoretical conclusions, gained on the basis of rather simple network models, are in agreement with the experimental results. In the current study of the nonlinear dynamics related with the opening process and elasticity features of DNA molecules our aim is to examine the energy exchange processes and the relaxation dynamics in DNA molecules after they have been brought into a non-equilibrium conformation by applied tension. Such an investigation is associated with recent mechanical experiments performed with single DNA molecules [3]-[9] forced away from their equilibrium conformations. Following force applications energy redistribution within the DNA molecule takes place such that a (new) equilibrium conformation is attained [2],[3],[21],[22]. We focus our interest on the elasticity properties and the relaxation dynamics of two different double-stranded DNA configurations, namely the planar untwisted (ladder-like) system and the twisted one arising when the two strands of the molecule are coiled around its molecular axis. In particular we address the questions in which way these two different DNA molecules react to imposed stress and how fast the equilibration process takes place for them. Furthermore, we compare the original equilibrium structures with those newly attained after release from the exerted mechanical stress. Regarding the biological functioning of DNA their elasticity properties are of importance for instance for the creation of a denatured bubble appearing prior to strand separation making the DNA accessible to the transcription and replication processes. Compared to previous theoretical studies of the unzipping dynamics of DNA induced by strong forces yielding complete strand separation [11],[12] we remark that for the current study we rather concentrate on mechanically stressed double-stranded DNA which gets forced away from the equilibrium configuration, however, only with beginning strand separation. From a biophysical point of view this also resembles the situation when the DNA molecule is inflicted to internal stress changes occurring in the living cell which alone is not strong enough to cause advanced bond breaking. Furthermore, the interaction between DNA and proteins such as enzymes can exert forces that initiate the local opening of the DNA molecule. The paper is organized as follows: In the second section we describe our network model for the structure of double-stranded DNA. The third section deals with the relaxation dynamics within DNA molecules forced into locally distorted configurations. We study the similarities and differences between the relaxation dynamics for a planar ladder-like DNA molecule and a twisted one. Finally, we summarize our results. 3 2 Network model for the double-stranded DNA Our DNA model takes the basic geometrical features of the double-stranded form of DNA, as a polymeric molecule composed of two strands of nucleotides, into account which are essential to model structural deformations generated by mechanical stress imposition and vibrational relaxation processes. Likewise the approach in [11],[19], we treat the double-stranded DNA as a network of coupled oscillators incorporating essential microscopic degrees of freedom of DNA and the inherent interactions between them. Such a network model approach is appealing because it can successfully describe the mechanical behavior on the microscopic scale relevant for biomolecular processes, but bears enough simplicity to be tractable with not too much computational efforts. The constituents of the oscillator network model represent the nucleotides which are regarded as single nondeformable entities. Thus, no inner dynamical degrees of freedom of the nucleotides are taken into account which is justified by the time scale separation between the small-amplitude and fast vibrational motions of the individual atoms and the slower and relatively large-amplitude motions of the atom groups constituting the nucleotides [23]. Concerning the structural components, each nucleotide is composed of a sugar, a phosphate and a base. The sugar-phosphate groups of neighboring nucleotides on the same strand are linked via covalent bonds establishing the rigid backbone to the strand. There is a base attached to every