We propose a model for DNA dynamics by introducing the helical structure through twist deformation in analogy with the structure of helimagnet and cholesteric liquid crystal system. The dynamics in this case is found to be governed by the completely integrable sine-Gordon equation which admits kink-antikink solitons with increased width representing a wide base pair opening configuration in DNA. The results show that the helicity introduces a length scale variation and thus provides a better representation of the base pair opening in DNA.
Deep Dive into Soliton-like base pair opening in a helicoidal DNA: An analogy with helimagnet and cholesterics.
We propose a model for DNA dynamics by introducing the helical structure through twist deformation in analogy with the structure of helimagnet and cholesteric liquid crystal system. The dynamics in this case is found to be governed by the completely integrable sine-Gordon equation which admits kink-antikink solitons with increased width representing a wide base pair opening configuration in DNA. The results show that the helicity introduces a length scale variation and thus provides a better representation of the base pair opening in DNA.
The B-form DNA double helix molecule is usually modeled by two parallel chains of nucleotides known as strands with linkage interms of dipole-dipole interaction along the strands and the two strands are coupled to each other through hydrogen bonds between the complementary bases [1]. Molecular excitations in DNA based on the above model is generally governed by nonlinear evolution equations [2,3,4] and in particular by the completely integrable sine-Gordon-type equations [5,6]. In the above studies, DNA is treated as two coupled linear chains without involving the helical character of its structure.
However, in nature DNA exists in a double helix form and recently there were attempts by few authors to study the dynamics by taking into account the helical character of the double helix through different forms of coupling. For instance, Gaeta [7,8,9], Dauxios [10] and Cadoni et al [11] assumed that the torsional coupling between the n th base on one strand and the (n + 4) th base on the complementary strand is the responsible force for the helical nature in DNA and found that the localized excitations are governed by solitons and breathers. Barbi et al [12,13] and Campa [14] however introduced the helicity through a proper choice of the coupling between the radial and the angular variables of the helix and obtained breathers and kinks. On the other hand, very recently, Takeno [15] introduced helicity in DNA through a helical transformation and obtained non-breathing compacton-like modes to represent base pair opening through numerical calculations.
In this, paper, we propose a model by introducing the helical character in each strand of the DNA molecule through a twist deformation of the chain in analogy with the twist in cholesteric liquid crystal [16] or orientation of spins in a helimagnet [17]. As an illustration in Fig. 1(a-c) we have presented a schematic representation of the arrangement of bases, spins and molecules respectively in a DNA double helical chain, in a helimagnet and in a cholesteric liquid crystal leading to the formation of helical structure. In Fig. 1 In a recent paper, one of the present authors studied the nonlinear spin dynamics of a helimagnet by incorporating the helicity interms of Frank free energy corresponding to the twist deformation which is responsible for helicity in a cholesteric liquid crystal system [17,18]. The Frank free energy density associated with the twist deformation in a cholesteric liquid crystal is given by [p • (∇ × p) -q 0 ] 2 where the unit vector p represents the director axis which corresponds to the average direction of orientation of the liquid crystal molecules, q 0 = 2π q is the pitch wave vector and q is the pitch of the helix. The discretised form of the above twist free energy is written as
where k is the unit vector along z-direction. In analogy with the above, we write down the free energy associated with the twist deformation in terms of spin vector as {[ k • (S n × S n+1 )] -q 0 } 2 . By taking into account the form of free energy the Heisenberg model of Hamiltonian for an anisotropic helimagnetic system is written as [17]
In Eq.( 1), S n = (S x n , S y n , S z n ) represents the spin vector at the n th site and the terms proportional to J and A respectively represent the ferromagnetic spin-spin exchange interaction and uniaxial magneto-crystalline anisotropy with the easy axis along z-direction. h de-notes the elastic constant associated with the twist deformation. We identify the above helical spin chain with one of the strands of the DNA double helical chain. Therefore, in a similar fashion we can write down the spin Hamiltonian H 2 for another helimagnetic system corresponding to the complementary strand with the spin vector S n replaced by
n . We assume that in the Hamiltonian the exchange, anisotropic and twist coefficients as well as the pitch in both the helimagnetic systems are equal. Now, for mapping the helimagnetic spin system with the DNA double helical chain we rewrite the Hamiltonian by writing the spin vectors as
are the angles of rotation of spins in the xy and xz-planes respectively. The new Hamiltonian corresponding to H 1 is written as
We now map the two helical spin systems with the two strands of the DNA double helix with the two angles θ n (θ ′ n ) and φ n (φ ′ n ) representing the angles of rotation of bases in the xz and xy-planes of the two strands respectively. A horizontal projection of the n th base of DNA in the xy and xz-planes is shown in Figs. 2(a) and 2(b). Here Q n and Q ′ n denote the tips of the n th bases attached to the strands R and R ′ at P n and P ′ n respectively. The DNA double helix chain is stabilised by stacking of bases through intrastrand dipole-dipole interaction and through hydrogen bonds (interstrand interaction) between complementary bases. The interstrand base-base interaction or hydrogen bonding energy between the complementary bases depends on the distance between them and using the
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