Soliton-like base pair opening in a helicoidal DNA: An analogy with helimagnet and cholesterics

Soliton-lik e base pair op ening in a helicoidal DNA : An analogy with helimagnet and c holesterics M. Daniel ∗ and V. V asumathi † Centr e for Nonline ar D ynamics, Scho ol of Physics, Bhar athidasan University, Tiruchir app al li - 620 024, India. (Dated: Marc h 11, 2022) Abstract W e prop ose a mo del for DNA dynamics b y introd ucing the helical structur e th rough twist de- formation in a nalogy with the structure o f helimagnet and c holesteric liquid crystal system. The dynamics in this case is foun d to b e go ve rned by the complete ly integ rable sine-Gordon equation whic h admits kink-an tikink solitons with increased width represent ing a wide b a se pair op ening configuration in DNA. Th e r esu lt s sho w that the helicit y introd uces a length scale v ariation and th us provides a b ett er r ep resen tation of the base pair open in g in DNA. P ACS nu mbers: 87.1 4 .Gg, 05.45.Yv, 02.30 .Jr ∗ Electronic addres s: daniel@cnld.b du.ac.in † Electronic a ddress: v asu@cnld.b du.ac.in 1 The B-fo rm D NA double helix mo lecule is usually mo deled by t w o parallel c hains of n ucleotides kno wn a s strands with link age in terms of dip ole-dip ole inte raction along the strands a nd the tw o strands a r e coupled to eac h other t hr o ugh h ydrogen b onds b et w een the complemen tary bases [1]. Molecular excitations in DNA based on t he ab o v e mo del is generally go v erned by nonlinear ev olution equations [2, 3, 4] and in particular by the completely in t egr a ble sine-Gordon-type equations [5, 6]. In t he ab o v e studies, D NA is treated a s tw o coupled linear c hains without in v olving the helical c haracter of its structure. Ho w ev er, in nat ure DNA exists in a double helix for m and recen tly there w ere attempts b y few a ut ho rs to study the dynamics b y taking in to account the helical c haracter o f the double helix through different forms of coupling. F or instance, Gaeta [7, 8, 9], Dauxios [10] and Cadoni et al [11 ] assumed that the torsional coupling b et w een the n th base on one strand and the ( n + 4) th base on the complemen tary strand is the resp onsible force for the helical nature in DNA and f o und that the lo calized excitations ar e go v erned by solitons and breathers. Barbi et al [12, 13 ] and Campa [14] how ev er in tro duced the helicit y through a prop e r c hoice of the coupling b e t w een the radial and the angular v ariables of the helix and obta ine d breathers and kinks . On the other hand, v ery recen tly , T ak eno [15] in tr o duced helicity in DNA thro ug h a helical transformation and obtained non-breathing compacton-lik e mo des to re presen t base pair op ening through n umerical calculations. In this, paper, w e prop ose a mo del b y in tro ducing the helical character in eac h strand of the DNA molecule through a twis t deformation of the c hain in analogy with the tw ist in c holesteric liquid crystal [16] or orientation o f spins in a helimagnet [17]. As an illus tration in Fig. 1(a-c) we hav e presen ted a sc hematic represen tation of the a r rangeme n t of bases, spins and molecules resp ectiv ely in a DNA double helical chain, in a helimagnet and in a c holesteric liquid crystal leading to the formation of helical structure. In Fig. 1(a) R and R ′ represen t the t w o c omplemen tary strands of the D NA double helix and the dots b et w een the arrow s represen t the hydrogen b onds b et w een the complemen tary bases. The arrows and short lines in Figs. 1(b) and 1(c) resp ective ly represen t the spins and molecules at differen t sites and planes in a helimagnet and in a c holesteric liquid crystal. When w e go along the z-direction, the or ie n tation of spins and molecules are tilted f rom one plane to the next through certain tilt angle. If w e join the tips of the arrow s represen ting the spin v ectors and also, the tips of the molecules they form a helix as show n in Figs. 1(b) and 1(c) resp e ctiv ely . 