A minimal model of peptide binding predicts ensemble properties of serum antibodies

Experimental setup

To study the impact of amino acid composition of random-sequence peptide libraries on measured signal intensity, serum samples from 15 BALB/c mice bred under specific pathogen-free (SPF) conditions were collected. These mice were infected with HB (Additional file 1, Figure S1). Further serum samples were collected at 10 dpi (days post infection; 15 samples), at 14 dpi (13 samples) and at 18 dpi (15 samples) totaling 58 serum samples. Microarrays of n_Pep = 255 random-sequence peptide probes (hereafter referred to as standard library) were incubated with the serum samples. The peptide arrays used have been shown to be suitable for serological diagnostics by Bongartz et al. [10]. Each probe consisted of l = 14 out of 20 proteinogenic amino acids. IgM and IgG antibody binding was simultaneously detected by means of isotype specific fluorochrome-labeled polyclonal secondary antibodies. In addition to serum samples, the peptide library was incubated separately with 13 different human monoclonal IgG antibodies.

The fluorescence signal intensities were read, log-transformed and corrected for the signal from the polyclonal secondary antibody binding directly to the peptide probes. Subsequently, the signal intensities were mean-centered and scaled to unit variance, which resulted in a normalized vector $\overrightarrow{s}$ for each IgM and IgG serum sample and for each of the 13 monoclonal antibodies. We use the terms signal intensity or antibody binding profile interchangeably to denote $\overrightarrow{s}$. Each signal intensity vector $\overrightarrow{s}$ has as many components as there are peptides in the standard random peptide libary. For brevity, our analysis focuses on the IgM data. The IgG data can be found in the Supporting Information (Additional file 2, Figure S2, Additional file 3, Figure S3, and Additional file 4, Figure S4). More details on the experimental setup and normalization procedures can be found in Methods.

A regression model based exclusively on peptide amino acid composition predicts antibody binding profiles

where $\overrightarrow{s}\left(255\times 1\right)$ is the signal intensity vector and X the amino acid composition matrix (AACM) of the peptide library. The X matrix is formed by counting the occurrences of each of the 20 amino acids in each peptide which results in a matrix with 20 columns and 255 rows. Importantly, X does not contain information about the position of an amino acid in a given peptide sequence.

The AAWS vector $\overrightarrow{w}\left(20\times 1\right)$ indicates the contribution of every amino acid to the measured signal intensity. Furthermore, the residual of the regression model, $\overrightarrow{\epsilon }$, captures the part of $\overrightarrow{s}$ which cannot be explained by X alone. AAWS and residuals were estimated by partial least squares regression (PLS) (see Methods for details on the data analysis).

Once the vector $\overrightarrow{w}$ has been estimated, we use the regression model to predict measured signal intensities given the peptides' amino acid composition. Figure 1 illustrates that the predicted signal intensities $\widehat{\overrightarrow{s}}=\mathbf{X}\overrightarrow{w}$ are in good agreement with signal intensities $\overrightarrow{s}$ measured for the serum of one healthy BALB/c mouse. In order to evaluate the performance of the regression model, we focus on the predictive performance, Q², which was determined by 10-fold cross-validation (Methods). The predictive performance equals 1 for perfect predictions and is close to zero for poor predictions.

All 58 BALB/c serum samples resulted in a median predictive performance of 0.39 (Figure 2).

A minimal model of antibody-peptide binding

We hypothesize that the high predictive performance of our regression model is due to properties of an antibody ensemble. We test this hypothesis with the help of a model that simulates binding between peptides and antibodies. In this model, the binding affinity of simulated monoclonal antibodies depends non-linearly on amino acid positions in the peptide sequences (Equations 2 and 4). The model we propose is similar to bit string models [23–26] in that it uses vectors as simple representations of peptides and antibodies. The peptide string is represented by unique real numbers taken from a vector of assigned AAWS, denoted $\overrightarrow{h}$, the twenty components of which were drawn from a uniform distribution on the closed interval 0[1]. A peptide $\overrightarrow{p^i}$ of l amino acids is thus represented by a vector of l numbers drawn from $\overrightarrow{h}$.

