# Protein Dielectrophoresis: I. Status of Experiments and an Empirical Theory

^{1}

^{2}

^{*}

## Abstract

**:**

^{21}V

^{2}/m

^{3}required, according to current DEP theory, to overcome the dispersive forces associated with Brownian motion. This failing results from the macroscopic Clausius–Mossotti (CM) factor being restricted to the range 1.0 > CM > −0.5. Current DEP theory precludes the protein’s permanent dipole moment (rather than the induced moment) from contributing to the DEP force. Based on the magnitude of the β-dispersion exhibited by globular proteins in the frequency range 1 kHz–50 MHz, an empirically derived molecular version of CM is obtained. This factor varies greatly in magnitude from protein to protein (e.g., ~37,000 for carboxypeptidase; ~190 for phospholipase) and when incorporated into the basic expression for the DEP force brings most of the reported protein DEP above the minimum required to overcome dispersive Brownian thermal effects. We believe this empirically-derived finding validates the theories currently being advanced by Matyushov and co-workers.

## 1. Introduction

^{6}V/m) were considered to be substantially lower than standard DEP theory predicts [2]. In fact, the word ‘substantially’ can be considered as an understatement of the situation. As reviewed elsewhere, at least 22 different globular proteins have now been investigated for their DEP responses [3,4,5,6,7]. In all the analyses by the authors of the cited studies, the so-called Clausius–Mossotti (CM) function has been invoked. However, the macroscopic electrostatic concepts and assumptions used in the theoretical derivation of CM arguably fail to describe the situation for nanoparticles, such as proteins, that possess a permanent dipole moment, interact with water dipoles of hydration, and possess other physico-chemical attributes at the molecular scale [6,7,8]. The fact that standard DEP theory does not provide a basis for understanding protein DEP is recognized as “a well-accepted paradigm, repeated in numerous studies” [6]. In another recent review it is correctly stated that protein DEP remains under development because due to their small size proteins “require greater magnitudes of electric field gradients to achieve manipulation” [7]. Put more bluntly, protein DEP is considered to not have a theoretical leg to stand on!

- (i)
- The internal electrical field E
_{i}induced in an uncharged (or uniformly charged) spherical particle, of radius R, located in an electric field E_{m}within a dielectric medium is given by:$${E}_{i}=\left(\frac{3{\epsilon}_{m}}{{\epsilon}_{p}+2{\epsilon}_{m}}\right){E}_{m}$$_{p}and ɛ_{m}the relative permittivity of the particle and surrounding medium, respectively. It is assumed that ɛ_{p}and ɛ_{m}are well defined. At the molecular scale this requires certain conditions to be met regarding dipole–dipole correlations. Boundary conditions also assume that the electric potential, current density and displacement flux are continuous across an infinitesimally thin surface at the sphere’s interface with the surrounding medium. Fine details such as those that occur, for example, at the molecular interface between a protein and its hydration sheath are not considered. - (ii)
- The induced polarization P
_{p}per unit volume of the sphere is given by:$${P}_{p}=\left({\epsilon}_{p}-{\epsilon}_{m}\right){\epsilon}_{o}{E}_{i}=3{\epsilon}_{o}{\epsilon}_{m}\left(\frac{{\epsilon}_{p}-{\epsilon}_{m}}{{\epsilon}_{p}+2{\epsilon}_{m}}\right){E}_{m}$$_{o}is the permittivity of vacuum. The macroscopic dielectric concepts involved in this equation and throughout this paper are summarized in Figure 1. It is assumed that the polarization P_{m}of the surrounding medium remains uniform right up to the particle–medium interface. This assumption requires examination at the molecular scale. - (iii)
- The dipole moment m of the sphere is the value of P
_{p}multiplied by the sphere’s volume:$$m=4\pi {R}^{3}{\epsilon}_{o}{\epsilon}_{m}\left(\frac{{\epsilon}_{p}-{\epsilon}_{m}}{{\epsilon}_{p}+2{\epsilon}_{m}}\right){E}_{m}$$_{p}and ɛ_{m}, CM is limited to values between +1.0 (ɛ_{p}_{>>}ɛ_{m}) and −0.5 (ɛ_{p}_{<<}ɛ_{m}). This represents a severe limitation, at the macroscopic scale, to the range of effective dipole moment densities that a particle can assume. - (iv)
- For the case where E
_{m}has a gradient, the particle experiences a DEP force given by:$${F}_{DEP}=\left(m\cdot \nabla \right){E}_{m}$$_{m}is assumed irrotational (i.e., ∇×E_{m}= 0). This assumption holds if E_{m}is said to be a conservative field. In our particular case of DEP, this means that moving a polarized particle from location a to b, and then back again to location a, will involve no net expenditure of work by the field. The actual path taken in moving from say a to z is of no relevance. In the language of thermodynamics each infinitesimal change in location is reversible. At the molecular level, the DEP motion of a protein involves the breaking (enthalpy absorbed and entropy increased) and remaking of hydrogen-bonded water networks at the hydrodynamic plane of shear. Some interesting variations of changes in Gibbs free energy (ΔG = ΔH − T ΔS) might occur. The response of an assembly of dipoles to an external electric field is basically a thermodynamically non-equilibrium process—the thermal energy is never equally distributed among the various degrees of motional freedom of the dipoles. Perhaps, at the molecular level, each infinitesimal change in location is not reversible?

