New mechanism of solution of the kT -problem in magnetobiology
aa r X i v : . [ phy s i c s . b i o - ph ] S e p New mechanism of solution of the kT -problem in magnetobiology Zakirjon Kanokov , ∗ , J¨urn W. P. Schmelzer , , Avazbek K. Nasirov , Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna, Russia ∗ Faculty of Physics, M. Ulugbek National University of Uzbekistan, Tashkent, Uzbekistan Institut f¨ur Physik, Universit¨at Rostock, Rostock, Germany and Institute of Nuclear Physics,Tashkent,Uzbekistan (Dated: November 13, 2018)
Abstract
The effect of ultralow-frequency or static magnetic and electric fields on biological processes isof huge interest for researchers due to the resonant change of the intensity of biochemical reactionsalthough the energy in such fields is small in comparison with the characteristic energy k B T ofthe chemical reactions. In the present work a simplified model to study the effect of the weakmagnetic and electrical fields on fluctuation of the random ionic currents in blood and to solvethe k B T problem in magnetobiology is suggested. The analytic expression for the kinetic energyof the molecules dissolved in certain liquid media is obtained. The values of the magnetic fieldleading to resonant effects in capillaries are estimated. The numerical estimates showed that theresonant values of the energy of molecular in the capillaries and aorta are different. These estimatesprove that under identical conditions a molecule of the aorta gets 10 − times less energy than themolecules in blood capillaries. So the capillaries are very sensitive to the resonant effect, with anapproach to the resonant value of the magnetic field strength, the average energy of the moleculelocalized in the capillary is increased by several orders of magnitude as compared to its thermalenergy, this value of the energy is sufficient for the deterioration of the chemical bonds. PACS numbers: 87.15.-v Biomolecules: structure and physical properties 82.35.Lr Physical properties ofpolymers ∗ Electronic address: [email protected] . INTRODUCTION One of key problems magnetobiology is the explanation of the mechanism of influence ofweak magnetic fields on biological objects. The experimental data show that influence ofweak magnetic fields on the test-system (both of animal [1, 2, 3] and botanical [4, 5] origin)are realized by the same mechanism. As a consequence, similar effects have to be expectedto be of significance for human beings as well. A large spectrum of data is known [6, 7],accumulated in biology, biophysics, ecology, and medicine showing that all ranges of thespectrum of electromagnetic radiation may influence health and working ability of people.The human organs are not capable to feel physically an electromagnetic field surroundinga human’s body; however, it may cause the decrease of immunity and working ability ofpeople, under its influence syndromes of chronic weariness may develop, and the risk ofdiseases increases. The action of electromagnetic radiation on children, teenagers, pregnantwomen and persons with weakened health is especially dangerous [6].The negative influenceof electromagnetic fields on human beings and other biological objects is directly proportionalto the field intensity and irradiation time. Hereby a negative effect of the electromagneticfield appears already at field strengths equal to 1000 V/m. In particular, the endocrinesystem, metabolic processes, functioning of head and a spinal cord etc. of people may beaffected [6].The absence of a theoretical explanation of the mechanism of action of weak magneticfields on biological objects is connected mainly with the so-called k B T problem. Here k B is the Boltzmann constant and T is the temperature of the medium. As it is noted in thereview [7], at present does not exist a comprehensive understanding from the point of viewof physics how weak low-frequency magnetic fields may affect living systems. In particular,it is not clear how low-frequency weak magnetic fields may lead to the resonant change ofthe rate of biochemical reactions although the impact energy is by ten orders of magnitudeless then k B T .At the same time, up to now there is no theory in the framework of the general physicalconcepts underlying magnetobiology and heliobiology, i.e. in the field of a science whereeffects of the weak and super-weak magnetic fields are studied, giving an answer to thesequestions. There are even no qualitative theoretical models explaining the interaction mech-anisms of fields with biological objects. From the point of view of physics, this situation is2onnected with complexity of the macroscopic open systems when the concepts of physics,biology, and