Synchronized Oscillations in Double-Helix B-DNA Molecules with Mirror-Symmetric Codons

Synchronized Oscillations in Double-Helix B-DNA Molecules with Mirror-Symmetric Codons Enrique Maciá 1 1 Departamento de Física de Materiales, Facultad CC. Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain; emaciaba@fis.ucm.es Version December 30, 2020 submitted to Symmetry Abstract: A fully analytical treatment of the base-pair and codon dynamics in double-stranded DNA 1 molecules is introduced, by means of a realistic treatment which considers different mass values 2 for G, A, T, and C nucleotides and takes into account the intrinsic three-dimensional, helicoidal 3 geometry of DNA in terms of a Hamitonian in cylindrical coordinates. Within the framework of the 4 Peyrard-Dauxois-Bishop model we consider the coupling between stretching and stacking radial 5 oscillations as well as the twisting motion of each base pair around the helix axis. By comparing 6 the linearized dynamical equations for the angular and radial variables when going from the bp 7 local scale to the longer triplet codon scale, we report an underlying hierarchical symmetry. The 8 existence of synchronized collective oscillations of the base-pairs and their related codon triplet 9 units are disclosed from the study of their coupled dynamical equations. The possible biological 10 role of these correlated, long-range oscillation effects in double standed DNA molecules containing 11 mirror-symmetric codons of the form XXX, XX’X, X’XX’, YXY, and XYX is discussed in terms of the 12 dynamical equations solutions and their related dispersion relations. 13


Introduction
The specific role of certain physical properties on the possible biological function of d n,n±1 = c 2 θ 2 n,n±1 + (R 0 + ρ n±1 ) 2 + (R 0 + ρ n ) 2 − 2 (R 0 + ρ n±1 ) (R 0 + ρ n ) cos θ n,n±1 (1) (d n,n±1 for the left strand is obtained by simply replacing ρ n → ρ n in Eq.(1)), where we have defined 83 θ n,n±1 = ± (ϕ n±1 − ϕ n ) as the relative angle between two neighboring bps, and ρ n = r n − R 0 as the 84 radial displacements about the equilibrium position (R 0 = 1 nm). In equilibrium, both distances 85 reduce to the value l 0 = d n,n±1 | eq. = d n,n±1 eq. = h 2 0 + 4R 2 0 sin 2 (θ 0 /2) 0.685 nm, where we have 86 adopted θ 0 = π/5.2 34.6º. We note that the distance between successive bps along the Z direction is 87 proportional to the twist angle, thereby preserving the helical structure during the dynamical evolution 88 [17]. It is further assumed that the dsDNA molecule does not move as a whole, so that the center 89 of mass is constant for each bp, and the radial displacements about the equilibrium position satisfy 90 ρ n = λ n ρ n , where λ n = m n /m n .

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Accordingly, we can write the dsDNA molecule lattice Hamiltonian as [8,18]: where n runs over the number N of bps, P ρ n and P ϕ n are the conjugate momenta of the nth bp radial and twist variables, respectively, and ξ = c 2 + R 2 0 1.147 nm is related to the helical geometry of the system, so that ξθ n,n±1 measures the helix arc length providing the shortest path between two points along a helical coil. In the limit of small radial and twist oscillations (r n R 0 , θ n,n+1 1) Eq.
(1) reads d n,n±1 = R 2 0 + c 2 θ n,n±1 ≡ ξθ n,n±1 , so that the Euclidean distance coincides with the helix arc length in this case [19]. The potential term: describes the harmonic interaction between neighboring bases along each backbone's strand, and the term: with u ± n,m = (1 + λ n ) ρ n ± (1 + λ m ) ρ m , describes the stacking interaction between adjacent bps, 95 whose role is to inhibit configurations with large relative radial displacements between neighboring 96 pairs. This interaction gives rise to local constraints in nucleotide motions, and it is characterized 97 by the exponential factor that effectively modulates an otherwise harmonic radial oscillation. This  Table 1.

