Physics-based surface patch analysis for prediction of hydrophobic contribution to viscosity of mAbs

ABSTRACT

The viscosity of monoclonal antibody solutions is critical in their biopharmaceutical application, as it directly influences the ease of subcutaneous injection. Although many descriptors have been developed to enable the in silico prediction of viscosity, they are typically based on electrostatic properties while neglecting hydrophobicity, or rely on AI-based approaches with limited generalizability, both rendering the models inadequate. Moreover, the scarcity of high-quality experimental datasets further limits the use of machine learning algorithms, necessitating interpretable analysis of protein–protein interactions. In this work, we combine computational modeling with experimental viscosity measurements for a set of monoclonal antibodies. We introduce an algorithm for surface patch analysis capable of quantifying the characteristics of hydrophobic patches. By calculating physically meaningful interaction energies, we can discern between the propensity for high and low viscosity due to the hydrophobic effect. Furthermore, by analyzing antibodies with problematic hydrophobic patches, we introduce a theory explaining their solubilization. This method is adaptable to any protein format and can be generalized for early in silico screening of viscosity in protein-based biopharmaceutical solutions.

Introduction

Protein-based therapeutics, such as monoclonal antibodies (mAbs), have become increasingly prominent in the pharmaceutical industry, offering targeted and effective treatments for a wide range of medical conditions.^1,2 The most used way to administer these drugs is through subcutaneous (SC) injection, a method that allows patients to self-administer the drugs at home, thereby extending access to these treatments.^3,4 This trend has a direct impact on how these drugs are formulated, often requiring high concentrations of protein in solution, resulting in challenges, such as protein aggregation during long-term storage and post-translational modifications that can affect potency. Additionally, viscosity tends to increase significantly at higher concentrations, making injections more difficult.^5–8

Traditional assessment of protein properties is done via wet-lab work. Viscosity is measured using a viscometer, which requires a comparatively large quantity of the protein material, typically not available in the discovery phase, where hundreds of molecules are considered as drug candidates. Relevant viscosity measurements are usually only available in the later stages of drug development, where the number of candidate molecules is already reduced to 10 or less. Viscosity therefore often causes problems later during development, because it cannot be one of the selection parameters early on. This can result in delayed drug commercialization. For these reasons, there is a growing trend in the biologics industry to develop computational methods for predicting protein properties.^9–12 Developing a computational method to predict viscosity based on the protein’s predicted structure would be highly beneficial, allowing for the early selection of viable candidates during the research process.

The deviation of protein solution viscosity from that of a simple colloidal solution stems from intermolecular interactions between protein molecules.^13,14 These interactions include Van der Waals (vdW),¹⁵ electrostatic (ES),¹⁶ and the hydrophobic interactions.^17–19 Since at standard formulation conditions (pH $\approx$ 6.0) monoclonal antibodies carry a net positive charge,²⁰ it would be expected that they would experience primarily repulsive interactions. However, as the solution passes through a syringe during injection, shear stress aligns the proteins with the flow direction.¹⁴ Their anisotropic surfaces, featuring oppositely charged and hydrophobic patches, promote energetically favorable interactions.²¹ These interactions cause the proteins to attract each other, which resists the flow and increases the viscosity of the solution. In addition to this, the shape of proteins also proves to be an important factor as it restricts the orientation of proteins during flow. Proteins with complex shapes permit only specific relative orientations at high concentrations ($\mathit{c}=150$ mg mL⁻¹) due to the excluded volume compared to those with a perfectly spherical shape, affecting molecular crowding effects as well.²² In summary, the contributions to viscosity are distributed approximately evenly across electrostatic, hydrophobic, hydrodynamic/hard-core, and vdW interactions.²³

Numerous descriptors have been developed to study physicochemical properties of proteins. One example, the Spatial Charge Map (SCM),²⁴ is a charge-based descriptor based on the principle that electrostatic forces significantly influence viscosity. It has been applied to classify monoclonal antibodies into high- and low-viscosity categories using an experimentally derived threshold value, assuming that exposed atoms with negative partial charges in the Fv variable region attract the predominantly positive Fc region. While the electrostatic contribution can therefore be quite accurately predicted,²⁴ the hydrophobic contribution remains difficult to quantify. Spatial aggregation propensity (SAP)²⁵ is a descriptor designed to visualize and calculate per-residue surface-exposed hydrophobicity, helping identify regions prone to aggregation. However, SAP has not been used to explain the magnitude of hydrophobic contributions to viscosity.

