A scaling relationship between thermodynamic and hydrodynamic interactions in protein solutions

Abstract

Weak protein interactions are associated with a broad array of biological functions and are often implicated in molecular dysfunction accompanying human disease. In addition, these interactions are a critical determinant in the effective manufacturing, stability, and administration of biotherapeutic proteins. Despite their prominence, much remains unknown about how molecular attributes influence the hydrodynamic and thermodynamic contributions to the overall interaction mechanism. To systematically probe these contributions, we have evaluated self-interaction in a diverse set of proteins that demonstrate a broad range of behaviors from attractive to repulsive. Analysis of the composite trending in the data provides a convenient interconversion among interaction parameters measured from the concentration dependence of the molecular weight, diffusion coefficient, and sedimentation coefficient, as well as insight into the relationship between thermodynamic and hydrodynamic interactions. We find relatively good agreement between our data and a model for interacting hard spheres in the range of weak self-association. In addition, we propose an empirically derived, general scaling relationship applicable across a broad range of self-association and repulsive behaviors.

Significance

Weak protein interactions are frequently associated with biological function and dysfunction and are of critical importance in the development of biotherapeutics. Despite their prominence, much remains unknown about their underlying molecular origins. Here, we probe the relative thermodynamic and hydrodynamic contributions to weak protein self-interaction in a diverse set of proteins displaying a range of self-interaction tendencies. We rationalize the data in the context of available theoretical models and find good agreement with an interacting hard sphere model where self-association is weak. We propose a general scaling relationship between thermodynamic and hydrodynamic interactions applicable across a broad range of self-association and repulsive behaviors.

Introduction

Weak interactions between macromolecules, with dissociation constant ($K_d$) > 10 μM (1), are of fundamental importance to both normal biological function and to dysfunction associated with human disease. Such interactions occurring between proteins have been implicated in processes as diverse as cellular adhesion (2) including T cell antigen recognition (3), immunoglobulin G (IgG) complement activation (4,5), phase condensation of membrane-less cellular structures (6), hemoglobin polymerization in sickle cell disease (7), crystallin aggregation in cataract formation (8), and amyloid-associated protein misfolding disorders (9). In addition to these biological aspects, it has become widely recognized that weak self-association in therapeutic protein formulations is at least in part responsible for a variety of poor solution behaviors including high viscosity (10), opalescence (11,12), liquid-liquid phase separation (13), and irreversible aggregation (14). On the other hand, repulsive interactions in dilute solution are a hallmark of well-behaved therapeutic antibodies (15). Given the challenges that poor solution behaviors present to the manufacturing, stability, and administration of biologic drugs (16,17,18,19), as well as the broad applicability across biological function and dysfunction, characterization of weak macromolecular self-interaction represents a staple of modern biochemistry.

In the context of their importance, it is not surprising that a tremendous body of research pertaining to the theoretical framework for understanding weak interactions as well as their empirical determination has been assembled over the past century. Of particular significance is the osmotic second virial coefficient ($B_2$), which embodies the sign and magnitude of the thermodynamic nonideality causing the solution osmotic pressure ($\mathrm{\Pi }$) to deviate from the expectations of the van ’t Hoff relation (20):

$$\mathrm{\Pi }=RTc\left(\frac{1}{M}+B_{2}c+\dots \right)$$ (1)

where $R$ is the gas constant, $T$ is the absolute temperature, $c$ is the macromolecular concentration, and $M$ is the molecular weight. From a molecular perspective, for isotropic interactions this nonideality can be understood from statistical thermodynamics as the pairwise interaction energy between molecules ($W$) at a given center-center distance ($r_{12}$) (21):

$$B_2{M}^2=-2\pi {\int }_0^{\infty }\left(e^{W/k_{B}T}-1\right)r_{12}^2{dr}_{12}$$ (2)

where $k_B$ is the Boltzmann constant. Therefore, in solution conditions where multibody interactions are of limited contribution (i.e., dilute solution), $B_2$ provides a valuable parameter for quantitatively describing thermodynamic interactions. Ultimately, this parameter arises from the sum of the excluded volume, electrostatic, and short-range intermolecular interactions that are derived from the molecular composition (22).

In addition to the thermodynamic aspects of weak interactions, considerable work has been devoted to hydrodynamic interactions starting with the analysis of the settling of uniform dispersions of particles in the early 20th century (23). In this case, particle concentration dependence arises from the velocity field generated by the transport of one molecule through the fluid as a result of the external gravitational field which in turn affects the transport of another. This dependence is given as a first order approximation in $c$ for the frictional coefficient at any given concentration ($f_c$) as:

$$f_c=f\left(1+k_{s}c\right)$$ (3)

where $f$ is frictional coefficient at negligible concentration. The virial coefficient, $k_s$, is the interaction parameter reflecting the concentration dependence of the sedimentation coefficient ($s$) for transport through the solution (which includes the contribution of the solute to the density of the fluid). A corresponding term, $k_s^{\prime }$, relates to sedimentation in pure solvent. In the limit of infinite dilution (24,25):

$$k_s=k_s^{\prime }-\overline{v}$$ (4)

where $\overline{v}$ is the partial specific volume. Since $\overline{v}$ is on the order of 0.7 mL/g for proteins, $k_s^{\prime }\approx k_s$ except in conditions of negligible or very weak concentration dependence. The interaction parameter can be measured for particle suspensions by gravitational settling rate analyses or alternatively by sedimentation velocity analyses in an analytical ultracentrifuge (AUC) for submicron suspensions and biomolecule solutions. In the latter, the convention has been to give the concentration dependence of $s$ as a first order approximation in $c$:

$$s=s_0\left(1-k_{s}c\right)$$ (5)

where $s_0$ is the sedimentation coefficient at negligible concentration. Thus, Eq. 5 is analogous to Eq. 3 and $k_s$ is the natural form for the measured interaction parameter since the sedimentation coefficient is determined by boundary transport through the fluid of solution density. Studies of the concentration dependence in $s$ are often posed against theoretical transport models. Of particular interest is the treatment by Batchelor (26), who showed the concentration dependence on the velocity, $\overline{U}$, of a spherical, impermeable particle free of thermodynamic interactions to be given by:

