Comparative Analysis of Protein Surface Hydrophobicity Maps Determined by Sparse Sampling INDUS and Spatial Aggregation Propensity.

Protein surface hydrophobicity plays a central role in various biological processes such as protein folding and aggregation, as well as in the design and manufacturing of biotherapeutics. While the hydrophobicity of protein surface patches has been linked to their constituent residue hydropathies, recent research has shown that protein surface hydrophobicity is more complex and characterized by the response of water to these surfaces. In this work, we employ water density perturbations to map the surface hydrophobicity of a set of model proteins using sparse indirect umbrella sampling simulations (SSI). This technique is used to identify hydrophobic surface patches for the set of model proteins, and the results are compared to those obtained from the widely adopted spatial aggregation propensity (SAP) technique. While SAP-based calculations show agreement with SSI in some cases, there are several examples of disagreement. We identify four general classes of difference in behavior and study factors that contribute to these differences. We find that the SAP method can sometimes mask the effect of weakly nonpolar or isolated nonpolar residues that can lead to strong hydrophobic patches on the protein surface. In addition, hydrophobic patches identified by SAP can exhibit shifts in both position and strength on the SSI map. Our results demonstrate that the combination of topography and chemical context controls the hydrophobicity of a given patch above and beyond the intrinsic polarity of the residues present on the patch surface. The availability of more accurate protein hydrophobicity maps in concert with new classes of hydrophobic molecular descriptors may create significant opportunities for in silico prediction of protein behavior for a range of applications, such as protein design, biomanufacturability, and downstream bioprocessing.

Connecting Non-Gaussian Water Density Fluctuations to the Lengthscale Dependent Crossover in Hydrophobic Hydration.

We connect density fluctuations in liquid water to lengthscale dependent crossover in hydrophobic hydration. Specifically, we employ indirect umbrella sampling (INDUS) simulations to characterize density fluctuations in observation volumes of various sizes and shapes in water and as a function of temperature and salt concentration. Consistent with previous observations, density fluctuations are Gaussian in small molecular scale volumes, but they display non-Gaussian "low-density fat tails" in larger volumes. These non-Gaussian tails are indicative of the proximity of water to its liquid to vapor phase transition and have implications on biomolecular interactions and function. We show that the onset of non-Gaussian fluctuations in large volumes is accompanied by the formation of a cavity in the observation volume. We develop a model that uses the physics of cavity-water interface formation as a key ingredient and show that it captures the nature of non-Gaussian density fluctuations over a broad region in water and in salt solutions. We discuss the limitations of this model in the very low density region of the distribution. Our calculations provide new insights into the origins of non-Gaussian density fluctuations in water and their connections to lengthscale dependent crossover in hydrophobic hydration.

Bridging Gaussian Density Fluctuations from Microscopic to Macroscopic Volumes: Applications to Non-Polar Solute Hydration Thermodynamics.

The hydration of hydrophobic solutes is intimately related to the spontaneous formation of cavities in water through ambient density fluctuations. Information theory-based modeling and simulations have shown that water density fluctuations in small volumes are approximately Gaussian. For limiting cases of microscopic and macroscopic volumes, water density fluctuations are known exactly and are rigorously related to the density and isothermal compressibility of water. Here, we develop a theory-interpolated gaussian fluctuation theory (IGFT)-that builds an analytical bridge to describe water density fluctuations from microscopic to molecular scales. This theory requires no detailed information about the water structure beyond the effective size of a water molecule and quantities that are readily obtained from water's equation-of-state-namely, the density and compressibility. Using simulations, we show that IGFT provides a good description of density fluctuations near the mean, that is, it characterizes the variance of occupancy fluctuations over all solute sizes. Moreover, when combined with the information theory, IGFT reproduces the well-known signatures of hydrophobic hydration, such as entropy convergence and solubility minima, for atomic-scale solutes smaller than the crossover length scale beyond which the Gaussian assumption breaks down. We further show that near hydrophobic and hydrophilic self-assembled monolayer surfaces in contact with water, the normalized solvent density fluctuations within observation volumes depend similarly on size as observed in the bulk, suggesting the feasibility of a modified version of IGFT for interfacial systems. Our work highlights the utility of a density fluctuation-based approach toward understanding and quantifying the solvation of non-polar solutes in water and the forces that drive them toward surfaces with different hydrophobicities.

