To determine the resistance of the genetic material to external influences, further studies were carried out on the effects of an alternating magnetic field (MF) on the appearance of oxidative DNA damage in conditionally healthy donors.

The degree of oxidative DNA damage was assessed by changes in the concentration of 8-oxoG in the blood serum of donors. The concentrations of 8-oxoG in DNA from the blood serum of healthy donors in the first and second treatment groups are shown in Fig. 2, for both before and after the in vitro treatment of blood samples with a magnetic field, at frequencies of 3, 30, and 50 Hz. These frequencies were selected based on previous studies [21, 41]. The initial level in the control samples, without exposure to the field, was 7.4 $\pm$ 1.9 ng/mL in the first group and 6.3 $\pm$ 1.5 ng/mL in the second group. After treatment of MF samples with frequencies (f) of 3, 30, and 50 Hz, a significant increase was observed in the levels of 8-oxoG in blood serum DNA in both treatment groups compared to the untreated blood samples. Thus, after exposure to an MF at a frequency of 3 Hz, the concentration of 8-oxoG increased in the first and second groups, reaching 14.8 $\pm$ 2.2 ng/mL and 24.9 $\pm$ 2.8 ng/mL, respectively. Exposure to an MF with a frequency of 30 Hz also resulted in an increase in the concentration of 8-oxoG in the first group, up to 14.1 $\pm$ 2.1 ng/mL, and in the second group, up to 22.0 $\pm$ 2.6 ng/mL. Finally, the level of the studied indicator in the blood samples of donors also turned out to be comparable in both groups after exposure to a magnetic field with a frequency of 50 Hz (Fig. 2). Therefore, the obtained results for the donors in the second group indicate the maximum modification susceptibility to oxidative DNA modifications under conditions associated with a normally functioning antioxidant protection.

As noted in a number of studies [6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 20, 23], magnetic fields affect animals and cells by promoting the generation of reactive oxygen species (ROS), potentially leading to oxidative stress at the cellular and systemic levels. An increased level of ROS generation, mediated by physical, chemical, or biological environmental factors, is the most genotoxic process affecting DNA [20, 22, 23].

To study the mechanisms involved in the process of magnetic fields causing the formation of free radicals in the blood, we developed a mathematical model. However, to simplify calculations, the model was used to study an aqueous solution of DNA isolated from the blood of apparently healthy donors.

In dilute solutions of biopolymers, the processes of ROS formation proceed as a chemical oscillator [42]. To describe the chemical oscillator in the conversion of hydrogen peroxide, the longest-lived ROS in an aqueous solution, we used chemical equations with known simplifications.

The initial stage is the initiation of the process, i.e., the attachment of an electron from e-c Rydberg excited levels of macromolecules (for example, DNA) [43], to free protons of water, to the formation of a hydrogen radical:

(1)$${\mathrm{H}}^{+}+\mathrm{e}^{-}{}_{\mathrm{Rg}}\to {\mathrm{H}}^{\mathrm{•}}$$

at neutral pH, the hydrogen radical interacts with dissolved oxygen molecules, the concentration of which is 3 orders of magnitude higher than the concentration of protons and hydroxide ions (10${}^{17}$ versus 10${}^{14}$ molecules/cm${}^3$) [44], to form a hydroperoxide radical:

(2)$${\mathrm{H}}^{\mathrm{•}}+{\mathrm{O}}_2\mathit{\hspace{1em}}\to \mathrm{HO}_{2-}{}^{\mathrm{•}}$$

further formation of peroxide occurs in two ways:

(3)$$\mathrm{HO}_{2-}{}^{\mathrm{•}}+{\mathrm{H}}^{\mathrm{•}}\to {\mathrm{H}}_2{\mathrm{O}}_2$$

or

(4)$$2\mathrm{H}\mathrm{O}_{2-}{}^{\mathrm{•}}\to {\mathrm{H}}_2{\mathrm{O}}_2\mathit{\hspace{1em}}+{}^1\mathrm{O}_2+hv$$

as the concentration of hydrogen peroxide increases, according to Eqns. 3–4, the pH of the solution increases. When the aqueous environment is at a neutral and alkaline pH, hydroperoxide radicals decompose to form superoxide ions:

