Cent. Eur. J. Phys. • 8(4) • 2010 • 667-671 DOI: 10.2478/s11534-009-0144-3

Central European Journal of Physics

New mechanism of solution of the kBT -problem in magnetobiology

Research Article

Zakirjon Kanokov12∗, Jürn W. P. Schmelzer13, Avazbek K. Nasirov14

1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia 2 Faculty of Physics, M. Ulugbek National University of Uzbekistan, Tashkent, Uzbekistan 3 Institut für Physik, Universität Rostock, Rostock, Germany 4 Institute of , Tashkent, Uzbekistan

Received 27 May 2009; accepted 26 August 2009

Abstract: The effects of ultralow-frequency or static magnetic and electric fields on biological processes is of huge interest for researchers due to the resonant change of the intensity of biochemical reactions, despite the energy in such fields being small in comparison with the characteristic energy kBT of the chemical reactions. In the present work, a simplified model to study the effects of weak static magnetic fields on fluctuations of the random ionic currents in blood is presented with a view to solving the kBT problem in magnetobiology. An analytic expression for the kinetic energy of the molecules dissolved in certain liquid media is obtained. The values of the magnetic field leading to resonant effects in capillaries are then estimated. The numerical estimates show that the resonant values of the energy of molecules in capillaries and the aorta are different. − These estimates prove that under identical conditions, a molecule in the aorta gets 10 9 times less energy than the molecules in blood capillaries. Therefore, the capillaries are very sensitive to the resonant effect. As the magnetic field approaches the resonant value, the average energy of a molecule localized in a capillary is increased by several orders of magnitude as compared to its thermal energy. This amount of energy is sufficient to cause deterioration of certain chemical bonds. PACS (2008): 87.15.-v; 87.50.-a Keywords: resonance • energy • chemical bonds • kT -problem • human capillaries © Versita Sp. z o.o.

1. Introduction show that the influence of weak magnetic fields on test- systems (both of animal [1–3] and botanical [4–7] origin) is caused by the same mechanism. As a consequence, similar effects are likely to be of significance for human beings. A One of the key problems in magnetobiology is the ex- large spectrum of data is known [8–10], accumulated in bi- planation of the mechanism which causes weak magnetic ology, biophysics, ecology, and medicine, showing that all fields to influence biological systems. Experimental data ranges of the spectrum of electromagnetic may influence health and working ability of people.

∗ E-mail: [email protected] Human organs are not capable of physically feeling an

667 New mechanism of solution of the kBT -problem in magnetobiology

electromagnetic field surrounding the body; however, elec- 2. The Langevin equation and its so- tromagnetic fields may cause a decrease in the immunity lution and working ability of a person. Under its influence, syn- dromes of chronic weariness may develop and the risk of 2.1. Basic equation diseases increases. The action of electromagnetic radia- tion on children, teenagers, pregnant women and persons More than 90% of biological tissues consist of polar with weakened health is especially dangerous [8]. The molecules of fibers, nucleinic acids, lipids, fats, carbohy- negative influence of electromagnetic fields on human be- drates and . The blood of the human being is a ings and other biological systems is directly proportional multi-component system consisting of plasma and blood to the field intensity and irradiation time. A negative ef- cells. As is known from the physiology of cardiovascular fect of the electromagnetic field on humans appears at system of people [13, 14], blood plasma is a water solution field strengths of 1000 V/m. In particular, the endocrine of electrolytes, nutrients, metabolites, fibers and vitamins. system, metabolic processes, functioning of head and a The electrolyte structure of plasma is somewhat similar to spinal cord etc. of people may be affected [8]. that of sea water and is connected to the evolution of life The absence of a theoretical explanation of the mechanism from the sea. of action of weak magnetic fields on biological objects is + 2+ − Concentrations of ions like Na , Ca and Cl in blood connected mainly with the so-called kBT problem. Here k T plasma are larger than in cytoplasm. On the contrary, the B is the Boltzmann constant and is the temperature of + 2+ concentration of ions K , Mg and phosphate in blood the medium. As it is noted in the review of Binhi [9, 10], at plasma are lower than that found in cells. These facts present, a comprehensive understanding from the point of allow us to consider the blood in blood vessels as a con- view of physics of how weak low-frequency magnetic fields ductor of an ionic electric current. may affect living systems does not exist. In particular, it It is well known that in all conductors, fluctuations of a is not clear how low-frequency weak magnetic fields can current occur because of their molecular structure. Such lead to the resonant change of the rate of biochemical effects have been measured experimentally by Johnson in reactions despite the impact energy being ten orders of 1928 and it is denoted as the Johnson noise [15]. The magnitude less then kBT . spectral density of the Johnson noise does not depend At the same time, up until now, there has been no theory on frequency and, therefore, it represents white electric in the framework of the general physical concepts under- noise. Johnson noise is observed in systems which are in, i.e. lying magnetobiology and heliobiology, in the field or close to, equilibrium states. of a science where effects of the weak and super-weak The concrete microscopic mechanisms for the occurrence magnetic fields are studied, giving an answer to these of Johnson noise can be different. However, in all cases, questions. There are even no qualitative theoretical mod- the Johnson noise is caused by a chaotic Brownian mo- els explaining the interaction mechanisms of fields with tion of the charged particles which possess two important biological objects. From the point of view of physics, this properties: fast casual changes of the direction and the situation is connected with the complexity of macroscopic basic opportunity of carrying a large charge through the open systems when the concepts of physics, biology, and section of a conductor. Thus, the geometrical shape of the chemistry are applied. This complexity is caused by the system considered is of no relevance for the process. The fact that being macroscopic they consist of many different Brownian character of the charge carriers’ motion remains objects being the elements of structure formation. the same. As the Brownian motion of ions is very poorly In connection with the absence of any standard theoreti- connected with fluctuations of their numbers, a disappear- cal interpretation on the interaction mechanisms of weak ance of sone particles and the creation of others does not and super-weak external fields on biological systems and, change the essence of the process analyzed. An arbitrar- k T especially with ”the B -problem”, in the papers [11, 12] ily large number of charge can be transferred by any path we proposed a simplified model to study the influence of through the given section inside a conductor [16, 17]. weak external magnetic and electric fields on the fluctu- This process is a stationary, Gaussian and Marko- ations of a stochastic ionic current in blood vessels. This vian, process. The random ionic current satisfies a lin- k T approach lead to a new method of solution of the B ear stochastic differential equation, namely the Langevin problem in magnetobiology. The present work is writ- equation. An element of the vascular system with a weak ten on the basis of results of works [11, 12] extending the random current is assumed to be described as a line el- analysis performed there. ement of random current with the length, L, located in a The aim of the present work is to estimate the energy of weak external static magnetic field, B. As it is well-known, molecules near to resonant value of a magnetic field. in this case the force acting on the element of random cur-

