On the ionic strength dependence of the electrophoretic mobility: From 2D to 3D slope-plots

Determining the charge and the nature (small ion, nanoparticle, or polyelectrolyte) of an unknown solute from its electrophoretic characteristics remains a challenging issue. In this work, we demonstrate that, if the knowledge of the effective electrophoretic mobility (μ_ep ) at a given ionic strength is not sufficient to characterize a given solute, the combination of this parameter with (i) the relative decrease of the electrophoretic mobility with the ionic strength (S), and (ii) the hydrodynamic radius (R_h ), is sufficient (in most cases) to deduce the nature of the solute and its charge. These three parameters are experimentally accessible by CZE and Taylor dispersion analysis performed on the same instrumentation. 3D representation of the three aforementioned parameters (μ_ep ; S and R_h ) is proposed to visualize the differences in the electrophoretic behavior between solutes according to their charge and nature. Surprisingly, such 3D slope plot in the case of small ions and nanoparticles looks like a "whale-tail," while polyelectrolyte contour plot represents a rather simple and monotonous map that is independent of solute size. This work also sets how to estimate the effective charge of a solute from a given experimental (S,Rh,μ ep 5 mM ) triplet, which is not possible to obtain unambiguously with only (Rh,μ ep 5 mM ) or (S,μ ep 5 mM ) doublet, where μ ep 5 mM is the effective electrophoretic mobility at 5 mM ionic strength.

Determination of effective charge of small ions, polyelectrolytes and nanoparticles by capillary electrophoresis

In this paper, a systematic and comparative study related to the effective charge determination of three kinds of solutes (small ions, polyelectrolytes and nanoparticles) was performed. Four approaches were compared regarding their conditions of validity and their advantages/disadvantages. Three of them allow the effective charge determination from the electrophoretic mobility and the hydrodynamic radius of the solutes using electrophoretic mobility modelings based on Nernst-Einstein (NE), O'Brien-White-Ohshima (OWO) and Yoon and Kim (YK) equations. Electrophoretic mobility and hydrodynamic radius were determined by capillary electrophoresis and Taylor dispersion analysis, respectively, using the same instrumentation in similar conditions, on a large set of samples. A fourth experimental approach based on the sensitivity of detection in indirect UV detection mode (IUV) was compared to the previously mentioned methods. OWO and YK modelings are well adapted for the effective charge determination of small ions and nanoparticles, while IUV is the only method adapted for polyelectrolytes.

Modeling the electrophoresis of oligolysines

CE is used to measure the electrophoretic mobility of low molecular mass oligo-L-lysines (n=1-8) in aqueous LiH₂PO₄ buffer, BGE, at pH 2.5 over a range of temperatures (25-50 °C) and ionic strengths (10-100 mM). Mobilities are corrected for Joule heating and under the conditions of the experiment, interaction of the peptides with the capillary walls can be ignored. A "coarse grained" bead modeling methodology (BMM) (H. Pei et al., J. Chromatogr. A 2009, 1216, 1908-1916) is used to model the mobilities. This model partially accounts for peptide conformation as well as the assumed form of its secondary structure. For highly charged oligolysines, it is necessary to properly account for the relaxation effect. In the present study, the BMM approach tends to overestimate oligolysine mobility and that effect tends to increase with increasing ionic strength and peptide length. It is proposed that association between the oligolysines and buffer components (H₂PO₄⁻ in this case) that go beyond classical electrostatic interactions are responsible for this discrepancy. A simple binding model is introduced that illustrates how this association can reconcile model and experiment.

Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium

The effective medium model [H.C. Brinkman, Appl. Sci. Res. A 1 (1947) 27] is used to calculate the electrophoretic mobility of spheres in a gel with uniform zeta potential on their surface. In the absence of a gel support medium or ion relaxation (the distortion of the ion atmosphere from equilibrium due to the presence of an external flow or electric field), our results reduce to those of Henry [D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106]. The relaxation effect can be ignored for weakly charged particles, or for particles with low absolute zeta potential. Using a procedure similar to that employed by O'Brien and White [R.W. O'Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607], the relaxation effect is accounted for in the present work and results are presented over a wide range of particle sizes, gel concentrations, and zeta potentials in KCl salt solutions. In the limit of no gel, our results reduce to those of earlier investigations. The procedure is then applied to the mobility of Au nanoparticles in agarose gels and model results are compared to recent experiments [D. Zanchet, C.M. Micheel, W.J. Parak, D. Gerion, S.C. Williams, A.P. Alivisatos, J. Phys. Chem. B 106 (2002) 11758; T. Pons, H.T. Uyeda, I.L. Medintz, H. Mattoussi, J. Phys. Chem. B 110 (2006) 20308]. Good agreement with experiment is found for reasonable choices of the model input parameters.

