International Journal of Polymer Science

Volume 2019, Article ID 6194674, 15 pages

https://doi.org/10.1155/2019/6194674

## Understanding Collaborative Effects between the Polymer Gel Structure and the Applied Electrical Field in Gel Electrophoresis Separation

^{1}Department of Chemical Engineering, Tennessee Technological University, Cookeville, TN 38505, USA^{2}Department of Chemical Engineering, California Baptist University, 8432 Magnolia Ave, Riverside, CA 92504, USA

Correspondence should be addressed to Pedro E. Arce; ude.hcetnt@ecrap

Received 24 July 2018; Accepted 9 January 2019; Published 17 April 2019

Academic Editor: Bernabé L. Rivas

Copyright © 2019 Jennifer A. Pascal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The collaborative effects between an applied orthogonal electrical field and the internal structure of polymer gels in gel electrophoresis is studied by using microscopic-based electrophoretic transport models that then are upscaled via the format of electro kinetics-hydrodynamics (EKHD). The interplay of the electrical field and internal gel morphology could impact the separation of biomolecules that, because of similar chemical properties, are usually difficult to separate. In this study, we focus on an irregular pore geometry of the polymer-gel structure by using an axially varying pore (i.e., an axially divergent section) and an orthogonal (to the main flow of solutes) applied electrical field. The microscopic-based conservation of species equation is formulated for the standard case of electrophoresis of charged particles within a geometrical domain, i.e., a pore, and upscaled to obtain macroscopic-based diffusion and mobility coefficients. These coefficients are then used in the calculation of the optimal time of separation to study the effect of the varying parameters of the pore structure under different values of the electrical field. The results are qualitatively consistent with those reported, in the literature, by using computational-based approaches as well as with experiments also reported in the literature, previously. The study shows the important collaborative effects between the applied electrical field and the internal geometry of the polymer gels that could lead to improving biomolecule separation in gel electrophoresis.

#### 1. Introduction, Motivation, and Previous Efforts

The separation of biomacromolecules can be accomplished in various types of porous/fibrous media with electrophoresis, including polymer gels, microfluidic and field flow fractionation devices, and capillary electrophoresis, just to name a few [1–3]. Separation of these molecules has many potential applications, including the development of new pharmaceuticals, tissue scaffolding, and clinical diagnostic tests [4, 5]. In spite of excellent advances made among many of the electro-driven separation techniques, gel electrophoresis and its variations (2D gel electrophoresis, SDS-PAGE electrophoresis, temperature gradient gel electrophoresis, etc.) remains the widest used separation technique for proteomics, DNA/RNA fragments, and other macromolecules and it is a multimillion dollar business (Apogee Electrophoresis, 2014). Therefore, improvements in the separation efficiency will have a highly beneficial impact on the technology.

The use of an orthogonal field in gel electrophoresis can improve the separation of biomolecules. In particular, when the physicochemical properties of these biomolecules are similar and the separation can become very challenging, the presence of an orthogonal field is helpful. Sauer et al. [6] were the first contribution to explain the beneficial role of orthogonal fields found in experimental trends [7], from fundamental principles. Oyanader and Arce et al. [8] further identified the role of geometrical scales of the gel pore matrix in controlling the separation and, in addition, determining optimal time of separations. Pascal et al. [3] extended the analysis to electro-field flow fractionation in Couette-type separation devices.

In addition to the applied electrical field, another key aspect controlling the separation efficiency in electrophoresis is the internal structure, or morphology, of the gel media. Since gels are polymer matrices with pores of different sizes, shapes, and geometries and that only one of these is not enough to capture the separation behavior of the system [9, 10], it is important to gain a deeper understanding of the role that the gel pore geometry can play on the electrophoresis separation efficiency of macromolecules and their optimal time of separation. Early successful studies reported explained the potential impact of the geometry of the pores in the gel morphology, on electrophoresis separation, and included the use of a sudden change in the axial direction of the cross-sectional area of a pore domain of a gel medium [11]. Furthermore, a Monte Carlo simulation approach was used in either an isotropic or a nonisotropic gel morphology [12, 13] to determine the effective diffusivity of “point molecules” inside a model gel matrix. All of these studies show the crucial role that the pore-gel morphology could play in dictating the values of the effective transport parameters of biomolecules that, in turn, control their optimal time of separations.