$$\overline{U}=U_0\left(1-6.55\varphi \right)$$ (6)

where $U_0$ is the velocity of the particle at infinite dilution, $\varphi$ is the volume fraction concentration, and the value of −6.55 arises from various contributions related to the motion of the particle and accompanying solvent flows that comprise the hydrodynamic interactions. This derivation considers transport through a fluid of solvent density and with particle concentration as volume fraction. Therefore, it is necessary to consider the relationship to corresponding interactions associated with sedimentation in a fluid of solution density and mass fraction concentration along with the associated nomenclature. The hydrodynamic interaction derived by Batchelor may be designated $K_s^{\prime }$, with the large $K$ denoting volume fraction and the apostrophe denoting solvent density. Conversion among mass and volume fraction derived interactions is possible given that $\varphi =c{v}_P$, where $v_P$ is the particle volume accounting for entrained and chemically bound solvent (25). Rowe (24) posed that this quantity could be given as:

$$v_P=\overline{v}{\left(f/f_0\right)}^3$$ (7)

Conversion between solvent and solution density derived interactions is possible in the dilute solution limit according to Eq. 4 (24,25). Therefore, (25):

$$K_s=K_s^{\prime }-\left(\frac{\overline{v}}{\overline{v}{\left(f//f_0\right)}^3}\right)=\left(\frac{k_s}{\overline{v}{\left(f/f_0\right)}^3}\right)$$ (8)

Given these fluid density and concentration considerations, the hydrodynamic interaction determined from the hard sphere model of Batchelor can be given as $K_s^{\prime }$ = 6.55 or $K_s$ = 5.55 (since $f/f_0$ = 1). Using a different approach, Rowe also derived the hydrodynamic interaction as (24):

$$k_s/\overline{v}=2\left(v_P/\overline{v}+{\left(f/f_0\right)}^3\right)$$ (9)

This model, which indicates that the hydrodynamic interaction increases with solvation and frictional ratio, reduces to $K_s^{\prime }$ = 5 or $K_s$ = 4 for the anhydrous hard spheres considered by Batchelor (26) (i.e., $v_P$ = $\overline{v}$ and $f/f_0$ = 1). More recently, Brady and Durlofsky derived an expression for the sedimentation rate of hard, noninteracting spheres valid for dilute to highly concentrated solutions (27). In the dilute limit, their expression reduces to $K_s^{\prime }$ = 5, significantly less than that of Batchelor (26) and more in line with that of Rowe (24). Numerous studies have been conducted to evaluate the validity of these theoretical models for real solutions, often using systems intended to mimic hard, noninteracting spheres. Among these, Buscall et al. (28) measured $K_s^{\prime }$ = 5.4 for 1.5-micron polystyrene latex spheres in sodium chloride solution by gravity sedimentation. A similar result of $K_s^{\prime }$ = 5.1 was obtained for submicron polystyrene latex particles using sedimentation velocity analyses by Cheng and Schachman (29). Tackie et al. (30) determined $K_s^{\prime }$ = 4.4 for bare silica particles in acidified water, whereas coated silica particles in cyclohexane have been measured at higher values more consistent with expectation of the Batchelor model, $K_s^{\prime }$ = 6 (31) and $K_s^{\prime }$ = 6.5 (32). Similar attempts have been made with biomolecule solutions. For instance, Newman et al. (33) determined $K_s^{\prime }$ = 6.7 for bacteriophage DNA in high concentration sodium citrate/sodium phosphate buffer, pH 8, consistent with the Batchelor model. Likewise, in sodium/potassium phosphate buffer of pH 7.8, Harding et al. (34) measured $K_s^{\prime }$ = 6.3 for the turnip yellow mosaic virus particle. However, a significant dependence of $K_s^{\prime }$ on buffer conditions was observed and attributed to thermodynamic interactions (34). Such dependencies have been consistently linked to hydrodynamic behavior (most typically through the effect on diffusivity) for other proteins including lysozyme (35), serum albumin (36), pancreatic trypsin inhibitor (37), γ-crystallin (38), and monoclonal antibodies (39), among others. This reflects the difficulty in identifying protein systems in which thermodynamic interactions are of sufficiently low magnitude or suitably balanced attractive and repulsive contribution to approximate the theoretical model.

Perhaps of more relevancy to such systems, hard sphere models that include weak, short-ranged attractive thermodynamic interactions have been developed (40,41,42). Batchelor gives the relationship as (42):

$$K_s^{\prime }=6.55+3.52\left(B_2^{\ast }-1\right)$$ (10)

where $B_2^{\ast }$ is the reduced osmotic second virial coefficient, given as $B_2/B_2^{ex}$ where $B_2^{ex}$ is the excluded volume contribution. For hard, noninteracting spheres, where $B_2=B_2^{ex}$, the expression reduces to $K_s^{\prime }$ = 6.55 in line with Eq. 6. This theoretical framework has been used as the starting point to understand how hydrodynamic interactions depend on the pairwise interaction potential, particularly as pertains to solution behaviors of practical interest, including phase separation (43,44) and aggregation/crystallization (45,46).

At the intersection of thermodynamic and hydrodynamic interactions is the diffusion coefficient ($D$). Diffusion is a dynamic property that is dependent on both chemical potential gradients and frictional forces (47). With the advent of dynamic light scattering (DLS) in the 1960s (48), it became possible to measure $D$ for molecules in dilute solution based on the fluctuations in the intensity of scattered light. At molecule concentrations where the interparticle distance is relatively small compared with Debye shielding length, scattering fluctuations from the diffusive motion are collective of the molecules and ions in solution (49). This collective diffusion ($D_c$), along the concentration gradient presents experimentally in a simple form, which can be modeled using a power law function in $D$ (50):

$$D_c=D\left(1+k_{1}c+k_2{c}^2+k_3{c}^3+\dots \right)$$ (11)

The constants ($k_1$, $k_2$, $k_3$, etc.) in the expansion reflect the attractive and repulsive interactions between molecules (51). Thus, a commonality between $B_2$, $k_s$, and the constant that describes the first order dependence of $D_c$ with concentration, termed the diffusion interaction parameter ($k_D$; represented by $k_1$ in Eq. 11) is immediately apparent. This relationship, as an expansion of the osmotic compressibility and hydrodynamic function, is given by (e.g., (52)):

$$k_D=2B_{2}M-k_s$$ (12)

The derivation of Harding and Johnson (25) includes an additional term, $\overline{v}$, in the form:

$$k_D=2B_{2}M-k_s-\overline{v}$$ (13)

which is exact for interaction parameters measured by sedimentation. They suggest equivalency with Eq. 12 if the hydrodynamic term in that case is taken as $k_s^{\prime }$ (25).

