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

An efficient kinetic Monte Carlo to study analyte capture by a nanopore: transients, boundary conditions and time-dependent fields

To better understand the capture process by a nanopore, we introduce an efficient Kinetic Monte Carlo (KMC) algorithm that can simulate long times and large system sizes by mapping the dynamic of a point-like particle in a 3D spherically symmetric system onto the 1D biased random walk. Our algorithm recovers the steady-state analytical solution and allows us to study time-dependent processes such as transients. Simulation results show that the steady-state depletion zone near pore is barely larger than the pore radius and narrows at higher field intensities; as a result, the time to reach steady-state is much smaller than the time required to empty a zone of the size of the capture radius λe. When the sample reservoir has a finite size, a second depletion region propagates inward from the outer wall, and the capture rate starts decreasing when it reaches the capture radius λe. We also note that the flatness of the electric field near the pore, which is often neglected, induces a traffic jam that can increase the transient time by several orders of magnitude. Finally, we propose a new proof-of-concept scheme to separate two analytes of the same mobility but different diffusion coefficients using time-varying fields.

Electric field induced changes in protein conformation

The effect of a low strength oscillating electric field on the conformation of Bovine Serum Albumin (BSA) and Lysozyme in solution has been measured. A purpose built cell has been used to measure the real time autofluorescence and Circular Dichroism of the protein solutions exposed to electric fields of differing strength and frequency. Exposure to the electric fields results in protein unfolding for both Lysozyme and BSA. The applied field strengths are extremely small compared to the protein inter-chain intra-molecular forces. We propose a model whereby the electrophoretic motion of the proteins leads to a frictional force that results in protein unfolding. For BSA and Lysozyme in the electric fields used in this study, the shear rates at the protein surface under electrophoretic motion are of order 10(3) and 10(4) s(-1) respectively. Prolonged electric field exposure results in significant frictional energy dissipation in the proteins. The energy dissipated in the proteins results in protein unfolding, which is a critical initial step for protein aggregation and potentially amyloid fibril formation.

Why are proteins charged? Networks of charge-charge interactions in proteins measured by charge ladders and capillary electrophoresis

Almost all proteins contain charged amino acids. While the function in catalysis or binding of individual charges in the active site can often be identified, it is less clear how to assign function to charges beyond this region. Are they necessary for solubility? For reasons other than solubility? Can manipulating these charges change the properties of proteins? A combination of capillary electrophoresis (CE) and protein charge ladders makes it possible to study the roles of charged residues on the surface of proteins outside the active site. This method involves chemical modification of those residues to generate a large number of derivatives of the protein that differ in charge. CE separates those derivatives into groups with the same number of modified charged groups. By studying the influence of charge on the properties of proteins using charge ladders, it is possible to estimate the net charge and hydrodynamic radius and to infer the role of charged residues in ligand binding and protein folding.

Electrophoresis of positioned nucleosomes

We present in this article an original approach to compute the electrophoretic mobility of rigid nucleo-protein complexes like nucleosomes. This model allows us to address theoretically the influence of complex position along DNA, as well as wrapped length of DNA on the electrophoretic mobility of the complex. The predictions of the model are in qualitative agreement with experimental results on mononucleosomes assembled on short DNA fragments (<400 bp). Influences of additional experimental parameters like gel concentration, ionic strength, and effective charges are also discussed in the framework of the model, and are found to be qualitatively consistent with experiments when available. Based on the present model, we propose a simple semi-empirical formula describing positioning of nucleosomes as seen through electrophoresis.

Modeling the electrophoresis of peptides and proteins: improvements in the "bead method" to include ion relaxation and "finite size effects"

A bead model methodology developed in our lab (Xin et al. J. Phys. Chem. B 2006, 110, 1038) and applicable to modeling the free solution electrophoretic mobility of peptides and proteins is generalized in two significant ways. First, an approximate account is taken of the relaxation effect, which makes the methodology applicable to more highly charged peptides and proteins than was previously possible. Second, a more accurate account is taken of the finite size of the beads making up the model structure. This improvement makes the method applicable at higher salt concentrations and/or to models consisting of larger sized subunits. The relaxation effect is accounted for by correcting "unrelaxed" mobilities on the basis of model size and average electrostatic surface, or zeta potential. Correction factors are estimated using those of spheres with the same hydrodynamic radius and zeta potential as the model structure. The correction factors of spheres are readily determined. The more general methodology is first applied to two sets of peptides (74 different peptides total) varying in size from 2 to 42 amino acids. The sets also cover a wide range of net charges. It is shown that accounting for finite bead size results in a small change in model mobilities under the conditions of the experiments (35 mM monovalent salt). The correction for ion relaxation, however, can be significant for highly charged peptides and improves agreement between model and experimental mobilities. Our correction procedure is also tested by examining the electrophoretic mobility of a particular protein "charge ladder" (Carbeck et al. J. Am. Chem. Soc. 1999, 121, 10,671), where the protein charge is varied over a wide range yet the conformation remains essentially constant. In summary, the effects of ion relaxation can be significant if the absolute electrophoretic mobility of a peptide exceeds approximately 0.20 cm2/(kV s).