chemistry are applied. This complexity is caused by the fact that being macro-scopic they consist of many different objects being the elements of a structure formation.In connection with the absence of any standard view on the interaction mechanisms ofweak and super-weak external fields with biological systems and, especially with ”a problem k B T ”, in the papers [8] we proposed a simplified model to study an influence of the weakexternal magnetic and electric fields on the fluctuation of a stochastic ionic current in bloodvessels and leading to a new method of solution of the k B T problem in magnetobiology. Thepresent work is written on the basis of results of works [8]extending the analysis performedthere.The aim of the present work is to estimate energy of molecules near to resonant value ofa magnetic field. II. THE LANGEVIN EQUATION AND ITS SOLUTIONA. Basic Equation
More than 90% of biological tissues consists of polar molecules of fibers, nucleinic acids,lipids, fats, carbohydrates and water. The blood of the human being is the multi-componentsystem consisting of plasma and blood cells. As it is known from the physiology of cardio-vascular system of people [9], plasma of blood is a water solution of electrolytes, nutrients,metabolites, fibers and vitamins. The electrolytes structure of plasma reminds the sea waterthat is connected with the evolution of a life form from the sea. Concentrations of ions likeNa + , Ca and Cl − in plasma of blood are larger than in cytoplasm. On the contrary, theconcentration of ions K + , Mg and phosphate in plasma of blood are lower than one incells. These facts allow us to consider the blood in blood vessels as a conductor of an ionicelectric current.It is well known that in all conductors the fluctuations of a current take place becauseof their molecular structure. Such effect has been experimentally measured by Johnson in1928 and it denoted as the Johnson noise [10]. The spectral density of the Johnson noisedoes not depend on a frequency and, therefore, it represents the white electric noise. TheJohnson noise is observed in the systems being in the equilibrium states or close to them.3he concrete microscopic mechanisms for the occurrence of the Johnson noise can bedifferent. However in all cases the Johnson noise is caused by the chaotic Brownian motionof the charged particles which possesses two important properties: fast casual change ofthe direction and the basic opportunity of carrying a large charge through the section ofa conductor. Thus the geometrical shape of the system considered is of no relevance forthe process. The Brownian character of the charge carriers’ motion remains the same. Asthe Brownian motion of ions is very poorly connected with fluctuations of their number:a disappearance of one particles and a creation of others does not change the essence ofthe process analyzed. An arbitrarily large number of charge can be transferred by any waythrough the given section inside of a conductor [11, 12].This process is stationary, Gaussian and Markovian process. The random ionic currentsatisfies a linear stochastic differential equation, namely the Langevin equation. An elementof the vascular system with a weak random current is assumed to be described as a lineelement of random current with the length L located in a weak external static magneticfield ( B ). As it is well-known, in this case the acting force onto the element of randomcurrent is the force F ( t ) = i ( t ) LB sin( α ), where α is between directions of the currentelement −→ L and magnetic field −→ B [13].In the subsequent analysis we consider the fluctuations of a scalar quantity, the magni-tude of the random current. The mentioned circumstances allow us to formulate the basicequations in the following form [8] di ( t ) dt = − Λ i ( t ) + f ( t ) . (1)Here Λ = λ − qn ch m B sin α , where m , q , n ch are the mass, charge and number of ions,respectively, in the volume V ; B is the induction of an external magnetic field, λ = k B TmD (2)is the friction coefficient, D = k B T πηr (3)is the diffusion coefficient, r -ionic radius, η is the viscosity coefficient of liquid; f ( t ) = qmL X i f i ( t ) , (4)4here f i ( t ) is the random force acting on the corresponding particle. It is the same for anyatom of the same type and it is not correlated with the random forces acting on other typeions [12, 14]: < f ( t ) > = 0 , < f ( t ) f ( t ′ ) > = γδ ( t − t ′ ) , (5)where γ = 2 k B T q n ch λmL (6)is the intensity of the Langevin source. B. Solution of the Basic Equation and General Analysis
