Calculation of weak protein-protein interactions: the pH dependence of the second virial coefficient

ATP and Tri-Polyphosphate (TPP) Suppress Protein Aggregate Growth by a Supercharging Mechanism

A common strategy to increase aggregation resistance is through rational mutagenesis to supercharge proteins, which leads to high colloidal stability, but often has the undesirable effect of lowering conformational stability. We show this trade-off can be overcome by using small multivalent polyphosphate ions, adenosine triphosphate (ATP) and tripolyphosphate (TPP) as excipients. These ions are equally effective at suppressing aggregation of ovalbumin and bovine serum albumin (BSA) upon thermal stress as monitored by dynamic and static light scattering. Monomer loss kinetic studies, combined with measurements of native state protein–protein interactions and ζ-potentials, indicate the ions reduce aggregate growth by increasing the protein colloidal stability through binding and overcharging the protein. Out of three additional proteins studied, ribonuclease A (RNaseA), α-chymotrypsinogen (α-Cgn), and lysozyme, we only observed a reduction in aggregate growth for RNaseA, although overcharging by the poly-phosphate ions still occurs for lysozyme and α-Cgn. Because the salts do not alter protein conformational stability, using them as excipients could be a promising strategy for stabilizing biopharmaceuticals once the protein structural factors that determine whether multivalent ion binding will increase colloidal stability are better elucidated. Our findings also have biological implications. Recently, it has been proposed that ATP also plays an important role in maintaining intracellular biological condensates and preventing protein aggregation in densely packed cellular environments. We expect electrostatic interactions are a significant factor in determining the stabilizing ability of ATP towards maintaining proteins in non-dispersed states in vivo.

Protein-polymer mixtures in the colloid limit: Aggregation, sedimentation, and crystallization

While proteins have been treated as particles with a spherically symmetric interaction, of course in reality, the situation is rather more complex. A simple step toward higher complexity is to treat the proteins as non-spherical particles and that is the approach we pursue here. We investigate the phase behavior of the enhanced green fluorescent protein (eGFP) under the addition of a non-adsorbing polymer, polyethylene glycol. From small angle x-ray scattering, we infer that the eGFP undergoes dimerization and we treat the dimers as spherocylinders with aspect ratio L/D - 1 = 1.05. Despite the complex nature of the proteins, we find that the phase behavior is similar to that of hard spherocylinders with an ideal polymer depletant, exhibiting aggregation and, in a small region of the phase diagram, crystallization. By comparing our measurements of the onset of aggregation with predictions for hard colloids and ideal polymers [S. V. Savenko and M. Dijkstra, J. Chem. Phys. 124, 234902 (2006) and Lo Verso et al., Phys. Rev. E 73, 061407 (2006)], we find good agreement, which suggests that the behavior of the eGFP is consistent with that of hard spherocylinders and ideal polymers.

Decorated networks of native proteins: nanomaterials with tunable mesoscopic domain size

Natural and artificial proteins with designer properties and functionalities offer unparalleled opportunity for functional nanoarchitectures formed through self-assembly. However, to exploit this potential we need to design the system such that assembly results in desired architecture forms while avoiding denaturation and therefore retaining protein functionality. Here we address this challenge with a model system of fluorescent proteins. By manipulating self-assembly using techniques inspired by soft matter where interactions between the components are controlled to yield the desired structure, we have developed a methodology to assemble networks of proteins of one species which we can decorate with another, whose coverage we can tune. Consequently, the interfaces between domains of each component can also be tuned, with potential applications for example in energy - or electron - transfer. Our model system of eGFP and mCherry with tuneable interactions reveals control over domain sizes in the resulting networks.

Cluster Percolation Causes Shear Thinning Behavior in Concentrated Solutions of Monoclonal Antibodies

High-concentration (>100 g/L) solutions of monoclonal antibodies (mAbs) are typically characterized by anomalously large solution viscosity and shear thinning behavior for strain rates ≥10³ s^-1. Here, the link between protein-protein interactions (PPIs) and the rheology of concentrated solutions of COE-03 and COE-19 mAbs is studied by means of static and dynamic light scattering and microfluidic rheometry. By comparing the experimental data with predictions based on the Baxter sticky hard-sphere model, we surprisingly find a connection between the observed shear thinning and the predicted percolation threshold. The longest shear relaxation time of mAbs was much larger than that of model sticky hard spheres within the same region of the phase diagram, which is attributed to the anisotropy of the mAb PPIs. Our results suggest that not only the strength but also the patchiness of short-range attractive PPIs should be explicitly accounted for by theoretical approaches aimed at predicting the shear rate-dependent viscosity of dense mAb solutions.

A coarse-grained xDLVO model for colloidal protein-protein interactions

Colloidal protein-protein interactions (PPIs) of attractive and repulsive nature modulate the solubility of proteins, their aggregation, precipitation and crystallization. Such interactions are very important for many biotechnological processes, but are complex and hard to control, therefore, difficult to be understood in terms of measurements alone. In diluted protein solutions, PPIs can be estimated from the osmotic second virial coefficient, B22, which has been calculated using different methods and levels of theory. The most popular approach is based on the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory and its extended versions, i.e. xDLVO. Despite much efforts, these models are not fully quantitative and must be fitted to experiments, which limits their predictive value. Here, we report an extended xDLVO-CG model, which extends existing models by a coarse-grained representation of proteins and the inclusion of an additional ion-protein dispersion interaction term. We demonstrate for four proteins, i.e. lysozyme (LYZ), subtilisin (Subs), bovine serum albumin (BSA) and immunoglobulin (IgG1), that semi-quantitative agreement with experimental values without the need to fit to experimental B22 values. While most likely not the final step in the nearly hundred years of research in PPIs, xDLVO-CG is a step towards predictive PPIs calculations that are transferable to different proteins.

