Hydrogen-Spin Networks in a Quantum Model of DNA

We have previously explained how DNA functions through a combination of biology, physics, and informatics. At the genetic level, each cell must solve two types of algorithmic problems: one involving information, which relates to coordinated energy processes, and another involving energy, which relates to informational processes. Resonance systems can serve as a common foundation for these dual-purpose algorithms by linking oscillatory energy to communication informatics. The concept of resonance genetics for hereditary information integrates biology, physics, and informatics, as resonance systems facilitate the reception and transmission of information (Petoukhov, 2016).

Moreover, biology and engineering share many intriguing design principles. The well-known phenomenon of vibro-transfer of energy among parts of an oscillatory system occurs when a rotary electromotor can operate stably even when disconnected from the power supply if it is placed on a vibro-platform with another rotary electromotor of similar resonant characteristics that is connected to a power source. This occurs through self-synchronization via resonant interactions (Peng et al., 2025; Zou et al., 2020). Through self-synchronization, only those motors that resonate with the base motor will work together as a single functional colony. The resonant integration of parts into a unified system relies on the fundamental physical principle of minimum energy (Hovgaard, 1923), each component of the system requires less energy to perform its work than it would when operating alone. Because the base stack is hydrophobic on the inside and its core is isolated from water, proteins, and other molecules in the surrounding environment, these molecules are excluded, making DNA an insulated wire and a linear (double-helical) crystal (Berlin et al., 2000). This should enable electron and proton charges to oscillate within the base stack without disturbing other atoms and, consequently, without much energy dissipation (Riera Aroche, Ortiz García, Martínez Arellano, et al., 2024).

Recently, synchronization phenomena have been studied for applications in communication systems and complex networks (Argyris et al., 2005). DNA sequences produce biologically active electromagnetic fields. Electromagnetic fields in both the extremely low-frequency and radio-frequency (RF) ranges have been shown to promote protein synthesis. This indicates that even a weak electromagnetic field can cause DNA to unwind and trigger protein synthesis (Cantu et al., 2023; Ivancsits et al., 2005; Kim et al., 2024; Liu et al., 2024; Sun et al., 2023). Because DNA is stabilized by hydrogen bonds and electromagnetic fields primarily interact with electrons, these fields likely impact electrons within the hydrogen bonds, possibly weakening them. Additionally, electrons in water’s flickering hydrogen bonds may be affected at certain frequencies. This process might create numerous temporary protons and electrons in and near the DNA, which can be energized by electromagnetic fields. As we explained in one of our recent works (Riera Aroche et al., 2025).

Drawing a parallel to these arguments, DNA has been characterized as an ideal quantum computer (Hood & Galas, 2003). Its scientific relevance stems from two main features: its digital structure and its complementarity, where each helix strand pairs precisely with its partner. The nitrogenous bases C, G, A, and T serve as carriers of resonant trait pairs, with resonant frequencies and oscillatory energies vital for encoding information (Petoukhov, 2016; Petoukhov & Petukhova, 2017), we discussed the quantum architecture of DNA, viewing each nitrogenous base as a superconductor owing to its aromatic nature, and each base pair as a two-level system (qubit) connected by a hydrogen bond functioning as a Josephson junction. Drawing an analogy to quantum cryptography, we examined a straightforward teleportation protocol that uses the four Bell states and classical information(Riera Aroche, Ortiz García, Martínez Arellano, et al., 2024). While we mathematically represented the states for each pair, our model did not specify which parts of the biological structure would actively participate, nor did we simulate a quantum operator.

DNA qubits resonate through shared oscillatory properties and wave-like interactions, much like musical notes. When one instrument produces a sound, it can cause another instrument’s string to vibrate if the two are tuned to the same pitch. Likewise, an analog radio’s tuning dial adjusts reception frequency by varying the capacitance in a resonant circuit, allowing it to select a specific station. We suggest that certain DNA sequences with comparable oscillatory patterns can also resonate, enabling synchronization and signaling. The oscillations of delocalized $\pi$-electron resonance clouds in the base stack depend on similar delocalized proton clouds of hydrogen bonds—these are the oscillations of electron and proton clouds. Noise-induced synchronization is predicted in many-body systems, leading to entangled collective oscillations. We also highlight how the transcriptional machinery’s resilience relates to the quantum synchronization phenomenon’s resistance to detuning the DNA qubits’ natural frequencies.