In either case, $k_D$ can be viewed as encompassing both the thermodynamic ($2B_{2}M$) and hydrodynamic ($-\left(k_s+\overline{v}\right)$ or $-k_s$) processes underlying self-interaction in dilute solution. A question then follows, in light of the hard sphere model predictions (40,41,42) and the solvation and shape dependencies of the hydrodynamic interaction predicted by Rowe (24), if these contributions should vary in a general and predictable manner relative to one another or if the relationship depends on molecular attributes (e.g., size and shape), solvent conditions (e.g., pH and ionic strength), or other factors. In support of the former, we and others, have observed a linear relationship between $B_2$ and $k_D$ in studies of single protein types and in narrow solvent ranges (15,53,54,55). This research suggests a simple linear scaling law, although the generality of this relationship across molecules of diverging physical properties remains to be established.

To gain greater insight into the thermodynamic and hydrodynamic contributions to protein self-interaction, we systematically characterized $B_2$, $k_D$, and $k_s$ for a diverse set of proteins with varied hydrodynamic properties (size and shape) using static and dynamic light scattering as well as sedimentation velocity analytical ultracentrifugation. The self-interaction tendencies for each protein were varied by preparing each in three buffers of differing pH and ionic strength. Using a data set of 40 different protein/buffer combinations with M ranging from ∼14 to 200 kDa and shapes (frictional ratio, $f/f_0$) ranging from ∼1.2 to 1.7, we observed a linear relationship between the thermodynamic and hydrodynamic interaction terms that was relatively consistent with the hard, interacting sphere model at low self-association potentials, despite the nonspherical shape and solvation of our model proteins. Significant deviations were observed for moderate self-association and repulsion in line with the expected limitations of the model. We propose our observed correlation as a general scaling relationship that describes the impact of the pairwise interaction potentials on the hydrodynamic interaction across a broad range.

Materials and methods

Proteins

Hen egg lysozyme (HEL) and carbonic anhydrase (CA) were purchased from Sigma. Ribonuclease A (RNAse) was purchased from Teknova. Nb1, a Nanobody molecule was produced at Sanofi using a microbial expression system. Fab1, an antigen binding fragment of an IgG, was produced using a Chinese hamster ovary (CHO) cell expression system at Sanofi. Recombinant human α-galactosidase A (aGal), an obligate homodimeric enzyme, was produced at Sanofi using a CHO expression system. Monoclonal antibodies were either purchased from Myoderm, or produced at Sanofi using a CHO expression system. MAb1, mAb2, and mAb3 are of the IgG4 subclass and mAb4, mAb5, and mAb6 are IgG1s. CODV (crossover dual variable domain) and TBTI (tetravalent bispecific tandem IgG) multispecific antibodies were produced at Sanofi using a CHO expression system. CODV1, CODV2, and CODV3 each contain three binding domains on either an IgG4 backbone (CODV1 and CODV2) or IgG1 backbone (CODV3). TBTI1 contains four binding domains on an IgG4 backbone. All samples used in this study are listed in Table 1. Crystal structure coordinates for each protein were obtained from the Protein Data Bank (HEL; PDB: 1LYZ (56), Nb1; PDB: 3DWT (57), CA; PDB: 1V9E (58), RNAse; PDB: 1FS3 (59), Fab; PDB: 1GHF (60), aGal; PDB: 1R47 (61), mAb; PDB: 1IGT (62), CODV; PDB: 5HCG (63), TBTI; PDB: 4HJJ (64)) and were used to render molecular illustrations using the online tool, ILLUSTRATE (65) (https://ccsb.scripps.edu/illustrate/). In cases where partial structures were available or multimeric assembly was required, the illustrations were assembled in approximate orientations. The illustrations are intended to present a simple comparative assessment of protein size and shape and are not intended to provide accurate structural detail.

Table 1.

Molecular attributes and purity of model proteins

Name	$M_{seq}$a (kDa)	$M$b (kDa)	$E$a (mL/mg·cm)	% main speciesc
RNAse	13.7	13.7	0.69	97.4
HEL	14.3	14.3	2.66	99.5
Nb1	27.9	27.9	1.55	97.7–99.5
CA	29.0	29.0	1.74	95.8–98.6
Fab	47.8	47.8	1.57	95.8–98.2
aGAL	90.7	100.0	2.45	97.7–99.6
mAb1	146.8	149.5	1.41	98.8–99.8
mAb2	144.4	147.0	1.35	90.8–94.6
mAb3	144.0	146.7	1.52	96.6–100.0
mAb4	145.9	148.6	1.51	96.7–99.4
mAb5	145.0	147.7	1.55	98.4–100.0
mAb6	145.6	148.2	1.49	99.8
CODV1	172.6	175.2	1.59	95.0–96.4
CODV2	172.0	174.6	1.59	98.3–99.5
CODV3	176.9	179.5	1.67	91.1–94.5
TBTI1	197.9	200.5	1.55	97.7–100.0

^aCalculated from amino acid composition.

^bAdjusted for glycosylation as indicated in the materials and methods.

^cDetermined by c(s) analyses of SV-AUC data for the lowest concentration prepared and given as the range observed among the buffer/pH systems used for each protein.

Calculated molecular properties

The sequence molecular weight ($M_{seq}$), extinction coefficient ($E$), and isoelectric point ($pI$) were calculated from amino acid sequence using SEDNTERP (66). To improve accuracy of molecular weight for calculations, 2.6 kDa was added to Fc-containing molecules (mAb1–6, CODV1–3, and TBTI1) to account for two G0 glycans. Similarly, aGal was assumed as 100 kDa (67) to account for glycosylation. For $pI$ calculations the following amino acid $pKa$ (where $K_a$ is the acid dissociation constant) values were assumed: Arg = 12, Asp = 4.5, Glu = 4.6, His = 6.2, Lys = 10.4, and Tyr = 9.7. All sulfhydryl side chains in Cys residues were assumed disulfide-bonded with no contributing $p{K}_a$. The Eisenberg hydrophobicity index was calculated according to the method of Eisenberg et al. (68).

