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Simulation of electromagnetic eigenmodes of biological cavity structures in COMSOL Multiphysics Michal CIFRA1,2 , Ondˇrej LAMPA1 1
2
Dept. of Electromagnetic Field, Czech Technical University, Technick´a 2, 166 27 Praha, Czech Republic Inst. of Photonics and Electronics, Academy of Sciences of the CR, Chabersk´a 57, 182 51, Praha, Czech Republic
[email protected],
[email protected]
Abstract. Electromagnetic field is generated in biological systems. Existence of eigenmodes in living cells is expected. Electromagnetic eigenmodes of the spherical cavity with conducting shell are treated analytically. COMSOL Multiphysics is used to model eigenmodes in spherical and elliptical cavity. It is shown that the certain modes may play an important role in the positioning of the centrosome in the cell. Biological electromagnetic field may contribute to the organization of the structures in the living cell. This paper includes some of the work being done in bachelor thesis of the second author.
Keywords bioelectromagnetism, endogenous electromagnetic field of biological systems, biological resonators, spherical resonator, COMSOL Multiphysics
in [13, 3, 14]. Resonances of microtubules were treated in [15]. None of the mentioned papers dealing with biological structures as resonators have used numerical methods for calculation and visualization of field distribution. To treat the biological cell as resonator we took zeroth approximation of the problem. We consider biological cell a spherical cavity with conducting wall and filled with homogeneous, lossless and isotropic dielectric medium. We provide here analytical treatment of eigenmodes in spherical cavity with conducting wall and homogeneous, lossless and isotropic dielectric medium. Further we employ COMSOL Multiphysics for the numerical calculation of the eigenfrequencies and field distribution in the spherical cell and dividing cell of elliptical shape. We discuss the significance of the results in the terms of organization in living cells in the discussion.
2. Theory 1. Introduction Biological systems generate electromagnetic field. This endogenous electromagnetic field of biological systems can be of broad frequency range. Electromagnetic activity in the range of kHz [1], MHz [2] and optical and UV region [3, 4] has been detected directly. Various experiments of indirect measurement of electromagnetic field of biosystems confirm its existence in the MHz [5] and optical and UV region [6, 7]. Other indirect measurements expect also existence of endogenous electromagnetic field in the GHz [8] and IR region [9, 10]. Fr¨ohlich postulated longitudinal vibrations of electrically polar structures in biological systems [11]. Vibration of electrically polar structure is the principle of the generation of electromagnetic field in biological systems. The field is assumed to participate on the transport of mass particles and molecules as well as spatio-temporal organization in the biosystems. If the structure is pumped with electromagnetic energy, field will distribute according the geometrical and material properties of the structure, fulfilling boundary conditions. Thus eigenmodes of the structure arise. The idea of the eigenmodes in a whole cell is treated in [12] and also mentioned
