Linear dielectric response of clustered living cells
aa r X i v : . [ phy s i c s . b i o - ph ] F e b Linear dielectric response of clustered living cells
Titus Sandu, Daniel Vrinceanu, and Eugen Gheorghiu International Center for Biodynamics, Bucharest, Romania ∗ Department of Physics, Texas Southern University, Houston, Texas 77004, USA (Dated: June 7, 2018)
Abstract
The dielectric behavior of a linear cluster of two or more living cells connected by tight junctionsis analyzed using a spectral method. The polarizability of this system is obtained as an expansionover the eigenmodes of the linear response operator, showing a clear separation of geometry fromelectric parameters. The eigenmode with the second largest eigenvalue dominates the expansion asthe junction between particles tightens, but only when the applied field is aligned with the clusteraxis. This effect explains a distinct low-frequency relaxation observed in the impedance spectrumof a suspension of linear clusters.
PACS numbers: 41.20.Cv, 87.19.rf, 87.50.C- ∗ Electronic address: [email protected] . INTRODUCTION Particle polarizability governs the electric response for many inhomogeneous systemsranging from biological cells to plasmonic nanoparticles and depends strongly on both itsdielectric and geometric properties. Analytical models have been reported [1, 2] only forspherical and ellipsoidal geometries, whereas more complex geometries have been approachedby direct numerical solution of the field equations using, for example, the finite differencemethods [3], the finite element method [4], the boundary element method [5, 6], or theboundary integral equation (BIE) [7].In a simplified representation, biological cells can be regarded as homogeneous particles(cores) covered by thin membranes (shells) of contrasting electric conductivities and permit-tivities. Complex geometries occur when cells are undergoing division cycles (e.g. buddingyeasts) or are coupled in functional tissues (e.g. lining epithelia or myocardial syncytia).In these cases, the dielectric/impedance analysis of cellular systems is far more complicatedthan previous models [8–10], which considered suspensions of spherical particles. Intriguingdielectric spectra [11] reveal distinct dielectric dispersions with time evolutions consistentlyrelated to tissue functioning or alteration, identifying a possible role of cell connectors (gapjunctions) in shaping the overall dielectric response.A direct relation between the microscopic parameters and experimental data can beanalytically derived only for dilute suspensions of particles of simple shapes, and is ratherchallenging for system with more realistic shapes, where only purely numerical solutions havebeen available. In this work we demonstrate that a spectral representation of a BIE providesthe analytical structure for the polarizability of particles with a wide range of shapes andstructures. The numerically calculated parameters encode particle’s geometry informationand are accessible by experiments.By using single and double-layer potentials [12], the Laplace equation for the fields insideand outside the particle is transformed into an integral equation. A spectral representationfor the solution of this equation is obtained providing the eigenvalue problem for the linearresponse operator is solved. Although not symmetric, this operator has a real spectrumbounded by -1/2 and 1/2 [13–15] and its eigenvectors are orthogonal to those of the conjugatedouble-layer operator. A matrix representation is obtained by using a finite basis of surfacefunctions. 2he true advantage of the spectral method is that the eigenvalues and eigenvectors of theintegral operator provide valuable insight into the dielectric behavior of clusters of biologicalcells. The eigenvectors are a measure of surface charge distributions due to a field. Onlyeigenvectors with a non-zero dipole moment contribute to the polarizability of the particle.We call these dipole-active eigenmodes. An effective separation of the geometric and mor-phologic properties from dielectric properties is therefore achieved [16]. We also show thatfor a particle covered by multiple confocal shells, the relaxation spectrum is a sum of Debyeterms with the number of relaxations equal to the number of interfaces times the numberof dipole-active eigenvalues. This is a generalization of a previous result [17] on cells ofarbitrary shape.Our method is related to another spectral approach which uses an eigenvalue differentialequation [18–21]. This method has been applied to biological problems by Lei et al. [22]and by Huang et al. [23]. These authors, however, considered homogeneous cells with muchsimpler expression for cell polarizability. The BIE spectral method seeks a solution on theboundary surface defining the particle, as opposed to the eigenvalue differential equation,where the solution is defined in the entire space.In a previous study on double (budding) cells it was shown that before cells separation anadditional dispersion occurs [24]. Moreover, in recent papers [3, 25] numerical experimentshave shown that the dielectric spectra of a suspension of dimer cells connected by tightjunctions exhibit an additional, distinct low-frequency relaxation. Our numerical calculationshows that the largest dipole-active eigenvalue approaches the value of 1/2 as the junctionbecome tighter. Although the coupling of this eigenmode with the electric field stimulusis relatively modest (the coupling weight is about 1-2 %), this eigenmode has a significantcontribution to the polarizability of clusters. Thus the eigenmodes close to 1/2 induce anadditional low-frequency relaxation in the dielectric spectra of clustered biological cells eventhough the coupling is quite small. Needle-like objects, such as elongated spheroids or longcylinders, have similar polarizability features.In this paper we consider rotationally symmetric linear clusters made of up to 4 identi-cal particles covered by thin insulating membranes and connected by junctions of variabletightness. Convenient and flexible representations for the surfaces describing these objectsare provided. The number of relaxations in the dielectric spectrum of the linear clusters,their time constants and their relative strengths are analyzed in terms of the eigenmodes of3he linear response operator specific to the given shape.