Sample preparation

Samples were exhaustively dialyzed against three buffers: 10 mM sodium acetate (pH 5) (identified as A5), 10 mM histidine/histidine hydrochloride (pH 6) (identified as H6), and 10 mM potassium phosphate (pH 7) (identified as P7). Buffers were produced using compendial grade chemicals and high purity water (Milli-Q, Millipore). The pH and conductivity of each buffer preparation were measured to assure consistency between batches and therefore substantiate comparison of measured parameters among samples (see Table S1). When present, surfactant was removed from the starting material before dialysis using DetergentOUT Tween spin-columns (G-Biosciences). Dialyzed samples were diluted to at least 10 concentrations using the appropriate buffer over the range 0.2–5.0 mg/mL. Samples were passed through 0.22-μm spin filters (Costar Spin-X) and the concentrations were measured using a Lunatic microfluidic spectrophotometer (Unchained Labs) and the appropriate extinction coefficient as indicated in Table 1.

Dynamic and static light scattering

Samples were analyzed in a fused silica cuvette using a Wyatt DynaPro NanoStar laser light scattering instrument. DLS and 90° static light scattering (SLS) data were acquired following equilibration at 20°C for 5 min as follows. For DLS, the laser power was automatically adjusted, and the detector was auto-attenuated. Data were acquired for 10 s, averaged over six acquisitions, and modeled using the method of Cumulants as implemented in the vendor software. The quality of the modeling was confirmed from the fit residuals for each accumulation and the sum of squares. For SLS, the detector was calibrated with toluene over a range of laser power from 10 to 100%. Over this range, the detector voltage varied linearly with laser power and linear regression was used to interpolate the appropriate reference voltage ($V_R$) at any given laser intensity. When measuring sample dilutions, the laser power was set such that the detector voltage was ∼0.9 V for the highest concentration and was held constant when measuring the lower concentrations. Detector voltages were averaged over 30 s and converted to the excess Rayleigh scattering ratio ($R_{\theta }$) as:

$$R_{\theta }=\frac{V_s\times n_0^2\times R_R}{V_R\times n_R^2}$$ (14)

where $V_s$ is the detector voltage from the sample, $n_0$ is the solvent refractive index (assumed 1.334), $R_R$ is the Rayleigh ratio of the toluene reference (assumed 1.11 × 10⁻⁵ cm⁻¹, determined from the data tabulated in Wu (69)), and $n_R$ is the refractive index of the toluene reference (assumed 1.492, extrapolated from tabulated data in CRC Handbook (70)). The quantity $Kc/R_{\theta }$ was then calculated from the measured sample concentration ($c$) and scattering constant $K$:

$$K=4{\pi }^2{\left(dn/dc\right)}^2{n}_0^2/N_A{\lambda }_0^4$$ (15)

where $dn/dc$ is the refractive increment (assumed 0.190 mL/g for all samples (71)), $N_A$ is Avogadro’s number (6.022137 × 10²³), and ${\lambda }_0$ is the instrument laser wavelength (6.58 × 10⁻⁵ cm).

Sedimentation velocity

Samples were loaded into AUC cells assembled with double sector charcoal-filled epon centerpieces and quartz windows. The cells were aligned in an AN-60 Ti analytical rotor and equilibrated at 20°C for 2 h before accelerating to 40,000 rpm. Radial absorbance scans were acquired continuously using a radial step size of 30 μm. Centerpiece width (optical pathlength) of 1.2 or 0.3 cm as well as absorbance band of 250, 280, or 300 nm were selected to attain detector response of 0.1–1.0 absorbance units. Samples of 0.2 mg/mL were analyzed by the continuous c(s) distribution model in SEDFIT (72) to determine sample quality with respect to low and high molecular weight contaminants. The appropriateness of the model was determined by the Z-score of the best fit. Distributions obtained at a resolution of N = 200 were regularized by maximum entropy at a level of p = 0.95 and integrated to determine the percentage of low and high molecular weight contaminant. All data sets were analyzed by the time derivative method as incorporated in DCDT+ (73,74). The apparent sedimentation coefficient was determined from the sedimenting boundary with a single component model of the g(s^∗) distribution.

Determination of hydrodynamic and thermodynamic properties

For each sample dilution series, the collective diffusion coefficient ($D_c$) obtained by DLS, the inverse of the sedimentation coefficient ($1/s$) obtained by sedimentation velocity, and $Kc/R_{\theta }$ obtained by SLS were each plotted as a function of measured concentration. Data were fit to linear models, the appropriateness of which was determined visually. In cases where linear models were insufficient to describe the data, the data set was removed from further analyses. From the linear models, the DLS data yielded the self-diffusion coefficient at infinite dilution ($D$) and the diffusion interaction parameter ($k_D$):

$$D_c=D\left(1+k_{D}c\right)$$ (16)

The sedimentation velocity data yielded the sedimentation coefficient in absence of self-interaction ($s_0$) and the sedimentation interaction parameter ($k_s^{\prime }$):

$$\frac{1}{s}=\frac{1}{s_0}\left(1+k_s^{\prime }c\right)$$ (17)

The measured $k_s^{\prime }$ values were converted to $k_s$ by Eq. 4. The SLS data yielded the molecular weight ($M$) and the osmotic second virial coefficient ($B_2$):

$$\frac{Kc}{R_{\theta }}=\frac{1}{M}+2B_{2}c$$ (18)

with $R_{\theta }$ and $K$ as defined in Eqs. 14 and 15, respectively. The form factor, $P\left(\theta \right)$, typically used to describe the angular dependence of the scattering intensity was assumed negligible. For all analyses, measurement uncertainty was calculated by propagation of the error statistics from the linear regressions. From the DLS data, the hydrodynamic radius was calculated as:

$$R_H=\frac{k_{B}T}{6\pi \eta D}$$ (19)

where $k_B$ is the Boltzmann constant (1.38 × 10⁻¹⁶ erg/K), $T$ is the experimental temperature (298 K), and $\eta$ is the solvent viscosity. The partial specific volume ($\overline{v}$) was calculated from the Svedberg equation as:

$$\overline{v}=\frac{RT{s}_0-MD}{-MD\rho }$$ (20)

where $R$ is the gas constant (8.314 × 10⁷ erg/K⋅mol), $\rho$ is the solvent density, and $M$ is the molecular weight. For this calculation, $M$ was obtained from amino acid composition and adjusted for glycan contribution (Table 1) was used. The frictional coefficient ($f$) was calculated as:

$$f=\frac{RT}{N_{A}D}$$ (21)

