Hydrodynamic Radius Coincides with the Slip Plane Position in the Electrokinetic Behavior of Lysozyme

Abstract

The zeta potential (ζ) is the effective charge energy of a solvated protein, describing the magnitude of electrostatic interactions in solution. It is commonly used in the assessment of adsorption processes and dispersion stability. Predicting ζ from molecular structure would be useful to the structure-based molecular design of drugs, proteins and other molecules that hold charge dependent function while remaining suspended in solution. One challenge in predicting ζ is identifying the location of the slip plane (X_SP), a distance from the protein surface where ζ is theoretically defined. This study tests the hypothesis that the X_SP can be estimated by the Stokes-Einstein hydrodynamic radius (R_h), using globular hen egg white lysozyme as a model system. Although the X_SP and R_h differ in their theoretical definitions, with the X_SP being the position of the ζ during electrokinetic phenomena (e.g. electrophoresis) and the R_h being a radius pertaining to the edge of solvation during diffusion, they both represent the point where water and ions no longer adhere to a molecule. This work identifies the limited range of ionic strengths in which the X_SP can be determined using diffusivity measurements and the Stokes-Einstein equation. In addition, a computational protocol is developed for determining the ζ from a protein crystal structure. At low ionic strengths a hyper-diffusivity regime exists, requiring direct measurement of electrophoretic mobility to determine ζ. This work, therefore, supports a basic tenant of EDL theory that the electric double layer during diffusion and electrophoresis are equivalent in the Stokes-Einstein regime.

INTRODUCTION

The zeta, or electrokinetic, potential (ζ) is the effective charge energy of a solvated solute. It can be used to assess how well dispersed colloids remain in solution [1, 2, 3], and to model the electrokinetic behavior in adsorption processes. Protein-based therapeutics may be formulated at high concentrations that are prone to aggregation. Experimentally measured ζ has been applied to optimize therapeutic antibodies and other proteins for formulation conditions that promote long-term dispersion stability [4, 5, 6, 7, 8, 9], and to study interactions between proteins with particles, materials and surfaces [10, 11].

The ability to predict ζ from the molecular structure of proteins would allow modeling of solubility as a parameter in computational protein design. In studies of protein self-assembly, unintuitive charge-dependent behavior may be observed due to a complicated balance between immediate and long-range electrostatic phenomena [5, 12] and between electrostatic and hydrodynamic processes. To model ζ, we treat the protein-solvent interface as an electric double layer (EDL), which is the collection of solvation layers that form around a protein in an attempt to neutralize its charge. Gouy and Chapman [13] developed an EDL model where a molecule with a uniform surface charge is neutralized by a region of diffusing ions that encompass the molecular surface. The propagation of the surface potential and ion concentrations from the surface are defined by the Poisson-Boltzmann equation (PBE). Figure 1 depicts key features of the EDL surrounding an idealized cationic spherical protein, and the electrostatic potential distribution extending into solution from the protein surface. A hydration layer extends from the surface to the slip plane position (X_SP), similar to the Stern layer of the Gouy-Chapman-Stern (GCS) EDL. The depicted EDL assumes that the propagation of electrostatic potential and ion concentrations within the hydration layer are defined by the nonlinear PBE, unlike the GCS EDL, which applies a modified PBE to consider ion size constraints [14]. ζ is weaker than the surface potential (ψ_o) and located at X_SP, which is somewhere in the cloud of diffusive ions less than a Debye length (κ⁻¹) away from the surface. The X_SP represents the cutoff of an immobile layer of solvent (referred to as “hydration layer” in this work) adhering to the molecule. It is only a few molecular-sized layers thick [15]. Ions adsorbed to the protein in this hydration layer can cause specific-ion effects that can be modeled with the GCS EDL [14, 16].

Figure 1. Idealized EDL around a spherical cation.

Plotted on the right, the surface potential (ψ_o) propagates outward into the cloud of ions treated as point charges immersed in a solvent with a constant relative dielectric. The zeta potential (ζ) is located at the slip plane, which is proposed to coincide with the hydrodynamic radius (R_h). R_p is the protein radius and κ⁻¹ is the Debye length.