II. THEORYA. Effective permittivity of a suspension
We consider a suspension of identical, randomly oriented particles of arbitrary shape anddielectric permittivity, ε , immersed in a dielectric medium of dielectric permittivity, ε .The dielectric permittivities are in general complex quantities and the theory described hereapplies also for time-dependent fields, providing that the size of a particle is much smallerthan the wavelength. When an applied uniform electric field interacts with the suspension,the response of the system is linear with the applied field and an effective permittivity forthe whole sample can be measured and is defined by [25–27]: ε sus = ε + f αε − f α . (1)This result is obtained in the limit of low concentration, weak intensity of the stimulus field,and using an effective medium theory within the dipole approximation. Here f = N V /V isthe volume fraction of all N particles, each of volume V , with respect to the total volumeof the suspension V . The averaged normalized polarizability α of a particle is defined as[25, 27, 28] α = 14 π V Z V Z Ω N (cid:18) ε − ε ε (cid:19) E ( N ) · N d Ω N dV (2)where E ( N ) is the electric field perturbation created inside the particle under a normalizedapplied electric excitation with direction N and d Ω N is the solid angle element generatedby that direction. The above normalized polarizability is dimensionless and is obtained bymultiplying the standard polarizability of a particle with the factor 4 π/V . In the followingwe will refer only to normalized polarizability, thus, without any confusion, the normalizedpolarizability α will be simply called polarizability. The directional average in (2) is equiv-alent to the averaged sum over three orthogonal axes due to the fact that the problem islinear with respect to the applied field. The latter is more convenient from computationalpoint of view.The electric field inside a particle is obtained by solving the following Laplace equation4or the electric potential Φ: ∆Φ ( x ) = 0 , x ∈ ℜ \ ΣΦ | + = Φ | − , x ∈ Σ ε ∂ Φ ∂ n (cid:12)(cid:12) + = ε ∂ Φ ∂ n (cid:12)(cid:12) − , x ∈ ΣΦ → − x · N , | x | → ∞ (3)where ℜ is the euclidian 3-dimensional space and Σ is the surface of the particle. Thederivatives are taken with respect to the normal vector n to the surface Σ.Due to the mismatch between the polarization inside and outside the object, electriccharges accumulate at the interface Σ and create an electric potential which counteracts theuniform electric field stimulus. The solution of the above Laplace problem (3) is thereforeformally given by Φ( x ) = − x · N + 14 π Z Σ µ ( y ) | x − y | d Σ( y ) (4)The single layer charge distribution µ induced by the normalized electric field is a solutionof the following BIE, obtained by inserting solution (4) in equations (3) µ ( x )2 λ − π Z Σ µ ( y ) n ( x ) · ( x − y ) | x − y | d Σ( y ) = n ( x ) · N (5)Here the parameter λ = ( ε − ε ) / ( ε + ε ) isolates all the information regarding the dielectricproperties for this problem.On using the linear response operator M that acts on the Hilbert space of integrablefunctions on the surface Σ, M [ µ ] = 14 π Z Σ µ ( y ) n ( x ) · ( x − y ) | x − y | d Σ( y ) , (6)the integral equation (5) is written as(1 / (2 λ ) − M ) µ = n · N . (7)The integral operator (6) is the electric field generated by the single layer charge distribution µ along the normal to the surface. It encodes the geometric information and has severalinteresting properties [13–15]. Its spectrum is discrete and it is not difficult to show thatall of its eigenvalues are bounded by the [-1/2, 1/2] interval. Although non-symmetric, theoperator (6) has real non-degenerate eigenvalues. The eigenvectors are biorthogonal, i. e.,5hey are not orthogonal among themselves, but orthogonal to the eigenvectors of the adjointoperator M † [ µ ] = 14 π Z Σ µ ( y ) n ( y ) · ( x − y ) | x − y | d Σ( y ) , (8)which is associated with the electric field generated by a surface distribution of electricdipoles (double layer charge distribution). Therefore, if | u k i is a right eigenvector of M corresponding to eigenvalue χ k , M | u k i = χ k | u k i and h v k ′ | is a left eigenvector correspondingto eigenvalue χ k ′ , h v k ′ | M † = χ k ′ h v k ′ | , then h v k ′ | u k i = δ k ′ k , (9)with the scalar product defined as the integral over the interface Σ, h f | f i = Z Σ f ∗ ( x ) f ( x ) d Σ( x ) . (10)The value 1/2 is always the largest eigenvalue of the operator M , regardless the geometryof the object. This is immediately seen if the object is considered to be conductor ( ε → ∞ ),and then the interior electric field has to be zero. In that case λ = 1, and the charge densitythat generates