Dielectrophoresis of Tetraselmis sp., a unicellular green ... for ThaiScience/Article/62/10027186.pdfOriginal Article Dielectrophoresis of Tetraselmis sp., a unicellular green alga,

Original Article

Dielectrophoresis of Tetraselmis sp., a unicellular green alga, in travellingelectric fields analyzed using the RC model for a spheroid

Sakshin Bunthawin1, Raymond J. Ritchie1 and Pikul Wanichapichart2

1 Biotechnology of Electromechanics Research Unit, Faculty of Technology and Environment,Prince of Songkla University, Kathu, Phuket, 83120 Thailand.

2 Membrane Science and Technology Research Centre, Department of Physics, Faculty of Science,Prince of Songkla University, Hat Yai, Songkhla, 90112 Thailand.

Received 24 May 2011; Accepted 29 August 2011

Abstract

Dielectrophoresis of a unicellular green alga, Tetraselmis sp., in a travelling electric field was analyzed using an RC(resistor-capacitor)-model, instead of the Laplace approach reported in our previous work. The model consists of resistor-capacitor pairs in series to represent the conductive and the capacitive properties of the shell and the inner part of the spheroid.The model is mathematically simpler than the Laplace model and the RC approach is experimentally superior because only thelower critical frequency [LCF] and cell translational speed are required to be measured experimentally. The effective compleximpedance of the spheroid was mathematically modeled to obtain the Clausius-Mossotti factor ([CMF]) as a function of celldielectric properties. Spectra of dielectrophoretic velocity and the lower critical frequency of the marine green alga, Tetraselmissp. were investigated to determine cell dielectric properties using a manual curve-fitting method. Effects of arsenic at differentconcentrations on the cell were examined to verify the model. Arsenic severely decreases cytoplasmic conductance (c)whereas it increases membrane conductance (m). Effects were easily observable even at the lowest concentration of arsenicused experimentally (1 ppm). The method offers a practical means of manipulating small plant cells and for rapid screeningfor effects on the dielectric properties of cells of various applied experimental treatments.

Keywords: dielectrophoresis, interdigitated electrodes, phytoplankton, RC model, travelling wave

Songklanakarin J. Sci. Technol.33 (5), 585-597, Sep. - Oct. 2011

1. Introduction

Manipulation of cells in travelling electric fieldsrelating two orthogonal linear motions can be used for cellcharacterization and separation (Wang et al., 1995; Talary etal., 1996; Jones, 2003; Fu et al., 2004). This has great bio-technological potential (Bunthawin et al., 2010). The transla-tional motions of a cell can be termed the dielectrophoreticvelocity and is a complex function of the frequency depen-dent Clausius-Mossotti factor ([CMF]), where the dielectro-

phoretic spectrum of a spheroidal cell is distinguished fromthat of a sphere (Bunthawin et al., 2010). Many more biologi-cal cells can be approximated by a spheroid than by a simplesphere. The relationship between the real and imaginary partsof the [CMF] reflects boundary frequencies of positive andnegative dielectrophoresis; the so-called “critical frequency”(Pohl, 1978; Gimsa et al., 1991; Gimsa and Wachner, 1998;Gimsa and Wachner, 1999; Bunthawin et al., 2007; Bunthawinet al., 2010). In the case of the real part [CMF] (Re[CMF])prevailing over its imaginary part (Im[CMF]), the spheroidwill be either attracted toward the tips or repelled from inter-digitated electrodes over AC field frequency ranges fromkHz to MHz (Bunthawin et al., 2010). The cell does not movewhen the imaginary part of the [CMF] predominates.