Proteins provide an excellent opportunity for EDL modeling where the molecular structure of their charged surfaces are known [17], and where changes in conformation can be studied experimentally or through simulation [18]. Hen egg white lysozyme, hereto referred to as lysozyme, is one such well-studied protein [19, 20, 21, 22, 23, 24] that can be used to evaluate EDL models.

An obstacle to using atomic structure of proteins to estimate ζ is the lack of general criteria for the location of the X_SP [2, 25, 26]. Other studies [7] have used the EDL edge defined by the Debye length, κ⁻¹, as X_SP for calculating ζ. However, at the κ⁻¹ motions of the ions are no longer determined by the surface potential, likely resulting in an underestimate of the calculated ζ. We hypothesize that a more accurate placement of X_SP is the radius of hydration (Fig. 1).

Theoretically, the R_h was derived as the radius of an uncharged sphere plus its immobile hydration layer undergoing diffusive motion [27]. The X_SP and R_h differ in their theoretical definitions, with the X_SP being the position of the ζ during electrokinetic phenomena (e.g. electrophoresis) [2] and the R_h being a radius pertaining to the edge of solvation during diffusion [27], defined by the Stokes-Einstein equation (Eq. 1).

$$R_h=\frac{k_{b}T}{6\pi \eta D}$$ (1)

where k_b is the Boltzmann constant, T is the absolute temperature, η is the pure solvent viscosity and D is the single particle diffusivity.

The idea that the X_SP coincides with R_h has been previously considered [19, 25, 26], but to our knowledge, has not been experimentally validated for proteins. The GCS model was shown to be valid for negatively-charged silica nanoparticles, where the X_SP was found to be one monolayer of water plus one hydrated counter-ion radius away from the surface [28]. This is consistent with GCS theory that places hydrated cations at the outer Helmholtz plane (OHP), which sits on top of a layer of water directly on the molecular surface [2]. With extensive studies on the hydration [22] and structure of lysozyme [29], we expect it to be a useful model system for the characterization of an EDL around a general globular protein.

In addition to the choice of protein, the role of the counter-ion in determining EDL structure must be considered. Our study involves the positively-charged protein, lysozyme, with weakly hydrated Cl counter-ions [30]. In GCS theory on positively-charged particles, the center of anions make up the inner Helmholtz plane, which is closer to the surface than the OHP, and can allow anions to sit on the molecular surface alongside water molecules [2]. The concentration of counter-ions at the protein surface is physically limited by their size as they pack with water to coat the protein [14]. In considering the finite size of ions, GCS theory is an improvement over GC theory that can model specific adsorption processes, which occur when an ion is attracted to a charged surface by more than just Coulombic forces. Multiple observations support that at pH 7, KCl is an indifferent electrolyte for lysozyme, dominated by Coulombic interactions [31]. For example, key features of an electrostatic-dominated process such as an isoelectric point independent of ion concentration, and concentration dependent counter-ion binding are both observed in this system [23]. Furthermore, small-angle X-ray scattering of lysozyme in KCl has also shown the Cl population in the nearest solvation layer increases with ion concentration [24]. Therefore, chloride-lysozyme interactions can be modeled Coulombically, simplifying our analysis. As will be shown in this work, a modified GC EDL model applied to averaged lysozyme conformations provides an accurate representation of its electrokinetic behavior.

We use the Stokes-Einstein equation to define the hydration layer within the modified GC EDL. Einstein originally derived Eq. 1 from the Navier-Stokes equation for dilute, non-charged spheres. Because lysozyme is charged at physiological pH, we must identify the effect that bearing a charge has on R_h. For example, at low ion concentrations, diffusivity is well known to show marked increases [20, 32, 33, 34] that lead to R_h being smaller than the physical size of the molecule itself – a hyper-diffusive regime. This increase in diffusivity is believed to result from long-range charge repulsion that accelerates diffusion as the κ⁻¹ increases. We will establish the Stokes-Einstein regime - a range of ionic strengths where Eq. 1 is valid for charge-bearing particle.

