CHOOSING THE OPTIMAL GEL MORPHOLOGY IN ELECTROPHORESIS SEPARATION BY A DIFFERENTIAL EVOLUTION APPROACH ( DEA )

This paper introduces an effective optimization approach to investigate the morphological effects of nanocomposite gel electrophoresis and operational parameters (see below) by integrating the numerical simulations based on finite element method and population-based search algorithm such as differential evolution. Simulations are performed to study the solute transport by Convection-Diffusion-Electromigration in a microvoid with axially varying cross-section. Morphological parameters such as channel shape and size, as well as operational parameters such as pressure gradient in the axial direction, and electric field in the orthogonal direction were considered and found to have considerable effects on the separation resolution in electrophoresis. Key observations on the most favorable hydrogel morphology for an efficient electrophoresis separation are presented.


INTRODUCTION
The global market for commercial biotechnology based separation systems reached $14.6 billion in 2011, and it will further grow to $26.7 billion at an estimated compound annual growth rate (CAGR) of 12.8% by 2016.Specifically, the electrophoresis based separations market is set to reach US $1.6 billion by the end of 2013 (Research, 2011).Traditionally, two-dimensional gel electrophoresis (2DE) have been used for proteomic analyses which separate proteins according to two distinct protein characteristics, size and charge (Kolch et al., 2005).The ongoing challenge, for proteome analysis by 2DE, is the reproducible separation of complex protein mixtures with a high degree of resolution and to retain native state of the protein during the separation (Hortin et al., 2006).Improvements in gel electro-phoresis will facilitate important advances in cancer diagnosis (Zhang et al., 2010), personalized medicine (Weston and Hood, 2004), and environmental proteomics (Keller and Hettich, 2009) to name just a few cases.
Gel morphology and charge effects have traditionally been manipulated by varying the copolymer composition (Stellawagen, 2009;Stellawagen and Stellawagen, 2009;Simhadri et al., 2010), but recent reports show that novel morphological changes can also lead to unique separations.Morphological changes can be induced in the gel using templating methods (Rill et al., 1996) and nanoparticle addition (Schexnailder and Schmidt, 2009).Of these various approaches to the hydrogel modification, the presence of nanoparticles can bring a significant change to the chemical physics of the gel morphology, and provide enhancements to other important properties They found that diverging channels give the best resolution when the injection width is relatively wide.On the other hand, when the injection width is small, then straight channels gave the best resolutions.Dennison et al. (1982) found that (macroscopically) electrophoretic resolutions are higher in conical/wedge shaped polyacrylamide gels than the regular gels.Protein transport in pores of different sizes have also been studied in membrane-based separations (Yu et al., 2003).Berezhkovskii and Dagdug (2011) and Berezhkovskii et al. (2010) derived formulas to show the dependence of the transport coefficients i.e., effective mobility and diffusivity on driving force and geometric parameters of the capillary based on mapping the particle random walk.Their results showed the potential impact of axiallyvarying shapes within a microvoid and found qualitative differences in the dependence of effective mobility and diffusion coefficient on the driving force in capillary domains formed by cylindrical and spherical compartments.Li et al. (2011) modeled dispersion of anisotropic particles moving in nanofilters operating in Ogston regime.They compared these transport parameters obtained from the 1-D analytical model (based on macrotransport theory) to the ones obtained from 2-D numerical model.Model of a microfabricated device having repeated arrays of alternating deep wells and shallow slits is used for the analysis.Under experimentally relevant electric fields, the results show the field and size dependences of mobility and diffusivity with maximum difference on the order of 10%.Dutta et al. (Dutta et al., 2006) reviewed the effect of dispersion due to fluid shear in pressure-driven transport of fluid and solute on the channel geometry in microfluidic devices.They analyzed dispersion in rectangular, elliptical, trapezoidal, and isotropically etched designs and proposed optimum cross-sectional designs which have been shown to reduce the dispersion arising from the presence of channel walls.