* Corresponding author.Email address: [email protected]

http://www.sjst.psu.ac.th

S. Bunthawin et al. / Songklanakarin J. Sci. Technol. 33 (5), 585-597, 2011586

Previously, dielectrophoretic velocity and two criticalfrequencies of a spheroid in a travelling electric field havebeen derived through the Laplace approach where the realpart of the [CMF] is zero (Gimsa and Wachner, 1998; Gimsaand Wachner, 1999; Bunthawin and Wanichapichart 2007;Bunthawin et al., 2007). Nonetheless, it is evident that thetwo critical frequencies (high critical frequency, HCF, and lowcritical frequency, LCF) deduced from such an approachwere rather mathematically complicated. Furthermore, a shellthickness of a spheroid simulated from the Laplace approachis non-uniform, the so-called “the confocal shell problem”(Asami et al., 1980; Gimsa and Wachner, 1998, 1999). An RC(resistor-capacitor)-model of a spheroid in a two-phaseelectric field has thus been proposed by Gimsa and hiscolleagues (Gimsa et al., 1991; Gimsa and Wachner, 1998)and later extended by Bunthawin et al. (2007) and Buntha-win and Wanichapichart (2007) as an impedance approach fora case of a travelling electric field to correct the ambiguitiesabout the shell thickness and the estimated dielectric proper-ties of a spheroidal cells. In the previous RC-model the im-pedance of the suspension was obtained by Kirchhoffs lawsand the “influential radius”, a parameter given by the charac-teristic length to which the field disturbance, was introducedby cell polarization (Gimsa and Wachner, 1998, 1999). Theinfluential radius also concerns the depolarization factorwhich does not depend on frequency but only on cell geo-metry (Asami, 2002; Bunthawin and Wanichapichart, 2007).

As reported by Bunthawin and Wanichapichart (2007),the higher critical frequency (HCF) of a marine green phyto-plankton alga could not usually be observed experimentally.For this reason, estimations of cell dielectric properties byusing a curve-fitting method could not be carried out. Thelack of experimental data at HCF raises some doubts aboutthe validity of the estimated cytoplasmic conductivity ob-tained from such methods (Bunthawin and Wanichapichart,2007; Bunthawin et al., 2007). Alternatively, the frequencymeasurement technique can be improved by means of mea-surement of the dielectrophoretic velocity, which allows oneto estimate the cytoplasmic conductivity and permittivity(Bunthawin et al., 2010). In cells undergoing dielectropho-resis in a travelling wave electric field, both positive andnegative dielectrophoretic velocities at LCF and HCF aremeasurable, making it possible to evaluate the dielectricproperties of the membrane and the cytoplasm, respectively.

The present study extends the RC model to measure-ments of dielectrophoretic velocity and critical frequency todetermine the dielectric properties of a shelled spheroid in atravelling wave electric field. The RC approach is mathemati-cally simpler than the Laplace model and the RC model isexperimentally superior because only the lower critical fre-quency (LCF) and cell translational speed are required to bemeasured experimentally. This is a critical advantage over theLaplace model. In this work, the marine green phytoplanktonalga Tetraselmis sp. is considered to be a single shelledspheroid. The dielectric properties of the cell membrane, cyto-plasm and the suspending medium are obtained from manual-

curve fitting of the lower critical frequency and cell transla-tional speed. Experimental results from the literature togetherwith the observations of the control and arsenic pre-treatedcells are also used to demonstrate the validity of the model.Comparisons will be made to findings in our previous studyon yeast cells (Bunthawin et al., 2010).

2. Theoretical Approach

2.1 The critical frequency

Estimates of the two critical frequencies and thedielectrophoretic velocity of a spheroid in a travelling electricfield are made based on the concept proposed by Bunthawinet al. (2010), where an RC model is employed instead of theLaplace approach. A spheroid with a single shell is assignedan equivalent RC-circuit (Figure 1), using a pair of resistorsand capacitors to represent the conductive and the capaci-tive properties of the shell, the inner part of the spheroid (thecytoplasm) and the suspending medium, respectively. Theeffective complex impedance of the spheroid is then calcu-lated from the Clausius-Mossotti factor ([CMF]) (Gimsa andWachner, 1998; 1999). At the boundary frequencies of thepositive and the negative dielectrophoresis (DEP), the zeroDEP force also relates to the value of the zero-real part of theClausius-Mossotti factor (Re[CMF]). Two critical frequencieswere then solved by using the frequency dependence ofRe[CMF] as a function of an angular frequency ( ω 2πf ).Dielectric dispersion of the spheroid’s compartments at lowerand higher frequency range is taken into account to extractLCF and HCF from the electric field frequencies (f) appearingin Re[CMF]. The modified [CMF] of the RC model is (Gimsaand Wachner, 1999):