An antibody binding site is represented by a vector $\overrightarrow{a^k}$ of length l. The binding strength of each position is given by a number between -1 and 1 that is drawn randomly from a uniform distribution and is scaled such that $\overrightarrow{{\left(a^k\right)}^{\mathsf{\text{T}}}}\overrightarrow{a^k}=1$. The binding association between peptide $\overrightarrow{p^i}$ and antibody $\overrightarrow{a^k}$ is computed as the dot product of the two vectors, $y_{i,k}={\overrightarrow{{(a}^k)}}^{\mathsf{\text{T}}}\overrightarrow{p^i}$. Thus, the binding association y _{i, k} depends explicitly on an amino acid's position in a given peptide sequence.

where [Ab] _k is the concentration of antibody k with ${\sum }_{k=1}^{n_{\mathsf{\text{Ab}}}}{\left[\mathsf{\text{Ab}}\right]}_k=1$. The thermodynamic equilibrium association constant for antibody k binding peptide i is defined as $K_{i,k}=\mathsf{\text{exp}}\left(-\frac{{\text{Δ}}_r{G}^{\mathsf{\text{o}}}}{RT}\right)$ with ${\text{Δ}}_{r}G=\mathsf{\text{exp}}\left(\frac{{\beta }_0+{\beta }_1{y}_{i,k}}{RT}\right)$. Logarithmizing the results of Equation 2, and centering them to zero and unit variance, we obtained a vector of normalized simulated signal intensities ${\overrightarrow{s}}_{\mathsf{\text{sim}}}$. A more detailed description of the mathematical model can be found in Methods.

Simulations show that the prediction of antibody binding profiles based exclusively on peptide amino acid composition improves with increasing antibody diversity

We first simulated signal intensities for n_Ab = 150 binding to a simulated peptide library of 255 14-mers. The peptide library used in the simulation determines the amino acid composition matrix X_sim. We estimated simulated intensities ${\overrightarrow{s}}_{\mathsf{\text{sim}}}$ (Figure 3A) and respective weights ${\overrightarrow{w}}_{\mathsf{\text{sim}}}$ (Figure 3B) using the linear regression model ${\widehat{\overrightarrow{s}}}_{\mathsf{\text{sim}}}={\mathbf{X}}_{\mathsf{\text{sim}}}{w}_{\mathsf{\text{sim}}}^{\to }$. Prediction of simulated signal intensities yielded a predictive performance (Q²) of 0.40, and the correlation between $\overrightarrow{h}$ and ${\overrightarrow{w}}_{\mathsf{\text{sim}}}$ was found to be r = 0.92 (Figure 3B), which indicates a very good recovery of $\overrightarrow{h}$. Recall that signal intensities were simulated in an amino acid position-dependent manner, while the composition-based regression model (Equation 1) relies on the amino acid position-independent matrix X_sim.

Further, our simulation framework enabled us to show in silico that the predictive performance increases with growing antibody diversity (Figure 4A). The same is true for the pairwise correlation of computed AAWS $\left({\overrightarrow{w}}_{\mathsf{\text{sim}}}^i\right)$, which nears perfection (r = 1) with increasing antibody diversity (Figure 4B), as does the correlation of AAWS with $\overrightarrow{h}$ (Additional file 5, Figure S5). Therefore, when using a position-independent linear statistical model for the prediction of antibody-peptide binding, high antibody diversity is a prerequisite for good predictive performance.

Predictive performance differs for monoclonal and serum-antibody binding profiles

In order to test our in silico-based prediction that predictive performance depends heavily on antibody diversity when only taking into account the peptide library's amino acid composition, we compared the predictive performance of the 58 BALB/c mouse serum samples (antibody diversity n_Ab >> 1) with that of the 13 human monoclonal IgG antibodies (antibody diversity n_Ab = 1). We found both a significantly higher predictive performance (Figure 2A, p < 0.001) and significantly higher pairwise correlations between AAWS for serum antibodies (Figure 2B, p < 0.001) than for monoclonal antibodies, which confirms the predictions of our mathematical model (Figure 4).