The Role of Ligand-Ligand Interactions in Multimodal Ligand Conformational Equilibria and Surface Pattern Formation.

Multimodal chromatography uses multiple modes of interaction such as charge, hydrophobic, or hydrogen bonding to separate proteins. Recently, we used molecular dynamics (MD) simulations to show that ligands immobilized on surfaces can interact and associate with neighboring ligands to form hydrophobic and charge patches, which may have important implications for the nature of protein-surface interactions. Here, we study interfacial systems of increasing complexity-from a single immobilized multimodal ligand to high density surfaces-to better understand how ligand behavior is affected by the presence of a surface and the presence of other ligands in the vicinity, and how this behavior scales to larger systems. We find that tethering a ligand to a surface restricts its conformations to a subset of those observed in free solution, yet the ligand maintains flexibility in the plane of the surface and can form contacts with neighboring ligands. We find that although the formation of a contact between two neighboring ligands is slightly unfavorable, three neighboring ligands exhibit a preference for the formation of a fully connected cluster. To explore how these trends in ligand association extend to a larger surface with high density of ligands, we performed coarse-grained Monte Carlo (MC) simulations of a 132-ligand surface using ligand interactions parametrized based on free energies obtained from the three-ligand MD simulations. Despite their simplicity, the coarse-grained simulations qualitatively capture the cluster size distribution of ligands observed in detailed MD simulations. Quantitative differences between the two suggest opportunities for improvements in the coarse-grained energy function for efficient predictions of cluster and pattern formations. Our approach presents a promising route to the engineering of multimodal patterns for future chromatographic resin design.

Formation of Ligand Clusters on Multimodal Chromatographic Surfaces.

Multimodal chromatography is a powerful tool which uses multiple modes of interaction, such as charge and hydrophobicity, to purify protein-based therapeutics. In this work, we performed molecular dynamics simulations of a series of multimodal cation-exchange ligands immobilized on a hydrophilic self-assembled monolayer surface at the commercially relevant surface density (1 ligand/nm2). We found that ligands that were flexible and terminated in a hydrophobic group had a propensity to aggregate on the surface, while less flexible ligands containing a hydrophobic group closer to the surface did not aggregate. For aggregating ligands, this resulted in the formation of a surface pattern that contained relatively large patches of hydrophobicity and charge whose sizes exceeded the length scale of the individual ligands. On the other hand, lowering the surface density to 1 ligand/3 nm2 reduced or eliminated this aggregation behavior. In addition, the introduction of a flexible linker (corresponding to the commercially available ligand) enhanced cluster formation and allowed aggregation to occur at lower surface densities. Further, the use of flexible linkers enabled hydrophobic groups to collapse to the surface, reducing their accessibility. Finally, we developed an approach for quantifying differences in the observed surface patterns by calculating distributions of the patch size and patch length. This clustering phenomenon is likely to play a key role in governing protein-surface interactions in multimodal chromatography. This new understanding of multimodal surfaces has important implications for developing improved predictive models and designing new classes of multimodal separation materials.

Conformational Equilibria of Multimodal Chromatography Ligands in Water and Bound to Protein Surfaces.

Multimodal chromatography uses small ligands with multiple modes of interaction, e.g., charged, hydrophobic or hydrogen bonding, to separate proteins from complex mixtures. The mechanism by which multimodal ligands interact with proteins is expected to be affected by ligand conformations, among other factors. Here, we study conformational equilibria of two commercially used multimodal cation exchange ligands, Capto MMC and Nuvia cPrime, in a range of solvents, a Lennard-Jones (LJ) liquid, ethanol, and water, using molecular dynamics (MD) simulations. By mapping ligand conformations onto two key torsion angles, ω and φ, in these solvents and in low and high dielectric media, we quantify the relative importance of intramolecular and solvent-mediated interactions. In a high dielectric medium, Capto MMC preferentially samples three conformations, which are stabilized by a combination of an intramolecular torsion potential (on ω) and LJ interactions. In an LJ liquid, solvent molecules compete with intramolecular interactions while simultaneously providing an osmotic force, stabilizing both closer and farther distances between ligand sites. This has the overall effect of "flattening out" the conformational landscape. Interestingly, in ethanol and water, hydrogen bonding between the amide hydrogen and solvent molecules stabilizes two additional conformations of Capto MMC in which ω takes on less favorable cis-like configurations. MD simulations of ligands in free solution with three therapeutic antibody fragments show that ligand conformational equilibria remain effectively unchanged upon binding to proteins. Although, there is 20-30% dehydration of the overall ligand upon binding, the hydrogen-bonding sites are dehydrated to a much smaller extent, particularly in cis-like configurations. Conformational preferences of Nuvia cPrime are similar to that of Capto MMC, except for the effect of symmetry arising from the absence of an alkyl thiol tail. Characterizing the conformational equilibria of these two ligands in free solution and bound to a protein provides a foundation for developing a mechanistic understanding of protein-multimodal ligand interactions.