Application of Affinity-Capture Self-Interaction Nanoparticle Spectroscopy in Predicting Protein Stability, Especially for Co-Formulated Antibodies

PURPOSE

From traditional monoclonal antibodies to more and more complex mAb-based formulations, biopharmaceutical faces one challenge after another. To avoid these issues, identification of therapeutic proteins in the initial discovery process that has high stability and low self-interaction would simplify the development of safe and effective antibody therapeutics.

METHOD

Affinity-capture self-interaction nanoparticle spectroscopy (AC-SINS) is a new prediction method capable of identifying mAbs with different self-association propensity. In this study, 10 formulated monoclonal antibody (mAb) therapeutics include different mAb isotypes and co-formulated antibodies were measured by AC-SINS and some biophysical methods to predict protein stability. The prediction results of all 10 mAbs were compared to their stability data (Δ%monomer and Δ%HMWs) at accelerated (25°C and 40°C) and long-term storage conditions (4°C) as measured by size exclusion chromatography.

RESULT

AC-SINS method has a good predictive correlation with each mAbs and co-formulated antibodies. There were no physicochemical, intermolecular, or biological interactions that occurred between the two components of co-formulated antibodies which confirmed by Analytical ultracentrifugation (AUC).

CONCLUSION

Here we discuss the correlation between each method and protein stability, and also use AC-SINS assay to predict the stability of co-formulated antibodies for the first time. This may be an effective way to predict the stability of these complex mAb-based formulations such as co-formulated mAbs.

Charge regulation of colloidal particles in aqueous solutions

We study the charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize a colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. On contact with water, these surface groups undergo proton transfer reactions, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. This theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.

Calculation of salt-dependent free energy of binding of β-lactoglobulin homodimer formation and mechanism of dimer formation using molecular dynamics simulation and three-dimensional reference interaction site model (3D-RISM): diffuse salt ions and non-polar interactions between the monomers favor the dimer formation

There are several important phenomena in chemistry, biology, and physics where molecules (or parts of a molecule) having charges of the same sign come closer together and become stable. DNA condensation, RNA folding, colloid-colloid interactions are some of the examples of this kind. In the current work, we have investigated how β-lactoglobulin, a protein found in milk, in spite of carrying +13 charge, favors the homodimer formation in the presence of salt. We have focussed on calculating the protein-protein binding free energy in the presence of salt and identifying the thermodynamic and microscopic mechanism of the process. Estimation of binding free energy of this salt-dependent process is done by combining molecular dynamics simulation with statistical mechanical theory of three-dimensional reference interaction site model (3D-RISM). Binding free energy is evaluated from the chemical potential of the solutes as opposed to potential of mean force calculation, which gives only a constrained free energy. Our calculated values semi-quantitatively match with the experimental results. By examining the different components of binding free energy, we have found that the role of salt ions (especially of Cl^-) is to shift the equilibrium towards the dimer. Non-polar (Lennard-Jones) interactions between the monomers is also favorable to the binding free energy. However, water slightly disfavors the dimer formation. For the microscopic mechanism, heterogeneous of both Na⁺ and Cl^- near the charged residues at the binding interface and change of this charge distribution on dimer formation contribute to the stability. A fine-tuning of enthalpic and entropic effects of salt ions is found to operate at different salt concentrations. Both thermodynamic and microscopic mechanism of dimer formation gives detailed insight into the complex electrostatics of charged protein-protein binding.

Determination of Protein-Protein Interactions in a Mixture of Two Monoclonal Antibodies

The coformulation of monoclonal antibody (mAb) mixtures provides an attractive route to achieving therapeutic efficacy where the targeting of multiple epitopes is necessary. Controlling and predicting the behavior of such mixtures requires elucidating the molecular basis for the self- and cross-protein-protein interactions and how they depend on solution variables. While self-interactions are now beginning to be well understood, systematic studies of cross-interactions between mAbs in solution do not exist. Here, we have used static light scattering to measure the set of self- and cross-osmotic second virial coefficients in a solution containing a mixture of two mAbs, mAbA and mAbB, as a function of ionic strength and pH. mAbB exhibits strong association at a low ionic strength, which is attributed to an electrostatic attraction that is enhanced by the presence of a strong short-ranged attraction of nonelectrostatic origin. Under all solution conditions, the measured cross-interactions are intermediate self-interactions and follow similar patterns of behavior. There is a strong electrostatic attraction at higher pH values, reflecting the behavior of mAbB. Protein-protein interactions become more attractive with an increasing pH due to reducing the overall protein net charges, an effect that is attenuated with an increasing ionic strength due to the screening of electrostatic interactions. Under moderate ionic strength conditions, the reduced cross-virial coefficient, which reflects only the energetic contribution to protein-protein interactions, is given by a geometric average of the corresponding self-coefficients. We show the relationship can be rationalized using a patchy sphere model, where the interaction energy between sites i and j is given by the arithmetic mean of the i-i and j-j interactions. The geometric mean does not necessarily apply to all mAb mixtures and is expected to break down at a lower ionic strength due to the nonadditivity of electrostatic interactions.

Calculation of Second Virial Coefficients of Atomistic Proteins Using Fast Fourier Transform

The second virial coefficient, B₂, measures a protein solution's deviation from ideal behavior. It is widely used to predict or explain solubility, crystallization condition, aggregation propensity, and critical temperature for liquid-liquid phase separation. B₂ is determined by the interaction energy between two protein molecules and, specifically, by the integration of the Mayer f-function in the relative configurational space (translation and rotation) of the two molecules. Simple theoretical models, such as one attributed to Derjaguin, Landau, Verwey, and Overbeek (DLVO), can fit the dependence of B₂ on salt concentrations. However, model parameters derived often are physically unrealistic and hardly transferable from protein to protein. Previous B₂ calculations incorporating atomistic details were done with limited sampling in the configurational space, due to enormous computational cost. Our FMAP method, based on fast Fourier transform, can considerably accelerate such calculations, and here we adapt it to calculate B₂ values for proteins represented at the atomic level in implicit solvent. After tuning of a single parameter in the energy function, FMAPB2 predicts well the B₂ values for lysozyme and other proteins over wide ranges of solvent conditions (salt concentration, pH, and temperature). The method is available as a web server at http://pipe.rcc.fsu.edu/fmapb2 .