As opposed to a sudden change in the cross-sectional area of flow in the gel-pore domain [11], Simhadri et al. [14] reported a computational analysis based on pores of rectangular geometry with an axially varying cross section. The current contribution examines the role of the same pore geometry, also with a rectangular diverging pore from an analytical point of view, i.e., *a continuously axially varying cross section* of the gel-pore domain (linear fashion) and its effect on the prediction of optimal separation times coupled with an orthogonal applied electrical field (constant). Through the use of the methodology known as “electrokinetic hydrodynamics,” or EKHD [15, 16], the relevant physics of the motion of the biomolecule inside the gel matrix is captured based on microscopic differential models of the gel-pore domain. Then, the model is upscaled to the macropore level by using the spatial-averaging method, a scaling technique, in conjunction with the fundamental principles of electrostatics and hydrodynamics. This approach is useful in obtaining the effective diffusivity and the effective mobility of the biomolecule (inside the gel matrix); then, these effective coefficients are used to compute optimal separation times as well as to assess the role of the axially varying morphology of the pore domain of the gel matrix on its separation performance. This can be accomplished by varying the various geometrical parameters of the diverging section, i.e., aspect ratio, angle of divergence, and axial position, and illustrating their impact on both the effective transport parameters as well as the optimal separation time as functions of the applied electrical field. One of the advantages of using the EKHD approach is that the universal or generalized dispersion coefficients (see Section 4) as well as the optimal separation time are *explicit functions* of the geometrical parameters controlling/affecting the morphology of the pore domain of the gel matrix and the applied orthogonal electrical field. This is an advantage with respect to computational simulations [14] where the relation of the effective transport coefficients, with geometrical parameters of the pore domain, is not explicit.

This study shows the important interplay among model scaling, pore morphology of the gel matrix, charge of the biomolecule, and the applied orthogonal electrical field in assessing or improving electrophoresis separations. In addition, the optimal times of separation that are obtained are relatively simple mathematical functions, and therefore, they lead to a tunable model that can predict behaviors of the system performance for a wide range of variables (i.e., magnitude of the applied orthogonal electrical field, angle of divergence, aspect ratio, axial position, etc.). The effects of these geometrical parameters as well as the molecule charges and the magnitude of the applied orthogonal electrical field on the optimal separation time are illustrated. Furthermore, a qualitative agreement between the predictions reported here with those published by Simhadri et al. [14] is observed. Both predictions (those communicated here and those by Simhadri et al. [14]) are in agreement with experimental evidences reported by Thompson et al. [17].

This contribution is organized as follows. First, a detailed introduction and reference to previous work on the subject is given in Section 1. Next, an overview of the related literature and remarks where this contribution differentiates from others is presented in Section 2. The formulation of the so-called fluid problem [15, 16] is introduced in Section 3 and its subsections on electrostatics and hydrodynamic aspects. The “solute problem” equations are developed in Section 4, and the upscaling of the model is performed here leading to the effective diffusivity and effective mobility. Asymptotic solutions and validation are discussed in Section 5. Parametric illustrations and discussions of results are presented in Section 6. Section 7 is a discussion of the implications of the finding for the practical aspects of the problems; in particular, calculations of the optimal time of separation and the effect of the different parameters are included. Finally, concluding remarks are given in Section 8 of the study.

#### 2. Overview of Relevant Literature

Biomacromolecules, in general, can often have similar physical or transport properties, i.e., electrophoretic mobility and/or diffusivity, causing the separation to become very challenging in certain cases [18]. In addition to the applied electrical field, for a given mixture of biomacromolecules, the internal gel structure, or morphology of the gel media, is one key property that can be exploited to enhance the separation efficiency of the material or device. Gels for electrophoresis are fibrous or porous matrices that can expose the biomacromolecules to a “differentiating” medium allowing the physical properties of the molecules, i.e., charge, size, shape, etc., to yield a unique motion for biomacromolecules. This unique motion is controlled by two different transport properties, i.e., effective diffusivity and effective mobility of the molecules inside the gel media. There are few experimental techniques that researchers have used to modify the morphology of the gel matrix. The traditional one is to change the concentration of the polymer cross-linker (see, for a review, Stellwagen [19]). A novel one is to use templating agents both within the polymer-chemical structure (molecular imprinting, see for a review Byrne and Salian [20]) and to manufacture gels by using molecular templates such as DNA and xanthan. For instance, Dharia et al. [21] and Rill et al. [22] proposed a modified architecture or morphology, by templating gels with DNA (or surfactant) templates that were removed from the gels once the polymerization took place, by electrophoresis action. The resulting material morphology, hypothesized as a “dual” porous structure [11, 13, 23, 24] displaying both small (of the order of the interfiber voids) and large (of the order of the templating agent) pores or voids, showed an improved efficiency in the electrophoretic separation of proteins. The most recent approach to modifying the morphology of the separation media is to add nanoparticles of different characteristics, i.e., nanocomposite gels, that could affect the separation efficiency (see, for example Thompson et al. [17, 23]). This subject has been reviewed by Simhadri et al. [25]; see also Texter [26].

