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. 2026 Feb 11;6(2):587–598. doi: 10.1021/acspolymersau.5c00177

Identifying Polymers that Bind or Reject Proteins with Machine Learning: Handling Categorical Features within a GPR Model

Ramindu De Silva †,‡,§, Wei Ge †,§, Carolin Bapp ∥, Ahmed Z Mustafa †, Robert Chapman ∥, Yanan Fan ‡,§,*, Scott A Sisson §,*, Martina H Stenzel †,*
PMCID: PMC13067167  PMID: 41971693

Abstract

Understanding the interaction between polymers and proteins is of interest for researchers in medicine, biology, food science, and water treatment, among other fields. The goal may be to create strong interactions with enzymes to improve their catalytic stability, while in nanomedicine and biomedical engineering, the focus is often on reducing protein adsorption on polymer surfaces. Researchers have developed libraries of polymers with various monomer combinations and tested their binding to different proteins to better understand these interactions. In this work, we aimed to identify the polymer with the highest or lowest binding affinity to all proteins, respectively, using Gaussian Process Regression (GPR). However, incorporating categorical features such as the type of monomer has not been widely applied in GPR. Here we compare a range of process models, which were coined Multiplicative kernel, Additive kernel, Easy to interpret Gaussian Process model (EzGP), Latent Variable Gaussian Processes (LVGP), and the Latent Map Gaussian Processes (LMGP) by their developers. The LVGP model was found to perform best on the polymer–protein data set, where the output for binding strength was given by Förster resonance energy transfer (FRET), which can be used to help generate large data sets for machine learning (ML). The polymer that had the highest affinity to glucose oxidase (GOx), uricase (Uri), casein (Cas), trypsin (Trp), carbonic anhydrase (CAn) and bovine serum albumin (BSA) carried positive charges as well as hydrophobic benzyl groups. Negatively charged monomers dominated the polymer that rejected the most proteins intermixed with some cationic units, reminiscent of zwitterionic polymers.

Keywords: machine learning, Gaussian process regression (GPR), categorical features, polymer, protein

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Introduction

Machine learning is now increasingly applied to problems in polymer science as it is now possible to generate large data sets with high-throughput techniques. − One of these questions is how polymers interact with proteins. Typically, a polymer is combined with a specific protein to extend the lifespan of therapeutic proteins or enzymes, aiming to stabilize their structure against denaturation and improve their activity. − Wrapping proteins in a layer of polymers was originally achieved by selecting a charged polymer that is complementary to the net charge of the protein. , However, binding is more complex as the surface of proteins is usually a combination of positive and negative charges, albeit usually with an excess, hydrophobic areas, and elaborately designed binding pockets. As a result, binding between polymer and protein is a combination of electrostatic interactions but also hydrogen bonds, hydrophobic interactions, as well as specific forces such as π–π interactions. The binding process is determined by the type of counterions, ionic strength, pH, and other parameters, which influence entropy and binding enthalpy in a complex interplay. , There is now an understanding that high binding strength between polymer and protein can only be achieved by tuning the polymer structure to include opposite charges as well as hydrophobic functionality. ,,− Sophisticated polymer design can now ensure that the polymer fits a specific protein like a glove to help maximize protein stabilization.

In numerous processes, we encounter mixtures of proteins, such as in blood serum, food, or sludge, which is the solid slurry produced in water treatment. When these protein mixtures come into contact with synthetic materials, such as polymers, rapid binding can occur between them, depending on the chemical structure. The addition of charged polymers to sludge is a widely used industrial wastewater treatment process that is designed to quickly remove proteins, among other impurities. In contrast, the adsorption of proteins to the surface of materials in the biomedical field is regarded as an undesirable process. Significant research efforts have been dedicated to reducing the adsorption of proteins.

We have recently developed a library of polymers based on cationic, anionic, hydrophobic, and neutral monomers and tested their interaction with various proteins of different sizes and isoelectric point (IEP) (Figure ). The strength of binding can directly be measured using Förster resonance energy transfer (FRET) as an output value. The challenge is now to identify which polymer is the most likely to bind to a range of proteins, while it is also of interest to find the polymer with the least overall affinity to the proteins. This universal polymer can be used for applications where mixtures of proteins are present, where the aim is to use one polymer to either attach or reject many proteins.

