Modeling the free solution electrophoretic mobility of short DNA fragments

Boundary element methods are used to model the free solution electrophoretic mobility of short DNA fragments. The Stern surfaces of the DNA fragments are modeled as plated cylinders that reproduce translational and rotational diffusion constants. The solvent-accessible and ion-accessible surfaces are taken to be coincident with the Stern surface. The mobilities are computed by solving simultaneously the coupled Navier–Stokes, Poisson, and ion-transport equations. The equilibrium electrostatics are treated at the level of the full Poisson–Boltzmann equation and ion relaxation is included. For polyions as highly charged as short DNA fragments, ion relaxation is substantial. At .11 M KCl, the simulated mobilities of a 20 base pair DNA fragment are in excellent agreement with experiment. At .04 M Tris acetate, pH = 8.0, the simulated mobilities are about 10–15% higher than experimental values and this discrepancy is attributed to the relatively large size of the Tris counterion. The length dependence of the mobility at .11 M KCl is also investigated. Earlier mobility studies on lysozyme are reexamined in view of the present findings. In addition to electrophoretic mobilities, the effective polyion charge measured in steady state electrophoresis and its relationship to the preferential interaction parameter γgG is briefly considered. © 1998 John Wiley & Sons, Inc. Biopoly 46: 359–373, 1998     

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.     

Calculation of the Electrophoretic Mobility of a Particle Bearing Bound Polyelectrolyte Using the Nonlinear Poisson-Boltzmann Equation

A numerical method for determining the electrophoretic mobility of a polyelectrolyte-coated particle is presented. The particle surface is modeled as having a permeable layer of polyelectrolyte molecules anchored to its surface. Fluid flow within the polyelectrolyte layer is subject to Stokes drag arising from the polyelectrolyte segments. The method allows arbitrary distribution of polymer segments and charge density normal to the surface to be used. The hydrodynamic plane of shear may also be varied. The potential profile is determined by a numerical solution to the nonlinearized Poisson-Boltzmann equation. The potential profile is then used in a numerical solution to the Navier-Stokes equation to give the required mobility. The use of the nonlinearized Poisson-Boltzmann equation extends the results to higher charge density/lower ionic strength conditions than previous treatments. The surface potentials and mobilities for three limiting charge distributions are compared for both the linear and nonlinear treatments to delimit the range of validity of the linear treatment. The utility of the numerical, nonlinear treatment is demonstrated by an improved fit to the electrophoretic mobility of human erythrocytes as a function of ionic strength in the range 10 to 150 mM.     

Adsorption of monovalent cations to bilayer membranes containing negative phospholipids.

The electrophoretic mobilities of multilamellar phosphatidylserine vesicles were measured in solutions containing monovalent cations, and the xi potentials, the electrostatic potentials at the hydrodynamic plane of shear, were calculated from the Helmholtz--Smoluchowski equation. In the presence of 0.1 M lithium, sodium, ammonium, potassium, rubidium, cesium, tetraethylammonium, and tetramethylammonium chloride, the xi potentials were -60, -62, -72, -73, -77, -80, -82, and -91 mV, respectively. Similar results were obtained with phosphatidylglycerol vesicles; different results were obtained with cardiolipin, phosphatidylinositol, and phosphatidic acid vesicles. The phosphatidylserine results are interpreted in terms of the Stern equation, a combination of the Gouy equation from the theory of the diffuse double layer, the Boltzmann relation, and the Langmuir adsorption isotherm. Evidence is presented that suggests the hydrodynamic plane of shear is 2 A from the surface of the membrane in solutions containing the alkali metal cations. With this assumption, the intrinsic association constants of the above monovalent cations with phosphatidylserine are 0.8, 0.6, 0.17, 0.15, 0.08, 0.05, 0.03, and 0 M-1, respectively. The validity of this approach was tested in two ways. First, the xi potentials of vesicles formed from mixtures of phosphatidylserine and a zwitterionic lipid, phosphatidylcholine, were measured in solutions containing different concentrations of sodium. All the data could be described by the Stern equation if the "relaxation" of the ionic atmosphere, which is predicted by classic electrostatic and hydrodynamic theory to occur at low salt concentrations and high potentials, was circumvented by using only large (diameter greater than 13 micrometers) vesicles for these measurements. Second, the fluorescent probe 2-(p-toluidinyl)naphthalene-6-sulfonate was used to estimate the potential at the surface of phosphatidylserine and phosphatidylglycerol vesicles sonicated in 0.1 M NaCl. Reasonable agreement with the predicted values of the surface potential was obtained.     