Considering DNA as a system in which each qubit is capacitively coupled to a resonator (the sugar-phosphate backbone), in the dispersive regime—where the DNA base pair qubit is detuned from the resonator’s frequency—the qubit’s state shifts the resonator’s frequency. This allows the RNA polymerase to determine the qubit’s state (Riera Aroche, Ortiz García, Martínez Arellano, et al., 2024; Riera Aroche, Ortiz García, Sánchez Moreno, et al., 2024). In other words, the qubit links its states to the resonator’s resonant frequency, enabling a non-destructive readout of the qubit’s state. Here, however, we complete the analysis of the quantum process of genetic information transmission, incorporating the role of the proton within the hydrogen bond as a biological quantum logic operator in DNA.

In the genetic code, the DNA alphabet is related to the 20 amino acids of the protein alphabet through a triplet code represented by codons, which are triplet of nitrogenous bases connected via phosphodiester bonds along the sugar-phosphate backbone. This model assumes each nitrogenous base is fixed to its corresponding transmission line, the sugar-phosphate backbone (Riera Aroche, Ortiz García, Sánchez Moreno, et al., 2024), as shown in Figure 1A. The hydrogen atom in a hydrogen bond oscillates harmonically under an elastic molecular force, modeled by Hooke’s law ($F=-kx$), treating the bond as a spring. Hydrogen bonds can facilitate strong electronic interactions within or between molecules by increasing resonance, delocalizing electrons, or flattening the conjugated backbone. A higher bond-strength constant, indicating a more stable bond, is associated with higher vibrational frequencies. Protons involved in hydrogen bonds (like O-H...O or N-H...O) are “unshielded,’ resulting in signals at higher chemical shift values (downfield) in the proton nuclear magnetic resonance spectrum. According to Hooke’s Law for simple harmonic motion:

Where $v$ represents the vibration frequency, $k$ the force constant related to the bond strength, and $\mu$ the reduced mass obtained from the bonded atoms.

If we identify each base pair using the proton spin coupling correlation, then codon ABC is represented by $H_A$, $H_B$, and $H_C$. Figure 1A. But base pairing in DNA involves two hydrogen bonds (A-T) or three (C-G) per pair. Therefore, $H_A$, $H_B$, and $H_C$, are the result of the hydrogen coupling of at least two hydrogen bonds.

Proton splitting occurs when a proton signal separates into multiple peaks $(n+1)$ due to magnetic effects from nearby non-equivalent protons, typically three bonds away. The resulting pattern follows an $n+1$ rule, where $n$ indicates the number of neighboring protons. The relative intensities of the peaks are governed by Pascal’s triangle. Spin-spin splitting results from considering all possible individual spins of the protons. This splitting is mutual; if one proton causes another to split, the reverse must also be true (Autschbach & Le Guennic, 2007; Edison et al., 1994).

Let the codon ABC be any DNA codon, and let $H_A$, $H_B$, and $H_C$ represent the coupling of protons in the hydrogen bonds of the nitrogenous base pairs that compose that codon. The coupling among the three — $H_A$, $H_B$, and $H_C$ — in the codon ABC results in a quadruple signal with eight possible spin permutations (Figure 1B). The central signals are three times larger than the outer ones because they correspond to three equivalent spin arrangements that absorb at the same frequency. Since the signals of $H_A$, $H_B$, or $H_C$ are affected by the number of protons ($n=2$ or $n=3$), we need to recalculate the relative areas based on Pascal’s triangle. For $n=2$, the relative areas will produce three signals, with the highest peak in the center representing two equivalent spin permutations $\left(\uparrow \uparrow , \downarrow \uparrow ,\uparrow \downarrow ,\downarrow \downarrow \right)$. Due to the two nuclear energy interactions, four new energy levels are created. Each nucleus can be aligned either with or against the applied field. One lower-energy level occurs when both nuclei are aligned with the field, and a higher-energy level when both are aligned against it. Between these two, additional energy levels correspond to configurations in which one nucleus is aligned with the field and the other is aligned against it.