_{o}. The form of CM (the term in brackets) shown in Equation (5) is valid at high frequencies (typically >50 MHz). At DC and below ~100 Hz

## 2. The Basic Problem to Be Empirically Resolved

_{DEP}are particle size and the magnitude of the field-parameter $\nabla {E}_{m}^{2}$. In a first approximation a cubic root relation is expected regarding the physical dimensions of a globular protein and its molecular weight. However, an empirical relationship, provided by Malvern Panalytical

^{®}in their calculator software, gives a good estimate of a protein’s hydrodynamic (Stokes) radius. This relationship is plotted in Figure 2, to show that those proteins reported to exhibit DEP responses have radii in the range 2–7 nm. Values of E

_{m}and of $\nabla {E}_{m}^{2}$ in the ranges 10

^{5}–10

^{8}V/m and 10

^{12}–10

^{24}V

^{2}/m

^{3}, respectively, are reported for the DEP translocation and trapping of protein molecules. We can ask to what extent these fields and their gradients are consistent with the expectations of the current theoretical model of DEP when applied to a globular protein molecule. This question can be addressed by considering both the time-averaged free energy (U

_{DEP}= −(mE

_{m})/2) of an electrically polarized particle and the work required to overcome the maximum dispersive (diffusional) force acting on it [4] (pp. 352–353). The first approach addresses the extent to which U

_{DEP}represents a sufficiently deep ‘trap’ to compete against thermal energy (3kT)/2 associated with Brownian motion. Using the relationship given for the dipole moment m in Equation (3):

_{T}of a polarized protein molecule is the sum of the Brownian thermal energy and U

_{DEP}:

_{T}must have a negative value—it should represent a sufficiently deep potential energy well for the molecule to be trapped for a time equivalent to the inverse of its probability to escape. For proteins such as bovine serum albumin (BSA) and avidin (R ≈ 3.5 nm, T = 300 K, and assigning CM = 0.5) suspended in an aqueous medium (i.e., ɛ

_{m}≈ 78) the required field is E

_{m}≥ 2.3 × 10

^{7}V/m. The maximum dispersive force per protein molecule is equal to -kT/2R. For R ≈ 3.5 nm and T = 300 K, this force is 1.2 × 10

^{−12}N. For the protein molecule to exhibit DEP it must oppose this dispersive force. With the expression for F

_{DEP}from Equation (5) or from Equation (7), using the definition ${F}_{DEP}=-\nabla {U}_{DEP}$, this implies the following condition must hold:

^{21}V

^{2}/m

^{3}. As discussed in Section 3.7, only two of the reported values of $\nabla {E}_{m}^{2}$ have exceeded this minimum value. In one reported DEP manipulation of BSA a value of 10

^{12}V

^{2}/m

^{3}is cited! There are also the interesting cases where both positive and negative DEP of BSA have been reported at DC and 1 kHz, and where DEP of opposite polarities have also been reported for the same protein types at DC.

