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 (µep ) 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 (Rh ), 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 (µep ; S and Rh ) 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.

Specific ion effects on the electrophoretic mobility of small, highly charged peptides: a modeling study.

In this work, we use coarse-grained modeling to study the free solution electrophoretic mobility of small highly charged peptides (lysine, arginine, and short oligos thereof (up to nonapeptides)) in NaCl and Na2SO4 aqueous solutions at neutral pH and room temperature. The experimental data are taken from the literature. A bead modeling methodology that treats the electrostatics at the level of the nonlinear Poisson Boltzmann equation developed previously in our laboratory is able to account for the mobility of all peptides in NaCl, but not Na2SO4. The peptide mobilities in Na2SO4 can be accounted for by including sulfate binding in the model and this is proposed as one possible explanation for the discrepancy. Oligo arginine peptides bind more sulfate than oligo lysines and sulfate binding increases with the oligo length.

The electrophoretic mobility of a weakly charged "soft" sphere in a charged hydrogel: application of the Lorentz reciprocal theorem.

The electrophoretic mobility of a dilute, weakly charged "soft" particle in a charged hydrogel modeled as an effective medium is investigated in this work. This is closely related to previous work (Li, F.; Allison, S. A.; Hill, R. J. J. Colloid Interface Sci. 2014, 423, 129-142) but approached in a different way using the Lorentz reciprocal theorem. Under the limiting conditions of the present work, it is possible to avoid numerical solution of differential equations. An analytical equation is derived for the mobility and applied to a number of cases.

Nanoparticle gel electrophoresis: soft spheres in polyelectrolyte hydrogels under the Debye-Hückel approximation.

A mathematical model for electrophoresis of polyelectrolyte coated nanoparticles (soft spheres) in polyelectrolyte hydrogels is proposed, and evaluated by comparison to literature models for bare-sphere gel electrophoresis and free-solution electrophoresis. The utilities of approximations based on the bare-particle electrophoretic mobility, free-solution mobility, and electroosmotic flow in hydrogels are explored. Noteworthy are the influences of the particle-core dielectric constant and overlap of the polyelectrolyte shell. The present theory, which neglects ion-concentration and charge-density perturbations, indicates that the gel electrophoretic mobilities of metallic-core nanoparticles in polyelectrolyte gels can be qualitatively different than for their non-metallic counterparts. These insights will be beneficial for interpreting nanoparticle gel-electrophoresis data, optimizing electrophoretic separations, and engineering nanoparticles for technological applications.

Coarse-grained modeling of the titration and conductance behavior of aqueous fullerene hexa malonic acid (FHMA) solutions.

The coarse-grained continuum primitive model is developed and used to characterize the titration and electrical conductance behavior of aqueous solutions of fullerene hexa malonic acid (FHMA). The spherical FHMA molecule, a highly charged electrolyte with an absolute valence charge as large as 12, is modeled as a dielectric sphere in Newtonian fluid, and electrostatics are treated numerically at the level of the non-linear Poisson-Boltzmann equation. Transport properties (electrophoretic mobilities and conductances) of the various charge states of FHMA are numerically computed using established numerical algorithms. For reasonable choices of the model parameters, good agreement between experiment (published literature) and modeling is achieved. In order to accomplish this, however, a moderate degree of specific binding of principal counterion and FHMA must be included in the modeling. It should be emphasized, however, that alternative explanations are possible. This comparison is made at 25 °C for both Na(+) and Ca(2+) principal counterions. The model is also used to characterize the different charge states and degree of counterion binding to those charge states as a function of pH.

Extracting information from the ionic strength dependence of electrophoretic mobility by use of the slope plot.

The effective mobility (µ(ep)) is the main parameter characterizing the electrophoretic behavior of a given solute. It is well-known that µ(ep) is a decreasing function of the ionic strength for all solutes. Nevertheless, the decrease depends strongly on the nature of the solute (small ions, polyelectrolyte, nanoparticles). Different electrophoretic models from the literature can describe this ionic strength dependence. However, the complexity of the ionic strength dependence with the solute characteristics and the variety of analytical expressions of the different existing models make the phenomenological ionic strength dependence difficult to comprehend. In this work, the ionic strength dependence of the effective mobility was systematically investigated on a set of different solutes [small mono- and multicharged ions, polyelectrolytes, and organic/inorganic (nano)particles]. The phenomenological decrease of electrophoretic mobility with ionic strength was experimentally described by calculating the relative electrophoretic mobility decrease per ionic strength decade (S) in the range of 0.005-0.1 M ionic strength. Interestingly, the "slope plot" displaying S as a function of the solute electrophoretic mobility at 5 mM ionic strength allows for defining different zones that are characteristic of the solute nature. This new representative approach should greatly help experimentalists to better understand the ionic strength dependence of analyte and may contribute to the characterization of unknown analytes via their ionic strength dependence of electrophoretic mobility.

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 highly charged peptides: application to oligolysines.