The energy of a given spin state is:

Where $m$ represents the positive value of the most stable state, and ℏ is Planck’s constant. Proton spin transitions generate radio-wave emissions that can be detected electronically. In systems with dipolar interactions among neighboring nuclear spins, decoupling influences signal intensity. These intensity changes depend on the distance $r$between the interacting spins. In this model of DNA function, RNA polymerase employs its CTD heptapeptide sequence: Y1S2P3T4S5P6S7 to separate signals and interpret each base pair within a codon. $H_0$ is the created RNA polymerase magnetic field, and $\gamma$ is the magnetogyric ratio (a proportionality constant describing the spin energies of the nucleus in an external magnetic field).

A nuclear spin functions as a tiny magnet, producing a local magnetic field. Two nearby hydrogen nuclei can interact directly through space (not through bonds), a phenomenon known as dipolar coupling or the Nuclear Overhauser Effect (NOE), involving the transfer of spin polarization. This led to changes in their signal intensities. When two nuclei interact at a distance $r$ and with an angle $\theta$ between the vector $\mathbb{r}$ and the magnetic field direction, the local magnetic field $H_{loc}$ caused by this dipolar interaction is expressed as (Ando et al., 2000):

The spacing between peaks offers valuable insights into molecular structure. This spacing, measured in Hertz, is known as the coupling constant between magnetically coupled protons. It is denoted as $\mathcal{J}$.

The overall position of a signal and the magnetic field strength needed for energy absorption are related to electron density and circulation within complexes, as shown in Figure 2. When subjected to a magnetic field, electrons follow particular paths and produce small magnetic fields, known as induced fields ($H_i$). These induced fields oppose the applied magnetic field, making the effective field ($H_{ef}$) less than the applied field itself:

$H_{ef}$ determines the resonance behavior of the nucleus, and $\sigma$ is the screening or protection constant. In nitrogenous bases, applying a magnetic field causes their $\pi$ electrons to move in loops, creating a secondary magnetic field with lines of force that close on themselves. The frequencies at which the response amplitude reaches a relative maximum are also known as the system’s resonant frequencies. Small periodic forces near a system’s resonant frequency can cause large amplitude oscillations by storing vibrational energy. Resonant systems can be used to generate vibrations at a specific frequency or to select specific frequencies from a complex vibration containing many frequencies.

A proton is a positively charged particle that has spin. Any rotating electric ($e$) charge produces a small magnetic moment aligned with its axis of rotation. For a nucleus to be magnetic, it must have a spin angular momentum. The nuclear magnetic moment $\mu$ is related to the angular momentum $\hslash I$ through the equation:

Where $I$ is the nuclear spin vector, $g_n$ is the nuclear Landé factor, and $m_H$ is the proton mass. The spin of a magnetic nucleus is quantized and can only have specific, well-defined values. When a magnetic field $H_0$ is applied, a nucleus with spin $I={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ can only align either with ($I=+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.})$ or against the field ($I=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.})$. According to Boltzmann’s Law, nuclei with $I=+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ predominate. The energy of the spin states can be determined by:

Where $\mu$ is the nuclear magnetic moment in the direction of the external field, $H_0$. The energy difference between the two spin states is: $\mu$ is given by ${\displaystyle \raisebox{1ex}{$\gamma hI$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.}$, where $\gamma$ is the magnetogyric ratio. It is defined in terms of the ratio of the magnetic moment to the angular momentum as ${\displaystyle \raisebox{1ex}{$2\pi \mu $}\!\left/ \!\raisebox{-1ex}{$Ih$}\right.}$. Please, see Figure 3

The expression for the interaction energy among $H_A, H_B$, and $H_C$, considering a set of quantized orientations with quantized energy levels for the nuclear spin is: Where:

$\mu$: the magnetic moment of $H_A, H_B$, and $H_C$

The 3 × 3 matrices, $K$, $D$, and C, show how the $x, y, z$ components of one nuclear spin magnetic moment interact with those of the other nuclear spin magnetic moment.