Measuring the effects of macromolecular crowding on antibody function with biolayer interferometry

Biotherapeutic proteins are commonly dosed at high concentrations into the blood, which is an inherently complex, crowded solution with substantial protein content. The effects of macromolecular crowding may lead to an appreciable level of non-specific hetero-association in this physiological environment. Therefore, developing a method to characterize the diverse consequences of non-specific interactions between proteins under such non-ideal, crowded conditions, which deviate substantially from those commonly employed for in vitro characterization, is vital to achieving a more complete picture of antibody function in a biological context. In this study, we investigated non-specific interactions between human serum albumin (HSA) and two monoclonal antibodies (mAbs) by static light scattering and determined these interactions are both ionic strength-dependent and mAb-dependent. Using biolayer interferometry (BLI), we assessed the effect of HSA on antigen binding by mAbs, demonstrating that these non-specific interactions have a functional impact on mAb:antigen interactions, particularly at low ionic strength. While this effect is mitigated at physiological ionic strength, our in vitro data support the notion that HSA in the blood may lead to non-specific interactions with mAbs in vivo, with a potential impact on their interactions with antigen. Furthermore, the BLI method offers a high-throughput advantage compared to orthogonal techniques such as analytical ultracentrifugation and is amenable to a greater variety of solution conditions compared to nuclear magnetic resonance spectroscopy. Our study demonstrates that BLI is a viable technology for examining the impact of non-specific interactions on specific biologically relevant interactions, providing a direct method to assess binding events in crowded conditions.

Controlling Phase Separation of Lysozyme with Polyvalent Anions

The ability of polyvalent anions to influence protein-protein interactions and protein net charge was investigated through solubility and turbidity experiments, determination of osmotic second virial coefficients ( B₂₂), and ζ-potential values for lysozyme solutions.  B₂₂ values showed that all anions reduce protein-protein repulsion between positively charged lysozyme molecules, and those anions with higher net valencies are more effective. The polyvalent anions pyrophosphate and tripolyphosphate were observed to induce protein reentrant condensation, which has been previously observed with negatively charged proteins in the presence of trivalent cations. Reentrant condensation is a phenomenon in which low concentrations of polyvalent ions induce protein precipitation, but further increasing polyvalent ion concentration causes the protein precipitate to resolubilize. Interestingly, citrate does not induce lysozyme reentrant condensation despite having a similar charge, size, and shape to pyrophosphate. We observe qualitative differences in protein behavior when compared against negatively charged proteins in solutions of trivalent cations. The polyphosphate ions induce a much stronger protein-protein attraction, which correlates with the occurrence of a liquid-gel transition that replaces the liquid-liquid transition observed with trivalent cations. The results indicate that solutions of polyphosphate ions provide a model system for exploring the link between the protein-phase diagram and model interaction potentials and also highlight the importance that ion-specific effects can have on protein solubility.

Viscosity Control of Protein Solution by Small Solutes: A Review

Viscosity of protein solution is one of the most troublesome issues for the high-concentration formulation of protein drugs. In this review, we summarize the practical methods that suppress the viscosity of protein solution using small molecular additives. The small amount of salts decreases the viscosity that results from electrostatic repulsion and attraction. The chaotrope suppresses the hydrophobic attraction and cluster formation, which can lower the solution viscosity. Arginine hydrochloride (ArgHCl) also suppresses the solution viscosity due to the hydrophobic and aromatic interactions between protein molecules. The small molecular additives are the simplest resolution of the high viscosity of protein solution as well as understanding of the primary cause in complex phenomena of protein interactions.

Charge Shielding Prevents Aggregation of Supercharged GFP Variants at High Protein Concentration

Understanding protein stability is central to combatting protein aggregation diseases and developing new protein therapeutics. At the high concentrations often present in biological systems, purified proteins can exhibit undesirable high solution viscosities and poor solubilities mediated by short-range electrostatic and hydrophobic protein-protein interactions. The interplay between protein amino acid sequence, protein structure, and solvent conditions to minimize protein-protein interactions is key to designing well-behaved pharmaceutical proteins. However, theoretical approaches have yet to yield a general framework to address these problems. Here, we analyzed the high concentration behavior of superfolder GFP (sfGFP) and two supercharged sfGFP variants engineered to have formal charges of -18 or +15. Under low cosolute conditions, sfGFP and the -18 variant formed a gel or phase separated at ∼10 mg/mL. Under conditions that screen surface charges, including formulations with high histidine or high NaCl concentrations, all three variants attained concentrations up to 250 mg/mL with moderate viscosities. Moreover, all three variants exhibited very similar viscosity-concentration profiles over this range. This effect was not mimicked by high sugar concentrations that exert excluded-volume effects without shielding charge. Collectively, these data demonstrate that charge shielding neutralizes not only long-range electrostatic interactions but also, surprisingly, short-range electrostatic effects due to surface charge anisotropy. This work shows that supercharged sfGFP behavior under high ionic strength is largely determined by particle geometry, a conclusion that is supported by colloid models and may be applicable to pharmaceutically relevant proteins.