Various machine learning (ML) and artificial intelligence (AI) approaches have emerged as tools for predicting protein viscosity. These range from simple decision tree algorithms and logistic regression models, which classify viscosity based on net charge and the number of hydrophobic amino acid residues,^26,27 to more complex models like 3D convolutional neural networks (CNNs) incorporating electrostatic potential calculations²⁸ and artificial neural networks (ANNs) that combine experimental and computational data to predict concentration-dependent viscosity.²⁹ Notably, to the best of our knowledge, all these models use training sets based on monoclonal antibodies, which poses limitations on their use for other protein types, i.e. beyond monoclonal antibodies.

In this work, we introduce a novel method for predicting the hydrophobic contribution to viscosity in protein solutions. Using an in-house developed algorithm, we can accurately define and quantify patches of, in principle, any projected physicochemical property on the protein’s surface. By experimentally measuring the viscosities of an internal dataset of nine developable antibodies and analyzing their hydrophobic patches, we investigate how the geometry and interaction strength of hydrophobic patches relate to solution viscosity, enabling the development of a predictive model. Although validated on antibodies, this method is modality-independent and could be applied to any protein format. Finally, this work represents a step toward rational control of viscosity in high-concentration protein solutions, which is crucial for therapeutic administration of biopharmaceuticals. The low computational cost makes this method widely applicable already in very early research stages of biopharmaceutical drug candidates.

Methods

Protein material

Protein solutions were prepared volumetrically in centrifuge tubes for nine different monoclonal antibodies (mAbs) provided by Novartis LLC. Stock solutions for excipients (sucrose, NaCl, NaOH, HCl – EMSURE grade) were prepared volumetrically. The required amount of each excipient was weighed on an analytical balance, transferred into an appropriate volumetric flask, and diluted to the mark with additionally purified water (APW). The stock solutions were homogenized by inverting the volumetric flask ten times. To prepare the final solutions, a concentrated mAb stock solution was first added to each tube. Next, a 1 M sucrose stock solution was introduced to achieve a final concentration of 220 mM. To investigate the effect of electrostatic screening, two solution conditions were prepared for each antibody: one without additional salt and one supplemented with 100 mM NaCl, using a 1 M NaCl stock solution. The pH was adjusted to 6.0 using dilute HCl or NaOH. Finally, APW was added to reach a final protein concentration of 150/. Protein concentration was measured spectrophotometrically after dilution to approximately 1/, using UV absorbance at 280, based on the calculated molar extinction coefficient (${\epsilon }_{280}$). Measurements were performed in triplicates to ensure accuracy.

Experimental methods

Viscosity measurements were performed using a RheoSense VROC (Viscometer-Rheometer-on-a-Chip) instrument, which utilizes microfluidic technology, with a rectangular slit microfluidic chip (2 $\times$ 50 $\times$ 13). The measurements were conducted at 25, a relevant temperature for biopharmaceutical applications, as subcutaneous administration of biologic drugs typically occurs at room temperature. The shear rate was adjusted between 2000 s⁻¹ and 6000 s⁻¹ to maintain a constant pressure within the channel. Prior to each measurement, the instrument’s accuracy was assessed using sucrose standards in the 10–15 cP range, with an estimated measurement error of 5%.

Computational details

The amino acid sequences of all monoclonal antibodies were used to construct the 3D structure of each antibody’s Fv region using IgFold.³⁰ The resulting Fv structures were grafted onto a model IgG1 framework, which served as the initial input for energy minimization. Molecular dynamics (MD) simulations were performed using the GROMACS package³¹ with the OPLS-AA force field.³² Each antibody was placed in a simulation box extending at least 2 beyond the protein in all directions and solvated with explicit SPC/E water molecules.³³ The system pH was set to 6.0 using PROPKA3,³⁴ and NaCl ions were added to maintain charge neutrality. Energy minimization was carried out using the steepest descent algorithm, followed by equilibration through successive NVT and NPT ensemble simulations for a total of 10 ns, ensuring potential energy convergence. The final antibody structure was extracted from the last frame of the simulation. Only an equilibration trajectory (no production run) was performed, as the purpose of the simulation was not to explore long-timescale conformational dynamics but to relax the predicted protein structure models. This equilibration ensures that steric clashes are removed and that the solvent accessible surface reaches a stable conformation for SAP/SCM projection (Figure S1).