## 3. The Status of Protein Dielectrophoresis (DEP) Experimentation

#### 3.1. Summary of Protein DEP

^{3}. The volume occupied per molecule is thus 1.66/C nm

^{3}for a C molar solution. The separation distances shown in Figure 6 were calculated by taking the cube root of the volume per molecule. A wide range of values for the field gradient factor ∇E

^{2}has been reported by the various investigators for a range of proteins, or has been estimated by Hayes [5] in his review. These values are shown in Figure 7 for both iDEP and eDEP studies, together with an indication of the minimum value of $\nabla {E}_{m}^{2}$ (3.5 × 10

^{21}V

^{2}/m

^{3}) calculated according to Equation (9), required to attain a DEP force that overcomes the dispersive forces of Brownian motion. The adjusted minimum value (~4 × 10

^{18}V

^{2}/m

^{3}) based on the empirical relationship described in Section 3.4 is also shown in Figure 7.

#### 3.2. Bovine Serum Albumin (BSA)

#### 3.3. The Dielectric β-Dispersion

^{−27}Cm) and 636 D (2.12 × 10

^{−27}Cm), respectively. The angle between the dipole moment and the long axis of the monomer was determined to be 50°. Moser et al. [53] performed dielectric and transient birefringence measurements on BSA solutions of concentrations in the range 0.2–1.4 mM and observed the effect of strong intermolecular interactions. In their measurement of the β-dispersion, Grant et al. [54] considered that the BSA concentrations (0.6–5.5 mM) were “high enough to permit molecular interaction”.

#### 3.4. Empirical Relationship Connecting Clausius–Mossotti (CM) and the β-Dispersion

_{v}, the dielectric increment Δɛ depicted in Figure 8, as well as the conductivity increment Δσ characterizing this dispersion in terms of the increase of conductivity of the suspension with increasing frequency, are given by:

_{v}and to derive multi-shell models for analyzing impedance and DEP measurements on cell suspensions [4]. However, as stated elsewhere [9] (without the following explanation), Equation (10) is not applicable to protein suspensions. According to the Maxwell–Wagner mixture theory for particle suspensions, the measured effective permittivity ε

_{eff}of a dilute particle suspension is given by [4] (pp. 222–223):

_{p}and ε

_{m}the particle and medium relative permittivity, respectively. The term effective permittivity is used to signify that a defined volume of a particle suspension may be replaced conceptually with an equal volume of a homogeneous medium of smeared-out bulk properties.

_{eff}≈ ε

_{m}, implying that for a sufficiently large observation scale a heterogeneous compound material can be considered as a homogeneous one. Inserting this approximation into the denominator of the left-hand side of Equation (11) leads to the expression for Δɛ in Equations (10). However, an instructive result is obtained if this is applied to form a relationship between the Clausius–Mossotti factor CM and the dielectric increments depicted in Figure 8. For a dilute protein suspension, this relationship should thus be of the form:

_{w}and C

_{p}, ρ

_{w}and ρ

_{p}, respectively, are the molar concentration and mass density of pure water and the protein, respectively. The concentration C

_{w}of pure water is taken as 55.5 M (1000 g/L divided by its molecular weight of 18 g/mol), and protein density values can be derived using the molecular-weight-depending function derived by Fischer et al. [55]. Equation (12) qualitatively predicts that in a frequency range where there is a dielectric increment $\Delta {\epsilon}^{+}$ a positive value for CM and a positive DEP response will result. As indicated in Figure 8 the opposite case should also hold for the high-frequency range where a dielectric decrement is exhibited. But can the CM-factor of Equation (12) simply be inserted into Equation (5) to describe protein DEP? Moser et al. [53] obtained $\Delta \epsilon /{C}_{p}$ = 1.11 per mM for a BSA monomer concentration, so that with ε

_{m,}= 78.4 and ρ

_{p}= 1.41 gm/cm

^{3}, Equation (12) yields the result CM = 369. This is not possible according to the definition and limited range of values (1.0 > CM > −0.5) of the macroscopic CM factor derived from Equation (2) for the induced polarization P

_{p}per unit volume of a particle. Furthermore, based on the work of Takashima and Asami [56] who obtained values for $\Delta \epsilon $ (per mM protein concentration) of 5.06 and 37.24 for cytochrome-c and carboxypeptidase, respectively, the corresponding values obtained for CM are 1745 and 12,480, respectively! This is the basis for stating [9] that the macroscopic theory leading to Equation (10) cannot be employed at the molecular level.