Predictive performance decreases in the course of an HB-infection

In order to quantify the influence of immune response stage during HB-infection on predictive performance, we divided the mouse serum samples into three groups: healthy, acute phase (10 and 14 dpi), and early chronic phase (18 dpi) [28]. We found that predictive performance (Figure 5A) and pairwise correlation of AAWS decrease significantly in the course of the immune response (Figure 5B).

In order to compare the experimental results with the mathematical model, we simulated signal intensities for 100 random mixtures of 16000 different antibodies (Figure 6A and 6B, case I) and found that, when multiplicative Gaussian noise is introduced into the simulated signal intensities, both predictive performance and pairwise correlation of AAWS decrease (Figure 6A and 6B, case II). By increasing the concentration of one monoclonal antibody (the dominant antibody) to a sufficiently high level (Figure 6A and 6B, cases III and IV), predictive performance is decreased.

Stages of murine immune response differ in their amino acid-associated weights

In order to test whether the AAWS determined for all 58 BALB/c mouse serum samples were systematically different from one another, we applied principal component analysis to them. Together, the first two principal components yield a strong separation of healthy and diseased mice. Also, acute and early chronic samples separate (Figure 7). Thus, during an immune response against HB, AAWS change in a systematic way.

Average amino acid-associated weights of healthy mice correlate with amino acid physico-chemical properties but not with widely used amino acid scales for epitope prediction

Because of both the good predictive performance and the high pairwise correlation of AAWS of healthy BALB/c mice, we considered their average AAWS as representative of healthy BALB/c mice (Figure 8). The differences between weights in Figure 8 indicate the difference in contribution to normalized signal intensity corresponding to an amino acid substitution. Tryptophan, phenylalanine and tyrosine, all of which have aromatic residues, contribute most to the signal intensity.

AAWS represent a priority scale for peptide-antibody binding assigning to every amino acid the importance of contribution to the measured (or simulated) signal intensity. In addition, analogously to QSAR modeling, AAWS can a posteriori be conceived of as a vector representing correlates of the respective amino acids' physico-chemical properties. We therefore correlated the average AAWS (Figure 8) with the z-scale developed by Sandberg and colleagues [29]. The z-scale aggregates in matrix form 26 physico-chemical amino acid properties for every one of the 20 examined amino acids (Additional file 6, Figure S6). The average AAWS yield an absolute correlation coefficient higher than 0.3 with the following physico-chemical properties: side chain van der Waals volume, alpha-polarizability, absolute electronegativity, number of hydrogen bond donors, total accessible molecular surface area, and indicator of negative charge in side chain.

In order to compare the average AAWS with other published amino acid-scales for epitope prediction, we correlated them with propensity scales published by Parker and colleagues [17] (hydrophilicity), Kolaskar and Tongaonkar [30] (antigenicity), Chou and Fasman [16] (secondary structure) and by Emini and colleagues [18] (accessibility) and found the resemblance with them to be poor (absolute values of correlation coefficients smaller than 0.22). Notably, the compared propensity scales also do not highly correlate (range of correlation coefficients: -0.61 to 0.67).

Amino acid-associated weights are a compact, information-preserving representation of serum-antibody binding profiles

A minimal linear regression model defines AAWS as predictors that are based solely on the amino acid composition of a given peptide. For serum antibodies of BALB/c mice, AAWS account for up to 50% of variation in antibody binding profiles, whereas monoclonal antibodies generally show poor predictive performance values. The regression model performs best for healthy mice (median Q² = 0.43, Figure 5). Furthermore, we find AAWS to be comparable across healthy BALB/c mouse serum samples (Figure 5B). During the immune response against HB, however, predictive performance decreases steadily. Accordingly, pairwise correlations of AAWS are highest for healthy mice and decrease during the immune response (Figure 5). Therefore, we hypothesize that the average AAWS for healthy mice, shown in Figure 8, are a signature of health. AAWS of infected mice, in turn, are systematically different from AAWS of healthy mice and can be separated by principal component analysis.