668 Zakirjon Kanokov, Jürn W. P. Schmelzer, Avazbek K. Nasirov

F t i t LB α α rent is ( ) = ( ) sin( ), where −→is the angle between The average value of the current may be determined from L −→directions of the current element and magnetic field Eqs. (5) and (7) as B [18]. − t In the subsequent analysis, we consider fluctuations of a hi(t)i = e Λ hi(0)i : (8) scalar quantity - the magnitude of the random current. The mentioned circumstances allow us to formulate the One can also easily compute the dispersion of the ran- basic equations in the following form [11, 12] dom current fluctuations, i.e. σ(t) = hi2(t)i − hi(t)i2. This quantity is given by di t ( ) − i t f t : = Λ ( ) + ( ) (1) Z t dt − t τ σ(t) = e 2Λ e2Λ γdτ : (9) 0 qnch Here, Λ = λ − m B sin α, where m, q, nch are the mass, charge and number of ions, respectively, in the volume V ; Taking the derivative of Eq. (9) with respect to time, we B obtain is the induction of an external magnetic field, dσ t ( ) − σ t γ : dt = 2Λ ( ) + (10) k T λ B = mD (2) The solution of Eq. (10) with the initial condition, σ(0) = 0 gets the form is the friction coefficient, − t σ(t) = σ(∞) 1 − e 2Λ ; (11)

kBT D = (3) 6πηr where γ r η σ ∞ σ t : is the diffusion coefficient, is the ionic radius and is ( ) =t lim→∞ ( ) =  qn B α  (12) λ − ch sin the viscosity; 2 m q X f t f t ; ( ) = mL i( ) (4) i As is evident from Eq. (12), the fluctuations of the random where fi(t) is the random force acting on the corresponding ionic electric current have a resonant character at particle. It is the same for any atom of the same type and qn B α λ ch sin : it is not correlated with the random forces acting on other = m (13) types of ions [17, 19]: The corresponding magnetic induction B is determined by 0 0 hf(t)i = 0 ; hf(t)f(t )i = γδ(t − t ) ; (5) expression λm B = : (14) qnch sin α where 2kBT q2nchλ γ = (6) mL2 3. Resonant energy of molecules is the intensity of the Langevin source. 3.1. Basic formulas 2.2. Solution of the basic equation and gen- eral analysis For any stochastic process i(t) the power spectrum i2(t) is defined as a function of the spectral density S(ω) by the As shown above, the random electric current may be de- relation [17, 19] scribed by a linear stochastic differential equation with Z ∞ white noise as the random source. The process under con- i2 t 1 S ω dω : ( ) = π ( ) (15) sideration is a process of Ornstein-Uhlenbeck type and 2 −∞ the formal solution of Eq. (1) may be presented in the form [11, 12, 17] For a scalar process with the real part of the relaxation rate Λ, the spectral density has the following form t Z γ i t e−Λti e−Λ(t−τ)f τ dτ : ( ) = (0) + ( ) (7) S(ω) = : (16) 0 ω2 + Λ2