a vanishing internal electric field obeys the equation (1 / − M ) µ = 0, andtherefore 1/2 is an eigenvalue of M . However, this eigenmode is not dipole-active and doesnot contribute to the total polarization of the object. The operator (6) is insensitive to ascale transformation, which means that its eigenvalue and eigenvectors depend only on theshape of the object and not on its size, or electrical properties.By employing the spectral representation of the resolvent of the operator M ( z − M ) − = X k ( z − χ k ) − | u k ih v k | , (11)the solution of equation (7) is obtained for z = 1 / (2 λ ) as µ = X k h v k | n · N i (1 / (2 λ ) − χ k ) − | u k i . (12)The polarizability of the homogeneous particle is obtained by using the distribution (12)to build the solution (4) of the Laplace equation and use it in equation (2). It has beenshown that, operationally, the polarizability is simply the dipole moment of the distribution(12) over unit volume [25, 27] 6 = 13 1 V X i,k h x · N i | u k i h v k | n · N i i / (2 λ ) − χ k , (13)where N i are three mutually orthogonal vectors (directions) of unit norm. The factor(1 / (2 λ ) − χ k ) − is a generalized Clausius-Mosotti factor. Each dipole-active eigenmodecontributes to α according to its weight p k =
13 1 V P i h x · N i | u k i h v k | n · N i i , which deter-mines the strength of coupling between the uniform electric field and the k -th eigenmodeand contains three components P k,i = h x · N i | u k i h v k | n · N i i /V . Equation (13) shows aclear separation of the electric properties, which are included only in λ , from the geometricproperties expressed by χ k and p k . B. Shelled particles
The polarizability of an object covered by a thin shell with permittivity ε S can be calcu-lated in a similar fashion. The electric field is now generated by two single layer distributions,and boundary conditions are imposed twice, for Σ and for Σ . The surface Σ is the outersurface of the shell and Σ is the interface between the particle and the shell. The solutionof a shelled particle in terms of single layer potentials has the form [28]Φ( x ) = − x · N + 14 π Z Σ µ ( y ) | x − y | d Σ( y ) + 14 π Z Σ µ ( y ) | x − y | d Σ( y ) , (14)where µ and µ are the densities defined on surface Σ and Σ , respectively. Four integraloperators M , M , M and M are defined, depending on which surface are variables x and y . For example M is defined when x and y are both on Σ , M is defined by x onΣ and y on Σ , and so on. Thus M ij [ µ j ] = 14 π Z Σ j µ j ( y ) n ( x ) · ( x − y ) | x − y | d Σ( y ) (15)for i, j = 1 ,
2. The equations obeyed by µ and µ are µ / (2 λ ) − M [ µ ] − M [ µ ] = n · N µ / (2 λ ) − M [ µ ] − M [ µ ] = n · N . (16)Here the electric parameters are: λ = ( ε S − ε ) / ( ε S + ε ), and λ = ( ε − ε S ) / ( ε + ε S ).We further assume a confocal geometry, i. e. the surface Σ is a slightly scaled versionof Σ , with a scaling factor η close to unity. This assumption does not provide constantthickness for the shell, but our main results should remain at least qualitatively valid [3, 6].7n the limit of very thin shells, and using the scaling properties of the operator M , onecan show [25, 27] that all four M operators are related to M = M M [ µ ] = η − ( µ/ M [ µ ]) M [ µ ] = − µ/ M [ µ ] M [ µ ] = M [ µ ] . (17)Equations (16) can then be arranged in a matrix form as / (2 λ − M ) (1 / M ) /η − / M / (2 λ ) − M µ µ = n · Nn · N . (18)By knowing the eigenvectors and the eigenvalues of M the charge densities µ and µ canbe found by inverting the matrix in (18). For example, µ is µ = X k η ( λ − χ k ) + (cid:0) + χ k (cid:1) η (cid:16) λ − χ k (cid:17) (cid:16) λ − χ k (cid:17) + (cid:0) + χ k (cid:1) (cid:0) − χ k (cid:1) h v k | n · N i| u k i . (19)The field generated by the the two distributions µ and µ outside the particle is thesame as the field generated by an equivalent single layer distribution µ e = X k h v k | n · N i (1 / (2˜ λ k ) − χ k ) − | u k i , (20)where ˜ λ k = (˜ ε k − ε ) / (˜ ε k + ε ) and the equivalent permittivity ˜ ε k is defined for each eigenmodeas: ˜ ε k = ε S (cid:18) ε − ε S ε S + δ (1 / − χ k ) ε + δ (1 / χ k ) ǫ S (cid:19) , (21)where δ = η − ≪