Translational diffusion constants of the amino acids: measurement by NMR and their use in modeling the transport of peptides

In this work, the translational self-diffusion constants, DT's, of 12 amino acids (Ala, Arg, Asn, Asp, Cys, Glu, His, Ile, Lys, Met, Phe, and Ser) are measured by field gradient NMR and extrapolated to infinite dilution. The experiments were carried out in D2O at 298 K at pD approximately =3.5 in 50 mM sodium phosphate buffer. Of these 12 amino acids, 6 are being reported for the first time (Asp, Cys, Glu, His, Lys, and Met) and the remaining 6 (Ala, Arg, Asn, Ile, Phe, and Ser) are compared with DT's from the literature. When corrected for differences in solvent viscosity and temperature, the discrepancy between DT's measured in the present work and those reported previously is always <8%, which is reasonable given the range of values reported previously by different groups. With the present work, DT's for all of the amino acids are now available. These diffusion constants are then used in modeling studies of the diffusion and free solution electrophoretic mobility, mu, of several model peptides. For this set of peptides, it is shown that modeling using revised input parameters results in improved agreement between model and experimental mobilities.

Review modeling the free solution and gel electrophoresis of biopolymers: The bead array-effective medium model

Free solution and gel electrophoresis is an extremely useful tool in the separation of biopolymers. The complex nature of biopolymers, coupled with the usefulness of electrophoretic methods, has stimulated the development of theoretical modeling over the last 30 years. In this work, these developments are first reviewed with emphasis on Boundary Element and bead methodologies that enable the investigator to design realistic models of biopolymers. In the present work, the bead methodology is generalized to include the presence of a gel through the Effective Medium model. The biopolymer is represented as a bead array. A peptide, for example, made up of N amino acids is modeled as 2N beads. Duplex DNA is modeled as a discrete wormlike chain consisting of touching beads. The technical details of the method are placed in three Appendices. To illustrate the accuracy and effectiveness of the approach, two applications are considered. Model studies on both the free solution mobility of 73 peptides ranging in size from 2 to 42 amino acids, and the mobility of short duplex DNA in dilute agarose gels are discussed.

Modeling the electrophoretic mobility and diffusion of weakly charged peptides

A bead model to determine the electrophoretic mobilities and translational diffusion constants of weakly charged peptides is developed that is based on a approximate structural model of peptides and is also grounded in electrohydrodynamic theory. A peptide made up of X amino acids is modeled as N=2X beads with 2 beads representing each amino acid in the chain. For the two beads representing a particular amino acid in a peptide, the radius of one bead is set to one-half the nearest neighbor Calpha-Calpha distance, and the radius of the other bead is chosen on the basis of the diffusion constant of the free amino acid. Peptide conformations, which are defined by a set of psi-phi dihedral angles, are randomly generated by using the transformation matrix approach of Flory (Flory, P. Statistical Mechanics of Chain Molecules; John Wiley: New York, 1969) and rejecting conformations which result in bead overlap. The mobility and diffusion constants are computed for each conformation and at least 100 independent conformations are examined for each peptide. In general, the mobility is found to depend only weakly on peptide conformation. Model and experimental mobilities are compared by examining the data of Janini and co-workers (Janini, G.; et al. J. Chromatogr. 1999, 848, 417-433). A total of 58 peptides consisting of from 2 to 39 amino acids are considered. The average relative error between experimental and model mobilities is found to be 1.0% and the rms relative error 7.7%. In specific cases, the discrepancy can be substantial and possible reasons for this are discussed. It should be emphasized that the input parameters of the peptide model are totally independent of experimental mobilities. It is hoped that the peptide model developed here will be useful in the prediction of peptide mobility as well as in using peptide mobilities to extract information about peptide structure, conformation, and charge. Finally, we show how simultaneous measurements of translational diffusion and mobility can be used to estimate peptide charge.