Physicochemical properties – surface hydrophobicity and charge – were computed at the atomic level using SAP²⁵ and SCM²⁴ scores, respectively, defined as

$$\mathrm{SA}{\mathrm{P}}_{\mathrm{i}}=\sum _{\begin{array}{c}\mathrm{side}\mathrm{chain}\\ \mathrm{atoms}\mathrm{j}\\ \mathrm{within}{\mathrm{R}}_{\mathrm{H}}=5\text{A}\mathit{A}\\ \mathrm{from}\mathrm{atom}\mathrm{i}\end{array}}\frac{\mathrm{SAS}{\mathrm{A}}_{\mathrm{atom}\mathrm{j}}}{\mathrm{SAS}{\mathrm{A}}_{\genfrac{}{}{0ex}{}{\mathrm{fully}\mathrm{exposed}}{\genfrac{}{}{0ex}{}{\mathrm{residue}\mathrm{containing}}{\mathrm{atom}\mathrm{j}}}}}\cdot \mathrm{H}{\mathrm{P}}_{\mathrm{residue}\mathrm{containing}\mathrm{atom}\mathrm{j}},$$ (1)

$$\mathrm{SC}{\mathrm{M}}_{\mathrm{i}}=\sum _{\begin{array}{c}\mathrm{side}\mathrm{chain}\\ \mathrm{atoms}\mathrm{j}\\ \\ \mathrm{within}{\mathrm{R}}_{\mathrm{C}}=10\mathit{A}\\ \mathrm{from}\mathrm{atom}\mathrm{i}\end{array}}{q}_{\mathrm{j}},$$ (2)

where HP refers to the per-residue hydrophobicity, and $\mathit{q}$ denotes the partial atomic charge. Hydrophobicity values were assigned based on the experimentally derived Wimley-White hydrophobicity scale,³⁵ chosen due to the consistently high correlation with experimental hydrophobicity measurements in diverse datasets.³⁶ It was normalized such that the most hydrophobic amino acid, tryptophan, is assigned $\mathrm{HP}=1$, ensuring consistency with the interaction potential definition defined later on. Partial charge was determined using the force field parameters from the molecular dynamics (MD) simulation setup, as detailed earlier.

Solvent-accessible surface area (SASA) was calculated using our custom script that distributes points evenly on the surface via the Fibonacci sphere method. Solvent-accessible points were identified based on the Shrake-Rupley “rolling probe” algorithm.³⁷ Fully exposed residue SASA values were assigned according to the scale proposed by Tien et al.³⁸

Surface patch identification algorithm

Our custom developed surface patch identification process is detailed in Figure 1. The algorithm begins by defining a fine-grained equidistantial mesh of points on the solvent excluded surface. For each of these points we calculate a descriptor – a smoothed projection of an arbitrary per-atom physicochemical property. Next, the surface points are sequentially evaluated against a predefined threshold value for the descriptor of interest. When a point exceeds this threshold, a new patch is initiated at that point, while other points are eliminated from the set. Subsequently, we iterate through neighboring points within a specified distance cutoff, applying the same evaluation criteria to either expand the patch or eliminate points. This process continues until no further significant points are identified, at which point the patch is finalized. The algorithm then proceeds to identify additional patches by repeating this process on the remaining unassigned points until all potential patches have been detected.

Figure 1.

Identifying and defining the geometry of hydrophobic patches. (A) Workflow of the surface patch identification algorithm. (B) Example of a descriptor on the protein surface with arbitrary values, and (C) the corresponding defined patch. Each point within the patch is assigned a uniform value equal to that of the patch itself. (D) Principal component analysis (PCA) is used to determine the best-fit plane (green), passing through the patch’s highest point along the surface normal, and to define (E) the effective interaction area (yellow, side view). Distances between surface points and their projections onto the effective interaction area are shown, reduced for clarity.

Surface point generation and triangulation were performed using MSMS software,³⁹ with a water probe radius of 1.4Å and a vertex density of 3vertex/Ų. For each surface point, coordinates, normal vector components, and nearest neighbor information were determined through triangulation. SAP and SCM scores for atoms within 3Å and 5Å, respectively, were then projected onto the surface using a weighting function, $\mathit{w}(\mathit{x})=1/(1+\mathit{x})$, where $\mathit{x}$ represents the distance from the atom’s center to the surface point.

