Measuring living cells using dielectric spectroscopy

EISEVIER

Bioelectrochemistry

and Bioenergetics40 (1996) I33- I39

Measuring living cells using dielectric spectroscopy Eugen Gheorghiu Institute of Biotechnology-BIOTEHNOS,

Biophysics Department, Dumbrava Rosie 18. 70254 Bucharest, Romania

Received 20 November 1995;accepted27 December 1995

Abstract The possible derivation of cellular parameters (living cell volume concentration, cell number distribution over cell cycle phases, complex permittivity of extracellular and intracellular media and morphological factors) using dielectric spectroscopy on (non)spherical cell suspensions is described. In this respect, several applications to biotechnology and associated areas, as well as their specific requirements, are emphasized. Recently developed models on the dielectric behaviour of suspensions of spherical and arbitrarily shaped cells are presented. Diffusive and shape effects are discussed in relation to specific a and p dispersions. Keywords: Biotechnology; Cell suspension;Cellular division cycle; Dielectric spectroscopy

1. Introduction

Rapid, in vivo impedance spectroscopy of living cell suspensions

The fast, non-invasive measurement of biological cell properties (volume concentration, electrical and morphological parameters) is an important goal when monitoring living cells. Many applications can be found in present-day biotechnology and related fields, e.g. biomass monitoring, sterilization control and quantitative evaluation of drug effects at the cellular level (which may be successfully used in pharmacological studies avoiding a range of preliminary tests on animals). We aim to derive non-sophisticated models (with a minimum number of parameters) able to provide reliable data, sensitive to the distinct biological events during the cellular division cycle. The electrical and morphological properties

of the cell

membrane

and intracellular

and

extracellular media are assumed to represent sensitive parameters of the cellular state. Whenever appropriate models are available, these parameters may be derived straightforwardly from cell suspension permittivity data. The non-invasive character of impedance measurements is shown by the negligible level of the field-induced membrane potential, less than 0.05% of the resting potential value, on application of a low intensity harmonic field (E, s 100 Vm-I, or= 1 Hz to 10 MHz). In order to measure living cells, the following steps are performed.

Processing

the complex

permittivity

the time

series of cellular

of the

to appropriparameters

to

derive the invariants under cellular dynamics (measures specific to the ergodic theory of chaos). Evaluating the cellular effect of different stimuli by means of the variance of these invariants with respect to standard growing conditions. A quantitative procedure to characterize the cell status and dynamics of synchronized (non)spherical cell (e.g. the yeast Saccharomyces cerevisiae) suspensions by fast dielectric spectroscopy is presented. This paper deals with the theoretical aspects of the dielectric behaviour of cell suspensions during cell cycle progression and does not emphasize the experimental problems of the method.

2. Dielectric behaviour of cell suspensions The measurement of a physical system requires a nonperturbing interaction, as well as the development of a model specific to both the system and the interaction field parameters. The electrical (dielectric) properties of biological systems are assumed to be intimately connected with

0302-4598/96/$1X00 Copyright 0 1996Published by Elsevier ScienceS.A. All rights reserved. P/I SO302-4598(96)05066-O

to derive

bulk. Fitting of the data (suspension permittivity) ate models.

134

E. Gheorghiu /Bioelectrochemistry

and Bioenergetics 40 (1996) 133-139

the state of the living system. Consequently, appropriate models to describe the dielectric behaviour of biological systemsmust be achieved. We have accomplisheddistinct treatmentsfor spherical and arbitrarily shapedcells. 2.1. Spherical cell suspension

L

Our aim has not been to determine a complete electrical model of a living cell, but rather a simplified model, comprising the major features of the biological cell and showing the samebehaviour on application of a harmonic, low intensity electric field (frequency range, 1 Hz to 10 MHz). Therefore the dielectric behaviour of a suspension of diffusive, conducting spheres, surrounded by low conducting shells with a negligible diffusion coefficient and

J

Fig. 1. Spherical particle surrounded by a membrane with fixed negative charge on the inner side.

a

6

4

2

Fig. 2. The suspension permittivity of (r and p dispersions.

2 6-

\

(3) ___-__-_-_-_-_-----.---. 1

_--.._--2

--.-3

:2%-,

_ 4

5

toPlo

Fig. 3. The (Y dispersion for resting potentials DV, = - 150 mV (11, DV, = - 10 mV (2) and DVO = 0 V (3).