Solution of electromagnetic field in spherical cavity with perfectly conducting walls and filled with homogeneous, lossless and isotropic dielectric material was found by MacDonald in 1902 [16]. He shown that the field exists in TEnml and TMnml modes in such a cavity. We will show the important steps in the solution of electromagnetic field in conducting sphere with radius a according to [17]. We start with Maxwell equations in the source-free space filled with homogeneous dielectric medium ~ ~ ×E ~ = − ∂B ∇ ∂t
(1)
~ ~ ×H ~ = ∂D ∇ ∂t
(2)
~ ·D ~ =0 ∇
(3)
~ ·B ~ =0 ∇
(4)
~ and D ~ are Constitutive relations for B ~ = µ0 µr H ~ B
~ = ε0 εr E ~ D
(5)
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M. CIFRA, O. LAMPA, BIOLOGICAL ELECTROMAGNETIC CAVITY RESONATORS
where µ0 and µr are permeability of the vacuum and relative permeability of the medium in resonator, respectively. Next in the Eq. 5, ε0 and εr are permittivity of the vacuum and relative permittivity of the medium in resonator, respectively. ~ and H ~ we can write Assuming ejωt time dependence of E
where Pnm (cos(ϑ)) is associated Legendr´ e polynomial √ and after suitable substitution B = R r into Eq. 12 and changing the variable r into kr
~ ∂X ~ = jω X ∂t
where J is Bessel equation of the n + (1/2) order. Thus ψ as solution of Eq. 9 is
(6)
~ is field quantity E ~ or H. ~ From Maxwell’s where X equations we can derive homogeneous wave equation, the Helmholtz equation ~ =0 (∇2 + k 2 )X
(7)
where k is propagation constant or wavenumber, k = ~ is either electric field E ~ or magnetic µ0 µr ε0 εr ω, and X ~ field H. √
Laplace operator ∇2 in spherical coordinate system is given by ∇2 =
1 ∂ 2 ∂ 1 ∂ ∂ (r )+ 2 (sin(ϑ) ) r2 ∂r ∂r r sin(ϑ) ∂ϑ ∂θ 1 ∂2 + 2 2 ( 2) r sin (ϑ) ∂ϕ
(10)
m2 sin2 (ϑ)
=0
(11)
sin(mϕ) cos(mϕ)
According to [17] electric field can be written in terms of some scalar function ψ which satisfies the Helmholtz equation Eq. 7 (18)
Thus we obtain for the field of TE modes Er = 0
Eθ = √
(19)
mA Jn+(1/2) (kr)Pnm (cos(ϑ))sin(mϕ) (20) rsin(ϑ)
A d Eφ = √ Jn+(1/2) (kr) {P m (cos(ϑ))} cos(mϕ) (21) dϑ n r From Maxwell’s equations written in components of spherical coordinates we get magnetic field of TE modes
Hr =
n(n + 1)A Jn+(1/2) (kr)Pnm (cos(ϑ))cos(mϕ) jωµ0 r3/2 (22)
Hθ =
¾
ª A d ©√ rJn+(1/2) (kr) jωµ0 r dr d × {P m (cos(ϑ))} cos(mϕ) dϑ n
(23)
(13) ª mA d ©√ rJn+(1/2) (kr) jωµ0 rsin(ϑ) dr × Pnm (cos(ϑ))sin(mϕ)
Hφ = − θ(ϑ) ∼ Pnm (cos(ϑ)) |m| ≤ n
(17)
(12)
where m, n are constants. The suitable solutions of the Eqs. 10,11,12 for our problem are respectively ½
~ =0 ~r.E
¾
½ ¾ 2R˙ n(n + 1) 2 ¨ R+ +R k − =0 r r2
φ(ϕ) ∼
where A is constant related to the amplitude of field, we chosen cos(mφ) from ejmϕ . It can be done without losing generality of solution. TE modes fulfill following condition
~ = ~r × ∇ψ ~ E
Scalar wave equation will then split into three equations. We will denote the first and second derivative of the function with respect to its corresponding variable with the one and two dots, respectively.
½ ˙ θ¨ + θcot(ϑ) + θ n(n + 1) −
A ψ(r, ϑ, ϕ) = √ Jn+(1/2) (kr)Pnm (cos(ϑ))cos(mϕ) (16) r
(9)
φ¨ + m2 = 0 φ
(15)
(8)
Helmholtz equation is vector equation. We can solve the wave equation by introduction of well-behaved scalar function ψ. To solve the wave equation, we will use the separation of variables. ψ(r, ϑ, ϕ) = R(r)θ(ϑ)φ(ϕ)
√ R(r) ∼ Jn+(1/2) (kr)/ r
(14)
(24)
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Similarly, for TM modes we obtain fnl = Er =
n(n + 1)A Jn+(1/2) (kr)Pnm (cos(ϑ))cos(mϕ) jωεr ε0 r3/2 (25)
Eθ =
ª A d ©√ rJn+(1/2) (kr) jωεr ε0 r dr d × {P m (cos(ϑ))} cos(mϕ) dϑ n