Hydrophobicity of proteins and nanostructured solutes is governed by topographical and chemical context.

Hydrophobic interactions drive many important biomolecular self-assembly phenomena. However, characterizing hydrophobicity at the nanoscale has remained a challenge due to its nontrivial dependence on the chemistry and topography of biomolecular surfaces. Here we use molecular simulations coupled with enhanced sampling methods to systematically displace water molecules from the hydration shells of nanostructured solutes and calculate the free energetics of interfacial water density fluctuations, which quantify the extent of solute-water adhesion, and therefore solute hydrophobicity. In particular, we characterize the hydrophobicity of curved graphene sheets, self-assembled monolayers (SAMs) with chemical patterns, and mutants of the protein hydrophobin-II. We find that water density fluctuations are enhanced near concave nonpolar surfaces compared with those near flat or convex ones, suggesting that concave surfaces are more hydrophobic. We also find that patterned SAMs and protein mutants, having the same number of nonpolar and polar sites but different geometrical arrangements, can display significantly different strengths of adhesion with water. Specifically, hydroxyl groups reduce the hydrophobicity of methyl-terminated SAMs most effectively not when they are clustered together but when they are separated by one methyl group. Hydrophobin-II mutants show that a charged amino acid reduces the hydrophobicity of a large nonpolar patch when placed at its center, rather than at its edge. Our results highlight the power of water density fluctuations-based measures to characterize the hydrophobicity of nanoscale surfaces and caution against the use of additive approximations, such as the commonly used surface area models or hydropathy scales for characterizing biomolecular hydrophobicity and the associated driving forces of assembly.

Single Molecule Force Spectroscopy and Molecular Dynamics Simulations as a Combined Platform for Probing Protein Face-Specific Binding.

Biomolecular interactions frequently occur in orientation-specific manner. For example, prior nuclear magnetic resonance spectroscopy experiments in our lab have suggested the presence of a group of strongly binding residues on a particular face of the protein ubiquitin for interactions with Capto MMC multimodal ligands ("Capto" ligands) (Srinivasan, K.; et al. Langmuir 2014, 30 (44), 13205-13216). We present a clear confirmation of those studies by performing single-molecule force spectroscopy (SMFS) measurements of unbinding complemented with molecular dynamics (MD) calculations of the adsorption free energy of ubiquitin in two distinct orientations with self-assembled monolayers (SAMs) functionalized with "Capto" ligands. These orientations were maintained in the SMFS experiments by tethering ubiquitin mutants to SAM surfaces through strategically located cysteines, thus exposing the desired faces of the protein. Analogous orientations were maintained in MD simulations using suitable constraining methods. Remarkably, despite differences between the finer details of experimental and simulation methodologies, they confirm a clear preference for the previously hypothesized binding face of ubiquitin. Furthermore, MD simulations provided significant insights into the mechanism of protein binding onto this multimodal surface. Because SMFS and MD simulations both directly probe protein-surface interactions, this work establishes a key link between experiments and simulations at molecular scale through the determination of protein face-specific binding energetics. Our approach may have direct applications in biophysical systems where face- or orientation-specific interactions are important, such as biomaterials, sensors, and biomanufacturing.

Arginine mutations in antibody complementarity-determining regions display context-dependent affinity/specificity trade-offs.