]CMF[ )(1*

*

s

fs

k UUU

L

, (1)

where sU and fU are the electrical potentials of the sus-pending medium and the cell-medium interface, respectively.The relationship between fU and the specific complex im-

pedances *( )Z is defined as inf.( )

c mf

c m s

Z ZU E rZ Z Z

,

where inf 1 k

rrL

and r represents the influential radius or

radial distance (see Figure 1). Lk is the depolarization factorwhich depends on the cell geometry: it has three componentsalong the x, y, and z axes as Lx, Ly and Lz, respectively. How-ever, for mathematical simplification, the length of the threeaxes a, b, and c of the spheroid are considered as a b c ,which it is relevant to a shape of Tetraselmis sp. (see Figure2). This analytical case of b = c is a prolate spheroid as pro-posed by Asami (2002), later extended by Bunthawin et al.(2007) and Bunthawin et al. (2010). For cell dimensions (see

Figures 1 and 2), the ratio 1ab , 2 2 3/ 2

11 ( 1)k

qLq q

587S. Bunthawin et al. / Songklanakarin J. Sci. Technol. 33 (5), 585-597, 2011

2 1/ 2( 1)n q q , 1

2x

y zLL L

, and 1kk

L , where

aqb

(Asami et al 1980). Combining these expressions,

[CMF] along any semi-axis (k) can be written as:

1[CMF ] 1( )(1 )

c mk

k c m s k

Z ZL Z Z Z L

, (2)

where the specific impedances of the compartments areexpressed as (Bunthawin and Wanichapichart, 2007; Bun-

thawin et al., 2007):

1 1

1 1, ,

1 1

( ) ( )( ) ( )( ) ( )

c c c o c

c m s m m m o m

s s s o s

Z jZ Z j

Z j

. (3)

It should be noted the specific impedances are gener-ally defined as a term of a complex impedance per length ( )and cross section area (A) as the ratio of / A (Gimsa andWachner, 1998). Nevertheless, for mathematical tractability,

Figure 1. RC model which is equivalent to the shelled polate spheroid (on the right) is shown in x- y plane with the zero potential )0( at the center of the spheroid. The conductive and capacitive properties of each branch are represented by their conductivity )(and dielectric constant )( .

Figure 2. Cell of (a) Tetraselmis sp. is considered as (b) a single shelled prolate spheroid with average dimension a = 10.0 µm and b = 8.0µm. Views of isometric (left) represents its three semi-axes along x-y-z plane where a > b = c with a constant shell thickness (an averaged value). All axes are measured from the center to the outer surface along their axis.


this ratio is set to be constant throughout this work so thatsubstituting Equation 3 into Equation 2 brings the cancella-tion of this ratio (Bunthawin et al., 2007 and Bunthawin et al.,2010). The conductive () and capacitive () properties ofthe cytoplasm (c), the membrane (m) and the suspendingmedium (s) are assigned as ,,c m s and ,,c m s , respec-tively, where the permittivity of a vacuum is 0 = 8.85 x 10-12

F.m-1 and 1j . Similarly, three orthogonal componentsof the [CMFk] may be written in tensor forms as:

[CMF ] 0 0[CMF ] 0 [CMF ] 0

0 0 [CMF ]

x

k y

z

, (4)

with

0 00 00 0

x

k y

z

LL L

L

. (5)

Substituting Equation 3 to 5 into Equation 2, for the longestaxis in x direction (dielectrophoretic force direction), then

2

( )[CMF ] .)( )

m s c s c m x c mx

m s c s c m x x

LL L

(6)

The real and imaginary part of the frequency depen-dence of [CMFx] can be solved directly using the relation

x[CMF ] Re[CMF ] Im[CMF ]x xj . By rearranging Equa-tion 6 in polynomial form of 1

1 0... 0n nn na a a

,where na , n are constant and 2 f , and setting f = LCFfor the lower and f = HCF for the higher critical frequency,then estimates of LCF and HCF are obtained as:

HCF2

ABC

, (7)

( )LCF

2E F G

D

, (8)

where,2( 2 ) ( 1)c s c s sA ,

2 2 2( ) 2 ( )s c s c s c sB ,

0C A ,

0mD B ,2 2 2(2 ) (2 2 ) ,c m s s c c s m c s m s m sE

2 2 2 2 2 2 2 2( ) 2 .m s c s c s m c m s m cF

The final two equations allow one to vary s at arbi-trary low values so that HCF and LCF can be measured.When reaches the maximum (critical) value, the two frequen-cies merge at the zero dielectrophoretic force. Equation 7 and8 are written in a more compact form and easily to derive thanthat of Laplace approach (see Appendix A).

