A Predictive Model of Antibody Binding in the Presence of IgG-Interacting Bacterial Surface Proteins

Many bacteria can interfere with how antibodies bind to their surfaces. This bacterial antibody targeting makes it challenging to predict the immunological function of bacteria-associated antibodies. The M and M-like proteins of group A streptococci (GAS) exhibit IgGFc-binding regions, which they use to reverse IgG binding orientation depending on the host environment. Unraveling the mechanism behind these binding characteristics may identify conditions under which bound IgG can drive an efficient immune response. Here, we have developed a biophysical model for describing these complex protein-antibody interactions. We show how the model can be used as a tool for studying the binding behavior of various IgG samples to M protein by performing in silico simulations and correlating this data with experimental measurements. Besides its use for mechanistic understanding, this model could potentially be used as a tool to aid in the development of antibody treatments. We illustrate this by simulating how IgG binding to GAS in serum is altered as specified amounts of monoclonal or pooled IgG is added. Phagocytosis experiments link this altered antibody binding to a physiological function and demonstrate that it is possible to predict the effect of an IgG treatment with our model. Our study gives a mechanistic understanding of bacterial antibody targeting and provides a tool for predicting the effect of antibody treatments in the presence of bacteria with IgG-modulating surface proteins.


INTRODUCTION
Many bacteria can interfere with how antibodies bind to their surfaces. Opsonization, the process in which antibodies target pathogens through binding to facilitate phagocytosis, is essential for the defense against infections. However, several significant bacterial pathogens express surface proteins that interfere with this process by binding antibodies outside the antigen-binding region. A common target is the Fragment crystallizable (Fc) region on IgG and proteins expressing IgGFc-binding regions include protein G from group C and G streptococci (1), protein A from Staphylococcus aureus (2), and M and M-like proteins from group A streptococci (GAS) (3)(4)(5).
By binding the Fc region on IgG the bacteria can reduce the ability of the immune response to detect and eliminate the bacteria from the host organism through Fc-mediated phagocytosis (6,7). This makes it difficult to predict the immunological effect of bacteria-associated antibodies.
According to our previous findings (8), the IgGFc-binding of M and M-like proteins is more prominent in low IgG concentration environments, corresponding to that of the human throat milieu. At the higher IgG concentrations in plasma, IgG is mostly bound to the surface proteins via Fab. This suggests that the antibody binding orientation is dependent on the IgG concentration. The mechanism for this behavior is however unclear.
Streptococcus pyogenes causes more than 700 million uncomplicated throat and skin infections annually and can be at the root of rare but very serious invasive infections such as sepsis (9). It is estimated that group A streptococcal infections lead to more than 0.5 million deaths annually (9). Monoclonal antibody treatments have shown potential for a wide spectrum of disease. Despite this, there are very few monoclonal antibodies (mAbs) clinically available for treatment of bacterial infections (10,11). Furthermore, pooled human IgG is used clinically for treatment of certain invasive GAS infections, but studies show no clear effect of this treatment (12-15). A better understanding of the binding characteristics of M protein can help identify conditions under which bound IgG can drive an efficient immune response.
While there exist models of antibody binding to antigens that build on receptor-ligand kinetics and structural data (16)(17)(18), there is no model that describes the complex interaction of multiple antibody clones with varying affinities and multivalency. Moreover, no model takes Fc binding into account. The binding of IgG to M protein in the host environment depends on total concentration of IgG, concentration of the different IgG clones, concentration of bacterial surface proteins, number of Fabbinding sites, location of epitopes, and binding energies to these locations ( Figure 1A).
Here, we have developed a biophysical model for describing all of the effects described above. Our implementation of the model is computationally efficient and we provide an easy to use, publicly available MATLAB-based software. The model can be applied to binding of ligands to any linear protein and is used here to investigate IgG binding to M protein. Computations using our new model shed light on the implications of the IgGFcbinding affinity and demonstrate under which circumstances an effective antibody binding is prominent. We show how our model can be used as a tool for studying the behavior of various IgG samples by performing in silico simulations in relation to experimentally measured binding values. We use the model here to study the behavior of specific Fragment antigen-binding (Fab) and Fc binding, pooled IgG and IgG in serum from healthy individual donors. The model is also used to simulate the effect of antibody treatment, by calculating how binding of IgG in serum is altered as specified amounts of monoclonal or pooled IgG is added. Phagocytosis experiments link the altered antibody binding to a physiological function, and demonstrate that it is possible to predict the effect of an IgG treatment of GAS with our model.