_{eff}≈ ε

_{m}, the identity ε

_{eff}= κε

_{m}is inserted into the denominator of the left-hand side of Equation (11) we obtain the relationship:

_{micro}= (κ + 2)CM

_{macro}

_{m}must be uniform right up to this boundary. Located within this mathematical boundary is the hydrodynamic plane of shear (defining the zeta-potential determined by electrophoresis) and the protein’s outer hydration sheath. It is tempting to propose a conceptual equivalence of Equation (14) in terms of the ratio of two polarizations and interfacial dipole moment free energies:

#### 3.5. The β-Dispersion and Dipole Moment Density

_{m}from zero (where D

_{m}= 0) to its final value (D

_{m}= ɛ

_{o}ɛ

_{r}E

_{m}) in the medium, that has a total volume V

_{m}; (ii) reduce D

_{m}by removing from the medium a cavity of volume v

_{p}large enough to contain the hydrated protein molecule; (iii) account for the incremental change (either positive or negative) of the medium polarization resulting from its interaction with the field of the protein’s induced and permanent dipole moment. These three actions can be expressed in the form [4] (pp. 87–89):

_{m}is very much larger than v

_{p}and so the first integral in Equation (16) represents a significantly larger contribution to ΔU than the second integral. The δU term thus plays a significant role. In the macroscopic derivation of the Maxwell–Wagner mixture theory that leads to Equation (10) the assumption is made that ε

_{eff}≈ ε

_{m}. This effectively removes the requirement for calculating the δU term in Equation (16), which at a molecular scale is a significant weakness. Evaluation of δU can conceptually, for our present purpose, be accomplished by assuming the applicability of the boundary condition regarding continuity of the normal component of displacement flux (D = ɛ

_{r}ɛ

_{o}E

_{m}) across the interface between the solvated protein and the bulk medium. The free energy change δU is then given by an integral of the following form [4] (p. 89), taken over the protein’s effective cavity volume υ

_{p}:

^{+}has a finite value in Figure 8, the protein’s effective permittivity ε

_{p}can be regarded as being greater than ε

_{m}. The integral in Equation (14) thus yields a negative value for δU. According to the work-energy theorem, for frequencies lower than f

_{xo}, work will be required on the particle by the field to withdraw it from the medium. Furthermore, this free energy is further reduced if the field E

_{m}increases. The protein monomer or dimer will attempt to minimize its electrostatic free energy by moving up a field gradient to a maximum value of this gradient. This describes the action of positive DEP. For frequencies lower than f

_{xo}(i.e., where the dielectric decrement Δɛ

^{−}has a finite value), the protein’s effective permittivity is less than that of the medium. The protein will move down a field gradient to search for a field minimum. Work is required by the field to insert the protein into the medium. This describes negative DEP. It is tempting to consider the cross-over of DEP polarity at 1–10 MHz for BSA, observed by Cao et al. [26], as experimental evidence for this scenario, because such cross-over is expected from inspection of the β-dispersion shown in Figure 8.

_{p}of the solvated protein:

_{p}will give the strength of the DEP force, whilst its polarity will also define the F

_{DEP}polarity. A negative value for M

_{p}will indicate it is directed against the direction of E

_{m}. Work will be required to insert the polarized particle into the field E

_{m}within the aqueous medium. This describes negative DEP. A polarized protein possessing a positive value for M

_{p}will be aligned with E

_{m}and exhibit positive DEP.

_{p}to be used in the integrals of Equations (17) and (15) is not straightforward. Different protein molecules have from 0.20 to 0.70 g strongly associated (bound) water per g protein, contributing to its effective radius of rotation by up to one to two water molecule diameters [63]. From their studies, Moser et al. [53] determined a hydration of 0.64 g of H

_{2}O per g of BSA. Grant et al. [54] confirmed the existence of a subsidiary dispersion (δ-dispersion) in the frequency range 200–2000 MHz, and concluded that this dispersion is probably due to the rotational relaxation of water ‘bound’ to the protein. The term ‘bound water’ is taken to mean water bound to the protein by bonding of greater strength than the water–water bonding that exists in pure bulk water. This characteristic water structure that is formed near the surfaces of solvated proteins arises not only through hydrogen bonding of the water molecules to available proton donor and proton acceptor sites on the protein surface, but also through electrostatic forces associated with the water molecule’s electric dipole moment. The protein molecule and the water around it thus form a strongly coupled system, involving mechanical damping of the protein motion by adsorbed water, together with a dynamic electrical coupling between the tumbling electric dipole of the protein and the fluctuating dipoles of the adsorbed and bulk water. With such heterogeneity of the dielectric medium and also possibly of E

_{i}within the effective volume υ

_{p}, computation of the integrals in Equations (17) and (18) thus involves some ‘interesting’ challenges. Not least of which is defining the effective volume υ

_{p}of the protein, and how the normal components of displacement flux D and polarization P vary within the heterogeneous boundary between the protein’s surface and the bulk aqueous medium.