Electrokinetic transport of rigid macroions in the thin double layer limit: A boundary element approach

A boundary element (BE) procedure is developed to numerically calculate the electrophoretic mobility of highly charged, rigid model macroions in the thin double layer regime based on the continuum primitive model. The procedure is based on that of O'Brien (R.W. O'Brien, J. Colloid Interface Sci. 92 (1983) 204). The advantage of the present procedure over existing BE methodologies that are applicable to rigid model macroions in general (S. Allison, Macromolecules 29 (1996) 7391) is that computationally time consuming integrations over a large number of volume elements that surround the model particle are completely avoided. The procedure is tested by comparing the mobilities derived from it with independent theory of the mobility of spheres of radius a in a salt solution with Debye-Huckel screening parameter, kappa. The procedure is shown to yield accurate mobilities provided (kappa)a exceeds approximately 50. The methodology is most relevant to model macroions of mean linear dimension, L, with 1000>(kappa)L>100 and reduced absolute zeta potential (q|zeta|/k(B)T) greater than 1.0. The procedure is then applied to the compact form of high molecular weight, duplex DNA that is formed in the presence of the trivalent counterion, spermidine, under low salt conditions. For T4 DNA (166,000 base pairs), the compact form is modeled as a sphere (diameter=600 nm) and as a toroid (largest linear dimension=600 nm). In order to reconcile experimental and model mobilities, approximately 95% of the DNA phosphates must be neutralized by bound counterions. This interpretation, based on electrokinetics, is consistent with independent studies.

Use of T4 lysozyme charge mutants to examine electrophoretic models

The electrophoretic mobility of a macro-ion is affected in a complex manner by a variety of forces that arise from the applied field. Coupling of the macro-ion and small-ion flows gives rise to non-conserved forces that are greater than those expected from ordinary hydrodynamic considerations. It is difficult to separate the steady-state hydrodynamic and electrodynamic contributions to the macro-ion mobility. Membrane-confined electrophoresis (MCE), a free solution technique, provides an experimental means by which to gain insight into these contributions. In this work we used MCE steady-state electrophoresis (SSE) of a series of T4 lysozyme charge mutants to investigate these effects and to examine the existing theoretical descriptions. These experiments isolate the effects of charge on electrophoretic mobility and permit a unique test of theories by Debye-Hückel-Henry, Booth and Allison. Our results show that for wild type (WT) T4, where divergence is expected to be greatest, the predicted results are within 15, 8 and 1%, respectively, of experimental SSE results. Parallel experiments using another free-solution technique, capillary electrophoresis, were in good agreement with MCE results. The theoretical predictions were within 20, 13 and 5% of CE mobilities for WT. Boundary element modeling by Allison and co-workers, using continuum hydrodynamics based on detailed structural information, provides predictions in excellent agreement with experimental results at ionic strengths of 0.11 M.

Modeling the gel electrophoresis of short duplex DNA by Brownian dynamics: cubic gel lattice with direct interaction

The technique of Brownian dynamics is used to model the electrophoretic mobility of spherical and rod-like particles in a three-dimensional cubic gel lattice. In addition to excluded volume interactions between the migrating particle and the gel, direct interactions are also included. The methodology is first applied to spherical particles in the absence of direct interactions and the resulting mobilities are shown to agree with independent studies. The methodology is then applied to rod-like models of short duplex DNA fragments 10-50 base pairs in length. In the absence of direct interactions between gel and DNA, calculated mobilities show a much weaker dependence on gel concentration than observed in experiments of DNA in Tris-acetate buffer and polyacrylamide gels. When an attractive interaction between gel and DNA of approximately -0.3 k(B)T per base pair at contact is included, good agreement between calculated and experimental mobilities is achieved.

Boundary element modeling of biomolecular transport

The boundary element (BE) methodology has emerged as a powerful tool in modeling a broad range of different transport phenomena of biomolecules in dilute solution. These include: sedimentation, diffusion (translational and rotational), intrinsic viscosity, and free solution electrophoresis. Modeling is carried out in the framework of the continuum primitive model where the biomolecule is modeled as an arbitrary array of solid platelets that contains fixed charges within. The surrounding fluid is modeled as a electrodynamic/hydrodynamic continuum which obeys the Poisson and low Reynolds number Navier-Stokes equations. Ion relaxation (the distortion of the ion atmosphere from equilibrium) can also be accounted for by solving the coupled ion transport equation (for each mobile ion species present), Poisson, and Navier-Stokes equations in tandem. Several examples are presented in this work. It is first applied to a detailed model of 20 bp DNA and it is concluded that it is not necessary to include a layer of bound water to reconcile experimental and model translational diffusion constants. With regards to diffusion, the BE approach is also applied to a 375-bp supercoiled DNA model (without ion relaxation), and also 20-60-bp DNA fragments with ion relaxation included in order to assess the magnitude of the electrolyte friction effect under a number of different salt/buffer conditions. Attention is then turned to modeling the electrophoretic mobility of three different cases. First of all, we consider a sphere with a central charge large enough in magnitude to insure that ion relaxation is significant. Excellent agreement with independent theory is obtained. Finally, it is applied to modeling short DNA fragments in KCl and Tris acetate salts. Quantitative agreement is achieved when the salt is KCl, but the calculated (absolute) mobility in Tris acetate is substantially higher than the experimental value. The interpretation of this is that there is an association between Tris(+) and DNA (perhaps hydrogen bonding) not accounted for in our modeling that is responsible for this discrepancy.