As shown above, the random electric current may be described by the linear stochasticdifferential equation with white noise as the random source. The process under considerationis a process of Ornstein-Uhlenbeck type and the formal solution of Eq.(1) may be presentedin the form [8, 12] i ( t ) = e − Λ t i (0) + Z t e − Λ( t − τ ) f ( τ ) dτ. (7)The average value of the current may be determined from Eqs. (5) and (7) as < i ( t ) > = e − Λ t < i (0) > . (8)One can also easily compute the dispersion of the random current fluctuations, i.e. σ ( t ) = < i ( t ) > − < i ( t ) > . This quantity is given by σ ( t ) = e − t Z t e τ γdτ. (9)Taking the derivative of Eq.(9) with respect to time, we obtain dσ ( t ) dt = − σ ( t ) + γ. (10)The solution of Eq. (10) with the initial condition σ (0) = 0 has the form σ ( t ) = σ ( ∞ )(1 − e − t ) , (11)where σ ( ∞ ) = lim t →∞ σ ( t ) = γ (cid:0) λ − qn ch B sin αm (cid:1) . (12)5s evident from Eq. (12), the fluctuations of the random ionic electric current have aresonant character at λ = qn ch B sin αm . (13)The corresponding magnetic induction B is determined by expression B = λmqn ch sin α . (14) III. RESONANT ENERGY OF MOLECULESA. Basic formulas
For any stochastic process i ( t ) the power spectrum i ( t )) is defined as a function of thespectral density S ( ω ) by the relation [12, 14] < i ( t ) > = 12 π Z ∞−∞ S ( ω ) dω. (15)For a scalar process with the real part of the relaxation rate Λ, the spectral density has thefollowing form S ( ω ) = γω + Λ . (16)For the fluctuation of an random ionic current the spectral density is connected with thepower P disseminated by the current at the given frequency as P = 1 π Z ∞ RS ( ω ) dω, (17)where R = mL λn ch q (18)is the electric resistance of a considered element of current with a length L [12]. We substitute(16) in (17) after integrating over ω , we obtain P = Rγ , (19)We substitute (18) into (19) and the resulting expression we multiply with the expositiontime t then we divide it by the total number of molecules in the considered volume V , i.e. n tot ≈ N · V ( N ≈ m − ). In this way, we obtain the average energy of a molecule [8] ε = P tn tot = k B T λ n tot Λ t. (20)6ecause i ( t ) is a stationary Gaussian process, Eqs.(7) and (11) are sufficient to completelydetermine the conditional density of probability P . It is taken from [12, 14] P ( i (0) | i ( t ) , t ) = 1 p πσ ( t ) exp (cid:20) − ( i ( t ) − i (0) exp( − Λ t )) σ ( t ) (cid:21) . (21)One can see that the width of the conditional probability distribution depends on σ ( t )and at the large values of σ ( t ) the density of probability goes to zero. B. Numerical Estimations
In order to estimate the resonant value of the magnetic induction as described by Eq.(14),we employ the following data [9]: for a person of 70 kg weight, we have an amount of 1.7 kgcalcium, 0.25 kg potassium, 0.07 kg sodium, 0.042 magnesium, 0.005 kg iron, 0.003 kg zinc.The effect of calcium in the organism of a human being is very significant. Its salts are apermanent constituent of the blood, of the cell and tissue fluids. Calcium is a componentpart of the cell nucleus and plays a major role in the processes of cell growth. 99% of thecalcium is concentrated in the bones, the remaining part in the blood system and tissues.The blood composes about 8.6% of the mass of a human body. Hereby the fraction ofthe blood located in the arteries is lower than 10% of its total amount. The same amountof blood is contained in the veins, the remaining 80% are contained in smaller units likethe microvasculature, arterioles, venues and capillaries. The typical values of the viscosityof blood plasma of a healthy human being at 37 ◦ C are 1 . · − Pa · s [9]. The density ofthe blood is of the order ρ = (1.06–1.064) · kg/m [9]. Knowing the radius of the ions,we may determine the diffusion coefficient which is estimated as D = (1.8–2.0) · − m /s.The friction coefficient is calculated by formula λ = 6 πηr/m obtained from formulas (2) and(4). Its value has been obtained: λ =(3–6) · s − . For example, for calcium ions we used r Ca = 10 − m [15], m Ca = 6 . − kg and q Ca = 2 · . · − C and we have obtained λ = 3 . · s − . The aorta can be considered as a canal with a diameter of (1.6–3.2) · − m and a cross section area of (2.0–3.5) · − m , which splits of step by step into anetwork of 10 capillaries each of them having a cross section area of about 7 . · − m with an average length of about 10 − m.The number of calcium ions in a volume V =(2–3.5) · − m of the aorta is equal to n ch =(0.8–1.4) · , in a volume V = 7 · − m of the capillary we have n ch = 2 . · .7ubstituting these values into Eq.(14), we get, at sin( α ) ≈