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Polymers, labeled with the cyanine dye Cy5, prepared using cationic monomers (D, Q), anionic monomers (C, S), neutral, hydrophobic monomers (F, Y, X, M) and neutral, hydrophilic monomers (N, A, H) were mixed with Cy3 labeled protein, with FRET being used as an output for binding strength. The neutral water-soluble polymers have been sorted according to reported logP values that are either experimental or calculated values (depending on availability): benzyl acrylate (F): 2.43; n-butyl acrylate (Y): 2.38; t-butyl acrylate (X): 2.32; methyl acrylate (M): 0.74; isopropylacrylamide (N): 0.19; poly­(ethylene glycol methyl ether acrylate (PEG): −0.05; 2-hydroxy ethyl acrylate (H): −0.21; acrylamide (A): −0.67).

The database that will be used here has continuous input features, such as the number of repeating units of each monomer, as well as categorical input features, such as the type of monomers. Examples for categorical inputs are polymer type, monomer identity, polymer architecture and processing conditions. These categorical features carry critical information and frequently influence polymer properties and performance. Accurate handling of categorical features can substantially enhance predictive reliability, allowing researchers to design polymer experiments more efficiently and confidently. Improved handling of categorical variables also supports deeper interpretability, providing clearer insights into how specific categorical features influence material properties. On the other hand, ignoring categorical features lowers the predictive accuracy of the model due to the loss of structural information.

Gaussian Process Regression (GPR), which has been prominent across many studies in polymer science due to its ability to perform well on small data sets, ,, model complex nonlinear relationships and provide uncertainty estimates around its predictions. Standard GPR formulations assume that all inputs are continuous, where the correlation between outputs is measured by a covariance (kernel) function. For continuous inputs, the covariance function is based on a distance metric such as the Euclidean distance between each observation. However, since there is no such notion of distance for categorical inputs, this poses a significant methodological challenge for effectively capturing the influence of categorical data.

Moreover, the current approaches followed in polymer informatics to handle categorical features contain numerous drawbacks. Using these approaches with a standard GPR kernel can lead to the covariance structure of the GPR model being fixed rather than being learnt by the model itself. , This can lead to inaccurate predictions being generated from the GPR model. Alternative approaches have been proposed to handle categorical features within a GPR model in the broader machine learning literature. However, none of these have been implemented in polymer informatics. Given the widespread usage of GPR in polymer informatics, the commonality of categorical features in polymer data sets, and the drawbacks of current approaches in handling categorical features within polymer informatics, it is important to introduce these alternative approaches to the domain.

In this work, we will compare the Multiplicative kernel method by Zhou et al., the Additive kernel method by Deng et al., the Easy to interpret Gaussian Process method (EzGP) by Xiao et al., the Latent Variable Gaussian Processes method (LVGP) by Zhang et al., and the Latent Map Gaussian Processes method (LMGP) by Oune and Bostanabad. We evaluate each method using extensive simulation studies based on scenarios that are commonly encountered in polymer science data sets such as (1) presence of many categorical input features, − (2) categorical inputs with many levels , and (3) categorical features that are correlated. , The approach that worked best for our system, LVGP, was then used to identify the polymer composition that has the highest binding to various proteins and the one that rejects proteins.

Experimental/Methods

The polymer–protein data set used in this study was generated using high throughput methods. , The polymers and proteins included in the data set are given in the following section of the manuscript (Data set).

The final data set used for modeling contains 96 polymers and 6 proteins which results in 576 polymer–protein combinations. Polymer inputs were represented using monomer composition variables (repeat-unit fractions). Protein information was included using VHSE descriptors. 50 amino-acid descriptors were computed and reduced using principal component analysis to 8 components (VHSE1–VHSE8), which were then treated as continuous inputs. As a secondary protein representation, we also evaluated ESM-2 embeddings (Supporting Information, Tables S9 and S10).

We benchmarked five Gaussian process regression approaches that incorporate categorical variables namely, Multiplicative kernel, Additive kernel, EzGP, LVGP and LMGP Continuous inputs were modeled using an RBF kernel and model parameters were estimated by maximum likelihood.

To compare methods under controlled conditions, we ran simulation studies spanning common polymer informatics scenarios such as (i) many categorical predictors, (ii) high-cardinality categorical levels, and (iii) correlation among categorical variables (details and synthetic functions are provided in the Supporting Information). Simulations were conducted with 100, 500, and 1000 total samples using 5-fold cross validation and 50 replications per setting.

For the real world polymer–protein data set, we evaluated predictive performance using 5-fold cross-validation repeated across 50 replications, reporting mean performance and standard errors. We additionally report an 80:20 train–test split analysis in the Supporting Information.