Antibodies commonly accumulate charged mutations in their complementarity-determining regions (CDRs) during affinity maturation to enhance electrostatic interactions. However, charged mutations can mediate non-specific interactions, and it is unclear to what extent CDRs can accumulate charged residues to increase antibody affinity without compromising specificity. This is especially concerning for positively charged CDR mutations that are linked to antibody polyspecificity. To better understand antibody affinity/specificity trade-offs, we have selected single-chain antibody fragments specific for the negatively charged and hydrophobic Alzheimer's amyloid ß peptide using weak and stringent selections for antibody specificity. Antibody variants isolated using weak selections for specificity were enriched in arginine CDR mutations and displayed low specificity. Alanine-scanning mutagenesis revealed that the affinities of these antibodies were strongly dependent on their arginine mutations. Antibody variants isolated using stringent selections for specificity were also enriched in arginine CDR mutations, but these antibodies possessed significant improvements in specificity. Importantly, the affinities of the most specific antibodies were much less dependent on their arginine mutations, suggesting that over-reliance on arginine for affinity leads to reduced specificity. Structural modeling and molecular simulations reveal unique hydrophobic environments near the arginine CDR mutations. The more specific antibodies contained arginine mutations in the most hydrophobic portions of the CDRs, whereas the less specific antibodies contained arginine mutations in more hydrophilic regions. These findings demonstrate that arginine mutations in antibody CDRs display context-dependent impacts on specificity and that affinity/specificity trade-offs are governed by the relative contribution of arginine CDR residues to the overall antibody affinity.

Efficient affinity maturation of antibody variable domains requires co-selection of compensatory mutations to maintain thermodynamic stability.

The ability of antibodies to accumulate affinity-enhancing mutations in their complementarity-determining regions (CDRs) without compromising thermodynamic stability is critical to their natural function. However, it is unclear if affinity mutations in the hypervariable CDRs generally impact antibody stability and to what extent additional compensatory mutations are required to maintain stability during affinity maturation. Here we have experimentally and computationally evaluated the functional contributions of mutations acquired by a human variable (VH) domain that was evolved using strong selections for enhanced stability and affinity for the Alzheimer's Aß42 peptide. Interestingly, half of the key affinity mutations in the CDRs were destabilizing. Moreover, the destabilizing effects of these mutations were compensated for by a subset of the affinity mutations that were also stabilizing. Our findings demonstrate that the accumulation of both affinity and stability mutations is necessary to maintain thermodynamic stability during extensive mutagenesis and affinity maturation in vitro, which is similar to findings for natural antibodies that are subjected to somatic hypermutation in vivo. These findings for diverse antibodies and antibody fragments specific for unrelated antigens suggest that the formation of the antigen-binding site is generally a destabilizing process and that co-enrichment for compensatory mutations is critical for maintaining thermodynamic stability.

Effect of guanidine and arginine on protein-ligand interactions in multimodal cation-exchange chromatography.

The addition of fluid phase modifiers provides significant opportunities for increasing the selectivity of multimodal chromatography. In order to optimize this selectivity, it is important to understand the fundamental interactions between proteins and these modifiers. To this end, molecular dynamics (MD) simulations were first performed to study the interactions of guanidine and arginine with three proteins. The simulation results showed that both guanidine and arginine interacted primarily with the negatively charged regions on the proteins and that these regions could be readily predicted using electrostatic potential maps. Protein surface characterization was then carried out using computationally efficient coarse-grained techniques for a broader set of proteins which exhibited interesting chromatographic retention behavior upon the addition of these modifiers. It was shown that proteins exhibiting an increased retention in the presence of guanidine possessed hydrophobic regions adjacent to negatively charged regions on their surfaces. In contrast, proteins which exhibited a decreased binding in the presence of guanidine did not have hydrophobic regions adjacent to negatively charged patches. These results indicated that the effect of guanidine could be described as a combination of competitive binding, charge neutralization and increased hydrophobic interactions for certain proteins. In contrast, arginine resulted in a significant decrease in protein retention times primarily due to competition for the resin and steric effects, with minimal accompanying increase in hydrophobic interactions. The approach presented in this paper which employs MD simulations to guide the application of coarse-grained approaches is expected to be extremely useful for methods development in downstream bioprocesses. © 2017 American Institute of Chemical Engineers Biotechnol. Prog., 33:435-447, 2017.

Understanding n-Octane Behavior near Graphene with Scaled Solvent-Solute Attractions.

