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

## Abstract

**:**

## 1. Introduction

## 2. Dynamical DNA Model Hamiltonian

## 3. Dynamical Equations of Motion

#### 3.1. General Expressions

#### 3.2. Dynamics of Homopolymer dsDNA Chains

#### 3.3. Dynamics of Poly(XX’)-Poly(X’X) dsDNA Molecules

#### 3.4. Dynamics of Poly(XYX)-Poly(X’Y’X’) dsDNA Molecules

## 4. Helical Waves Related Dispersion Relations

## 5. Orchestrated Codon Oscillations

## 6. Environmental Effects in the DNA Dynamics

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

dsDNA | double-stranded DNA |

bp | base pair |

A | adenine |

C | cytosine |

G | guanine |

T | thymine |

PDB | Peyrard–Dauxois–Bishop |

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**Figure 1.**On the

**left**, sketch of the dsDNA 3D lattice model showing the harmonic bonds between adjacent base pairs through the sugar-phosphate backbone and between complementary bases due to H-bonds. The right strand runs in the 5’ → 3’ direction, whereas the left strand runs in the antiparallel 3’ → 5’ direction. On the

**right**are details of one of the bp planes, showing the effect the mass difference has on radial displacements, shifting the center of mass outside the axis of the helix. Reprinted from Physics Letters A, 376, Torrellas, G.; Maciá, E., Twist–radial normal mode analysis in double-stranded DNA chains, 3407–3410, Copyright (2012), with permission from Elsevier.

**Figure 2.**(Main frame) Optical phonon branches for polyG-polyC (black), poly(GC)-poly(CG) (magenta), poly(AT)-poly(TA) (siena), poly(GAG)-poly(CTC) (green), and poly(AGA)-poly(TCT) (light blue) dsDNA polymers, where q is measured in nm${}^{-1}$, and we have used the values of the the model parameters and characteristic frequencies listed in Table 1 and Table 2, respectively. (Low inset) Close up view of the optical branch curves shown in the main frame in the interval $1.5\le q\le 4$. (Top inset) Acoustic ${\nu}_{-}\left(q\right)$ and optical ${\nu}_{+}\left(q\right)$ phonon branches for polyG-polyC dsDNA homopolymers. (Reprinted from Maciá, E. Base-Pairs’ Correlated Oscillation Effects on the Charge Transfer in Double-Helix B-DNA Molecules. Materials 2020, 13, 5119).

**Table 1.**Geometrical, dynamical, and potential model parameters adopted in the dsDNA model studied in this work. The same effective Morse potential is used to describe H-bonding in both GC and AT bps. The spring constant ${k}_{B}$ is difficult to estimate, and different possible values, ranging from $0.04$ to $0.5$ eV Å${}^{-2}$, have been reported in the literature [24,25,26,27,28,29].

Geometrical | Dynamical | Potential |
---|---|---|

${\theta}_{0}=\pi /5.2$ rad | $M=653.5$ amu | $b=0.5$ Å${}^{-1}$ [15,30] |

${h}_{0}=0.34$ nm | ${\lambda}_{GC}=1.130$ | $\alpha =5$ Å${}^{-1}$ [14,30,31] |

${l}_{0}=0.68$ nm | ${\lambda}_{AT}=1.028$ | $D=0.15$ eV [21,30,31] |

${R}_{0}=1.00$ nm | ${\mu}_{GC}=163.06$ amu | ${k}_{S}=0.7$ eV Å${}^{-2}$ [21] |

$\xi =1.15$ nm | ${\mu}_{AT}=163.15$ amu | ${k}_{B}=0.04$ eV Å${}^{-2}$ [30] |

**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 $\lambda ={\lambda}_{GC}$ and ${\lambda}^{*}={\lambda}_{AT}$. The values for the alternative choice $\lambda ={\lambda}_{AT}$ and ${\lambda}^{*}={\lambda}_{GC}$ slightly differ by just a few GHz. The frequencies ${\omega}_{\phi H}$, ${\Gamma}_{\pm}$, ${\Gamma}_{YXY}$, and ${\Gamma}_{XYX}$ are related to the orchestrated codon oscillations described in Section 5.

Oscillation | Codon | ${\mathit{\omega}}_{\mathit{k}}$ | (${10}^{12}$ rad s${}^{-1}$) | ${\mathit{\nu}}_{\mathit{k}}$ (THz) | ${\mathit{\tau}}_{\mathit{k}}$ (ps) |
---|---|---|---|---|---|

Twist | ${\omega}_{\phi}$ | $1.067$ | $0.170$ | $5.874$ | |

Stacking | ${\omega}_{S}$ | $3.260$ | $0.519$ | $1.927$ | |

Stretching | ${\omega}_{H}$ | $5.335$ | $0.849$ | $1.178$ | |

Twist-Stacking | X’XX’ | ${}^{-}{\Omega}_{\phi S}$ | $6.122$ | $0.974$ | $1.026$ |

YXY | ${\overline{\Omega}}_{\phi S}$ | $6.362$ | $1.013$ | $0.988$ | |

XXX | ${\Omega}_{\phi S}$ | $6.507$ | $1.036$ | $0.966$ | |

XYX | ${\tilde{\Omega}}_{\phi S}$ | $6.670$ | $1.062$ | $0.942$ | |

XX’X | ${}^{+}{\Omega}_{\phi S}$ | $6.918$ | $1.101$ | $0.908$ | |

Twist-Stretching | XXX | ${\omega}_{\phi H}$ | $10.685$ | $1.701$ | $0.588$ |

Twist-Stacking-Stretching | XX’X | ${\Gamma}_{+}$ | $10.425$ | $1.659$ | $0.603$ |

XYX | ${\Gamma}_{XYX}$ | $10.498$ | $1.671$ | $0.599$ | |

YXY | ${\Gamma}_{YXY}$ | $10.810$ | $1.728$ | $0.579$ | |

X’XX’ | ${\Gamma}_{-}$ | $10.911$ | $1.737$ | $0.576$ | |

${\omega}_{\phi SH}$ | $12.511$ | $1.991$ | $0.502$ |

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**MDPI and ACS Style**

Maciá, E.
Synchronized Oscillations in Double-Helix B-DNA Molecules with Mirror-Symmetric Codons. *Symmetry* **2021**, *13*, 241.
https://doi.org/10.3390/sym13020241

**AMA Style**

Maciá E.
Synchronized Oscillations in Double-Helix B-DNA Molecules with Mirror-Symmetric Codons. *Symmetry*. 2021; 13(2):241.
https://doi.org/10.3390/sym13020241

**Chicago/Turabian Style**

Maciá, Enrique.
2021. "Synchronized Oscillations in Double-Helix B-DNA Molecules with Mirror-Symmetric Codons" *Symmetry* 13, no. 2: 241.
https://doi.org/10.3390/sym13020241