cellular morphology
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BioSystems 109 (2012) 356– 366
Contents lists available at SciVerse ScienceDirect
BioSystems
jo ur nal homep age : www.elsev ier .com/ locate /b iosystems
lectrodynamic eigenmodes in cellular morphology
. Cifranstitute of Photonics and Electronics, Academy of Sciences of the Czech Republic, Czech Republic
r t i c l e i n f o
rticle history:eceived 29 March 2012eceived in revised form 14 June 2012ccepted 15 June 2012
eywords:elllectromagnetic eigenmodesavity resonator
a b s t r a c t
Eigenmodes of the spherical and ellipsoidal dielectric electromagnetic resonator have been analysed. Thesizes and shape of the resonators have been chosen to represent the shape of the interphase and dividinganimal cell. Electromagnetic modes that have shape exactly suitable for positioning of the sufficientlylarge organelles in cell (centrosome, nucleus) have been identified. We analysed direction and magnitudeof dielectrophoretic force exerted on large organelles by electric field of the modes. We found that theTM1m1 mode in spherical resonator acts by centripetal force which drags the large organelles which havehigher permittivity than the cytosol to the center of the cell. TM-kind of mode in the ellipsoidal resonatoracts by force on large polarizable organelles in a direction that corresponds to the movement of thecentrosomes (also nucleus) observed during the cell division, i.e. to the foci of the ellipsoidal cell.
Minimal required force (10−16 N), gradient of squared electric field and corresponding energy (10−16 J)
of the mode have been calculated to have biological significance within the periods on the order of timerequired for cell division. Minimal required energy of the mode, in order to have biological significance,can be lower in the case of resonance of organelle with the field of the cellular resonator mode.In case of sufficient energy in the biologically relevant mode, electromagnetic field of the mode willact as a positioning or steering mechanism for centrosome and nucleus in the cell, thus contribute to thespatial and dynamical self-organization in biological systems.
. Introduction
Morphogenesis is long-standing and intriguing problem iniology. Based on great amount of experience of embryologists,evelopmental biologists and cell biologists in microscope obser-ations, it is apparent that the shape of the organism, be it single cellr multicellular embryo, is regulated as a whole (Beloussov, 2008).uch a regulation requires that the behavior of one spatial regionf the organism is intimately connected to the behavior of otheregion. Looking for concepts which could represent physical naturef such holistic regulation, the concept of field lends itself directly.ield concept of certain physical quantity can be simply imaginedn a picture in which quantity has certain magnitude (scalar field)r certain magnitude and direction (vector field)1 in every point ofpace and time. Fields are governed, depending on their type, byundamental laws which can be described by mathematical equa-ions. The nature of the field, materials as well as material interfaces
described by boundary conditions) present in spatial domain inhich the field is active restrict the possible field shapes.E-mail address: [email protected] Example of the scalar field is temperature field, example of vector field is electriceld.
303-2647/$ – see front matter © 2012 Elsevier Ireland Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.biosystems.2012.06.003
© 2012 Elsevier Ireland Ltd. All rights reserved.
In electrodynamics and also in physics in general, under the term“mode”, we understand a specific shape of the field having specificfrequency. Eigenmodes2 of the structure are set of modes whichcan be supported by given structure. Concept of eigenmodes is fas-cinating when one realizes that mere structure gives the form tothe input energy. In other words, when the structure is excitedwith (electromagnetic) energy, it will distribute according to thegeometrical and material properties of the structure into the eigen-modes. In technical physics, we usually use solid state materials tosupport modes of interest. Therefore the shape of the structure isnot influenced by the energy of the eigenmode in such rigid nondeveloping (non growing) systems. However, if the system is struc-turally dynamic (growing or dynamically reorganizing) it may beable to respond to the eigenmode shape.
Biological systems have been predicted and observed to gen-erate electromagnetic field in multiple frequency bands from verylow frequency of few Hz up to the optical region as reviewed by Cifraet al. (2011). Electrodynamic forces generated by cellular skeletonhave been predicted to play role in morphogenesis by Kucera and
Havelka (2012). Here we are wondering which shapes can electro-magnetic field assume on cellular level based on the knowledge ofelectromagnetic and geometric properties of biological cells.2 “eigen” – from German meaning “own”, “self”.
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Above mentioned characteristics, when specified, uniquelydetermine set of eigenmodes of the resonator. Eigenmodes of thespherical and spheroidal resonators we treat in this paper can becategorized into TEnml and TMnml modes.5
M. Cifra / BioSyste
Biological cell exhibits rich variety of structures. The idea thatellular cavity structures could function as waveguides and res-nators of electromagnetic field is not new. Actually, there are manyembrane bound organelles (where membrane could impose nec-
ssary boundary conditions) which attracted attention of severalesearchers.
Popp and Rattemeyer treated nucleus with DNA as a resonat-ng structure (Rattemeyer, 1978; Popp et al., 1989; Van Wijk andhen, 2005; Popp and Beloussov, 2003) for interpretation of photonmission from biological systems (biophoton emission).
Cope (1973) treated mitochondria as coaxial resonators for IRegion. The lipid bilayer membrane could serve as a dielectricetween inner and outer conductor (water regions surroundingembrane are rather lossy in IR region). It was proposed that theave guiding property of mitochondrial membrane could serve
or transport of the energy along mitochondrion by means of IRhotons.
Possible resonances of microtubules were treated by Jelínek andokorny (2001). Cut-off frequency for the first waveguide modes oficrotubule lies in soft X-ray region. It was speculated that this
ind of modes could interact with biologically significant atoms oficrotubules and contribute to their dynamic behavior.Pietak just recently studied geometrical correlation of selected
lectromagnetic eigenmode patterns in leaves (Pietak, 2011a,b)nd developing fruits (Pietak, 2012) with the morphology of thoseissues. Frequencies of the eigenmodes were found approximatelyn the EHF region – extremely high frequencies (30–300 GHz). Strik-ng geometric correlations of eigenmode shapes with the tissue
orphology were considered a structural evidence for the role elec-romagnetic eigenmodes in plant and fruit tissue morphogenesis.