(26)
ª mA d ©√ rJn+(1/2) (kr) jωεr ε0 rsin(ϑ) dr × Pnm (cos(ϑ))sin(mϕ) Hr = 0
Hθ = − √
(27)
(28)
mA Jn+(1/2) (kr)Pnm (cos(ϑ))sin(mϕ) rsin(ϑ) (29)
A d Hφ = − √ Jn+(1/2) (kr) {P m (cos(ϑ))} cos(mϕ) dϑ n r (30) Having the field of TE and TM modes we can derive characteristic equations according to boundary conditions, which say that on the metal surface r = a, the tangetial ~ is equal to 0. For TM modes component of E ©√
ª rJn+(1/2) (kr) r=a = 0
(31)
(35)
where n is the order of the function and l is the order of the root of the Eqs. 33 and 34. Dominant mode, i.e. mode with lowest cut-off frequency, is TM1m1 mode. Since |m| ≤ n and n = 1, m can be −1, 0, 1. Thus the dominant mode is triply degenerated, i.e. three field distributions have same cut off frequency. It is due to the symmetry of the sphere. Cut-off frequency for the dominant mode is fnl =
Eφ = −
c αnl √ 2π εr a
c 2.7437 √ 2π εr a
(36)
For the radius a = 5µm and εr = 1.5 the dominant mode has cut-off frequency 2.139.1013 Hz Eigenmodes of elliptical cavity are not so often treated in literature as spherical cavity eigenmodes. They can be solved analytically via Mathieu functions. We are not going to treat elliptical eigenmodes here. Basically, there will be splitting of degenerated modes compared to spherical eigenmodes. In some cases we can assign eigenmode of spherical cavity to a corresponding elliptical eigenmode. However, field distribution may differ.
3. Simulation We used COMSOL Multiphysics for the numerical calculation of eigenmodes in cavity resonators. Numerical approximation of problems in COMSOL Multiphysics is based on Finite Element Method.
3.1. Spherical cavity Spherical conducting cavity of 5µm radius filled with dielectric medium of εr = 1.5 was modeled.
and for TE modes Jn+(1/2) (kr)|r=a = 0
(32)
Using suitable identities we can rewrite Eqs. 31 and 32 using spherical Bessel functions of the n-th order jn
ka{jn+1 (ka) − jn−1 (ka)} − jn (ka) = 0
(33)
jn (ka) = 0
(34)
TE TM respecand αnl Roots of the Eqs. 33 and 34 are αnl tively. They can be found in mathematical tables or publications dealing with the problem of resonances and eigenmodes in spherical cavities, e.g. in [17]. The resonant frequencies are defined by following equation
Dominant mode TM1m1 electric field distribution of spherical conducting cavity is shown on the Fig. 1. It corresponds to the analytical expression. Highest intensity of the electric field is in the center of the sphere. We are not interested here in absolute values of the field intensity since it depends on the excitation. Excitation is normalized for all modes displayed. Therefore we have not included color bar showing the linear intensity scale. Field lines density was set to be drawn in intensity dependent manner. Dominant mode TM1m1 is triply degenerated. Other two modes with same n and l, but with different m are perpendicular to the depicted mode but with the same cut-off frequency. Numerical estimation of the dominant frequency mode is 2.142.1013 Hz, what differ slightly from analytical calculation 2.139.1013 Hz. It is acceptable difference for our initial purposes. The error in eigenmode frequency is likely to be caused by finite mesh discretization.
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M. CIFRA, O. LAMPA, BIOLOGICAL ELECTROMAGNETIC CAVITY RESONATORS
4. Discussion of results
Fig. 1. Dominant mode TM1m1 electric field distribution of spherical conducting cavity.