Structurally, gel matrices can be viewed as consisting of pores of different geometries, sizes, shapes, etc. This situation leads to the fact that, realistically, only one *idealized geometry*, i.e., rectangular, cylindrical, or annular, for instance, is not enough to capture the internal architecture of the media [9, 10]. Instead, a collection of these may represent different limiting physical situations that can, *collectively*, aid to understanding the transport and separations of biomacromolecules. In addition, and based on the use of relatively simple geometrical models, Trinh et al. [11–13] were able to show that the geometry of the pores or capillary channels could play a significant role in the determination of the effective transport properties of biomacromolecules.

The use of pores or capillary channels with *axially varying cross sections* is not uncommon in studies of transport phenomena. For example, the effect of an axially varying cross section on the transport of solute in a cylindrical capillary was studied by Horner [27] in order to predict the behavior of atherosclerosis in blood vessels. In this study, the lubrication approximation for the hydrodynamics in conjunction with the spatial averaging approach (Cwirko and Carbonell [28]; Paine et al. [29]; Slattery [30]; Whitaker [31]; Zanotti and Carbonell [32–34]) was used in order to obtain both effective transport parameters as well as concentration profiles of the solute within the artery. Xuan et al. [35] further studied, experimentally, the effect of pressure on the movement of polystyrene particles in a converging diverging microchannel. The ratio of the particle velocity in the throat to that in a straight channel was studied in terms of pressure gradient, particle size, and particle trajectory.

Several contributions have been reported in the literature related to the electrokinetic transport in rectangular channels of varying cross section for applications in bioseparations and microfluidics. For instance, Ghosal [36] studied the transport of a polyelectrolyte through a nanopore with variable axial cross section and examined the effects of an (*axially*) applied electrical field. Through the use of electrostatics and hydrodynamics within a continuum mechanics framework, expressions for the electrophoretic speed of the polyelectrolyte, representative of a typical strand of DNA often used in gel electrophoresis, were determined. Four different modes of transport of the polyelectrolyte were found as a function of the dimensionless height of the channel. It was noted that the use of continuum hydrodynamics with water as the buffer is a reasonable approach that is valid for pores of sizes as low as tenths of a nanometer. In addition, Ghosal [37] also examined electroosmotic flow through a microfluidic channel of varying axial cross section and arbitrary zeta potential distribution. Supported by the use of Green’s function associated with the problem, a number of asymptotic cases for the electrohydrodynamic profile were studied as a function of the aspect ratio, , including cases for the long channel approximation (). Although many of these studies are more relevant for electrokinetic-based separations than for electrophoresis, one general conclusion is that the variable axial cross section increases mixing, worsening the resolution for an electro-based separation.

Opposite to the observation made above, Ross [38] examined electrokinetic separations in a diverging microchannel and found, for a specific case, that the diverging channel could actually lead to a better separation than a straight channel could. Yariv and Dorfman [39] studied the electrophoretic transport in rectangular channels of periodically varying cross sections. Macrotransport theory, as described by Brenner and Edwards [1], was used to obtain expressions for the effective velocity and effective dispersion within the channel. The analysis yielded asymptotic results for the macroscopic transport parameters (i.e., effective mobility and effective diffusivity) for both small and large Peclet numbers. Xuan et al. [35] examined the electrophoretic motion and separation of *particles* within a converging-diverging microchannel fabricated with poly(dimethylsiloxane) chips. The dependence of the separation on electrical field strength, particle size, and morphology was studied by using the ratio of the velocity of the particle at the throat of the device to that in a straight microchannel as the main parameter. Ghosal [37] presented an analysis related to channels of varying cross sections; he examined that electroosmotic flow and sophisticated mathematical techniques were employed to obtain volume fluxes for various geometries and zeta potential distributions. In addition, Yariv and Dorfman [39] also computed expressions for dispersion and effective velocity but only for very large and small Peclet number values. None of these contributions, however, have reported any study of the role of the orthogonal electrical fields in conjunction with the characteristics of the pore or the separation domain nor have they shown any calculation of the optimal time of separation.