In our earlier work (Cifra and Lampa, 2008; Cifra, 2010, 2009),e proposed that certain cellular electromagnetic eigenmodes maylay role in positioning of large organelles. We analysed the eigen-odes of cell under approximation of cell as a cavity with perfectly
onducting boundary and filled with homogeneous, lossless andsotropic dielectric medium. This analysis enabled us to find thehapes of electromagnetic cellular eigenmodes. However, approx-mation with perfectly conducting boundary did not correspond
ith the electric properties of the living cell and did not enable uso determine the quality factor of the modes in a real cell, because
odel of resonator with perfectly conducting boundary and loss-ess medium leads to infinite quality factor.
In the first part of this paper, we extend our earlier work and pro-ide here analytical treatment of eigenmodes in spherical dielectricesonator with lossy and isotropic dielectric medium. Further wemploy COMSOL Multiphysics3 for the numerical calculation of theigenmodes and field distribution in the ellipsoidal resonator. Ana-ytical methods used for analysis of the field in spherical resonatorre standard in classical electrodynamics and described in manyextbooks and papers such as (Balanis, 1989; Gastine et al., 1967;ulien and Guillon, 1986; Yadav and Singh, 2004). In the secondart of this paper, we analyze the results in terms of contributiono organization in living cells.
. Positioning of Centrosome and Nucleus
Centrosome is organelle present in many types of cells ands responsible for nucleation of microtubule growth. Therefore
entrosome behaves as microtubule organizing center. Sinceicrotubule cytoskeleton is important for organization of struc-ures and processes within the cell, the position of centrosome asicrotubule nucleator is crucial for the cell function. There is well
3 www.comsol.com.
(2012) 356– 366 357
known observation that the centrosome always tends to be posi-tioned near to or in the center of the nondividing cell and associatednear the nucleus (Alberts et al., 2008; Euteneuer and Schliwa, 1992)as depicted in Fig. 1.
A model for the centrosome and nucleus positioning behaviorbased on force exerted by microtubule polymerization in confinedspace has been proposed to explain the centrosome and nucleuscentering by Reinsch and Gonczy (1998), Holy et al. (1997) andDaga et al. (2006), yet the exact mechanisms enabling this in vivoare not fully elucidated.
During the cell division the mitotic spindle, crucial structure forthe correct distribution of the identical copies of chromosomes tonew daughter cells (Alberts et al., 2008), is formed. In animal cells,mitotic spindle (assembled from microtubules) nucleates from thetwo divided centrosomes. During the creation of mitotic spindle,both centrosomes move to foci of the dividing cell. Dividing animalcell can be roughly approximated by ellipsoid, see Fig. 2.
Here we propose that certain eigenmodes of the cell as cav-ity electromagnetic resonator play role in positioning of organellessuch as nucleus and centrosomes.
3. Fundamentals of Cavity Resonator Electromagnetics
Physically, following three characteristics are crucial in the anal-ysis of electromagnetic cavity resonator:
• Inner bulk material in the volume of the resonator. This is usuallydescribed in optics by index of refraction p = √
εr�r . εr is relative4
permittivity (also termed as dielectric constant) and �r is rela-tive permeability, this pair of quantities is preferentially used inradiofrequency physics. These quantities are frequency depen-dent in general. �r describes interaction of the electric componentof the electromagnetic field with matter and �r the interaction ofthe magnetic component of the electromagnetic field with mat-ter. Both εr and �r of the material can be complex numbers. Theirimaginary parts indicate that electromagnetic field is absorbedby the medium and transformed to another form of energy (e.g.motion of electrically charged particles, which, if it is disordered,is macroscopically observed as heat). The ratio of imaginary andreal part of permittivity is often expressed as tan ı = ε′′/ε′, ε′′ beingimaginary part and ε′ being real part of the permittivity. Highertan ı means higher losses of the material
• Boundary of the resonator. Electromagnetic field obeys bound-ary conditions at the interface of the two materials (e.g.metal–dielectric, dielectric–dielectric). Boundary conditionsstem from the fact that certain field components must to be con-tinuous (without the “jump” in the value) across the interfacein order for the field to be physically real. Boundary conditionsused in calculation significantly influence the field shape in theresonator and need to be carefully treated in order to describephysical reality.
• Size and shape of the resonator. They both influence the frequen-cies and shape of the field in the resonator.
4 Relative to the permittivity of the vacuum.5 TE stands for Transverse Electric (EM field has electric component transversal
to the radius vector, i.e. Er = 0), TM stands for Transverse Magnetic (EM field hasmagnetic component transversal to the radius vector, i.e. Hr = 0), subscripts n, m,and l are found in the field equations (e.g. Gastine et al., 1967). Basically, the higherthe mode numbers nml, the higher is number of field maxima and minima along
358 M. Cifra / BioSystems 109 (2012) 356– 366
Fig. 1. (a) Human osteosarcoma cell fixed with methanol and stained with polyclonal antibody to alpha-tubulin (green) and monoclonal antibody TU-30 to gamma-tubulin(red). DNA is stained by DAPI. (Photography E. Draberova, IMG AS CR, Prague) (b) Schematically depicted location of the centrosome in the interphase of the model sphericalcell. Two perpendicular rectangles are centrioles, which nucleate growth of microtubules in animal cells. Lines represent microtubules. Nucleus is not depicted here.
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ig. 2. (a) Human osteosarcoma cell fixed with methanol and stained with polyclonred). DNA is stained by DAPI. (Photography E. Draberova, IMG AS CR, Prague) (b) Scivision of the cell. Two perpendicular rectangles depict centrioles, which nucleate
Eigenmodes in the resonators have three important parameters:
shape of the field (given by field equations)frequency at which they can be excitedquality factor Q - it is ratio of how much energy can be storedin the resonator compared to input energy rate, in other words,ratio of energy stored in the resonator and energy lost per cycletimes the frequency, see Eq. (1).