2.2 Relation of travelling wave dielectrophoretic force anddielectrophoretic velocity

The time-averaged dielectrophoretic force ( tF

) (Wanget al., 1995) acting on a shelled spheroid in a travelling elec-tric field is written in the form (Equation 4, Bunthawin et al.,2010):

20

23t sF ab

2 2 2 2Re[CMF] Im[CMF]E E E E a

2 2 2 2 ˆRe[CMF] Im[CMF] x x y y z z kE E E E a (9)

where ˆka (k=x,y,z) are unit vectors in the Cartesian coordi-nate frame, a and b are semi-axes of the spheroid along thex and y axis, E is the root mean square (rms) value of theelectric field strength and ,x y and z are the electricalphases addressing the electrode along x, y and z direction,respectively. The magnitude of the conventional forcedepends on both Re[CMF] and 2E (gradient of the squareof the electric field strength) whereas the direction dependson the sign of the Re[CMF]. In the case of the spheroidal celltravelling with a constant speed through a viscous medium( ) , the cell velocity can be written as (Equation 14, Bun-thawin et al., 2010):

2 20 Re[CMF] ˆ

9s

cDEP xb Ev a

K

, (10)

where K is the shape factor that relates with the geometry ofthe cell (Equation 11, Bunthawin et al., 2010) and ˆ xa is aunit vector along the x axis. It should be noted that if is nega-tive, the cell will be repelled from the electrode with negativeforce and the velocity, with reference to the electrode, will beminus.

3. Experimental

3.1 Cell preparation

Cells of Tetraselmis sp., obtained from the NationalInstitute for Coastal Aquaculture (NICA), Thailand, wereconsidered as a spheroid with average dimensions a =10.0±0.7 µm and b = 8.0±0.5 µm (meanSD) (Figure 2). Hence inFigure 1 the long spheroidal axis (a) was taken as 10 mm andthe short axis (b) as 8 mm. The cells were cultured in Sato andSerikawa’s artificial seawater (Sato and Sarikawa, 1978), andharvested when the cells were in log-growth phase. The cellswere centrifuged at 7,000 rpm for 2 min and re-suspendedtwice in 0.5 M sorbitol solution as described by Wanicha-pichart et al. (2007). The solution conductivity ( )s wasmeasured by using a conductivity meter (Tetracon 325, LF318), and it was adjusted to be between 3 and 300 mS.m-1 byadding 0.1 M KCl solution, using a micropipette (Nichipet,model 5000DG). Non-viable cells were prepared by boilingthe cells at 80°C for 10 min, cooling down to room tempera-ture, centrifuging and re–suspending in the same experimen-tal solution as was used for the live cells (Wanichapichart et


al., 2007). Arsenic pre-treated cells were prepared by addingarsenic solution (NaAsO2, MW 129.9) into the cell culture atconcentrations varying from 1 to 150 ppm and leaving themfor 24 hrs before being used in an experiment.

3.2 Electrode and electrical setup

Diluted suspensions of cells were used to measuremovement in a travelling electric field. About 100 µl of cellsuspensions with 9.2106 and 4.0105 cells.ml-1 were pipettedonto the glass slide between the electrodes. The interdigi-tated electrode was fabricated on commercially availablestandard microscope glass slides of dimension 80301 mm(Marienfeld, Germany). Each bar was made of gold 200 µm inlength, 100 µm in width (d1), and 0.2 µm in thickness (t). Theseparation of the adjacent bars on the same array (d2) was100 µm and that for the central channel (d3) was 300 µm (Fig-ure 3).

