Methods

Article selection.

The subject of the search, which is called here “HMF effect,” is expressed in literature in many different forms, which means that scientific terminology in this area is still evolving. A targeted search for relevant publications on the HMF biological effect was, for this reason, a challenge.

First, we used a scientific information aggregator The Global Science Gateway, or WorldWideScience [worldwidescience.org], to select papers that contained “hypomagnetic” in their titles. The aggregator has an access to 106 databases all over the world. This gave, after a subsequent semantic control, 14 articles that included original results suitable for further investigation. At this stage and later, we took into account only peer-reviewed articles and did not restrict the date range of documents.

PubMed database of the National Center for Biotechnology Information [pubmed.gov] provides an advanced search capability for word combinations. We have used “zero magnetic,” “null magnetic,” “magnetic deprivation,” “magnetic field compensation,” and “magnetic shielding.” This gave about 40 more articles.

Separately, we have studied the contents of a specialized scientific journal, Bioelectromagnetics, and found a few articles that could not be selected based on keywords, because the HMF effect was not their main subject.

Relevant papers in Russian have been found in the open bibliographic database of Russian scientific periodicals eLIBRARY [elibrary.ru]. As a rule, articles in Russian include an abstract in English. The same keywords as above have been used therefore, which resulted in about 20 items in the domain of biology. We also reviewed the contents of “Biofizika,” a specialized journal, for the past five years, and separated about 10 more articles.

Further search for the relevant articles both in English and in other languages continued through examination of the lists of references in each of the articles that had been identified. Many articles that contain results on the HMF effect are problematic for the targeted search since their titles, as for example “Reduction of the background magnetic field …” and “… geomagnetic field screening,” are related to the subject of search only by their meaning. For this reason, examining the lists of references of the already selected papers was most productive. This type of search has been repeated with every new relevant article iteratively until no new articles could be detected. This gave us more than 160 additional references that referred both to the original experimental data and to reviews, collections, and books.

On completion of the search, we had the list of about 250 bibliographic sources, of which about 80 were not available to us primarily because of the lack of a resource in the local library and its absence in the Internet. Nearly 170 full-text items have been carefully studied. However about 30 of them have been found unsuitable for further analysis, because they did pass one or more of the exclusion criteria listed below. Finally, 137 articles are used for further numerical analysis.

In selecting these articles, we thus applied one inclusion criterion: The text contains information about a biological effects of weak MF that is considerably less than the GMF. Next, four qualitative exclusion criteria were used: (i) The text is an abstract of conference or a non-peer-reviewed article. (ii) The text is a review with no original data. (iii) The text of article does not contain quantitative information on the MF magnitude of the exposure. (iv) The numerical results on the HMF effect in the article have been published by these authors in their other article that has already been considered relevant.

We did not assess the quality of experimental work and this was not an exclusion criterion. Instead, we addressed any deficiency in the completeness of the description of experiments in Table 1.

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Table 1. Data on the effects of hypomagnetic field.

https://doi.org/10.1371/journal.pone.0179340.t001

Results

The results (sorted by year from oldest to newest) are summarized in Table 1. For each reference, the Table shows the biological test system, taxonomic group, measured characteristic, the relative magnitude of the HMF effect, the type and size of the exposure system, HMF values inside and MF outside the system, the duration of exposure, and bibliographic source. Table 1 references 137 publications, which reflect an unbiased reporting of research results in this area.

The reported effects in Table 1 are almost all statistically significant at p < 0.05 or less as noted in each reference. In a few cases, the effects were not statistically significant, which is indicated with a double asterix superscript.

As said above, only the maximal magnetic effect was taken into account, if a few effects have been reported in the same article. It is also possible that studies in which the HMF effect was not observed may not have been published as it may be more difficult to publish null effects. Given these confounders, the ability to analyze this Table is limited. Nevertheless, a correlation analysis of the data was reasonable, and we were able to derive some generalizations.

Fig 1 shows two correlation diagrams, where each point represents a study in which the relative effect E has been observed after exposure to HMF of a given value for a given time. The studies are divided into two groups, depending on the manner in which HMF has been obtained—by compensation (Co, red points) of the geomagnetic field, and by its shielding (Sh, blue triangles).

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Fig 1. Correlation diagrams.

Diagrams that show no correlation between the magnitude of a magnetic effect and the HMF value—A, or the duration of the HMF exposure—B, in groups Co (red) and Sh (blue). Pearson’s coefficients are in all cases less than 0.1 in absolute value, being negative, −0.07, for diagram A.

https://doi.org/10.1371/journal.pone.0179340.g001

As can be seen, the relative magnitude of the effects is in the range of 1–1,000%, mainly 10–200%. There is no statistically significant correlation of the effect size with the HMF value or the exposure time. The effect size also does not correlate with their product, or “dose,” and with the size of exposure system that usually tells one about MF heterogeneity. This suggests that HMF and the duration of exposure can influence on the measured value only on par with many other hidden physiological factors, the effect of which should be considered mostly random in the set of various organisms.

Distributions of the HMF effect magnitudes in groups Co and Sh are shown in Fig 2. They are close to log-normal one, and their means are not differ significantly. However, their variances are distinct at p = 0.03. Perhaps, this is the consequence of a closer similarity between shielding devices than compensating Helmholtz systems.

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Fig 2. Distributions of the magnitudes of HMF effects.

Distributions in groups of HMF obtained by compensating and shielding.

https://doi.org/10.1371/journal.pone.0179340.g002

Other physical factors—temperature, pressure, optical and ionizing radiation, many chemicals—may have approximately the same range of effect magnitudes as that for MF effects. However, such effects are all more specific—they appear about the same way in different organisms. The absence of specificity in HMF biological effects is clearly seen in Fig 1, and this provides additional information about the nature of these effects. We suggest that the fact of no specificity is in a good agreement with our conjecture that there is no biophysical MF target common for all organisms. Only a physical target can be a general one.

Each work reviewed is unique and, therefore, not replicated by an independent scientific group. Rare exclusion is HMF nociception studies reported in [71, 81, 92, 97] and [112, 114, 132, 152]. Non-reproducibility is a common problem in laboratory magnetobiology. At the same time, this is also a fact that deserves attention. It agrees well with the statement that many physiological factors with their occasional contributions are involved in the formation of the HMF effect. This makes the replication an accidental coincidence rather than a regularity.

As over 95% of the studies referenced in Table 1 report significant effects of HMF, this biological effect is very well established but the mechanism remains unknown. Against the background of the diversity of observed magnetic effects, the primary MF target has not yet been identified. However, some generalizations can still be attempted. According to the data of the Table, closest to the primary physical level are studies that report on the involvement of gene expression in magnetoreception, e.g. [80, 108, 120, 123] and [139, 144, 156, 162, 166, 173], with different genes expressing in different situations. This suggests that the MF targets are rather physical and distributed, but they operate as magnetic receptors only in some specific conditions. Interestingly, it has been shown in [174] that adding/removing small segments containing a specific sequence into two different promoters switched the ability of a weak EMF to induce their gene expression.

A wide range of exposure times required for the appearance of the magnetic effect—from minutes to months—is another fact, indicating that there is no a single type of specialized biomolecules responsible for magnetoreception. Different biomolecules with very different characteristic speeds of functioning may become MF targets—the mediators of a magnetic signal transduction from the objects of the physical level to downstream systems and reactions.

As follows from the “Questions” column, there arise many methodological questions that are explained in a complete version of the Table available as supporting information “S1 Table”. Most important methodological problems are addressed in a special subsection “Methodological comments” in Discussion.

The conclusion drawn on the basis of Fig 1, that there cannot be a biophysical magnetoreceptor common to all organisms, is not completely obvious, and is a supposition rather than a logical inference. Since this supposition seems, however, to be significant both scientifically and methodologically, we present below details of this reasoning.