Results and Discussion

Data Set

The input features of the data set included the monomer repeat units of acrylamide (A), hydroxyethyl acrylate (H), carboxyethyl acrylate (C), N-[3-(dimethylamino)­propyl] acrylamide (D), methyl acrylate (M), polyethylene glycol (PEG), benzyl acrylate (F), and butyl acrylate (Y). The proteins used included glucose oxidase (GOx), uricase (Uri), casein (Cas), trypsin (Trp), carbonic anhydrase (CAn), and bovine serum albumin (BSA) (Table ). The differences between these proteins are their molecular weight, but more importantly, their isoelectric point (IEP). To quantify binding between polymer and proteins, the polymer is labeled with the cyanine dye Cy5 while the protein is conjugated with Cy3. If polymer and proteins bind, Förster resonance energy transfer (FRET) takes place, which can be used as a measure of binding strength (ESI, Figures S1 and S2). , The reader is referred to earlier publications explaining details on the data set and the use of FRET. In the initial study by Mustafa et al., FRET was used as a measure of the binding strength between polymers and proteins, and the FRET output was confirmed by isothermal calorimetry (ITC). The subsequent study by Bapp et al. expanded the study to a range of proteins and large polymer data sets, which are now analyzed here using machine learning (ML). The chosen polymers, prepared in aqueous solution, are water-soluble by visual inspection; however, it is possible that some form loose, nanoscale aggregates upon addition of hydrophobic monomers, which will not affect the interaction with the protein, as the overall hydrophilicity ensures sufficient mobility.

1. Characteristics of the Proteins Used in This Study.

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The copolymers in this data set were synthesized via a high-throughput workflow using a consistent protocol. Consequently, polymer architecture, topology and microsequence distribution were not independently controlled variables in the library. We therefore represent each polymer using its monomer repeat-unit composition (fractions), which is a composition descriptor. This representation captures the fractional presence of each monomer type and thus proxies key interaction drivers relevant to polymer–protein binding such as net charge density, hydrophilicity/hydrophobicity balance, hydrogen-bonding capacity, and aromaticity. Because the data set does not include explicit measurements of sequence distribution, branching/architecture, or dispersity metrics, these effects cannot be modeled directly. This is a limitation of the study. This featurization choice aligns with prior high-throughput polymer protein design studies that successfully use composition-based polymer feature vectors in GPR models.

Protein crystal structures were retrieved from The Protein Data Bank. Electrostatic potential and hydrophobicity were calculated and visualized using the UCSF ChimeraX program. ESP ranges from red for negative potential through white to blue for positive potential. MLP maps range from dark cyan (most hydrophilic) to white to dark goldenrod (most lipophilic). No crystal structure is available for κ-casein. Instead, sequence-obtained molecular modeling shows that it consists of two separate domains: a hydrophilic domain carrying a negative charge and a hydrophobic domain carrying a positive charge. These data were obtained with permission from ref .

Converting Categorical Features of Polymers and Proteins

In polymer science data sets, categorical variables are handled either through one-hot encoding , or by using domain-specific polymer descriptors. −

One-hot encoding maps each category to a separate binary feature (1 = present, 0 = absent). In polymers, a copolymer sequence can be encoded as 1/0 for moieties A/B, with sequence length reflecting chain length. − Larger polymer libraries are encoded by their repeating units. However, one-hot vectors inflate dimensionality, harming generalization and accuracy , and in Bayesian Optimization they cause naive GPR surrogates to interpolate nonsensically between categories, yielding suboptimal designs.

On the other hand, domain specific descriptors convert categories to physiochemical numbers. ,− This ties GPR performance to the chosen representation which can be adversely affected by poor or task-mismatched descriptors, bias, omitted key factors, and can transfer poorly across problems. Some domain-specific encodings (Morgan fingerprints) produce very high-dimensional vectors which can result in similar high-dimensional issues as one-hot encoding. Effectively addressing the limitations of current approaches in handling categorical features within a GPR model holds significant practical value for the domain of polymer science.

Recent polymer informatics studies make use of polymer language models and graph embeddings to introduce chemically meaningful representations of polymers into data sets. − These representations can improve predictions when chemical diversity and data set size support learning in a higher-dimensional feature space. In this work, the polymers were included in the data set by featurizing them following a similar method to Tamasi et al. through the monomer repeating units. We expect that richer polymer descriptors could increase absolute predictive performance, but they may also alter the relative performance of the kernels that handle categorical features by converting inputs into continuous high-dimensional embeddings. Evaluating these representations in combination with the five GPR variants that we compare in this study is an important direction for future work.