Organizing structure of the eukaryotic cell is cytoskeleton. It is composed of actin filaments, intermediate filaments and microtubules. Microtubules are organizers and transport ”highways” of the cell due to microtubule motor proteins, which almost literally ”walk” on the microtubules. Motor proteins are able to transfer macromolecules and organelles. Organizing and nucleating center of the microtubules growth is centrosome, which is relatively large organelle always located near the nucleus, i.e. center of the cell. We want to estimate what is the effect of the endogenous electromagnetic field in cells on the location of the centrosome. We consider centrosome a dielectric neutral particle which has higher permittivity than surrounding cytosol. We see that the field in the treated cavities is nonuniform. Nonuniform field can act on dielectric neutral particles by force due to polarization of particle. This was thoroughly described by Pohl [18]. This effect is called dielectrophoresis. It is principally described by following equation:
3.2. Elliptical cavity We are also interested in the field distribution in elliptical cavity. It is our zeroth approximation for the geometry of dividing cell. Elliptical conducting cavity with semiaxis x = 6µm, y = 4µm, x = 4µm filled with dielectric medium of εr = 1.5 was modeled. Dominant mode of the elliptical cavity is similar to the one of the spherical cavity (not shown here), with the highest intensity in the center of the ellipsoid. It is only doubly degenerated due to rotational symmetry. Its cut-off frequency is 2.29.1013 Hz. Next mode with 2.6.1013 Hz cut-off frequency with similar geometry was split from degeneracy due to longer x semi-axis. Higher TM modes are of special interest for us. They have highest intensity in the focal points, Fig. 2.
F~ ' V (εp − εs )∇E 2
(37)
where F~ is force acting on the dielectric particle, V is volume of the particle, εp is the permittivity of the particle, εs is the permittivity of the surrounding media, E is electric field. The dielectric particle with higher permittivity than its surroundings is forced to move in the direction of gradient of the squared intensity of electric field. It is due the fact that it will have lowest potential energy in the place of highest intensity of electric field.
Fig. 3. Schematically depicted location of the centrosome in the interphase of the cell. Two perpendicular rectangles are centrioles, which nucleate growth of microtubules. Lines represent microtubules.
We can see on the Fig. 3 that the centrosome is located in the middle of the cell, what fulfills our assumptions. During the cell division, the mitotic spindle is formed. It is crucial for the correct distribution of the identical copies of chromosomes to new daughter cells [19]. During the creation of mitotic spindle, both centrosomes moves to foci of the dividing cell. Dividing cell can be roughly approximated by ellipsoid, see Fig. 4. Fig. 2. Electric field distribution of the third TM mode in elliptical electrical conducting cavity. Arrows show direction of electric field vector.
It seems be that electric field acts on the centrosomes by force and helps them to move in the suitable location in the cell.
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Fig. 4. Schematically depicted location of the centrosomes and shape of mitotic spindle during the division of the cell. Two perpendicular rectangles depict centrioles, which nucleate growth of microtubules. Lines represent microtubules.
5. Conclusion Electromagnetic field of biological systems may excite structural eigenmodes in the cell. Nonuniform electric field acts by dieletrophoretic force on the neutral dielectric particles. Position of sufficiently large organelles in cell may be influenced in this manner. It may contribute to organization in the living cell. Further work should employ the lossy dielectric resonators as approximation of the biological cell. It will be more realistic approximation but also more demanding on the computing power. Various perturbations (conducting or dielectric objects, slight deformations) in the resonator can be still treated analytically. They may be introduced into the resonator model to create a more accurate model of the biological cell. More complicated structures and perturbations can be simulated in COMSOL Multiphysics. COMSOL also enables multiphysical simulations, which could include temperature dependence of the molecular processes, analysis of the vibrational states via stressstrain and acoustics expansions of the software. Analytical calculation of the dielectric force acting on larger organelles, e.g. centrosome, is currently being carried out.
Acknowledgements First author thanks to supervisor prof. Vrba for support in PhD program. Dr. Pokorn´y of IPE of AS of the CR is acknowledged for discussions and specialized supervision. Research is supported by Czech Grant Agency under grant No. 102/08/H081 and FRVS grant No. 2300/2008. HUMUSOFT company is acknowledged for the support with the COMSOL Multiphysics.
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About Authors. . . Michal CIFRA graduated in Biomedical Engineering at the ˇ University of Zilina, Slovakia in 2006. Currently he works on his Ph.D. thesis at the Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech Technical University in Prague in cooperation with the Institute of Photonics and Electronics, Academy of Sciences of the Czech Republic. Ondˇrej LAMPA started his higher education studies in 2005 at the Faculty of Electrical Engineering, Czech Technical University in Prague. Presently he works there on his bachelor thesis Cell Structures as Electromagnetic Cavity Waveguides and Resonators at the Department of Electromagnetic Field.