Quality factor Q for dielectric resonator is defined as
= ωenergy stored
energy lost per cycle= ω
W
Pd + Pr(1)
here ω = 2�f, f being frequency, W is energy stored in resonator,denotes dielectric losses and Pr denotes losses due to radiation
df energy from the resonator.
adial and angular directions. Note that the convention in literature is to denote bothrincipal mode number and index of refraction is denoted as n and both imaginaryart of index of refraction and Boltzmann’s constant as k. To avoid the confusion weill use p to denote index of refraction and q to denote the imaginary part of index
f refraction.
ibody to alpha-tubulin (green) and monoclonal antibody TU-30 to gamma-tubulinically depicted location of the centrosomes and shape of mitotic spindle during theh of microtubules in animal cells. Lines represent microtubules.
To combine the contribution of losses using correspondingQ factors, for the dielectric resonator (resonator with dielectricboundaries) we can also write
1Q
= 1Qd
+ 1Qr
(2)
Q with subscripts d, r are related to power losses due to the dielec-tric loss inside the resonator and radiative loss, respectively. Forany resonator and method used, once the frequency f = f′ + if′′ of themode is found, it enables one to calculate the quality factor directlyas
Q = f ′
2f ′′ (3)
Resonator modes (or any oscillatory process in general) can becategorized into three regimes based on its Q factor:
• overdamped, Q < 1/2• critically damped, Q = 1/2, energy stored in the mode is dissipated
into other form approx. within one period of oscillation• underdamped, Q > 1/2.
In the current work, underdamped modes are of our interest,because they can store energy and thus enable formation of theirfield shape.
M. Cifra / BioSystems 109 (2012) 356– 366 359
Table 1Permittivity of the dry biomaterial, water and biomaterial for three frequencies. * - Imaginary part of refractive index q was calculated from absorption coefficient �a asq = �a(�/4�), where � is free space wavelength, ♦ calculated using Eq. (B.1), 75% of material volume being water and 25% dry biomaterial.
Frequency Permittivity
Dry biomaterial Water Biomaterial
1.6 THz p = 2 + i0.2* p = 2.15 + i0.09*↓ ↓ε = 3.94 + i0.96 Wilmink et al. (2011), Fig. 4 ε = 4.61 + i0.41 (muscle tissue,
Wilmink et al. (2011), Fig. 6)
24 THz p = 1.52 + i0.1 ε = 1.2 + i0.55 Ellison(2007), Fig. 25
ε = 1.44 + i0.51♦
↓ε = 2.3 + i0.23lyophilized bacterialcells, Cooper andPowell (1986), 11 �mwavelength (27.3 THz),Fig. 2
50 THz p = 1.5 + i0.1 p = 1.25 + i0.11 ε = 1.7 + i0.29♦
↓ ↓ε = 2.24 + i0.3 see text ε = 1.54 + i0.28
(Wieliczka et al., 1989),−1
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. Studied Models
We created two model resonator geometries to approximatenterphase nondividing cell (spherical model) and dividing cellprolate ellipsoid model). The diameter of size of the modelnterphase cell is a = 7 �m and semiaxes of prolate ellipsoid are
= 9.1726 �m, b = c = 6.1151 �m. Ellipsoidal dividing cell has theame volume as the spherical interphase cell. Capability of spheri-al cells of small (a = 3.3 �m) and large radius (a = 65 �m) to behaves dielectric resonator is also analysed in order to assess generalityf the model.
.1. Dielectric Properties of Resonator
Permittivity of any material is generally dependent on fre-uency. Dielectric properties of the material inside the resonatorwhich represents inner volume of the cell) used in our calcula-ions are based on experimental findings from measurements ofulk biological material. The is no single publication which covershe permittivity data of biomaterial in the whole range of frequen-ies (wavelengths) of interest (1–50 THz). Furthermore, data fromry biomaterials only are often presented.
Generally, water is the most abundant component of the bio-ogical cell, therefore it plays the crucial role in the effectiveermittivity of the cell. Water is strong absorber of THz radiationith two major peaks at 6 and 19.5 THz (Fig. 5 in Arnone et al.,
000). Frequency dependence of the permittivity of the water cane modelled by few relaxation and resonance processes (Ellison,007). For more data on water complex permittivity in THz and IRegion, (see Downing and Williams, 1975; Ellison, 2007; Segelstein,981; Wieliczka et al., 1989).
In the frequency range where no direct data on naturally wetiological material are present, theory of mixture of dielectricaterials is used, see Appendix B, assuming that water volume
raction in cell is 75%. We gathered data on dielectric propertiesf water, biomaterial or dry biomaterial and water for the threerequency ranges: low, middle, high, see Table 1. It was not pos-
ible to obtain the permittivity data for biomaterial in the highrequency range of interest (50 THz, mid IR range) from the litera-ure. Therefore we try to estimate the permittivity values based onhe properties in optical region. It is well known that biomoleculeshave higher refractive index than water in optical region. Pro-tein refractive index is usually in the range of p = 1.5–1.6 (Hand,1935; McMeekin et al., 1962) while extreme case such as tubulinhas p = 2.9 (2002-Mershin-Biosytems). Lipid bilayers have p = 1.5–2(Ohki, 1968). Carbohydrates have p = 1.4–1.5 (Aas, 1996 – Fig. 3).Based on the dielectric relaxation theory, permittivity at lowerfrequencies can assume only higher values, although resonancebehavior may locally perturb this trend. It needs to be noted thatthe permittivity in the mid IR region has rich behavior because itinvolves frequencies of many molecular vibration modes. Based onthe refractive index and permittivity data in the optical region, wemade an estimation of dielectric properties of biological materialin the mid IR (50 THz): p = 1.5 + 0.1i, ε = 2.24 + i0.3.
It needs to be noted that the data on permittivity values comefrom macroscopic measurement of biological material. It may rep-resent some averaged level of losses because field configurationused in technical macroscopic measurement is rather uniform andtherefore substantially different from field configuration in cav-ity resonator. In such macroscopic measurement, field “sees” thebiomaterial oriented rather randomly and that makes it likely toaverage out the losses. In the case non-random orientation of themolecules with respect to the field on the sub-cellular level lossescould be substantially higher or lower.
4.2. Spherical Dielectric Resonator
For the resonator with dielectric boundary, dielectric propertiesof surrounding material are crucially important for the resonatorbehavior. We assume that a watery medium is surrounding the cellresonator. Water refractive index and permittivity has been alreadydiscussed in the previous section.