sugar. Since, for simplicity, we do not distinguish between the four different types of bases, the nucleotides are considered as identical objects of fixed mass. Two bases on opposite strands are linked via hydrogen bonds holding the two strands of DNA together. For computational convenience we cast the double helix structure in a Cartesian coordinate system whose z−axis coincides with the central helix axis as sketched in Fig. 1. The base pairs are situated in planes perpendicular to the central helix axis and the vertical distance between two consecutive planes is given by h. For the equilibrium configuration each base has equilibrium coordinates x(0) n,i,y(0) n,i and z(0) n,i . The index pair (n, i) labels the n-th base on the i−th strand with i= 1,2 and 1 ≦n≦N, where Nis the number of base pairs considered. The equilibrium distance between two bases within a base pair, d0, is determined by d0=q(dx n)2+ ( dy n)2,(1) where dx n=x(0) n,1−x(0) n,2and dy n=y(0) n,1−y(0) n,2are the projections of the line connecting the two bases on the x, y−axes of the coordinate system. The deviations dnfrom d0through displacements, xn,i,yn,i and zn,i, of the bases from their equilibrium positions, x(0) n,i,y(0) n,i and z(0) n,i , are expressed as dn=q(dx n+xn,1−xn,2)2+ ( dy n+yn,1−yn,2)2+ ( zn,1−zn,2)2−d0.(2) The relative twist between two consecutive base pairs, being rotated around the 4 l0+ln,1 d0+dn R0 (n−1)θ0+θn l0 θ0 Figure 1: Left: Geometry of the twisted double-stranded DNA model showing the variables. Right: The DNA molecule at the end of the pulling process with exaggerated distortions. central axis, is given by an angle θ0. For later use we introduce the quantity θn= arctan dy n+yn,1−yn,2 dx n+xn,1−xn,2 + 2mπ , (3) as the angle between the x−axis (as the reference direction) and the line connecting two (displaced) bases of a base pair measuring the alignment of the associated H-bridge. mis an integer to assure monotonicity of θnwith respect to n. The three-dimensional equilibrium distance between two adjacent bases on the same strand is given by l0=rx(0) n,i −x(0) n−1,i2+y(0) n,i −y(0) n−1,i2+h2,(4) and deviations from l0are determined by ln,i =nLx n,i +xn,i −xn−1,i2+Ly n,i +yn,i −yn−1,i2 + ( h+zn,i −zn−1,i )2o1/2 −l0,(5) with Lx n,i =x(0) n,i −x(0) n−1,i and Ly n,i =y(0) n,i −y(0) n−1,i. Our model Hamiltonian is then given by H=Ekin +Vh+Vc+Vs+Vl.(6) 5 Vhand Vcrepresent the potential energy part for the H-bonds and the covalent bonds, respectively. The deformations of the hydrogen bonds corresponding to deviations dn(t) from the equilibrium bond length d0are described by vibrations in a Morse potential which is given by Uh=DX i=1,2 N X n=1 [ exp (−α dn)−1 ]2.(7) D is the depth of the Morse potential and αis the range parameter. For the sake of simplicity we do not distinguish between the two different pairings in DNA, namely the G-C and the A-T pairs. The former pair involves three hydrogen bonds while the latter involves only two. Thus our model applies to pure poly(dG-dC) and poly(dA-dT) DNA sequences. The covalent bonds between the sugar-phosphate groups of neighboring nucleotides on a strand are rather strong and rigid (with bond energies of the order of 2 −10 eV ) in comparison with the weak and flexible hydrogen bonds (with bond energies of the order of 0.04 −0.3eV ). Therefore, it seems justified to treat the potential of the covalent bonds, simulated by elastic rods, in the harmonic approximation given by Uc=KX i=1,2 N X n=1 l2 n,i ,(8) where Kis the elasticity coefficient. The potential term Vstakes stacking effects into account which impede that one base slides over another [23]. For the form of Vswe adopt the one used in [11] Vs=S 2X n,i (dn,i −dn−1,i)2exp[−β(dn,i +dn−1,i)] .(9) The decrease of the molecular packing with substantially increased base pair opening is accounted for by the exponential factor with attenuation parameter β. As the changes of the axial distance between adjacent base planes is concerned we assume that the supposedly small longitudinal helix deformations can be modeled by a harmonic elasticity potential term given by Vl=C 2X n,i (zn,i −zn−1,i)2.(10) The kinetic energy is determined by Ekin =1 2mX i=1,2 N X n=1 p(x) n,i 2+p(y) n,i2+p(z) n,i2,(11) where mis the mass of a base and p(x,y,z) n,i denote the (x, y, z)−component of its momentum. 