Optimization Methodologies in Designing New Morphological Structures
In order to design a new nanocomposite gel there will be several morphological-based possibilities as well as operational-based parameters that need to be considered.In such a multiparameter system, simul-taneously changing all the parameter values increases the number of possibilities that can affect the electrophoresis separation efficiency.Therefore, a systematic optimization-based approach is appealing because an optimal separation of practical use can be achieved efficiently from a complete understanding of all possibilities.This (overall) morphology-based optimization could be helpful in assisting potential new designs that work towards systematically generating a multidimensional structure having broader tunability (Thompson et al., 2008) for a given type of molecule.
The concept of automation through optimization techniques with the objective of increasing the separation quality of biomolecules is becoming increasingly important in the field of electrophoresis.For example, McGuffin and Tavares (1997) developed a computer-based program for the systematic optimization of separations in capillary zone electrophoretic systems; this was accomplished based on a combination of regression and theoretical models which requires no experimental data for implementation.They calculated both the separation resolution between the adjacent species and the chromatographic resolution static (CRS), a response function used for the chromatographic separations to assess the overall quality of the separation.The main advantage of this optimization program is that by carefully adjusting the input parameters such as capillary dimensions, and buffer composition, the instrumental parameters for optimal experimental conditions can be predicted.Pfeiffer et al. (2004) implemented both heuristic and numerical optimization techniques for the design of microchip-based capillary electrophoresis systems which have greater system performance while simultaneously occupying less area.Buchholz et al. (2002) used capillary electrophoresis in combination with genetic algorithms with the application to increase the separation quality of nucleotide and nucleotide sugar separation.There are several optimization algorithms that have been applied to other electrophoresis (Buchholz et al., 2002;McGuffin and Tavares, 1997) techniques, but the application to gel electrophoresis seems to be missing.In this contribution, a systematic-optimization-based approach is introduced and its first successful application to study the effect of a rectangular and diverging microvoid (i.e., an axially varying cross section) on the separation performance of the material is presented.
Global optimization techniques can be classified as deterministic and stochastic methods.Deterministic methods require a first or second order derivative of the objective function to exist and also will increase the difficulty in searching optimum solution (Rangaiah, 2010).Stochastic methods for global optimization rely, on the other hand, on probabilistic approaches.They are easy to implement and require no transformation of original problem (treated as a black box) so that it can be linked to any other software in which the dynamic model of the electrophoretic system has been implemented.One such stochastic method for global optimization of interest to the current work is the Differential Evolution (DE) method presented by Storn and Price (1997).It is a stochastic population-based optimization algorithm.Since its introduction, DE has been applied for the optimization of reactive distillation processes, thermal cracking operation, heat exchanger equipment design to name a few in the field of engineering (Babu and Sastry, 1999).