To generate the travelling electric field, the electrodewas energized with four sinusoidal signals (a quadraturephase) of 1.4, 2.8, and 7.1 V (rms) in phase sequence asdescribed in Wang et al. (1995). A synthesized functiongenerator (Standford Research Systems, Model DS345, Cali-fornia) connecting with a lab-made phase shift unit (PSU)and the inter-junction unit (IJU) was employed to control thephase sequence. As the field frequency of the applied signalswas gradually increased, from the lower to the upper limit ofthe function generator (10 Hz-30 MHz), two critical frequen-cies LCF and HCF were observed. While cells were under-going dielectrophoresis, dielectrophoretic velocities ( DEPv )were recorded through CCD camera (Sony SLV-Japan)connecting with a microcomputer (Acer, Aspire 4310). TheWinfast PVR™ program was employed to store and displaythe recording files to determine the velocities using a digitalstop-watch function of the Winfast program. Experimentalmeasurements of LCF and HCF and DEPv on the cells took

Figure 3. Configuration of the octa-pair interdigitated electrode and the quadrature phase sequence. (a) Three–dimensional view of theelectrode tips on the glass slide (not to scale). (b) Diagram of the electrode configuration with exaggerated scale in top view whereeach bar was made of gold 200 m in length ( ), 100 m in width (d1), and 0.2 in thickness (t). The separation of the adjacentbars on the same array (d2) was 100 and that for the central channel (d3) was 300.


less than five minutes. Electric field strengths were calcu-lated from the Quick Field™ program version 5.5.

4. Results and Discussion

4.1 The control cells

Suspensions of the control Tetraselmis sp. cell (9.2106 and 4.0105 cell.ml-1) subjected to the field strength 28kV.m-1 (equivalent to the signal 1.4 V) underwent positivedielectrophoresis between 50 kHz to 0.5 MHz, where themedium conductivity (s) was ranged from 0.01 to 0.10 S.m-1,respectively (see Figure 4). The theoretical curves were fittedbased on Tetraselmis cell dimensions a = 10 m, b = 8 mand a standard membrane thickness () of 13 nm (Bunthawinet al., 2010). At increasing s (Figure 5a), the magnitudes ofthe spectra were reduced significantly and the lower criticalfrequency (LCF) was shifted upwards to a higher frequency.In the case of s = 0.01 S.m-1, the spectrum showed a promi-nent peak of 9.4 m.s-1 at 200 kHz, where the lower criticalfrequency was at 50 kHz. For s = 0.03, 0.06, and 0.10 S.m-1,the spectra were transformed into a bell shape with plateaus

at heights of 5.2, 4.4, and 3.5 µm.s-1, respectively. The fre-quency dependency plateaus occupied the range 500 kHz to10 MHz. The lower critical frequencies (LCF) of the final threes values used were 200, 300, and 500 kHz, respectively.

As is seen from Figure 5b, electric field strengthswere proportional to the magnitudes of the spectra but didnot affect the two critical frequencies. For s = 0.01 S.m-1 and9.2106 cell.ml-1, increasing the field strengths from 28 to 143kV.m-1 brought the peaks up to 29 µm.s-1. For electric fieldstrengths of 28, 57, and 143 kV.m-1 the maximum values of thepeaks were 9.4, 17.2, and 29.0 µm.s-1, respectively, with allpeaks fixed at 300 kHz. However, it is interesting that therewere no differences in the dielectrophoretic spectra resultingfrom using 4.0105 cell.ml-1 or 9.2106 cell.ml-1 indicating thatthe technique is not strongly affected by the numerical den-sity of cells used (Figure 5c).

4.2 Arsenic pre-treated and boiled cells

Arsenic had significant effects upon the dielectro-phoretic velocity spectra (Figure 6). At increasing concen-trations, amplitudes of the spectra and the higher criticalfrequency were not affected but the lower critical frequencyfell in value (Figure 7). Plots of critical frequencies for thecontrol and the arsenic pre-treated cells are shown in Figure7. It should be noted that electric field strengths did notaffect the spectra. For the case of 1, 5, 10, 50, and 150 ppmarsenic pre-treated cells, the lower critical frequencies (LCF)were reduced from 35 kHz to 15 kHz, while the peaks of eachspectra remained at the critical conductivity value of 0.25S.m-1. The spectra for the boiled and the control cells arecompared in Figure 8.