6 In order to assign values to the various parameters we note that the geometrical parameters of the equilibrium configuration are well known [23]. The rotation angle for the twisted configuration is θ0= 36◦, the distance between base pair planes is h= 3.4˚ A , and the inter-base distance is d0= 20 ˚ A. (For the twisted configuration d0corresponds to the diameter of the helix.) The ladder-like system is completely untwisted, viz. θ0= 0. For the average mass of one nucleotide we use M= 4.982 ×10−25kg. Like Barbi et al [19] we set α= 4.45 ˚ A−1,D= 0.04 eV , and K= 1.0eV ˚ A−2. Concerning the parameters Cand Sthere is little experimental evidence from which an estimate of them could be inferred. In [20] the value of the parameter Shas been set to S= 2K which we adopt here. (Alternatively, in a model approach of DNA denaturation dynamics [11] the parameters Sand Cwere specified to fit the melting temperature of certain DNA polymers.) The parameter Cis treated as adjustable. Furthermore we use for the value of the attenuation parameter β= 0.5˚ A−1[11]. With a suitable time scaling t→pDα2/m t and dividing the original Hamiltonian by Done passes to a dimensionless formulation with quantities: ˜xn,i =αxn,i ,˜yn,i =αyn,i ,˜zn,i =αzn,i (12) ˜p(x) n,i =p(x) n,i √mD ,˜p(y) n,i =p(y) n,i √mD ,˜p(z) n,i =p(z) n,i √mD ,(13) ˜ C=C α2D,˜ K=K α2D,˜ S=S α2D,(14) ˜ dn=α dn,˜r0=α r0,˜ h=α h , ˜ β=β α.(15) In the following, we omit the tildes. The equations of motion are derived from the Hamiltonian presented in Eqs. (6)-(11) and read as ˙xn,i =p(x) n,i ,(16) ˙p(x) n,i = 2 [ exp(−dn)−1 ] exp(−dn)∂dn ∂xn,i −2Kln,i ∂ln,i ∂xn,i +ln+1,i ∂ln+1,i ∂xn,i  −S1−β 2(dn−dn−1)(dn−dn−1) exp[−β(dn+dn−1)] −1 + β 2(dn+1 −dn)(dn+1 −dn) exp[−β(dn+1 +dn)] ×∂dn ∂xn,i ,(17) ˙yn,i =p(y) n,i ,(18) ˙p(y) n,i = 2 [ exp(−dn)−1 ] exp(−dn)∂dn ∂yn,i 7 −2Kln,i ∂ln,i ∂yn,i +ln+1,i ∂ln+1,i ∂yn,i  −S1−β 2(dn−dn−1)(dn−dn−1) exp[−β(dn+dn−1)] −1 + β 2(dn+1 −dn)(dn+1 −dn) exp[−β(dn+1 +dn)] ×∂dn ∂yn,i ,(19) ˙zn,i =p(z) n,i ,(20) ˙p(z) n,i = 2 [ exp(−dn)−1 ] exp(−dn)∂dn ∂zn,i −2Kln,i ∂ln,i ∂zn,i +ln+1,i ∂ln+1,i ∂zn,i  −S1−β 2(dn−dn−1)(dn−dn−1) exp[−β(dn+dn−1)] −1 + β 2(dn+1 −dn)(dn+1 −dn) exp[−β(dn+1 +dn)] ×∂dn ∂zn,i −C( 2 zn,i −zn+1,i −zn−1,i ),(21) with the derivatives ∂dn ∂xn,i =(−1)i+1 (dx n+xn,1−xn,2) dn+d0 ,(22) ∂ln,i ∂xn,i =Lx+xn,i −xn−1,i ln,i +l0 ,(23) and the equivalent expressions for ∂dn/∂yn,i,∂ln,i/∂yn,i,∂dn/∂zn,i and ∂ln,i/∂zn,i. The values of the scaled parameters are given by K= 0.683, r0= 44.50, h= 15.13 and l0= 31.39. One time unit of the scaled time corresponds to 0.198 ps of the physical time. 3 Relaxation dynamics of mechanically stressed DNA 3.1 Pulling process in the ladder and twisted configuration We study the energy exchange process taking place after the double-stranded DNA has been forced mechanically into a non-equilibrium configuration. In more detail, we consider the situation when at one end (referred to as the pulled 8 end) of short DNA molecules consisting of 10 up to 30 base pairs the two strands get pulled laterally in opposite directions by an external force (exerted e.g. when the two strands of one end of the molecule are attached to two different solid supports that are progressively separated [8]) so that they are dragged apart leading to gradual opening of the double-stranded chain. The pulled end is formed by the base pairs with index n= 1 and the rest of the molecule with n > 1 is located above it in the zdirection. Therefore a positive value of zn,i −zn−1,i will represent a local z-stretching of the strand i. In order to mimic the pulling process we assume that the pulling force is directed solely along the x−direction and the x−coordinates of the terminal bases are shifted uniformly with a pulling velocity of vp= 20µm/s while the corresponding yand zcoordinates are kept fixed at y1,i =z1,i = 0. Moreover, due to the constraint imposed by the backbone rigidity there result deformations not only at the pulled end but also for the rest of the helix. In general, the extent of lateral deformations subsides along the strands from the pulled towards the free end. The motion of the pulling unit is assumed to terminate when the lateral elongation at