For this analysis lets simplify the equation for E. In Eq (2) lets neglect the contribution associated with the standard errors s that was shown do not exceed 0.1v in most cases. Recall that, with the exception of a few cases, the values of E_i, i = 1, 2, … 137, represent statistically significant magnetic effects, and standard errors s_i refer to the corresponding mean values v_i. If we express not in percentages and we omit the index i, then E ≈ |v − v₀|/v₀. Now suppose that a biophysical magnetoreceptor common to all organisms exists, and its response to MF at the level of MF-induced biophysical events is given by H-dependence in the form f(H), where f(H ≳ H_g) ≈ 0. Since the observed magnetic effect v − v₀ is linked to the primary biophysical magnetic response by a chain of low-predictable events at the biochemical and the following levels in different species, the linear statistical model of the observed magnetic effect is the following equation (3) where a is a random variable that maps the scaling of biophysical events to the observed biological level, and is distributed, due to a variety of concomitant factors, normally with zero mean. Then E ∼ (a/v₀)f(H), and the covariance of E and H can be written, in a rough approximation, as follows where ρ(H) is the idealized density distribution of the magnetic fields used in the experiments, and the line above the symbols denotes the mean value. For simplicity, suppose that the HMF effect is described by f(H, H_th) = 0 for H > H_th and f(H, H_th) = 1 for H ≤ H_th, where H_th is a threshold level of MF, cf. Figs of sections “Universal physical mechanism” and “Intraprotein rotations.” Then the covariance takes a simple form Hence it is clear that both for H_th → 0 and for H_th → ∞, covariance tends to zero, i.e., there is no statistical relationship between E and H. However, in the interval 0 < H_th < ∞, the covariance takes negative values, which could be observed in the experiment.

To show how this works in the case of quantities close to the experimentally observed, we performed a numerical simulation of the results presented in Table 1.

First, it was established that the arrays of MF H values used in the experiments and of the measured control values v₀ have approximately a lognormal distribution logN(μ, σ) with parameters μ = 5, σ = 2 and μ = 0.2, σ = 0.7, respectively. Using these distributions, sets of corresponding quantities were generated. To simulate the array of the effect values, Eq (2) has been applied, in which v − v₀ was replaced by af(H, H_th) according to the Model (3). The standard deviation of the normal distribution for the amplitude a of magnetic effects, σ = 1, was selected so that the array of the effects generated for the case f(H, H_th) = 1 formed an average value and a deviation the same as in Fig 1A. The result of 1000 simulations is shown in Fig 3A. Since here we have put f ≡ 1, the effects in this case are considered only for selecting a correct amplitude distribution of a and cannot be used for conclusions about the character of the functional dependence f(H, H_th).

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Fig 3. A numerical simulation of HMF effects.

Effects that are expected in 1000 different biological species under the following model assumptions: A—unconditional HMF effects at f ≡ 1, B—HMF effects at f = f(H, H_th) with a fixed H_th = 100 nT, C—HMF effects at f = f(H, H_th) with a widely distributed H_th.

https://doi.org/10.1371/journal.pone.0179340.g003

Recall that our assumption was that there is one and the same biophysical MF target for all organisms. This means that there is one and the same function f(H, H_th) with the same parameter H_th for all organisms. We have chosen H_th = 100 nT—the middle of the logarithmic interval of the fields in the experiments, used v − v₀ = af(H, H_th) and again performed a thousand simulations, Fig 3B. As is seen, there is a negative covariance, which corresponds to the above analytical reasoning.

An alternative hypothesis is that the MF targets in organisms are different and form largely a random set. This means, in terms of the above analysis, a significant difference in the threshold field H_th in different organisms. A wide distribution of H_th means the appearance of a significant subset, for which the covariance tends to zero. In other words, the expected picture is a significant decrease in the statistical relationship between E and H. For simulated values of H_th, we used the absolute values of the realizations of a normal, log-widely distributed random variable with N(0, 10⁴). In this case, the correlation diagram E(H) had the form shown in Fig 3C. Apparently, indeed, there is a significant decrease in the statistical relationship of the magnetic effects with the magnitude of the MFs used.

Comparing Fig 3B and 3C with Fig 1A, one can conclude that the alternative hypothesis of a wide distribution of the threshold fields is more in line with the empirically observed pattern. What is the physical meaning of the wide distribution of threshold fields? As shown in this article, the equality, in the order of magnitude, γHτ ∼ 1 controls the appearance of the HMF effects. Consequently, the threshold value is the combination of the gyromagnetic ratio and the thermal relaxation time: H_th ∼ 1/γτ. A wide distribution of H_th means that at least one of the two values, γ or τ, does not have a fixed value in different organisms. In other words, either the magnetic moments themselves, i.e., primary MF physical targets, are different, or very different are local conditions that affect their relaxation. In both cases, this means that the biophysical targets are different, and therefore there is no biophysical MF target common for all organisms.

This analysis, of course, does not prove the assumption of nonexistence of a general biophysical target. However, this assumption does not contradict the experimental data of Fig 1A and, as one can see, has a rational justification.

We note that the simulated E(H) pattern with negative covariance is quite stable to the variation of model parameters; the traces of negative covariance are observed even in the case of logarithmically wide distribution of threshold fields H_th, Fig 3C. This indicates the existence of a large number of targets, not only different, but also not obeying the statistical Model (3). Consideration of such targets will be carried out in a separate study.

From the viewpoint of theoretical physics, nothing but the very existence of the magnetic vacuum effect can be drawn from the current experiments, until sufficiently detailed MF-dependences of the effect are measured. Information on the nature of MF targets can be extracted from the MF-dependences only on the basis of comparison of experimental data and a theory. Experimental methods for the identification of the physical MF targets in the body require the parallel development of the theory of magnetoreception in terms of probability, physical properties of the primary MF target, and MF parameters.

Below, we will present several mechanisms of magnetoreception that are known from literature and often discussed with respect to specific and nonspecific magnetic biological effects. We will study their applicability in the limit of small dc MFs and show that not all of these are equally suitable for the explanation of the nonspecific biological effects of HMF.

Among those mechanisms of magnetoreception, there is one that is valid for both the ac MFs and HMFs. The mechanism does not depend on the characteristics of intermediate biophysical/biochemical stage of magnetoreception, and therefore is universally applicable to any magnetic targets of molecular nature. This fact makes it reasonable to attempt experimental verification of this mechanism.

Radical pair mechanism

Some chemical reactions are known to change their rate in the MF due to formation of the intermediary magnetically sensitive state—a spin-correlated pair of radicals, e.g. [183, 184]. This is one of the most actively discussed magnetoreception mechanisms that is called Radical Pair Mechanism (RPM). The total spin of the radicals or the relative orientation of the spins affects the probability of the product formation. For example, probability of recombination of the newly formed radicals in a homolytic reaction depends on the relative orientation of their spins. A constant MF changes the likelihood of a favorable orientation and, thereby, able to shift the chemical balance. This mechanism does not possess a frequency selectivity in the low-frequency range, since the evolution of the magnetosensitive spin state occurs within a very short lifetime of the pair usually of the order of 10⁻⁹–10⁻⁷ s.

Spin-chemical magnetoreception has certain difficulties. Known effects in spin chemistry appear in a relatively strong MF, and their possible involvement in magnetoreception is not yet established for sure. Maximum values of only about 0.1% are estimated for the effects of MFs like the geomagnetic field, even in the favorable case of large thermal relaxation time of the electron spin states [185]. Finally, there is no laboratory example where an MF less than the geomagnetic field would significantly change the rate of a biradical biochemical reaction: one cannot trace the result of the action of an MF of the order of units of μT on such a reaction in vitro.