AN EFFECTIVE SOLUTION METHODOLOGY: BRIEF DESCRIPTION
The methodology described in this section will be used for the exploration of (optimal) potential morphological effects and favorable operation conditions applicable, potentially, to enhance electrophoresisbased species separations by using, for example, nanocomposite gels.The combination of the dynamic numerical model in COMSOL software with the Differential Evolution (DE) (Storn and Price, 1997)-one of the most robust population-based search algorithms available is proposed for the first time to study the species transport in electrophoretic systems.As known (COMSOL-Multiphysics, v3.5a), COMSOL is a software for the simulation of multiphysics problems and it is based on finite element method (FEM).The main advantage of the use of COMSOL lies in its capabilities to solve the potential highly nonlinear characteristics of physicochemical problems without over constraining their solution with simplifying assumptions.The modeling and analysis of the different model applications are conducted through modules within the COMSOL Multiphysics software interface.The module created in COMSOL, for handling electrophoretic transport problems such as the one of interest in this work is used.This module is efficient for formulating and solving the convection-electromigration-diffusion transport involved.In addition, Differential Evolution (Storn and Price, 1997) is comparatively a recent technique in the class of population based search heuristics and it has emerged as one of the most favored techniques by engineers for solving continuous optimization problems (Ali et al., 2009;Babu and Angira, 2003).
For the current paper, the transport characteristics of solute are studied in an expanding (or diverging) domains by modeling the axially-expanding domain using convective-diffusive-electromigrative transport equations based on continuum mechanics.There are several parameters that need to be considered related to the study of the effect of the divergent angle, α, of the idealized microvoid domain shown in Figure 1.Indeed, several discrete combinations of morphological (such as height and length of the domain) and operational parameter (electric field, flow rate) values are possible if approached in an empirical manner.In order to reduce the time and cost necessary to potentially design the internal structure of gels for improving separation quality, optimization methods such as DE can be used as a tool to predict or suggest optimal design conditions that, in addition, will guide the selection of optimal conditions for the experimentalist.The flowchart shown in Figure 2 illustrates the steps carried out for the entire optimization procedure for its application to electrophoresis in nanocomposite gel used in this study.The functionality of COMSOL's integrating capabilities with MATLAB (MATLAB, 2008) (M) has been very advantageous for this work so that the COMSOL-generating M-file can be converted into a black box (optimization method guided by the value of the electrophoresis resolution) for executing the DE optimization algorithm.Application of DE in the optimization processes require electrophoretic spatial resolution values only and hence identification of parameters using this strategy is possible without a need for a mechanistic understanding of the discrete combinations of parameters (Price, 1999).The first step in the optimization is the generation of randomized set of parameter values of each design variable.The parameter values are given to the FEM package where the parametrically defined FEM model of the electrophoretic system is updated according to the parameter values set of the microvoid domain.Then the separation quality of the given sample mixture is judged by evaluating the objective function value (specified by electrophoretic resolution, see Equation ( 4)) of all the individuals of population (design parameter set) and finding out the best parameter values of the current generation in the computational iteration.The individual parameter values obtained from the first generation forms the basis set of parameter values for the next generation.The cycle of DE generating new values for the desired parameters to be optimized, and COMSOL simulating the electrophoresis model while keeping other parameters constant is repeated until a best separation criterion (as given by the electrophoresis resolution, Equation ( 4 Each charg icrovoid mov on, and elec adient applie olar species ecies in the ux is given by hattacharjee,    Sauer et al., 1995).The hydrodynamic velocity of the solvent of the system is determined by using the Navier-Stokes equation with suitable boundary conditions.

Initial and Boundary Conditions
A mixture of two sample test solutes A and B (point particles), each having a fixed valence (z), mobility, diffusivity, and a total initial concentration are injected at the center of the inlet wall of microvoid.The mathematical representation of initial and boundary conditions are given below.
The walls of the microvoid are assumed to be impermeable to the solute transport, and thus the noflux boundary condition is applied and is given as At the initial time, t=0 a Gaussian plug of solute is injected at the center of the microvoid that is mathematically given by, 2 2 2 0 2 0 0 ( , , 0) 2 where, 0  is the initial injection standard deviation, and C 0 is the initial concentration of solute.

Resolution
The most important parameter that measures the performance of the system is the resolution; this parameter helps to understand how the solute bands are separated sufficiently so that separation can take place.Giddings (1991) mathematically defined the resolution as a ratio of distance between the two adjacent solute bands to the function of band dispersion as In the above equation x 1 and x 2 can be obtained from the first moments whereas the average standard deviation of the two bands can be calculated from the second moments for the solute bands approximated as Gaussian distributions (Giddings, 1991).Typical desirable resolution is 1.5 which corresponds to "baseline" resolution.The resolution can be impeded due to dispersion in the microvoid which is a result of diffusion, microvoid geometry, and flow conditions, among other parameters.
Numerical illustration of this model to show the optimized parameters for separation will be discussed in the next section.