Different thresholds for the significant SAP value per surface point (0.005 kcal − 0.015 kcal) and cutoff distances (1 Å − 5 Å) were evaluated (Fig. S2–S4). The optimal parameters were selected based on their ability to best define the interaction potential threshold, $\mathit{U}$, which serves as a classifier in the final viscosity prediction. Prediction accuracy was calculated as

$$\mathrm{Accuracy}=\frac{\mathit{TP}+\mathit{TN}}{\mathit{TP}+\mathit{TN}+\mathit{FP}+\mathit{FN}}.$$ (3)

We used Euclidean distance as the metric for neighboring point search cutoff. Alternatively, we also considered using the geodesic distance or limiting to connected points from triangulation. These alternatives were ultimately rejected as they fail to adequately capture the spatial decay of forces and potential energy while the triangulated surface might also contain singularities or discontinuities in the projected property values.

Interaction surface

We define the interaction surface between two identical patches as the best-fit plane through the points of each patch, positioned to pass through the highest point along the plane’s surface normal (Figure 1). To determine this best-fit plane, we apply Principal Component Analysis (PCA) to the surface points, which calculates the dominant directions of variation in the data.⁴⁰ PCA is a statistical method that transforms a set of correlated variables into a set of linearly uncorrelated principal components, ordered by their variance. In the context of surface fitting, the component with the least variance corresponds to the normal vector of the plane, as it represents the direction with the smallest spread in the data. Given a set of N points $\mathbf{X}=\{{\mathbf{x}}_i\}$ in three-dimensional space, where each point ${\mathbf{x}}_i=(x_i,y_i,z_i)$ represents a coordinate on the surface, we first standardize the dataset to ensure a mean of zero and a standard deviation of one:

$$\mathbf{x}{ }_i^{\prime }=\frac{{\mathbf{x}}_i-\mathbf{c}}{\mathit{\sigma }},$$

where the centroid $\mathbf{c}$ and standard deviation $\mathit{\sigma }$ are defined as:

$$\mathbf{c}=\frac{1}{N}\sum _{\mathit{i}=1}^N{\mathbf{x}}_i,\mathit{\sigma }=\sqrt{\frac{1}{N}\sum _{\mathit{i}=1}^N{({\mathbf{x}}_i-\mathbf{c})}^2}.$$

Next, we compute the covariance matrix:

$$\mathbf{C}=\frac{1}{N}\sum _{\mathit{i}=1}^N\mathbf{x}{ }_i^{\prime }{{\mathbf{x}}_i^{\prime }}^{\mathit{T}}.$$

And perform the eigenvalue decomposition:

$$\mathbf{C}{\mathbf{v}}_j={\lambda }_j{\mathbf{v}}_j,$$

which gives us the eigenvalues ${\lambda }_j$ and their corresponding eigenvectors ${\mathbf{v}}_j$, sorted such that ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$. The first two eigenvectors, ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$, represent the principal directions of the surface, while the third eigenvector, ${\mathbf{v}}_3$, associated with the smallest eigenvalue, defines the normal vector to the best-fit plane. This assumption holds well for small, uniform surface patches and is further validated through visual assessment. The best-fit plane is defined to pass through the point that lies farthest along the normal vector direction. This choice accounts for steric hindrance between opposing patches, which constrains the interaction geometry. This point is given by:

$${\mathbf{x}}_{max}=arg\underset{{\mathbf{x}}_i\in \mathbf{X}}{max}({\mathbf{x}}_i\cdot {\mathbf{v}}_3).$$

The equation of the best-fit plane is then:

$${\mathbf{v}}_3\cdot (\mathbf{x}-{\mathbf{x}}_{max})=0,$$ (4)

where ${\mathbf{v}}_3$ defines the plane’s normal. This method ensures that the estimated plane minimizes the least-squares perpendicular distance to the surface points while preventing overlap between interacting patches.

Hydrophobic interaction potential calculation

To describe the interaction between hydrophobic patches, we use the surface interaction potential density $W_{\mathrm{H}}$, developed by Donaldson et al.¹⁹ (see also Results). This quantity represents the interaction potential per unit area:

$$W_{\mathrm{H}}=\frac{\mathit{dU}}{\mathit{dA}}=-2\mathit{\gamma }{H}_{\mathrm{y}}{e}^{-\mathit{d}/{\mathit{d}}_{\mathrm{H}}},$$

where $\mathit{U}$ is the interaction potential, $\mathit{A}$ the interacting surface area, $\mathit{\gamma }$ the hydrophobic-water interfacial tension (set to 50 mJ/m ${ }^2$), $H_{\mathrm{y}}$ the Hydra parameter quantifying hydrophobicity, $\mathit{d}$ the distance between interacting surfaces, and $d_{\mathrm{H}}$ the hydrophobic decay length (set to 3Å). It therefore follows that the differential form of $\mathit{U}$ is given by

$$\mathit{dU}({\mathbf{x}}_i)=W_{\mathrm{H}}({\mathbf{x}}_i)\mathit{dA}=-2{\gamma }_{\mathrm{i}}{H}_{\mathrm{y}}({\mathbf{x}}_i)e^{-\mathit{d}({\mathbf{x}}_i)/d_{\mathrm{H}}}\mathit{dA}.$$ (5)