It is very likely, however, that RPM actually works in the case of specific magnetoreception in migratory animals. It has long been conjectured that photochemical reactions that generate spin-correlated biradicals underlie a compass sense in migrant species [186]. This idea is in good agreement with the fact that the eyes of birds often mediate their magnetic reactions. It was found that formation of biradicals takes place in the cryptochrome molecules in the retina of the bird’s eye. Since there is an orderly arrangement of the immobile photo-receptors with cryptochromes in the retina, they form an oriented array of N elements that can provide an enhanced sensitivity. The sensitivity of the visual system as a whole to MF changes may increase in proportion to . The fact that magnetic susceptibility in some birds depends on the spectral range of the optical radiation and on its intensity supports this hypothesis. The fact that a hypothetical RPM-based magnetic sensor reacts equally to the oppositely directed MFs is also in accord with a significant part of observations in animal magnetic navigation.

With regard to non-specific magnetoreception, RPM is certainly possible, but unlikely due to (i) the lack of frequency selectivity, (ii) small magnitudes of the effect, and (iii) lack of the laboratory examples of its fitness. Experiments [187] regarding the MF influence on the growth and gene expression in Arabidopsis thaliana in MFs of 0.05–100 mT had to prove the involvement of cryptochromes in magnetoreception in plants; however, one has failed to confirm these results [188]. In [162], it is suggested that the RPM cannot while the quantum interference mechanism [189] can explain weak MF-dependences of gene expression in plants that are observed in this study. These failures of the RPM are probably due to irregular arrangement of the cryptochrome flavoproteins in plants, as opposed to the bird’s eye. Perhaps, the biradical mechanism is just one of the necessary elements of magnetoreception.

For example, a combined mechanism that considers biradical reactions in the presence of magnetic nanoparticles eliminates the difficulties of nonspecific biradical magnetoreception. Magnetic nanoparticles have a magnetic moment, and therefore they are sources of the endogenous MF. This field is nonuniform and rather strong, of the order of 1–100 mT near the nanoparticles [31]. In contrast to weak external MFs of the geomagnetic level, this strong MF can lead to a significant shift in the rate of biradical reactions by the suggested mechanism of S-T mixing in strongly inhomogeneous MF [31], while the nanoparticles themselves are rotated under the weak external MFs. This concept has been further developed in [32, 190, 191].

Can biradical mechanism explain the effect of a zero-MF? Apparently, no. Judging by the experimental MF-dependences of the biradical reactions rate, which are available in the original literature, there are no peculiarities in the MF close to zero, see references in [192]. In the fields that are less than 0.1 mT, the dependences are very smooth, which means that the zero-MF effect as defined above is absent. This is not by chance.

The natural state of a biradical is a superposition of the singlet S (↑↓) and triplet T (↑↑) states. Leaving aside many details, one can say that the quantum levels of these states are separated by an interval of the order of electron energy in MF H, i.e., Δε = μ_BH, where μ_B is the Bohr magneton. The magnetic effect is associated with transitions in these states. There are general restrictions on observation of S-T transitions and, therefore, on the magnetic effects in biradical reactions.

One of the results of quantum mechanics is the following statement. It is impossible to measure the change in energy Δε of a quantum system in a time less than Δt ∼ ℏ/Δε, because the act of measuring itself makes the measured value less certain in the value of the order of ℏ/Δt [193] p. 157–159. A biradical must keep coherence for at least Δt in order to allow measuring its energy changes. In the biradical reactions, the decay of the coherent state of the pair to form a whole molecule or a pair of free radicals should be considered an act of measuring. Accordingly, the duration of measurement is the biradical lifetime that is limited by its thermal relaxation time τ, among many other factors. In other words, for observation of electronic chemical magnetic effects in weak MFs, it is necessary that inequality τ > ħ/μ_BH be fulfilled, or τ > 100 ns for MFs of about the geomagnetic field. The existence of electronic states with so a large thermal relaxation time is questionable. There are other difficulties, which are mainly the insignificant magnitude of magnetic effects and the absence of a characteristic threshold MF value in the range of HMFs.

For many biradical reactions, characteristic is that the yield of free radicals increases with the MF growing from zero to the fields on the order of the hyperfine interaction, i.e., 1–100 mT. This is the so-called Low Field Effect (LFE) [192, 194]. Experimentally observed changes do not exceed a few percent. In [192], the canonical case has been studied of the S-T conversion in a pair of radicals, one of which had a magnetic core with spin 1/2 (a proton). The Zeeman energy of the electrons and the isotropic hyperfine interaction of one of the electrons to its nucleus were taken into account. Other interactions and spatial dynamics of the radicals were not considered. An approximate analytical solution for the LFE has been found: (4) where γ_e is the electron gyromagnetic ratio, τ_e is the lifetime of the coherent state of electron pair. Let one assume that τ_e approaches the necessary value of τ of about 100 ns, even though several other physicochemical factors also limit the lifetime. Even with this incredible condition, the critical MF that corresponds to a half-maximum effect, as follows from Eq (4), is H ∼ 2/γ_eτ_e ∼ 1 mT. It is very far from 1 μT required by the HMF effect.

A recent review [23], after many preceding original publications, supports that RPM underlies the compass sense, i.e., a spin-chemical cryptochrome reaction in a bird eye depends on the MF direction with respect to the reactant molecules. Due to the anisotropic caracter of the hyperfine interaction responsible for that magnetic effect, the effect of changing the MF direction and that of the MF switching on/off are of the same order of magnitude. This means that the above restrictions are valid for this case also. Indeed, Fig 7 of that review presents relative values of the magnetic effect that have been numerically calculated at different coherent spin evolution times that, of course, are always not greater than the thermal relaxation time. The values for changing MF of 50 μT have been 6.7% for τ = 5 μs and 0.6% for 0.5 μs. As can be estimated, for the ultimate value τ = 100 ns, the effect would be about 0.1% at 50 μT and 0.002% at 1 μT, which is at obvious variance with the HMF effect data.

To be fair, a few other theoretical models with sets of interactions which could be viewed as more realistic than that in [192] predict somewhat higher sensitivity. However, their results are also strongly dependent on time constants, the choice of which are not always indubitable. In the model [195] that describes a coupled chain of flavin–tryptophan biradicals, the relative reaction yield at the RP lifetime of the order of 100 ns has been numerically calculated to be about 5% in a MF of 100 μT, however with almost linear H-dependence in this range of small MFs. This, of course, excludes the possibility of applying this model to the HMF effects. Moreover, the model is hardly applied where its implicit assumption is not valid, that the electron thermal relaxation time is greater than the chemical time constants. Even greater magnetic effect of about 18% per 100 μT is calculated in [196] for a hypothetical cryptochrome RP of the flavin cofactor with the superoxide radical. However, again, the largest chemical time constant has been chosen to be as large as 1 μs, while the thermal decoherence has not been addressed. It is stated that such long lifetimes are characteristic for biological systems, but no direct experimental evidence in vitro has been adduced in the form of a MF-dependence.

Experimentally observed MF sensitivity, e.g., in cryptochromes in vitro, is about 1% per mT, e.g. [197]. In this latter study, a chemical amplification of the magnetic effect in cryptochrom biradicals have been found. This nearly tenfold amplification is suggested to support a cryptochrome-based magnetic compass sensor in animal navigation. However, as is seen from the above estimate, this is still far from enough to explain nonspecific HMF effect.

It follows that RPM is of little use to explain nonspecific biological effects of zero MFs, although its involvement in the specific magnetic sense of birds and other species is plausible. This conclusion could be challenged if a mechanism of biological amplification of smallest primary MF-induced changes, other than the eye cryptochrome immobility and ordering, will be revealed in the future.

Universal physical mechanism

In [36], it has been considered a nonuniform precession of magnetic moment under the action of the MF, whose direction is unchanged while the value changes. This MF does not cause quantum transitions; hence the classical model of the Larmor precession is sufficient. It turns out that this precession mode has interesting properties, similar to those of non-specific magnetoreception.

The precession of magnetic moments in MF precedes any biophysical or biochemical mechanism of magnetoreception, and largely determines the spectral and non-linear characteristics of the biological response. Objects in biological cells that possess a precessing magnetic moment are unpaired electrons, paramagnetic ions, protons and other magnetic nuclei. Protein-bound ions and the rotations of molecular groups with a distributed electric charge may also have a virtual magnetic moment.