2 z x y k ^ (a) (b) (c) z ^ x y k | R R z k ^ FIG. 1: A s c hematic r epresen tation of (a) a DNA double helical c hain (b) a helimagnet a nd (c ) a c h olesteric liquid crystal system In a recen t pap er, one of the presen t authors studied the nonlinear spin dynamics of a helimagnet by incorp orating the helicity in t erms of F rank free energy corresponding to the t wist deformation whic h is res p onsible for helicit y in a c holesteric liquid crystal system [17, 1 8]. The F ra nk free energy densit y asso ciated with the twist deformation in a c ho lesteric liquid crystal is giv en b y [ p · ( ∇ × p ) − q 0 ] 2 where t he unit v ector p repres en ts the director axis whic h corr esp onds t o the a v era g e direction of orientation of the liquid crystal molecules, q 0 = 2 π q is the pitc h w a v e v ector and q is the pitc h of the helix . The discre tised form of the ab o ve t wist free energy is written as { [ ˆ k · ( p n × p n + 1 )] − q 0 } 2 where ˆ k is the unit v ector along z-direction. In analogy with the ab o ve, we write do wn the free energy asso ciated with the t wist deformatio n in terms of spin vec tor as { [ ˆ k · ( S n × S n + 1 )] − q 0 } 2 . By taking into account the f orm of free energy the Heisen b erg mo del of Hamiltonian for an anisotro pic helimagnetic system is written as [17] H 1 = X n  − J ( S n · S n + 1 ) + A ( S n z ) 2 + h { [ ˆ k · ( S n × S n + 1 )] − q 0 } 2 i . (1) In Eq.(1), S n = ( S x n , S y n , S z n ) represen ts the spin vec tor at the n th site and the terms prop or- tional t o J and A resp ectiv ely represen t the ferromagnetic spin-spin exc hange in teraction and uniaxial magneto-crystalline anisotropy with t he easy axis along z-direction. h de- 3 notes the elastic constan t asso ciated with the t wist deformatio n. W e iden tify the ab o v e helical spin c hain with one of the strands of the DNA double helical ch ain. Therefore, in a similar fashion w e can write dow n the spin Hamiltonian H 2 for another helimagnetic system corresp onding to the complemen tary strand with the spin v ector S n replaced b y S ′ n . W e assume that in the Hamiltonian the exc ha nge, anisotropic and twist co efficien ts as w ell as the pitc h in b oth the helimagnetic systems ar e equal. Now, f or mapping the helimagnetic spin system with the DNA double helical c hain we rewrite the Hamilto nian b y writing the spin v ectors as S n ≡ ( S x n , S y n , S z n ) = (sin θ n cos φ n , sin θ n sin φ n , cos θ n ) and S ′ n ≡ ( S ′ x n , S ′ y n , S ′ z n ) = (sin θ ′ n cos φ ′ n , sin θ ′ n sin φ ′ n , cos θ ′ n ), where θ n ( θ ′ n ) and φ n ( φ ′ n ) ar e the angles of rotatio n of spin s in the xy and xz-planes r e sp ectiv ely . The new Ha milto nian cor- resp onding to H 1 is written as H 1 = X n [ − J { sin θ n sin θ n +1 cos( φ n +1 − φ n ) + cos θ n cos θ n +1 } + A cos θ 2 n + h { sin θ n × sin θ n +1 sin( φ n +1 − φ n ) − q 0 } 2  , (2) W e n o w m ap the t w o helical spin systems with the tw o s trands of the DNA double helix with the t w o angles θ n ( θ ′ n ) and φ n ( φ ′ n ) represen ting the angles of rotatio n of ba ses in the xz and xy-planes of the t w o strands resp ec tiv ely . A horizontal pro jection 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 attac hed to the strands R and R ′ at P n and P ′ n resp e ctiv ely . The DNA double helix chain is stabilised by stac king of ba ses through in trastrand dip ole-dip ole in teraction and through h ydrogen b onds (in terstrand interaction) betw een complemen tary bases. The in terstrand base-base interaction or h ydrogen b o ndin g energy betw een the compleme n tary bases dep end s on the distance b et w een them and using the simple geometry in F ig s . 