The protein information was included using descriptors. We included numerical descriptors for each peptide based on the Vectors of Hydrophobic, Steric, and Electronic (VHSE) method. In this method, protein information is captured through 50 numerical descriptors that represent hydrophobic, steric, and electronic properties of proteins. The method then utilizes principal component analysis to reduce the number of features to 8. Within the final 8 features, the first two are related to hydrophobic properties of proteins (VHSE1 and VHSE2), the next two are related to the steric properties of proteins (VHSE3 and VHSE4) and the final four are related to the electronic properties of proteins (VHSE5, VHSE6, VHSE7 and VHSE8). The peptides module in Python was used to obtain corresponding descriptors for each peptide.

The peptide descriptors were treated as continuous input features. These descriptors were standardized using min–max scaling so that they were on the same scale. The data set contained 96 polymers and 6 peptides which resulted in a total of 576 polymer–protein combinations. The data set using all polymer and their repeating units as well as the numerical descriptors for the peptides (VHSE1 – VHSE8) can be found in the Supporting Information.

Apart from the VHSE method described above, we separately encoded the proteins in the data set Evolutionary Scale Modeling v2 (ESM-2) a transformer based protein language model. ESM-2 captures deeper evolutionary patterns from protein sequences which allows for deeper understanding of protein structures. This method takes in the raw protein sequences and uses an encoder trained on a large protein database. For each protein sequence, we extracted the final-layer residue embeddings from ESM-2 and computed a fixed-length sequence representation. We then applied principal component analysis (PCA) to the resulting protein-level embedding vectors to reduce the high-dimensional ESM embeddings down to 6 components (since only 6 proteins in the data set). The resulting protein descriptors along with the results for each of the GPR models when trained on this descriptor set are provided in the Supporting Information (Tables S9 and S10).

Generic Methods to Handle Categorical Variables within a GPR Model

A range of models have been described in the statistics and machine learning literature that can process categorical features, including the Multiplicative kernel method by Zhou et al., the Additive kernel method by Deng et al., the Easy to interpret Gaussian Process method (EzGP) by Xiao et al., the Latent Variable Gaussian Processes method (LVGP) by Zhang et al., and the Latent Map Gaussian Processes method (LMGP) by Oune and Bostanabad. Table provides a quick comparison between these generic methods, detailed descriptions of each method are available in the Supporting Information. More information on the formulation of GPR models can be found in the Supporting Information of this paper.

2. Comparison among Generic Methods to Handle Categorical Inputs in a GPR.

method nature of GP when the method is most suitable pros cons
Zhou et al. (Multiplicative) single GP surface over continuous and categorical features underlying data has a single smoothness single GP fit implies simplicity and lower number of parameters to estimate 1. does not vary smoothness of the GP surface based on the levels of categorical variables
        2. Kernel is multiplicative in nature. Can be zero for certain observations if any τ is zero in matrix T
Deng et al. (Additive) fits a GP surface for each categorical feature. This is enabled by specifying the model in an additive manner problems with many qualitative factors where only a subset is informative or where smoothness over inputs truly differs by factor 1. smoothness of GP surface over each categorical feature adds flexibility 1. cross category interactions are not considered
      2. avoid kernel collapsing to zero 2. in the presence of many input features or many levels parameter estimation can be complex
EzGP fits a base GP with only continuous features and GP surfaces over each level of each categorical variable when the underlying surfaces differ in kind (linear vs sinusoidal vs cubic) and not just in degree 1. base GP for continuous features avoids covariance collapsing to zero even if effect of categorical features is zero 1. more parameters need to be estimated than Deng et al. even though angles need not be estimated
      2. higher flexibility due to level specific length scales 2. if the number of levels within a categorical feature is high the number of parameters to estimate increases significantly
        3. high computational time
LVGP maps categorical features into latent coordinates and fits a single GP surface over the continuous and learned latent coordinates 1. when flexible correlations among the levels of the categorical features are important 1. matches physical interpretation of categorical features arising due to physical processes 1. requires careful optimization to map categorical features into latent coordinates
  learns a separate latent space per factor with a fixed set of levels for all observations 2. large number of levels in categorical features but the number of data points is relatively small 2. low computational time  
      3. allows different smoothness for different factors  
LMGP fits a single GP surface over a shared latent space for all categorical features when categorical inputs are conditional or variable in length supports conditional or variable length categorical inputs by working with level combination vectors ζ(t) 1. if the number of levels of categorical features are high the latent map explodes in size and requires more computational resources
        2. careful optimization is needed when estimating entries of the latent map
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Before attempting to analyze our data set, we conducted simulation studies to assess the performance of the methods listed in Table (multiplicative kernel, additive kernel, EzGP, LVGP, and LMGP). The simulations were carried out to assess the performance of the methods in three different scenarios: multiple categorical features (unrelated features); many levels within the categorical variables (related features such as monomer types); and correlation between categorical features (expressed in statistical features). The simulations were carried out for both nominal (no intrinsic order) and ordinal (ranked features) categorical variables (see Supporting Information). The synthetic functions used for the simulations are provided in Table S1 of the Supporting Information.