4.3 Electrical properties of the cells

As would be predicted from Equation 8 and the realpart of the [CMF] in Equation 10, the lower critical frequencyand the dielectrophoretic velocity are solely dependent onthe capacitive and conductive properties of the cytoplasm(c, c), the membrane (m, m), and the suspending medium(s, s), respectively. These parameters generally affected dif-ferent characteristic features of the spectrum. It is clear thatthe dielectric properties of the membrane relate to the lowercritical frequency (LCF) and the magnitude of the negativevelocity spectra. For the cytoplasm, its conductivity and per-mittivity affect the magnitude of the positive velocity and thehigher value the HCF value, respectively. Cell size and themembrane thickness will also have effects and so differenttypes of cells would be predicted to behave differently. How-ever, qualitatively the results found for Tetraselmis (Table 1)are similar to those we found in yeast (Bunthawin et al.,2010). For example, the cytoplasmic conductivity (c) in bothcell types is about 0.3 S m-1 and membrane conductivity (m)is about 0.15 S m-1.

Combination of these sensitivities was employed toevaluate the dielectric properties of the control cells and the

Figure 4. Positive dielectrophoresis induction of Tetraselmis sp. intravelling wave electric field for 1.5 min (a-c), using thesignal of 50 kHz, 10 Vpp and s = 0.01 S.m-1.


Figure 5. Dielectrophoretic spectrum of the normal cells are affected by (a) the conductivities of the suspending medium (b) the electricfield strengths while it is not clearly affected by cell densities (c). Theoretical curves (the solid lines) were plotted by using thedielectric parameters shown in Table 1. The errors in fitting the experimental data to the theoretical expression are less than 10%(Bunthawin et al., 2010).


Figure 6. Plots of theoretical and experimental values of dielectrophoretic velocity as a function of field frequency. The theoretical lineswere plotted using the following parameters: 1S.m01.0 s , 16 S.m103 m , c = 1S.m37.0 , 48c , 78s , a= 10, b = c = 8 m, and = 13 nm.

Figure 7. Effect of concentrations of arsenic solution on the lower critical frequency spectra of the arsenic pretreated cells compared withthe normal cells. Experimental data (dots) were obtained from cell suspensions subjected to field strength of 28 kV.m-1. Theore-tical curves (lines) were simulated by using the dielectric parameters in Table 1.


arsenic pre-treated cells (Table 1). Arsenic was found toseverely affect both the conductance of both the cytoplasmand cell membrane (c and m) but arsenic decreases cwhereas it increases m. Effects are easily observable evenat the lowest concentration of arsenic used experimentally(1 ppm), suggesting that arsenic poisoning of cells could bedetectable at very low concentrations using the technique.

5. Conclusion

This study proposes the complete RC-model re-presenting a spheroidal cell in a travelling electric field as asolution to the confocal shell problem for a spheroid, i.e. alayer of non-constant thickness can be applied as well to

unicellular algal cells as it can to yeast (Bunthawin et al.,2010). The model consists of three resistor-capacitor pairsdescribing the conductive and capacitive of the cytoplasm,the cell membrane and the suspending solution with noapproximations being necessary. To describe a cell transla-tion in a travelling electric field, the dielectrophoretic velocityand two critical frequencies (LCF and HCF) were identified.Both are expressed in terms of the real part of the Clausius-Mossotti factor, which is used to describe cell dielectricproperties.

The RC approach provides a simpler to apply modelfor biological cells compared with the Laplace approach (seeAppendix A) because only the lower critical frequency [LCF]and cell translational speed are required to be measured ex-

Figure 8. Plot of the dielectrophoretic spectrum of the boiled cells as compared with the control cells. Theoretical curves (the solid andinterrupt lines) were plotted by using the dielectric parameters as shown in Table 1.

Table 1. Summary of the conductivities and dielectric constants of the cytoplasm, the cell membrane, and thesuspending medium for normal and arsenic pre-treated cells.