The theoretical frictional coefficient of an ideal sphere of corresponding molecular weight ($f_0$) was calculated as:

$$f_0=\left(6\pi \eta \right){\left(\frac{3M\overline{v}}{4\pi N_A}\right)}^{\frac{1}{3}}$$ (22)

For this calculation, $M$ was determined as indicated above. The radius of an equivalent sphere ($R_0$) was calculated as (75):

$$R_0=\sqrt[3]{\frac{3V_0}{4\pi }}$$ (23)

where $V_0$ is the anhydrous molecular volume given as (75):

$$V_0=\frac{M\overline{v}}{N_A}$$ (24)

where $M$ is the molecular weight, $\overline{v}$ is the partial specific volume, and $N_A$ is Avogadro’s number. The osmotic second virial coefficient was calculated according to either Eq. 12 or Eq. 13 (25).

Results

To evaluate the quantitative relationship between thermodynamic and hydrodynamic contributions to protein self-interaction, we prepared a set of 16 different standard proteins and proteins of therapeutic interest (Table 1) in 3 different buffer/pH systems (Table S1), yielding 48 potential self-interaction data sets. Dilution series ranging from 0.2 to 5.0 mg/mL were analyzed by DLS (Fig. S1), sedimentation velocity AUC (SV-AUC) (Fig. S2), and SLS (Fig. S3). Several of these protein-buffer/pH combinations were not amenable to analysis and were excluded from this work. Specifically, RNAse demonstrated multimodal DLS decay functions in 10 mM sodium acetate (pH 5) (A5 buffer) and 10 mM potassium phosphate (pH 7) (P7 buffer). Similar multimodal distributions were observed for HEL in A5 buffer and 10 mM histidine hydrochloride (pH 6) (H6 buffer). Precipitation was observed during dialysis of α-Galactosidase against A5 buffer and TBTI1 against H6 buffer. Nonlinearity in the concentration dependence of the diffusion and sedimentation coefficients for mAb6 over the range tested was observed in H6 and P7 buffers. These 8 protein-buffer/pH combinations were removed from the overall analyses. In total, DLS, SLS, and SV-AUC data from dilution series of 40 different protein-buffer/pH combinations were included in the overall analyses.

To ensure the generality of the self-interaction data set, the proteins were selected due to their variation in molecular size and anticipated shape (Fig. 1). Moreover, the buffer/pH systems were selected in order to produce a range of self-interaction behaviors for each protein by providing different regimes in pH and ionic strength. The diversity of this data set is reflected by the relatively broad ranges in calculated and measured molecular properties (Table S3). The calculated isoelectric point ($\mathit{pI}$) (Fig. 2 A) and Eisenberg hydrophobicity index (EHI, Fig. 2 B) ranged from 5.5 to 10.6 and −29 to 48, respectively. These calculated parameters reflect the amino acid sequence diversity among the proteins arising from side chain pKa values and chemical composition. But they may not accurately recapitulate pI or hydrophobicity in solution as the calculations are only based on amino acid composition. The measured hydrodynamic radius ($R_H$, Fig. 2 C) ranged from 1.95 to 6.34 nm. The measured partial specific volume ($\overline{v}$, Fig. 2 D) and frictional ratio ($f/f_0$, Fig. 2 E) ranged from 0.700–0.745 mL/g and 1.20–1.65, respectively. We used an indirect method to determine $\overline{v}$ from the assumed $M$ and measured sedimentation and diffusion coefficients (see materials and methods). We note that the values obtained for RNAse, HEL, and CA are in agreement with those determined previously using more direct measurements (Table S2).

Figure 1.

Size range and molecular configuration of model proteins. Illustrations of the type of molecules used in this study were constructed using crystal structure coordinates. The approximate molecular weight ($M$) of each is indicated.

Figure 2.

Molecular diversity of the data set. Boxplots indicating the mean (solid horizontal line) for the calculated isoelectric point (A) and Eisenberg hydrophobic index (B) as well as the measured hydrodynamic radius (C), partial specific volume (D), and frictional ratio (E). Data points in (C‒E) are the mean values for each protein determined among measurements in the three buffer/pH systems.

In addition to molecular diversity, the accuracy of the data was evaluated using the orthogonal information content of the methods employed. First, the measured $R_H$ was observed to increase with increasing molecular weight in a manner consistent with a simple predictive hydrodynamic model within the bounds of $\overline{v}$ and $f/f_0$ values observed in the data set (Fig. 3 A; see materials and methods). We also observed that the molecular weight measured by SLS was in relatively good agreement with that expected from amino acid sequence (Fig. 3 B). Systematically higher measured values (on average 14.8% greater) may reflect signal contributions from high molecular weight contaminants, which contribute a greater fraction by signal than by mass due to the proportional dependence of signal intensity on molecular weight. This seems reasonable given the small amount of contaminating high molecular weight species (presumed to be aggregates) observed by sedimentation velocity for many of the samples (Table 1). Despite this systematic deviation, self-interaction assessed through the osmotic second virial coefficient ($B_2$) was well correlated ($R^2$ > 0.94) with $B_2$ calculated from DLS and SV-AUC measurements via Eq. 13 (Fig. 3 C). Although, we cannot exclude the possibility that the low amounts of contaminating aggregates in some samples may influence the measured interactions. The coincidence of the results from the two methods was further supported by the overlay of the data with the line of unity (Fig. 3 C).

Figure 3.