Simulations were performed with a single categorical feature with 5 and 10 levels; two categorical features with 8 and 6 levels; four categorical features with 4, 6, 6, and 8 levels; and ten categorical features with 2, 3, 3, 5, 13, 10, 8, 6, 8, and 3 levels, respectively. The probabilities used in each of these simulations are provided in Table S2 of the Supporting Information. The impact of correlated categorical features was assessed by comparing two correlated categorical features; four correlated categorical features; and ten correlated categorical features against their independent counterparts.

For each model the continuous features were modeled using the Radial Basis Function (RBF kernel) due to its flexibility. The model parameters were optimized using maximum likelihood estimation.

Simulations were performed with the total data points of the synthetic data sets being 100, 500 and 1000. 50 replications were performed for each simulation. 5-fold cross validation was implemented for all simulations, and the performance (R 2) along with the corresponding variability (standard error) were recorded. The simulations were run using the Katana high performance computing facility at UNSW Sydney. Results of the simulation studies are provided in Tables S3–S8 of the Supporting Information.

Overall, our simulation studies show that, given our experimental setup of generating data, whether the categorical inputs are nominal or ordinal, the LVGP approach consistently matches or outperforms competing methods and remains the most stable option for capturing the effects of multiple categorical features, high-cardinality levels, and correlation among categories. Under our experimental setups, LVGP outperformed the other methods because it replaces each categorical level with one or two-dimensional latent numerical coordinates and then fits a standard kernel in this latent space. This gives a flexible correlation model and avoids the rigid assumptions of multiplicative or additive encodings. It remains stable even when the number of features or levels is large. Performance patterns were broadly similar for nominal and ordinal setups, indicating that under these designs the relative ranking depended on how the categorical features were handled within the GPR kernel function.

The multiplicative method assumes that the local variation over all inputs is the same across all level combinations. In most cases different levels have different smoothness. LVGP permits category-specific differences in smoothness through distances in latent space. But if the underlying data does have a single smoothness, then the multiplicative method can perform well.

These models were then used to analyze our real data set of polymer–protein mixtures. The 8 monomer units displayed on the left of Figure are treated as 8 ordinal input features in the data set. Table provides the results of the performance of each method on this data set. We evaluated the performance (R 2) of all five methods (multiplicative, additive, EzGP, LVGP and LMGP) based on 5-fold cross validation across 50 replications, and the corresponding standard errors in performance were also recorded.

3. Mean R 2(± Standard Error) Across 50 Replications for the Five Methods Fit to the Real Polymer-Protein Dataset .

multiplicative additive EzGP LVGP LMGP
0.653 ± 0.005 0.695 ± 0.004 0.731 ± 0.002 0.871 ± 0.003 0.779 ± 0.03
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a

Best mean performance is highlighted in bold.

Similar to the simulation studies, the LVGP method outperformed all other methods. The results align with that of the simulation study as the performance for the multiplicative, additive, and EzGP methods is worse than the LVGP and LMGP methods due to the presence of a large number of categorical features.

We also perform an independent 80:20 train test split on the data and refit each of our models on the train set and obtain predictions on the test set. The actual vs predicted values are plotted for each of the models and these plots are included in Figure S3 of the Supporting Information.

We subsequently only report results using the LVGP method, as it performed the best on the test data. Figure illustrates the binding of all polymer–protein combinations in the data set based on the predicted values from the GPR model. The rows are sorted according to the mean predicted FRET value over all proteins, from darkest (high FRET) at the top to lightest (low FRET) at the bottom. A polymer that is listed in the top rows in Figure , is a more promising binder to these proteins on average, while a polymer lower on the list is, on average, a weaker binder.