Using field equations inside and outside the spherical dielec-tric resonator (Gastine et al., 1967) and general electromagneticboundary conditions (see. e.g. Balanis, 1989) lead to characteristicequation for TM and TE modes which is both complex and tran-scendental.
Following characteristic equation is valid for TM modes
Jn−1/2(k1a)
Jn+1/2(k1a)− n
k1a=
√ε1
ε2·
H(2)n−1/2(k2a)
H(2)n+1/2(k2a)
− ε1n
ε2k1a(4)
3 ms 109 (2012) 356– 366
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nd for TE modes
Jn−1/2(k1a)
Jn+1/2(k1a)=
√ε2
ε1·
H(2)n−1/2(k2a)
H(2)n+1/2(k2a)
(5)
is Bessel function and H(2) is Hankel function of the second kind,heir subscript indices denote the order of the functions. Relativeermittivity of resonator is denoted by ε1, relative permittivity ofhe surrounding medium is denoted as ε2. k1 = √
ε1k0, k2 = √ε2k0,
ave constant k0 = 2�f/c0, f is frequency, c0 velocity of light in vac-um, a radius of the spherical resonator. Roots of the equations areomplex frequencies which correspond to the frequencies of theesonator modes. This enables us to calculate the frequency of theodes and further using Eq. (3) also the quality factor of the modes.Additionally to analytical calculations, COMSOL Multiphysics
oftware has been used to calculate the field and also as a tool foreld visualization. COMSOL uses Finite Element Method, a numer-
cal method which can be used also for the calculation of thelectromagnetic field. Eigenmode solver of COMSOL enables cal-ulation of modes of any structure. Since the method is numericalnd the computer memory is limited, there is a limit for numberf points in the space where the field can be calculated. Therefore,he domain where the calculation is to be performed needs to bepatially limited using suitable boundary conditions. We can usempedance boundary condition if we can find the correct value ofhe impedance of the field at the boundary. Wave impedance inpherical coordinates is defined as ratio of perpendicular (to radiusector) components of electric and magnetic fields. Since fields ofhe dielectric resonator are determined analytically by equationsf electric and magnetic field (found e.g. in Gastine et al., 1967) wean use following general equation to calculate impedance Z at theoundary r = a, a being radius of the spherical resonator.
= E⊥H⊥
(6)
here E⊥ is component of the electric field perpendicular to radiusector (i.e. � or in spherical coordinates) and H⊥ is perpendicularomponent of magnetic field.
However, in general, boundary impedance of modes in dielectricesonator differs from mode to mode based on their type (TM or TE)nd mode number n. Therefore, when we set specific numeric valuef boundary impedance in COMSOL or other numerical software, itill be valid only for the mode for which it has been calculated.
f other modes are found by the numerical eigenmode solver, theyay not correspond to any other real mode, i.e. they may not repre-
ent any mode which would be present in physically real dielectricesonator.
.3. Ellipsoid Resonator
Analytical ellipsoid resonator treatment involves spheroidalunctions (Mathieu functions) and is studied in substantiallymaller amount in the literature (for example by Li et al.,002, 2003; Kokkorakis and Roumeliotis, 1997; Mehl, 2009) thanhe spherical case. Definition and implementation of generalpheroidal functions is therefore not standard and requires signifi-ant amount of programming computational efforts such as by Vega2003) and is beyond the scope of this paper as it is partially activeesearch topic on its own. However, numerical methods such asinite Element Method enable analysis of the field in any resonatorhape in principle. Therefore we apply COMSOL Multiphysics eigen-ode solver to calculate the field in ellipsoid resonator.
Boundary impedance for the mode of dielectric ellipsoid res-nator was estimated based on the boundary impedance for theorresponding mode of spherical dielectric resonator radius of
= 8 �m. This radius was chosen to be between the radius of
Fig. 3. Dependence of quality factor due to radiative losses, Qr , of TM1m1 mode onthe permittivity contrast εINT/εEXT .
sphere with volume equivalent to the ellipsoid (a = 7 �m) and the9.1726 �m length of the major axis and 6.1151 �m length of twominor axes of ellipsoid resonator. Estimated boundary impedancefor the mode of interest was Z = 299.96 + i44.913 . When using thismethod, it needs to be noted that the field shape and the frequencyof the modes in ellipsoidal resonator with dielectric boundary isnot exact but it describes major geometric features.
5. Cellular Electromagnetic Modes of Biological Interest
Regarding possible biological function, we found the resonatormodes which have shape spatially correlated with cell structurescommonly observed under microscope. We assume that the patternof the field in the cell determines the occurring structure namelyby attraction of large cellular structures to the region of maximaof electric field intensity as will be explained later. For now onehas to keep in mind that we search for modes having maxima ofelectric field intensity in the location of centrosomes and/or nucleusas depicted in Figs. 1 and 2.
5.1. Spherical Resonator – Nondividing Cell
Spherical resonator represents a nondividing (interphase) cells.In the case of dielectric boundary resonator a new classificationof modes additional to TE and TM modes arises compared to res-onator with perfectly conducting boundary which traps the fieldcompletely inside the resonator. Modes of the dielectric resonatorcan be classified to external and internal modes, as done by Gastineet al. (1967), due to the nature of dielectric resonator. It is typical forexternal modes to have significant portion of energy in field outsidethe dielectric resonator. This is reflected in low quality factor of theexternal modes. It may be educative to find the frequencies andquality factors of external modes, but the external modes cannotstore much energy (their Q factor is ≤ the order of 0.1) due to theirphysical nature and therefore are not very interesting regardingpossible biological role.
In the dielectric spherical resonator, the discovered mode ofinterest is TM1m1. Quality factor due to radiative losses, Qr, is purelydependent on the permittivity contrast εINT/εEXT, where εINT is abso-lute value of permittivity of the resonator dielectric and εEXT isabsolute value of permittivity outside the resonator. See Fig. 3 for
Qr for TM1m1 for various values of permittivity contrasts. This figurewas obtained by calculating the Q factor of TM1m1 from complex fre-quency obtained from Eq. (4) for various values of permittivities.The Qr values do not depend directly on the resonator diameter.