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.

Electrophoretic mobility equation for protein with molecular shape and charge multipole effects

We derive a simple formula for the free solution electrophoretic mobility of protein by including both molecular shape and charge distribution effects. The molecular shape of protein is described by a deformed sphere model, while the charge distribution is represented in terms of net charge, charge dipole, and charge quadrupole. The deformed sphere model approximates the radial coordinate of the protein surface as a simple quadratic equation based on the atomic coordinate data. Charge dipole does not affect the mobility of protein. Combined with the quadratic coefficients of the surface equation, charge quadrupole affects the mobility. When the charge quadrupole contribution is negligible, the mobility equation simplifies to the Henry equation in which the sphere radius is replaced with the hydrodynamic radius of protein. The deformed sphere model predicts correctly the hydrodynamic radius of protein from the atomic coordinate data. The hydrodynamic radius is not the radius of sphere of equal volume but the effective radius that correlates with the translational diffusivity of protein. To illustrate the utility of our mobility equation we study the electrophoresis of lysozyme and compare our results with previously published works.

Valence and anion binding of bovine ribonuclease A between pH 6 and 8

Several studies have shown that divalent anion binding to ribonuclease A (RNase A) contributes to RNase A folding and stability. However, there are conflicting reports about whether chloride binds to or stabilizes RNase A. Two broad-zone experimental approaches, membrane-confined electrophoresis and analytical ultracentrifugation, were used to examine the electrostatic and electrohydrodynamic characteristics of aqueous solutions of bovine RNase A in the presence of 100 mM KCl and 10 mM Bis-Tris propane over a pH range of 6.00-8.00. The results of data analysis using a Debye-Huckel-Henry model, compared with expectations based on pK(A) values, are consistent with the binding of two chlorides by RNase A. The decreased protein valence resulting from anion binding contributes 2-3 kJ/mol to protein stabilization. This work demonstrates the utility of first-principle valence determinations to detect protein solution properties that might otherwise remain undetected.

The primary electroviscous effect, free solution electrophoretic mobility, and diffusion of dilute prolate ellipsoid particles (minor axis = 3 nm) in monovalent salt solution

The principal objective of the present work is the modeling of the primary electroviscous effect of charged prolate ellipsoid models of low axial ratio. Other transport properties examined include (free solution) electrophoretic mobilities and translational diffusion constants. A numerical boundary element method is employed to solve the coupled Poisson, low Reynolds number Navier-Stokes, and ion transport equations. The methodology is first applied to the primary electroviscous effect of spheres with a centrosymmetric charge distribution and excellent agreement with independent theory is obtained. Specific model studies are also carried out for prolate ellipsoid models with axial ratios less than 4 and a minor axis equal to 3 nm. Most studies are carried out in aqueous NaCl solution (2 to 50 mM) at 20 degrees C for a range of different particle charges, although limited results are also presented in LiCl and KCl solution. The primary electroviscous effect for weakly charged prolate ellipsoids is smaller than that of a sphere under similar conditions. These studies are also carried out at high absolute particle charge. A comparison is made between the primary electroviscous effect and electrophoretic mobilities of prolate ellipsoids and corresponding spherical models.

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.

Active site control of myosin cross-bridge zeta potential

The electrical properties of contractile proteins contribute to muscle structure and perhaps function but have not been characterized adequately. Electrophoretic mobility, mu(e), is sensitive to the net electric charge and hydrodynamic size of a molecule in solution. Zeta potential, zeta, particle charge, Q(e), and particle charge-to-mass ratio are proportional to mu(e). We measured mu(e) for nucleotide complexes of skeletal muscle heavy meromyosin (HMM) and subfragment 1 (S1). The results indicate that mu(e) for HMM changes depending on the ligand bound in the active site. The changes in electric charge appear to occur mainly on the S1 moieties. For HMM(MgATPgammaS)(2) and HMM(MgADP.P(i))(2) the values of mu(e) are -0.077 and -0.17 (microm/s)/(V/cm), respectively. For these complexes, mu(e) is independent of [ATP], [ADP], and [P(i)]. When P(i) dissociates from HMM(MgADP.P(i))(2) to form HMM(MgADP)(2), mu(e) decreases to -0.61 (microm/s)/(V/cm). This large decrease in mu(e) is independent of free [ADP] or [ATP]. Increasing [P(i)], on the other hand, increases mu(e) for HMM(MgADP)(2) to values near those observed for the steady-state intermediate. For HMM, mu(e) = -0.34 and is independent of P(i). MgADP binding to HMM decreases mu(e) to -0.57 (microm/s)/(V/cm), and the dissociation constant is 9 microM. Taken together, these data indicate that mu(e) and, thus, zeta are controlled by ligand binding to the active site. The magnitudes of the particle charge-to-mass ratios for the HMM complexes are all in a range that falls within published values determined for a variety of other proteins. Possible roles that the observed nucleotide-dependent changes in cross-bridge electric charge might have in the contractile cycle in muscle are considered.