The "coarse-grained" bead modeling methodology, BMM, is generalized to treat electrostatics at the level of the nonlinear Poisson-Boltzmann equation. This improvement makes it more applicable to the important class of highly charged macroions and highly charged peptides in particular. In the present study, the new nonlinear Poisson-Boltzmann, NLPB-BMM procedure is applied to the free solution electrophoretic mobility of low molecular mass oligolysines (degree of polymerization 1-8) in lithium phosphate buffer at pH 2.5. The ionic strength is varied from 0.01 to 0.10 M) and the temperature is varied from 25 to 50°C. In order to obtain quantitative agreement between modeling and experiment, a small amount of specific phosphate binding must be included in modeling. This binding is predicted to increase with increasing temperature and ionic strength.

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.

Numerical solution of the nonlinear Poisson-Boltzmann equation for a macroion modeled as an array of non overlapping beads.

In this work, an approximate numerical procedure, AB, is developed to solve the nonlinear Poisson-Boltzmann equation around a macroion modeled as an array of non overlapping beads containing charges placed at their centers. The bead radii, their charges, and the relative bead configuration are arbitrary. In the limit of a single bead of arbitrary charge, the AB procedure is exact. For dimers and other bead arrays, it is possible to compare the approximate potentials derived using the AB procedure with exact potentials obtained by a boundary element, BE, procedure. Average surface or "zeta" potentials are examined for dimers, trimers, and tetramers. The bead size and charges are chosen to ensure that nonlinear charge effects are significant. In these test cases, the AB procedure is accurate to better than 5% over a wide range of ionic strength. Finally, the electrostatics of short duplex DNA (≤100 base pairs) are examined using both "smooth cylinder" and "touching bead" models. It is concluded that the modeling DNA as a string of touching beads using the AB procedure yields "zeta" potentials that are accurate to better than 10%.

Modeling the electrophoresis and transport of peptides: the effective sphere model and complex formation.

Modeling the electrophoretic mobility of peptides is examined in this study using a "coarse grained" bead model, or B model for short 8 and also a simpler "effective sphere" (ES) model. A comparison between the B and ES models is carried out for peptide models covering a broad range of ionic strength, peptide charge, and peptide length. At ionic strengths lower than approximately 0.013 M, the B and ES models agree to within a few percent. The ES model is much simpler than the B model and is of particular value in certain applications such as complex formation between peptide and other species in the BGE. The mobility behavior of oligoglycine in a borate buffer at high pH can be accounted for when complex formation is included in modeling.

Modeling the electrophoresis of oligoglycines.

The electrophoretic mobility of low molecular mass oligoglycines is examined in this study using a "coarse-grained" bead modeling methodology [Pei, H., Allison, S. A., J. Chromatogr. A 2009, 1216, 1908-1916]. The advantage of focusing on these peptides is that their charge state is well known [Plasson, R., Cottet, H., Anal. Chem. 2006, 78, 5394-5402] and extensive electrophoretic mobility data are also available in different buffers [Survay, M. A., Goodall, D. M., Wren, S. A. C., Rowe, R. C., J. Chromatogr. A 1996, 741, 99-113] and over a broad range of temperatures [Plasson, R., Cottet, H., Anal Chem. 2005, 77, 6047-6054]. Except for assumptions about peptide secondary structure, the B model has no adjustable parameters. It is concluded that the oligoglycines adopt a random configuration at high temperature (50 degrees C and higher), but more compact conformations at lower temperature. It is proposed that triglycine through pentaglycine adopt compact cyclic structures at low temperature (up to about 25 degrees C) in aqueous solution. At 25 degrees C, buffer interactions are also examined and may or may not influence peptide conformation depending on the buffer species. In a borate buffer at high pH, the mobility data are consistent with complex formation between the oligoglycine and borate anion.

The dependence of the electrophoretic mobility of small organic ions on ionic strength and complex formation.

The ionic strength dependence of the electrophoretic mobility of small organic anions with valencies up to -3 is investigated in this study. Provided the anions are not too aspherical, it is argued that shape and charge distribution have little influence on mobility. To a good approximation, the electrophoretic mobility of a small particle should be equal to that of a model sphere with the same hydrodynamic radius and same net charge. For small ions, the relaxation effect (distortion of the ion atmosphere from equilibrium due to external electric and flow fields) is significant even for monovalent ions. Alternative procedures of accounting for the relaxation effect are examined. In order to account for the ionic strength dependence of a specific set of nonaromatic and aromatic anions in aqueous solution, it is necessary to include complex formation between the anion with species in the BGE. A number of possible complexes are considered. When the BGE is Tris-acetate, the most important of these involves the complex formed between anion and Tris, the principle cation in the BGE. When the BGE is sodium borate, an anion-anion (borate) complex appears to be important, at least when the organic anion is monovalent. An algorithm is developed to analyze the ionic strength dependence of the electrophoretic mobility. This algorithm is applied to two sets of organic anions from two independent research groups.

Viscosity of dilute model bead arrays at low shear: inclusion of short range solute-solvent interactions.