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M. Cifra / BioSyste
or getting the total Q factor of the mode of dielectric resonator,e can use Eq. (2), where we need to know Qr and Qd. Once we
now the value of permittivity of the material inside and outsidehe resonator, we can find the Qr from Fig. 3 and Qd from Eq. (A.2).his estimation does not take into account the fact that the dielec-ric losses (tan ı) in the material inside and outside the resonatoran be different. However, dielectric losses inside and outside theesonator are of the same order of magnitude in our case and mostf the energy of the field is stored inside the resonator so no greateviation from Qd estimation based on the losses in the resonator
s expected.For the spherical cell radius a = 7 �m and permittivity in the mid-
le frequency range (24 THz), TM1m1 has frequency of f = 24.67 THz,uality factor Q = 0.903 (from Eq. (4) and Eq. (3)) and bound-ry impedance Z = 305.04 + i72.718 (from Eq. (6)). Calculationith COMSOL yielded very similar frequency f = 24.81 THz and
= 0.9104. The small deviation from analytical value from Eq. (4)ay be caused by the error of numerical method used (FEM – finite
lement method) which may stem from the finite density of dis-rete tetrahedral mesh. For the depiction of electric field of TM1m1ode see Fig. 4.For the small cell with radius a = 3.3 �m), TM1m1 mode fre-
uency is f = 1.67 THz and Q = 1.3362 and for large radius cella = 65 �m) the TM1m1 has frequency 44.89 THz and Q = 0.8737. Iteems that the TM1m1 modes can be supported within the wholeange of cell sizes. Actually, it is interesting to observe that the res-nances appear to be feasible starting from cca. few THz, i.e. wherehe permittivity ratio εCELL/εWATER crosses the unity and rises abovet.
The mode TM1m1 electric field distribution in dielectric res-nator is shown in Fig. 4. Highest intensity of the electric field isn the center of the sphere. We are not interested now in absolutealues of the field intensity since it depends on the excitation. Exci-ation is normalized for all modes displayed. Therefore we have notncluded color bar showing the linear intensity scale. Mode TM1m1s triply degenerated.6 Other two modes with same n and l, but
ith different m (not depicted here) are orthogonal to the depictedode but with the same resonant frequency. These two degener-
ted modes have maxima of electric field intensity in the center ofhe resonator as well. Actually, external mode TM′
1m1 of dielectricesonator perfectly resembles the shape of TM1m1 mode of con-uctive boundary resonator (see e.g. Cifra, 2010, 2009; Cifra andampa, 2008 for study of field distribution of spherical resonatorith conductive boundary). If we did not accept categorization of
he dielectric resonator modes into internal and external, externalode TM′
1m1 would be denoted as TM1m1 and the internal modee call now TM1m1 would be denoted as TM1m2. The l = 2 woulderfectly explain why we observe rise in intensity along the radiusector after some distance from center because l = 2 means we useecond root of the Bessel function, i.e. second local maximum alongadial direction has to occur within the resonator volume. However,his approach to mode renumbering, although useful for under-tanding in this case, could not be used fully generally becausexternal modes are of different nature than internal modes. Namelyrequency and quality factor of external modes, in contrast to inter-al modes, are rather weakly dependent on dielectric propertiesf inner resonator medium, as explained and comprehensivelyescribed by Gastine et al. (1967).
6 The term degeneracy means that the modes have same resonant frequency butifferent mode shape. This stems from the fact that the resonant frequency does notepend on modal number m, but m does influence shape number and can assume aertain range of integer values limited by n.
(2012) 356– 366 361
5.2. Ellipsoidal Resonator – Model of Dividing Cell
Prolate ellipsoidal resonator represents dividing cell. The modeof interest is that with highest intensity in the both focal pointsof ellipsoid. They have double degeneracy. Distribution of elec-tric field intensity in the dielectric boundary resonator is depictedin Fig. 5. For the ellipsoid of 9.1726 �m length of the major axisand 6.1151 �m length of the minor axes, having permittivity cor-responding to the middle frequency range (Table 1), the discoveredmode of interest has frequency f = 30.108 THz and Q = 1.5589.
6. Effect of the Electric Field on Organelle Positioning
We theoretically quantify the effect of electric field of the modesof interest on cellular morphology in this section. Effect of the fieldon the matter can be mediated by transfer of energy or information.Here needs to be noted that also information needs to be carriedby some minimum amount of energy. Thus, both transfer of energyand information involve force of the field acting on the matter.
6.1. Mechanisms
6.1.1. Electric Field Acting on ChargeElectric field E can act by force F on charge Q as
F = QE (7)
At high frequencies as those involved in discovered modes ofinterest, there will be no net translational force of electric fieldacting on a charge, since the field is oscillatory.
6.1.2. Electric Field Acting on DipoleNonuniform electric field can act by translational force on elec-
tric dipole p which is common electric approximation of many polarmolecules or macromolecular structures and organelles:
F = ∇(p · E) (8)
6.1.3. Electric Field Acting in Induced Dipole – DielectrophoresisEven if the cellular organelle does not have significant electric
dipole moment itself, nonuniform electric field can polarize theorganelle if the organelle permittivity is different from that of thesurrounding medium, thus creating an induced dipole moment. Thephenomenon of force of nonuniform electric field acting on neutraldielectric particle is called dielectrophoresis and is described thor-oughly by Pohl (1978). We consider cellular organelles a dielectricneutral particles which have higher permittivity than the surround-ing cytosol. This can be expected for the high frequencies in theregion of tens of THz calculated for the cellular eigenmodes, sinceorganelles contain proteins and lipids which have higher opticalpermittivity than water.
The dielectrophoretic force is approximately given by
F � V(εp − εs)∇E2 (9)
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 thesurrounding media; ∇ denotes the gradient; E is electric field.
The dielectric particle with higher permittivity than its sur-roundings is forced to move in the direction of gradient of the
squared intensity of electric field. It is due to the fact that the par-ticle will have lowest potential energy in the location of highestintensity of electric field. In the following, we will analyse the effectof dielectrophoretic force on the organelle positioning.
362 M. Cifra / BioSystems 109 (2012) 356– 366
Fig. 4. TM1m1 mode electric field distribution in dielectric spherical cavity resonator, radius a = 7 �m, f = 24.67 THz (Q = 0.903) (a) plane x = 0. White arrows depict electricfield direction. (b) Quasi 3D view in three orthogonal cutting planes.