In order to connect hydrodynamic and electrokinetic phenomena, we define an effective hydrodynamic radius during electrophoresis (i.e. the electrophoretic radius) (R_e). Eq. 2a shows the relation between R_e, protein radius (R_p) and X_SP. Henry derived an equation for electrophoretic mobility (u_e) accounting for electrophoretic retardation [2] from the Poisson-Boltzmann and Navier-Stokes equations while assuming the ionic atmosphere surrounding the charged particle to remain in its equilibrium state [35] (Eq. 2b). Henry’s equation has been experimentally tested on nanometer to micron-scale polystyrene, gamboge and silica spheres [36, 37]. Eq. 2c expresses this relationship in terms of the protein net surface charge (Qe) [38], knowing u_e, the net valence of the protein (Q), and the pure solvent viscosity (η). Q is determined from controlling the solution pH and knowing the pKa values of the charged surface residues [23, 39, 40, 41]. η can be measured by a rheometer (see Table S1); however, much data already exists on the viscosity of aqueous electrolyte solutions [42] and thus we can use an empirical relationship [43] (see Eq. S3). The Henry correction factor for electrophoretic retardation (f(κR_e)) varies between 1 and 1.5 [1], allowing us to calculate it as shown in Eq. 2d.

$$R_e=R_p+X_{SP}$$ (2a)

$$u_e=\frac{2{\epsilon }_o{\epsilon }_r\zeta f(\kappa R_e)}{3\eta \left(\frac{f}{f_0}\right)}$$ (2b)

$$R_e=\frac{\mathrm{Qef}(\kappa R_e)}{6\pi \eta u_e(1+\kappa R_e)\left(\frac{f}{f_0}\right)}$$ (2c)

$$f(\kappa R_e)\approx 1+\frac{1}{2{\left(1+\frac{\delta }{\kappa R_e}\right)}^3},\delta =\frac{2.5}{1+2e^{-\kappa R_e}}$$ (2d)

where κ is the inverse Debye length and $\left(\frac{f}{f_0}\right)$ is a shape factor (1.17 for lysozyme [38]).

The central hypothesis is that X_SP, R_e, R_h coincide, relating the position of the ζ and the edge of solvation. We assess this claim in two ways: first we experimentally determine R_e and R_h to assess the similarity of the EDL during electrophoresis and diffusion. Second, in the Stokes-Einstein regime, we use diffusivity alone to specify X_SP, and compute the ζ from the molecular structure of lysozyme. This assessment assumes the modified GC EDL model to be accurate for a protein in solution.

MATERIALS AND METHODS

Lysozyme Sample Preparation.

Salt-free hen egg white lysozyme (LYSF, Worthington Biochemical Corporation, Lakewood, NJ) was dissolved in deionized water and filtered through a 20 nm pore-size Anotop syringe filter (GE Whatman, Pittsburgh, PA) to separate monomers from aggregates. pH was measured with a model 14002-850 sympHony pH electrode (VWR, Randnor, PA) following calibration. The measured pH of the deionized water used to dissolve lysozyme was 6.55. Post-filtration lysozyme concentrations were determined with an Aviv Model 14DS spectrophotometer (Lakewood, NJ) by UV absorption at 280 nm using α₂₈₀=2.64 mL/mg cm [45]. Three protein concentrations were prepared considering the balance between allowing the solutions to remain concentrated enough for accurate light scattering measurements but dilute enough to ensure protein-protein interactions were negligible [46]. Solutions were mixed with 99.995% pure KCl (7447-40-7, VWR, Radnor, PA) to allow a wide range of ion concentrations (10⁻⁶ to 1M) at three different protein concentrations.

Electrophoretic Mobility Measurement.

Filtered protein solutions were allowed five minutes of thermal equilibration at 25°C before the addition of the KCl solution, and measured immediately to minimize dimerization. Electrophoretic mobilities were measured at 25°C by phase analysis light scattering (PALS) [47] using a Malvern Zetasizer Nano ZS. Our apparatus used forward scattering of a 4 mW He-Ne laser at λ = 633 nm at an angle of 17° through a 2 mm spaced palladium plated dip cell. A minimum of twenty technical replicates were performed for each KCl concentration (see supplementary methods for details). Samples were checked for monodispersity by dynamic light scattering. Code for determining R_e based on previous work [48] is provided as a supplementary file.

Diffusivity Measurement.