the pulled end has reached a certain value, taken as d= 2.31 ˚ A, so that the corresponding terminal hydrogen bond is broken. Moreover, the next-neighboring hydrogen bond is broken too and the third one is already considerably stretched. The typical resulting deformation pattern of the double stranded DNA is depicted in Figs. 2 and 3 for the ladder and twisted configuration, respectively. (In the Figures energy is expressed in terms of eV , the unit for length is ˚ Aand the time is given in ns.) Comparing the resulting DNA deformations we note that for both, the ladder and the twisted configuration, virtually equal lateral elongations are obtained. However, unlike the ladder configuration the twisted one experiences additional changes of the twist angles. There arises an overall decrease in the twist angle and the further a base pair is apart from the pulled end the more is the corresponding twist angle reduced (cf. Fig. 3 (b)) leading to an untwist of the helix. Nevertheless, with growing distance from the pulled end the difference between two consecutive angle changes becomes smaller. Thus the relative twist angle between two adjacent base planes diminishes towards the free end. While for the twisted system the axial deformations are negligible (see Fig. 3 (c)) we note that for the ladder configuration there results noticeable diminution of the axial equilibrium distances between neighboring base planes which is strongest at the free end and weakens gradually towards the pulled end (see Fig. 2 (b)). The relations between the partial deformation energies for the twisted system are shown in Fig. 4. The major contributions stem from the Morse potential energy associated with the H-bridge deformations and the stacking interaction energy both exceeding the energetic content of the covalent bond and axial distance change energy, respectively by four orders of magnitude. Typical for DNA molecules is that there exist different elasticity regimes, that is the degree of the deformation depends not overall linearly on the force (see e.g. [3]) which is reflected here in the nonlinear character of the temporal behavior of the deformation energies. 9 Figure 8: Relaxation dynamics: Time-evolution of the structural changes of the twisted system. Superimposed is the temporal behavior of the deformations during the pulling process with an inversed time. (a) Mean lateral deformations ¯ d(t). (b) Mean changes of the distances between consecutive base planes ¯ l(t). (c) Mean angular deformation ¯ φ(t). than in the twisted one. Hence the ladder system is more rigid with respect to spatial deformations than its twisted counterpart which with its additional angular degrees of freedom seems to possess more flexibility but less adaptability. In particular we observe that, with regard to the lateral deformations the twisted double-stranded system almost reverts to the original non-distorted equilibrium shape on the expense of substantial fluctuations in axial direction. In contrast, the ladder system is eventually left in comparatively strong lateral deformation but without significant axial distance fluctuations pointing to strong backbone rigidity. These results demonstrate that the relaxation process does not take place as the complete reversal of the pulling process. Similar behavior has been observed for the force-induced melting of the DNA double helix and its subsequent reannealing [3]. Finally we remark that the velocity of the complete recombination of the DNA molecule is lower than the velocity imposed by the forcing unit not uncommon for the opening-closing cycle of DNA molecules (see e.g. [8],[24]). 