669 New mechanism of solution of the kBT -problem in magnetobiology

For the fluctuation of a random ionic current the spectral The blood composes about 8.6% of the mass of a human density is connected with the power P disseminated by body. Approximately 10% of the blood is located in the the current at the given frequency as arteries. A similar fraction of blood is contained in the veins and the remaining 80% is contained in smaller units Z ∞ like the microvasculature, arterioles, venues and capillar- P 1 RS ω dω ; = π ( ) (17) ies. The typical values of the viscosity of blood plasma of 0 − a healthy human being at 37°C are 1:2×10 3 Pa·s [13, 14]. ρ : ÷ where The density of the blood is of the order = (1 06 mL2λ 1:064) × 103 kg/m3 [13, 14]. Knowing the radius of the R = (18) nchq2 ions, we may determine the diffusion coefficient which is − estimated as D = (1:8 ÷ 2:0) × 10 9 m2/s. The friction is the electric resistance of a considered element of current coefficient is calculated by formula λ = 6πηr/m obtained L with a length [17]. from formulas (2) and (3). Its value has been obtained ω − We substitute (16) in (17) and after integrating over , we as λ = (3 ÷ 6) × 1013 s 1. For example, for calcium obtain r −10 m : −26 Rγ ions we used Ca = 10 m [20], Ca = 6 6810 kg q · : × −19 P = : (19) and Ca = 2 1 6 10 C and we have obtained 2Λ − λ = 3:52 × 1013 s 1. The aorta can be considered as − Further, we substitute (18) into (19) and the resulting ex- a canal with a diameter of (1:6 ÷ 3:2) × 10 2 m and a − pression we multiply with the exposition time t before di- cross section area of (2:0 ÷ 3:5) × 10 4 m2, which splits viding it by the total number of molecules in the consid- of step by step into a network of 109 capillaries each of − − ered volume V , i.e. ntot ≈ N · V (N ≈ 1028 m 3). In this them having a cross section area of about 7:01 × 10 12 m2 − way, we obtain the average energy of a molecule [11, 12] with an average length of about 10 3 m. The number of calcium ions in a volume V = (2 ÷ 3:5) × −6 3 n : ÷ : × 19 Pt kBT λ2 10 m of the aorta is equal to ch = (0 8 1 4) 10 , ε = = t : (20) V × −15 3 ntot ntotΛ in a volume = 7 10 m of the capillary we have nch = 2:7 × 1010. Substituting these values into Eq. (14), −12 i t we get, at sin(α) ≈ 1, for the aorta B ≈ 0:5 × 10 T and Because ( ) is a stationary Gaussian process, Eqs.(7) B ≈ and (11) are sufficient to completely determine the condi- for the capillary 270 µT. P For numerical estimates of the average energy of a tional density of probability 2. It is taken from [17, 19] molecule, we express the parameter in the following form

P i | i t ; t  ω B  2 ( (0) ( ) ) = λ − ( ) ; " # Λ = 1 λ (22) 1 (i(t) − i(0) exp(−Λt))2 p exp − : (21) 2πσ(t) 2σ(t) where qn ω B ch B: ( ) = m One can see that the width of the conditional probability distribution depends on σ(t) and at the large values of Substituting into Eq. (20) the values of the total number − σ(t) the density of probability tends to zero. of molecules in a volume V = 7 × 10 15 m3 for capillary ntot ≈ 1013 and total number of molecule in a volume − 3.2. Numerical estimations V = (2 ÷ 3:5) × 10 6 m3 for the aorta ntot ≈ 1022. The energy received by a molecule in a capillary during In order to estimate the resonant value of the magnetic in- t = 1s was calculated for the values of the parameters duction as described by Eq. (14), we employ the following Λ = 0:5λ, 0.05λ, and 0.005λ which correspond to values of data [13, 14]: for a person of 70 kg mass, there is 1.7 kg the induction of an external magnetic field B = 135 µT, calcium, 0.25 kg potassium, 0.07 kg sodium, 0.042 kg mag- 256.5 µT and 268.65 µT, respectively, because as it was nesium, 0.005 kg iron, 0.003 kg zinc. The effect of calcium mentioned above B = 270 µT is the resonant value for in the organism of a human being is very significant. Its a capillary. At these values of B we obtain the follow- salts are a permanent constituent of the blood, of the cell ing estimates for the energy received by a molecule in and tissue fluids. Calcium is a component part of the cell a capillary during t = 1 s: ε ≈ 2kBT , ε ≈ 20kBT nucleus and plays a major role in the processes of cell and ε ≈ 200kBT . Similar estimations for the parameters growth. 99% of the calcium is concentrated in the bones, Λ = 0:5λ, 0.05λ, and 0.005λ for a molecule in the aorta − − the remaining part in the blood system and tissues. led us to values ε ≈ 2 × 10 9kBT , ε ≈ 20 × 10 9kBT and