(5)$$\mathrm{HO}_{2-}{}^{\mathrm{•}}\to {\mathrm{H}}^{+}+\mathrm{O}_2{}^{-}{}_{-}{}^{\mathrm{•}}\mathit{\hspace{1em}}({\mathrm{pK}}_{\mathrm{a}}\sim 4,8)$$

the released proton can again interact with the Rydberg e-${}_{\text{Rg}}$ of the macromolecule. Then, with an increase in hydrogen peroxide, the superoxide ions contribute to its decomposition through the formation of hydroxyl radicals [45]:

(6)$${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{O}_2{}^{-}{}_{-}{}^{\mathrm{•}}\to {\mathrm{OH}}^{-}+{\mathrm{OH}}^{\mathrm{•}}+{\mathrm{O}}_2$$

the presence of a singlet oxygen in a neutral and weakly alkaline medium leads to the further formation of superoxide ions:

(7)$${\mathrm{OH}}^{-}+{}^1\mathrm{O}_2\to {\mathrm{OH}}^{\mathrm{•}}+\mathrm{O}_2{}^{-}{}_{-}{}^{\mathrm{•}}$$

dismutation of the hydroxyl radical leads to the formation of hydrogen peroxide:

(8)$$2\mathrm{O}{\mathrm{H}}^{\mathrm{•}}\to {\mathrm{H}}_2{\mathrm{O}}_2$$

where the hydroxyl radical can interact with the resulting hydrogen peroxide:

(9)$${}^{\mathrm{•}}\mathrm{OH}+{\mathrm{H}}_2{\mathrm{O}}_2={\mathrm{H}}_2\mathrm{O}+{\mathrm{H}}^{+}+{}^{-}\mathrm{O}_2{}^{\mathrm{•}}$$

As the pH increases, peroxide is able to spontaneously decompose with the acidification of the medium:

(10)$${\mathrm{H}}_2{\mathrm{O}}_2\to {\mathrm{H}}^{+}+\mathrm{HO}_{2-}{}^{-}\mathit{\hspace{1em}}({\mathrm{pK}}_{\mathrm{a}}\sim 11,5)$$

or with the formation of singlet oxygen:

(11)$${\mathrm{H}}_2{\mathrm{O}}_2\to 2{\mathrm{H}}_2\mathrm{O}+{}^1\mathrm{O}_2$$

self-decomposition of peroxide, according to for the rate constants of Eqn. 10, has an order of magnitude from 1 to 2 and leads to the formation of singlet oxygen ${}^1$O${}_2$, which again participates in the cycle of ROS interconversions.

By introducing the notation k${}_i$ for the rate constants of reactions (1)–(11), we can obtain the corresponding kinetic equations for the rates of formation of various ROS biopolymers in dilute aqueous solutions. Using the condition of constancy for the intermediate products in the system (1)–(11) and simplifying the kinetic model of production and loss of hydrogen peroxide, we can obtain an expression for the rate of hydrogen peroxide formation:

(12)$${\displaystyle \frac{d}{dt}}\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]={\displaystyle \frac{k1}{2}}\left[\mathrm{OH}_{-}{}^{-}\right]\left[\mathrm{e}^{-}{}_{\mathrm{Rg}}\right]+$$ $${\displaystyle \frac{k4}{2}}{\left({\displaystyle \frac{k2}{k3}}\right)}^2{\left[{\mathrm{O}}_2\right]}^2-k_{10}\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]$$

from Eqn. 12, it follows that in dilute solutions of biopolymers, the rate of accumulation of hydrogen peroxide depends on the pH of the solution, the concentration of electrons $\mathrm{e}^{-}{}_{\mathrm{Rg}}$ at the Rydberg excited levels of macromolecules, and the concentration of dissolved oxygen, which coincides with the conclusions presented in [46]. Periodic accumulation and decomposition of hydrogen peroxide determine the frequency of the entire cycle of interconversions in the chemical oscillator of ROS.