1, for the aorta B ≈ . · − T and for the capillary B ≈ µ T.For numerical estimates average energy of a molecule, we express the parameter in thefollowing form Λ = λ (cid:18) − ω ( B ) λ (cid:19) , where ω ( B ) = qn ch m B. (22)Substituting into Eq.(20) the values of the total number of molecule in a volume V = 7 · − m for capillary n tot ≈ and total number of molecule in a volume V =(2–3.5) · − m for the aorta n tot ≈ .The energy received by a molecule in a capillary during t = 1s was calculated for theΛ = 0 . λ , 0.05 λ , and 0.005 λ which correspond to values of the induction of an externalmagnetic field B =135 µ T, 256.5 µ T and 268.65 µ T, respectively, because as it was mentionedabove B =270 µ T is the resonant value for a capillary. At these values of B we obtain thefollowing estimates for the energy received by a molecule in a capillary during t = 1s: ε ≈ k B T , ε ≈ k B T and ε ≈ k B T . The similar estimations for the Λ = 0 . λ , 0.05 λ ,and 0.005 λ for a molecule in the aorta led us to values ε ≈ · − k B T , ε ≈ · − k B T and ε ≈ · − k B T . Apparently, it follows from these estimates that under identical conditionsa molecule of the aorta has 10 − times less energy than the molecules of capillaries.At Λ = 0 . · − λ for the molecule of the aorta we get: ε ≈ k B T , but according toEq.(21) the probability of such a process is close to zero. These estimates show that largevasculatures are more sensitive to ultra-weak field and capillaries are sensitive to weak andmoderate magnetic fields. IV. CONCLUSIONS
The numerical estimations showed that the resonant values of the energy of molecularmotion in the capillaries and aorta are different. These estimations proved further thatunder identical conditions a molecule of the aorta gets 10 − times less energy than themolecules of the capillaries. The capillaries are very sensitive to the resonant effect, withan approach to the resonant value of the magnetic field strength, the average energy of themolecule localized in the capillary increases by several orders of magnitude as compared toits thermal energy, this value of the energy is sufficient for the deterioration of the chemicalbonds. Even if the magnetic field has values not so near to the resonant values, with an8ncrease of the time of exposition to the magnetic field a significant effect can be reached.A series of experiments are desirable to check the suggested mechanism of an action of theweak magnetic fields on the biological objects, especially, “a k B T problem”. Acknowledgments
Authors thank Drs. G. G. Adamian and N. V. Anfonenko for valuable discussions andcomments. Z. Kanokov is grateful to the Deutsche Forschungsgemeinschaft (DFG 436 RUS113/705/0-3) for the financial support. [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., V. 31, No 3., 295 (1999).[3] G. C. Moses, A. H. Martin, Biochem. Mol. Biol. International., 29, 757 (1993).[4] F. Bersani (Ed.), Electricity and Magnetism in Biology and Medicine (Kluwer, Acad./PlenumPubl., New York, 1999).[5] V. V. Lednev, Biophysics, 41, No. 1, 224 (1996); V.V. Lednev, L. K. Srebnitskaya, E. N.Ilyasova, Z. E. Rozhdestvenskaya, A. A. Klimov, N.A. Belova, Kh. P. Tipas, Biophysics, 41,No. 4, 815 (1996); V.V. Lednev, L. K. Srebnitskaya, E. N. Ilyasova, Z. E. Rozhdestvenskaya,A. A. Klimov, Kh. P. Tipas, Doklady Akademii Nauk SSSR, 348, No. 6, 830 (1996).[6] N. G. Ptitsyna, G. Villoresi, L. I. Dorman, N. Lucci, M.I. Tyasto Physics-Uspekhi, 41, 687(1998). N.G. Ptitsyna a, G. Villoresi b, L.I. Dorman c, N. Iucci d, a[7] V. N. Binhi and A. V. Savin, Physics-Uspekhi, 46, 259 (2003); V. N. Binhi, A. B. Rubin,Electromagn. Biol. Med. 26, 45 (2007).[8] Zakirjon Kanokov, Jurn W. P. Schmelzer, Avazbek K. Nasirov arXiv:0904.1198v1;arXiv:0905.2669v1.[9] D. Marmon, L. Heller, Phisiologiya serdechno-sosudistoy sistemy (Izdatelstvo ”Piter”,S.Petersburg, 2002) (In Russian); Yu. N. Kukushkin, Khimiya vokrug nas (Vysshaya shkola,Moscow, 1992) (In Russian).[10] J. B. Johnson, Phys. Rev. 32, 97 (1928).
11] G.N. Bochkov, Yu.E. Kuzovlev, Uspehi Phys. Nauk. 141, 151 (1983) (In Russian).[12] J. Keizer, Statistical Thermodynamics of Non-Equilibrium Processes (Springer, Berlin, 1987).[13] Edward M. Purcell., Electricity and magnetism. Berkeley physics course., V.2 (Mcgraw-hillbook company, 1984).[14] N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster-dam, 1981).[15] R. D. Shannon, Acta Cryst. A 32, 751 (1976).11] G.N. Bochkov, Yu.E. Kuzovlev, Uspehi Phys. Nauk. 141, 151 (1983) (In Russian).[12] J. Keizer, Statistical Thermodynamics of Non-Equilibrium Processes (Springer, Berlin, 1987).[13] Edward M. Purcell., Electricity and magnetism. Berkeley physics course., V.2 (Mcgraw-hillbook company, 1984).[14] N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster-dam, 1981).[15] R. D. Shannon, Acta Cryst. A 32, 751 (1976).