670 Zakirjon Kanokov, Jürn W. P. Schmelzer, Avazbek K. Nasirov

− ε ≈ 200 × 10 9kBT . Apparently, it follows from these es- [4] V. N. Binhi, In: F. Bersani (Ed.), Electricity timates that under identical conditions a molecule of the and in Biology and Medicine (Kluwer, − aorta has 10 9 times less energy than the molecules of Acad./Plenum Publ., New York, 1999) capillaries. [5] V. V. Lednev, Biophys.-USSR 41, 224 (1996) − At Λ = 0:5 × 10 9λ for the molecule of the aorta we get: [6] V. V. Lednev et al., Biophys.-USSR 41, 815 (1996) ε ≈ 2kBT , but according to Eq. (21) the probability of [7] V. V. Lednev et al., Dokl. Akad. Nauk SSSR+ 348, such a process is close to zero. These estimates show 830 (1996) that large vasculatures are more sensitive to ultra-weak [8] N. G. Ptitsyna, M. I. Tyasto, G. Villoresi, L. I. Dorman, field and capillaries are sensitive to weak and moderate N. Lucci, Phys.-Usp.+ 41, 687 (1998) magnetic fields. [9] V. N. Binhi, A. V. Savin, Phys.-Usp.+ 46, 259 (2003) [10] V. N. Binhi, A. B. Rubin, Electromagn. Biol. Med. 26, 45 (2007) 4. Conclusions [11] Z. Kanokov, J. W. P. Schmelzer, A. K. Nasirov, arXiv:0904.1198v1 The numerical estimations show that the resonant values [12] Z. Kanokov, J. W. P. Schmelzer, A. K. Nasirov, of the energy of molecular motion in the capillaries and arXiv:0905.2669v1 aorta are different. These estimations prove that under [13] D. Marmon, L. Heller, Phisiologiya serdechno- − identical conditions, a molecule of the aorta gets 10 9 sosudistoy sistemy (Izdatelstvo Piter, St. Petersburg, times less energy than the molecules of the capillaries. 2002) (in Russian) The capillaries are very sensitive to the resonant effect, [14] Yu. N. Kukushkin, Khimiya vokrug nas (Vysshaya with an approach to the resonant value of the magnetic shkola, Moscow, 1992) (in Russian) field strength, the average energy of a molecule localized [15] J. B. Johnson, Phys. Rev. 32, 97 (1928) in the capillary increases by several orders of magnitude [16] G. N. Bochkov, Yu. E. Kuzovlev, Usp. Fiz. Nauk.+ 141, as compared to its thermal energy. This amount of energy 151 (1983) (in Russian) is sufficient for the deterioration of the chemical bonds. [17] J. Keizer, Statistical Thermodynamics of Non- Even if the magnetic field has values away from the res- Equilibrium Processes (Springer, Berlin, 1987) onant values, with an increase in the exposure time to a [18] E. M. Purcell, Electricity and magnetism. Berkeley magnetic field a significant effect can occur. physics course, Vol. 2 (Mc Graw-Hill book company, A series of experiments are desirable to be performed in 1984) order to verify the suggested mechanism of the action of [19] N. G. van Kampen, Stochastic Processes in Physics weak magnetic fields on the biological objects, especially, and Chemistry (North-Holland, Amsterdam, 1981) in order to resolve finally “the kBT problem” based on the [20] R. D. Shannon, Acta Crystallogr. A 32, 751 (1976) theoretical concepts developed here.

Acknowledgements

The authors thank Drs. G. G. Adamian and N. V. Antonenko for valuable discussions and comments. Z. Kanokov is grateful to the Deutsche Forschungsge- meinschaft (DFG 436 RUS 113/705/0-3) for the financial support.

References

[1] W. E. Koch, B. A. Koch, A. N. Martin, G. C. Moses, Comp. Biochem. Physiol. A 105, 617 (1993) [2] J. Harland, S. Eugstrom, R. Liburdy, Cell Biochem. Biophys. 31, 295 (1999) [3] G. C. Moses, A. H. Martin, Biochem. Mol. Biol. Int. 29, 757 (1993)

671