2(a,b), w e can write the distance b et w een the t ips of bases a s [6], ( Q n Q ′ n ) 2 ≈ 2 [sin θ n sin θ ′ n (cos φ n cos φ ′ n + sin φ n sin φ ′ n ) − cos θ n cos θ ′ n ] . (3) No w, the ab o v e equation represe n ts the h ydrogen bonding energy betw een complemen tary bases and w e can write do wn the Ha milto nian for the inters trand in teraction o r hydrogen 4 | | | y x O n n P φ n φ n n n P Q Q | | | | | O P n θ n n Q n Z n Z n Q θ n n P X Z (a) (b) FIG. 2: A horizon tal pro jection of the n th base pair in a DNA double helix (a) in the x y -p la ne and (b) in the xz -plane. b onds is written as H 12 = η [sin θ n sin θ ′ n (cos φ n cos φ ′ n + sin φ n sin φ ′ n ) − cos θ n cos θ ′ n ] , (4) where η is a constant. The total Hamiltonian for our helicoidal mo del of DNA in terms of the angles of rotation of bases using the ab ov e Hamiltonians is written as H = H 1 + H 2 + H 12 = X n [ − J { sin θ n sin θ n +1 cos( φ n +1 − φ n ) + cos θ n × cos θ n +1 + sin θ ′ n sin θ ′ n +1 cos( φ ′ n +1 − φ ′ n ) + cos θ ′ n cos θ ′ n +1 } + h { [ sinθ n sin θ n +1 × sin( φ n +1 − φ n ) − q 0 ] 2 + [sin θ ′ n sin θ ′ n +1 × sin( φ ′ n +1 − φ ′ n ) − q 0 ] 2 } + η { sin θ n sin θ ′ n × (cos φ n cos φ ′ n + sin φ n sin φ ′ n ) − cos θ n cos θ ′ n } + A (cos 2 θ n + cos 2 θ ′ n )  . (5) Using the equation of motion for the corresp onding quasi-spin mo del [19 ] in the limit A >> J, η , h , w e obtain ˙ φ n = 2 A cos θ n and ˙ φ ′ n = 2 A ′ cos θ ′ n . Hence, under absolute minima of 5 p oten tial the Hamiltonian (5 ) b e comes H = X n  I 2 ( ˙ φ 2 n + ˙ φ ′ 2 n ) + J [2 − cos( φ n +1 − φ n ) − cos( φ ′ n +1 − φ ′ n )] − η [1 − cos( φ n − φ ′ n )] + h { 2 q 2 0 − [sin( φ n +1 − φ n ) − q 0 ] 2 − [sin( φ ′ n +1 − φ ′ n ) − q 0 ] 2 }  , (6) where I = 1 2 A 2 is the momen t of inertia of the bases around the axes at p n ( p ′ n ). While rewriting the Hamiltonian in the a b ov e for m, w e ha v e restricted that the bases are rotating in the plane w hic h is normal to the helical axis. In otherw o rds , w e hav e now restricted our problem to a plane base rot a tor mo del [6] b y assuming θ = θ ′ = π / 2. Ha ving formed the Hamiltonia n, t he dynamics of the D NA double helix molecule can b e understo o d b y constructing the Hamilton’s equations of mo t io n corresp onding to the Hamiltonian (6) a s I ¨ φ n = [ J + 2 h cos( φ n +1 − φ n )] sin( φ n +1 − φ n ) − [ J + 2 h cos( φ n − φ n − 1 )] sin( φ n − φ n − 1 ) + η sin( φ n − φ ′ n ) − 2 hq 0 [cos( φ n +1 − φ n ) − cos( φ n − φ n − 1 )] , (7a) I ¨ φ ′ n =  J + 2 h cos( φ ′ n +1 − φ ′ n )  sin( φ ′ n +1 − φ ′ n ) −  J + 2 h cos( φ ′ n − φ ′ n − 1 )  sin( φ ′ n − φ ′ n − 1 ) + η sin( φ ′ n − φ n ) − 2 hq 0 [cos( φ ′ n +1 − φ ′ n ) − cos( φ ′ n − φ ′ n − 1 )] , (7b) where o verdot represen ts deriv ativ e with resp ectiv e to time. Eqs. (7a) and (7b) describ e the dynamics of the DNA double helix at the discrete lev el when the helical nature of the molecule is represen ted in the form of a t wist-lik e deformation. It is expected that the difference in angular rotation of bases with resp e ct to neighbouring bases along the t w o strands is small [5, 20]. Also, v ery recen tly Gaeta [8, 9] prop osed that the helical structure of DNA will in tr o duce q ualitativ e c hang es only in the small amplitude 6 regime. Hence, under the small a ngle approximation, in the con tinuum limit, the discrete equations of mot ion (7a,b) after suitable res caling of