Simulated unbiased antibody mixtures show ensemble properties

In order to interpret the reported experimental results, we built a mathematical model based on the law of mass action. We defined a property vector $\overrightarrow{h}$ that characterizes each peptide's amino acid binding strength. In this model, the binding signals for a given simulated monoclonal antibody depend on the amino acid's position in a given peptide.

For a single simulated antibody, AAWS calculated by the amino acid composition-based linear regression model generally yield neither good predictive performance nor a high correlation with assigned AAWS $\overrightarrow{h}$. However, highly diverse antibody mixtures with random--in the sense of an independent identically distributed--repertoire, and no dominant antibodies, show both a close to perfect predictive performance and recovery of assigned AAWS $\overrightarrow{h}$ (Figure 4 and Additional file 5, Figure S5). Our mathematical model thus predicts that high predictive performance and high correlation of estimated AAWS and $\overrightarrow{h}$ are ensemble properties of such antibody mixtures: the average affinity of these mixtures does not depend on the epitope's amino acid position anymore. In contrast to that, the monoclonal antibody-epitope affinities do. We call random and highly diverse antibody mixtures that are not biased by dominant antibodies "unbiased". In fact, introducing, in simulations, a dominant antibody by increasing the concentration of a single antibody decreases predictive performance (Figure 6A). In addition, we showed that noise reduces predictive performance (Figure 6A).

Serum samples of healthy BALB/c mice show signs of unbiased antibody mixtures

As shown in our mathematical model, unbiased antibody mixtures are characterized by high predictive performance values. In view of the relatively high predictive performance of antibody binding profiles of serum samples from healthy BALB/c mice, we postulate that these sera exhibit properties of unbiased antibody mixtures.

The first prerequisite for an unbiased mixture is high diversity. This requirement seems to be met. The potential antibody diversity is very high [31], and the functional diversity is estimated to be of the order of 10⁴ [32]. However, fulfillment of the second requirement, the independent identical distribution of antibody binding sites, is harder to claim. On the one hand, the antibody repertoire is composed of preexisting gene segments and shaped by clonal selection, but on the other hand, V(D)J recombination and--in later stages of an immune response--somatic hypermutation arrange and mutate these segments in a largely random fashion [1]. Our results suggest that randomness in fact prevails. This is consistent with the hypothesis that antibody repertoires can potentially recognize the entire antigenic universe [33, 34].

The predictive performance of healthy BALB/c mice is not perfect but amounts to a median of 0.43. This can be due to both noise and the fact that serum violates the assumptions of randomness to a certain degree. Noise may be caused by varying peptide spot quality on microarrays and by the experimental procedure itself. It is known that during a primary acute immune response, antibodies of a certain specificity for the antigen are produced in high abundance [35, 36]. Therefore, it can be expected that sera of infected mice deviate from the properties of an unbiased mixture and would have reduced predictive performance values. Indeed, this is corroborated by experimental results (Figures 5 and 7).

Unbiased mixtures represent a special case for which the use of propensity scales for epitope prediction is justified

The prediction of linear B-cell epitopes was first done by using propensity scales [19, 37, 38]. These scales assign a propensity value to every amino acid based on a priori studies of their physico-chemical properties. We found that our average AAWS, a posteriori termed signature of health (Figure 8), are not significantly correlated to widely used propensity scales.

Blythe and Flower tested 484 amino acid propensity scales on a set of 50 epitope-mapped proteins. They found that even the best set of scales perform only marginally better than random [39]. We show that unbiased mixtures represent a special case for which the converse holds true: antibody binding profiles of unbiased mixtures can be predicted based on AAWS. We show that the use of amino acid scales becomes increasingly less justified with increasing dominance of antibodies in a serum. In fact, each of Blythe and Flower's experiments used polyclonal antibodies raised against the whole protein [39]. We conjecture that the used polyclonal antibody mixtures were biased, that is, they contained dominant antibodies. In this regard, our study provides a possible explanation to Blythe and Flower's survey. More generally, our work suggests that results obtained with polyclonal antibody mixtures tend to be skewed by the inherent ensemble properties, which obscure the affinities of epitope-specific antibodies.