The intrinsic viscosity, [eta], of certain polymer-solvent systems, such as alkanes in benzene, are "anomalous" in the sense that [eta] for low molecular weight fractions are low and in certain cases negative (Dewan, K. K.; Bloomfield, V. A.; Berget, P. G.; J. Phys. Chem. 1974, 75, 3120). In this work, the theory of the viscosity of a dilute suspension of macromolecules at low shear is formulated that accounts for possible solute-solvent interactions. In doing so, we show that negative intrinsic viscosities are possible and are able to reproduce quite well the known length dependence of [eta] for alkanes in benzene. The coarse grained, solvent continuum, bead model developed here is an extension of previous work (Allison, S. A.; Pei, H. J. Phys. Chem. B 2009, 113, 8056). Following Fixman (Fixman, M. J. Chem. Phys. 1990, 92, 6858), we assume that solute-solvent interactions are short-range in character and can be separated from long-range hydrodynamic interactions between different beads. These interactions are accounted for by introducing three adjustable parameters specific to the transport of small "monomeric" solutes in the solvent of interest. Long range hydrodynamic interactions are accounted for to order a(J)(2)/r(IJ)(3) (a(J) is a bead radius and r(IJ) is an interbead distance). In modeling a macromolecule as an arbitrary array of N beads, the transport of the array is examined numerically in 5 different elementary shear fields. The most computationally demanding component of the procedure involves the inversion of a 12N by 12N matrix. In the present work, we restrict ourselves to systems with a maximum N of about 100. Our procedure is first applied to short rods and rings of from 2 to 10 beads which can be compared with independent results from the literature. Agreement is found to be better than 5%. Modeling macromolecules as wormlike chains, the procedure is then applied first to duplex DNA and then to alkanes in benzene. In both cases, it is possible to obtain excellent agreement between modeling and experiment.

Translational diffusion constants of short peptides: measurement by NMR and their use in structural studies of peptides.

In this work, pulsed field gradient NMR is used to measure the translational self-diffusion constants (D(T)'s) of five simple peptides (GG, GR, GGR, GGNA, and GGRA) as well as glycine, G, at low concentration. The experiments were carried out in D(2)O at 298 K at pD = 3.5 in 80 mM sodium phosphate buffer. Of the five peptides, four are being reported for the first time (all except GG) and the results of G and GG are compared with D(T)'s from the literature. When corrected for differences in solvent viscosity and temperature, the discrepancy between D(T)'s of G and GG measured in the present work are lower than previously published values by several percent. Given the range of values reported in the literature for specific values of the amino acids by different groups, this discrepancy is regarded as reasonable. Diffusion constants can provide useful information about molecular size and conformation. Modeling a peptide made up of N amino acids as 2N beads (2 for each amino acid present in the peptide), we examine the diffusion constants of the above-mentioned peptides and conclude they are consistent with unfolded or random conformations in solution. Also, by comparing the diffusion constants of G and GG, an estimate of the change in solvation volume due to the loss of a water molecule can be estimated.

Viscosity of dilute suspensions of rigid bead arrays at low shear: accounting for the variation in hydrodynamic stress over the bead surfaces.

In this work, we examine the viscosity of a dilute suspension of irregularly shaped particles at low shear. A particle is modeled as a rigid array of nonoverlapping beads of variable size and geometry. Starting from a boundary element formalism, approximate account is taken of the variation in hydrodynamic stress over the surface of the individual beads. For a touching dimer of two identical beads, the predicted viscosity is lower than the exact value by 5.2%. The methodology is then applied to several other model systems including tetramers of variable conformation and linear strings of touching beads. An analysis is also carried out of the viscosity and translational diffusion of several dilute amino acids and diglycine in water. It is concluded that continuum hydrodynamic modeling with stick boundary conditions is unable to account for the experimental viscosity and diffusion data simultaneously. A model intermediate between "stick" and "slip" could possibly reconcile theory and experiment.

Using electrophoretic mobility and bead modeling to characterize the charge and secondary structure of peptides.

Using a modeling methodology developed in our laboratory previously, the free solution electrophoretic mobilities of several peptides are examined to see what they can tell us about: (i) the pK(a)s of specific side groups, and (ii) possible secondary structure. Modeling is first applied to mobility versus pH data of several small peptides (Messana, I. et al., J. Chromatogr. B 1997, 699, 149) where the only adjustable parameter associated with the charge state of the peptide is the pK(a )of the C-terminal. In addition to examining this parameter, the question of possible secondary structure is addressed. For two of the peptides considered, GGNA and GGQA, it is possible to account for the observed mobilities using "random" models with little restriction on the allowed range of Phi-Psi angles. For GGRA and RPPGF, "compact" models (possibly involving an I-turn) must be used to match modeling mobilities with experiment. Finally, three more complicated peptides ranging in size from 15 to 20 amino acids are also examined and characterized (Sitaram, B. R. et al., J. Chromatogr. A 1999, 857, 263). Here also, we find evidence of I-turns or some other "compact" structure in two of the three peptides examined.

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.

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.