A commentary on the screened-Oseen, counterion-condensation formalism of polyion electrophoresis

The use of linear theory, in particular, counterion condensation (CC) theory, in describing electrophoresis of polyelectrolyte chains, is criticized on several grounds. First, there are problems with CC theory in describing the equilibrium distribution of ions around polyelectrolytes. Second, CC theory is used to treat ion relaxation in a linear theory with respect to the polyion charge despite the fact that ion relaxation arises as a consequence of nonlinear charge effects. This nonlinearity has been well established by several investigators over the last 70 years for spherical, cylindrical, and arbitrarily shaped model polyions. Third, current use of CC theory ignores the electrophoretic hindrance as well as the ion relaxation for condensed counterions and only includes such interactions for uncondensed counterions. Because most of the condensed counterions lie outside the shear surface of the polyion (in the example of DNA), the assumption of ion condensation is artificial and unphysical. Fourth, the singular solution, based on a screened Oseen tensor, currently used in the above mentioned theories is simply wrong and fails to account for the incompressibility of the solvent. The actual singular solution, which has long been available, is discussed. In conclusion, it is pointed out that numerical alternatives based on classic electrophoresis theory (J.T.G. Overbeek, Kolloid-Beih, 1943, 54:287-364) are now available.

Visualizing ion relaxation in the transport of short DNA fragments

Ion relaxation plays an important role in a wide range of phenomena involving the transport of charged biomolecules. Ion relaxation is responsible for reducing sedimentation and diffusion constants, reducing electrophoretic mobilities, increasing intrinsic viscosities, and, for biomolecules that lack a permanent electric dipole moment, provides a mechanism for orienting them in an external electric field. Recently, a numerical boundary element method was developed to solve the coupled Navier-Stokes, Poisson, and ion transport equations for a polyion modeled as a rigid body of arbitrary size, shape, and charge distribution. This method has subsequently been used to compute the electrophoretic mobilities and intrinsic viscosities of a number of model proteins and DNA fragments. The primary purpose of the present work is to examine the effect of ion relaxation on the ion density and fluid velocity fields around short DNA fragments (20 and 40 bp). Contour density as well as vector field diagrams of the various scalar and vector fields are presented and discussed at monovalent salt concentrations of 0.03 and 0.11 M. In addition, the net charge current fluxes in the vicinity of the DNA fragments at low and high salt concentrations are briefly examined and discussed.

Modeling the electrophoresis of lysozyme. II. Inclusion of ion relaxation

In this work, boundary element methods are used to model the electrophoretic mobility of lysozyme over the pH range 2-6. The model treats the protein as a rigid body of arbitrary shape and charge distribution derived from the crystal structure. Extending earlier studies, the present work treats the equilibrium electrostatic potential at the level of the full Poisson-Boltzmann (PB) equation and accounts for ion relaxation. This is achieved by solving simultaneously the Poisson, ion transport, and Navier-Stokes equations by an iterative boundary element procedure. Treating the equilibrium electrostatics at the level of the full rather than the linear PB equation, but leaving relaxation out, does improve agreement between experimental and simulated mobilities, including ion relaxation improves it even more. The effects of nonlinear electrostatics and ion relaxation are greatest at low pH, where the net charge on lysozyme is greatest. In the absence of relaxation, a linear dependence of mobility and average polyion surface potential, (lambda zero)s, is observed, and the mobility is well described by the equation [formula: see text] where epsilon 0 is the dielectric constant of the solvent, and eta is the solvent viscosity. This breaks down, however, when ion relaxation is included and the mobility is less than predicted by the above equation. Whether or not ion relaxation is included, the mobility is found to be fairly insensitive to the charge distribution within the lysozyme model or the internal dielectric constant.