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Heat map of FRET values for all polymer–peptide combinations obtained using the LVGP method. The rows are sorted in descending order from top to bottom according to the average binding ability of polymers. The columns are sorted in descending order from left to right according to the binding ability of peptides. The top 10 and bottom 10 polymers are enlarged and displayed on the left side of the figure.

Accordingly, poly­(A70-co-D20-co-F10) has the highest average binding among all polymers in the library. This polymer is not unexpected, as it exhibits a variety of forces similar to those found in proteins. Acrylamide (A) is a water-soluble monomer that also displays strong hydrogen bonding, which has been used earlier to create capsules around proteins. N-[3-(dimethylamino)­propyl]­acrylamide (D), a positively charged repeating unit, binds to predominantly negatively charged proteins via electrostatic interactions. Although one would expect that more positively charged repeating units would improve performance, it should be noted that our protein selection also includes proteins with a positive net charge. This would mean that changes in protein composition, such as enlarging the protein data set to include more proteins with IEP > 7, would likely shift this composition. Important for strong binding is the presence of hydrophobic units, such as the benzyl group present in F, to enable binding to hydrophobic pockets.

The polymer poly­(C90-co-D10) has the lowest average binding among all polymers of the library, which is not surprising, as polymers with negative and positive charges are known to have low protein adsorption. It is interesting that it is not the copolymer with an equal distribution of C and D poly­(C50-co-D50) that has the lowest binding, but the polymer with an excess of the negatively charged C. The reason is the negative net charge of the proteins used (Table ). This is not coincidental, as most proteins in nature have negative charges, which is the result of the synthesis process in the cell, where strongly positively charged peptides would interact with the negatively charged ribosomes, delaying their release. Poly­(C90-co-D10), which is predominantly negatively charged, is therefore easily repelled by most proteins.

The rows of FRET ratios shown in Figure display a clear color gradient. Most polymers show strongest FRET with GOx, then Cas and then Uri and CAn, respectively. BSA and Trp display consistently weaker FRET. Certain polymers that are in the middle portion of Figure display higher binding to CAs than to GOx. These differences are in part due to differences in binding strength, but in part also due to differences in Cy3-labeling efficiency between the proteins. While we tried to optimize conditions such as the protein concentration and labeling conditions for each protein, this means that the FRET ratios for a given polymer on two different proteins cannot be directly compared as a read out of relative binding strength. In the case of BSA and Trp which have systematically weaker FRET than GOx despite these optimization efforts it is fair to say that most polymers find it more challenging to bind to these proteins.

As mentioned previously, the LVGP method maps each categorical feature into 2D latent coordinates, and the Gaussian Process correlation depends on the Euclidean distance in that latent space. We make use of this aspect of LVGP to further explain the reasons for poly­(A70-co-D20-co-F10) to have the highest binding. Figure visualizes the learned 2D latent coordinates for monomers A, D, and F.

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2D latent maps from the fitted LVGP model for monomers A, D and F. The color gradient represents the marginal mean predicted FRET value. A single point in each individual figure represents a value that each can have based on our data set.

The axes of the graphs are not meaningful as they are only identifiable up to rotation or translation. However, the distances and clustering between points are meaningful as they reflect the response similarity (mean predicted FRET) across different monomer compositions. Colors represent the marginal mean predicted FRET for each monomer composition across the data set and therefore they are used to interpret level similarity and broad trends, not to compute the optimal multivariate composition. For poly­(A70-co-D20-co-F10), the latent maps for both A and F indicate that the respective monomer compositions (A70 and F10) lie in the higher binding regions. But the latent map for D indicates a weaker binding. Thus, D contributes in a more interaction dependent manner with A and F carrying the polymer toward a higher FRET.

Each predicted value obtained from the GPR model has an associated uncertainty estimate (a credibility interval). An informal method often used to decide whether two values are the same can be based on whether the two credibility intervals overlap: if they do not, then the probabilities that the two predicted values are the same is very small. We make use of this to identify those polymers that are different from the polymers that have highest and lowest average binding. Figure depicts the polymers ordered in descending order from highest average binding to lowest average binding along with their uncertainty estimates. We draw a vertical line at the lower bound of the uncertainty estimate for the polymer with highest average binding. Polymers whose uncertainty estimates that do not overlap with this are denoted in red. It can be said that the binding of the polymers given in red are substantially different from the polymer with the highest binding.

