Electro – Mechanical contributions to low frequency dielectric responses of biological cells in Colloidal Suspension.

We investigate electro-mechanical contributions to the low frequency dielectric response of biological cells in colloidal suspension. Prior simulations of biological cells in colloidal suspension yield maximum dielectric constant values about 103 in magnitude as the frequency of applied electric fields drops below the kHz range. Experimentally measured relative dielectric values in yeast cells , on the other hand, have maximal values up to 107 - 108 . We consider both electrical and mechanical energy stored in cellular suspension and show that low frequency mechanical contributions can give rise to dielectric constant values of this magnitude.


Introduction
Biological cells in colloidal suspension are often modeled as having primarily electromagnetic interactions with an external ac electric field. Except for electro-rotation, there has been no discussion of mechanical effects in the α dispersion range. Prior numerical simulations [2] used formalism appropriate for β dispersion effects, i.e. Maxwell -Wagner based dispersion models [1] Experimental values for the low frequency differ from what is predicted using Maxwell-Wagner [3] based calculations. We argue that this discrepancy is due to electro -mechanical effects which are not significant in higher frequencies because of inertial effects. We show that the mechanical contributions in the α dispersion range can result in effective dielectric constant values up to 7 10 - 8 10 , whereas β dispersion effects only give maximum dielectric constant values of about 3 10 in magnitude. In this paper we examine how electromechanical rotation of cells can contribute to a more realistic dielectric models of cells in colloidal suspension .

Model
Experimentally , it is found that the complex dielectric constant * ε for N cells in a colloidal suspension of volume V has distinct dispersion regions denoted by α , β , and γ . One of the most recent models for complex dielectric response [5] , is given by : ε is a very high frequency contribution to the dielectric , and is about ten to twenty hertz in value. This is for γ dispersion and simply is ignored in β and α dispersion regimes when we consider lower frequency dispersion effects. The second and third terms are for β dispersion and have a real valued magnitude of about 10 3 which is in turn negated when we look at the real part of the fourth ( last ) term due to α dispersive effects with a real valued magnitude of about 10 8 in upper value. The We should take into consideration that We shall now attempt to make a general derivation of cell ω so as to give a detailed experimentally accessible formulation of how angular velocity of a cell influences formation of actual dielectric values, using equation 2.7 above.

Rotational Spectra of Biological Cells in Electric Field
We are , here , setting up a time independent average value of the frequency of rotation ( actually the angular velocity ), which we will call ( ) θ ω , 2 t cell which is a spatial and time averaged quantity. In order to do this, we will set up a relationship between a polarization vector with regards to net charge in the cell, and the external electric field impinging upon the cell, to get a net torque, and then from there to set up a differential equation relating the net torque with angular velocity, cell moment of inertia, and an added damping coefficient we will call D in order to set up a general expression for Let us examine an external field torque upon a cell, with an equation of : Here, we have that we have an external electric field 0 E which is at a given angle θ with respect to a dipole moment p of the cell in colloidal suspension. We shall be comparing this torque with moment of inertia of the cell times the time derivative of rotational frequency plus an additional term which is composed of damping coefficient D times the rotational frequency of the cell. A critical assumption for making this work is that the frequency of rotational movement of the cell , cell ω , is far smaller than that of the applied AC electric field, 0 ω . If we do this, we have that any time we can set we may treat the frequency of the cell as a different quantity than the frequency of the applied electric field. Also ,if Should we be not be making this assumption, we would be writing, This assumes that 0 ω ω ≈ cell in a resonance condition. We are assuming otherwise , here. First , a ring of cell shell space an angle θ from an axis of rotation of the cell, with a radius distance a from the center of the sphere , and a thickness of the shell as θ d leads to a net torque on that particular shell of the cell we may write as : We have that λ is the thickness of the 'shell' . For our purposes, we set a ⋅ = 2 .
λ . Also we have that = ⋅ ⋅ λ ω θ sin a gradient of the 'velocity' of the ring 'surface' of the cell, and that the surface area of the 'ring' is given by viscosity of the ring as its net ' friction' with respect to the medium the cell is in colloidal suspension with is given by η . Here, by dimensional analysis, we have that The area A is times a net velocity change, divided by the assumed shell thickness of the cell. . We can then get , if we integrate over the entire sphere , a total torque of As an example, we derived , for a general dipole moment of the cell  (3.14) where angle θ is between the applied electric field to the cell and the net dipole value written up in equation 3.13. And, = a cell radius which can be varied as one sees fit.
which as we will see in the next section varies wildly as we change the radius of cells in the colloidal suspension . Furthermore, we have that we may set up a maximum value for the cell dielectric constant which is dependent upon the radius of the cell and the angle α which is measuring the impact of charge distribution on the cell ends, i.e. : Here, the variable f refers to the frequency of the applied AC electric field impinging directly upon the cell in colloidal suspension. We also can take this angular cellular velocity and then put it directly into a given dielectric constant of the cell , as

IV. Basic results from the above relationships of section III.
We should now discuss some of the basic implications of our model and what we can expect experimentally, if the following predictions are true. First of all, we managed to find a way to duplicate the curve for α dispersion as a function of AC applied electric field frequency rates. This is assuming  [7] in 1998 gave a more typical representation of an electro rotational spectra of biological cells in colloidal suspension. We shall write it here and compare it with what we wrote for equation 3.7 above. In addition, we shall also refer to some issues affecting the onset of electrorotation which will be to show how non uniform charge distribution in cell structure will lead to torque allowing us to consider a rotational model along the lines we wrote above .We should note that the electro rotation we are working with is not

V. Conclusion
Asami [2] and other authors actually calculated realistic dielectric values for cells in colloidal suspension for the β region of dispersion values. Those papers correctly calculate the electromagnetic contribution to the low frequency dielectric constant (as well as conductivity!). But cell anisotropies and inhomogeneities result in a polarization vector that is not parallel to E. Note in our calculation we assumed that P was PERPENDICULAR to E. Of course this represents and extreme case and in general the angle between P and E can vary. So our model is somewhat idealized. Also we need to mention in the discussion that brownian motion and the elastic energy stored in some cells may also give a significant contribution to the low frequency dielectric constant.
(The elastic contribution may be significant in tissue for example) Also we hope that this work will motivate experiments to investigate mechanical contribution to the dielectric response in the alpha range Our paper gives a useful start in outlining the importance of what we refer to as electromechanical effects in the calculation of a net dielectric value sus ε when we are considering when we have applied an electric field to cells in suspension in the low hertz limit for α dispersion effects. This approach gives order of magnitude agreement with some experimental data sets. . One paper actually claims to be able to link both α and β dispersion [8], by use of charge mobility. This may be appropriate for some biological systems, but it neglects what we think is an unexplored effect which has been seen experimentally. . Another paper [9] is interesting, but is heavily weighed toward adjustment of what they call geometrical parameters in order to obtain dielectric values for biological cells considerably below our maximal values. Both of these mentioned approaches have been extensively utilized in β dispersion , but do not make sense when very low frequency AC electric fields are applied to biological cells in colloidal suspension. Additional work needs to be done to consider a range of effects , i.e. possible interaction effects between biological cells in low frequency AC electric fields .
However, we believe that the methodology outlined is a necessary beginning to start a systematic investigation of α dispersion effects with biological cells . .