The total interaction potential is obtained by integrating over the patch area. As the vertex density of surface points is approximately uniform, this integral can be approximated as a Riemann sum over the discrete surface points ${\mathbf{x}}_i$:

$$\mathit{U}={\int }_A\mathit{dU}\approx -2{\gamma }_{\mathrm{i}}\sum _{\mathit{i}=1}^N{H}_{\mathrm{y}}({\mathbf{x}}_i)e^{-\mathit{d}({\mathbf{x}}_i)/d_{\mathrm{H}}}\mathrm{\Delta }\mathit{A},$$ (6)

where $\mathrm{\Delta A}$ is equal to the inverse of the vertex density.

Here, $H_{\mathrm{y}}({\mathbf{x}}_i)$ is the surface hydrophobicity at surface point ${\mathbf{x}}_i$, defined as the SAP projection of atoms within 3Å. $\mathit{d}({\mathbf{x}}_i)$ is the distance between interacting points from the two interacting patches, equal to twice the distance from the surface point to its projection on the best-fit plane, given by

$$\mathit{d}({\mathbf{x}}_i)=2\left|({\mathbf{x}}_i-{\mathbf{x}}_{max})\cdot {\mathbf{v}}_3\right|.$$

Results and discussion

Our approach to predicting the hydrophobic contribution to protein solution viscosity connects surface hydrophobicity, patch geometry, and viscosity. Starting from a computationally predicted 3D protein structure (see Methods), we define the solvent-excluded surface and use a custom developed surface patch identification algorithm to rigorously and quantitatively define patches with significant hydrophobic character. For each identified patch, we calculate its size, hydrophobicity, and analyze its geometry. To validate our predictions, we measure viscosity of nine antibodies while screening electrostatic intermolecular interactions through the addition of ionic excipients (e.g., NaCl). Using an interaction potential developed for hydrophobic interactions,¹⁹ we calculate the average interaction potential energy for each protein. By correlating these energies with measured viscosities, we establish meaningful liability thresholds.

Patch–patch interaction calculation for mAbs

We assume that the dominant way the hydrophobic effect influences the solution viscosity is due to the attractive force between two strongest hydrophobic patches on the two interacting proteins. This is a particularly appropriate assumption for highly concentrated antibody solutions, where many molecules are in close contact and can undergo cluster and network formation.⁴¹ For the identification of hydrophobic patches, we use our developed algorithm (see Methods, Figure 1). Notably, unlike comparable patch recognition algorithms implemented elsewhere, our approach provides a more fine-grained and continuous patch definition and eliminates irregularities (Figure 1). It captures the combined effects of multiple atoms and also considers their spatial decay on the protein–protein interaction surface.

For calculation of the interaction potential, we utilize the equation for hydrophobic surface interaction potential density, derived by Donaldson et al.:¹⁹

$$W_{\mathrm{H}}=-2\mathit{\gamma }{H}_{\mathrm{y}}{e}^{-\mathit{d}/{\mathit{d}}_{\mathrm{H}}}$$ (7)

describing the attraction or repulsion between hydrophobic and hydrophilic surfaces, respectively. It depends on the hydrophobic-water interfacial tension $\mathit{\gamma }$ ($\simeq$ 50 mJ/m ${ }^2$), Hydra parameter $H_{\mathrm{y}}$ describing the intensity of hydrophobicity ($>$0 for hydrophobic and $<$0 for hydrophilic surfaces) and hydrophobic decay length $d_{\mathrm{H}}$ ($\simeq$ 3Å).