1. The distribution (20) is similar with the distribution (12) obtained fora homogeneous particle, except that λ has to be replaced for each mode with an equivalentquantity ˜ λ k . Equation (21) can be applied recursively for a multi-shelled structure. Thestrict separation of electric and geometric properties is weakened in this case, because theshape-dependent eigenvalue χ k appears now in the electric equivalent quantity ˜ λ k .The polarizability of the shelled particle is obtained by using the distribution (20) tobuild the solution (4) of the Laplace equation and use it in equation (2), to get α = 13 1 V X i,k h x · N i | u k i h v k | n · N i i / (2˜ λ k ) − χ k . (22)8quation (22) is obtained by replacing λ with ˜ λ k in Eq. (13). The parameter V in Eq. (22)is the total volume of the cell (the core and the shells). In the limit of a dilute suspensionof identical shelled particles, with a low volume fraction f , the effective permittivity (1) is ε sus = ε f X k p k ˜ ε k − ε (1 / χ k ) ε + (1 / − χ k )˜ ε k ! . (23) C. The Debye relaxation expansion
In general, the effective permittivity ǫ sus of a suspension of objects with m shells will have m +1 Debye relaxation terms for each dipole active eigenmode. The proof is recursive and isbased on partial fraction expansion with respect to variable iω of equations (22) and (23),provided that the complex permittivity of various dielectric phases is ǫ = ε − i σ/ ( ωε vac )where i = √− ε vac is the permittivity of the free space (8 . × − F/m). Thus thefirst Debye term comes out from (23) and the remaining m Debye terms result from (21)by the homogenization process described for shelled particles. Hence, a suspension of cellswith m shells (and m + 1 interfaces) has a dielectric spectrum containing a number of Debyeterms equal to m + 1 times the number of dipole-active eigenvalues.The suspension effective permittivity ǫ sus has the expansion ǫ sus = ǫ f + X k,j ∆ ε kj / (1 + i ω T kj ) (24)where ǫ f = ε hf − i σ lf / ( ωε vac ), ε hf is the high-frequency permittivity, and σ lf is the low-frequency conductivity; ∆ ε kj and T kj are the dielectric decrement and the relaxation timeof the kj Debye term, respectively; index k enumerates the dipole-active eigenmodes andindex j enumerates interfaces.Although the measurable bulk quantities in equation (24) are directly correlated with themicroscopic (electric and shape) parameters, a solution of the inverse problem, which aimsat obtaining the microscopic information non-intrusively, from the effective permittivity, isin general difficult, if not impossible for the general multi-shell structure. However, biolog-ical cell has a thin and almost non-conductive membrane, and several simplifications andapproximations can be made. Two Debye relaxation terms in the effective permittivity ǫ sus are expected for each dipole-active eigenmode, corresponding to the two interfaces whichdefine the membrane. 9he first relaxation is derived from the equivalent permittivity (21) which can be writtenalso as Debye relaxation terms: ˜ ǫ k = ε + ∆ ε/ (1 + iωT ) . (25)The relaxation time T that is given by the poles of ˜ ǫ k in (21) is a quite good approximationof the first relaxation time T k T = ε vac (1 + δ/ δχ k ) ε S + δ (1 / − χ k ) ε (1 + δ/ δχ k ) σ S + δ (1 / − χ k ) σ ≈ T k . (26)The main reason is as follows. At frequencies close to 1 /T there is a huge change in ˜ ǫ k oforder ε S / ( δ (1 / − χ k )), and consequently a significant change in the total permittivity ǫ sus given by (23). Therefore T provides an approximate value for the relaxation time T k of thesuspension effective permittivity ǫ sus .For a non-conductive shell σ S ≈
0, or more precisely when σ S ≪ δ (1 / − χ k ) σ , therelaxation time (26) is: T k ≈ ε vac ε S / ( δ · σ (1 / − χ k )) (27)showing a strong dependence on the thickness of the shell and on the shape of the particle,through the eigenvalue χ k . Due to the small parameter δ in (27) the first relaxation (i. e.,membrane relaxation) tends to have a lower frequency than the second relaxation, which ispresent even for particles with no shell (see the discussion below). In addition, cumbersomebut straightforward calculations provide the dielectric decrement ∆ ε k in (24)∆ ε k ≈ f p k ε S ( δ (1 / χ k ) (1 / − χ k )) − (28)that is very large due to the same strong dependence on the thickness of the shell. Theeffect is even more dramatic when the second largest eigenvalue is very close to the largesteigenvalue, (1 / − χ ) →
0, like in the case of two cells connected by tight junctions.For a suspension of shelled spheres η = 1 + ∆ R/R and δ = η − ≈ R/R , where∆ R is the thickness of the membrane, R is inner radius, and R + ∆ R is the total radius.Thus both T k and ∆ ε k are proportional to R and ε S and inverse proportional to ∆ R likein the Pauly-Schwan theory [1, 8]. Moreover, the dielectric decrement ∆ ε k in (28) is ageneralization of equation (54a) in Ref. [8]. In the same time, the relaxation time (27)differs with respect to equation (56a) in Ref. [8] only by the conductivity term. We will10how elsewhere that a more appropriate treatment of the relaxation times recovers also therelaxation time given by equation (54a) in Ref. [8].Thus, a