Accuracy of the data set features. Correlation plots of the hydrodynamic radius and molecular weight calculated from amino acid sequence (A), the molecular weight measured by static light scattering and the calculated molecular weight (B), and the osmotic second virial coefficient measured by static light scattering as well as by sedimentation velocity and dynamic light scattering (C). Data points in (A) and (B) are the mean values for each protein determined among measurements in the three buffer/pH systems. The curves in (A) are from an anhydrous equivalent sphere model with the mean values of $\overline{v}$ = 0.720 mL/g and $f/f_0$ = 1.48 (solid black line). The gray dashed lines are the upper bounds ($\overline{v}$ = 0.745 mL/g and $f/f_0$ = 1.63) and lower bounds ($\overline{v}$ = 0.700 mL/g and $f/f_0$ = 1.22). Error bars in (B) are given as the standard error of the mean. The dashed lines in (B) and (C) represent the lines of unity between the y and x axes.

With the diversity and accuracy of the data set assured, the self-interactions were then scrutinized. In line with the diverse molecular properties, a wide range of self-interaction behaviors were observed through measurement of $k_D$ and $k_s$ (Figs. 4 A, S1, and S2; Table S4). Measured $k_D$ values ranged from −42.6 ± 1.6 mL/g (CODV3 in P7 buffer) to 77.3 ± 1.7 mL/g (mAb5 in A5 buffer). In most cases, the self-interaction tendencies varied significantly for each protein in the different buffer/pH systems. For instance, CODV2 demonstrated high positive $k_D$ in A5 buffer (49.9 ± 0.9 mL/g), with weaker repulsion in H6 buffer (11.0 ± 1.6 mL/g) and attraction in P7 buffer (−38.9 ± 2.0 mL/g). This broad range in self-interaction tendency was apparent when $k_D$ was plotted against ${2B}_{2}M$ measured by SLS (Fig. 4 B). Here, regression analysis indicated a high degree of linear correlation ($R^2$ >0.96), suggesting that the two quantities can be interconverted using the parameters of the linear regression. Linear least squares regression analyses indicated y-intercept of −7.58 (±1.14) and slope of 0.61 (±0.02). The linear form of this relationship has been suggested previously with data sets consisting of a single type of protein, namely mAbs (15,53,54,55). Remarkably, the composite of the results from these studies and other studies that contained companion $k_D$ and $B_2$ measurements overlays with the linear regression from our present data set (Fig. 4 C), validating our analysis. To further probe the interconversion of interaction parameters, measured $k_D$ was also plotted as a function of $2B_{2}M$ calculated from measured $k_D$ and $k_s$ via Eqs. 12 and 13 (Fig. 4 D). Linear least squares regression indicated y-intercept of −5.19 (±0.60) and −5.63 (±0.60), respectively, and slope of 0.61 (±0.01) for both. Therefore, the data do not distinguish between Eqs. 12 and 13 across the broad range of interaction potentials in the data set. In addition, even though the comparability of the measured and calculated thermodynamic interaction is apparent via the overlay of the data in Fig. 4 D with the linear regression from Fig. 4 B, the linear least squares regression analyses fall short of statistical equivalency in the y-intercept. Comparability of measured $k_s$ with that calculated from measured $k_D$ and $B_2$ via Eqs. 12 and 13 (Fig. 4 D) is also apparent, although the correlation is relatively low ($R^2$ = 0.74) due to the low precision in the data. Linear least squares regression indicated y-intercept of −0.86 (±2.21) and −0.26 (±2.17), respectively, and slope of 0.84 (0.08). As above, these data do not distinguish between calculations made with Eqs. 12 or 13. It should be noted that the regression of the data falls slightly short of statistical equivalency with the line of unity. The cause of this discrepancy is uncertain given the qualitative consistency of measured $2B_{2}M$ with that calculated with $k_s$ and $k_D$ (Fig. 4, B and D). We posit that imprecision in the SLS measurement may bias the regression analyses in Fig. 4, B and E. However, we cannot exclude the possibility of subtle inconsistencies among the interaction parameters on the basis of our data alone.

Figure 4.

Self-interaction of model and therapeutic proteins in different buffer/pH systems. Self-interaction measured by dynamic light scattering ($k_D$; A, top panel) and sedimentation velocity ($k_s$; A, bottom panel). Measurements in A5 buffer are indicated with orange bars, H6 with blue bars, and P7 with green bars. Conditions excluded from the overall analyses are indicated with an “X” of matching color. Error bars were determined by propagation of error from the linear least squares regression of the data. The $k_D$ data from A were plotted as a function of the osmotic second virial coefficient measured by SLS (B) and fit by linear least squares to a linear model (solid line). The resulting model was overlayed with $k_D$ and $B_2$ data reported in the literature (14,15,53,54,55,76) (C). The $k_D$ data from (A) were also plotted as a function of $2B_{2}M$ calculated from Eq. 12 (solid blue symbols) and 13 (open red symbols) and overlayed with the best-fit linear regression from (B) (solid line) (D). Measured $k_s$ was plotted as a function of calculated $k_s$ (E) using either Eq. 12 (solid blue symbols) or 13 (open red symbols). The line of unity is indicated by the dashed line.

Our data support the linear relationship between hydrodynamic and thermodynamic interactions as anticipated from theoretical models of hard, interacting spheres (40,41,42). To probe this relationship further, we evaluated the dependence of the hydrodynamic interaction on the pairwise interaction potential in the context of Eq. 10. To define $B_2^{\ast }$, one must define the excluded volume ($V_e$) and the effect on the osmotic second virial coefficient ($B_2^{ex}M$). In the case of anhydrous spheres, $V_e$ is given as (77):

$$V_e=\left(32/3\right)\pi r^3=8M\overline{v}/N_A$$ (25)

where $r$ is the radius of the sphere and $N_A$ is Avogadro’s number. Likewise, $B_2^{ex}M$ is (77):

$$B_2^{ex}M=\left(16/3\right)\pi r^3{N}_A/M=4\overline{v}$$ (26)