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Average binding and corresponding 95% uncertainty estimate (credibility interval) for polymers predicted by the fitted LVGP model sorted in descending order. Polymers in red have no uncertainty interval overlap with the polymer with the highest average binding.

Figure S4 in the Supporting Information depicts the same but where the polymers are ordered in ascending order of average binding from top to bottom, and where we wish to identify polymers that have a substantially different average binding compared to the polymer with the lowest average binding.

The main purpose of this work was to explore methods for using categorical input variables in GPR. The various approaches were tested on a data set that linked the binding between polymers and proteins using the FRET ratio between Cy3 and Cy5 as output data. In an earlier study using a few polymers and GOx only, the FRET ratio was compared to ITC data to confirm that FRET can indeed be a valid output value for polymer–protein binding. A clear advantage of using the FRET ratio as a measure of binding strength is that analysis can be completed on a wellplate using hundreds of polymers. Combined with a liquid handling robot, it would be possible to obtain thousands of data values. This contrasts techniques such as ITC or fluorescence correlation spectroscopy (FCS), which are significantly more time-consuming. While it is possible to generate a large data set using FRET analysis, ML analysis can only be as good as the quality of its data set. It should be noted that dye attachment can sometimes influence the binding process, as dyes are often charged or hydrophobic. The FRET efficiency is also affected by the interaction of the dye with the environment, and therefore, different polymer–protein systems could result in different FRET ratios despite having comparable interactions. This issue becomes evident when comparing the polymers adjacent to the best- and worst-binding sites, which are quite different in chemical composition. Either there is truly a group of diverse polymers that interact with proteins the same way, or the input data has errors. The results obtained by ML, therefore, should ideally be complemented by other techniques.

As discussed earlier, the polymers identified as good or bad protein binders appear reasonable given our understanding of this process. However, the results are based on our polymer data sets, which, for example, do not include truly zwitterionic polymers or polymers that are too hydrophobic and thus have low water solubility. Moreover, the results would change if the polymers are soluble or tightly anchored to surfaces. The results also reflect the protein mixtures used. We have used a combination of proteins with positive or negative net charge. Other features may dominate actual protein mixtures; therefore, the results may shift. What we have achieved, however, is the development of ML techniques based on GPR that can handle non-numerical input features.

Conclusions

Categorical features regularly appear in the data sets generated in polymer science. This is exemplified when trying to identify the composition of a polymer, and therefore, the type of chemically distinct monomers that bind well to a large variety of proteins. Here, we examined statistically principled approaches for modeling categorical inputs in GPR within the polymer informatics context. We compared five different methods proposed in the literature to model categorical features using a GPR model. We evaluated the performance of all five methods based on extensive simulation studies conducted on both nominal and ordinal categorical features. The simulation studies were designed primarily around three common scenarios encountered with categorical features in polymer science data sets. They were (1) in the presence of many categorical features, (2) in the presence of many levels in categorical features, and (3) in the existence of correlation between categorical features. We also implemented all five methods on a real-world data set based on polymer–protein binding and derived important insights on this data set.

Across our simulation designs, LVGP consistently matched or outperformed the alternative methods and remained stable as complexity increased. Further, the LVGP method also performed the best on the real-world polymer–protein data set. We found using the LVGP method that a statistical polymer based on neutral (A), cationic (D) and aromatic repeating units (F) (A70-co-D20-co-F10) had the highest average binding across a large array of proteins. The polymer with the lowest binding poly­(C90-co-D10), which is composed of mainly negatively charged units (C), but also a few positively charged repeating blocks (D).

From the results of the simulation study and the real data application it is evident that LVGP is a competitive model to rely on when handling categorical features within a GPR model. However, it should be noted that the other methods also perform quite well. Further, while LVGP was the strongest overall in our study, these conclusions are design specific. Under different data generating regimes other methods can be preferable. For example, the multiplicative method may be preferred when smoothness over quantitative inputs is truly shared across categories; the additive method when factor contributions are largely independent and factor specific smoothness is present; EzGP when level combinations induce qualitatively different response surfaces; and LMGP when categorical inputs are conditional or variable length. Accordingly, method choice should be guided by prior knowledge of the system and exploratory diagnostics rather than by a single default decision.

Supplementary Material

lg5c00177_si_001.pdf (1.4MB, pdf)
lg5c00177_si_002.xlsx (50.6KB, xlsx)

The data set used in our polymer–protein study is provided in Table S11 of the Supporting Information. Code for the simulation study and the polymer protein study are available in this repository (https://github.com/z5492642/Handling-categorical-features-within-a-GPR-model-A-polymer-protein-study).