Let us consider the formation of a chemical oscillator involved in ROS interconversions in aqueous solutions of biopolymers. Fig. 3 shows a simplified schematic of H${}_2$O${}_2$ transformations, in the course of chemical reactions (1)–(11), in aqueous solutions of biopolymers (the main chemical oscillator). Taking into account the fact that several more oscillators of hydrogen peroxide can exist in the solution in addition to the main oscillator, the reactions involved in the formation of hydrogen peroxide, which proceed according to Eqns. 1–11, will be divided into three groups. Each of which has feedback: the first group is the initiation of the cycle and the initial formation of H${}_2$O${}_2$, which is described by Eqns. 1–5, in Fig. 3, marked I; the second group are reactions involving dissolved and singlet oxygen and are described by Eqns. 2,4, and 6–7, in Fig. 3, marked II; the third group are reactions involving the hydroxide radical, which are described by Eqns. 6–9, and are designated by III in Fig. 3. A separate block, depicted by IV, highlights the processes involved in hydrogen peroxide decomposition, illustrated by reactions (10)–(11), which proceed efficiently when a sufficient amount of H${}_2$O${}_2$ is accumulated in the solution: k${}_0$ represents the degree of hydrogen peroxide decomposition. Within each of the selected groups, hydrogen peroxide transformations occur within their own periods of oscillation. In solutions of biopolymers, several parallel chemical oscillators of H${}_2$O${}_2$ interconversions can exist simultaneously, each of which can have its own frequency of fluctuations in the concentration of hydrogen peroxide. Thus, we have a system of coupled oscillators with positive feedback.

Let us introduce the value $\tau$—the time involved in increasing the concentration of (H${}_2$O${}_2$) in each cycle of reactions by e times. The value of $\tau$ considers both the time of accumulation of hydrogen peroxide and the lifetime and diffusion of the corresponding ions and radicals in each group of reactions. Then, the rate of change in the concentration of hydrogen peroxide in the chemical oscillator can be written as below, while the schematical depiction is also shown in Fig. 3:

(13)$$\begin{array}{c}\hfill \frac{d{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{II}}{dt}+\frac{1}{{\tau }_1}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{II}=\frac{1}{{\tau }_1}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^I\hfill \end{array}$$

(14)$$\frac{d}{dt}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{III}+\frac{1}{{\tau }_2}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{III}=\frac{1}{{\tau }_2}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{II}$$

(15)$$\begin{array}{c}\hfill \frac{d}{dt}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{IV}+\frac{1}{{\tau }_3}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{IV}=\frac{1}{{\tau }_3}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{III}\hfill \end{array}$$

(16)$${\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^I=-k_0{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}^{IV}$$

taking into account the occurrence of hydrogen peroxide decomposition reactions and a decrease in its concentration, the index k${}_0$ in Eqn. 16 has the “–” sign. Transforming the system for Eqns. 13–16 and discarding the term containing the third derivative, due to its smallness, we can obtain the final equation for changing the concentration of hydrogen peroxide in an aqueous solution of a biopolymer:

(17)$${\displaystyle \frac{d^2}{d{t}^2}}\left[H_2{O}_2\right]+{\displaystyle \frac{{\tau }_1+{\tau }_2+{\tau }_3}{{\tau }_1{\tau }_2+{\tau }_1{\tau }_3+{\tau }_2{\tau }_3}}{\displaystyle \frac{d}{dt}}\left[H_2{O}_2\right]+$$ $${\displaystyle \frac{k_0+1}{{\tau }_1{\tau }_2+{\tau }_1{\tau }_3+{\tau }_2{\tau }_3}}\left[H_2{O}_2\right]=0$$

Eqn. 17 is a differential equation for free oscillations that arise in a harmonic oscillator [47]. The expression in front of [H${}_2$O${}_2$] is the square of the frequency of natural oscillations in the chemical oscillator:

$${\omega }_0^2=\frac{k_0+1}{{\tau }_1{\tau }_2+{\tau }_1{\tau }_3+{\tau }_2{\tau }_3},$$

and the expression before $\frac{d}{dt}\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]$ is the damping index of the fluctuations in hydrogen peroxide concentrations in a chemical oscillator.

$$\delta =\frac{{\tau }_1+{\tau }_2+{\tau }_3}{{\tau }_1{\tau }_2+{\tau }_1{\tau }_3+{\tau }_2{\tau }_3}.$$

Then, Eqn. 17 will take the form:

(18)$$\frac{d^2}{d{t}^2}\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]+\delta \frac{d}{dt}\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]+{\omega }_0^2\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]=0$$

Let us consider the influence of a low-intensity and low-frequency magnetic field, which sine changes and is described by the equation H = H${}_m$sin$\omega$t, on the yield of hydrogen peroxide in a dilute aqueous solution of biopolymers.