The "Sticky Patch" Model of Crystallization and Modification of Proteins for Enhanced Crystallizability

Crystallization of macromolecules has long been perceived as a stochastic process, which cannot be predicted or controlled. This is consistent with another popular notion that the interactions of molecules within the crystal, i.e., crystal contacts, are essentially random and devoid of specific physicochemical features. In contrast, functionally relevant surfaces, such as oligomerization interfaces and specific protein-protein interaction sites, are under evolutionary pressures so their amino acid composition, structure, and topology are distinct. However, current theoretical and experimental studies are significantly changing our understanding of the nature of crystallization. The increasingly popular "sticky patch" model, derived from soft matter physics, describes crystallization as a process driven by interactions between select, specific surface patches, with properties thermodynamically favorable for cohesive interactions. Independent support for this model comes from various sources including structural studies and bioinformatics. Proteins that are recalcitrant to crystallization can be modified for enhanced crystallizability through chemical or mutational modification of their surface to effectively engineer "sticky patches" which would drive crystallization. Here, we discuss the current state of knowledge of the relationship between the microscopic properties of the target macromolecule and its crystallizability, focusing on the "sticky patch" model. We discuss state-of-the-art in silico methods that evaluate the propensity of a given target protein to form crystals based on these relationships, with the objective to design variants with modified molecular surface properties and enhanced crystallization propensity. We illustrate this discussion with specific cases where these approaches allowed to generate crystals suitable for structural analysis.

Critical examination of the colloidal particle model of globular proteins

Recent studies of globular protein solutions have uniformly adopted a colloidal view of proteins as particles, a perspective that neglects the polymeric primary structure of these biological macromolecules, their intrinsic flexibility, and their ability to sample a large configurational space. While the colloidal perspective often serves as a useful idealization in many cases, the macromolecular identity of proteins must reveal itself under thermodynamic conditions in which the native state is no longer stable, such as denaturing solvents and high protein concentrations where macromolecules tend to have screened excluded volume, charge, and hydrodynamic interactions. Under extreme pH conditions, charge repulsion interactions within the protein chain can overcome the attractive hydrogen-bonding interactions, holding it in its native globular state. Conformational changes can therefore be expected to have great significance on the shear viscosity and other rheological properties of protein solutions. These changes are not envisioned in conventional colloidal protein models and we have initiated an investigation of the scattering and rheological properties of model proteins. We initiate this effort by considering bovine serum albumin because it is a globular protein whose solution properties have also been extensively investigated as a function of pH, temperature, ionic strength, and concentration. As we anticipated, near-ultraviolet circular dichroism measurements and intrinsic viscosity measurements clearly indicate that the bovine serum albumin tertiary structure changes as protein concentration and pH are varied. Our findings point to limited validity of the colloidal protein model and to the need for further consideration and quantification of the effects of conformational changes on protein solution viscosity, protein association, and the phase behavior. Small-angle Neutron Scattering measurements have allowed us to assess how these conformational changes influence protein size, shape, and interprotein interaction strength.

From osmotic second virial coefficient (B22 ) to phase behavior of a monoclonal antibody

Antibodies are complex macromolecules and their phase behavior as well as interactions within different solvents and precipitants are still not understood. To shed some light into the processes on a molecular dimension, the occurring self-interactions between antibody molecules were analyzed by means of the osmotic second virial coefficient (B22 ). The determined B22 follows qualitatively the phenomenological Hofmeister series describing the aggregation probability of antibodies for the various solvent compositions. However, a direct correlation between crystallization probability and B22 in form of a crystallization slot does not seem to be feasible for antibodies since the phase behavior is strongly dependent on their anisotropy. Kinetic parameters have to be taken into account due to the molecular size and complexity of the molecules. This is confirmed by a comparison of experimental data with a theoretical phase diagram. On the other hand the solubility is thermodynamically driven and therefore the B22 could be used to establish a universal solubility line for the monoclonal antibody mAb04c and different solvent compositions by using thermodynamic models.

COFFDROP: A Coarse-Grained Nonbonded Force Field for Proteins Derived from All-Atom Explicit-Solvent Molecular Dynamics Simulations of Amino Acids

We describe the derivation of a set of bonded and nonbonded coarse-grained (CG) potential functions for use in implicit-solvent Brownian dynamics (BD) simulations of proteins derived from all-atom explicit-solvent molecular dynamics (MD) simulations of amino acids. Bonded potential functions were derived from 1 μs MD simulations of each of the 20 canonical amino acids, with histidine modeled in both its protonated and neutral forms; nonbonded potential functions were derived from 1 μs MD simulations of every possible pairing of the amino acids (231 different systems). The angle and dihedral probability distributions and radial distribution functions sampled during MD were used to optimize a set of CG potential functions through use of the iterative Boltzmann inversion (IBI) method. The optimized set of potential functions—which we term COFFDROP (COarse-grained Force Field for Dynamic Representation Of Proteins)—quantitatively reproduced all of the “target” MD distributions. In a first test of the force field, it was used to predict the clustering behavior of concentrated amino acid solutions; the predictions were directly compared with the results of corresponding all-atom explicit-solvent MD simulations and found to be in excellent agreement. In a second test, BD simulations of the small protein villin headpiece were carried out at concentrations that have recently been studied in all-atom explicit-solvent MD simulations by Petrov and Zagrovic (PLoS Comput. Biol.2014, 5, e1003638). The anomalously strong intermolecular interactions seen in the MD study were reproduced in the COFFDROP simulations; a simple scaling of COFFDROP’s nonbonded parameters, however, produced results in better accordance with experiment. Overall, our results suggest that potential functions derived from simulations of pairwise amino acid interactions might be of quite broad applicability, with COFFDROP likely to be especially useful for modeling unfolded or intrinsically disordered proteins.

Weak self-interactions of globular proteins studied by small-angle X-ray scattering and structure-based modeling

We investigate protein-protein interactions in solution by small-angle X-ray scattering (SAXS) and theoretical modeling. The structure factor for solutions of bovine pancreatic trypsin inhibitor (BPTI), myoglobin (Mb), and intestinal fatty acid-binding protein (IFABP) is determined from SAXS measurements at multiple concentrations, from Monte Carlo simulations with a coarse-grained structure-based interaction model, and from analytic approximate solutions of two idealized colloidal interaction models without adjustable parameters. By combining these approaches, we find that the structure factor is essentially determined by hard-core and screened electrostatic interactions. Other soft short-ranged interactions (van der Waals and solvation-related) are either individually insignificant or tend to cancel out. The structure factor is also not significantly affected by charge fluctuations. For Mb and IFABP, with a small net charge and relatively symmetric charge distribution, the structure factor is well described by a hard-sphere model. For BPTI, with a larger net charge, screened electrostatic repulsion is also important, but the asymmetry of the charge distribution reduces the repulsion from that predicted by a charged hard-sphere model with the same net charge. Such charge asymmetry may also amplify the effect of shape asymmetry on the protein-protein potential of mean force.