E. GheorRhiu/Bioelectrochrmistry

und Bioener~etic.s

regular surface distribution of fixed charges on the inner side, has been discussed [I]. We consider (as in Fig. 1) a spherical particle with a complex permittivity

1.35

40 (IS?%) 133-139

The expression for the membrane potential has the following form [1,2]

(1) Neglecting the effect of active transport and cytoplasmic non-homogeneities on the dielectric behaviour, this model is assumed to exhibit a strong similarity to spherical biological cell suspensions of (Y and p dispersions (which occur at 100 Hz to 10 kHz and 100 kHz to 10 MHz respectively). The equations for the potential are considered in the quasi-static approximation (Poisson equation for media 1 and 3 and Laplace equation for the shell).

where DV, is the membrane potential when no field is applied and DV, is the field-induced membrane potential. 2.1.1. Equivalent permittivity of suspension The complex permittivity aS of a homogeneous sphere is derived as part of the Maxwell-Wagner formalism [3,4]

2.4' 2.3.

2.i' 2 . 1.9'

1.8’ b

6

6.5

7

7.5

log[ *II I

oative charge on the inner side of the inner shell; Fig. 4. (a) Spherical particle surrounded by a diffusive membrane bounded by non-diffusive shells, with ne, (b) p dispersion for cells with nucleus (I) (R = I pm, d = 1 pm) and without nucleus (2) (R = 0 Frn, d = 0 km).

136

E. Gheorghiu E

experimental

E theoretical

/ Bioelectrochemisrry

(.

(-

and Bioenergefics

40 (1996)

133-139

)

1

Fig. 5. Fitting of the experimental data to the model (Eq. (3)).

using the condition that the whole suspension has the same dielectric behaviour as a homogeneous sphere suspended in medium 3. If p is the volume concentration of spherical particles in the suspension and H is the coefficient of the dipolar term of the potential in the extracellular medium [ 11, the following expression for the suspension equivalent complex permittivity is obtained

3P[H/qWo)]

&g =Eg3 I + I--p[H/(R;E,)]

i

Fg=&+-’

CT iws,’

g =- 5 g i 6x0

2.1.2. Discussion Eq. (3) for es enables the description of both (Y and B dispersions as shown in Fig. 2. This approach emphasizes the relationship between (Y dispersions and the field-induced displacement of the

lnitiatian rat DNA Synthesis 0

\

DNA Synthesis

Fig. 6. Landmarks of SaccIwromyces cereuisiae

@

charge on the external side of the membrane. Consequently, the resting potential of biological cells can be derived by (Y dispersion measurements (Fig. 3) [ll. This model has recently been developed by considering the diffusion effects of the presence of the membrane and nucleus on the dielectric behaviour of a spherical cell suspension (Fig. 4(a)) [5]. The diffusive shell in the membrane is introduced to simulate the effect of the presence of ionic channels on the dielectric behaviour of a biological cell suspension. The additional non-diffusive shells model the cellular ability to prevent system dissipation. By taking into consideration the diffusive phenomena, as well as the presence of the membrane charge densities, important effects are observed in the cx dispersion range, whereas in the B range the related effects are rather small. It should be noted that, although membrane diffusion affects the LYdispersion, no significant effect of the presence of the cell nucleus on the suspension permittivity of the (Y or B dispersion is observed (Fig. 4(b)). It should be noted that, when dealing with cell suspensions with shapes that can be regarded as spheres, we managed to derive a fast, robust fitting procedure enabling the accurate determination of the cell concentration, intracellular permittivity and conductivity (Fig. 5). For this purpose, the extracellular and suspension complex permittivities must be measured independently.

Bud emergence

cell division.

Fig. 7. Suspension of non-spherical bodies.

E. Gheorghiu/Bioelectrochemistry

i” i7 #

E’ 0

Vs 5 i.z Y e ES

S

SS

Fig. 8. A non-spherical shelled body.

2.2. Non-spherical cell suspensions As shown in Fig. 6, during the cell division cycle, the cell shape is usually non-spherical. Consequently, we were interested in taking into consideration the shape effect on the dielectric behaviour of the cell suspension. Let us consider a homogeneous medium with a volume V and complex permittivity eO (eg3) in which randomly oriented, (non)spherical bodies (with complex permittivity ei) are suspended (Fig. 7). A uniform electric field j!$ = E,@ is applied to the entire ensemble. The suspension permittivity E,, (eg) relates the volume average of the displacement to the volume average of the electric field 6= q,,z

(4) The suspension permittivity is derived using the mean field method by taking into account the mutual dipole-dipole interaction between suspended bodies. We consider a regu-

and

Bioenergetics

40 11996)

133-139

137

lar distribution of particles, i.e. the suspension is equivalent to a rectangular lattice containing one cell (of volume V,) in the centre of each element. The electric field has the same distribution regardless of the lattice unit emplacement. The total electric field is given by means of two sources E=E,+E,~(E,)=(E)-(E,)

(5)