The mechanism considered is based on the following points: the external MF acts on a magnetic moment; the magnetic moment precesses and undergoes thermal relaxation. A biological effect occurs if, during the relaxation time or faster, the MF imparts a significant disturbance to the dynamics of the magnetic moment. A measure of the disturbance is a deviation from the state of the unperturbed uniform precession in the GMF. Idealizations of the model: 1) uniform precession is a natural background for the microscopic events in organisms, 2) the events of the next level—biophysical or biochemical events—form a nonhomogeneous Poisson process, the rate of which varies periodically with the precession phase in the local coordinate system of the target, 3) biological effect is associated with the disturbance of precession of the magnetic moments of the same type—the disturbance being averaged over time and over the realizations of random precession phase. Of course, the biological effect is observed only when the biophysical-level changes go through the stages of transformation at the biochemical, physiological and biological levels of the system.

In this scenario, the average probability of biophysical events P(H, h, Ω, γ, τ, β) depends on six values. Three of these are MF variables: the constant MF component H and the amplitude h and frequency Ω of the variable component. Three others are MF target parameters: the gyromagnetic ratio γ and the parameters τ (relaxation time) and β, describing respectively the thermal and the “signal” interaction of the magnetic moment with its immediate environment.

The change in average probability under exposure to an ac/dc MF, i.e., ΔP ≡ P(H, h, …) − P(H, 0, …) has the following approximate form [36]: (5) Here and below, in order to significantly simplify mathematical equations, the arguments of functions are either omitted or given in an abbreviated form—only those arguments are displayed that are relevant to the context of a given equation. At h = 0, under exposure to a dc MF that is decreasing from the geomagnetic field H_g to an HMF H, the probability change, i.e., ΔP_dc ≡ P(H, 0, …) − P(H_g, 0, …), is equal to (6) where it is assumed that sinc²(γH_gτ) ∼ 0. This Eq (6) is for the HMF effect that we associated above with E(H).

It is seen that the effects of the ac MF at H = 0 depends on γh, while the HMF effect depends on γHτ. The difference can be used to extract information about the gyromagnetic factor and the thermal relaxation time separately from experimental MF-dependencies. This can be done as follows.

Consider two dependencies: 1) ΔP_dc(H) and 2) ΔP(Ω) at H = 0 and at a certain value of h that will be given below.

The first dependence is shown in Fig 4A. The argument γHτ ≈ 2.8, at which the curve reaches the middle between the initial and final levels, does not depend on β. Having found the corresponding value H′ from an experiment, one can define the product of the primary target parameters: γτ = 2.8/H′. Further, we assume that the value of η ≡ 2.8/H′ is known, so that γ = η/τ.

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Fig 4. The change in the probability of primary reaction as a function of different variables.

(A) The probability change with a decrease in the constant MF, i.e. the “zero-field” effect, Eq (6). (B) The probability change with a decrease in frequency of the alternating MF at ηh = 20, 1—according Eq (7) and 2—according Eq (8).

https://doi.org/10.1371/journal.pone.0179340.g004

In the second case, at H = 0 and with account of the relation γ = η/τ, Eq (5)—that now can be conveniently denoted as ΔP_ac—takes the following form: (7) For sufficiently large values of ηh = 2.8h/H′ > 10, Eq (7) can be simplified considerably. Note that because H′ is mostly less than 1 μT, the “large” values of h for large ηh begin with a few μT. Due to the properties of the Bessel functions, only those terms of the sum contribute significantly in ΔP, for which the argument of the Bessel functions is of the order of unity. That is, the values of Ωτ must also be as large as ηh. But then, by virtue of the properties of the sinc-function, only members of the sum with n = 0 make a major contribution. Hence, relation Eq (7) can be written as (8) Functions (7) and (8) that are shown in Fig 4B imply that the approximate Function (8) can be used to determine the value of the argument Ω′τ at which there is a strong change in ΔP_ac. This value is determined by the position of the first extremum in the derivative of the function , i.e., x ≈ 1.1. Hence we find ηh/Ω′τ = 1.1 and, finally, using the definition η = γτ = 2.8/H′, (9) where h is the amplitude of the ac MF used in the experiment. We rounded the numbers up to one significant figure: only the orders of magnitude make sense, as this theory relates with the experiment only through less predictable MF-signal transduction at the biophysical, biochemical, and higher levels.

Thus it is clear how to determine the gyromagnetic ratio and the thermal relaxation time of the precessing primary magnetic moment. First, on the basis of preliminary dc-experiments, one should set a level of the static MF H_min, where the biological response is almost unchanged with further reduction of H. Second, it is necessary to obtain experimental dependences of the biological effect (i) E(H) at h < H_min in a dc-experiment, and (ii) E(Ω) at H < H_min and h > 10H_min in a ac/dc-experiment. Inflection points H′ and Ω′ of these dependences will identify the constants γ and τ according to relations Eq (9).

The above-described HMF effect in the dc MF and the effect in the ac MF are useful. If the HMF effect is due to the non-uniform precession of magnetic moments, then 1) the effect is about three times greater in magnitude [36] and much more likely than the frequency selective ac/dc magnetic effect, since magnetic moments of all types respond to the HMF, and thus those associated with a biological measurand will respond inevitably; 2) revealing of the HMF effect does not require selection of the MF frequency, see Fig 4A; 3) the measurements allow (for the first time) to determine two parameters of the precessing moments at once—their gyromagnetic ratio and thermal relaxation time.

As an example, H-dependences of the gravitropic reaction in watercress roots [96] are shown in Fig 5. Experimental data of a few measured parameters were approximated by the Levenberg–Marquardt algorithm. An approximating function, in accordance with Eq (6), was the function k sinc²(k′H) + k″, where k, k′, and k^′′—coefficients that minimize the standard deviation of experimental points from the curve. In other words, the fitting was achieved by scaling and offset of the squared sinc-function along the ordinate, and by the abscissa scaling. Then, the obtained curve, already together with the experimental points, was subjected to an inverse transform. Therefore, fitting functions of all the measured parameters were reduced to the same H-dependence, −sinc²(x) with x = γHτ/2. This is convenient for visualizing the overall of dependencies. Thus, the subjective factor was totally excluded from the comparison procedure of the experimental H-dependencies with a common motive −sinc²(x).

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Fig 5. Approximation of curve −sinc²(x) with x = γHτ/2 by experimental data [96].

Designations: the number of roots growing horizontally ▫ or vertically •; growth angle of the remaining roots ○.

https://doi.org/10.1371/journal.pone.0179340.g005

This physical mechanism predicts that deprivation of both dc and ac MFs is equally important for observing the magnetic vacuum effect. Removal of only one of these MFs would be insufficient. Experimentally, the interference of the HMF effects, at dc MF compensation, with the effects of geomagnetic storms has been observed in [91] and with the effects of the background EM noise in [136, 137, 198].

Interestingly, the universal mechanism predicts the absence of the primary effect in strong MFs at the level of Tesla. This is due to a 2π-rotational symmetry of the phase of magnetic moments precessing in their local molecular environment. The characteristic precession and magnetic resonance frequencies change proportionate to the MF value. As shown in [36], observation of the nonuniform precession effects (that are proportional to approximately at Ω = γH) becomes more difficult in a strong constant MF, although observation of the resonance is facilitated. In a strong MF, the resonance can be observed at a lower ac MF amplitudes, but the local effects of the inhomogeneous precession are only possible at larger amplitudes. Therefore, in a strong constant MF, the possible effects of inhomogeneous precession simply disappear, which is well seen in Fig 4A. Perhaps this explains the relative safety of a short-term exposure of organisms to the MFs of the diagnostic procedure know as magnetic resonance imaging. However, other biological effects can develop due to the growing contribution of spin-correlated biradicals.