The above hydrophobic interaction potential model assumes that the two interacting surfaces are flat and geometrically complementary, meaning they align perfectly. However, proteins exhibit complex, non-planar geometries with limited surface complementarity. To address this, for each surface patch, we utilize principal component analysis (PCA) to calculate a best-fit plane for the constituent 3D surface points (computational details in Methods). This approach acknowledges that while a concave surface patch might interact favorably with a similarly sized convex patch, it would interact weakly with an identically concave patch. Consequently, we assume that effectively, the two patches of two interacting proteins interact at the level of this best-fit plane. The projection of surface points onto this plane represents the average interaction surface without accounting for specific surface complementarity effects, thereby providing a simplified but transferable model of protein–protein interactions in solution (Figure 1). This model is valid for relatively simple patch shapes and sizes.

SCM algorithm failed to correctly rank the antibodies (see Figure S5). In fact, they have all been classified as “non-viscous,” pointing out that electrostatic interactions are not the main driver of viscosity in the tested protein solutions. Experimentally, to isolate the hydrophobic effect on viscosity, we screen the charge–charge interactions with the addition of an ionic excipient in solution, which also affects the manifestation of hydrophobicity. With high ionic strength, we effectively suppress the electrostatic (hydrophilic) interactions and thus promote hydrophobic interactions (ionic strength of 50 mM is already enough to screen ES interactions⁴²). Other hydrodynamic/hard-core and vdW interactions are assumed to be similar across the antibodies in our dataset due to their overall sequence and shape similarity.

To quantify hydrophobicity, we use a modified version of the Spatial Aggregation Propensity (SAP) score,²⁵ calculated per atom and projected onto the molecular surface (see Methods). However, we modify the calculation of SAP to involve only the positive (hydrophobic) entries in the amino acid hydrophobicity scale, setting all others (hydrophilic) to zero. This adaptation provides a more accurate representation of hydrophobic patches after electrostatic screening. We construct the Hydra parameter $H_{\mathrm{y}}$ from the hydrophobic interaction potential equation (Equation 7) as the projection of SAP on the surface. The projection is normalized so that the maximum value of $H_{\mathrm{y}}=1$ corresponds to the most hydrophobic amino-acid in the hydrophobicity scale. The distance $\mathit{d}$ in the Equation 7 represents twice the distance of each point from the average interaction surface (see Methods).

Finally, the hydrophobic contribution to the protein solution viscosity is described by the calculated interaction potential of the strongest hydrophobic patch on the protein’s surface with a corresponding patch on the interacting protein. This assumption is particularly valid for monoclonal antibodies, where the two most hydrophobic patches are typically located on opposite Fab arms, specifically in the Fv – CDR region, and are of comparable size and shape due to identical sequences. Other hydrophobic patches, such as those in the hinge region, are generally either inaccessible or negligible (see Figure S6). We calculate the interaction potentials $\mathit{U}$ for the antibodies in our dataset, in units of $k_B\mathit{T}$ at 298, and show them in Table 1. The SAP threshold (see Figure 1) used for the hydrophobic patch identification was set to 0.0075kcal, based on having the best viscosity determination accuracy for our dataset (Figure 2(A)). However, the model and algorithm are shown to be robust, showing a high degree of accuracy in a wide range of SAP thresholds and physically meaningful cutoff values (Figure 2(C)).

Table 1.

Viscosities (full and with ES screening) and interaction potentials ($\mathit{U}$). Shown are the measured viscosity of mAbs in the dataset for with (ES screening) and without the addition of 100 mM NaCl in the solution and the calculated interaction potentials ($\mathit{U}$). The interaction potential is calculated with the SAP threshold of 0.0075kcal and cut-off 3Å.

	Viscosity	Viscosity	Interaction
mAb	(cP)	(ES interactions	potential $\mathit{U}$
		screened) (cP)	($k_B\mathit{T}$)
1	18.9	51.9	−2.23
2	11.9	8.0	−1.21
3	31.1	34.3	−4.83
4	10.7	5.5	−0.35
5	9.5	6.4	−1.89
6	8.0	7.4	−0.58
7	14.0	8.0	−1.71
8	15.6	12.7	−1.89
9	75.5	34.2	−3.20

Figure 2.

Comparison between interaction potential and SAP score for viscosity ranking and the model robustness. Viscosity (ES interactions screened) is plotted against (A) SAP score and (B) interaction potential ($\mathit{U}$), for the threshold of 0.0075 kcal and cut-off 3Å. (C) Robustness of model performance is shown by considering different patch identification thresholds and cut-offs and assessing their accuracy.

Experimental

We measured the viscosities of nine mAbs, formulated at 150 mg mL⁻¹, at room temperature (25 ${ }^{\circ }\mathrm{C}$) and calculated their patch–patch interaction energies, as seen in Table 1 (see Methods). We isolated the hydrophobic contribution to viscosity by screening the charge–charge interactions via the addition of 100 mM NaCl in solution.