Theory of the electrokinetic behavior of human erythrocytes

We develop a theory of electrophoresis of human erythrocytes that predicts mobilities significantly smaller than those based on the classical Smoluchowski relation. In the classical treatment the charge is assumed to be spread uniformly on the hydrodynamic surface. The present model takes into account that most of the charge, due mainly to sialic acid, is contained in the glycocalyx. The glycocalyx is modeled as a permeable layer of polyelectrolyte molecules anchored to the cell membrane. The charge is assumed to be uniformly distributed throughout this layer. The fluid flow in the layer is treated as being dominated by Stokes friction arising from idealized polymer segments. The Navier-Stokes equations are solved to give the dependence of electroosomotic velocity with distance from the cell surface. An expression for the electrophoretic mobility is obtained which contains two parameters (a) the thickness of the glycocalyx and (b) the mean polymer segment radius. The best fit to experimental data is obtained if these are given the values 75 A and 7 A, respectively. Deviation from experimental data at low ionic strength (less than 0.05 M) occurs. However, this deviation is in the direction one would expect if at low ionic strength the polyelectrolyte layer expands slightly due to decreased charge shielding.     

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.     

Sterically stabilized liposomes. Reduction in electrophoretic mobility but not electrostatic surface potential.

The electrophoretic mobility of liposomes containing a negatively charged derivative of phosphatidylethanolamine with a large headgroup composed of the hydrophilic polymer polyethylene glycol (PEG-PE) was determined by Doppler electrophoretic light scattering. The results show that this method is improved by the use of measurements at multiple angles to eliminate artifacts and that very small mobilities can be measured. The electrophoretic mobility of liposomes with 5 to 10 mol% PEG-PE is approximately -0.5 mu ms-1/Vcm-1 regardless of PEG-PE content compared with approximately -2 mu ms-1/Vcm-1 for similar liposomes but containing 7.5% phosphatidylglycerol (PG) instead of PEG-PE. Measurements of surface potential by distribution of an anionic fluorescent probe show that the PEG-PE imparts a negative charge identical to that by PG, consistent with the expectation of similar locations of the ionized phosphate responsible for the charge. The reduced mobility imparted by the surface bound PEG is attributed to a mechanism similar to that described for colloidal steric stabilization: hydrodynamic drag moves the hydrodynamic plane of shear, or the hydrodynamic radius, away from the charge-bearing plane, that of the phosphate moities. An extended length of approximately 50 A for the 2,000 molecular weight PEG is estimated from the reduction in electrophoretic mobility.     

Computation of the electrophoretic mobility of proteins.

A scheme is presented for computing the electrophoretic mobility of proteins in free solution, accounting for the details of the protein shape and charge distribution. The method of Teubner is implemented using a boundary integral formulation within which the velocity distribution, the equilibrium electrical potential around the molecule, and the potential distribution due to the applied field are solved for numerically using the boundary element method. Good agreement of the numerical result is obtained for spheres with the corresponding semi-analytical specialization of Henry's analysis. For protein systems, the method is applied to lysozyme and ribonuclease A. In both cases, the predicted mobility tensors are fairly isotropic, with the resulting scalar mobilities being significantly smaller than for spheres of equal volume and net charge. Comparisons with previously published experimental results for ribonuclease show agreement to be excellent in the presence of a net charge, but poorer at the point of zero charge. The approach may be useful for evaluating approximate methods for estimating protein electrophoretic mobilities and for using electrophoretic measurements to obtain insight into charge distributions on proteins.     

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.     