16 3.5 Guided relaxation We have also studied the elastic response of the stretched DNA double chains for the twisted system when at the end of the pulling process the molecule is not immediately exempted from the forcing unit. The starting point is an unopened molecule whose terminal end is pulled laterally with a velocity vp= 20 µms−1. When a certain lateral extension of the terminal base pair is achieved the direction of the pulling motion is reversed (without waiting time) which returns the forcing unit to the position it had before opening allowing for guided relaxation. We monitored the mean lateral extension ¯ d(t) for different values of the velocity of the backward motion, namely vp= 20 µms−1,vp= 10 µms−1and vp= 5 µms−1. The results regarding the structural changes are shown in Fig. 9 for backward velocity vb= 20 µms−1. As expected, at the end of the backward guidance of the terminal base pair the whole of the double helix restores virtually the original (undistorted) shape. Analogously, the associated deformation energies decrease steadily in the course of the relaxation process and adopt their initial zero content which they had before the pulling operation. However, we observe an asymmetry between the forward and backward elongation curves. Particularly, in an early stage of the retraction of the pulling unit the corresponding relaxation curves are in advance of their counterparts for the pulling process which seems to present a typical behavior of DNA molecules (see e.g. [2],[3],[13]) With lowered backward velocity the process till complete relaxation is achieved takes correspondingly longer times. 3.6 Parameter dependence For a parameter study we have varied the stacking interaction strength Sand simulated the pulling process to follow afterwards the relaxation dynamics. As the main result we found that the larger Sis the more rapid proceeds the relaxation process, viz. the faster is the quasi-equilibrium regime attained. Furthermore, it holds in general that the energy migration takes more time in the case of the ladder system than for its twisted counterpart. With view to a dependence on the molecule length N, i.e. the number of base pairs, we remark that the time scale for the relaxation process remains basically the same when Nis varied in the range from 10 to 30 providing strong evidence for the enormous elasticity of DNA molecules on the lower amplitude scale. In contrast, for large-amplitude elongations the dissociation time of DNA molecules increases from a fraction of a second to hours over the range of N= 10 to 20 base pairs [12]. In addition, with regard to asymmetry we investigated also cases when the ends of the strands are pulled laterally with unequal forces (in the extreme case only one strand is pulled) resulting in elongation patterns being asymmetric with respect to the central helix axis. We observed then that the systems behave during the subsequent relaxation dynamics like in the symmetric cases discussed above and attain finally quasi-equilibrium regimes being not far away 17 Figure 9: The relaxation dynamics for the twisted system for backward velocity vb= 20µm/s. (a) Mean lateral deformations ¯ d(t). (b) Mean changes of the distances between consecutive base planes ¯ l(t). (c) Mean angular deformation ¯ φ(t). from the respective original equilibrium configurations with fluctuating structural coordinates. 4 Summary In this paper we have demonstrated that within the frame of an oscillator network model we are able to study numerically the mechanical stability and elasticity properties of DNA molecules. Our model of coupled oscillators takes essential microscopic degrees of freedom of DNA and the inherent interactions between them into account. We have focused our interest on the opening-closing dynamics of double-stranded DNA molecules. The opening of a DNA molecule has been forced by mechanical stress imposed at a terminal end of the molecule bringing it into a non-equilibrium state. We have followed the subsequent relaxation towards an equilibrium state with recombination to its double-stranded conformation. The similarities and differences between the relaxation dynamics for a planar ladder-like DNA molecule and a twisted one have been discussed. In particular we have shown that the attainment of a quasi-equilibrium regime proceeds faster in the case of the twisted DNA form than for its ladder coun18 terpart. This has led us to the conclusion that in the twisted form the DNA molecule is more flexible with respect to conformational changes than in the planar ladder-like version. There are some limitations in our model approach and for an improvement one should certainly consider the dependence on the DNA sequence, the impact of the chemical environment and the influence of temperature on the mechanical stability. 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