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The time scale of the bp twist Eq.(5) is completely determined by the characteristic frequency ω ϕ , whereas the radial Eq.(6) involves three different characteristic frequencies, namely, ω ϕ , ω ϕS,n,n±1 (related to twist-stacking coupling), and ω ϕSH (fully coupling twist, stacking, and stretching interactions). The set of coupled Eqs.(5)-(6) describes the dynamics of general dsDNA molecules, where two kinds of bps can be arranged either periodically or aperiodically [32-36], and their mathematical structure clearly indicates the correlated nature of next-neighboring bps dynamics. It is then convenient to zoom out our perspective and consider the dynamics of consecutive triplets of bps, which are closely related to the so-called codon units in genomics. To this end, we properly add up the dynamical equations of consecutive bps corresponding to sites n − 1, n, and n + 1, grouping the resulting expression in terms of the collective variables x n ≡ 2ϕ n − ϕ n−1 − ϕ n+1 and y n ≡ 2ρ n − ρ n−1 − ρ n+1 , to obtain:ẍ with: and:ÿ n + ω 2 ϕSH y n − 1 2 (ω 2 ϕS,n−1,n−2 y n−1 + ω 2 ϕS,n+1,n+2 with: where: where By comparing Eqs.(10) and (12), which describe the codon dynamics as a whole, with Eqs.(5) 118 and (6), respectively describing the motion of their constituent bps, we can appreciate they exhibit a 119 closely related algebraic structure, which is further highlighted when the auxiliary functions H x , H y , 120 and G y simultaneously vanish, so that Eqs.(10) and (12) become formally identical to Eqs.(5)-(6) upon 121 the variable exchange ϕ n ↔ x n and ρ n ↔ y n , respectively. In order to get H x = 0 the bps mass ratios λ k must take on correlated values of the form λ n−2 = λ n+2 and λ n±2 = λ n±1 . Such a relationship There exist four kinds of homopolymer dsDNA molecules, namely, polyG-polyC, polyA-polyT, 130 polyT-polyA, and polyC-polyG chains, all of them satisfying λ k ≡ λ, ∀k, so that H x = H y = G y = 0 131 and Eqs.(5), (10), (6) and (12) respectively adopt the form: and: where is about an order of magnitude smaller than ω 2 S (see 135   Table 2). We note that this characteristic frequency remains exactly the same upon the exchange  Table 2. Characteristic frequencies and their related scale times in the lattice dynamics of polyX-polyX', poly(XX')-poly(X'X) and poly(XYX)-poly(X'Y'X') dsDNA molecules with λ = λ GC and λ * = λ AT . The values for the alternative choice λ = λ AT and λ * = λ GC slightly differ by just a few GHz. The frequencies ω ϕH , Γ ± , Γ YXY , and Γ XYX are related to the orchestrated codon oscillations described in Section 5.

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Completely analogous expressions are obtained if we choose the reading frame so that the Another instance of a dsDNA system exhibiting a suitable local mirror symmetry is provided by periodic DNA chains whose unit cell includes codons of the form XYX, so that the local bps arrangement around Y type bases reads 5'...XXYXX...3'. If we label the bps in such a way that the Y base occupies the n reference site, then we have the relationships λ n±2 = λ n±1 ≡ λ, and λ n ≡ λ * , ∀n, so that we get H x = 0. Therefore, the bps and codon dynamical equations corresponding to the twist variables just coincide with Eqs.(17)-(18) previously obtained for the homopolymer case. On the other hand, we have: where Ω 2 ± ≡ λ ±1 Ξ ϕS + ω 2 S . Thus, the radial motion equations read: .20, and: terms, hence they do not coincide with each other upon the transformation ρ n ↔ y n .

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Making use of the model parameters listed in Table 1 we obtain the values listed in Table 2

Helical waves related dispersion relations 189
Motivated by previous results [19,39], we look for solutions of the form ϕ n = ϕ 0 e i(ωt−nqξ) and ρ n = ρ 0 e i(ωt−nqξ) , describing a helical wave propagating throughout the dsDNA with frequency ω and wave vector q, where ϕ 0 8º= 0.14 rad, and ρ 0 0.05 nm, are the twist and radial oscillation amplitudes at ambient temperature, respectively [40]. In so doing, Eqs. (17) where G(q) ≡ 4ω 2 ϕ sin 2 (qξ/2) and H(q) ≡ ω 2 ϕSH − Ω 2 ϕS cos(qξ). The solution to Eq.(28) requires the matrix determinant to identically vanish, thereby leading to a biquadratic equation whose solutions yield the dispersion relations for the acoustic and optical phonon branches given by: Since Λb 2 B ω 2 ϕ is much smaller than both Λω 2 S and Λω 2 H terms, the exact dispersion relations given by Eq.(29) can be very well approximated by the simpler expressions: in agreement with previously reported results [15,21].

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A close look at the bottom inset reveals that the wider the q = 0 bandgap the narrower the maximum 210 bandgap at q * for the considered dsDNA chains.