We employ molecular dynamics simulations of n-octane near a layered graphene surface to study the related phenomena of solvation, density fluctuations, wettability, and structure and dynamics of n-octane molecules in the inhomogeneous interfacial environment. That solvation in bulk n-octane displays a lengthscale-dependent crossover similar to that of hydrophobic solvation in water is known. Here we show that, near an extended graphene interface having attractive interactions with n-octane, lengthscale-dependent solvation is similar to that in the bulk and displays a small to large crossover. However, as the n-octane-graphene interactions are reduced to make the surface increasingly solvophobic, the crossover behavior is modulated and essentially absent near the most solvophobic surfaces, similar to that in water near hydrophobic interfaces. We show that the macroscopic measure of wettability, namely, the contact angle, characterizes n-octane-graphene coupling over a limited range of attractions. In contrast, molecular measures such as the free energy of cavity formation or the local compressibility in the interfacial region provide an effective measure of this coupling over a broader range of attractions. Finally, as n-octane-graphene attractions are increased, the n-octane liquid displays a wetting transition and corresponding change from sigmoidal to layered density profile. Analysis of the local structure shows that n-octane molecules prefer approximately linear conformations and surface-parallel orientations near the graphene surface, and their translational dynamics slow down with increasing n-octane-graphene attractions. Our study highlights molecular scale behavior of n-octane molecules that is relevant to understanding nanoparticle-solvent coupling in composite materials with enhanced mechanical or thermal properties.

Interactions of Multimodal Ligands with Proteins: Insights into Selectivity Using Molecular Dynamics Simulations.

Fundamental understanding of protein-ligand interactions is important to the development of efficient bioseparations in multimodal chromatography. Here we employ molecular dynamics (MD) simulations to investigate the interactions of three different proteins--ubiquitin, cytochrome C, and α-chymotrypsinogen A, sampling a range of charge from +1e to +9e--with two multimodal chromatographic ligands containing similar chemical moieties--aromatic, carboxyl, and amide--in different structural arrangements. We use a spherical harmonic expansion to analyze ligand and individual moiety density profiles around the proteins. We find that the Capto MMC ligand, which contains an additional aliphatic group, displays stronger interactions than Nuvia CPrime ligand with all three proteins. Studying the ligand densities at the moiety level suggests that hydrophobic interactions play a major role in determining the locations of high ligand densities. Finally, the greater structural flexibility of the Capto MMC ligand compared to that of the Nuvia cPrime ligand allows for stronger structural complementarity and enables stronger hydrophobic interactions. These subtle and not-so-subtle differences in binding affinities and modalities for multimodal ligands can result in significantly different binding behavior towards proteins with important implications for bioprocessing.

Pathways to dewetting in hydrophobic confinement.

Liquid water can become metastable with respect to its vapor in hydrophobic confinement. The resulting dewetting transitions are often impeded by large kinetic barriers. According to macroscopic theory, such barriers arise from the free energy required to nucleate a critical vapor tube that spans the region between two hydrophobic surfaces--tubes with smaller radii collapse, whereas larger ones grow to dry the entire confined region. Using extensive molecular simulations of water between two nanoscopic hydrophobic surfaces, in conjunction with advanced sampling techniques, here we show that for intersurface separations that thermodynamically favor dewetting, the barrier to dewetting does not correspond to the formation of a (classical) critical vapor tube. Instead, it corresponds to an abrupt transition from an isolated cavity adjacent to one of the confining surfaces to a gap-spanning vapor tube that is already larger than the critical vapor tube anticipated by macroscopic theory. Correspondingly, the barrier to dewetting is also smaller than the classical expectation. We show that the peculiar nature of water density fluctuations adjacent to extended hydrophobic surfaces--namely, the enhanced likelihood of observing low-density fluctuations relative to Gaussian statistics--facilitates this nonclassical behavior. By stabilizing isolated cavities relative to vapor tubes, enhanced water density fluctuations thus stabilize novel pathways, which circumvent the classical barriers and offer diminished resistance to dewetting. Our results thus suggest a key role for fluctuations in speeding up the kinetics of numerous phenomena ranging from Cassie-Wenzel transitions on superhydrophobic surfaces, to hydrophobically driven biomolecular folding and assembly.

Role of arginine in mediating protein-carbon nanotube interactions.