non-conductive and thin shell/membrane produces a large relaxation of the com-plex permittivity of the suspension [31]. The experimental evidences further support thesetheoretical facts: when attacking the membrane with a membrane disrupting compound ( forexample a detergent) the relaxation almost vanishes as the cellular membrane is permeated[32].For frequencies higher than 1 /T k the cell permittivity is essentially determined by thedielectric properties of the cytoplasm, and does not depend on membrane’s properties. Thesecond Debye relaxation occurs at higher frequencies than the first (membrane) relaxation,and has the relaxation time T k ≈ ε vac (1 / χ k ) ε + (1 / − χ k ) ε (1 / χ k ) σ + (1 / − χ k ) σ (29)derived from the pole of equation (23). The corresponding dielectric decrement is∆ ε k ≈ f p k (1 / − χ k )( ε σ − ε σ ) × ((1 / χ k ) ε + (1 / − χ k ) ε ) − × ((1 / χ k ) σ + (1 / − χ k ) σ ) − The last two equations are similar to the ones that are given for spherical particles in Ref.[8] (equations (46) and (49) in the aforementioned reference). The relaxation given by T k is basically the relaxation of a homogenous particle embedded in a dielectric environmentand was also discussed in Ref. [22] by a closely related spectral method. If the conductivityof the cytoplasm is comparable to the conductivity of the outer medium, the decrement ofthe second relaxation is small such that it cannot be distinguished in the spectrum. On thecontrary, if the conductivity of the outer medium is much greater or smaller that that ofcytoplasm, than a second observable relaxation occurs. Unlike the membrane relaxation,this second relaxation depends only weakly on the shape. By assuming that σ ≪ σ and by using a finite-difference method, this resonance was also obtained in [3] and it wasinstrumental in explaining the experimental data on the fission of yeast cells of Asami et al. [33] by Lei et al. [22].The shape of the particle is important because it affects the number of dipole-activeeigenvalues and their strengths. In principle, each dipole-active eigenvalue introduces a new11elaxation in the dielectric spectrum, providing this relaxation is well separated from theothers. A cluster with complex geometry can have several dipole-active eigenvalues, butunless the cluster is larger in one dimension then in the others, or there are tight junctions,the relaxations overlap to create broad features in the spectrum. An extra relaxation isintroduced when the particles are covered by thin membranes. In addition, if (1 / − χ k ) → III. RESULTSA. Numerical procedure
The calculation of the effective permittivity for a suspension uses equations (1) or (23),and reduces then to finding the eigenvalues χ k and eigenvectors | u k i and | v k i of the linearresponse operator M . This problem is solved by employing a finite basis of NB functionsdefined on the surface Σ. A natural basis for a surface not far from a sphere is the generalizedhyperspherical harmonics functions˜ Y lm ( x ) = 1 p s ( x ) Y lm ( θ ( x ) , ϕ ( x )) , (30)where s ( x ) is related to the surface element through d Σ = s ( x ) d Ω x and d Ω x is the solidangle element.Another choice could be based on Chebyshev polynomials of the first kind [29]˜ T lm ( x ) = 1 p s ( x ) T l ( θ ( x )) e imϕ ( x ) . (31)Both bases are complete and orthogonal in the Hilbert space of square integrable functionsdefined on Σ.In this paper, we model the linear cluster of particles as an object with axial symmetry.We seek to find a surface of revolution for which the thickness of the interparticle joints canbe varied without perturbing the overall shape of the object. We use two representations12or the surface Σ: (A) for clusters of two particles we use spherical coordinates { x, y, z } = { r ( θ ) sin θ cos φ, r ( θ ) sin θ sin φ, r ( θ ) cos θ } , and (B) for clusters with more than two particleswe specify the surface in terms of a function g ( z ) as { x, y, z } = { g ( z ) cos φ, g ( z ) sin φ, z } .In the case B, the surface element is d Σ = g ( z ) p g ′ ( z ) dzdϕ, (32)and the normal to surface Σ is n = 1 p g ′ ( z ) cos ϕ sin ϕ − g ′ ( z ) . (33)In the basis of generalized hyperspherical harmonics the operator M has matrix elementsgiven by M lm ; l ′ m ′ = δ mm ′ π Z Z z max Z Z z min A ( z, z ′ , ϕ − ϕ ′ ) P ml (cos θ ( z )) P m ′ l ′ (cos θ ( z ′ )) e im ( ϕ − ϕ ′ ) G ( z, z ′ ) dz dz ′ dϕ dϕ ′ (34)where G ( z, z ′ ) = q g ( z ) g ( z ′ ) p (1 + g ′ ( z ))(1 + g ′ ( z ′ )) − , (35)and A ( z, z ′ , φ ) = ( g ( z ) − g ( z ′ )) cos φ − ( z − z ′ ) g ′ ( z )[ g ( z ) + g ( z ′ ) − g ( z ) g ( z ′ ) cos φ + ( z − z ′ ) ] / (36)After the angle integration in equation (34) and by using the