The hydrogen nuclei are energetically excited by a combination of the $H_0$ and RF-radiation. When the energy supplied to $A, B$, and $C$ matches the energy difference between the spin states, resonance occurs, allowing all the hydrogen nuclei to absorb energy as they transition between them. In the codon ABC, three wave vectors are chosen so that one serves as the main mode and the other two function as sidebands, creating a triplet under a particular resonance condition. RNA polymerase acts as a multi-frequency array harvester. An electromagnetic harvester with an array of the highly conserved heptapeptide sequence: Y1S2P3T4S5P6S7 that generates steady power across a uniform spectrum of natural frequencies. Tuning the harvester’s resonant frequency to match the dominant vibration frequency improves the vibration energy harvester’s efficiency (Dowlati et al., 2024).

The transition follows the Bohr condition, with energy transfer occurring when $∆E$ precisely equals $h\upsilon$, where $\upsilon$ represents the radiation frequency in the RF range. Consequently, the relation for a nuclear transition is given by:

Where $\omega$ is the angular frequency.

The coupling of protons at $H_A, H_B,H_C$ on the codon is not the only way nitrogenous bases are coupled and synchronized to a frequency. The codon ABC can be viewed as a system of three coupled oscillators. We previously modeled DNA as a combination of $n$-circuits with the base pair represented by nine Radio-Frequency Superconducting Quantum Interference Devices (SQUIDs) inside (Riera Aroche, Ortiz García, Sánchez Moreno, et al., 2024). We investigated the theoretical possibility of using networks of base-pair-coupled oscillators as data storage systems. Let’s consider the oscillation frequency of a codon ABC as:

Where ${\omega }_0$ is the frequency of one base pair and $a_n$ is the amplitude of the three oscillators (a codon). Then, if oscillators are synchronized by the first and the third harmonic, they should be described by the system of non-linear differential equations:

Where, ${\dot{X}}_A={\omega }_1∆v$ for the first oscillator,${ \dot{X}}_B={\omega }_2∆v$ for the second, and ${\dot{X}}_C={\omega }_3∆v$ for the third. ${\delta }_0$ and ${\delta }_2$ are defined by approximating $∆i\left(∆v\right)$ by a third-order polynomial. $∆v=v_1-v_0$. $∆i=i_1-i_2$ and represents the differential current. ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ are the frequency of the first, second, and the third oscillator, respectively; and ${\alpha }_1$, ${\alpha }_2$ the coupling coefficient and shows the influence of the systems A and B, while ${\alpha }_3$ is the coupling coefficient and shows the influence of the system C on the second one.

The solution to Equation 11 lies in $X_i=A_i\cos \left(\omega t\right)$; $i=A,B,C$. Where $A_i$ is the amplitude, $X_i$ represents the elongations in function of time, and $\omega$ is the angular frequency. We can derive once: ${\dot{X}}_i=-\omega A_i\sin \left(\omega t\right)$, and then: ${\ddot{X}}_i=-{\omega }^2{A}_i\cos \left(\omega t\right)$. Then we substitute into the system of three equations, transforming it into an algebraic system with a coefficient matrix, enabling us to calculate each oscillator’s frequency.

Kuramoto’s schematic phase model was used to demonstrate how small residual forces can induce collective synchronization in the motion of many coupled oscillators (Acebrón et al., 2005; Gupta et al., 2014). Here, ABC are considered three oscillators, each with its own natural frequency ${\omega }_i$ (Figure 1A). With $K$ representing the coupling force among A, B, and C, and assuming the phases rotate at a frequency $\Omega$, we can analyze the system’s evolution:

With $i=A,B,C$

The behavior of the phases can be visualized as three points moving around the unit circle in the complex plane of the phase space diagram. The complex order parameter $r$of this group can be described as the magnitude and phase of its centroid. If the three base-pair oscillators move together, we will have $r \approx 1$, and they will act as a single large oscillator entity.