Finally, this primary physical mechanism, while predicting the maximum effect in ac/dc MF at h = 1.8H and Ω = γH, is in agreement with numerous experimental data, a review of which is available in e.g. [18] p. 307–314.

A particular characteristic of this mechanism is that it predicts the form of MF-dependencies for HMF and ac/dc MF effects in the same biological system. Therefore, its most effective validation would be to compare these dependencies. How to validate this theory is discussed in detail in [36]. We note that in terms of quantum mechanics, this mechanism is not connected with quantum transitions. In opposite, these are a hindrance to its implementation [199, 200], and therefore a significant deviation from parallelism of the fields h and H destroys the ac/dc MF effect.

Intraprotein rotations: Molecular gyroscope mechanism

Interestingly, the magnetoreception mechanisms that are presented above involve, in one way or another, a variety of rotations: the precession of magnetic moments of different nature and the rotations of nanoparticles that have their own intrinsic magnetic moment. The presence of rotations in the models of magnetoreception is not by chance. This is a consequence of the specificity of the MF interaction with matter. This link is carried out only through the interaction with magnetic moments. The magnetic moments, on the one hand, experience a torque in the MF, and, on the other hand, are bound with spins or can be generated by rotations of electric charges.

The following briefly shows a mechanism [180, 201] developed from study [202]—the so-called molecular gyroscope—and derives its properties in the HMF.

The essence of the mechanism is a rotation of large fragments of macromolecules or amino acid residues with distributed electric charge. Due to this rotation, a magnetic moment appears that interacts with an external MF. In the quantum description, the rotation of fragments is an interference of their angular states with a nonzero magnetic quantum number.

How can the molecular gyroscopes arise? The folding of long protein chains that are synthesized by ribosomes results in the formation of protein globules. In the process of the folding/maturation, their occurs an evolutionary-determined and workable conformation of a protein molecule. Incorrect folding can break or make impossible a specific function of the protein. The characteristic time of folding is highly dependent on the length of the protein chain, ranging from microseconds to seconds [203, 204]. At some stages of the folding, virtual cavities free of water molecules may occur within the protein. Indeed, hydrophobic cavities of the order of 1 nm or less push out water molecules [205]. In these cavities, the amino acid residues (molecular gyroscopes) rotate within milliseconds whilst looking for the best position. That would be enough for the action of ELF MFs on such rotations and, therefore, on the path and the result of the protein folding.

The magnetoreception mechanism that utilizes the interference of quantum states of a molecular gyroscope is based on the fact that the main rotational degree of freedom of a gyroscope can long remain non-thermalized, or “cold.” Rotations of small molecules that are fixed inside virtual cavities are well protected from the thermal vibrations of the cavity walls and, therefore, have a relatively long lifetime. In combination with quantum interference effects, these coherent long-lived rotations allow one to explain magnetoreception. In this way the molecular gyroscope mechanism resolves the kT problem [206].

What does the gyroscopic mechanism predict regarding the zero-MF biological effects?

We would like to recall the following. The fact that the best position of an amino acid residue has been found in the process of folding is considered the result of some reaction. The reaction yield, or the number of gyroscopes having entered this reaction, or alternatively, the equilibrium number of gyroscopes is associated with a biological effect.

The general result of the gyroscopic model of magnetoreception is a relation for the time-averaged reaction probability P under exposure to a dc MF H and a collinear ac MF of the amplitude h and frequency Ω, (10) where p is MF-independent part of the probability, w is the rate of new gyroscope creation events, τ is the thermal relaxation time, m or m′ is the magnetic quantum number, σ_mm′ are the density matrix elements that describe the initial state of a gyroscope, γ is its gyromagnetic ratio, J_n is the n-th order Bessel function of the first kind, and In this notation, ω_mm′ = ℏ(m² − m^′2)/2I, where I is the moment of inertia of the gyroscope. In the current work, we will study a special case of HMF, that is, h = 0. Note that if m ≠ m′ or m ≠ −m′ then P = p, because in this case τω_mm′ ≫ 1 and the sinh group in Eq (10) turns to zero. Evidently, one should take only those terms of the sum in Eq (10) where n = 0 because h = 0, and m′ = ±m. After some algebra, we obtain the formula (11) where k ≡ sinh²(1) ≈ 1.38.

A rotator in the form of an amino acid residue has a significant moment of inertia, in a quantum scale. For this reason, Boltzmann statistics is used to assess the populations of the gyroscope rotational states. The energies of the rotational states are ε_m = ℏ² m²/2I − ℏγHm. Since the Zeeman splitting ℏγH in this case is much less (∼ 10^-6) than the energy of rotational levels, this splitting can be ignored in the Boltzmann distribution. Thus, σ_m,±m = exp(−αm²)/Z, where α ≡ ℏ²/(2IkT) is a coefficient and Z = ∑_m exp(−αm²) is the statistical sum. Further, one will need a notation After its substitution in Eq (11), we obtain (12) The above series can be simplified by using the integral representation of the sums: and .

To link P, which now reads p + awτ(1 + s), to an observable, we write the first-order kinetic equation for the number N of gyroscopes per unit of tissue volume, . This gives N = w/P in stationery conditions. We would like to know the relative change of N under the HMF exposure, as MF drops from H_g to smaller values H. This is the relative number of gyroscopes ρ ≡ 1 − N/N₀, where N₀ = w/P₀ is a solution for the geomagnetic field. Substituting P in ρ and s in P, and given γH_gτ ≫ 1 and hence s(H_g) ≈ 0, we finally arrive at the relation for observable ρ in HMF: (13) where τ_k ≡ p/(aw) is a kinetic time constant and s(H) is given by Eq (12).

Function (13) is shown in Fig 6 for the case of α = 10⁻⁴ and 10⁻⁵ that corresponds in the order of magnitude to I ∼ 10⁻³⁷ and I ∼ 10⁻³⁶ g⋅cm²—moments of inertia of Glu residue at rotation around the long axis and the peptide bond respectively. As is seen, the critical MF is about 0.03/γτ.

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Fig 6. A magnetic vacuum effect.

The number of molecular gyroscopes in HMF at different ratio of the thermal relaxation time and the characteristic time of chemical kinetics: τ ≫ τ_k, α = 10⁻⁴ (1), α = 10⁻⁵ (2); and τ = 0.1τ_k, α = 10⁻⁴ (3).

https://doi.org/10.1371/journal.pone.0179340.g006

Apparently, the quantum gyroscopic mechanism can explain biological effects of HMF, the relative magnitude of the magnetic effect reaching, theoretically, impressive 50% for small residues and about 10–15% for larger ones in the case of large thermal relaxation times. Recall that in the RPM, the electronic states lose their coherence in time intervals less than 10⁻⁷ s. In contrast, as calculated in [180], the rotational states of molecular gyroscopes can live milliseconds, reaching 100 ms for Glu-like object rotating in a cavity of radius 1.5 nm. As far as the gyromagnetic ratio is of the order of e/Mc, where e is the elementary charge and M is the mass of Glu, the critical MF for this case is about 0.5 μT. Some difficulty is that the critical MF quickly grows for smaller cavities, becoming about 8 mT in a 1-nm cavity. On the other hand, this hypothetical mechanism is reinforced by the fact that quantum processes associated with whole large molecular fragments must play a crucial role in solving the problem of protein folding [207].

Stable magnetic nanoparticles

Magnetic nanoparticles—tiny magnets—have been found in the body of many animals [208]. Often, magnetic nanoparticles in organisms are single-domain magnetite Fe₃O₄ crystals. Depending on their size, these particles are mainly divided into two classes: superparamagnetic (SP) and stable ferrimagnetic particles. The latter particles have a permanent magnetic moment that is rigidly bound to the particle’s geometry: the magnetic moment is said to be blocked. The magnetic moment of SP particles is not blocked, and is capable of changing its direction more or less randomly under the action of heat. In many studies, these facts are drawn to explain magnetic susceptibility of organisms, e.g. [30, 209].