elliptic integrals given inthe Appendix, the matrix elements are obtained by numerical evaluation of the resulting ( z , z ′ ) double integral using an NQ -point Gauss-Legendre quadrature [29, 30]. Because of theintegrable singularity apparent in the kernel of the operator M in equation (6), the meshof z must be shifted from the mesh of z ′ by a transformation which insures that there is nooverlap between the two meshes.The delta symbols δ mm ′ in equation (34) reflects the fact that we consider only objectswith rotational symmetry in this paper. Moreover, for fields parallel with the cluster axis m = 0, while m = 1 for perpendicular fields.The convergence of the results is obtained in two steps. First, the number NQ of quadra-ture points is increased until the matrix elements of M converge, and then the size NB of13he basis set is increased until the relevant eigenvalues χ k and their corresponding weights p k have acquired the desired accuracy. A necessary test for convergence is the fulfillment ofthe sum rules P k P k,i = 1 and P i,k χ k P k,i = 1 / /
2. For a sphere there isjust one dipole-active eigenmode which has eigenvalue χ = 1 / p = 1, while foran ellipsoid there is one dipole-active eigenmode along each axis. Fast and accurate solutionsare achieved for spheroids with a basis size of NB =20 and with NQ = 64 quadrature points.In general, the size of the basis and the number of the quadrature points increase with thenumber of cells in the cluster and with the decreasing of the junction size. Thus, for ournumerical examples a basis with NB =35-40 and NQ = 128 quadrature points are enoughfor a converged solution in the case of the dimers and NB =50 and NQ = 200 quadraturepoints in the case of the clusters with up to four cells. B. Two cells joined by tight junction
The equation r ( θ ) = ( h +cos θ ) / (1 − a cos θ ) describes the shape of a two-particle cluster.Parameter h controls the tightness of the inter-particle junction and parameter a measuresthe deviation from a spherical shape. More precisely, h is the radius of the smallest circleat around the thinnest part of the junction.Figure 1 shows the effective permittivity for a suspension of particles with the followingparameters: ε = 70, σ = 0 .
25 S/m, ε S = 6, σ S = 0, ε = 81, σ = 0 .
374 S/m, volumefraction f = 0 .
05, membrane thickness δ = 0 . a = 0 .
2. The effective per-mittivity does not depend on the thickness parameter h when the stimulus electric field isperpendicular to the cluster axis. However, a new relaxation becomes apparent as h → χ k and their weights P k, for a field parallelto the z axis. As the junction become tighter ( h →
0) more eigenvalues become dipole-active. While all eigenvalues are important in shaping the dielectric spectrum, the secondlargest eigenvalue χ is crucial to explaining the occurrence of an additional relaxation atlow frequencies, as observed for small h in [3, 25]. Although its weight P , also decreases for14 pa r , pe r p f(Hz) h=0.5 h=0.1 h=0.01 h=0.001 perp FIG. 1: (Color online) The spectrum of the effective permittivity of a suspension of dimers withvarious junction thickness h , and with parallel and perpendicular field configurations. The sus-pension permittivity for an electric field perpendicular to cluster axis does not depend on h and ispointed by an arrow. small h , this dipole active eigenmode approaches 1/2 as the junction becomes tighter. Thus,according to equations (27) and (28) the effect of χ is “enhanced” due to the presence of anonconductive shell (as the case for biological particles analyzed in [3, 25]). Moreover, thedecrease of P , is compensated by the increase of 1 / (1 / − χ ).The presence of a new relaxation at low frequency along with its relationship with thesize of h has been already singled out in Gheorghiu et al. [25] by using the same methodbut without the analysis of dipole-active eigenmodes. Using a finite discrete model [3],the relaxation was observed before the segregation during cell division, while other papers[34, 35] fail to relate the size of h to the new relaxation, even though one of them [34]employs essentially the same method as the one outlined in the present work.Figure 3 shows the charge distribution associated with the first four eigenvalues for twodistinct values of h . The second eigenmode is an antisymmetric combination of net chargedistributions (monopoles) on each particle of the dimer. The third charge distribution is anantisymmetric combination of charge distributions with a dipole moment on each part of the15 .00.20.40.6 10 -3 -2 -1 . - k h P k , h FIG. 2: (Color online) The largest seven eigenvalues and their weights for a binary cluster withparameter a = 0 .