By using this parameter and setting the imaginary parts equal in equation 12, Kuramoto derived the following expression:

Therefore, even though A, B, and C possess their own autonomy, they all communicate via the magnitudes $r$and $\psi$. This interaction tends to pull the phase ${\theta }_i$ towards the average phase $\psi$. Additionally, the effective coupling force, $Kr$, depends on the coherence $r$. For the coupling matrix $Kij$, we consider the ABC nearest-neighbor coupling model (Acebrón et al., 2005).

A, B, and C base pairs exhibit, in addition to quantum beating oscillations at individual frequencies ${\omega }_i$, two collective coherent quantum oscillations characterized by two frequencies, ${\omega }_1=\overline{∆}/\hslash$, and ${\omega }_2={\tilde{\omega }}_R$. $\overline{∆}$ represents the average energy level difference across a codon. The frequency of individual coherent quantum-mechanical oscillations that can be excited in a single base pair is ${\omega }_1$, while ${\tilde{\omega }}_R$ is the sugar-phosphate backbone frequency renormalized by interaction. The collective shift of base pair frequencies is resonantly enhanced when the backbone’s frequency, ${\tilde{\omega }}_R$, matches the frequency’s average value (the amplitude of these oscillations is strongly enhanced in the resonant case as ${\omega }_1\simeq {\omega }_2$). Please see the schematic representation in Figure 1A.

The description of DNA as a circuit, in which each strand of the sugar-phosphate backbone constituted a resonator, was discussed in detail in the reference (Riera Aroche, Ortiz García, Sánchez Moreno, et al., 2024). The sugar-phosphate backbone is characterized by two parameters, $L_0$ and $C_0$, which are the inductance and capacitance per unit length, as described in (Riera Aroche, Ortiz García, Sánchez Moreno, et al., 2024). Please see Figure 4. The frequencies are expressed as ${\omega }_R=c{k}_n$, where $c=1∕\sqrt{L_0{C}_0}$, $k_n=\pi n∕\mathcal{l}$, and $\mathcal{l}$ is the size (n = 1,2 . . .). In the reference (Asadi et al., 2018), the equations of motion for a mechanical resonator with two modes of response are detailed.

A particular nucleotide quantum gate is defined by the electron spin direction. Quantum logical operations in DNA are performed through each nucleotide’s ability to act as a quantum logic gate. These individual nucleotide quantum gates are maintained in coherence by the hydrogen proton in the hydrogen bond. A quantum logic system must include a pathway for coherence, allowing qubits of superimposed information to be transmitted and maintained in superposition. It also requires a mechanism for quantum decision-making to disrupt this coherence, and a quantum gate that enables these qubits to interact coherently. These elements of a quantum logic system are properties of DNA, thereby supporting the DNA molecule’s quantum logic function (Mihelic, 2013).

High-power RF pulses are applied to a sample to rotate the equilibrium longitudinal magnetization, and the free-induction decay of the transverse component is then detected as a time-dependent signal. In Figure 5A, applying an RF pulse at the same frequency as the Larmor frequency ${\omega }_0$ for the static field $H_0$ produces an RF magnetic field $H_i$ along the x-axis in the rotating frame of ${\omega }_0$. Then, M is rotated around the x-axis into the y-z plane. The rate at which M changes over time in the rotating flame of ${\omega }_0$ is given by: and the rotation angle is provided by:

With $t_p$ refers to the duration of the RF pulse application.

In DNA, the Hamiltonian of a coupled hydrogen-nucleus system in the codon ABC subjected to a magnetic field of intensity $H_0$, which represents the total energy of all spin systems, is defined as:

Where:

$I\left(A\right)$: is the component of the nuclei’ spin I in the $H_0$ direction

${\gamma }_A$: is the magnetrogyric ratio of A nuclei

${\sigma }_A$: is a screening coefficient of A nuclei

$J_{ABC}$: is a coupling constant between nuclei A, B, and C

When all nuclei have a spin to $I={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, the only options for the quantum number are $+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ or $-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. Therefore, the system is either in the state $\alpha$ or $\beta$. The spin functions for the three nuclei of a codon with spins in the individual states $\alpha$ or $\beta$ can be written as:

For $I={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, there are $2^n$ product functions. Thus, the matrix elements could be calculated using the basic product functions.