We assume that for solvated, nonspherical proteins, the excluded volume can be approximated with an equivalent sphere of $r=R_H$. This assumption has been supported by coarse-grained modeling of molecules of varying size and shape (78). Furthermore, as suggested by Rowe (24), a corresponding adjustment to $\overline{v}$ can be given as $\overline{v}{\left(f/f_0\right)}^3$. Thus, in the limit of this equivalent spherical model, $V_e$ is given as:

$$V_e=\left(32/3\right)\pi {R_H}^3=8M\overline{v}{\left(f/f_0\right)}^3/N_A$$ (27)

and $B_2^{ex}M$ as:

$$B_2^{ex}M=\left(16/3\right)\pi {R_H}^3{N}_A/M=4\overline{v}{\left(f/f_0\right)}^3$$ (28)

Within our data set, $B_2^{ex}M$ calculated from $R_H$ and $f/f_0$ were consistent (Table S5). We further calculated the hydrodynamic interaction in terms of $k_s$, $K_s^{\prime }$, and $K_s$ from the measured $k_s^{\prime }$ values (Table S5) and evaluated this as a function of $B_2^{\ast }$ measured by SLS. The solvent density and volume fraction derived hydrodynamic interaction, $K_s^{\prime }$, was used for consistency with the theoretical model. The data (Fig. 5) indicate a different trend than predicted by the Batchelor model (42). Linear least squares regression of our data indicated that this difference is statistically significant, indicating an increase of 2.96 (±0.23) in $K_s^{\prime }$ for every unit increase in $B_2^{\ast }$ and a $K_s^{\prime }$ value of 2.18 (±0.59) at $B_2^{\ast }$ = 0. This contrasts with the values of 3.52 and 3.03, respectively, that are expected from the Batchelor model. In the condition where thermodynamic interactions are represented by only the excluded volume effect ($B_2^{\ast }$ = 1), we observe $K_s^{\prime }$ = 5.14 in contrast to $K_s^{\prime }$ = 6.55 as anticipated from the models of Batchelor (26,42). Our observed value is in line with the expected values from the models of Rowe (24) and Brady and Durlofsky (27) and consistent with the results of several sedimentation studies using polystyrene latex spheres (28,29). While the trending of our data yields a statistically significant difference with the theoretical expectation, the relatively high variability arising from the SLS measurements suggests caution. To provide additional support, identical regression analyses using $B_2$ calculated from SV-AUC and DLS measurements via Eq. 13 were conducted. These data (Fig. S4) provided greater precision with equivalent regression statistics, showing an increase of 3.07 (±0.12) in $K_s^{\prime }$ for every unit increase in $B_2^{\ast }$ and a $K_s^{\prime }$ value of 2.45 (±0.31) at $B_2^{\ast }$ = 0. Despite the observed differences with the Batchelor model (42), it is important to note that intersection of our data with the model occurs at $B_2^{\ast }$ = −1.52, with close agreement (within 1 unit in $K_s^{\prime }$) in the range of −3.3 ${<B}_2^{\ast }<$ 0.2. Therefore, the model reasonably approximates our experimental data within the range of negligible to slightly self-associating pairwise interaction potentials.

Figure 5.

Effect of pairwise interaction potential on hydrodynamic interactions. Correlation of the hydrodynamic interaction ($K_s^{\prime }$) measured by SV-AUC with the reduced osmotic second virial coefficient ($B_2^{\ast }=B_2/B_2^{ex}$), where $B_2$ was measured by SLS. The data were fit with a linear least squares regression model (solid black line) and compared with the hard, interacting sphere model of Batchelor (solid blue line) (42).

Discussion

We have characterized the thermodynamic and hydrodynamic contributions to dilute-solution self-interactions for a diverse set of proteins in three different buffer/pH systems. Herein, we draw general conclusions by analyzing the collective behavior among the proteins and conditions studied. Therefore, implicit in this analysis is that the results for each protein are congruent with those of the others in the data set, an assumption that is supported by the remarkable linearity of the data (Fig. 4 B) and the consistency with which this linearity is extended to data from other unrelated studies of mAbs and HEL (Fig. 4 C). Moreover, our focus is on the magnitude of the interaction parameters, which reflect the composite of all underlying intermolecular and hydrodynamic interactions, rather than the mechanisms by which the proteins interact. Such an approach is supported by the observation that the magnitude of these parameters often relates to observed solution behaviors. For instance, solution conditions resulting in a relatively narrow range of negative $B_2$ values have been shown to promote protein crystallization (the crystallization slot) whereas conditions resulting in more negative values promote amorphous aggregate formation (79). Similarly, moderate to highly positive $k_D$ values (e.g., greater than 20 mL/g) have been associated with favorable mAb solution behavior (15). While these parameters provide rich information in and of themselves, alternative approaches that probe the underlying mechanism of self-assembly may provide useful complementary information. For instance, whole-boundary modeling of sedimentation velocity data (80) has been used to extract self-association mechanisms from data sets containing significant hydrodynamic interactions (81,82). By analyzing the composite trending of our data set in a manner agnostic to interaction mechanism, we are able to determine the impact of the pairwise interaction potential on the hydrodynamic interaction across a broad range of interaction magnitude as well as the hydrodynamic interaction that occurs at $B_2^{\ast }=1$. By comparing our data to expectations for interacting and noninteracting hard sphere models, we are able to provide insight into how molecular attributes may dictate the way that proteins self-interact in dilute solution.