2, as a function of h . The inset shows the shape of the dimer. dimer and the forth distribution is antisymmetric combination of charge distributions witha quadrupole moment on each particle. At small h (tight junctions), charge accumulates inthe vicinity of the junction [36]. 16 u () u () u () u () FIG. 3: The first four eigenvectors for a dimer given by equation r ( θ ) ; a = 0 . h = 0 . h = 0 .
01 (solid line).
C. Clusters of more than two particles
Smoother yet tight junctions would bring (1 / − χ ) closer to 0 than sharp and tight junc-tions. The reason is simple: smoother junctions have the two parts of the dimer farther apart.17e have analyzed linear clusters of cells connected by smooth and tight junctions by usinga ( z , φ ) parameterization which describes a surface by { x = g ( z ) cos ϕ, y = g ( z ) sin ϕ, z } .The construction starts from a dimer shape that resembles the shape of the epithelial cellslike MDCK (Madin-Darby Canine Kidney) cells. An example of such shape, displayed infigure 4, extends from − z max to z max and it can be decomposed in three parts: the left cap( − z max ≤ z ≤ − z ), the central part ( − z ≤ z ≤ z ), and the right cap ( z ≤ z ≤ z max ).At position ± z the shape function has its maximum. An m -cell linear cluster is ob-tained by repeating the central part m − − L m to L m , where L m = z max + ( m − z . Mathematically, the shape is described by: g m ( z ) = g ( z + ( m − z ) , for − L m ≤ z ≤ − L m + z max g (mod( z + (1 + ( − m ) z / , z ) − z ) , for − L m + z max ≤ z ≤ L m − z max g ( z − ( m − z ) , for L m ≤ z ≤ L m − z max , (37)where mod( x , y ) is the remainder of the division of x by y . For the examples consideredhere, the dimer shape function is: g ( z ) = 0 .
01 + 2 . z − . z + 40 . z − . z + 79 . z − . z + 21 . z − . z + 0 . z − . z , (38)with z max = 1 . z = z max / ABLE I: Most representative dipole-active eigenmodes and their weights for the trimer in parallelfield. χ k P k, χ k P k, -z max g ( z ) -3z -2z -z +z max z -z z z max -z max g ( z ) FIG. 4: Smooth construction of a cluster (lower panel) from a dimer (upper panel). The partsdetermined by z ∈ [ − z , z ] are “glued” together with the ends of the dimer. The arrows showwhere the junctions will be placed in the cluster. In Figure 6 we plot P k, / (1 / − χ k ) versus (1 / − χ k ), which shows that the number ofdipole-active eigenmodes increases with the number of particles in the cluster. Accordingto (27) and (28), Figure 6 shows in fact the dielectric decrement versus its correspondingrelaxation frequency for each dipole-active eigenmode of the given clusters. For clustersof two or three particles, there is one important active eigenmode close to 1 /
2, while forclusters of four particles there are two active eigenmodes.It can be conjectured that for a general linear cluster made of m particles, there are m − /
2, of which the largest one is always dipole-active and has the largest20 pa r , pe r p f(Hz) perp FIG. 5: (Color online) Effective permittivity for clusters (shown in the inset) of one, two, three,and four cells connected by tight and smooth junctions. The field is either parallel (solid lineswith symbols) or perpendicular (solid lines only) to the cluster axis. The effective permittivityeither increases strongly with the number of cells for parallel geometry, or does not change for aperpendicular geometry. weight. In fact one can show that for two cells connected by smooth and tight junctioncharacterized by parameter h , (1 / − χ ) ∝ h when h →
0, or more precisely (1 / − χ )is proportional with the solid angle encompassed by the missing part of a cell when it isconnected with other cell in the dimer. The proof is based on the theorem of the solid angle[12]. The generalization to a finite cluster is also straightforward to (1 / − χ ) ∝ h /m (inthat case the solid angle encompassed by the middle junction is proportional to h /m . Theweight of the second eigenmode is P , = h x · N | u ih v | n · N i /V . If we consider thatthe surface of the cluster is determined by the function g ( z ) then, up to a constant factor, h v | n · N i ≈ g (0) = h for two cells connected by smooth and tight junctions. The proofconsiders that the second eigenfunction of M † is an antisymmetric combination of constantdistributions on each part of the dimer. This assertion is confirmed in Figure 7. Moreover, h x · N | u i /V is weakly dependent on h . Therefore, for a parallel setting of the field stimulus, P , / (1 / − χ ), which is the measure of the dielectric decrement of low-frequency relaxation,21 - - - - c e ll c e ll s c e ll s c e ll s P k , / ( / -) / - FIG. 6: (Color online) P k, / (1 / − χ k ) versus (1 / − χ k ) for clusters of up to four cells. The secondeigenvalue has the largest contribution to intensity of relaxation. is finite and it increases when the number of cells is increased. The increase of relaxationdecrement when m → ∞ is physically limited by σ S ≪ δ (1 / − χ k ) σ , since the membraneconductivity is not strictly 0.Due to cluster’s shape and membrane properties, the variation of P k, / (1 / − χ k ) and(1 / − χ k ) with respect the eigenmode k determines a low frequency relaxation when thedipole-active eigenvalue χ k is close to 1/2. We note here that for dipole-active eigenmodes ofellipsoids the term (1 / − χ k ) is called the depolarization factor and has analytical expression[37]. Prolate spheroids with longitudinal axis much larger than the transverse axis (needles)have the longitudinal depolarization factor approaching 0 and the transverse depolarizationfactor approaching 1/2. More precisely, for a long prolate spheroid, the longitudinal depo-larization factor scales as (1 / − χ ) ∝ a x /a z , a z > a x = a y , as a x →