The role of electrostatics in protein-protein interactions of a monoclonal antibody

Understanding how protein-protein interactions depend on the choice of buffer, salt, ionic strength, and pH is needed to have better control over protein solution behavior. Here, we have characterized the pH and ionic strength dependence of protein-protein interactions in terms of an interaction parameter kD obtained from dynamic light scattering and the osmotic second virial coefficient B22 measured by static light scattering. A simplified protein-protein interaction model based on a Baxter adhesive potential and an electric double layer force is used to separate out the contributions of longer-ranged electrostatic interactions from short-ranged attractive forces. The ionic strength dependence of protein-protein interactions for solutions at pH 6.5 and below can be accurately captured using a Deryaguin-Landau-Verwey-Overbeek (DLVO) potential to describe the double layer forces. In solutions at pH 9, attractive electrostatics occur over the ionic strength range of 5-275 mM. At intermediate pH values (7.25 to 8.5), there is a crossover effect characterized by a nonmonotonic ionic strength dependence of protein-protein interactions, which can be rationalized by the competing effects of long-ranged repulsive double layer forces at low ionic strength and a shorter ranged electrostatic attraction, which dominates above a critical ionic strength. The change of interactions from repulsive to attractive indicates a concomitant change in the angular dependence of protein-protein interaction from isotropic to anisotropic. In the second part of the paper, we show how the Baxter adhesive potential can be used to predict values of kD from fitting to B22 measurements, thus providing a molecular basis for the linear correlation between the two protein-protein interaction parameters.

Long range Debye-Hückel correction for computation of grid-based electrostatic forces between biomacromolecules

Background

Brownian dynamics (BD) simulations can be used to study very large molecular systems, such as models of the intracellular environment, using atomic-detail structures. Such simulations require strategies to contain the computational costs, especially for the computation of interaction forces and energies. A common approach is to compute interaction forces between macromolecules by precomputing their interaction potentials on three-dimensional discretized grids. For long-range interactions, such as electrostatics, grid-based methods are subject to finite size errors. We describe here the implementation of a Debye-Hückel correction to the grid-based electrostatic potential used in the SDA BD simulation software that was applied to simulate solutions of bovine serum albumin and of hen egg white lysozyme.

Results

We found that the inclusion of the long-range electrostatic correction increased the accuracy of both the protein-protein interaction profiles and the protein diffusion coefficients at low ionic strength.

Conclusions

An advantage of this method is the low additional computational cost required to treat long-range electrostatic interactions in large biomacromolecular systems. Moreover, the implementation described here for BD simulations of protein solutions can also be applied in implicit solvent molecular dynamics simulations that make use of gridded interaction potentials.

A simplified representation of anisotropic charge distributions within proteins

Effective coarse-grained representations of protein-protein interaction potentials are vital in the modeling of large scale systems. We develop a method to fit an arbitrary number of effective charges to approximate the electrostatic potential of a protein at a given pH in an ionic solution. We find that the effective charges can reproduce an input potential calculated from a high resolution Poisson-Boltzmann calculation. Since the effective charges used in this model are not constrained to the locations of the original charged groups, the extra degrees of freedom allows us to reproduce the field anisotropy with fewer charges. The fitting procedure uses a number of approximations in the charge magnitudes, initial conditions, and multipoles to speed convergence. The most significant gains are found by fitting the multipole moments of the effective charge potential to the moments of the original field. We show that the Yukawa potential is not only sufficient as a pairwise summation in reproducing the potential, but comes naturally from the linearized expansion of the Poisson-Boltzmann equation. We compute interaction energies and find excellent agreement to the original potential. From the effective charge model we compute the electrostatic contribution to the second virial coefficient.

Toward optimized potential functions for protein-protein interactions in aqueous solutions: osmotic second virial coefficient calculations using the MARTINI coarse-grained force field

Coarse-grained (CG) simulation methods are now widely used to model the structure and dynamics of large biomolecular systems. One important issue for using such methods - especially with regard to using them to model, for example, intracellular environments - is to demonstrate that they can reproduce experimental data on the thermodynamics of protein-protein interactions in aqueous solutions. To examine this issue, we describe here simulations performed using the popular coarse-grained MARTINI force field, aimed at computing the thermodynamics of lysozyme and chymotrypsinogen self-interactions in aqueous solution. Using molecular dynamics simulations to compute potentials of mean force between a pair of protein molecules, we show that the original parameterization of the MARTINI force field is likely to significantly overestimate the strength of protein-protein interactions to the extent that the computed osmotic second virial coefficients are orders of magnitude more negative than experimental estimates. We then show that a simple down-scaling of the van der Waals parameters that describe the interactions between protein pseudo-atoms can bring the simulated thermodynamics into much closer agreement with experiment. Overall, the work shows that it is feasible to test explicit-solvent CG force fields directly against thermodynamic data for proteins in aqueous solutions, and highlights the potential usefulness of osmotic second virial coefficient measurements for fully parameterizing such force fields.

Charge regulation in biomolecular solution

Proteins and other biomolecules contain acidic and basic titratable groups that give rise to intricate charge distributions and control electrostatic interactions. ‘Charge regulation’ concerns how the proton equilibria of these sites are perturbed when approached by alien molecular matter such as other proteins, surfaces and membranes, DNA, polyelectrolytes etc. Importantly, this perturbation generates a charge response that leads to attractive intermolecular interactions that can be conveniently described by a single molecular property – the charge capacitance. The capacitance quantifies molecular charge fluctuations, i.e. it is the variance of the mean charge and is an intrinsic property on par with the net charge and the dipole moment. It directly enters the free energy expression for intermolecular interactions and can be obtained experimentally from the derivative of the titration curve or theoretically from simulations. In this review, we focus on the capacitance concept as a predictive parameter for charge regulation and demonstrate how it can be used to estimate the interaction of a protein with other proteins, polyelectrolytes, membranes as well as with ligands.