We recall that mechanisms that could provide only a non-specific response to a magnetic stimulus are considered. For this reason, some proposed mechanisms are not commented on. For example, micrometer-sized ferromagnetic crystals, which could naturally grow in a few places in the beak of homing pigeons [28], are likely to exert a mechanical pressure on bird’s ophtalmic nerve [210]. However it is clear that regular occurrence of so large crystals would be an evolutionary formed property that provided a specific function in some species only.

Stable nanoparticles in double-well potential.

Single magnetoreceptors that are highly sensitive to the MF variations on the background of the geomagnetic field have been proposed in [212]. In this model, a nanoparticle with blocked magnetic moment is in a rotational potential created by the cytoskeleton filaments and the MF. In general case, this potential is composed of two wells separated by a barrier. The MF can change the height of the barrier. The particle undergoes stochastic Brownian rotations under the influence of thermal perturbations and is able to overcome the barrier. The transition probability depends exponentially on the barrier height. For this reason, even small MF variations might influence the rate of transitions and, thereby, cause a biological response. The minimal level of such variations is calculated to be of the order of 100–200 nT in a reference nanoparticle.

What are the properties of this model in the zero MF mode, when H → 0?

In general, the stationary orientation of the nanoparticle magnetic moment does not follow the direction of the constant MF. This orientation is determined by the balance of the elastic torque of cytoskeleton filaments and that of magnetic forces. Let x axis determines the equilibrium direction of the magnetic moment of the particle in the absence of MF and thermal disturbances, Fig 7.

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Fig 7. Notation for variables.

Relative position of the MF vector H and the magnetic moment m of a nanoparticle. Note that m cannot follow H where φ₀ is about π as far as a deviation of m from direction n produces a restoring torque of magnitude −κφ; hence, two potential minima occur over φ.

https://doi.org/10.1371/journal.pone.0179340.g007

Rotational potential of the nanoparticle then equals where φ is the angle of particle (or its magnetic moment) deviation from the equilibrium direction x, κ is the cytoskeleton elasticity coefficient, φ₀ is the angle between the MF and the equilibrium direction. For particles that are mainly oriented against the direction of the constant MF, i.e., when φ₀ ∼ π (or η₀ ∼ 0), the potential has a double-well form, Fig 8A.

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Fig 8. Characteristics of the random rotational oscillations of a magnetic nanoparticle.

(A) The potential function of the particle with different values of the parameter a = mH/2kT that is proportional to the MF, b = 0.5, and φ₀ = 1.1π. (B) The averaged standard deviation of the particle oscillations under the action of thermal disturbances at different values of the elasticity of the cytoskeleton b = κ/2kT.

https://doi.org/10.1371/journal.pone.0179340.g008

Further, we assume that nanoparticles are mobile enough to significantly change their orientation in the geomagnetic field under the heat. In other words, mechanical elasticity κ, associated with the bending of the filaments, magnetic elasticity mH, and the scale of the thermal perturbations kT differ within one–two orders of magnitude only. Furthermore, we assume that each nanoparticle is fixed in such a manner that it can mainly revolve around a single axis.

Fig 8A shows that in some MF H = 4kT/m, or at a ≡ mH/2kT = 2, the particle motion in the lower well occurs within a narrow range 1 that is determined by the width of the well at the level kT, while a reduction of the MF to 2kT/m leads to a significant expansion of the area of possible motion to the size 2.

The average amplitude of the oscillations within the well can be defined as twice the standard deviation of a particle moving in a viscous medium under the influence of thermal disturbances. The Langevin equation for the rotation angle of a particle is where ξ(t) is a random moment with the correlation function 〈ξ(t)ξ(t + τ)〉 = δ(τ) and δ(τ) is the Dirac delta-function. By means of substitutions a = mH/2kT, b = κ/2kT, and t = t′γ/2kT it can be reduced to a dimensionless form , where , and can be integrated numerically, for example, by the Runge-Kutta method. Results are shown in Fig 8B.

Shown are dependences of the standard deviation, averaged over the uniformly distributed angle φ₀, on the parameter a = mH/2kT for various b. It is seen that the scope of random rotations increases significantly with decreasing MF, especially for the least fixed particles. Increasing of the amplitude of random oscillations takes place in a logarithmically wide range of parameters.

In other words, in the geomagnetic field, a significant portion of nanoparticles are “locked” in one of the two narrow potential wells. After reducing the level of the MF, the potential barrier between the wells disappears, and the particles under the heat begin rotating more freely. We recall that larger oscillations supposedly cause the subsequent downstream biophysical and biological effects. In other words, in this model, the magnetic effect is associated with the magnitude of Brownian rotational deviations (15) This MF-dependence is shown in Fig 8B.

Nanoparticles produce their own MF. In their vicinity, this MF exceeds the geomagnetic field by several orders [31]. In the zero MF, the increase in oscillations of the large own MF could influence the biradical reactions. In contrast to the direct action of the zero MF on biradical reactions in organisms, the indirect action through the rotations of nanoparticles does not require the assumption on the incredibly long lifetime of the spin-correlated biradicals.

Indeed, a small enough, but still stable particle, in a HMF, rotates about π/2 or larger, under thermal perturbation. Because of the great nonuniformity of the particle’s MF, a local MF seen by a radical pair can fluctuate in a few mT as the particle rotates, which is enough to shift the RP reaction together with the MF fluctuations. If the external MF grew to GMF, the same particle would not rotate, there would be only unchanged local MF near the RP, and no signal would propagate from it. This means that in the presence of magnetic nanoparticles, MF switching between GMF and HMF would cause a fluctuating biochemical signal through the adjacent RP reaction. Unlike this, if no magnetic particle existed, the RP could only see that the external MF changed from HMF to GMF, rather than in a few mT. Note that the RP could detect GMF to HMF changes in this case, however this would require the electron pair to be a slowly thermally relaxed object, which is mostly impossible, as shown in Section “Radical pair mechanism.”

Returning to the alternative hypothesis based on stable magnetic particles, one should notice, nevertheless, that numerical values for the critical MF (16) are not promising for the HMF effect to be explained. In order to explain the HMF effect, the deflection point in Fig 8B should correspond to about 1 μT or less. This required value can be gained for the particles of radius greater than 160 nm, which is well above 70 nm—the size where a magnetite particle becomes a multidomain one and significantly reduces its magnetic moment. For a reference particle of radius 50 nm, the critical field, as we saw above, is about 30 μT.

Thus, the rotational dynamics of blocked single-domain nanoparticles, while remaining suitable for detecting small geomagnetic variations [212], could not be involved into the magnetic vacuum effect.

Superparamagnetic nanoparticles

SP nanoparticles can occur in the body due to oxidation of ferrihydrites in ferritin protein and subsequent crystallization of iron oxides [213]. Depending on the species and individual characteristics of the organism, these processes may have a status from normal to abnormal and be accompanied by the appearance of significant amounts of SP nanoparticles. Magnetite/maghemite concentration in human tissues varies from tens to hundreds ng per gram [214]. The equilibrium density and distribution of SP nanoparticles in tissues are not yet known in those organisms that do not possess specific magnetoreception.

Some possibilities to explain the HMF biological effect could be associated with SP particles. There are a few modes of the magnetization state in which one could examine the effect of MF.

Protons in water

In a series of recent studies made by the group of A. Konovalov and reviewed in [151], there have been investigated a number of physical and chemical properties of aqueous solutions of different chemical compounds, for example, phenosan and α-tocopherol [136]. Studied were dependences of the properties on the concentration of dissolved substances in the range of 10⁻²⁰–10⁻² M. Physico-chemical properties of solutions in these low concentrations changed significantly with deprivation of the geomagnetic field. Biological properties, i.e., the effect of these solutions on some organisms, were also dependent on the preliminary exposure of the solutions to HMFs. These data, together with previous results of other authors, e.g. [221–224], reinforce the assumption that a possible mechanism of biological action of weak MFs comprises the intermediate stage of modifying the properties of the aqueous medium in cells.