ELECTROKINETIC PHENOMENA : X. ELECTRIC MOBILITY AND CHARGE OF PROTEINS IN ALCOHOL-WATER MIXTURES

1. The electrophoretic velocities of gelatin-, egg-albumin-, and gliadin-covered quartz particles in various alcohol-water solutions are, within the limits employed in usual experimental procedures, proportional to the field strength. 2. The electrophoretic mobilities of small, irregularly shaped quartz particles covered with an adsorbed film of protein in alcohol-water solutions are equal to the electroosmotic mobilities of the liquid past similarly coated flat surfaces. Hence the size and shape of such particles does not influence their mobilities, which depend entirely on the protein film. 3. The corrected mobility and hence presumably the charge of gelatin-covered quartz particles in solutions containing 35 per cent ethyl alcohol is proportional to the combining power of the gelatin; therefore the gelatin is adsorbed with the active groups oriented toward the liquid. The same is true in 60 per cent alcohol. 4. The charge calculated by means of the Debye-Henry approximation from the mobility of gelatin in solutions containing up to 35 per cent ethyl alcohol is, in the neighborhood of the isoelectric point, proportional to the combining power of the gelatin. Therefore the dielectric constant and the viscosity of the bulk of the medium may be used in the Debye-Henry approximation Q = 6 π η r v_m (1 + κ r) to predict changes in charge from mobility. 5. In the neighborhood of the isoelectric point gelatin is probably completely ionized in buffered ethyl alcohol-water mixtures up to 60 per cent alcohol. 6. In the presence of ethyl alcohol the isoelectric point of gelatin is shifted toward smaller hydrogen ion activities. This shift, like that caused by alcohol in the isoelectric points of certain amino acids, is approximately linearly related to the dielectric constant of the medium.     

A general method for the evaluation of diffusion constants,dilute-solution viscoelasticity,and the complex dielectric constant of a rigid macromolecule with an arbitrary configuration. II

As a continuation of a previous paper [Biopolymers 16 , (1977)], in which we described a general method for the evaluation of the diffusion constants of a rigid macromolecule with an arbitrary configuration, this paper presents a theory for evaluating the relaxational behavior of the molecule in solution. The diffusion equation of a molecule subjected to hydrodynamic or electric forces is solved and both a complex viscosity and a complex dielectric constant are obtained by assuming that the molecule is composed of Tokes spheres and, in the dielectric property, the molecule possesses a fixed charge distribution. The viscoelasticity of a rigid rod is calculated and compared with the analytical results.     

Effect of solution pH on protein transport through ultrafiltration membranes

Although a number of previous studies have demonstrated that solution pH can have a dramatic effect on protein transport through ultrafiltration membranes, the exact origin of this behavior has been unclear. Experimental data were obtained for the transport of a broad range of proteins with different surface charge and molecular weight. The effective hydrodynamic size of the proteins was evaluated using size‐exclusion chromatography. The membrane charge, both before and after exposure to a given protein, was evaluated using streaming potential measurements. In most cases, the electrostatic interactions were dominated by the distortion of the electrical double layer surrounding the protein, leading to a distinct maximum in protein transmission at the protein isoelectric point. Attractive electrostatic interactions did occur when the protein and membrane had a large opposite charge, causing a second maximum in transmission at a pH between the isoelectric points of the protein and membrane. The sieving data were in good agreement with theoretical calculations based on available models for the partitioning of charged solutes in cylindrical pores. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 64: 27–37, 1999.     

Medium reorganization energy and enzymatic reaction activation energy

Reorganization and activation energies for charge transfer reactions occurring inside a dielectric sphere have been calculated by solving the problem of polar medium reorganization within and outside a dielectric sphere placed in another infinite dielectric. The dielectric sphere is assumed to simulate a protein globule, i.e. an enzyme molecule. It has been shown that for some reaction types the activation energy tends to decrease as the globule radius increases and that for each of the reaction types considered there is an optimal globule radius an increase of which does not bring about any tangible activation energy reduction. The calculated optimal radii for different processes are in good agreement with the increasing molecular sizes in the series: ribonuclease less than or equal to lysozyme less than serine proteinases approximately equal to cysteine proteinases less than NAD-dependent dehydrogenases. The calculated radii are usually about 1.5 to 1.7 times (and molecular masses about 4-5 times) smaller than the experimental ones. The reasons for this discrepancy are discussed and it has been suggested that the approximate nature of the treatment of a protein globule as a structureless dielectric is the main reason. It is shown that charge transfer at an acute angle to the globule surface is the optimum process. For endoergonic reaction stages it is the net charge transfer towards the periphery and for exoergonic ones that in the reverse direction which are advantageous. These conclusions are consistent with the data about the structure of the above-mentioned enzymes.     