With respect to the impact of the pairwise interaction potential on the hydrodynamic interaction, we observe a linear increase in $K_s^{\prime }$ with increasing $B_2^{\ast }$. This general behavior is consistent with models for interacting hard spheres (40,41,42), where short-range attractive forces (e.g., those arising from anisotropic charge distribution, hydrophobic interaction, etc.) lead to increased proximity of neighboring proteins and enhanced exposure to solvent downflow velocity fields. This effect corresponds to decreasing $K_s^{\prime }$ values as $B_2^{\ast }$ becomes increasingly negative. On the other hand, net repulsion driven by dominating electrostatic interactions reduces proximity effects, and in the dominating solvent backflow leads to increasingly positive $K_s^{\prime }$ values as $B_2^{\ast }$ becomes greater than zero. Our data suggest that these effects vary linearly over a broad range of net attractive and repulsive thermodynamic interactions (at least −2 ${<B}_2^{\ast }<$ 6). Moreover, in the narrow range of negligible to slightly self-associating pairwise interaction potentials (−3.3 ${<B}_2^{\ast }<$ 0.2) our data are reasonably well described by the interacting hard sphere model of Batchelor (42). This is perhaps surprising given that the model proteins used herein are solvated and nonspherical. Although beyond this narrow range of interaction potentials the agreement with the model fails, including at the condition in which the pairwise interaction is given solely by the effect of excluded volume ($B_2^{\ast }$ = 1). Here, we observe a significantly lower hydrodynamic interaction ($K_s^{\prime }$ = 5.14) than is anticipated from the model ($K_s^{\prime }$ = 6.55). Equivalency may be anticipated even though in our case (and in contrast to the theoretical model) $B_2^{\ast }$ = 1 does not necessarily equate to zero thermodynamic interaction. However, we assume that the balanced attractive and repulsive interactions resulting in $B_2^{\ast }$ = 1 may approximate that condition (e.g., molecular proximity and the resulting hydrodynamic interaction are dominated by the excluded volume effect with all other intermolecular interactions resulting in negligible net contribution). We cannot conclude that this discrepancy between our data and the theoretical model is due to the asymmetrical shape (given as $f/f_0$) or solvation (given as $v_P/\overline{v}$) of the proteins in our data set as the spherical model is expected to be the limiting case (Eq. 9) and we observe a lower value than predicted. The reason for the discrepancy is uncertain. However, we note that our data are consistent with the models of Rowe (24) and Brady and Durlofsky (27), who both derived $K_s^{\prime }$ = 5 for the limiting case of noninteracting spherical particles. It may also be surprising, given the dependence of the hydrodynamic interaction on $f/f_0$ (Eq. 9), that congruency in our data was observed despite the range of hydrodynamic properties of the proteins in our data set, with M ranging from ∼14 to 200 kDa and $f/f_0$ ranging from ∼1.2 to 1.7. Based on our data, it does not appear that $f/f_0$ is a strong determinant in the hydrodynamic interaction, at least within the range tested. However, we cannot exclude the possibility that proteins of higher $f/f_0$ may deviate from the trend that we observed in our data set. In addition, it should also be noted that $B_2^{ex}$ is dependent upon $f/f_0$ (or $R_H$) in the equivalent sphere model that we used in our analyses. Therefore, the shape dependency is apparent in the thermodynamic as well as hydrodynamic component of the overall interaction. The larger excluded volume of more asymmetric proteins will result in a diminished $B_2^{\ast }$ compared with corresponding proteins with equivalent $B_2$ but of more globular shape, while $K_s^{\prime }$ is expected to be greater. However, it should also be noted that the validity of the $B_2^{ex}$ model employed may not yield a good approximation of excluded volume effect for some proteins. For these reasons, caution is warranted in interpreting the applicability of our data to proteins that are dissimilar to those used in our analyses. However, at least for proteins similar to those used, which encompasses a large proportion of properties shared by proteins of biological significance and of therapeutic interest, we can propose a general relationship between the hydrodynamic interaction and pairwise interaction potential based on our data:

$$K_s^{\prime }=5.1+3.0\left(B_2^{\ast }-1\right)$$ (29)

This empirically derived relationship is approximately equivalent to Eq. 10 over the limited range of −3.3 ${<B}_2^{\ast }<$ 0.2. Further work is needed to establish the extent of applicability of Eq. 29 among proteins with properties that diverge significantly from the proteins in our data set.

While the condition of $B_2^{\ast }$ = 1, is of great interest in that it represents the reference point for $K_s^{\prime }$ of theoretical noninteracting molecules, the observed trend in our data as well as prediction from theory suggests a corresponding hydrodynamic equivalent occurs at $K_s^{\prime }$ = 0. Our data indicate that this occurs at $B_2^{\ast }$ = −0.74. This falls within the narrow range of approximate agreement of our data and the Batchelor model (42), which gives this condition at $B_2^{\ast }$ = −0.86. While the molecular origins of this condition are not clear, it stands to reason that, hypothetically, a condition should be attainable in which a specific magnitude of self-interaction leads to a proximity effect with balanced downflow and backflow contributions. We are not aware of any research linking this hydrodynamic equivalent of the theta condition to protein solution behavior. However, given the importance of weak interactions in biological processes and the ubiquitous presence of the external gravitational field, we posit that hydrodynamic interactions may provide a selective pressure in protein evolution. The framework for protein evolution is typically given in terms of conformational stability, dynamics, and function (83), including the strong self-association related to allostery (84). We note that in the case of the mAbs in this study ($B_2^{ex}\approx$ 6 × 10⁻⁵ to 9 × 10⁻⁵ mL mol/g²), and given the precision of our analysis, balanced net hydrodynamic interactions are achieved at ${2B}_{2}M\approx$ −13 to −19 mL/g or alternatively at $k_D$ values of −15 to −19 mL/g (as determined via the regression in Fig. 4 B). Interestingly, studies of mAb self-association in phosphate-buffered saline often yield results overlapping this range (for instance, see Mieczkowski et al. (85), Hopkins et al. (86), and Yang et al. (87)). Since phosphate-buffered saline is often used as a buffer to approximate physiological pH and salt conditions, the natural tendency of IgG to weakly self-associate in this buffer has been ascribed significance to in vivo function (for instance, see Diebolder et al. (4) and Yang et al. (5)). Considering the present data, a potential relationship between IgG attributes favoring balanced net downflow and backflow contributions to the hydrodynamic interaction and in vivo functional mechanisms involving weak self-interaction is suggested, although not established beyond coincidence of the self-association trends. Further work is needed to address the physical basis of the condition where $K_s^{\prime }$ = 0 as well as the physicochemical determinants of this state in IgG and other protein solutions and to explore potential links to in vitro solution behavior and/or biochemical function.

Author contributions

J.S.K. and Y.R.G. designed the research. J.S.K. carried out the experiments. All authors interpreted the data and wrote the manuscript.

Declaration of interests

The authors were employees of Sanofi during the work described in this manuscript and may hold shares and/or stock options in the company. In response to reasonable requests, noncommercially available materials and experimental protocols, that Sanofi has the right to provide, will be made available to not-for-profit or academic requesters upon completion of a material transfer agreement. Requests may be made by contacting the corresponding author.

Editor: Jeremy Schmit.