Effect of Solvent on Stretching and Twisting of DNA

We extend the Peyrard-Bishop model into helicoidal DNA case in presence of solvent. Our model focuses on twisting and stretching DNA base pair motions. The dynamics of this system are drawn from the solitary wave solution by using multiple scale expansion to the case of the vectorial lattice. The breather soliton is the solution for the stretching motion and the kink soliton is the solution for the twisting motion. A positive value of the solvent strength factor stretches the hydrogen bond and the strand arc length which makes the DNA more winded.


Introduction
The conformational changes of DNA due to the environment is not fully understood yet. The complex dynamics of DNA have been investigated. Peyrard-Bishop (PB) presented simple model focusing on the denaturation which is initialized by the bubble formation, without considering the DNA helicity in a vacuum [1]. In this model, PB interpret the hydrogen bond as a Morse potential. Next, Dauxois extended the PB model by considering the DNA helicity, assumed the bases in a different strand do not only interact in transversal direction but also in longitudinal direction [2]. Still, in helicoidal DNA case, Barbi et al [3] extended the PB model by generalizing the curvature of the DNA double helix, assuming the helicity can be presented by observing the interaction of three body diks (nucleotida). Because of that, Barbi uses two degrees of freedom, radial and angular (polar coordinates) and modifies the Hamiltonian PB model by adding the helicoidal term [3].
In this paper, we include the effect of the environment, in our case is the solvent, to realistically model the dynamics of DNA. PB, Dauxois, and Barbi have modeled the DNA dynamics but still cannot explain the real DNA condition. In fact, DNA is immersed in solvent, one of the environment factors affecting the dynamics of DNA. According to Zoli, solvent will stabilize the open state or the base pairs opening [4]. Solvation causes weakening and lengthening of the hydrogen bond from base pairs [5] while experimentally it is the transition of DNA from B-DNA to A-DNA due to increased solvent concentration [6]. Some experimental facts of DNA-solvent interaction cannot be fully explained. Because of that, we present a model of DNA in solvent for helicoidal DNA by adopting the helicoidal model of Barbi by adding the solvent potential term in Lagrangian.

Lagrangian Model
The helicoidal model of DNA is described in polar coordinate system with two degrees of freedom; r n dan ϕ n , radial dan angular variable, both of them will produce the stable helical (variable coupled) structure. The distance of a base from the hydrogen bond center in equilibrium condition (B-DNA) is denoted by R 0 . The distance between the neighbouring base pairs denoted by h. In our model, the adjacent site height h is assumed constant. The length of backbone between disks in one strand when in equilibrium state is denoted which is larger than h. The twist angle between base pair in equilibrium state is ϕ n − ϕ n−1 = ±Θ 0 . The values of h, R 0 , and Θ 0 adopted from B-DNA parameters [3]. When the opening base pair processing, the backbone interpreted by a spring will stretch, L d . The strands are hydrophobic, making the water molecules around base pairs eliminated from the core of DNA, by forming helicoidal backbone. The competition between the rigidity of backbone and the stacking interaction of the neighboring base pairs stabilizes the double helix structure of DNA. To get the effect of helicity, we should take three body curvature, and adding to the Lagrangian in third terms (1), where G 0 is a curvature constant. The DNA-solvent interaction described by the solvent potential, in the last terms of the Lagrangian (1) with f s is the solvent strength, and l s is the width of solvent potential [4] that expresses the effective range. Thus the Lagrangian of our model is The mass of a disk, m, the depth of the Morse potential, D and the width of the Morse potential, α, are adopted from the PB model [3], m = 300 u.m.a., D = 0.04 eV, α =4.45Å −1 , while K = 1.0 eVÅ −2 and G 0 = KR 2 0 /2 are adopted from the helicoidal DNA model [3]. The value of parameter geometry for B-DNA are R 0 ≈ 10.0Å, Θ 0 = 36 • and h = 3.4Å. In this model, the assumed constant h means that before and after solvation h is constant.
To simplify the Lagrangian, we present some dimensionless variables, r n = αr n , t = For simplicity hereafter, the new variables are written without prime. The shift of base, y n , or the arc length y n = r n − R 0 and the length of the disk position in angular, φ n = R 0 (ϕ n − nΘ 0 ).
The stacking interaction (fourth term) is expanded to second order around the equilibrium position, yielding The Morse potential and the solvent potential are expanded to forth order, so we get the equation of motion: andφ The where . K yy and K φφ are the effective elastic constant from the shifting or stretching of disk and twist rotation, K yφ is the coupling constant between the stretching and twisting motion.