Filtered protein solutions were allowed five minutes of thermal equilibration at 25°C before the addition of salt. Then diffusivity measurements were taken immediately after adding salt-containing solutions. Final KCl concentrations were 1.0 μM, 10.0μM, 100.0 μM, 1.0 mM, 5.0 mM, 10.0 mM, 50.0 mM, 100.0 mM, 500.0 mM and 1.0 M. Diffusivities were measured at 25°C by dynamic light scattering (DLS) using a Malvern Zetasizer Nano ZS (Malvern Instruments, Worcestershire, U.K.) with a 4 mW He-Ne laser at λ = 633 nm. Measurements were performed using back-scattering at an angle of 173°. The temperature was maintained to within ±0.1°C by a built-in Peltier element. Six measured autocorrelation functions with acquisition times of about 150 seconds were analyzed by cumulants analysis and averaged to determine the diffusivity of each sample.

Viscosity Measurement.

To ensure the empirical viscosity relationships used in this work (see Eq. S3) were accurate, viscosities were measured with a Malvern Kinexus ultra+ rotational rheometer (Malvern Instruments, Worcestershire, U.K.) and are shown in Table S1. During measurement, 25μL of each solution was held between a 20 mm diameter flat plate upper geometry and 65 mm diameter flat plate lower geometry spaced with a 0.1 mm gap size. A shear rate of 11980 1/s was used as it was found to provide reasonably close values of pure water at 25°C. After allowing 5 minutes between the plates to reach thermal equilibrium, viscosities were measured for 60 seconds. Only the last 55 seconds were averaged to allow a 5 second delay to reach the machine’s steady state.

ζ Calculation from Structure.

Computation of the ζ of lysozyme from its crystal structure (PDB id: 6LYZ [29]) involves six main steps that account for the structural motions of the solvated protein, its atomic charge distribution, its electric potential distribution into solution, and the distance from the protein, where solvent no longer adheres (Fig. 2). First, molecular dynamics (MD) simulations were performed in AMBER 2015 to produce an ensemble of near-native conformations [18]. Simulations were performed in explicit water extending 20 Å from the molecular surface, using the ff14SB force field [49], and were comprised of three parts: energy minimization, thermal excitation, and simulation in solution. Energy minimization was performed to optimize the initial geometry of the crystal structure in solution for 3000 cycles. Thermal excitation gradually heated the minimized structure from 0 K to 298.15 K over a period of 100 ps with a 2 fs interval to induce thermal motion of the solvent and protein. The heated structure was then allowed to undergo simulation for 100 ns with a 2 fs integration step. Twenty molecular conformations were sampled from the last 20 ns of simulation, one per nanosecond. This ensemble was intended to capture local conformational variation upon simulation equilibration, and not large-scale conformational changes.

Figure 2. Computational ζ prediction from protein structure.

Predicting the ζ of a molecular structure involves six main steps as shown above. Each step provides a necessary piece of information accounting for the structural motions of the solvated molecule, its atomic charge distribution, its electric potential distribution into solution, and the distance from the molecule, where water and ions no longer adhere. The slip plane position in the final step is exaggerated for visual clarity.

The ζ was computed for each of the twenty conformations. PDB2PQR [39, 40] was used to assign atomic charges and radii. The pKa predictor, PROPKA 3.0, was used to identify the protonation state of titratable residues at pH 7.0 [41]. The net +8 valence of lysozyme was in agreement with previous experimental data [23]. Electrostatic potential distributions of the prepared structures were computed by solving the non-linear Poisson-Boltzmann equation over each conformation with the adaptive Poisson-Boltzmann solver (APBS) [50]. A dielectric of 4.0 was used for the protein [51] and the dielectric of water at 25°C, 78.285 [52], was used for the solvent.

Hydrodynamic and protein radii were computed for each conformation. Protein radii (R_p) were calculated as the average distance between the center of mass and a solvent-excluded surface (SES) generated by MSMS [53]. The SES generated used a probe radius of water (1.4 Å). Hydrodynamic radii (R_h) were calculated by Eq. 1 using translational diffusivities computed by HYDROPRO [54]. For lysozyme, HYDROPRO used a bead-per-atom model with an atomic element radius of 2.6 Å. Computed radii were used to estimate the slip plane position by subtracting the protein radius from the hydrodynamic radius of each conformation. In the final step, the coordinates of the edge of the hydrodynamic radius (i.e. slip plane positon) were obtained and used to capture the electrostatic potentials at each point. This involves once again generating a SES on each structure using MSMS [53]. The SES generated is composed of Cartesian coordinates and their normal vectors directed away from the protein surface. Each SES is inflated by translating its initial coordinates to the slip plane position along their respective normal vector. The APBS tool, multivalue, uses the APBS calculated potentials and the inflated coordinates to capture the electric potentials at each point. A zeta potential value for each conformation (ζ_c) was computed by averaging the captured potentials at the inflated SES as shown below in Eq. 3. Modeled ζ values for lysozyme were computed by averaging the different ζ_c values of each conformation.