0. On the otherhand, extensive numerical calculations support the fact that cylinders with the same aspectratio behave similarly to prolate spheroids [37]. Thus, it is not hard to observe that thelow-frequency relaxation of linear clusters of cells connected by tight junctions is similarto that of a needle or a thin cylinder as long as the cluster and as thick as the junction.22
303 -1 0 1-0.60.00.6 v ( z ) , v ( z ) u ( z ) , u ( z ) z FIG. 7: The first (dotted line) and the second (solid line) eigenfunction of M (upper panel) and M † (lower panel) for a dimer whose shape is depicted by dashed line in the lower panel. Thesecond eigenfunction of M † is an antisymmetric combination of almost constant distributions oneach part of the dimer.
23n the other hand, the high-frequency relaxation of the cluster shows the relaxation of asuspension of spheroids with the same volume as the volume of a single cell. Therefore thedielectric spectrum for a suspension of clusters is the same as the spectrum of a two speciessuspension made of thin cylinders and spheroids.
IV. CONCLUSIONS
We present a theoretical framework based on a spectral representation of BIE and able tocalculate the dielectric behavior of linear clusters with a wide range of shapes and dielectricstructures. The theory agrees with the results of Pauly and Schwan for a sphere coveredwith a shell [1, 8]. In fact, for spheroids, our theory is the same as the analytical resultsof Asami et al. [2]. We present extensive calculations of clusters with shapes resemblingMDCK cells.A practical numerical recipe to compute the effective permittivity of linear clusters witharbitrary number of cells is provided. Examples are given for cluster with shapes described as r ( θ ) in spherical coordinates or using ( z , ϕ ) parameters as { x = g ( z ) cos ϕ, y = g ( z ) sin ϕ, z } .Other studies in the literature used only spherical coordinates representation [25, 27, 35]. Adirect relation between the geometry and dielectric parameters of the cells and their dielectricbehavior described by a Debye representation has been formulated for the first time. Otherwork [22], which is based on a closely related spectral method [18–21], found a direct relationlinking the geometry and electric parameters to the dielectric behavior only for homogenousparticles. Moreover, the method used in [22] treats only particles with spheroidal geometry.We show that the spectral representation provides a straightforward evaluation of thecharacteristic time constants and dielectric decrements of the relaxations induced by cellmembrane. We prove that the effective permittivity is sensitive to the shape of the embed-ded particles, specially when the linear response operator has strong dipole active modes(with large weights p k ). A low-frequency and distinct relaxation occurs when the largestdipole-active eigenvalue is very close to 1 /
2. Clusters of living cells connected by tightjunctions or very long cells have such an eigenvalue. Our results also shed a new light onthe understanding of recent numerical calculations [38] performed with a boundary elementmethod on clustered cells where the low-frequency relaxation is attributed to the tight (gap)junctions connecting the cells. The method used in [38] does not use the confocal geometry24ssumption .The present work has several implications and applications. We emphasize the capabilitiesof dielectric spectroscopy to monitor the dynamics of cellular systems, e.g., cells during cellcycle division, using synchronized yeast cells [3, 11, 39], or monolayers of interconnectedcells [40, 41]. Also the method is able to assess the dielectric behavior of linear aggregatesor rouleaux of erythrocytes, where the ellipsoidal or cylindrical approximations are notadequate [42, 43].The proposed representation is a powerful alternative to finite element or other purelynumerical approaches, because it provides the analytical framework to explain and predictthe complex dielectric spectra occurring in bioengineering applications. Extension of thismethod to other surfaces of revolution, for example linear clusters with more than 4 particles,is straightforward providing an adequate parametric equation is available. Finally, in manycases (e.g. shapes with high symmetry) the method is faster, offers accurate solutions andlast but not least can be integrated in fitting procedures to analyze experimental spectra.
Acknowledgments
This work has been supported by Romanian Project “Ideas” No.120/2007 and FP 7Nanomagma No.214107/2008.
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