In this contribution, and for the particular case of gel-based electrophoretic applications, attention is given to the analysis of orthogonal (The word “orthogonal” here implies perpendicular to the axial (varying) direction of the capillary channel as it was used in Oyanader and Arce [8].) applied electric fields to potentially enhance the separation. As mentioned in Section 1, above, Sauer et al. [7] studied this effect on the transport of solute in a Poiseuille flow regime using a capillary channel or pore of rectangular geometry. Before the work reported by Sauer et al. [6], only ad hoc expressions were known for dispersion under the influence of an applied electrical field (Eringen and Maugin [40]; Hiemenz [41]; Hunter [42]) and the effects of an orthogonal electrical field were only known experimentally (Lee et al. [7]; see also Gajdos and Brenner [43], Heller et al. [44], and Hiemenz [41]). Sauer et al. [6] introduced a mathematically nontrivial and fundamental modeling approach as a tool for the analysis of the system under study. This effort was the first that *systematically* explained experimental trends (Lee et al. [7]) in enhancing separation efficiency when an orthogonal field is applied in electrophoretic separations. One limitation of the analysis is that the predicting formulae for the effective transport parameters are in terms of integrals that require numerical solution. Motivated by this limitation, and attempting to further apply and/or develop the methodology, Oyanader and Arce [8] have revisited this contribution and have derived simpler and mathematically friendlier expressions for the effective transport parameters that clearly show the dependence on the orthogonal electrical field and the scaling. More specifically, Oyanader and Arce [8] have shown that these expressions depend critically on the scaling of the geometrical parameters of the device. The present analysis is concerned with the effect of an applied orthogonal electrical field on a rectangular pore with (continuously) axially varying cross sections. To the best of our knowledge, this is the first analytical-based contribution in this subject. A computational-based approach was reported by Simhadri et al. [14].

In general, the effects of transverse or orthogonal applied electrical fields have not been *extensively* studied regarding the separation of biomacromolecules, except for the case of electro-FFF (see, for example, Pascal et al. [3]). However, Baldessari and Santiago [45] in a review related to electrophoresis in nanochannels stated that the presence of the thick electrical double layer (EDL) in nanochannels can produce transverse electrical fields. Because of this observation, the dispersion in nanochannels must then be a function of the applied orthogonal electrical field. Therefore, in addition to the previous work by Sauer et al. [6] and Oyanader and Arce [8], Baldessari and Santiago’s [45] work suggests that further examining the role of applied orthogonal electrical fields in the separation of biomacromolecules in various systems is important.

#### 3. Problem Formulation: General Framework and Methodology

The system under study in this contribution is depicted in Figure 1. This figure is a general representation of the different scales presented in a gel-electrophoresis system. The transport associated with the motion of biomolecules in gel electrophoresis is a multiscale problem: *level 1* (see Figure 1) describes the *macroscopic* scale in the system where in general, pores of different geometries, sizes, shapes, etc., are located at a smaller scale. *Level 2* in the figure shows the pore-scale domain or *microscopic* scale where for example, an axially varying cross-section pore domain can be easily located in the gel matrix. In general, these domains could be oriented in many different ways and have different sizes and shapes. In this contribution, we are interested in a single type of pore orientation with potentially different shapes, all controlled by the so-called “shape factor” (). Level 2 is where microscopic transport phenomena based on continuum mechanics can be described. *Level 3* is, therefore, a single-pore domain (representing all the pores of the gel matrix (A distribution of pore orientation can be used to analyze the effect of the microdomain on the macroscale level transport. However, this is beyond the scope of the present work.)) with the Cartesian coordinate system and other important dimensions anchored to the domain.