#### 3.6. Interfacial Polarizations

_{m}is assumed to be irrotational. Boundaries of the form depicted in Figure 10b between dielectrics of different permittivity have been shown, through theory and classical molecular dynamics simulations of hydrated cytochrome c, to exist in the hydration shells of proteins [68]. The large dispersion strength (Δɛ ~ 2400) shown in Figure 10c for a suspension of polystyrene microspheres was analyzed and determined not to arise from classical Maxwell–Wagner interfacial polarization, electrophoretic particle acceleration, or the presence of a frequency-independent surface conductance [69]. The most likely origin was considered to be a frequency-dependent surface conductance that varies with the ionic strength of the suspending aqueous electrolyte. Interfacial polarizations of these types should be included in the exercise to find a molecular-based DEP theory. It is also pertinent to mention that excised samples of biological tissue can exhibit large Δɛ values [52], a good example being skeletal muscle with measured relative permittivity ε

_{r}≈ 10

^{7}at 10 Hz [70]. This is known as the α-dispersion and, according to the convention used in assigning Greek letters to dielectric dispersions, occurs in a frequency range below that of the β-dispersion.

#### 3.7. Protein Dipole Polarization

_{DEP}and F

_{DEP}given by Equation (7). In the absence of an electric field, the orientations of the dipole moments of proteins in solution will on average be distributed with the same probability over all directions. On average their net dipole moment in any specific direction is zero. On application of a field each dipole will experience a field alignment torque m × E

_{i}, so that net polarization results. The electrical potential energy U of each dipole is given by U = −(mE

_{i}cosθ), where θ is the angle between the dipole moment and the local field vector E

_{i}. From Boltzmann–Maxwell statistics the probability of finding a dipole oriented in an element of solid angle dΩ is proportional to exp(−U/kT), with k the Boltzmann constant and T in kelvin. A moment pointing in the same direction as dΩ has a component (mcosθ) in the direction of E

_{i}. As detailed elsewhere [4,9] the thermal average of cosθ is given by the derivation of the so-called Langevin function:

_{i}≈ 3 × 10

^{5}V/m (e.g., Lapizco-Encinas et al. [17], assuming E

_{i}≈ E

_{o}) the factor (mE

_{i}/kT) ≈ 0.01. So, to a good approximation $m\langle \mathrm{cos}\theta \rangle ={m}^{2}{E}_{i}/3kT$. For E

_{i}> 3 × 10

^{7}V/m (e.g., Cao et al. [26]) the full expression for the thermal average of cosθ should be used. With an applied field less than 10

^{6}V/m, then through Equation (7) the average orientational DEP force (F

_{oDEP}) acting on a protein’s dipole is given by:

_{oDEP}, which also follows from Equation (4), has two important features. The first is that the DEP force exerted on a polarized protein molecule possessing a permanent dipole moment is directly proportional to ∇E

^{2}. Previously, one of the authors [9] has concluded that for frequencies below fxo (see Figure 8) proteins with a permanent dipole should exhibit positive DEP directly proportional to ∇E, and not ∇E

^{2}, whereas negative DEP should be expected solely above fxo and have a ∇E

^{2}dependence. This conclusion is only valid for a ‘rigid’ protein molecule whose dipole is constrained from responding to the alignment torque m×E

_{i}, or where the relaxation time of the protein’s permanent dipole is too slow to respond to a high-frequency oscillating field. Based on Equation (20) the ratio of the DEP force exerted on an orientationally polarized dipole moment to that on an induced dipole moment (Equation (5)) is:

^{−27}Cm; R = 3.5 nm), and assuming E

_{i}= E

_{m}, this gives near equality of F

_{oDEP}and F

_{DEP}(F

_{oDEP}= 0.71 F

_{DEP}). This result indicates that unless E

_{i}>> E

_{m}, Equation (20) does not offer a theoretical basis to explain why the majority of experimental ∇E