F ric cav( view
6
dmtB(
F
Dpa
o
D
t
ig. 5. Electric field distribution of the TM kind of mode in prolate ellipsoidal dielectQ = 1.5589) (a) Plane z = 0. White arrows depict electric field direction (b) Quasi 3D
.2. Minimal Required Force
In our current study, in the case when the organelle or particleoes not expend its own energy sources for translational move-ent, the force exerted by electric field needs to reach certain
hreshold above which it will “overcome” effective force of therownian motion.7 The threshold force is given by Hölzel et al.2005) as
th =√
2D �t
kT (10)
is the diffusion constant of the organelle (particle); �t is timeeriod of the acting force; k and T are Stefan–Boltzmann constantnd temperature.
Diffusion constant for rigid sphere (approximation of the
rganelle) is given by= kT
viscous drag= kT
6��r(11)
7 Stronger the deterministic force, smaller the deviation of probability distribu-ion of particle localization from the mean position.
ity resonator, major axis a = 9.1726 �m, minor axes b = c = 6.1151 �m. f = 30.108 THzin three orthogonal cutting planes.
� is dynamic viscosity (we take that of water (1 mPa s) as approx-imation) r is radius of the particle (we take 150 nm as reasonableapproximation of centrosome radius (Ueda et al., 1999), for nucleusit would be about 500–3000 nm)
For the temperature of 293 K the threshold force to overcomeBrownian motion is
Fth = 1.513 × 10−15 N for�t = 10 s (12)
Fth = 4.783 × 10−16 N for�t = 100 s (13)
Fth = 1.513 × 10−16 N for�t = 1000 s (14)
The dielectrophoretic force on the spherical particle in inhomo-geneous electric field is given exactly by Pohl (1978) and Hölzelet al. (2005)
FDEP = 2�r3Re{εm × CM factor}∇E2 (15)
where CM factor is Clausius–Mossotti factor
CM factor = εp − εm
εp + 2εm(16)
εp and εm being complex permittivity of particle and medium,respectively.
For example, at 24 THz taking εp = 2.3 + i0.23 (permittivityof protein rich particle) and εm = 1.44 + i0.51 (permittivity of
ms 109 (2012) 356– 366 363
mtfs1
6
fiaye1bb
c
W
w
u
wSivW
lr
6
me
u
mEimetelficmbibdtoA
m
Fig. 6. Spectral distribution of thermal energy, frequency dependence. Frequency
M. Cifra / BioSyste
aterial inside the resonator), we get 2�r3Re{εm × CM fac-or} = 4.7856 × 10−32. To reach and overcome the threshold forceor �t (duration of dielectrophoretic force) of 100 s8 gradient ofquared intensity of electric field needs to assume the value of016 V2 m3.
.3. Energy in the Cell
In order to know possible intensity of electric field and electriceld gradient of the modes, we first need to assess the energy avail-ble in the cell. The total thermal power produced by the cell (ofeast cell type which has typical radius a = 4 �m) was determinedxperimentally using calorimetry by Lamprecht (1980) to be about0−13 W. That means that the energy of about 10−13 J is producedy cellular chemistry per second. This can be considered an upperoundary of possible energy production rate in the cell.
Thermal electromagnetic energy based on Planck’s law can bealculated using equation
EMtherm = bT4V (17)
here V is volume and T is temperature in Kelvin.
= bT4 = 8�5k4
15(hc)3T4 = 7.5502 × 10−16T4 (18)
here u is thermal radiation energy density [J/m3], k istefan–Boltzmann’s constant, h is Planck’s constant and c is veloc-ty of light in vacuum. For the cell with radius a = 7 �m, i.e.olume V = 4
3 �a3 m3 in room temperature (T = 296.15 K = 23 ◦C)EMtherm = 8.34 × 10−21 J.Thermal electromagnetic energy WEMtherm ∼ 10−20 J forms a
ower boundary of the range of possible energy available in theesonator.
.3.1. Spectral Properties of the Excitation EnergyThe most trivial possible case of energy distribution is ther-
al Planck spectral energy distribution depicted in Fig. 6 based onquation
(f )df = df8�hf 3
c3
1ehf /kT − 1
(19)
It is interesting to observe that spectral distribution of ther-al electromagnetic energy which is omnipresent under common
arth temperature conditions (0–37 ◦C) has peak frequency exactlyn the range of first electromagnetic resonant modes of cells of com-
on sizes. In other words, the wavelength of omnipresent thermallectromagnetic radiation on Earth is just the same as the diame-er of most cells (Fig. 6). What could be other possible sources oflectromagnetic field in the cells for the modes of interest whichie in frequency region of 1–50 THz (i.e. at the border of middle andar infrared region)? From molecular point of view, stretch, bend-ng, rotation of molecular groups lie in this region, as is also thease in proteins as reviewed by Barth (2007). Compared to otherolecules, proteins are special in that they can convert chemically
ound energy from phosphate bonds of nucleotides as ATP and GTPnto the other types of energy. IR generation and radiation fromiological systems of above thermal level is poorly explored, partlyue to the experimental difficulties, because it can be hard to dis-
inguish the IR radiation just due to the macroscopic temperaturef the object and due to processes which generate IR nonthermally.s only work known to the author, Fraser and Frey (1968) detected8 100 s is reasonably short time compared to duration of the cell division whichay take from tens of minutes to hours
axis is in logarithmic scale with base 10, i.e. 13 correspond to 1013 Hz = 10 THz (freespace wavelength 30 �m) and 13.5 corresponds to 1013.5 Hz = 31.6 THz (free spacewavelength 10.53 �m).
IR radiation from active crab nerve in the region of 1–10 �m. Emis-sion intensity was 2 orders of magnitude higher than that of thethermal radiation. This detected intensity of IR radiation is proba-bly specific to the neuron information transfer function but may beused as a guiding value for biologically feasible IR electromagneticfield intensities. Thus, keeping in mind the total thermal energy inthe volume of the cell 10−20 J and that the detected IR radiationfrom nerve cells is 2-3 orders of magnitude higher than that, wemay assume that the energy content in the cell in infrared spectrumof approximately 10−18 ÷ 10−16 J is within quite feasible range.