Arginine-rich proteins (e.g., lysozyme) or poly-L-arginine peptides have been suggested as solvating and dispersing agents for single-wall carbon nanotubes (CNTs) in water. In addition, protein structure-function in porous and hydrophobic materials is of broad interest. The amino acid residue, arginine (Arg(+)), has been implicated as an important mediator of protein/peptide-CNT interactions. To understand the structural and thermodynamic aspects of this interaction at the molecular level, we employ molecular dynamics (MD) simulations of the protein lysozyme in the interior of a CNT, as well as of free solutions of Arg(+) in the presence of a CNT. To dissect the Arg(+)-CNT interaction further, we also perform simulations of aqueous solutions of the guanidinium ion (Gdm(+)) and the norvaline (Nva) residue in the presence of a CNT. We show that the interactions of lysozyme with the CNT are mediated by the surface Arg(+) residues. The strong interaction of Arg(+) residue with the CNT is primarily driven by the favorable interactions of the Gdm(+) group with the CNT wall. The Gdm(+) group is not as well-hydrated on its flat sides, which binds to the CNT wall. This is consistent with a similar binding of Gdm(+) ions to a hydrophobic polymer. In contrast, the Nva residue, which lacks the Gdm(+) group, binds to the CNT weakly. We present details of the free energy of binding, molecular structure, and dynamics of these solutes on the CNT surface. Our results highlight the important role of Arg(+) residues in protein-CNT or protein-carbon-based material interactions. Such interactions could be manipulated precisely through protein engineering, thereby offering control over protein orientation and structure on CNTs, graphene, or other hydrophobic interfaces.

Lengthscale-Dependent Solvation and Density Fluctuations in n-Octane.

Much attention has been focused on the solvation and density fluctuations in water over the past decade. These studies have brought to light interesting physical features of solvation in condensed media, especially the dependence of solvation on the solute lengthscale, which may be general to many fluids. Here, we focus on the lengthscale-dependent solvation and density fluctuations in n-octane, a simple organic liquid. Using extensive molecular simulations, we show a crossover in the solvation of solvophobic solutes with increasing size in n-octane, with the specifics of the crossover depending on the shape of the solute. Large lengthscale solvation, which is dominated by interface formation, emerges over subnanoscopic lengthscales. The crossover in n-octane occurs at smaller lengthscales than that in water. We connect the lengthscale of crossover to the range of attractive interactions in the fluid. The onset of the crossover is accompanied by the emergence of non-Gaussian tails in density fluctuations in solute shaped observation volumes. Simulations over a range of temperatures highlight a corresponding thermodynamic crossover in solvation. Qualitative similarities between lengthscale-dependent solvation in water, n-octane, and Lennard-Jones fluids highlight the generality of the underlying physics of solvation.

Binding, structure, and dynamics of hydrophobic polymers near patterned self-assembled monolayer surfaces.

We use molecular dynamics simulations to study the binding, conformations, and dynamics of a flexible 25-mer hydrophobic polymer near well-defined patterned self-assembled monolayers containing a hydrophobic strip (with -CH3 head-groups) having different widths in a hydrophilic (-OH) background. We show that the polymer binds favorably to hydrophobic strips of all widths, including the subnanometer ones comprising 3, 2, or even 1 row of -CH3 head-groups, with the binding strength varying from about 107 to 25 kJ/mol for the widest to the narrowest strip. Near wide hydrophobic patches containing 5 or more -CH3 rows, pancakelike conformations are dominant, whereas hairpinlike structures become preferred ones near the narrower strips. In the vicinity of the narrowest 1-row strip, the polymer folds into semiglobular conformations, thus maintaining sufficient contact with the strip while sequestering its hydrophobic groups away from water. We also show that the confinement makes the translational dynamics of the polymer anisotropic as well as conformational dependent. Our results may help to understand and manipulate the self-assembly and dynamics of soft matter, such as polymers, peptides, and proteins, at inhomogeneous patterned surfaces.

Application of a spherical harmonics expansion approach for calculating ligand density distributions around proteins.

Protein-ligand interactions are central to many biological applications, including molecular recognition, protein formulations, and bioseparations. Complex, multisite ligands can have affinities for different locations on a protein's surface, depending on the chemical and topographical complementarity. We employ an approach based on the spherical harmonic expansion to calculate spatially resolved three-dimensional atomic density profiles of water and ligands in the vicinity of macromolecules. To illustrate the approach, we first study the hydration of model C180 buckyball solutes, with nonspherical patterns of hydrophobicity/-philicity on their surface. We extend the approach to calculate density profiles of increasingly complex ligands and their constituent groups around a protein (ubiquitin) in aqueous solution. Analysis of density profiles provides information about the binding face of the protein and the preferred orientations of ligands on the binding surface. Our results highlight that the spherical harmonic expansion based approach is easy to implement and efficient for calculation and visualization of three-dimensional density profiles around spherically nonsymmetric and topographically and chemically complex solutes.