In the absence of couplings ($J_{AB}=J_{AC}=J_{BC}=0$), the Hamiltonian can be separated into three terms, each depending on different spin coordinates (A, B, and C), and its eigenfunctions can be expressed as products of the eigenfunctions of each individual term. Since there are two specific functions for each nucleus $\alpha$ and $\beta$, the sum of $\left(A\right) + \left(B\right) + \left(C\right)$, will have$2^3 = 8$ eigenfunctions which will constitute an orthonormal basis of the spin function space of the system:

$\alpha \alpha \alpha \equiv {\mathsf{\Phi }}_1$	$\alpha \alpha \beta \equiv {\mathsf{\Phi }}_2$	$\alpha \beta \alpha \equiv {\mathsf{\Phi }}_3$	$\beta \alpha \alpha \equiv {\mathsf{\Phi }}_4$
$\alpha \beta \beta \equiv {\mathsf{\Phi }}_5$	$\beta \alpha \beta \equiv {\mathsf{\Phi }}_6$	$\beta \beta \alpha \equiv {\mathsf{\Phi }}_7$	$\beta \beta \beta \equiv {\mathsf{\Phi }}_8$

This basis allows us to express the eigenvalue equation of the Hamiltonian with couplings:

Using the Bloch representation, we can describe all quantum gates as rotations of the Bloch vector of a DNA qubit’s state (Figure 5 B). To illustrate this, we will consider the Pauli operators and their action on a qubit’s state vector. From these, we will define rotations about an axis on the Bloch sphere. Every qubit is represented as a point on the Bloch sphere, where the poles represent the eigenstates. The colatitude angle, θ, ranges from 0 to π, while the azimuthal angle, ϕ, spans from 0 to 2π. The effect of a quantum gate on a single qubit is a rotation on the Bloch sphere, resulting in a new qubit state. On the Bloch sphere, we visualize the action of different logic gates or the temporal evolution of a two-level system state under a given Hamiltonian. An axis and a rotation angle define a rotation operator that rotates a point on the sphere around the axis by the angle.

In DNA as a quantum computer, we have two systems: the first consists of two spins ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ A and T, or two spins ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ C and G, prepared in a known pure state. Additionally, we have the system U, which consists of a spin ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ in an unknown pure state.

The initial state of a base pair (system $|\psi ⟩X_i$) is (Riera Aroche, Ortiz García, Martínez Arellano, et al., 2024):

Where the state of the system $U$ is represented in the form $|U⟩=\alpha |\uparrow ⟩+\beta |\downarrow ⟩$, and is equivalent to $|H⟩=\alpha |\uparrow ⟩+\beta |\downarrow ⟩$.

$\alpha$ and $\beta$ satisfy $\alpha *\alpha +\beta *\beta =1$.

$|H⟩$ is the proton spin state

Then, when $H_0$ with a magnitude ${\displaystyle \frac{\upsilon }{\hslash }}$ is applied in the direction of the $x\left(z\right)$ axis on the spin $X_i\left(H\right)$, the operator $E=exp\left(-i{t}_1{H}_{X_i\left(H\right)}∕\hslash \right)=i^{-1/2}{E}_{CNot}$ multiplies the state by an irrelevant global phase $i^{-1/2}$ and reverses the orientation of the spin at $X_i$ if the spin of $|\uparrow ⟩H$, and leave it in its original orientation if $|\downarrow ⟩H$.

With $H_{X_i\left(H\right)}$ the Hamiltonian $H_{X_i\left(H\right)}=H_0\left(S_x^{X_i}-S_z^H+{\displaystyle \frac{\upsilon }{\hslash }}{S}_x^{X_i}{S}_z^H\right)$.

RNA phosphorylation of amino acids in the CTD, according to the bibliographies (Riera Aroche et al., 2025; Riera Aroche, Ortiz García, Sánchez Moreno, et al., 2024), produces RF pulses at a frequency of ${\omega }_i$, creating $H_0$.