6.4. Electric Field Gradients of Biologically Important Modes
Now we know the range of possible energy content in the cell.Energy of the electric field E in the volume V can be calculated as
W = 12
∫ ∫ ∫V
εEE∗ dV (20)
where ε is relative permittivity, E, and E* is electric field intensityand its complex conjugate, V is the volume of the resonator.
We are able to calculate total energy of electric field of everymode numerically in COMSOL (based on Eq. (20)). From that wecan deduce the intensity of electric field E and gradients of electricfield and vice versa. We have found that to obtain electric field gra-dients sufficient to overcome Brownian motion within 100 s (usingassumptions in Section 6.2) the energy of the mode has to be atleast on the order of 10−16 J. Gradient of the electric field squared(important parameter for dielectrophoretic force) has been calcu-lated in COMSOL for the energy of mode of 10−16 J and is depictedin Fig. 7.
In the case of spherical resonator the electric field of mode ofinterest acts by dielectrophoretic force to position centrosome orother protein rich (high permittivity), large organelles to the centreof the cell, i.e. the force is centripetal. This location of centrosomeand nucleus is observed exactly in nondividing cell schematically
depicted in Fig. 1. In the case of ellipsoidal resonator, the electricfield of mode of interest acts by dielectrophoretic force to posi-tion centrosome or other protein rich (high permittivity), largeorganelles to the foci of the cell. This location of centrosomes and
364 M. Cifra / BioSystems 109 (2012) 356– 366
Fig. 7. (a) Direction (black arrows) and relative magnitude of dielectrophoretic force given by the gradient of squared electric field �F∼∇E2. White color indicates zones where∇ g. 4, x
niodm
tcsHmzf
7
pnid
oepctbd<st1cpafE
F
w
m(
E2 drops below 1016 V2 m3. (a) For spherical dielectric resonator mode (see also Fi
uclei is observed exactly in dividing cell schematically depictedn Fig. 2. Smaller organelles or macromolecules will be affectednly by stronger fields (gradients) since the dielectrophoretic forceepends on the volume of the particle it acts on, here organelle oracromolecule.In both spherical and ellipsoidal dielectric resonator (Fig. 7),
here are zones within the cell volume where the force acts inentrifugal directions, i.e. away from the centre of the nondividingpherical cell or away from the foci of the ellipsoidal dividing cell.owever, the force is rather small in those regions. Therefore, ther-al forces or other mechanisms can move the organelle from the
one of centrifugal dielectrophoretic force to the zone of centripetalorce.
. Discussion
The mechanism of large organelle (centrosome and/or nucleus)ositioning by force exerted by electric field of specific electromag-etic modes of the cell was explained. However, there are few more
ssues which could not be covered in current paper and should beiscussed.
First is the issue of minimum required energy in the mode inrder for it to exert sufficient force on the organelle through thelectric field. Expression for dielectrophoretic force on sphericalarticle studied in current paper is given by Eq. (15). To obtain suffi-ient force, either the gradient of the squared electric field (relatedo total energy in the cell) or the Clausius–Mossotti factor has toe large enough. For spherical organelle (particle) on which theielectrophoretic force is acting, CM factor is limited to the range−0.5, 1> (Morgan and Green, 1997). If the particle does not havepherical shape, or its polarization is not isotropic, CM factor (nowermed polarization factor K(ω)) is not limited by interval <−0.5,> as for spherical particle but may assume values higher than 1. Inase of highly anisotropic polarization or prolonged shape (such asrolate ellipsoid with semi-axes a » b = c) and long axis of particle orxis with high polarization parallel to the electric field, polarizationactor K(ω) reduces to (εp − εm)/εm)) (Morgan and Green, 1997), i.e.q. (15) becomes
DEP = 2�abc
3Re{εmK(ω)}∇E2 (21)
here K(ω) = εp − εm/εm is polarization factorIf permittivity of the particle (εp) is much higher than the per-
ittivity of the medium (εm), gradient of squared electric field∇E2) may be smaller than those described as minimal required at
= 0 plane. (b) For prolate ellipsoid dielectric resonator mode Fig. 5, z = 0 plane.
the end of Section 6.2 while the dielectrophoretic force will be suf-ficient to exert positioning effect on organelles within reasonabletime scale. Large difference (εp − εm) can occur in the case of highquality (Q »1) resonance of particle with external field. In technicalphysics, metamaterials exhibit resonances which increase effectiverelative permittivity of the material to the order of 104 as measuredby Krupka (2008). However, if such analogies of technical metama-terials exist in biology is just currently being researched, e.g. byGiakos (2010). If such resonant behavior of organelle or macro-molecule occurred, this would decrease minimal required energycontent in the electromagnetic mode of interest (also gradient ofsquared electric field) to have biological significance. According toEq. (21), if K(ω) = N, the dielectrophoretic force FDEP acting on ellip-soid is greater by factor N compared to the spherical particle withsame volume and with CM factor = 1, while the gradient of squaredelectric field ∇E2 is kept same.
Second issue to discuss is the coherence of electromagneticmodes. In principle, also incoherent field is capable of excitation ofelectromagnetic resonator modes. For the mode of interest in non-dividing spherical cell, even if the excitation of mode is incoherentbut at the correct frequency, the maximum of intensity of electricwill be always located in the center of the cell and the dielec-trophoretic force will always act in centripetal direction. Same isvalid for the mode of interest in the case of diving ellipsoidal cell.Coherence of the excitation would be very important if the inputenergy rate (pumping) of the mode was small and the quality factorof the mode was high so that the energy could sum up. Excitationshould be coherent for efficient summing up of the energy. Other-wise, if the phase relationships are random (incoherent excitation),energy would not sum up effectively.