As pre-treated (ppm) Parameters Control Boiled cells

1 5 10 50 1501( . )c S m 0.37 0.030 0.030 0.028 0.025 0.013 0.06

1( . )m S m 0.17 3.0 3.1 3.4 3.7 3.8 301( . )c F m 48 48 48 48 48 48 911( . )m F m 8 10 12 16 21 32 20

1( . )s F m 78 78 78 78 78 78 78Cell dimension (Fig. 1) ( )a m 10 10 10 10 10 10 10Cell dimension (Fig. 1) ( )b c m 8 8 8 8 8 8 8Membrane thickness ( )nm 13 13 13 13 13 13 13


perimentally. This is a critical advantage over the Laplaceapproach (Gimsa and Wachner, 1998 and 1999; Bunthawinand Wanichapichart, 2007; Bunthawin et al., 2007) becausethe lower critical frequency [LCF] and translational speedsare easily measured experimentally on living cells, the highcritical frequency [HCF] is often not measureable on cells.By increasing the solution conductivity (s), two frequenciesconverge and join as soon as the s reaches a critical value(ct). Under this critical conductivity the cell experiences azero force, and the conductivity of the cytoplasm (c) can bepredicted. The RC model allowed the evaluation of permitti-vity and conductivity of the membrane and the cytoplasm ofcontrol and arsenic pre-treated cells. Since significant differ-ences could be detected between control and arsenic treatedcells the RC model offers the possibility that dielectrophoreticmethods could be used for separation of different types ofcells or even cells differing in metabolic status such assuccessfully fertilized cells from non viable egg cells in IVFprograms and cancerous from normal cells and for screeningcells such as genetically modified plant protoplasts in bio-technological applications. We found experimentally that thenumerical density of the cells in the cell suspension does notappear to be critical. This observation has important implica-tions for practical applications of the technique.

Acknowledgements

This work was partially supported by Faculty ofTechnology and Environment (TE) and Prince of SongklaUniversity (PSU), Thailand, through Biotechnology of Elec-tromechanics Research Unit. Thanks are also extended toNICA for providing the Tetraselmis sp. cell cultures and A.Tuantranont and his colleges (MEMS-NECTEC) for support-ing the microelectrode array.

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Abbreviations

a, long spheroidal cell axis (mm);b, short spheroidal cell axis (mm); , membrane thickness (nm);[CMF], Clausius-Mossotti Factor;Re[CMF], real part of C-M factor;Im[CMF] imaginary part of C-M factor;[HCF], High Critical Frequency;[LCF], Low Critical Frequency;

Lk is the depolarization factor which has threecomponents on axes x, y & z (Lx, Ly & Lz);

RC-model, Resistor-Capacitor model;c electrical conductance of the cytoplasm;m conductance of the cell membrane;

s conductance of the suspending medium; capaci-tive and conductive properties of the cytoplasm( smc ,, ), capacitance of membrane, cytoplasmand external medium respectively.

Appendix (A)

Two critical frequencies of Laplace approach

The effective complex dielectric of a homogeneous spheroid is defined as (Asami et al., 1980)

* * ** *

* * *

( ) (1 )( ) (1 )

m c m k keff m

m c m k

L v LL v

, (A1)

where k = x, y, z and 2

1 1a b

. The complex dielectrics of cytoplasm, membrane and suspending medium are

defined as cc c

o

j

, m

m mo

j

and s

s so

j

, respectively. Substituting these complex forms

into Equation A1 yields

* 0 0

0

0 0

( ) ( ) (1 )( ) ,

( ) ( ) (1 )

m c mm c m k k

meff m

m c mm c m k

j j L v Lj

j j L v

(A2)

where its real and imaginary parts are

2 20 0 0

2 20 0

1Re[ ] ,m m m meff

A B C D B A

C D

(A3)

2 2 2 20 0

2 20 0

1Im[ ] .m m m meff

C B A D A B

C D

(A4)

The constants A, B, C and D stand for

( ),( ) ,

( ),( ) ,

m c m

m c m

m c m

m c m

ABC bD

where (1 ), (1 ).k k kL v L L v


Landua and Lifschitz (1985) defined the Clausius-Mossotti factor ([CMF]) as [CMF]( )

eff s

s eff s kL

. For a

spheroid with the volume of 243

V ab (a case of Tetraselmis sp.), the real and the imaginary parts of the [CMF] are thenexpressed as