t ime and redefinition of the parameter η reduce to φ tt = ( J + 2 h ) I φ z z + η sin( φ − φ ′ ) , (8a) φ ′ tt = ( J + 2 h ) I φ ′ z z + η sin( φ ′ − φ ) . (8b) Adding a nd subtracting Eqs. (8a) and (8b) and after suitable rescaling of the v ariable z , w e obtain Ψ tt − Ψ z z + sin Ψ = 0 , (9) where Ψ = 2 φ a nd w e hav e further chos en 2 η = − 1. Also, while deriving Eq. (9), w e hav e c hosen φ ′ = − φ , b ecause among the p ossible rotations of bases, rotation of compleme n tary bases in opp osite direction easily facilitate an open state configuratio n. Eq. ( 9) is the com- pletely integrable sine-Go rdon equation whic h w as originally solv ed for N-soliton solutions in terms of kink and antik ink using the most celebrated Inv erse Scattering T ransform metho d [21]. F or instance, the kink and an tikink one soliton solution o f the sine-Gordon e quation is written in terms of the o r ig inal v ariables as φ ( z , t ) = 2 arctan[ exp [ ± 1 √ 1 − v 2 × s I ( J + 2 h ) ( z − v t )]] , (10) where + and − represe n t the kink and an tikink soliton solutions respectiv ely and v is t he v elo cit y o f the soliton. In Fig. 3(a ) w e hav e plotted the angular rotation of bases φ in terms of the kink-antikink one soliton solution a s given in Eq. (10) b y choosing the stac king, helicit y , momen t of inertia and v elo cit y parameters resp ectiv ely as J = 1 . 5 eV , h = 3 . 0 eV , I = 1 . 3 × 10 − 36 g cm 2 and v = 0 . 4 cm s − 1 [10, 1 5 ]. The kink-an tikink soliton solution whic h can propagate infinite distance and time describ es an op en state configuration in the DNA double helix whic h is sc hematically represen ted in Fig . 3(b). In order to understand the effect of helicity on the op en state configuration, in Fig. 3(c), w e hav e a ls o plott e d the kink and an tikink one soliton solution of the sine-Go r don equation in the absence of helicity , that is b y c ho osing h = 0 (k eeping all other parameters v alues the same), in the solution giv en 7 (a) (b) z t t z φ φ Kink Antikink Antikink Kink Base pair - 10 0 10 0 1 2 3 4 5 0 1 2 3 4 5 - 10 0 10 0 1 2 3 4 5 - 10 0 10 0 1 2 3 4 5 0 1 2 3 4 5 - 10 0 10 0 1 2 3 4 5 (c) z t t z φ φ (d) z Kink Antikink Kink Antikink Base pair - 10 0 10 0 1 2 3 4 5 0 1 2 3 4 5 - 10 0 10 0 1 2 3 4 5 - 10 0 10 0 1 2 3 4 5 0 1 2 3 4 5 - 10 0 10 0 1 2 3 4 5 FIG. 3: (a) Kink and an tikink one soliton solutio ns (Eq. (10)) of the sine-Gordon equation when helicit y is present ( h 6 = 0). (b) A sket c h of the formation of op en state configuration in terms of kink-an tikink solito ns in a DNA double h e lix when helicit y is present ( h 6 = 0). (c) Kink and an tikink one s o liton solutions of the sine-Go rdon equation when helicit y is absent (Eq. (10) wh e n h = 0). (d) A sk etc h of the formation of op en state configuration in terms of kink-an tikink solitons in DNA double h elix when helicit y is absent ( h = 0). 0 0.5 1 1.5 2 2.5 3 3.5 -8 -6 -4 -2 0 2 4 6 8 φ z h=0 h=1 h=2 h=3 h=4 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 Width Helicity (a) (b) FIG. 4: (a) T h e kink one soliton (Eq. (10)) represent ing base pair o p ening at t = 1 for different v alues of helicit y . (b) V ariation of wid th of the kink so liton against h eli cit y . in Eq.(10 ). F rom Figs. 3(a) and 3(c), w e observ e that when there is helicit y in the mo del ( h 6 = 0), the kink-an tikink soliton is getting broadened. In other w ords, helicit y in DNA mak es more num b er of base pairs to participate in the for ma t io n of op en stat e configuration without intro duc ing any qualitative c hange in the dynamics. This is a ls o sc hematically represen ted in Fig. 3(d) whic h lo oks eviden t on comparing Fig. 3(b). In order to highligh t the ab o v e fact, we hav e separately plotted the kink one solito n solution at a giv en time 8 ( t = 1) for differen t v alues of helicit y b y choosing h = 0 , 1 , 2 , 3 , 4 in F ig. 4(a). The increase in width aga inst helicity is explicitly represen ted in Fig . 