Third issue is the mode selection. In radioengineering, mode isselected by running the excitation current along the electric fieldline of the mode to be excited and preferentially in the location ofmaximum of electric field of the mode. It may be the case that fieldgenerators in cell have orientation which prefer the specific modedependent on the cell cycle. It also may be that the high excita-tion of specific mode is triggered only in certain cell cycle phasesor under specific situations to steer the organelle motion. Syner-gistic mechanisms of these processes, if exist at all, remain to bediscovered. For future theoretical work, the higher modes of thebiological cell dielectric resonator are of the interest. One reason is
that these modes lie in the optical region and it is well documentedthat metabolically active cells are source of the optical radiation(Popp et al., 1992; Popp and Beloussov, 1996, 2003; Beloussov et al.,2000, 2007; Musumeci et al., 2003; Van Wijk and Shen, 2005; Cifra
ms 109
esbmrrihttitrtttmtdoVdao
etotanccipnofi(B
8
drpei
etwtmoit
ic(iS
M. Cifra / BioSyste
t al., 2011; Rastogi and Pospísil, 2010; Prasad and Pospísil, 2011)o there is no doubt about the source and excitation of those modesased on energetic reasons. Further, it is well known that the higherodes (WGM – Whispering Gallery Modes) in spheroidal dielectric
esonators may reach very high quality factors (Q � 103 ÷ 1013) aseviewed e.g. by Chiasera et al. (2010) because they are only lim-ted by dielectric losses and scattering within the resonator. Theigh Q factors of WGM can be easily understood using ray propaga-ion approach. Higher modes in spheroidal resonators concentrateheir energy near the boundary which corresponds to wave whichs incident on the boundary well beyond the critical angle so theotal internal reflection occurs.9 The WGM reaching the visibleegion could realistically have high Q stemming from the WGMype of propagation and low absorption loss in watery cytosol inhe visible spectral region. They could store the energy and effec-ively transmit it together with the information along the cellular
embrane. Microtubules can be bound to membrane connectinghe centrosome which is located near the nucleus. Nucleus has theimensions to support the dominant electromagnetic modes in theptical region as studied by (Rattemeyer, 1978; Popp et al., 1989;an Wijk and Shen, 2005; Popp and Beloussov, 2003). Mitochon-ria which are situated along the microtubules have been studieds optical waveguides by Thar and Kühl (2004) and are also sourcef photon emission (Batyanov, 1984; Stauff and Ostrowski, 1967).
Electric field of the specific cellular eigenmodes does not nec-ssarily need to organize the cell organelles only by exertinghe force on the them which results in net translation. Complexrganelles may be sensitive, especially in the case of centrosome,o the electric field gradient which could provide the informationbout the position. In such a case, organelles must possess mecha-isms for the processing of the information while the heavy workould be exerted by molecular mechanical means by dynamicalytoskeleton movement which involves motor protein action. Thisdea is congruent with the findings of Albrecht-Buehler, who pro-osed based on his experiments that motile cells have rudimentaryear infrared vision, where the detectors could be perpendicularlyriented centrioles in cell centrosomes. Infrared electromagneticelds was found to play a role in mutual cellular interactionsAlbrecht-Buehler, 1992, 1991, 2005). For review see (Albrecht-uehler, 2010).
. Conclusion
Eigenmodes of the spherical and ellipsoidal electromagneticielectric resonator have been analysed. The size and shape of theesonators have been chosen to represent the shape of the inter-hase and dividing cell. Electromagnetic modes that have shapexactly suitable for positioning of the sufficiently large organellesn cell (centrosome, nucleus) have been identified.
We analysed direction and magnitude of dielectrophoretic forcexerted on large organelles by electric field of the modes. We foundhat the TM1m1 mode in spherical resonator acts by centripetal forcehich drags the large organelles with the higher permittivity than
hat of the cytosol to the centre of the cell. Specific TM-kind of
ode in the ellipsoidal resonator acts by force on large polarizableranelles in a direction that corresponds to the movement observedn the movement of the centrosomes during the cell division, i.e. tohe foci of the ellipsoid cell.
9 This kind of propagation (along the curved boundary with very small losses)s used to explain the effective propagation of the weak sound (whisper) along theurved walls of galleries. Hence, they are often termed as Whispering Gallery ModesWGM). They were first observed in acoustic domain. Lord Rayleigh was the first tonterpret them in terms of efficient reflection on the curved walls of the dome ofaint Paul’s Cathedral in London.
(2012) 356– 366 365
Energy supply for the excitation of the eigenmodes has beentheoretically analysed. Minimal required force (10−16 N), gradientof squared electric field and corresponding energy (10−16 J) of themode that would have biological significance within the time ofthe order magnitude of time required for cell division have beendetermined. Minimal required energy of the mode for the biolog-ical action to take place can be lower in the case of resonance oforganelle with the field of the cellular resonator mode.
In case of sufficient energy in the biologically relevant mode,electromagnetic field of the mode will act as a positioning orsteering mechanism for centrosome and nucelus in the cell, thuscontributes to spatial and dynamical self-organization in biologicalsystems.
Acknowledgment
Author acknowledges support from Czech Science Agency,projects n. P102/10/P454 and P102/11/0649. Discussions with JiríPokorny are deeply appreciated.
Appendix A. Power Losses in Resonator and Related QFactors
The power Pd dissipated in the resonator in the medium due todielectric losses can be written for both TE and TM modes as
Pd = d
∫ ∫ ∫V
EE∗ dV = ωW tan ı (A.1)
where d is effective conductivity due to dielectric losses and tan ıis the loss tangent. RHS of the equation stems from the usage of Eq.(20) and writing d = ωε tan ı
Resonator quality factor assuming only dielectric losses can bethus written as
Qd = ωW
Pd= 1
tan ı= ε′
ε′′ (A.2)
Appendix B. Mixture of Dielectric Materials
We assume simple model of the mixture of two materials: thehost and the included material (Sihvola, 2000, 2005). Relative per-mittivity of the more abundant host material is denoted as εe andthe permittivity of the material included is denoted as εi. The frac-tional volume occupied by the included material is f. Then thefractional volume occupied by the host material is 1 − f. It is fur-ther assumed that the particles of included material are sphericalwhile they need not be of the same size if only all of them are smallcompared to the wavelength. Then, the Maxwell Garnett mixingrule can be used:
εeff = εe + 3fεeεi − εe
εi + 2εe − f (εi − εe)(B.1)
where εeff is effective permittivity of the mixture of the two mate-rials.
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