Gel electrophoresis: Importance of concentration-dependent permittivity and double-layer polarization
Graphical abstract
Highlights
▸ A gel-concentration dependent permittivity model is proposed for gel electrophoresis. ▸ Gel structure reduces the solution permittivity and the degree of double-layer polarization. ▸ Electrophoretic behavior different qualitatively from that in conventional electrophoresis.
Introduction
Gel electrophoresis, the electrophoresis where the dispersion phase contains polymer gel, is often adopted to analyze/separate biocolloids such as proteins and nucleic acids (Hames, 1998). Because it is simple, efficient, highly sensitive, and easy to control, gel electrophoresis is also widely used in biochemistry, colloidal science and biotechnology (Calladine et al., 1991, Hames, 1998, Salieb-Beugelaar et al., 2009, Viovy, 2000).
The liquid phase in conventional electrophoresis is an aqueous electrolyte solution, and particles are free to migrate in all possible directions. In contrast, that in gel electrophoresis contains a cross linking, three-dimensional polymer structure, and particles need to migrate through the pores of that structure. Compared with conventional electrophoresis, gel electrophoresis has the following advantages. (i) Because the gel introduced has higher specific heat than water, the problems associated with Joule heating effect during electrophoresis can be alleviated. (ii) The polymer structure formed by gel serves as molecular sieve, raising separation stability and, therefore, improving the resolution of analysis, especially to entities such as proteins and nucleic acids. (iii) The presence of gel retards both the convective and diffusive movement of ionic species, thereby improving the accuracy through narrowing the entities (e.g., protein) zones (Hames, 1998).
The movement of a particle in conventional electrophoresis yields hydrodynamic drag acting on it, known as long range effect (Stigter, 2000). An extra hydrodynamic drag coming from the hydrodynamic interaction between the particle and gel polymer structure, called steric or short range effect, needs be considered in gel electrophoresis, making the analysis more complicated (Ogston, 1958). Depending upon the conditions considered, one of these two effects may dominate. For example, if the relative particle size is large, then because it is uneasy for it to pass through the pore of the polymer structure, the steric effect dominates. The ratio of (pore size/radius of gyration of particle) determines how the particle passes through the pore (Salieb-Beugelaar et al. (2009)). If this ratio is very small, the particle must uncoil before passing through the pore like a snake, known as reptation movement. The electrophoretic behavior of a particle of large radius of gyration can be described successfully by a reptation theory (Calladine et al., 1991, Salieb-Beugelaar et al., 2009). On the other hand, if that ratio is sufficiently large, it can pass easily through the pore. In this case, the migration of the particle is similar to that in conventional electrophoresis with an extra molecular sieving or steric effect. The concept of (molecular) sieving is based on a Poisson statistics brought up by Ogston (1958), developed further by the others (Calladine et al., 1991, Salieb-Beugelaar et al., 2009, Pyell, 2010), and is now adopted in the analysis of gel electrophoresis. However, because the hydrodynamic effect is not considered in those theories the corresponding gel electrophoresis theory fails to explain satisfactorily experimental data, in general.
Several attempts have been made on the theoretical gel electrophoresis analysis. Ogston (1958) and Ogston et al. (1973), for example, considered the diffusion of suspended spherical particles in polymer gel by assuming that the gel fiber structure is uniform, of negligible thickness, and has constant length. Neglecting the hydrodynamic effect, he obtained an analytical expression for the dependence of the diffusivity of macromolecules on the gel volume fraction and the relative particle size. However, as pointed out by Johnson et al. (1996) and Pluen et al. (1999), that expression might overestimate the diffusivity and underestimate the hydrodynamic drag acting on the particle. The analysis of Ogston was modified by Phillips et al. (1989) through adopting an effective medium (EM) approach, where the polymer gel is modeled by an effective Brinkman fluid (Brinkman, 1947, Debye and Bueche, 1948), with the hydrodynamic drag arising from the porous structure of polymer gel lumped into a friction coefficient term. The resulting model fitted well with the experimental data for dilute gel concentration. Unfortunately, its performance becomes unsatisfactory as gel concentration exceeds ca 6%. Brady (1994) proposed an expression for the dependence of the macromolecular diffusivity on both the hydrodynamic effect and the steric effect. Based on the electrophoresis of various molecules in agarose gels for gel concentration ranging from 0.5 to 6%, Johnson et al. (1996) concluded that the performance of Brady’s EM model is much better than that of Ogston (1958), Ogston et al. (1973) and Phillips et al. (1989). Hill (2006) investigated the electrophoresis of rigid spheres in an electrolyte saturated Brinkman medium. Allison et al. (2007) studied the eletrophoresis of a sphere of uniform surface potential in an infinite gel medium.
Recently, Tsai and Lee (2011) modeled the gel electrophoresis of a concentrated dispersion of particles of constant surface potential by adopting a cell model approach. Although their analytical result is capable of describing the experimental data of Park and Hamad-Schifferli (2008), in general, an appreciable difference between the two occurs at relatively high gel concentration. In an attempt to explain that difference, their analysis is extended in this study to take account of the dependence of the liquid permittivity on gel concentration.
Section snippets
Theory
Fig. 1 shows schematically the problem considered: the electrophoresis a spherical particle of radius a and surface Πp immersed in a gel solution subject to an applied uniform electric field E of strength E; r, θ, φ are the spherical coordinates adopted with the origin at the center of a large spherical computational domain of radius b and boundary Πw. For convenience, a z axis is also defined with its origin coincides with that of the spherical coordinates, and directed to the direction
Dependence of ε and λ on gel concentration w
Prior to solving the present problem, the dependence of the permittivity of the liquid phase ε and the hydrodynamic friction coefficient per unit volume λ due to the presence of gel in the liquid phase on the gel concentration w (%) need be estimated. For illustration, we assume that the gel is agarose, which is often adopted in gel electrophoresis. Based on the correlation relationship between Darcy permeability, Kp, and the volume fraction of gel, Hgel, obtained by Jackson and James (1986)
Conclusions
Gel electrophoresis is analyzed theoretically taking account of both the hydrodynamic and the steric effects coming from the gel polymer structure. In particular, the influence of gel concentration on the permittivity of the liquid phase, which is overlooked in all of the previous analyses, is considered. Through fitting the available experimental data, we propose a cubic relationship for the dependence of the permittivity on the gel concentration and, based on that, the accuracy of previous
Nomenclatures
- a
radius of spherical particle [nm]
- as
averaged fiber radius of polymer structure [nm]
- b
radius of computational domain [nm]
- Dj
diffusivity of ionic species j [m2/s]
- E
applied uniform electric field [mV/cm2]
- E
strength of E [mV/cm2]
- e
elementary charge [C]
- ez
unit vector in the z direction
- FE
electrical force [N]
- FH
hydrodynamic force [N]
- gj
hypothetical perturbed potential function [mV]
- Hgel
volume fraction of gel
- kB
Boltzmann constant [J/K]
- Kp
Darcy permeability [m2]
- n
magnitude of unit normal vector n
- nj
number
Acknowledgment
This work is supported by the National Science Council of the Republic of China.
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