$${\zeta }_c=\frac{1}{N}{\sum }_{r\in S}\phi (r)$$ (3)

where N is the number of elements in the set S containing the position vectors composing the inflated SES and ϕ(r) is the electrostatic potential at position r.

RESULTS AND DISCUSSION

Assessing the Feasibility of Lysozyme for EDL Analysis.

The lysozyme-KCl solution interface at pH 7 provides an appropriate and well-studied system for assessing the relation between diffusivity, hydration and ζ with varying ionic strength [20, 22, 23, 26]. As a structure, lysozyme is highly spherical holding asphericity [55] and shape parameter [55] values indicative that the molecule can be represented by a sphere. The average asphericity from the 20 conformations produced by molecular dynamics was 0.0514 ± 0.03 and the average value for the shape parameter was 0.0196 ± 0.02 (0 is a perfect sphere for both values [55]). This supports our analysis of the hydration of lysozyme by subtracting the R_p from the R_h. Also, the net valence of the protein remains at +8 and is independent of ion concentration [23] indicating the surface charge distribution provides a comparable EDL foundation for the different ionic strengths.

Although lysozyme can form dimers at pH 7 [23], we ensured monomers were present following filtration as described in the supplement. DLS measurements alone were not sensitive to dimerization (see supplementary methods). Therefore, we determined the oligomerization state using PALS electrophoretic mobility measurements (see Fig. S1). All measurements were performed immediately upon the addition of salt solutions and routinely checked by DLS to ensure monodispersity.

The hydration of lysozyme studied by NMR and X-ray diffraction indicates solvent mostly forms a monolayer over the surface with ordered water structures extending no more than ~4.5 Å [22]. This hydration layer thickness is consistent with our measurements of R_h determined from experimental diffusivities and R_e determined from electrophoretic mobilities. The Stokes-Einstein equation (Eq. 1) accurately connects diffusivity and hydration for our system, where it is valid. As the hydration layer thickness is the distance from the surface to where solvent no longer adheres to the protein, it is theoretically similar to the X_SP, where the ζ exists. To assess how well the Stokes-Einstein relation provides an estimate of the hydration layer thickness, and thus the X_SP, it is first necessary to determine the range of ionic strength in which Eq. 1 is valid. We do this through a combined analysis of the diffusive and electrophoretic behavior.

Electrophoretic Behavior of Lysozyme in KCl.

Two approaches for calculating electrophoretic mobility – using the Henry model Eq. 2c, or using explicit protein structure – were compared to experimental values measured for multiple protein concentrations. Lysozyme concentrations were sufficiently low to allow negligible protein-protein interactions. Experimental u_e consistently decreased with increasing ion concentration, which is expected of GC EDL behavior [56]. Theoretical mobility values were calculated by rearrangement of the Henry model (Eq. 2c) using pure solvent viscosity values of KCl (Eq. S3) and a constant R_e value of lysozyme plus a monolayer of water (1.62+0.284 nm = 1.904 nm) [21, 30]. Structure-derived mobility values using our ζ model were calculated from the average of MD simulations of the lysozyme structure (see methods), and converted into electrophoretic mobilities (Eq. 2b). The Henry equation was the appropriate electrokinetic model under our experimental conditions (i.e. only electrophoretic retardation is considered [2], no EDL polarization [57, 2], and no surface conductivity [57]) [58, 59]. The ζ model used a constant hydrodynamic radius calculated from HYDROPRO (2.02 nm) to estimate the X_SP. A comparison of the two calculated and the experimental u_e are shown in Fig. 3.

Figure 3. Comparison of experimental and calculated electrophoretic mobilities, u_e.