Not all of the physical properties of water are determined by that undoubted fact that liquid water is a collection of water molecules. For example, water conducts electricity as a result of the dissociation of water molecules into hydroxyl and hydronium ions. Due to the specific Grotthuss mechanism of their mobility, these ions are called ionic defects in the structure of water. In addition to ionic defects, there are so-called Bjerrum’s orientation defects. All these defects break a structure rule according to which every oxygen atom covalently binds two hydrogen atoms, or protons, and every proton binds two oxygen atoms by means of a hydrogen bond.

From the viewpoint of electrokinetic properties, water is a collection of the defects that move by means of jumps of protons. Protons detach from their water molecules and form to some extent an independent system—a proton subsystem of water.

The proton subsystem of water is attractive in explaining nonspecific magnetoreception and searching the molecular target of weak MFs. There are the following reasons for that.

The spin of a proton is the same as that of an electron. Electron spins play a crucial role in chemical reactions by participating in a quantum exchange interaction. The exchange interaction of protons is much smaller in magnitude than that of electrons, but it could influence the mobility of water structure defects and contribute to the kinetic phenomena in aqueous solutions.

A proton has a magnetic moment associated with its spin. The impact of the MF on the magnetic moment means some control over the proton spin and, therefore, over the progress of the chemical process with the participation of the proton, provided the exchange interaction is of sufficient magnitude. The effect of proton exchange interaction on proton transport is a new water related mechanism to be considered.

As is known, a water molecule occurs in two isomeric forms. A water molecule can be in the ortho or para state, depending on the relative orientation of the spins of its protons—unidirectional or opposite-directional, respectively. In the gas phase, the equilibrium amounts of these isomers are at the ratio of 3:1, due to quantum mechanical considerations. Studies [222] and [224] show that liquid water can produce non-equilibrium amounts of water spin isomers in a significantly different ratio. This indicates that in liquid water, a metastable local spin ordering exists that can be magnetically sensitive by virtue of the connection between spin and magnetic moment.

Unlike free radical pairs, that are conjectured to be a possible MF target also in nonspecific magnetoreception, the number of protons in biological water medium is enormous, and they participate in many biochemical reactions.

It is essential that the de Broglie wavelength of protons in liquid water at room temperature is 2.5 Å, while their jump length is about 0.8 Å. So a proton in these processes is a quantum object and should be described by the wave function.

Spin effects in proton tunneling can occur due to the proton wave functions covering the nearby potential wells in the network of hydrogen bonds. If a neighboring well is empty, the tail of the wave function that covers it, determines the probability of a quantum transition of the proton into this well. However, if the neighboring well is occupied by another proton, e.g., as in the orientation defect of D-type in Fig 9, then, due to the overlapping of the wave functions of two protons, the exchange interaction occurs between them, which depends on the mutual orientation of their spins. The exchange interaction leads to a coupling between the coordinate of the proton and the spin state of the system. The coupling can be controlled by an external magnetic field through the proton magnetic moments. Thus, probability of the proton jumps between wells varies. Recent studies [225] show that many electrical, dielectric, and even chemical properties of water can be explained by proton jumps rather than water molecule rotations.

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Fig 9. Motion of a D-defect due to the tunneling of a proton to the neighboring H-bond along the reaction coordinate x.

White circles are protons, gray circles are oxygens. Shown are the exchange interaction potential U(x) of the tunneling proton and proton spin states.

https://doi.org/10.1371/journal.pone.0179340.g009

In addition to the interaction of protons and electrons, which forms hydrogen bonds, the protons react with each other. The interaction of protons, the Coulomb repulsion, includes as part the exchange energy that depends on the relative orientation of spins.

Most of the hydrogen bonds in water contain only one proton. To describe the orientation defects in the grid of hydrogen bonds, it is assumed that a part of the bonds contain two protons to form so-called D-defects, Fig 9, or do not contain them at all—L-defect.

Fig 9 shows a proton potential along the reaction coordinate—when moving between bonds with different spin orientations of their protons. The potential is asymmetric for the spins in opposite orientation. If the asymmetry is large enough, then a jump to the well with higher energy may not occur until the well is occupied by a proton in the suitable spin state. Evidently, this impedes relaxation of the local spin equilibrium and affects the mobility of water structure defects.

Thus, the wave functions of two protons that are localized in adjacent wells of a D-defect overlap providing their exchange interaction. Since the MF affects spin magnetic moments, it affects the exchange interaction, and hence the mobility of structural defects. Mobility of defects, in turn, contributes to the proton activity in biochemical reactions.

The proposed protonic mechanism of magnetoreception looks like that kind of the biradical mechanism, in which the evolution of electron spins—the singlet-triplet transitions—is affected by the MF of the magnetic moment of a third particle, such as a proton. In the protonic mechanism, all three particles are protons, and the external MF affects the evolution of the whole system.

Most protons in water solutions relax thermally very slowly, in a matter of seconds; so even a weak ac MF h can significantly change the state of these protons. An estimate of the ac MF in the order of magnitude can be obtained from a consistency between the inverse value of proton relaxation time τ_p and the Rabi frequency γ_ph—the frequency with which the projection of spin on a constant MF changes, where γ_p is the proton gyromagnetic ratio. Hence, at a resonance frequency, the MF amplitude that can overturn a spin during its life time is h ∼ 1/γ_pτ_p ∼ 1 nT. On the other hand, we saw that condition γHτ ∼ 1 plays an important role in HMF magnetoreception and determines the level of dc MF, in the order of magnitude, at which the HMF effect is possible: H ∼ 1/γτ. Thus, in the RP mechanism, this level 1/γ_eτ_e at τ_e ∼ 10⁻⁷ s is 57 μT and is too large to explain the HMF effect. Spin relaxation time of water protons is about 3 seconds, which again gives (18) This, of course, does not mean that changes in the spin subsystem must be accompanied by readily observable effects. A great relaxation time means a feeble link of spin subsystem with molecular vibrations, which in turn makes it difficult to convert the MF signal into the state of the molecules responsible for chemical reactions.

We note that the energy of the MF interaction with the proton magnetic moment is rather small, of the order of 10⁻¹⁰ kT (that does not dismiss the RPM). On the other hand, due to the rigid connection of spin and magnetic moment, this weak magnetic interaction controls the relatively strong exchange interaction that can even be greater than kT for weakly bound protons [181]. In other words, the MF affects the probability of proton jumps by means of the Pauli exclusion principle; this modifies the mobility of water structure defects, and possibly modulates the rate of some biochemical reactions.

Comparison of the HMF effect models

Several mechanisms could qualify for an explanation of the effects of non-specific magnetoreception. Table 2 presents their properties: type of the model—quantum mechanical or within classical dynamics, a formula for the critical MF, its expected value, and a link to the corresponding equation in the article.

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Table 2. Putative mechanisms of nonspecific magnetoreception.

https://doi.org/10.1371/journal.pone.0179340.t002

It can be seen that the first two mechanisms—widely discussed in the literature the radical pair and the magnetite-based mechanisms—are unlikely to explain the HMF effects. The critical MF, where a magnetic response could be expected, is too large as compared with the experimentally established one of the order of a μT. Due to the common nature of the HMF effects and those of ac MFs of about H_g, these mechanisms are also unlikely with regard to the ac MF effects that show frequency selectivity.

Two mechanisms of nonspecific magnetoreception are more likely: the universal physical and the gyroscopic mechanisms.

The universal, or general physical, mechanism is interesting because its predictions do not depend on the nature of the molecular magnetic moments. It is assumed only that they precess and relax. Therefore, there is an attractive possibility to measure the MF target parameters, the gyromagnetic ratio and the thermal relaxation time, separately. The mechanism predicts the maximum effect of approximately 12% and the existence of HMF and ac/dc MF effects simultaneously in one and the same organism.