Electrophoretic Motion of a Slightly Deformed Sphere with a Nonuniform Zeta Potential Distribution

Electrophoretic motion is analyzed for a rigid, slightly deformed sphere with a nonuniform zeta potential distribution. Hydrodynamics and electrostatics solutions for the deformed sphere with an arbitrary double-layer thickness are determined by using the domain perturbation method. The surface shape and the zeta potential distribution for the deformed sphere are expressed by using the multipole expansion representation. In terms of monopole, dipole, and quadrupole moments of the surface shape and the zeta potential distribution, explicit expressions are obtained for the translational and rotational electrophoretic mobility tensors. The ensemble average for the mobility of the deformed sphere with a uniform orientation distribution is also derived. The utility of the general mobility expression is demonstrated by studying the electrophoretic motion of axisymmetric and ellipsoidal particles. The translational and rotational mobilities of axisymmetric particles are both affected by the monopole, dipole, and quadrupole moments of the zeta potential. For ellipsoidal particles, however, the dipole moment of the zeta potential does not affect the translational mobility, while the rotational mobility depends only on the dipole moment. The mobility of the deformed sphere with either a thick or a thin double layer is also derived.

Analysis by capillary electrophoresis of the kinetics of charge ladder formation for bovine carbonic anhydrase

A series of charge ladders of bovine carbonic anhydrase II were synthesized and the relative abundances of the rungs analyzed by capillary electrophoresis as a function of the quantity of acylating agent used. A simulation that models the kinetics of formation of the members of the charge ladders is described. The observed rate constants decreased as the extent of acylation increased. These rate constants correlated adequately with theoretical rate constants calculated using Debye-Hückel theory. The data are compatible with, but do not demand, a model for the formation of this charge ladder in which all unacetylated amino groups in each rung have indistinguishable reactivity and in which the reactivity of the amines in each rung decreases as the net charge on the protein increases; in this model, decreased reactivity is due to increased extent of protonation. This agreement between experiment and model suggests that the charge shielding that results from an ionic strength of 130 mM is not sufficient to suppress the influence of the increasingly negative charge of the protein with acetylation on the extent of protonation of Lys epsilon-NH2 groups.

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.

Quantitative calculations of antibody--antigen binding: steroid--DB3 binding energies by the linear interaction energy method

Linear interaction energy/molecular dynamics calculations have been used to compute steroid/antibody binding energies. The absolute binding affinities of 10 steroids to antibody DB3 and of a hapten to catalytic antibody 1E9 are computed and compared to experiment. A detailed analysis of the molecular origins of the observed binding patterns is provided. The binding energy of an untested steroid is predicted.

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.

An apparatus for membrane-confined analytical electrophoresis

A membrane-confined analytical electrophoresis apparatus for measuring the solution charge of macromolecules has been described previously (T. M. Laue et al., Anal. Biochem. 1989, 182, 377-382). Presented here is a design for this apparatus, which permits the on-line acquisition and display of absorbance data from up to 512 positions along an analysis chamber. Concentration distributions of macromolecules in solution can be monitored in the chamber to provide steady-state electrophoresis, electrophoretic mobility and diffusion measurements. Buffer chambers press semipermeable membranes against the open ends of a fused-silica cuvette to form the analysis chamber. This configuration permits both the flow of buffer and the establishment of an electric field across the cuvette, while retaining macromolecules in the field of view. Though a gel may be included in the analysis chamber, none is required for gradient stabilization. The volume of sample required for analysis is 8 microL, most of which is recoverable. Experimental conditions can be varied during study by simply changing the circulating buffer and/or the electric field. The analysis and buffer chambers are held in an aluminum housing that sits in an aluminum water jacket. The water jacket provides temperature control, shielding from external electrical noise and also serves as an optical mask. Plans for the cell assembly, optical system and the computer interface for data acquisition are provided. The assembly and operation of the apparatus and the analysis of data are described.