^{2}values shown in Figure 7 fall well below the minimum requirement of ∇E

^{2}> 3.5 × 10

^{21}V

^{2}/m

^{3}. It also implies that we require a better understanding of the relationship between the DEP force and the β-dispersion shown in Figure 8. Qualitatively, a protein molecule will exhibit positive DEP if the polarization per unit volume (i.e., total dipole moment) of the bulk water it displaces is less than that of the protein and its associated water molecules of solvation. A quantitative understanding should include a molecular-level description of short- and long-range interactions of the dipoles (protein–water and water–water) and the nature of the interfacial and/or dipole charges that can create the situation E

_{i}>> E

_{m}. A route to this might be offered through the suggested empirical relationship given in Equation (15), that relates the protein’s local cavity field and its polarization to the large values of the effective polarization factor (κ + 2)CM given in Table 1. Of the proteins listed in Table 1, only three (BSA, concanavalin, ribonuclease) appear to have been investigated for their DEP characteristics. It is of interest to compare the locations of these proteins in the ∇E

^{2}‘ranking’ of Figure 7, with their relative values of (κ + 2) CM given in Table 1 (~1000: BSA; ~11,000: ribonuclease; ~ 15,000: concanavalin). If the macroscopic CM factor is replaced by the proposed microscopic version (κ + 1)CM in Equation (9), then the DEP results cited for ribonuclease and concanavalin lie well above the ‘minimum required’ level in Figure 7.

#### 3.8. Protein Stability

^{5}V/m) could be employed, and might also provide insights into the DEP behavior of a test sample as it makes the transition from the molecular- to the macro-scale. In their studies, Zhang et al. [25] employed low sample concentrations (0.78 μM) but high conductivities (0.1 S/m). The likelihood of molecular interactions and self-aggregation was thus low (Figure 6) but with such a high ionic strength the precipitation of the BSA was likely.

#### 3.9. Other Experimental Details

_{2}from the environment can lead to an increased conductivity and lowered pH value. In DC-DEP, artificial pH gradients might also be generated in a way similar to the preparation of pH gradients for isoelectric focusing. Due to adsorption at the surface of the measuring chamber, as well as within fluidic tubing, solute concentrations can decrease even in the course of the actual experiment. Often, counter-measures are taken using buffers or surface modifications [19,32]. Published results should thus be compared and interpreted carefully.

^{2}are calculated. Both values are given for only a few of the studies cited here [16,20,38]. The spatial distribution of just |E| is given by Agastin et al. [18], that of ∇ |E|

^{2}in rather more cases [20,22,26,28,32,37] and sometimes the distribution of both values is given [22,26,38]. Owing to experimental limitations actual measurements of |E| or ∇ |E|

^{2}have not been carried out in any of these works. All calculations have been performed numerically by commercial software based on finite-element-methods (FEM). Although this is not specified by any of the authors, it is very probable that the spatial models of these simulations were based on simple geometrical bodies like cuboids and cylinders. This means that in essence the edges are modelled with infinitesimal radius of curvature. This should lead to infinite values of both |E| and ∇ ∇ |E|

^{2}since both are calculated as spatial derivatives of the potential distribution. In practice, this is not the case because the calculations are performed on a mesh or grid with finite resolution. This means that the field distributions are qualitatively correct. However, the maximal values of |E| and ∇ |E|

^{2}are now dependent on the spatial resolution of the mesh. It might well be that in several cases the resolution is not known because the software automatically adapts the mesh locally. In only two reports have the resolutions been given—namely, values of 50 nm³ [38] and 100 nm³ [23]. In order to determine the impact of the chosen resolution we have calculated the field distribution for two basic electrode arrangements, i.e., for co-planar interdigitated electrodes and for arrays of cylindrical pins. Using the FEM software Maze (Field Precision, Albuquerque, USA) the resolution of the Cartesian grid was varied from 120 nm down to 12 nm. This produced a roughly linear increase of both |E| and ∇ |E|

^{2}(data not shown) with resolution (i.e., with the inverse of the linear voxel dimensions). For interdigitated electrodes |E| and ∇ |E|

^{2}increase by a factor of 4 and 10, respectively, whilst for cylindrical arrays these factors amount to 2 and 60, respectively. As a consequence, the currently available data on |E| and ∇ |E|

^{2}should only serve as a more or less rough estimate when comparing them with physical theory.