The experimental mobility of lysozyme is well described by both the GC EDL, and by the structure-based modified GC EDL ζ model.

The electrophoretic mobilities of lysozyme are well described by the Henry model indicating the lysozyme-KCl EDL behaves like a GC EDL (Fig. 3). The Henry model [35] treats the protein as a sphere with a uniform surface charge and provides a standard for theoretical comparison with our detailed structure-based ζ model. Considering proteins are not perfectly spherical and hold a hydration layer that dampens their surface charge, measured protein mobilities were expected to be slightly lower than those predicted by the Henry model. Both models adequately represent the electrokinetic behavior of lysozyme in KCl, which indicates the modified GC EDL (structure model) is consistent with GC EDL theory (Henry model). This is significant as we have presented a theoretical validation of the GC EDL on an experimental crystal structure. It is important to note, dimerization can occur rapidly at the higher ionic strengths, and thus mobility values at the higher ion concentrations most likely represents a mixture of monomers and dimers. We sought to minimize these effects as described in our measurement protocol (Fig. S1).

Diffusive Behavior of Lysozyme in KCl.

Diffusivities of three concentrations of lysozyme were measured by DLS at a series of ionic strengths from 1.0 μM to 1.0 M KCl (Fig. 4). Diffusivities are in agreement with previous work [20]. Diffusion behavior transitions between two different regimes with increasing ion concentration. The minimum KCl concentration defining the onset of the Stokes-Einstein regime where Eq. 1 is valid, denoted C^SE, was determined by comparison of R_h from diffusivity and R_e from electrophoretic mobility measurements (Fig. 5). Based on this analysis, the C^SE for KCl was interpolated to occur at 6.6 mM.

Figure 4. Measured Diffusivities of Lysozyme at Various KCl and Protein Concentrations.

Low and high salt concentrations induce different regimes of diffusive behavior. The Stokes-Einstein regime was found to occur at KCl concentrations at or above 6.6 mM (marked as C^SE) based on an analysis presented in the next figure. Below this concentration, diffusion transitions to a hyper-diffusive regime, where diffusivity exceeds D_max, which is estimated based on the R_p of lysozyme.

Figure 5. Comparison of Experimental and Computational Radii of Hydration.

(A) Hydration radii for the entire tested range of ionic strengths. (B) Expanded view of the Stokes-Einstein regime.

At low ion concentrations, the hyper-diffusive regime exists in which diffusivity is enhanced by inter-particle electrostatic phenomena. Another possible cause for this enhancement could be a change in structure; however, previous work [60] has shown lysozyme structure is retained at the low KCl concentrations used here. Consequently, the higher diffusivities at lower ion concentrations deviate from Stokes-Einstein theory and despite extensive study still remain difficult to theoretically model [20, 32, 61, 34]. An in-depth discussion of the hyper-diffusivity regime can be found in the work of Muthukumar and colleagues [61]. As counter-ions become incorporated in the EDL, they neutralize the electrostatic enhancement causing a transition. Once enough ions have become incorporated in the EDL to allow each lysozyme molecule to appear neutral to its neighbors (i.e. electroneutrality at the EDL edge), the Stokes-Einstein regime begins. In the Stokes-Einstein regime (i.e. [KCl] > C^SE), diffusivity and effective size can be related by the Stokes-Einstein equation (Eq. 1). In addition to the effects of ionic strength on the electrophoretic and diffusive behaviors through long-range electrostatic effects, we expect changes at the local scale in the nearest solvation layer around lysozyme in the Stokes Einstein regime.

EDL Contraction Affects Solvation in the Stokes-Einstein Regime.