Identification of small-molecule binding pockets in the soluble monomeric form of the Aβ42 peptide

The aggregation of intrinsically disordered peptides and proteins is associated with a wide range of highly debilitating neurological and systemic disorders. In this work we explored the potential of a structure-based drug discovery procedure to target one such system, the soluble monomeric form of the Aβ42 peptide. We utilised for this purpose a set of structures of the Aβ42 peptide selected from clusters of conformations within an ensemble generated by molecular dynamics simulations. Using these structures we carried out fragment mapping calculations to identify binding "hot spots" on the monomeric form of the Aβ42 peptide. This procedure provided a set of hot spots with ligand efficiencies comparable to those observed for structured proteins, and clustered into binding pockets. Such binding pockets exhibited a propensity to bind small molecules known to interact with the Aβ42 peptide. Taken together these results provide an initial indication that fragment-based drug discovery may represent a potential therapeutic strategy for diseases associated with the aggregation of intrinsically disordered proteins.

Small-angle neutron scattering study of structure and interaction during salt-induced liquid-liquid phase transition in protein solutions

The liquid-liquid phase transition (LLPT) in aqueous salt solutions of lysozyme protein has been studied by small-angle neutron scattering. Measurements have been carried out on fixed protein concentration with varying salt concentration approaching LLPT. The data are fitted considering protein interaction by the two Yukawa (2Y) potential which combines short-range attraction and long-range repulsion. We show that LLPT arises because of enhancement of non-DLVO (Derjaguin-Landau-Verwey-Overbeek) short-range attraction without any conformational structural change of the protein. The salt concentration required for LLPT as well as corresponding short-range attraction decreases significantly with increase in protein concentration.

Structure based descriptors for the estimation of colloidal interactions and protein aggregation propensities

The control of protein aggregation is an important requirement in the development of bio-pharmaceutical formulations. Here a simple protein model is proposed that was used in molecular dynamics simulations to obtain a quantitative assessment of the relative contributions of proteins' net-charges, dipole-moments, and the size of hydrophobic or charged surface patches to their colloidal interactions. The results demonstrate that the strength of these interactions correlate with net-charge and dipole moment. Variation of both these descriptors within ranges typical for globular proteins have a comparable effect. By comparison no clear trends can be observed upon varying the size of hydrophobic or charged patches while keeping the other parameters constant. The results are discussed in the context of experimental literature data on protein aggregation. They provide a clear guide line for the development of improved algorithms for the prediction of aggregation propensities.

Second Osmotic Virial Coefficients and Aggregation of Monoclonal Antibodies by Static Laser Light Scattering

The second osmotic virial coefficient and the apparent molar mass of two human and one mouse monoclonal antibodies were measured in different aequeous buffer solutions which also contained sodium chloride or ammonium sulfate, respectively, by static laser light scattering in batch mode. The apparent molar mass indicates aggregation. At a constant pH value of 6.5 the sodium chloride concentration was varied from 0 to 2 M and the ammonium sulfate concentration from 0 to 0.8 M, respectively. A 20 mM sodium-phosphate buffer was used for all experiments. Furthermore the pH value was varied without adding additional salt from 4.5 to 10. The results of the salt dependency are in line with the Hofmeister-series. The results of the pH dependency correspond to the net charge of the molecules.

Effects of ionic strength and sugars on the aggregation propensity of monoclonal antibodies: influence of colloidal and conformational stabilities

PURPOSE

To develop a general strategy for optimizing monoclonal antibody (MAb) formulations.

METHODS

Colloidal stabilities of four representative MAbs solutions were assessed based on the second virial coefficient (B 2) at 20°C and 40°C, and net charges at different NaCl concentrations, and/or in the presence of sugars. Conformational stabilities were evaluated from the unfolding temperatures. The aggregation propensities were determined at 40°C and after freeze-thawing. The electrostatic potential of antibody surfaces was simulated for the development of rational formulations.

RESULTS

Similar B 2 values were obtained at 20°C and 40°C, implying little dependence on temperature. B 2 correlated quantitatively with aggregation propensities at 40°C. The net charge partly correlated with colloidal stability. Salts stabilized or destabilized MAbs, depending on repulsive or attractive interactions. Sugars improved the aggregation propensity under freeze-thaw stress through improved conformational stability. Uneven and even distributions of potential surfaces were attributed to attractive and strong repulsive electrostatic interactions.

CONCLUSIONS

Assessment of colloidal stability at the lowest ionic strength is particularly effective for the development of formulations. If necessary, salts are added to enhance the colloidal stability. Sugars further improved aggregation propensities by enhancing conformational stability. These behaviors are rationally predictable according to the surface potentials of MAbs.

Anisotropy of the Coulomb interaction between folded proteins: consequences for mesoscopic aggregation of lysozyme

Toward quantitative description of protein aggregation, we develop a computationally efficient method to evaluate the potential of mean force between two folded protein molecules that allows for complete sampling of their mutual orientation. Our model is valid at moderate ionic strengths and accounts for the actual charge distribution on the surface of the molecules, the dielectric discontinuity at the protein-solvent interface, and the possibility of protonation or deprotonation of surface residues induced by the electric field due to the other protein molecule. We apply the model to the protein lysozyme, whose solutions exhibit both mesoscopic clusters of protein-rich liquid and liquid-liquid separation; the former requires that protein form complexes with typical lifetimes of approximately milliseconds. We find the electrostatic repulsion is typically lower than the prediction of the Derjaguin-Landau-Verwey-Overbeek theory. The Coulomb interaction in the lowest-energy docking configuration is nonrepulsive, despite the high positive charge on the molecules. Typical docking configurations barely involve protonation or deprotonation of surface residues. The obtained potential of mean force between folded lysozyme molecules is consistent with the location of the liquid-liquid coexistence, but produces dimers that are too short-lived for clusters to exist, suggesting lysozyme undergoes conformational changes during cluster formation.