Correlation of electrophoretic mobilities from capillary electrophoresis with physicochemical properties of proteins and peptides

Excellent correlation was observed for the electrophoretic mobilities measured by capillary zone electrophoresis versus q/MW2/3, where q is the calculated charge and MW is the molecular weight. Mobilities of a set of 33 diverse peptides from enzymatic digests and 10 intact proteins were measured for separations at pH 2.35, 8.0, and 8.15 with constant ionic strength, temperature, and viscosity. The correlation suggests that the frictional drag is proportional to the surface area of a sphere that has a volume proportional to the MW. The correlation of electrophoretic mobility with physicochemical properties will facilitate the elucidation of optimum separation strategies for protein and peptide mixtures.     

Surface pH controls purple-to-blue transition of bacteriorhodopsin. A theoretical model of purple membrane surface.

We have developed a surface model of purple membrane and applied it in an analysis of the purple-to-blue color change of bacteriorhodopsin which is induced by acidification or deionization. The model is based on dissociation and double layer theory and the known membrane structure. We calculated surface pH, ion concentrations, charge density, and potential as a function of bulk pH and concentration of mono- and divalent cations. At low salt concentrations, the surface pH is significantly lower than the bulk pH and it becomes independent of bulk pH in the deionized membrane suspension. Using an experimental acid titration curve for neutral, lipid-depleted membrane, we converted surface pH into absorption values. The calculated bacteriohodopsin color changes for acidification of purple, and titrations of deionized blue membrane with cations or base agree well with experimental results. No chemical binding is required to reproduce the experimental curves. Surface charge and potential changes in acid, base and cation titrations are calculated and their relation to the color change is discussed. Consistent with structural data, 10 primary phosphate and two basic surface groups per bacteriorhodopsin are sufficient to obtain good agreement between all calculated and experimental curves. The results provide a theoretical basis for our earlier conclusion that the purple-to-blue transition must be attributed to surface phenomena and not to cation binding at specific sites in the protein.     

Analysis of polypeptide molecular weights by electrophoresis in urea

Ten proteins of differing disulfide contents and isoionic points were subjected to disc gel electrophoresis in the presence of 8 urea-0.9 acetic acid to evaluate the use of this technique in determining polypeptide molecular weights. Comparison of the electrophoretic mobilities before and after reduction of the proteins' disulfide bonds demonstrated that only after all disulfide bonds were broken, could their molecular weights be estimated with any degree of accuracy. The expression of the electrophoretic mobilities as a function of the proteins' effective hydrodynamic sizes, thereby taking into account the extent of constraint by disulfide bonds, allowed a comparison of disulfide cross-linked and linear forms of the protein polypeptides. The extent to which intrinsic charge affects a protein's electrophoretic mobility was estimated by comparing alpha-lactalbumin and lysozyme, two proteins of identical size but vastly different isoionic points. They exhibited a 20% difference in mobilities. An apparent slow reduction of disulfide bonds was observed to occur when proteins were exposed to reducing agent at low pH in 8 urea.     