The magnetic moment of a molecular gyroscope does not precess. The moment occurs due to the rotation of charges that are distributed over the gyroscope—a rotating molecule with its ends temporarily fixed. Gyroscopic mechanism is attractive by the high value of the expected effects, up to 50%, and a relatively small value, 0.03, of the critical parameter γHτ. However, (i) the critical MF is highly dependent on the size of a virtual space for rotation, and (ii) the existence of the long-, or coherently, rotating parts of a protein chain in the process of folding has not yet been confirmed in any experiments besides magnetobiological ones.

Is it possible to distinguish between the precession and gyroscopic mechanisms in the experiment? It is not clear yet. Perhaps, for this to be done, one needs first to determine the parameters γ and τ of the MF target. As shown in [189, 226], molecular mechanisms are sensitive to the rotation of samples. Any pequliarity in H-dependence should shift proportionally to the speed of rotation and inversely proportionally to the gyromagnetic ratio.

The proton-exchange mechanism requires very precise technique for its validation—a magnetic exposure system that could eliminate any extraneous MF variations exceeding 1 nT. Available observations of 10 to 1000 greater critical MFs do not support the mechanism. In addition, this has not been mathematically developed to a level that would allow one to make experimentally verifiable predictions. Further, even if the mobility of a portion of water protons was really changed in 1-nT MFs, it is unclear how this change could affect the rate of biochemical reaction.

Besides all listed above, there are no other microscopic magnetic moments to consider that could be involved in magnetoreception.

Methodological comments

As follows from the data provided by the table column “Questions,” many articles contain certain omissions that impede unambiguous interpretation of the results. It is useful therefore to formulate some general methodological comments that could improve the situation in the future.

Currently, there are about two hundred publications, documenting the fact that the magnetic vacuum causes a reaction of the organisms. However, the MF-dependence of the effect has not yet been established. There were only a few studies that investigated dependences of the biological effects on the MF value with its gradual decrease, and this does not allow one to compare experiments with theory fruitfully. At the same time, there is a theory exposed in Section “Universal physical mechanism” that predicts the kind of the functional MF-dependence and provides the method of extracting information about the nature of magnetic targets from such dependencies. Under these conditions, one could expect new experimental works that fill the gap. It is important to pay attention to the logarithmic character of the expected dependency, see e.g. Fig 4A, and thus the need in a logarithmic MF increment in the experiment. The number of points in the MF-dependence of a measured parameter should provide stability of interpolation. Another significant methodological factor of the experimental design is the mode by which the MF is switched on or off. If such a switch is simply a closure of a circuit, which starts or stops flowing current, it is accompanied by a rapid change in the magnetic field. In turn, a pulse of the induced electric field and, accordingly, an electric current pulse in a test sample tissue occurs. For standard laboratory systems of the magnetic exposure, this current may be greater than the biologically endogenous currents by several orders of magnitude, thus causing a biological response with an aftereffect. Then the response will be mistakenly associated with the MF action. This factor of nonreproducibility can be easily removed by smoothing the MF switching process, which will preclude the emergence of uncontrolled current pulses.

Table 1 shows that only two types of exposure systems are used in the study of the biological effects of HMF: magnetic shielding and passive compensators of the GMF. Both methods have advantages and disadvantages that have not been properly addressed in almost all reports.

For compensation the advantages are that the non-electromagnetic environment is not disturbed so that the interference of other senses such as sight and smell are preserved between exposure and sham conditions. However in almost all cases the compensations are fixed in time so do not correct for variations in the ambient static and ELF magnetic field. Also there is no correction for confounding effects of electric fields and RF fields, what is crucial for obtaining reproducibility in different laboratories, where such fields may vary significantly. For example, the recent work [12] indicates that the animal orientation effects of a static geomagnetic field can be disturbed by effects of man made RF noise. It could be better if in this compensation approach, systems capable of active correct were used; however this technology is limited to relatively low frequencies and would not address the RF issue.

For the shielding approach it is important to have a number of controls. It is necessary to correctly address electrical, thermal, acoustic, humidity, and lighting conditions. A sham shielded container that affected light and smell but did not attenuate any ambient EM radiation is needed. A second control container is needed that only shields electric fields “identical” to the electric field shielded in the experimental container. Then if light is to be introduced, it must be done so such that there is no increase in temperature or introduction of EM fields. Finally if magnetic fields are introduced as suggested in testing the universal mechanism hypothesis, then they have to be introduced in such a way that they do not introduce electric fields and are not distorted by proximity to the shielding material. Such care in the shielding experiments have been reported [4] but are rare.

The selection of compensation versus shielding should also be matched to the question being investigated. For example, if one wanted to look at the effects of HMF in space travel, the environment of both the sham and exposure systems should reflect as much as possible other unique exposure conditions such as light, sound, electric fields, and RF fields. Since these confounds are bound to be very different then in the laboratory on earth doing experiments in a shielded facility and then introducing the other stimuli may be the best approach. Of course, a major confound would be the presence of gravity.

Not infrequently, the articles provide no information on the background variable electromagnetic fields and slow variations of the GMF. However, such fields can significantly alter the magnitude of the observed biological effects.

Sometimes, researchers comment on the effects of weak MFs, less than 1 μT, based on the results obtained from experiments with a relatively strong MF of the order of mT and more. Since the molecular mechanisms of the action of weak and strong MFs can be substantially different, such comments are not convincing.

As we suggested in the Introduction, there are two types of magnetoreception in weak MFs: specific and nonspecific ones. Specific magnetoreception is a genetically-mediated and elaborated in the course of biological evolution ability of many species to perceive the slight geomagnetic variations of the order of 10 nT and utilize these in order to survive. Nonspecific magnetoreception is mostly a random response to MF changes—one that is not due to specific localized sensitive biophysical structures, is not used by organisms to survive, but having the nature of the adaptation syndrome to a stress-factor. The mechanisms of these two types of magnetoreception have a common physical basis in the interaction of the magnetic moments with the MF, but differ in the next level of biophysical/biochemical events that are initiated by peculiarities of the motion of the magnetic moments. Nonspecific magnetoreception could be of fundamental importance in terms of health risks caused by a chronic EM exposure of humans and biosphere.

Two dimensionless quantities presumably control the occurrence of non-specific effects of the molecular nature produced by uniaxial ac/dc MFs, which has been discussed repeatedly in the literature. This is the ratio of the ac MF frequency to the cyclotron frequency of the proposed target–particle, and the ratio of the amplitude of the MF to its permanent component. These quantities may affect the occurrence of magnetic effects only along with other physico-chemical and physiological factors. These factors are difficult to control; therefore, the occurrence of non-specific effects on the whole is less predictable. However, such effects are more likely to occur under the following conditions (19) (20) which is in agreement with the experimental data and theoretical models. In the first equation γ is the gyromagnetic factor. In the case of a target–particle with charge q and mass M, this is γ = q/Mc, where c is the speed of light. Indeed, there are no other combinations of target parameters with the dimension of frequency besides γH. This equation requires one to select the ac MF frequency approximately proportionate to the dc MF magnitude. The second equation requires to choose ac MF amplitude approximately equal to the dc MF magnitude.

As follows from this article, in addition to the relations Eqs (19) and (20), the dimensionless quantity (21) controls the occurrence of the HMF effects, see Eqs (4), (6), (12) and(13), and Figs 4A, 5 and 6. In order for an HMF effect to arise, this equation requires the MF to be reduced to the order of 1/γτ. Eq (21) is another general ratio in magnetobiology that defines the thermal relaxation time of the MF targets to be a parameter directly measurable in the experiment, as shown in Section “Universal physical mechanism.”

All these relations Eqs (19)–(21) give the scope for interpretation of a variety of experimental curves. All what is needed is a fast and inexpensive experimental model of nonspecific magnetoreception with an enhanced level of reproducibility. Perhaps, such a model is bacterial or plant cell cultures in conjunction with the parallel sequencing in gene expression that would allow one to choose some responding genes for subsequent quantitative PCR.