Determination of the dipole moments of RNAse SA wild type and a basic mutant

In this study, we report the effects of acidic to basic residue point mutations (5K) on the dipole moment of RNAse SA at different pHs. Dipole moments were determined by measuring solution capacitance of the wild type (WT) and the 5K mutant with an impedance analyzer. The dipole moments were then (1) compared with theoretically calculated dipole moments, (2) analyzed to determine the effect of the point mutations, and (3) analyzed for their contribution to overall protein-protein interactions (PPI) in solution as quantitated by experimentally derived second virial coefficients. We determined that experimental and calculated dipoles were in reasonable agreement. Differences are likely due to local motions of residue side chains, which are not accounted for by the calculated dipole. We observed that the proteins' dipole moments increase as the pH is shifted further from their isoelectric points and that the wild-type dipole moments were greater than those of the 5K. This is likely due to an increase in the proportion of one charge (either negative or positive) relative to the other. A greater charge disparity corresponded to a larger dipole moment. Finally, the larger dipole moments of the WT resulted in greater attractive overall PPI for that protein as compared to the 5K.

Viscosity of concentrated therapeutic protein compositions

The use of monoclonal antibodies as therapeutic agents has been increasing steadily over the last decade for the treatment of various conditions. There is often a need to deliver a large dose of the protein, so there is a trend toward developing commercially viable liquid formulations of highly concentrated antibodies. Such concentrated solutions are associated with a number of challenges, including optimization of production processes, plus chemical and physical stability of the final product where solution viscosity becomes a critical quality attribute. Assessment of the rheological characteristics of concentrated compositions is essential as are development strategies to reduce the viscosity. This review covers the state-of-the-art rheology measurement techniques, focusing particularly on concentrated protein solutions. Current understanding of the mechanisms leading to high viscosity and control by formulation parameters is discussed.

Brownian dynamics simulation of protein solutions: structural and dynamical properties

The study of solutions of biomacromolecules provides an important basis for understanding the behavior of many fundamental cellular processes, such as protein folding, self-assembly, biochemical reactions, and signal transduction. Here, we describe a Brownian dynamics simulation procedure and its validation for the study of the dynamic and structural properties of protein solutions. In the model used, the proteins are treated as atomically detailed rigid bodies moving in a continuum solvent. The protein-protein interaction forces are described by the sum of electrostatic interaction, electrostatic desolvation, nonpolar desolvation, and soft-core repulsion terms. The linearized Poisson-Boltzmann equation is solved to compute electrostatic terms. Simulations of homogeneous solutions of three different proteins with varying concentrations, pH, and ionic strength were performed. The results were compared to experimental data and theoretical values in terms of long-time self-diffusion coefficients, second virial coefficients, and structure factors. The results agree with the experimental trends and, in many cases, experimental values are reproduced quantitatively. There are no parameters specific to certain protein types in the interaction model, and hence the model should be applicable to the simulation of the behavior of mixtures of macromolecules in cell-like crowded environments.

Diffusion in crowded biological environments: applications of Brownian dynamics

Biochemical reactions in living systems occur in complex, heterogeneous media with total concentrations of macromolecules in the range of 50 - 400 mgml. Molecular species occupy a significant fraction of the immersing medium, up to 40% of volume. Such complex and volume-occupied environments are generally termed 'crowded' and/or 'confined'. In crowded conditions non-specific interactions between macromolecules may hinder diffusion - a major process determining metabolism, transport, and signaling. Also, the crowded media can alter, both qualitatively and quantitatively, the reactions in vivo in comparison with their in vitro counterparts. This review focuses on recent developments in particle-based Brownian dynamics algorithms, their applications to model diffusive transport in crowded systems, and their abilities to reproduce and predict the behavior of macromolecules under in vivo conditions.

Electrostatic chameleons in biological systems

Due to large equilibrium fluctuations of protons at physiological pH, the orthophosphate ion as well as the imidazole group on histidine substantially regulate their charge upon approaching charged interfaces. This implies that these-and comparable-ions function as electostatic "proximity switches" when interacting with lipid membranes, DNA, proteins, etc. Using straightforward statistical thermodynamics as well as mesoscopic computer simulations we quantify the charge regulation mechanism and argue that it is important in a range of biological as well as technical processes.

The stabilities of protein crystals

We describe a model for protein crystallization equilibria. The model includes four terms, (1) protein translational entropy opposes crystallization, (2) proteins are attracted to each other by a nonelectrostatic contact free energy favoring crystallization, (3) proteins in the crystal repel each other but, to a greater extent, attract counterions sequestered in the crystal, which favors crystallization, and (4) the translational entropy of the counterions opposes their sequestration into the crystal, opposing crystallization. We treat the electrostatics using the nonlinear Poisson-Boltzmann equation, and we use unit cell information from native protein crystals to determine the boundary conditions. This model predicts the stabilities of protein crystals as functions of temperature, pH, and salt concentrations, in good agreement with the data of Pusey et al. on tetragonal and orthorhombic crystal forms of lysozyme. The experiments show a weak dependence of crystal solubility on pH. According to the model, this is because the entropic cost to neutralize the crystal is compensated by favorable protein-salt interactions. Experiments also show that adding salt stabilizes the crystal. Cohn's empirical law predicts that the logarithm of solubility should be a linear function of salt. The present theory predicts nonlinearity, in better agreement with the experiments. The model shows that the salting out phenomena is not due to more counterion shielding but to lowered counterion translational entropy. Models of this type may help guide faster and better ways to crystallize proteins.

