Protein Dielectrophoresis: A Tale of Two Clausius-Mossottis-Or Something Else?

Abstract

Standard DEP theory, based on the Clausius-Mossotti (CM) factor derived from solving the boundary-value problem of macroscopic electrostatics, fails to describe the dielectrophoresis (DEP) data obtained for 22 different globular proteins over the past three decades. The calculated DEP force appears far too small to overcome the dispersive forces associated with Brownian motion. An empirical theory, employing the equivalent of a molecular version of the macroscopic CM-factor, predicts a protein's DEP response from the magnitude of the dielectric β-dispersion produced by its relaxing permanent dipole moment. A new theory, supported by molecular dynamics simulations, replaces the macroscopic boundary-value problem with calculation of the cross-correlation between the protein and water dipoles of its hydration shell. The empirical and formal theory predicts a positive DEP response for protein molecules up to MHz frequencies, a result consistently reported by electrode-based (eDEP) experiments. However, insulator-based (iDEP) experiments have reported negative DEP responses. This could result from crystallization or aggregation of the proteins (for which standard DEP theory predicts negative DEP) or the dominating influences of electrothermal and other electrokinetic (some non-linear) forces now being considered in iDEP theory.

Keywords: Clausius–Mossotti function; Lorentz cavity; Maxwell cavity; dielectric beta-dispersion; dielectrophoresis; electrokinetics; molecular dynamics simulations; proteins.

Figure 5

The β-dispersion exhibited by a 0.18 mM (i.e., p = 0.0084) BSA solution [40] is shown, together with that of a suspension of phospholipid vesicles (radius 13.5 nm, volume fraction p = 0.14) [42]. The δ-dispersion is associated with the protein hydration sheath [41,44,45]. Positive DEP can be expected for both particle types in the frequency range where their relative permittivity, ε_r, exceeds that (~78) of pure bulk water.

Figure 8

(a) Major contributions to the permanent dipole moment M_p of a protein are its peptide groups that carry a charge q. (b) H₂0 dipoles near charged groups can act to enhance M_p, corresponding to a correlation coefficient g_k > 1. Reversal of the polarities shown for the water dipoles gives anti-correlation, with g_k < 1. (c) Brownian (kT) changes of location and magnitude of q produce fluctuations with time of M_p about its mean value $\langle {\mathrm{M}}_p⟩$. The dashed red line indicates the mean square $\langle {\mathrm{M}}_p^2⟩$ of these fluctuations. (d) Without an applied electric field, orientations of the dipoles in a sample of polar fluid are random. The mean value, $\langle {\mathrm{M}}_T⟩{}_0$, of the total moment of these dipoles is thus zero. However, due to random fluctuations, its mean square value $\langle {\mathrm{M}}_T^2⟩{}_0$ is finite and corresponds to spontaneous polarization.

Figure 9

The average orientation of a single dipole along the direction of an applied field is given by the Langevin function (L(x) = coth x − 1/x) [18,19]. Fields of ~10⁶ to 10⁷ V/m are used in protein DEP experiments [4,5]. Polar molecules typically have dipole moments less than 5 debye units (e.g., 1.8 D for H₂O) so that x << 1. Globular proteins have large dipole moments and as shown for BSA (m = 710 D [54]) saturation of its polarization can occur in DEP studies.

Figure 1

Standard DEP theory employs macroscopic electrostatics to calculate an induced dipole moment and the Clausius–Mossotti factor, [CM]_macroscopic. The DEP of particles possessing a permanent dipole moment is better formulated within the context of the Clausius–Mossotti law of molecular dielectrics ([CM]_molecular). The new theory [1] holds the key to transitioning between the two [CM]s, whose origins trace back to Green [7], Faraday [8], Mossotti [9] and Clausius [10].

Figure 2

Reported values of $\nabla {\mathrm{E}}_m^2$ employed in DEP experiments mapped against the effective (hydrodynamic) radius of the test protein molecule. The straight-line plots show the predicted threshold values of $\nabla {\mathrm{E}}_{Th}^2$ required to overcome Brownian forces, derived using Equation (13) for prolate and oblate spheroids. (BSA: bovine serum albumin; CHY: chymotrypsinogen; IgG: immunoglobulin G.)

Figure 3

The contribution to the DEP force that a rigid dipole moment makes relative to that of the induced moment is shown for various proteins, as calculated using Equation (15). Values of the hydrodynamic radii were derived using an empirical relationship between protein size and molecular weight (Malvern Panalytical^®—Zetasizer Nano ZS) and permanent dipole moment values were derived from the literature [4,24].

Figure 4

Protein solutions exhibit dielectric dispersions Δε, designated α, β, δ and γ, due to relaxations of the protein’s electrical double-layer; permanent dipole; hydration shell; bulk water dipoles, respectively [16]. The magnitude of Δε(β), unlike Δε(δ), greatly exceeds that predicted by Equation (16) [4]. Δε(γ) exhibited by pure water arises from correlations of its dipole moments (Kirkwood factor g_k = 2.67), where g_k = 1 corresponds to no correlation.

Figure 6

The frequency-dependence of the empirical factor $\left(\kappa +2\right){\left[CM\right]}^{empirical}$ derived from Equations (17) and (20) for selected proteins. The frequency, f_xo, marking the transition between the dielectric increment and decrement of the β-dispersion is also given.

Figure 7

BSA, avidin and prostate specific antigen (PSA) have been reported to exhibit a DEP cross-over at MHz frequencies. This is consistent with the permittivity of the protein solution falling below the value ε_r = 78 for pure water, as shown here for BSA (from data of Figure 6).

Figure 10

Frequency dependence of the DEP susceptibility factor Re[K] for ubiquitin and lysozyme [1], together with plots for their empirical factors (κ + 2)[CM] [4].

Figure 11

(a) BSA samples at the same buffer conductivity and pH exhibit negative iDEP at a concentration of 0.15 mM [21], but positive iDEP at 7 nM [26,98]. The low magnitude of $\nabla {\mathrm{E}}_m^2$ associated with the negative DEP result suggests minimal influence of a non-linear electrokinetic or electrothermal effect. (b) A simple protein phase diagram [99,100] to show how crystallization might occur during the DEP of a protein sample in a metastable state.