ILLUSTRATIVE RESULTS AND DISCUSSION
This section includes graphical illustrations showing the numerical results obtained from the simulation of rectangular diverging microvoid domain (as sketched in Figure 1) of a single microcavity in, for example, a nanocomposite gel material.
The numerical approach is incorporated into the optimization routine as shown in Figure 2. In total five parameters i.e., microvoid size (L/H), shape (given by inclination angle α), dimensionless orthogonal potential (Ω), flow rate of the buffer (given by Pe) and mobility ratio of two species were considered to study the influence on the separation quality and the selected parameter values are motivated by the analysis of electric field effects given in Oyanader and Arce (2005).Even though numerous parameters are available the strategy of optimizing the effect of microvoid shape (α), and operational parameters (Ω, Pe) is utilized while fixing the other parameters.
The selected parameters, their ranges and resulting optimized parameters are shown Table 1 and Table 2 for aspect ratios of 9 and 15, respectively.Overall, from both the tables it can be observed that microdomains which have smaller values of the divergent angle are able to produce high separation resolution compared to higher angle values.As the species band travels down the axially varying cross section, it broadens or disperses and this dispersion can be attributed to an increase of diffusion over convection and microvoid morphology.This increased dispersion (over convection) smears the species bands, leading to a lower separation efficiency of the electrophoresis separation; this result is clearly due to changing the competition between diffusivetransport with respect to the convective-transport.As the mobility ratio increases the optimized values of the operational parameter i.e., Ω values are lower than for the case of lower mobility even though the resolutions are similar in values for example, for fixed Pe=2, α=0°, L/H=9 the highest resolution of 2 is obtained at Ω=4.8 when µ b /µ a =2, but the same resolution value is obtained when Ω=3.9 for a µ b /µ a =5.From this it can thus be said that it appears to require a smaller magnitude of orthogonal electric field when the difference in species mobility becomes higher.However, as the Pe is increased (for the fixed L/H, and model solution run time) the resolution appears to decrease and also it requires higher values of orthogonal electric field.For example, for a fixed time, α=0°, µ b /µ a =2, L/H=9 when Pe=2 the resolution is 2.0 when Ω=4.8, but the resolution reduced is to 1.6 for Pe=6, and further reduced to 1.2 for Pe=10 and also the values of orthogonal field needed are also higher as the Pe increases.Similar trends are observed for an aspect ratio of L/H=15 but however the values of resolution and Ω are different from those of the L/H=9 case which means that the change in the size parameter of the microvoid has an important effect on the separation efficiency.

SUMMARY AND CONCLUDING REMARKS
In this contribution, an effective differential evolution approach (DEA) is introduced and implemented to obtain the optimal morphological and values of the operational parameters describing gel electrophoresis separation.The hydrogel's structure is described by using a capillary model and, in particular, a rectangular geometry of their microvoid is used.The parameters considered include the microvoid size, L/H and the microvoid shape, α (morphological); the flow rate of the buffer given by the Pe and the dimensionless orthogonal electric field, Ω (operational); and the mobility ratio of species (transport).From the analysis, we can infer the following key observations: 1-All the parameters mentioned above had an effect on the separation resolution for electrophoresis; however, the DEA does an excellent performance in handling the multiparameter space to suggest optimal values; 2-The highest possible separation resolution can be obtained when the angle of deviation of the microvoid domain is relatively small; and 3-Imperfections in microvoids, i.e. deviations from an ideal type of geometry such as rectilinear channels or cells (obtained by a magnetic orientation of the nanoparticle in the gels, see Thompson et al., 2012) should be avoided in order to increase separation efficiency.
In particular, and as an overall conclusion, this study suggests that the magnetic alignment of nanoparticles within the gels achieved by Thompson et al. (2012) and that showed potential for improving separation has now a possibly fundamental explanation based on the study reported here.This study also suggests (for the first time and to the best of our knowledge) a possible useful approach to choose both a favorable hydrogel morphology as well as optimal values of the operating parameters for a given type of separation medium The use of nanoparticles (Thompson et al., 2010;Thompson et al., 2012) such as ceramic, metallic or polymer types might be an excellent way to achieve this objective by orienting them during the gels manufacture with the use of magnetic fields, for example.

ACKNOWLEDGMENTS
Dr. J. Simhadri is grateful to research assistanceships provided by the Center for Manufacturing Research (CMR) and the Center for the Management, Utilization and Protection of Water Resources from Tennessee Technological University, Cookeville, Tennessee-USA.
Figure 2: No ects of nano rating the n lement meth earch algorit