1 3 2 42 2

3 4

Re[CMF] [( )],A A A AA A

(A5)

1 3 2 42 2

3 4

Im[CMF] [( )],A A A AA A

(A6)

where

1 0

2

3 0

4

Re[ ] ,

Im[ ] ,

(1 ) Re[ ] ,

Im[ ] ( 1),

eff s

eff s

s k eff k

eff k s k

A

A

A L L

A L L

with 120 8.85 10 F.m-1, 2 f (rad.s-1), f in Hertz Re[ ]eff

and and the Im[ ]eff are from Equation A3 and A4,A4,

respectively.To obtain an express for two critical frequencies via Equation A5 and A6 are quite complex. The lower [LCF] and the

higher [HCF] critical frequencies of the model can, however, be separately derived from the condition Re[CMF]=0. Approxi-mations are made by using the principle of dielectric dispersion of a cell as suggested by Schwan (1988), i.e. the dielectricproperties of the cell membrane are prominent in the lower frequencies range (, dispersions). The complex dielectric formof each compartment becomes

cc

o

j

and ,c c

(A7)

mm m

o

j

and ,m m m oj (A8)

ss

o

j

and .s s

(A9)

On the other hand, those of the cytoplasm are dominant in the higher frequency range providing

cc c

o

j

and ,c c c oj (A10)

m m and 0,m

(A11)

ss s

o

j

and .s s s oj (A12)

Substituting Equation A7 to A9 and A10 to A12 to Equation A5 yields the angular frequency () dependence of theRe[CMF] for [LCF] and [HCF], respectively. For the lower case, the Re[CMF] is expressed as

21 3 2 4 5

2 2 23 4 5

( )( )Re[CMF] ,( )

sow

Z Z Z Z ZZ Z Z

(A13)

where


*1 0 ( )

* *2 0 ( )

* *3 0 ( )

* *4 0 ( )

*5

Re[ ],

Im[ ],

Re[ ],

Im[ ],

( 1),

eff ow

eff ow

k eff ow

k eff ow

k s

Z

Z

Z L

Z L

Z L

and2

* 6 7 8 9( ) 2 2 2

7 9

Re[ ] ,eff owZ Z Z Z

Z Z

(A14)

* 6 9 7 8( ) 2 2 2

7 9

( )Im[ ] ,eff owZ Z Z Z

Z Z

(A15)

where

26 0

*7 0

*8*9

[ ( )],[ ( )],

,

,

m m c m

m c m

m c

c

Z aZ b

Z a

Z b

with2

(1 ), (1 ), 1 1 .k k ka L v L b L v va b

Setting Eq. (A13) = 0 and then solving for , yields the functions

1 2 30 6

1 1 2 ,2ow

kZ L

(A16)

since 2 [LCF],ow then

1 2 30 6

1 1[LCF] 2 ,4 kZ L

(A17)

where

2 2 2

1 0 7 8 7 0 6 9 0

2 2 2 32 0 7 6 7 8 9 0 6 9

4 4 3 3 2 3 27 0 7 8 0 7 8

2 23 0 7 7 8 0

2 (1 2 ) 2 (1 ) 2 (1 2 ) 2 ,

(6 1) 2 2 (1 2 )

( 2) 2 (2 1) 4 ,

6

s k s k s k k

s k s k k

s k k s k s k

s k s

Z Z L Z L Z Z L L

Z L Z Z Z Z L Z Z L

Z L L Z Z L Z Z L

L Z Z Z

2 20 6 7 8 9

2 2 30 7 6 7 9 0 8 0 7 6 9

2 2 2 2 2 4 4 20 6 9 7 0

8 (1 )

2 ( ) 2 (2 3)

( ) .

k s k k

s s k s k k

s k s k

L L Z Z Z Z L

Z Z Z Z Z L Z Z Z L L

Z Z L Z L

A similar procedure was applied for the higher critical frequency, yielding

1 2 30 6

1 1[HCF] 2 .4 kZ L

(A18)

Theoretical plots of the lower and the higher critical frequencies are the same result as the RC-model but the latter ismore easily to derive.

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