Diffusion, crowding & protein stability in a dynamic molecular model of the bacterial cytoplasm

A longstanding question in molecular biology is the extent to which the behavior of macromolecules observed in vitro accurately reflects their behavior in vivo. A number of sophisticated experimental techniques now allow the behavior of individual types of macromolecule to be studied directly in vivo; none, however, allow a wide range of molecule types to be observed simultaneously. In order to tackle this issue we have adopted a computational perspective, and, having selected the model prokaryote Escherichia coli as a test system, have assembled an atomically detailed model of its cytoplasmic environment that includes 50 of the most abundant types of macromolecules at experimentally measured concentrations. Brownian dynamics (BD) simulations of the cytoplasm model have been calibrated to reproduce the translational diffusion coefficients of Green Fluorescent Protein (GFP) observed in vivo, and “snapshots” of the simulation trajectories have been used to compute the cytoplasm's effects on the thermodynamics of protein folding, association and aggregation events. The simulation model successfully describes the relative thermodynamic stabilities of proteins measured in E. coli, and shows that effects additional to the commonly cited “crowding” effect must be included in attempts to understand macromolecular behavior in vivo.

The interior of a typical bacterial cell is a highly crowded place in which molecules must jostle and compete with each other in order to carry out their biological functions. The conditions under which such molecules are typically studied in vitro, however, are usually quite different: one or a few different types of molecules are studied as they freely diffuse in a dilute, aqueous solution. There is therefore a significant disconnect between the conditions under which molecules can be most usefully studied and the conditions under which such molecules usually “live”, and developing ways to bridge this gap is likely to be important for properly understanding molecular behavior in vivo. Toward this end, we show in this work that computer simulations can be used to model the interior of bacterial cells at a near atomic level of detail: the rates of diffusion of proteins are matched to known experimental values, and their thermodynamic stabilities are found to be in good agreement with the few measurements that have so far been performed in vivo. While the simulation approach is certainly not free of assumptions, it offers a potentially important complement to experimental techniques and provides a vivid illustration of molecular behavior inside a biological cell that is likely to be of significant educational value.

Long- and short-range electrostatic interactions affect the rheology of highly concentrated antibody solutions

PURPOSE

To explain the differences in protein-protein interactions (PPI) of concentrated versus dilute formulations of a model antibody.

METHODS

High frequency rheological measurements from pH 3.0 to 12.0 quantitated viscoelasticity and PPI at high concentrations. Dynamic light scattering (DLS) characterized PPI in dilute solutions.

RESULTS

For concentrated solutions at low ionic strength, the storage modulus, a viscosity component and a measure of PPI, is highest at the isoelectric point (pH 9.0) and lowest at pH 5.4. This profile flattens at higher ionic strength but not completely, indicating PPI consist of long-range electrostatics and other short-range attractions. At low concentrations, PPI are near zero at pI but become repulsive as the pH is shifted. Higher salt concentrations completely flatten this profile to zero, indicating that these PPI are mainly electrostatic.

CONCLUSIONS

This discrepancy occurs because long-range interactions are significant at low concentrations, whereas both long- and short-range interactions are significant at higher concentrations. Computer modeling was used to calculate antibody properties responsible for long- and short-range interactions, i.e. net charge and dipole moment. Charge-charge interactions are repulsive while dipole-dipole interactions are attractive. Their net effect correlated with the storage modulus profile. However, only charge-charge repulsions correlated with PPI determined by DLS.

On the contributions of diffusion and thermal activation to electron transfer between Phormidium laminosum plastocyanin and cytochrome f: Brownian dynamics simulations with explicit modeling of nonpolar desolvation interactions and electron transfer events

The factors that determine the extent to which diffusion and thermal activation processes govern electron transfer (ET) between proteins are debated. The process of ET between plastocyanin (PC) and cytochrome f (CytF) from the cyanobacterium Phormidium laminosum was initially thought to be diffusion-controlled but later was found to be under activation control (Schlarb-Ridley, B. G.; et al. Biochemistry 2005, 44, 6232). Here we describe Brownian dynamics simulations of the diffusional association of PC and CytF, from which ET rates were computed using a detailed model of ET events that was applied to all of the generated protein configurations. The proteins were modeled as rigid bodies represented in atomic detail. In addition to electrostatic forces, which were modeled as in our previous simulations of protein-protein association, the proteins interacted by a nonpolar desolvation (hydrophobic) force whose derivation is described here. The simulations yielded close to realistic residence times of transient protein-protein encounter complexes of up to tens of microseconds. The activation barrier for individual ET events derived from the simulations was positive. Whereas the electrostatic interactions between P. laminosum PC and CytF are weak, simulations for a second cyanobacterial PC-CytF pair, that from Nostoc sp. PCC 7119, revealed ET rates influenced by stronger electrostatic interactions. In both cases, the simulations imply significant contributions to ET from both diffusion and thermal activation processes.

Association and electrostatic steering of alpha-lactalbumin-lysozyme heterodimers

The salt and pH dependent association of hen egg white lysozyme with alpha-lactalbumin whey proteins has been studied using molecular level Monte Carlo simulations. A highly uneven charge distribution of alpha-lactalbumin leads to strongly ordered heterodimers that may facilitate the formation of structured, mesoscopic aggregates. This electrostatic steering gives rise to 80% alignment at 5 mM 1 : 1 salt which, due to screening, diminishes to 60% at 100 mM salt. The free energy of interaction minima, dominated by electrostatics, ranges between -9 kT at 1 mM salt to -2 kT at 100 mM (neutral pH). Calculated osmotic second virial cross coefficients indicate complexation in the pH interval 6-10. Multivalent ions are found to effectively destabilize the protein complex and, at constant ionic strength, the order is La(3+) > Ca(2+) > Mg(2+) > Na(+). Upon binding of calcium to alpha-lactalbumin both the interaction and orientational alignment with lysozyme are reduced due to induced changes in the whey protein charge distribution. This potentially explains the experimentally observed absence of supramolecular structuring for the calcium loaded holo alpha-lactalbumin. Where available, good agreement is found with experimental data.