According to the stochastic local operations and classical communication (SLOCC) classification (Li, 2018), any three-qubit pure state $|\psi ⟩$ may be fully separable, biseparable, or genuinely entangled. By definition, the state is fully separable or pure product if it can be written as a tensor product of three single-qubit pure states: $|\psi ⟩=|{\varnothing }_1⟩⨂|{\varphi }_2⟩⨂|{\varphi }_3⟩$. A state that is not entirely separable but becomes separable under a specific bipartition of the three qubits is called a biseparable state. Biseparable states are classified into three types according to the partition. For instance, a state that is biseparable with respect to the first qubit and the other two qubits has the form $|\psi ⟩=|{\varnothing }_1⟩⨂|{\varphi }_{23}⟩$. Here, $|{\varphi }_{23}⟩$ is an entangled state of the second and third qubits. If a state is neither entirely separable nor biseparable, it is referred to as genuinely entangled. Under the SLOCC classification, there are only two types of these genuinely entangled states: GHZ and W states: $|GHZ⟩={\displaystyle \frac{1}{\sqrt{2}}}\left(|000⟩+|111⟩\right)$ and $|W⟩={\displaystyle \frac{1}{\sqrt{3}}}\left(|001⟩+|010⟩+|100⟩\right)$.

For methionine, the codon ABC is T-A, A-T, and C-G. Then, if RNA polymerase reads TAC:

If you introduce the resonant frequency of each core when there are no couplings (the resonance frequency of a shielded core): $\upsilon ={\displaystyle \frac{H_0\left(1-\sigma \right)\gamma }{2\pi }}$, then:

If $J_{TA}=J_{Tc}=J_{AC}=0$, there is no coupling, and this Hamiltonian would be separable into three terms dependent on different spin coordinates. The orthonormal basis of the spin function space of the system allows expressing the eigenvalue equation of the Hamiltonian with couplings:

Each column in a matrix (eigenvector) contains the coefficients that specify an eigenfunction.

The spin Hamiltonian for each nucleus in TAC during RNA polymerase exerting an $H_0$ in the z direction axis:

Using $\alpha \equiv {\varnothing }_{{\displaystyle \frac{1}{2}}}$ and $\beta \equiv {\varnothing }_{-{\displaystyle \frac{1}{2}}}$:

$\alpha \alpha \alpha \equiv {\mathsf{\Phi }}_{TAC1}$	$\alpha \alpha \beta \equiv {\mathsf{\Phi }}_{TAC2}$	$\alpha \beta \alpha \equiv {\mathsf{\Phi }}_{TAC3}$	$\beta \alpha \alpha \equiv {\mathsf{\Phi }}_{TAC4}$
$\alpha \beta \beta \equiv {\mathsf{\Phi }}_{TAC5}$	$\beta \alpha \beta \equiv {\mathsf{\Phi }}_{TAC6}$	$\beta \beta \alpha \equiv {\mathsf{\Phi }}_{TAC7}$	$\beta \beta \beta \equiv {\mathsf{\Phi }}_{TAC8}$

Then, the operator associated with the z component of the total spin angular momentum has the value $F_z=I_{zT}+I_{zA}+I_{zC}$:

The selection rules are described using the quantum number: $M_F\equiv {\displaystyle \raisebox{1ex}{$F_z$}\!\left/ \!\raisebox{-1ex}{$\hslash $}\right.}$. Thus, the following $M_F$ values ​​are obtained for each basis function: ${\displaystyle \frac{3}{2}}$ for ${\mathsf{\Phi }}_1$, ${\displaystyle \frac{1}{2}}$ for ${\mathsf{\Phi }}_{2, \mathrm{3,4}}$, $-{\displaystyle \frac{1}{2}}$ for ${\mathsf{\Phi }}_{\mathrm{5,6},7}$, and $-{\displaystyle \frac{3}{2}}$ for ${\mathsf{\Phi }}_8$.

Now, by analyzing the structure of DNA and its hydrogen bonds, we can clearly understand that the functions ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_3$ are not possible. And if we assume that RNA polymerase reads only one strand, then the most likely spin combination of the total coupling hydrogen bond will correspond to the direction of the magnetic field. Therefore, using the teleport protocol of section 3, for TAC (Equations 19,20, and 22):