We justify our approach of defining and calculating interaction potentials, by also accounting for patch shape at the level of the best-fit plane using PCA, by comparing ranking using both approaches in Figure 2.

For each patch identification threshold and cutoff value considered (see Methods, Figure S2), we searched for the optimal interaction potential threshold, $\mathit{U}$, by scanning through uniformly spaced consecutive values of $\mathit{U}$ and calculating the classification accuracy for separating antibodies into highly viscous ($\ge$15 cP) and non-viscous ($<$15 cP) groups. The $\mathit{U}$ threshold yielding the highest accuracy was then selected, and the corresponding model performance and its parameters were recorded (threshold of 0.0075kcal and cutoff 3Å, Figure 2).

We can see that using the interaction potential values can serve as a better classifier than just the SAP value of the patch. Non-problematic antibodies span a broad range of SAP scores. In contrast, their interaction potentials remain within a narrow range, while only antibodies with high viscosity exhibit higher interaction potentials. Defining a threshold for the SAP score is in itself challenging, as the highest SAP score of a non-problematic antibody is significantly greater than the lowest SAP score of a problematic one. Additionally, since the SAP score lacks direct physical interpretability, any chosen threshold, such as $\approx$ 90 in our case, is somewhat arbitrary and may be specific to this dataset and this set of algorithm hyperparameters (for the robustness of the model using only SAP score, see Figure S3). The interaction potential $\mathit{U}$, by contrast, has a clear physical basis and can be interpreted in the context of protein–protein interactions. These results further support the idea that the interaction potential offers a better and physically meaningful framework for explaining differences in antibody behavior under varying experimental conditions. Importantly, we also find that classification accuracy remains highly consistent across different patch identification thresholds and cutoff values, showing the robustness of the approach.

In Figure 3 we plot all the antibodies, ordered by their hydrophobic interaction potentials ($\mathit{U}$), and color them according to whether their viscosity exceeds the threshold that limits the feasibility of subcutaneous administration. Viscosity values between 10 cP and 50 cP are considered as threshold values, above which administration becomes problematic, depending on the injection device.^43,44 Here, we use the threshold of 15 cP. We can see that there exists a strong correlation between the hydrophobic contribution to viscosity of the protein and the interaction potential. Clearly, we see the trend that for antibodies whose strongest patch–patch hydrophobic interaction energy is higher than cca −2.1 $k_B\mathit{T}$, viscosity with NaCl is higher than 15 cP.

Figure 3.

Viscosity ranking plot by antibody. Antibodies are plotted in order of increasing interaction potential, given in units of $k_B\mathit{T}$, at 298. Green color indicates antibody has a viscosity lower than 15 cP, while red indicates a viscosity greater than or equal to 15 cP. Red horizontal line is positioned at a determined threshold (−2.1 $k_B\mathit{T}$), to classify antibodies into viscous and not viscous.

Interestingly, our measurements and the order of magnitude of our calculations well agree with the previously identified interaction potential threshold of 2.5 $k_B\mathit{T}$, which marks the transition from negligible to significant ultra-weak protein–protein interactions.⁴⁵ It has been shown that in this regime proteins begin to form dynamic transient clusters, with increased coordination numbers and an order-of-magnitude increase in cluster dissociation times, while not leading to irreversible gelation. This agreement suggests there is a general and physically meaningful threshold of $\approx$ 2–3 $k_B\mathit{T}$ when weak protein–protein interactions begin to have measurable biological and rheological consequences.

While we have focused mostly on hydrophobic patches in this work, it is also important to consider the spatial position and interplay between the hydrophobic and electrostatic patches (described through the SCM descriptor). Under typical biopharmaceutical formulation conditions, electrostatic and hydrophobic interactions act simultaneously. The range of electrostatic interactions under such conditions without added VRAs (10–50 mM ionic strength) can be approximated by the Debye length of $\sim$ 5–10Å (Figure 4). When electrostatic and hydrophobic patches coexist within such a distance, the electrostatic repulsion dominates. Further increasing the ionic strength to beyond 100 mM decreases the Debye length below the hydrophobic interaction range ($\sim$ 3Ź⁹), allowing hydrophobic attraction to dominate. This must be taken into account when interpreting the solution viscosity. Our data illustrates this. In Table 1 we observe that while increased ionic strength generally lowers viscosity below the threshold of 15 cP by screening attractive electrostatic interactions, it leads to an increase in viscosity for mAb 1 and mAb 3, whereas for mAb 9 the viscosity decreases slightly but remains high. For those antibodies, the electrostatic effect is coupled with hydrophobicity. Screening local electrostatic repulsions facilitates immediate self-association, as reducing the electrostatic range with increasing ionic strength allows the hydrophobic effect to dominate at short distances.