EDL contraction refers to the disintegration of the outer solvation layers with increasing ionic strength. This effect can be theoretically quantified with the Debye length, representing the EDL edge from the protein surface [2]. As shown in Fig. 5A, analysis of protein diffusivity is only physically meaningful in the Stokes-Einstein regime. To estimate where the Stokes-Einstein regime becomes valid, we identified the ion concentration, where R_h and R_e first coincide. Experimental R_h values were calculated with Eq. 1 using experimentally determined single particle diffusivities (see Fig. S3) and the pure solvent viscosity (Eq. S3). Experimental R_e values were calculated with Eq. 2c using experimentally determined electrophoretic mobilities and the pure solvent viscosity. For comparison, we used the HYDROPRO [54] software to model single particle diffusivities based on the ensemble of lysozyme structures sampled by molecular dynamics. The structure of lysozyme during molecular dynamics remains compact with an average radius of 16.27 ± 0.16 Å, calculated from the average center of mass to solvent excluded surface. This value represents the R_p, and is in agreement with past experimental findings [21]. As the EDL contracts with increasing ion concentration in the Stokes-Einstein regime, experimental R_h values decrease (Fig. 5B). Experimental R_h decreases from 2.17 nm to 1.81 nm. The computed hydrodynamic radii from HYDROPRO diffusivities remain constant at 2.02 nm, which is close to R_h for the Stokes-Einstein regime, and R_e values across the entire range of ionic strength.

In the Stokes-Einstein regime, calculated radii are all within error indicating agreement in the methods for determining molecular size [27, 48] and thus the hydration layer thickness. This connection indicates the EDL of lysozyme is the same under both electrophoretic and diffusive conditions, supporting our hypothesis that the X_SP coincides with R_h. To estimate the slip plane position from experimental, we subtracted the protein radius (R_p) from the measured R_h values in the Stokes-Einstein regime and R_e values outside of this regime (Eq. 4).

$$X_{SP}=\{\begin{array}{cc}{R}_e-R_{p^{,}}\hfill & C_i<C^{SE}\hfill \\ R_h-R_{p^{,}}\hfill & C_i\ge C^{SE}\hfill \end{array}\phantom{\}}$$ (4)

where X_SP is the slip plane position relative to the protein surface, R_e is the electrophoretic radius (Eq. 2c), R_p is protein radius, C_i is the ion concentration, C^SE is the ion concentration at which the Stokes-Einstein regime begins, and R_h is the hydrodynamic radius (Eq. 1).

R_e is approximately equal to the radius of lysozyme plus a water molecule (1.62+0.284 nm), indicating a single layer of water of solvation. However, R_h decreases with increasing ionic strength in the Stokes-Einstein regime, implying a shrinking hydration layer. This begs the question: is the X_SP constant or variable with regard to ionic strength? To assess this question, we apply our protein structure derived ζ model using either a constant or a variable slip plane position.

ζ Analysis of the Slip Plane Estimates.

Previously, we demonstrated that an electrophoretic mobility, u_e, determined from molecular structure with a calculated X_SP using HYDROPRO correlates quite well with experimentally measured u_e (Fig. 3). However, this calculated X_SP is constant, in contrast to observed changes in hydrodynamic radii based on diffusivity measurements. If we use X_SP values derived from experiment (Eq. 4) combined with a structure-based ζ model, we see little improvement in the correlation with directly observed u_e values (Fig. 6). This suggests that for the given resolution of our instrument for determining u_e, there is no advantage to including a variable X_SP. X_SP can be represented by the R_h experimentally and, for computational purposes, the X_SP can be approximated as constant over a wide range of ionic strengths.

Figure 6. Comparison of measured and computed slip-plane estimates.

CONCLUSIONS

This work demonstrates that the slip plane of the lysozyme-KCl interface is not just a theoretical boundary for estimating the ζ, but also the physical interface between bulk and constrained waters along the protein surface. Future work will test the coincidence of the slip plane and hydrodynamic radius on other proteins. X_SP can be estimated experimentally by diffusivity measurements in the Stokes-Einstein regime, thus connecting diffusivity, hydration and ζ. In the absence of diffusivity or mobility measurements, it is possible to accurately predict ζ directly from experimental structures or atomic-resolution models. Our data and ζ model were unable to establish a relationship between X_SP and ionic strength. Relative to KCl, stronger kosmotropes or chaotropes along the Hofmeister series (e.g. KH₂PO₄, KNO₃), may alter the hydration layer [62, 63]. Such effects have been systematically studied for lysozyme [45]. It would be valuable to examine how effective the computational protocol developed in this work could recapitulate the effects of ion types on hydration and electrophoretic mobility. The ζ prediction protocol proposed here is an important first step in developing solution conditions for optimizing protein dispersion stability.

ABBREVIATIONS:

EDL

electric double layer

GC

Gouy Chapman

GCS

Gouy-Chapman-Stern

PBE

Poisson-Boltzmann equation