According to Eqn. 12, in biopolymer solutions, the rate of hydrogen peroxide accumulation depends on, among other things, the concentration of electrons at the Rydberg excited levels of macromolecules. A low-intensity magnetic field can affect the magnetic moments of these electrons. In addition, in aqueous solutions of biopolymers, there are radical pairs with different g-factors of partners (for example, ${\mathrm{H}}^{\mathrm{•}}$, $\mathrm{HO}_{2-}{}^{\mathrm{•}}$, and ${\mathrm{OH}}^{\mathrm{•}}$). The magnetic field apparently orients the spins of biradical pairs (for example, ${\mathrm{H}}^{\mathrm{•}}\text{и}{\mathrm{O}}_2$), changing their mobility, which brings us back to the theory of the interaction of triplet–singlet pairs [33].

All of the above suggests that under the influence of a magnetic field, the increment in the concentration of hydrogen peroxide changes in proportion to its strength:

$$\mathrm{\Delta }\left[H_2{O}_2\right]=\alpha H_m\sin \omega t$$

where $\alpha$ is the coefficient of proportionality.

As a result, the oscillations arising in the chemical oscillator will become forced, causing the following equation to become valid for them:

(19)$$\frac{d^2}{d{t}^2}\left[H_2{O}_2\right]+\delta \frac{d}{dt}\left[H_2{O}_2\right]+{\omega }_0^2\left[H_2{O}_2\right]=\alpha H_m\sin \omega t$$

the solution of such an inhomogeneous equation is equal to the sum of the general solution of a homogeneous equation that characterizes the decaying process of hydrogen peroxide formation, which occurs after the application of an external influence, as well as a particular solution that describes a steady oscillatory process under the action of an applied driving force.

If the losses in the chemical oscillator are small, and the frequency of the driving force differs slightly from the natural frequency of the oscillations in the hydrogen peroxide concentration in the chemical oscillator, then, we can obtain the solution for Eqn. 19 in the following form:

(20)$$\left[H_2{O}_2\right]={\left[H_2{O}_2\right]}_{\mathrm{M0}}{\mathrm{e}}^{-\delta \mathrm{t}}\sin \left({\omega }_{0}t+\phi \right)+$$ $${\left[H_2{O}_2\right]}_{\mathrm{M}}\sin (\omega t+\phi )$$

additionally, the value of the amplitude of the forced fluctuations in the concentration of hydrogen peroxide under the action of an alternating MF is described by the expression:

(21)$${\left[H_2{O}_2\right]}_{\mathrm{M}}=\alpha H_m{\left({\left({\omega }_0^2-{\omega }^2\right)}^2+4{\delta }^2{\omega }^2\right)}^{-1/2}$$

thus, it follows from Eqn. 20 that natural oscillations in the hydrogen peroxide concentration decay with time in the chemical oscillator, and only forced oscillations in the H${}_2$O${}_2$ concentration are present; the amplitude of which depends on the frequency of the external magnetic field. The process has a quasi-resonance character, whereby resonance occurs when the frequency of the external field coincides with the natural frequency of the chemical oscillator of fluctuations in the concentration of hydrogen peroxide in solution. Thus, it is possible to, first, convert a weak magnetic field signal into a chemical one, and then, in a complex system, into a biochemical response.

Theoretical dependences for the content of hydrogen peroxide in an aqueous solution of DNA relying on the frequency of an alternating magnetic field have been constructed. The calculation considered the strength of the magnetic field –450 A/m and the attenuation index of the fluctuations in the hydrogen peroxide concentrations in the chemical oscillator was 0.7–2.5 (Fig. 4).

To verify the model, the theoretically calculated and experimentally obtained values for the hydrogen peroxide concentrations were compared after treatment of aqueous DNA solutions with a low-intensity MF at the corresponding frequencies. The results are shown in Fig. 5. The H${}_2$O${}_2$ content in the aquatic environment without MF treatment was less than 7.0 µM/dm${}^3$. In the control DNA solution, at a concentration of 2.5 µg/mL, the content of H${}_2$O${}_2$ reached 94.5 $\pm$ 8.3 µM/dm${}^3$. The DNA control solution was kept in the measuring module for 30 min without MF treatment. As can be seen from Fig. 5, a significant increase (3–5 times) in the content of H${}_2$O${}_2$ in the DNA solutions can be observed after their treatment with variable MF frequencies: f = 3, 4, 8, and 50 Hz.