Technological features may bias amino acid-associated weights

We have shown that antibody mixtures exhibit ensemble properties. Resulting AAWS were shown to be consistent across healthy mice and qualitatively different from AAWS of infected mice (Figure 7). We have also provided a possible explanation for the difference between AAWS of healthy and infected mice: dominant antibodies in the course of the immune response.

However, the actual signature of health values shown in Figure 8 should be interpreted with caution. In addition to being indicative of both amino acid antibody binding preferences and physico-chemical properties (Figure 8 and Additional file 6, Figure S6), signal intensity may also be influenced by two other factors: (i) the accessibility of peptides and (ii) a possible interaction of aromatic amino acids and aromatic labeling dyes.

Accessibility may bias the resulting signal intensities systematically. For example, we find that cysteine contributes negatively to the signal intensity. This could partly be due to its ability to form disulfide bonds, which causes increased aggregation of cystein-containing peptides, and diminishes their surface exposure. This would lead to reduced antibody-peptide binding and accordingly to reduced signal intensity. Furthermore, it cannot be ruled out that aromatic amino acids interact via π-stacking with the aromatic labeling dyes Alexa Fluor 546 and 647 which are coupled to the secondary antibodies. Indeed, it has recently been found that TAMRA, another aromatic dye, cross-reacts with individual amino acids in a peptide sequence [40]. In order to minimize this effect, we performed secondary antibody correction on the log-transformed signal intensities.

Ethics Statement

Animals were housed and handled following national guidelines and as approved by our animal ethics committee.

Mice

BALB/c mice were bred and maintained under specific pathogen-free (SPF) conditions by the Department of Molecular Parasitology, Humboldt University Berlin, Berlin, Germany. Infection of mice with HB was carried out by oral gavage with 200 L3 stage larvae in distilled water.

Sera

Mice were narcotized and bled either by cardiac or retro-orbital puncture at the age of 8 weeks. Blood samples were collected from healthy SPF-BALB/c mice (n = 15), which were then infected with HB. Blood was collected at three time points post infection (dpi): at 10 dpi (n = 15), 14 dpi (n = 13) and 18 dpi (n = 15). The blood was allowed to clot at room temperature and centrifuged. The supernatant was stored at -20°C.

Monoclonal antibodies

The 13 human monoclonal antibodies were kindly provided by the group of Hedda Wardemann (Max Planck Institute for Infection Biology, Berlin, Germany). Ten different Ig gene sequences of IgG⁺ memory B cells from 2 healthy human donors, PN and VB, (PN115, PN138, PN16, PN89, VB1, VB142, VB161, VB176, VB18, VB4) [42] and three further ones from 3 other human donors ED38 [43], eiJB40 and mGO53 [44] were expressed as detailed in [45].

Random peptide library

The peptide library consists of 255 different 14-mer peptides. Their sequence was designed with a random generator. Repetitions of three or more consecutive amino acids were not allowed.

Peptide synthesis and microarray design

The peptide library was displayed in five identical sub-arrays on each slide purchased from JPT Peptide Technologies GmbH, Berlin, Germany. Furthermore, TAMRA-derived peptides, as internal fluorescence control, and mouse-IgM, mouse-IgG, human-IgM and human-IgG as secondary antibody controls, were included on each sub-array. Peptide microarrays were stored at 4°C.