E, denotes the effective field due to the external sources (including the dipoles of the particles outside the respective lattice unit), E, represents the electric field of the induced dipole by cell polarization in the effective field and the angular brackets indicate averaging over the lattice unit volume V (which is equivalent to averaging over the entire suspension volume). Considering the field-induced polarization, we have (6) P denotes the polarization due to the effective field in the centre of the lattice element (I’> = ~,aE,1,;;

(E,) = -+)

where C is the interaction constant; in the dipolar approximation, C = l/3 for spheres in a periodic lattice. For low concentrations of suspended particles, we assume that C has the same value regardless of the particle shape. Consequently, the suspension permittivity is given by a&O & sus

=&o+q

al 0.0

0.2

0.4

0.6

0.8

1 .o

1.2

1.4

Fig. 9. Cell shapes simulated by JZq.(IO).

-p(cz/3)

(8)

138

E. Gheorghiu / Bioelecrrochemistry and Bioenergerics 40 f 1996) 133-139

Fig. 10. The shape effect on the dielectric behaviour of a suspension of shelled bodies: 1, spheres; 2, oblate spheroids.

where p represents the volume ratio of the bodies (p = VJV). It is worth noting that, when dealing with spheres, Eq. (3) is equivalent to the classical Maxwell-Wagner equation & S”S- &o 8,“s + 2&O

=

3. Conclusions

Ei - Fo

P

Ei + 2E0

= P( 43)

(9)

The problem remains to determine a method to compute the polarizability of an arbitrarily shaped particle using Eq. (8). To this end, we must derive the field distribution inside the body in relation to the effective electric field [6]. Fig. 8 represents the geometry of a prototype body. By considering the following expression r(e)=ao+a,cose+Cos*e

where the respective polarizabilities are determined from data on synchronized cell progression.

(‘0)

the whole range of cell shapes during cell cycle progression can be simulated (Fig. 9). When applying our formalism to the dielectric behaviour of a suspension of shelled ellipsoids, the same shape dependence has been derived [7] (Fig. 10). The use of this study to reveal cell cycle progression is based on the assignment, to each cell state (of cellular division), a specific set of shape factors and electrical properties, i.e. a distinct polarizability a j. The result of synchronization loss can be regarded as a superposition of these states, with different weights kj, which can be derived by fitting the data to the following expression

(11)

When analysing the contribution to the suspension permittivity of the cellular parameters (cell volume ratio p, cell/membrane volume fraction, complex permittivity of the membrane, intracellular/extracellular media and shape factors) via Eq. (1 l), the calibration curves directly relate the suspension permittivity at different (few) frequencies to the living cell concentration p and the contributions of other factors are ignored; consequently, important errors (larger than 10%) occur. Whenever applications do not require a high accuracy, the direct derivation of the amount of biomass by means of calibration curves can be successfully used, especially when dealing with (almost) spherical cells and when the cell/membrane volume ratio is preserved. The rapid measurement of quantitative data on living cell distributions over the cell cycle phases (e.g. characterization of a drug effect at the cellular level) can be supported by fitting the experimental data to an expression similar to Eq. (11). Using present-day equipment, impedance measurements enable the derivation of: * (time series of) the resting potential by means of in vivo measurements; . viable cell concentration (enabling biomass monitoring and sterilization control); * cell number distribution over the cell cycle phases.

E. Gheorghiu/

Bioelectrochemi.wry

In addition, a quantitative evaluation of the cellular effect of different (chemical, physical or biological) stimuli can be achieved. Our method is not restricted to suspensions of .%&aromyces cereuisiae, but can be applied to any type of normal or neoplastic cell exhibiting cellular division.

References [I] E. Gheorghiu, The resting potential in relation to the equivalent complex permittivity of a spherical cell suspension, Phys. Med. Biol., 38 (1993) 979.

and Bioenerptics

40 11996) 133-139

139

[2] E. Gheorghiu, The dielectric behaviour of suspensions of spherical cells: a unitary approach, J. Phys. A: Math Gen.. 27 (1994) 3883. [3] J.C. Maxwell, Electricity and Magnetism, Vol 1 Clarendon, Oxford, 1892. [4] K.W. Wagner, Arch. Electrotech., 2 (1914) 371. [5] E. Gheorghiu, Dielectric behaviour of spherical cell suspensions in relation to diffusion effects and nucleus presence, Bioelectrochem. Bioenerg., 38 (1995) 123-127. [6] D. Vrinceanu and E. Gheorghiu, The dielectric behaviour of nonspherical cell suspensions, Bioelectrochem. Bioenerg., 00 (1996) OCO. [7] K. Asami, T. Hanai and N. Koizumi, Jpn. J. Appl. Phys., 19 (1980) 359.