## 4. Concluding Comments

^{21}V

^{2}/m

^{3}required, according to Equation (9), to overcome the dispersive forces associated with the Brownian motion of the protein molecules. All of the other studies fell far short of this requirement. In one reported DEP manipulation of BSA, a value of 10

^{12}V

^{2}/m

^{3}is cited [17].

^{+}, the protein’s effective permittivity ε

_{p}can be regarded as being greater than the value ε

_{m}for the surrounding medium. This should result in a positive DEP response. A negative DEP response should then be exhibited on increasing the field frequency to the part of the β-dispersion where a dielectric decrement occurs, as shown in Figure 8. There are three examples where a DEP cross-over (transition from positive to negative DEP with increasing frequency) has been observed at 1–10 MHz, namely: that reported for BSA by Cao et al. [26] as shown in Figure 3; for avidin (Bakewell et al. [27]) and PSA (Kim et al. [37]) as shown in Figure 5. This is consistent with the DEP responses of these proteins resulting from polarization of their permanent dipole moment, and not only as the result of an induced dipole moment.

^{2}. This corrects a previous conclusion [9], based on the presumption of a rigid rather than rotationally free permanent dipole, that the DEP force arising from a permanent dipole would be proportional to ∇E. However, as shown by Equation (21), the contribution of the DEP force expected for a BSA from its permanent dipole moment is predicted (according to current accepted theory) to be slightly less than the contribution of its induced moment. This indicates that, unless the ‘cavity’ field experienced by the protein molecule is very much larger than the field existing within the surrounding bulk medium, we have is no explanation in terms of the standard DEP theory (even if modified to encompass both an induced plus a permanent dipole moment) why the majority of experimental ∇E

^{2}values shown in Figure 7 fall well below the minimum requirement of ∇E

^{2}> 3.5 × 10

^{21}V

^{2}/m

^{3}to overcome thermal dispersion effects. As shown in Figure 7, the minimum required ∇E

^{2}value is lowered by a factor of ~1000-fold for BSA, if the macroscopic CM-factor is replaced in Equations (5) and (9) by the empirically based molecular version CM

_{micro}= (κ + 2)CM

_{macro}formulated in Section 3.4, and tabulated for various proteins in Table 1. Of the proteins listed in Table 1, only three (BSA, concanavalin, ribonuclease) are cited in Figure 7. The location of these proteins in the ∇E

^{2}‘ranking’ of Figure 7 is significant. Their relative values of (κ + 2)CM given in Table 1, namely: ~11,000 for ribonuclease and ~15,000 for concanavalin, would place them above the minimum requirement level indicated in Figure 7 for BSA. It would clearly be of value to populate Table 1 with as yet unavailable dielectric spectroscopy data for the other proteins cited in Figure 5, and vice versa. With this information protocols could be developed to spatially manipulate or selectively sort targeted protein molecules, so bringing protein DEP in line with the achievements and promise enjoyed by the more established DEP of cells and bacteria, for example [75].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A dielectric of relative permittivity ɛ

_{m}is shown partly inserted between two electrified electrodes. ‘Free’ charge density σ on the electrodes creates the Maxwell field E and electric displacement D (both = σ/ɛ

_{o}). ‘Bound’ charge density Δσ created by polarization (charge displacement) of the dielectric generates the polarization vector P ($\mathsf{\Delta}\sigma ={P}_{s}\cdot \hat{n}=P$), and equates to the number density of polarized molecules—i.e., the dielectric’s dipole moment M per unit volume. These relationships give D = E + P/ɛ

_{o}, and P = ɛ

_{o}(ɛ

_{m}− 1)E

_{o}= χ

_{m}ɛ

_{o}E

_{o}.

**Figure 2.**Globular proteins studied for their dielectrophoresis (DEP) response, with their hydrodynamic (Stokes) radii located on the empirical relationship between protein size and molecular weight (dotted curve) provided by Malvern Panalytical

^{®}(Zetasizer Nano ZS).

**Figure 4.**A range of aqueous solvent conductivity has been used in DEP studies of BSA. The experimental factors associated with the two cases [17,25] of negative iDEP and the cross-over of polarity between 1~10 MHz [26] are discussed in Section 3.2.

**Figure 5.**A summary of the DEP responses, at specific frequencies of the applied field, for proteins other than BSA. (HRP: horse radish peroxidase; PSA: prostate specific antigen; eGFP: enhanced green fluorescent protein; TnI-Ab: troponin I antibody).