Antibody binding assays

The microarrays were briefly immersed in 100% v/v ethanol, washed three times with T-PBS (phosphate buffered saline containing 0.05% w/v Tween20), three times with deionized water and dried by centrifugation. Since the microarray surfaces had been pre-treated to minimize unspecific binding of the target antibodies, no blocking step was required prior to incubation. All incubations were performed using a five-well adhesive incubation chamber (Multiwell GeneFrameTM, ABgene Germany, Hamburg, Germany) with a total assay volume of 45μ L per well. Serum was diluted 1:10 in T-PBS and monoclonal antibodies were applied in a concentration of 10μ g/mL. We showed in a technological case study that approximately 10μ g/ml of antibody are best for reliable signal intensity measurements [14]. The concentration of IgM in in the serum of healthy SPF BALB/c mice was found to be around 0.50 mg/ml [46], which yields 50μ g/ml for a 1:10 dilution. The diluted sera are thus within the optimal binding range. After incubation for four hours at room temperature, the microarrays were washed three times with T-PBS and three times with deionized water. Serum-antibody binding was detected with polyclonal goat anti-mouse IgM-Alexa Fluor 546 and polyclonal goat anti-mouse IgG-Alexa Fluor 647 (Invitrogen Ltd, Paisley, UK), simultaneously.

Monoclonal antibody binding was detected with polyclonal goat anti-human IgG Alexa Fluor 647 (Invitrogen Ltd, Paisley, UK). Secondary antibodies were diluted in T-PBS (20μ g/mL, 300μ L) and incubated for one hour at room temperature. The microarrays were washed three times with T-PBS, three times with deionized water, rinsed with running deionized water and dried by centrifugation. Water, ethanol and PBS were filtered.

Signal detection

By PLS-based computation of the intercepts, β₀ and β₁, we replaced log(I) with the resulting PLS-computed, mean-centered and scaled-to-unit variance residuals ε for further analysis. The results reported in the main text of this paper are based exclusively on the calculated normalized residuals.

Statistics

The two-sided, non-paired Wilcoxon rank sum test was used to compute all p-values. P-values were regarded as significant when p < 0.05. Association between variables was assessed by Pearson correlation (r) unless otherwise stated.

Generation of simulated signal intensities with a mathematical model

Peptides and antibody binding sites were modeled as strings. Binding strengths between antibodies and the various amino acid residues of a peptide, referred to as assigned AAWS $\overrightarrow{h}$, were sampled from the uniform distribution on the closed interval 0[1]. A binding site on an antibody $\overrightarrow{a^k}$ was simulated in a similar fashion with a random number from the closed interval [-1, 1] for every sequential position and scaled such that ${\overrightarrow{\left(a^k\right)}}^{\mathsf{\text{T}}}\overrightarrow{a^k}=1$. The binding association between peptide $\overrightarrow{p^i}$ and antibody $\overrightarrow{a^k}$ was calculated by $y_{i,k}={\overrightarrow{{(a}^k)}}^{\mathsf{\text{T}}}\overrightarrow{p^i}$.

Similar to a bit string model approach in [47], our approach to calculating K _{i, k} assumes additivity in free energy of binding, an assumption that is supported by experimental results [48, 49]. The signal intensity that we measure on the array is assumed to be proportional to the ratio of bound-to-total surface of the peptide spot, S _i . An expression for this quantity, based on the law of mass action, can be obtained from classical Langmuir adsorption theory [27] resulting in Equation 2 with R = 8.314472, T = 273.15 + 25, β₀ = 0 and β₁ = RT.

At last, signal intensities were log-transformed, mean-centered, and scaled to unit variance. If Gaussian noise ($\mathcal{N}$(μ = 0, σ = 0.01)) was introduced into simulated signal intensities, the noise term was introduced before logarithmic transformation of the data. We showed that, for monoclonal antibodies, visibly fluorescent spots have at least a K-value of 10⁷M^-1 [14].

Partial least squares regression

All calculations involving PLS were carried out with the pls package [50] for the R statistical programming environment [51].

Model diagnostics

The vector ${\overrightarrow{s}}_{\mathsf{\text{Leftout}}}$ is the left-out test data set, the signal intensity of which is predicted (${\widehat{\overrightarrow{s}}}_{\mathsf{\text{Leftout}}}$) from the remaining training data set. The left-out test data represented randomly chosen 10% of the total data set.

Principal component analysis

Principal component analysis was performed using the pcaMethods R-package [52].