Figure 4.

Close bordering of hydrophobic and electrostatic patches. Hydrophobic patches (green), electrostatic patches (red/blue), and their close bordering (purple) is shown for the three example antibodies (visualized only for the Fv region): mAb 1, mAb 3, and mAb 9. Comparing the sizes of the patches with the approximate ranges of interactions, we see that the proximity of the patches is significant enough to cause competing effects.

Figure 4 shows the strongest hydrophobic patch and the nearest significant electrostatic patch, both within the Fv region, for the mAbs 1, 3, and 9. It reveals how close they border each other, interpreted as the overlap of points with significant SAP and SCM scores (see Methods), as quantified in Table S1. Notably, the nearest electrostatic patch is positively charged, and because the antibody surface is predominantly positive at the experimental pH, it is unlikely to favor any other attractive interactions. Consequently, without electrostatic screening, long-range repulsive forces would to an extent prevent the hydrophobic patch from interacting with its counterpart patch. This has significant practical applications in formulation development, since such patch positioning determines the effectiveness of different viscosity reducing additives (VRAs) to the formulation. Addition of a charged VRA that primarily screens electrostatic interactions, such as arginine or NaCl, is often ineffective in such cases, as evident from Table 1. A non-charged, hydrophobic additive like proline would have to be tested in such a case – in fact, proline was tested on mAb 1, where it successfully reduced the viscosity below the threshold of 15 cP (data not shown).

We note here that due to the limited size of our dataset, we could not separate it into train, test and validation sets. In lieu, we show the algorithm robustness and the physically meaningful interaction potential threshold, and we see further validation as an important part of our future work.

Conclusions

In this work, we present a novel in silico computational framework for quantitatively predicting antibody solution viscosity based on molecular hydrophobicity. Using the molecular 3D structure and the spatial distribution of hydrophobicity on the surface, we connect interaction potential with viscosity. Importantly, we developed an algorithm for patch recognition on the molecular surface, which accounts for the spatial decay of interaction potential, enabling the identification of relevant hydrophobic regions. Using this approach, we successfully differentiate between high- and low-viscosity antibodies in our experimental dataset of nine monoclonal antibodies.

Notably, in this work, we focus on the contribution of hydrophobicity to viscosity. This has historically been either neglected with respect to electrostatic interactions, only taken into account phenomenologically,²³ or considered for relative ranking in the context of structure optimization.⁴⁶ Here, we show that hydrophobicity impacts and explains viscosity, where electrostatic descriptors alone cannot predict whether an antibody will be highly viscous. We also observe how the close bordering of electrostatic and hydrophobic patches significantly influences the solution viscosity, with direct impact on the biopharmaceutical formulation development.

Our analysis was conducted on an experimental dataset comprising only monoclonal antibodies, but since it is physics based, it is generalizable to other protein formats. Future work will extend this approach to more complex therapeutic proteins.

Overall, this study demonstrates how computational analysis of protein–protein interaction interfaces can inform the rational design of therapeutic proteins and high-concentration protein formulations, a critical challenge in biopharmaceutical development.

Funding Statement

This work has been supported by the Slovenian Research Agency [Program Grant P1-0099] and the European Research Council under the Horizon 2020 Research and Innovation Program of the European Union [Program agreement 884928-LOGOS].

Abbreviations

AI

Artificial intelligence

ANN

Artificial neural network

APW

Additionally purified water

CNN

Convolutional neural network

ES

Electrostatic

HP

Hydrophobicity parameter (Wimley – White)

MD

Molecular dynamics

ML

Machine learning

mAb

Monoclonal antibody

OPLS-AA

Optimized Potentials for Liquid Simulations – All-Atoms

PCA

Principal component analysis

SAP

Spatial Aggregation Propensity

SASA

Solvent-accessible surface area

SC

Subcutaneous

SCM

Spatial Charge Map

SPC/E

Extended simple point charge water model

VRA

Viscosity-reducing additive

vdW

Van der Waals

Disclosure statement

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Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/19420862.2026.2614767