4(b). F rom the figure it may b e noted that the increase in helicit y slo ws dow n the rotation of bases and mak es more and more n um b er of base pairs to part icipate in the op e n state configuration, th us pr oviding a b etter represen tation of base pair op ening in D NA. Th us, helicit y introduces a length scale v a riation in the base pair op ening. Similar results h a v e also b een o bserv ed b y Da ux ios [10] thro ugh a p erturbation a nalys is on his helicoidal mo del of DNA and obta ine d soliton with a muc h broader width. In order to ha v e a mor e realistic mo del, dissipativ e (viscous effect) and noise terms should b e added to the equations of motion. Experimentally , the life time of soliton in this case is shown to b e o f few nano seconds at ro om temp erature (see for e.g. Ref.[22]). Also, in a recen t pap er, Y akushevic h et al [23 ] through a n umerical analysis, sho w ed that when the viscosit y is strong the soliton mo v es a length o f only few c hain links and it will stop after that. On the other ha nd, when the viscosit y is low the soliton passes more than 3000 c hain links like a hea vy Bro wnian particle whic h is found to b e stable with resp ect to thermal oscillation. When the a bov e t w o effects a r e tak en in to accoun t Eq. ( 9 ) tak es the form Ψ tt − Ψ z z + sin Ψ = ǫ [ α Ψ t + β  ( z , t )] where the terms prop ortional to α and β are related to viscous surrounding a nd thermal force s re sp ectiv ely . A solito n p erturbation ana lysis [24] on the ab o v e equation sho ws that when the viscosit y is high the soliton mov es for a small distance and then stops. But when the viscosit y is lo w the soliton mov es for a long distance along the c hain. The detailed analytical calculations of the ab o ve study will b e separately published elsewhe re. In summary , we prop osed a new helicoidal mo del to study D NA dynamics b y introducing the helical ch aracter in analogy with the twist deformation in a c holesteric liquid crystal system and the spin arrangemen t in a helimagnet. The nonlinear dynamics of DNA under the presen t helicoidal mo del is found to b e g ov erned by the completely inte grable sine- Gordon equation in the con tinu um limit whic h admits kink and an tikink soliton solutions. F ro m the nature of solitons, we observ e that helicit y in tro duces a length scale v aria tion without causing an y change in the shap e of the soliton. Due to this scaling v ariation, the width of the solito n increases and hence w e obtain bro a de r kink-an tikinks. In otherw ords a large num b er of bases are inv olv ed in the base pa ir op ening, th us leading to a b etter represen tation. This broadened base pair o pening may act as a b etter energetic activ ato r 9 in the case of RNA-P o lymerase transp ort during transcription pro cess in DNA. As the con tinuum helicoidal mo del do es not intro duc e qualitativ e c ha ng es in the DNA dynamics , w e prop ose to study the full nonlinear dynamics of the helicoidal model of DNA (without making small angle approx imation) b y solv ing equations (7a) and (7b) nume rically and the results will b e publishe d elsewhere. The w ork of M . D a nd V.V form pa rt of a ma jor D ST pro ject. [1] L. V. Y akushevic h, Nonlinear Physics of D NA (Wiley-V CH, Berlin, 2004). [2] S. W. Englander, N. R. Kallen b a nc h, A. J. Heeger, J. A. Krumh a nsl and S. Lit win, Proc. Natl. Acad. Sci. 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