**Figure 6.**Mean distance between BSA and streptavidin molecules for the reported sample concentrations, estimated as the cube root of the volume occupied per protein molecule.

**Figure 7.**Values of the field gradient factor $\nabla {E}_{m}^{2}$ as reported by the investigators or estimated by Hayes [5]. The minimum value of $\nabla {E}_{m}^{2}$ required to compete against Brownian diffusive effects, calculated according to Equation (9), is shown for the case of BSA. The adjusted value shown for this is based on the empirical relationship described in Section 3.4.

**Figure 8.**The β-dispersion and δ-dispersion, arising from orientation polarization of the protein and protein-bound water, respectively, exhibited by 0.18 mM BSA (based on Moser et al. [50] and Grant el al. [51]). The radian frequency of orientation relaxation for BSA is given by the reciprocal of its relaxation time τ. For frequencies below fxo (~1 MHz) the relative permittivity ɛ

_{r}of the BSA solution exceeds that of pure water, and is less than this above fxo. According to Equation (11) the dielectric increment $\Delta {\epsilon}^{+}$ and decrement $\Delta {\epsilon}^{-}$, respectively, specify the frequency ranges where positive and negative DEP, respectively, should be observed for monomer BSA in aqueous solution.

**Figure 9.**Schematic of a ‘Russian doll’ model for a protein with a permanent dipole moment M

_{p}, which occupies the innermost cavity together with its strongly bound water molecules. The protein’s dipole field extends beyond an outer macroscopic, ‘mathematical’, surface where the classical boundary conditions of electrostatics can be applied. Within this mathematical surface is a boundary that contains the protein’s outer hydration sheath, and the hydrodynamic plane of shear that defines the zeta-potential within the protein’s diffuse electrical double-layer.

**Figure 10.**(

**a**) Schematic of a dipole formed at the site of a structural defect in a molecular lattice. An example could be the disruption of the hydrogen bond network in bulk water at a protein–water interface—with the possible creation of ferroelectric nanodomains [66]. (

**b**) Dipole polarization at a boundary of dielectric inhomogeneity. A solvated protein, with its bound water and surrounding bulk water, represents an inhomogeneous dielectric [68]. (

**c**) Dielectric dispersion exhibited by an aqueous suspension of polystyrene nanospheres (R = 94 nm) (based on Schwan et al. [69]).

**Table 1.**Values of the factor (κ + 2) CM given by Equation (13) for various globular proteins, derived from reported Δɛ and corresponding protein concentration values. The protein density values were derived from the weight-depending function given by Fischer et al. [55].

Protein | Mol. Wt. | Density (g/cm ^{3}) | Δɛ/c_{p}(c _{p}: mM) | (κ + 2)CM Equation (13) | Reference |
---|---|---|---|---|---|

Ubiquitin | 8600 | 1.49 | 3.82 | 4020 | [58] |

RNAse SA | 10,500 | 1.48 | 15.00 | 15,720 | [57] |

Phospholipase | 13,000 | 1.46 | 1.82 | 189 | [56] |

Cytochrome-c | 13,000 | 1.46 | 5.06 | 5240 | [56] |

Ribonuclease | 13,700 | 1.46 | 11.0 | 11,400 | [59] |

7.12 | 7350 | [56] | |||

Lysozyme | 14,300 | 1.46 | 1.34 | 1390 | [56] |

Myoglobin | 17,000 | 1.45 | 0.07 | 2090 | [60] |

1.79 | 1440 | [61] | |||

Trypsin | 23,000 | 1.43 | 6.74 | 6810 | [56] |

Carboxypeptidase | 34,000 | 1.42 | 37.24 | 37,440 | [56] |

Hemoglobin | 64,000 | 1.41 | 1.29 | 1290 | [62] |

BSA | 66,000 | 1.41 | 1.11 | 1110 | [53] |

Concanavaline | 102,000 | 1.41 | 15.31 | 15,270 | [56] |

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Hölzel, R.; Pethig, R.
Protein Dielectrophoresis: I. Status of Experiments and an Empirical Theory. *Micromachines* **2020**, *11*, 533.
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Hölzel R, Pethig R.
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2020. "Protein Dielectrophoresis: I. Status of Experiments and an Empirical Theory" *Micromachines* 11, no. 5: 533.
https://doi.org/10.3390/mi11050533