Orchestrated codon oscillations
In addition to the synchronized helical waves, for which the intertwined angular and radial variables share the same ω value, we can also disclose long-range correlated motions of the angular and radial variables in time, orchestrated in such a way that they simultaneoulsy satisfy the conditions ϕ n−1 (t) = ϕ n+1 (t) and ρ n−1 (t) = ρ n+1 (t) ∀n. In that case, the collective variables read x n = 2(ϕ n − ϕ n±1 ) = ∓2θ n,n±1 and y n = 2(ρ n − ρ n±1 ) = ∓2p n,n±1 , where the collective motion variable p n,n±1 plays for radial oscillations a role completely analogous to that played by the variable θ n,n±1 for twist ones. In terms of the revamped collective variables the codon dynamical Eqs.(10) and (12) can be respectively rewritten in the form: and: The condition ρ n−1 (t) = ρ n+1 (t) guarantees that H x = 0 for polyXX'-polyX'X dsDNA molecules (see Section 3.2). In addition, polyX-polyX' and poly XYX-poly X'Y'X' polymers satisfy the relationships λ n−1 = λ n+1 and λ n−2 = λ n+2 in a natural way. Accordingly, codon Eqs.(34)-(35) can be properly simplified to read:θ n,n±1 + 4ω 2 ϕ θ n,n±1 = 0,p n,n±1 + Γ 2 u p n,n±1 ± where Γ 2 u ≡ ω 2 ϕSH − ω 2 ϕS,n±1,n±2 (u labels the DNA unit cell type), so that its precise value depends on the nature of the codons present in the considered dsDNA molecule. Thus, for polyX-polyX' polymers Γ 2 ϕH . This frequency accounts for the collective radial oscillations of XXX (alternatively, X'X'X') codons, describing a long-range oscillation where stacking interactions become ineffective, giving rise to a coupled twist-stretching mode which cannot be observed at the bp scale. Since homopolymers satisfy the H y = 0 condition, the resulting dynamical equations of motion (36) become effectively decoupled in terms of simple harmonic motions with two different natural frequencies, namely, 2ω ϕ for twist, and ω ϕH for radial motions, respectively. Solving the twist harmonic expression θ n,n±1 (t) = θ 0 cos(2ω ϕ t + δ 0 ) for the time variable, and plugging it into the radial harmonic expression p n,n±1 (t) = p 0 cos(ω ϕH t + δ 0 ), where the phases δ 0 and δ 0 are determined from the initial conditions, we get the relationship: describing the spatial pattern of the long-range correlated motion in terms of the collective variables 213 θ n,n±1 and p n,n±1 .

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Making use of Eq.(13)-(16) into Eq.(36) we obtain the following expressions for the radial codon equations corresponding to polyXX'-polyXX', and polyXYX-polyX'Y'X' chains, respectively: where Γ 2 ± = ω 2 ϕSH − λ ±1 ω 2 ϕS , hence Γ 2 + − Γ 2 − = (λ −1 − λ)ω 2 ϕS , and: where: Completely analogous expressions are obtained for poly(YXY)-poly(Y'X'Y') polymers by simply permuting λ * ↔ λ in Eqs.(38)-(40). The obtained dynamical equations indicate that for the resonance conditions Γ + = Γ − , for polyXX'-polyXX' chains, and λ Ω 2 + = −2λΩ 2 − for polyXYX-polyX'Y'X' chains, the twist and radial equations decouple from each other leading to two harmonic motions in a way completely analogous to that previously described for the homopolymers case. In the polyXX'-polyX'X case, the relationship given by Eq.(37) holds by simply replacing ω ϕH → Γ ± . It is important to note that this resonance can only take place for dsDNA molecules satisfying the mismatch condition λ = 1, so that it may be regarded as signaling the presence of local alterations in the due bp sequencing. Alternatively, within the framework of epigenetic processes, one may consider the attachment of a small molecule with the required mass value (i. e., ∆m = m n − m n ) to the lighter pyrimidine nucleotide in order to get the required resonance condition λ = 1. In the polyXYX-polyX'Y'X' case the resonance condition leads to the expression: which depends on the mass ratio values λ and λ * , along with the twist and stacking characteristic 215 frequencies which, in turn, can be expressed in terms of model parameters.

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The general solution to the dynamical equations for polyXX'-polyX'X polymers can be obtained from the knowledge of the general solution to the homogeneous version of Eq.(38) given by π n = A 0 cos(Γ ± t + δ 0 ), plus the particular solution ansatz ρ n = −p n,n±1 , leading to the diffential equation p n,n±1 + Γ 2 ∓ p n,n±1 = 0, whose solution readsπ n = B 0 cos(Γ ∓ t + δ 0 ). For the sake of simplicity we adopt the initial conditionsρ n (0) =ρ n±1 (0) = 0, and ρ n (0) = −p n,n±1 (0), so that we get: describing a modulated oscillation where γ − = 39 GHz and γ + = 1.698 THz. Since ρ n,n±1 = 217 ρ n ± p n,n±1 = 2ρ n we see that all the bps in the codon move in phase. A similar solution describing a 218 modulated oscillation in opposite phase is obtained by choosing ρ n = p n,n±1 as the particular solution 219 to the inhomogeneous differential Eq.(38). In this case, we get a larger high frequency value γ + = 1.774 220 THz. By following a completely analogous procedure Eq.(39) for polyXYX-polyX'Y'X' polymers can 221 be solved in a similar way.  of frequencies ω ϕ , ω H , and ω S , along with their related coupled frequencies. In addition, the reduction 246 of the twist frequency value makes the acoustic dispersion relation slope to decline, so that the sound 247 speed is reduced as well, ultimately leading to a lower thermal conductivity around the place where 248 the small molecule has been bonded. Furthermore, a smaller ω ϕ value will wide the gap between the 249 acoustic and optical branches (see Figure 2).