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acspolymersau.5c00177.

  • S1: Gaussian Process Regression (GPR) S2. Multiplicative kernel method S3. Additive kernel method S4. Easy to interpret Gaussian Process model (EzGP) method S5. Latent variable Gaussian Processes (LVGP) method S6. Latent map Gaussian Processes (LMGP) method S7. Simulation study for nominal categorical features S8. Simulation study for ordinal categorical features S9. Results for Simulation Study S10. Results for ordinal feature simulation Table S1. Functions used to generate continuous features for simulations. Table S2. probability values used for nominal feature simulations. Table S3. Mean R 2(±standard error) across 50 replications for the five methods fit to synthetic data sets with nominal categorical features (n = 100) under three scenarios: (i) increasing numbers of nominal predictors, (ii) increasing within-predictor cardinality (levels), and (iii) presence vs absence of correlation among nominal predictors. Best mean performance within each subscenario is highlighted in bold. Table S4. Mean R 2(±standard error) across 50 replications for the five methods fit to synthetic data sets with nominal categorical features (n = 500) under three scenarios: (i) increasing numbers of nominal predictors, (ii) increasing within-predictor cardinality (levels), and (iii) presence vs absence of correlation among nominal predictors. Best mean performance within each subscenario is highlighted in bold. Table S5. Mean R 2(±standard error) across 50 replications for the five methods fit to synthetic data sets with nominal categorical features (n = 1000) under three scenarios: (i) increasing numbers of nominal predictors, (ii) increasing within-predictor cardinality (levels), and (iii) presence vs absence of correlation among nominal predictors. Best mean performance within each subscenario is highlighted in bold. Table S6. Mean R 2(±standard error) across 50 replications for the five methods fit to synthetic data sets with ordinal categorical features (n = 100) under three scenarios: (i) increasing numbers of nominal predictors, (ii) increasing within-predictor cardinality (levels), and (iii) presence vs absence of correlation among nominal predictors. Best mean performance within each subscenario is highlighted in bold. Table S7. Mean R 2(±standard error) across 50 replications for the five methods fit to synthetic data sets with ordinal categorical features (n = 500) under three scenarios: (i) increasing numbers of nominal predictors, (ii) increasing within-predictor cardinality (levels), and (iii) presence vs absence of correlation among nominal predictors. Best mean performance within each subscenario is highlighted in bold. Table S8. Mean R 2(±standard error) across 50 replications for the five methods fit to synthetic data sets with ordinal categorical features (n = 1000) under three scenarios: (i) increasing numbers of nominal predictors, (ii) increasing within-predictor cardinality (levels), and (iii) presence vs absence of correlation among nominal predictors. Best mean performance within each subscenario is highlighted in bold. Table S9. Descriptors for proteins obtained after PCA using the ESM-2 method Table S10. Mean R 2(±standard error) across 50 replications for the five methods fit to the real polymer–protein data set where protein descriptors are based on the ESM-2 method. Best mean performance is highlighted in bold. Table S11. Data set for polymer–protein binding application. Figure S1. Fluorescence spectra showing FRET Figure S2. Histogram of FRET values Figure S3. Average binding and corresponding uncertainty estimate for polymers predicted by the fitted LVGP model sorted in ascending order. Polymers in red are statistically significantly different from the polymer with the lowest average bin Figure S4. Actual vs Predicted FRET values in the test set for each method based on an 80:20 train-test split (PDF)

  • Table S7 (XLSX)

CRediT: Ramindu De Silva conceptualization, formal analysis, methodology, software, validation, writing - original draft; Wei Ge validation, writing - review & editing; Carolin Bapp data curation, writing - review & editing; Ahmed Mustafa data curation; Robert Chapman data curation, writing - review & editing; Yanan Fan formal analysis, supervision, writing - review & editing; Scott Sisson conceptualization, supervision, writing - review & editing; Martina H. Stenzel conceptualization, funding acquisition, project administration, supervision, writing - review & editing.

The authors declare no competing financial interest.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

lg5c00177_si_001.pdf (1.4MB, pdf)
lg5c00177_si_002.xlsx (50.6KB, xlsx)

Data Availability Statement

The data set used in our polymer–protein study is provided in Table S11 of the Supporting Information. Code for the simulation study and the polymer protein study are available in this repository (https://github.com/z5492642/Handling-categorical-features-within-a-GPR-model-A-polymer-protein-study).

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