The envelope soliton solution for small amplitude
The soliton solution is obtained by using the analitycal method developed by Cocco in [7]. We use the well-known technic to get the solution of envelope soliton for small amplitude use the Multiple Scale Expansion (MSE), based on the perturbation theory. The model of helicoidal DNA in solvent has two component wave vectors E; y n and φ n . First, we solve the linear order.
Here the wave is packed with a weak dispersion. From (2) and (3) the equation of motion is written: Now, we take care of the nonlinearity of the system to get the small amplitude envelope soliton. To increase the accuracy of the nonlinear term gradually, we present an expansion parameter . We introduce the coordinates x 1 = x, t 1 = t, t 2 = 2 t. The nonlinearty solution form is are the slowly varying amplitudes that envelope the wave packet. The velocity of the envelope is smaller than the velocity of the carrier wave. The carrier wave is e i(q 0 n 0 −ω + t 0 ) in first order and e 2i(q 0 n 0 −ω + t 0 ) in second order. The wave packet is interpreted by first order O( ), the second term arise because the linear system produce the nonzero solution on the force constant matricsĴ (0) at q 0 = 0,  for γ, is γ = γ c A 2 . The solution of the constant term (e 0 ), µ and σ, are obtain from the second order and the third order. Checking the second order, we get −2K yφ ∂ ∂x 1 σ 1 = 0 and σ 1 = const.. From the non oscillating term of the third order we get the solution for µ 2 can be written we get the solution µ 1 and σ 2 , µ 1 = µ 1c |A| 2 , σ 2 = σ 2c |A| 2 dx 1 . Now we are done with the envelopes, η, σ, γ, µ, unless those are dependent on A. The nonlinear term O( 3 ) in exp i (q 0 n 0 − ω + t 0 ) gives the equation for A in form of Nonlinear Schrödinger (NLS). We found the NLS equation for envelope expression A in moving frame with velocity ω (1) This simplifies to P A s 1 s 1 + iA t 2 + HiA s 1 +IA + Q |A| 2 A = 0, with P = E F , H = H F , I = I F , Q = Q F . When P Q > 0, the bright soliton case, the envelope solution will be [8] where u e is the velocity of the envelope and u c is the velocity of carrier, while the amplitude is where L e is the inverse width of the envelope. The complete solution is and where V e = ω   Effect of solvent on the twisting motion The solvation affects DNA gets more winded ( Figure 2) and increasingly steep of the kinksoliton solution. The arc length of the base pair for x = 0 is φ 0 = 0, then the arc length of the neighbours have the same value but have an opposite direction.  Figure 2. The analytical solution for arc length φ n with the variation of f s .

Conclusion
The solvent effect on the helicoidal DNA dynamics especially for stretching motion changes the stability of the open state, it is showed in the shifting of the minimum point to a positive value, y n > 0. The shifting is caused by the constant force due to the existence of solvent. For twisting motion, the solvation caused the arc length of the disk to be more positive in value and more negative in value with the initial condition at x = 0. This means that DNA after solvation is more winded for the more positive value of the solvent barrier factor. The hydrophobic effect of DNA strand forces the structure to be double helix. If the environment of DNA consist of water effect the backbone winding. Because of the behaviour of the base pair, it can be concluded that DNA will be more winded if there is some water around the DNA. Related to the analytical solution of this model, that the positive value of f s lets the DNA more winded, we get can suggest that a positive f s correspond to the situation around DNA with high water content.