Physicochemical characterization of γ-crystallins from bovine lens—Hydrodynamic and biochemical properties

A detailed hydrodynamic study has been made on the γ-crystallin of the bovine lens. Sedimentation study indicates that γ-crystallin shows a nearly gaussian peak throughout the course of sedimentation at high speed, using a synthetic boundary cell. The diffusion and sedimentation coefficients are 10.3×10^?7 cm²/sec and 2.51 S, respectively. The weight-average molecular weight of the unfractionated γ-crystallin calculated from sedimentation equilibrium is 21,800. The four major subfractions of γ-crystallin show similar hydrodynamic properties with an intrinsic viscosity of 2.50 ml/g and a Stokes radius of 21 Å. The distinct electrophoretic mobilities exhibited by the four subfractions show gel-concentration dependence and similar slopes in the Ferguson plot, indicative of being charge isomers of the same molecular species. Amino acid analysis of these four subfractions corroborated the conclusions that these γ-crystallin polypeptides are closely related and comprise a multigene family of crystallins. Based on the sedimentation and intrinsic viscosity data, γ-crystallin can be modeled as a prolate ellipsoid with an axial ratio of approximately 3.0 and a hydration factor of 0.27 g water per gram protein. The circular dichroism data for γ-crystallins showed a minimum at about 217 nm, characteristic of a β-sheet conformation. These structural characteristics are in good accord with those derived from X-ray diffraction data for γ-crystallin II.     

Impact of chemical and structural anisotropy on the electrophoretic mobility of spherical soft multilayer particles: the case of bacteriophage MS2

We report a theoretical investigation of the electrohydrodynamic properties of spherical soft particles composed of permeable concentric layers that differ in thickness, soft material density, chemical composition, and flow penetration degree. Starting from a recent numerical scheme developed for the computation of the direct-current electrophoretic mobility (μ) of diffuse soft bioparticles, the dependence of μ on the electrolyte concentration and solution pH is evaluated taking the known three-layered structure of bacteriophage MS2 as a supporting model system (bulk RNA, RNA-protein bound layer, and coat protein). The electrokinetic results are discussed for various layer thicknesses, hydrodynamic flow penetration degrees, and chemical compositions, and are discussed on the basis of the equilibrium electrostatic potential and hydrodynamic flow field profiles that develop within and around the structured particle. This study allows for identifying the cases where the electrophoretic mobility is a function of the inner structural and chemical specificity of the particle and not only of its outer surface properties. Along these lines, we demonstrate the general inapplicability of the notions of zeta potential (ζ) and surface charge for quantitatively interpreting electrokinetic data collected for such systems. We further shed some light on the physical meaning of the isoelectric point. In particular, numerical and analytical simulations performed on structured soft layers in indifferent electrolytic solution demonstrate that the isoelectric point is a complex ionic strength-dependent signature of the flow permeation properties and of the chemical and structural details of the particle. Finally, the electrophoretic mobilities of the MS2 virus measured at various ionic strength levels and pH values are interpreted on the basis of the theoretical formalism aforementioned. It is shown that the electrokinetic features of MS2 are to a large extent determined not only by the external proteic capsid but also by the chemical composition and hydrodynamic flow permeation of/within the inner RNA-protein bound layer and bulk RNA part of the bacteriophage. The impact of virus aggregation, as revealed by decreasing diffusion coefficients for decreasing pH values, is also discussed.     

Physico-chemical characterization of proteins by capillary electrophoresis

The electrophoretic mobility of proteins was successfully determined by means of capillary electrophoresis (CE) with various background electrolytes (BGEs). The objective was focused on the variation in BGE physico-chemical composition and the consequential impact on the observed protein charge. Experimental and calculated mobilities, according to Henry's equation, versus ionic strength have been compared. For positively-charged lysozyme, a good agreement between observed and calculated mobilities was observed using triethanolamine chloride at pH 7.0 as the BGE. Mobility close to zero was shown using borate (pH 8.0) and phosphate (pH 7.0) at a low ionic strength of about 20 mmol l⁻¹, and as a consequence, specific adsorption of oxyanions was evidenced. Lysozyme retention in the case of reversed-phase high-performance liquid chromatography (RP-HPLC) was decreased by the presence of phosphate ions. CE and HPLC are complementary tools for characterizing the behaviour of lysozyme. On the other hand, the mobility of the negatively-charged α-lactalbumin remained constant as regards phosphate at pH 7.0 in the 20–200 mmol l⁻¹ range, contrary to the decrease that had been expected with the increasing ionic strength. β-Lactoglobulin exhibited increasingly lower mobilities than those expected of boric acid/borate at pH 7.0 and 8.0 (I=20 mmol l⁻¹).