Biophysical Modeling of Competitive Antibody Binding to M Protein
Here we introduce a statistical-physics-based theoretical model for the interaction between antibodies and bacterial surface proteins, and show in silico computations with this model in relation to measured binding curves and known affinities. The model describes the binding of IgG to M protein, taking into consideration the binding competition between the Fab and Fc regions as well between different IgG clones. Different IgG binding possibilities on M protein can be described as statistical weights, making it possible to derive the probability of binding to occur at a specific site, and whether it is Fab-or Fc-mediated.
We approached the modeling by implementing the computationally efficient multi-ligand transfer matrix method for competitive binding as described by Nilsson et al. and implemented for dual ligand binding to DNA molecules (19)(20)(21). M and M-like proteins are fairly straight and rigid (22). We therefore model M protein as a one dimensional lattice ( Figure 1B). For this linear-protein approximation, M protein is divided into equally large sites i. Each antibody is ascribed a size given in a number of sites representing the width it covers on M protein when bound. A polyclonal IgG sample is characterized by its number of clones, a range of affinities to specific sites and the proportion of the bulk concentration of each clone. We can thereby describe a polyclonal IgG sample as a distribution of affinities (23). Each antibody in this distribution is assigned binding sites on the M protein. Additionally, the Fc binding site is defined on the M protein. Every binding possibility has a certain statistical weight that depends on the binding site as well as the affinity and concentration of the IgG clone. The goal is to calculate the probability of a type of binding to occur at a specific site. The probability is given by Equation (1). Equation (1) is numerically evaluated by applying the transfer matrix method. A transfer matrix that includes all states, is defined for each site on M protein. The elements of the transfer matrix describe the statistical weight for a type of binding to occur at a specific site i, provided that the adjacent site i + 1 is in a certain state. Through a series of calculations the mean binding probability on M protein for any given set of parameters can then be determined. The parameters of this model are shown in Table 1 and the adopted values for the model implementation are described in Implementation of Model. A more detailed theoretical description is presented in section Competitive Antibody Binding Model and we have made our software publicly available. The computational time required for calculating the binding probability curves such as the ones shown in Figure 1 is <10 s on a standard laptop computer. Altogether we define a system in which a number of antibodies, characterized by their site-dependent affinities, compete for binding to different sites via both Fab and Fc. All possible binding states are identified and the probability of Fab and Fc binding can be calculated using the transfer matrix method.
To test the model, we show experimentally measured independent Fab and Fc binding of pooled human IgG with its corresponding theoretical fit calculated with the model It describes the probability for a type of binding to occur at this specific site i, provided that the adjacent site i + 1 is in a certain state. 5. Through a series of calculations the total binding probability on M protein for any given set of parameters is then determined. See section Competitive Antibody Binding Model for more details on the mathematical framework. (C) Competitive Fab and Fc binding can be predicted with the model using non-competitive experimental values. As an example of how this model can be applied, an affinity determination is performed on independently measured Fab and Fc binding of human pooled IgG to S. pyogenes strain SF370. Data is from three experimental repeats and the error bars indicate standard deviation (n = 3, mean ± SD). The dependent binding is thereafter predicted with the model (right panel). (D) Modeling of monoligand Fab-and Fc-specific binding. By performing calculations with the model for monoclonal antibody fragments with determined binding sites, we show that the model predicts binding as expected. The plots show normalized binding as a function of total IgG concentration. The experimental values were measured using flow cytometry. Data is from three experimental repeats and the error bars indicate standard deviation (n = 3, mean ± SD). SF370 with a GFP plasmid was added to a serial dilution of either the IdeS-cleaved monoclonal antibody αM or Xolair and thereafter stained with Fab or Fc specific AlexaFluor647-conjugated F(ab') 2 fragments, respectively. The best fit to the measured binding curves were calculated with the model. Binding affinities extracted from this calculation are given in each of the plots and the ideal binding curve fit serves as a control for the model output. The confidence interval of the affinity estimate is calculated through bootstrapping. ( Figure 1C). The affinities extracted from this model fitting were used to predict the competitive Fab and Fc binding ( Figure 1C).
To assess IgG interaction with M protein in its native form, the experimental values were measured using flow cytometry measurements of IgG bound to bacteria. This also allows the model to be applied to bacterial surface proteins without the need of experimental parameter determination with purified proteins. The pooled IgG has been cleaved at the hinge region using the IgG-cleaving enzyme IdeS (24), separating the divalent antibody fragments (F(ab') 2 ) from the Fc fragments prior to being added to the bacterial samples. The measured binding is thus non-competitive between the Fab and Fc region. Independent fits with the model yield affinity estimates for both Fab and Fc binding and binding curves of the competitive binding have been calculated using these. According to this calculation, the Fc binding is out-competed at high total IgG concentrations, which is in line with published findings (8). By performing calculations with the model for monoclonal antibody fragments with determined binding sites, we show that the model predicts binding as expected. To study Fab and Fc binding independently, IdeS cleaved monoclonal antibodies are used. The Fab binding is studied using a specific antibody to M protein [αM, (25)] and a human IgG clone with no specificity to M protein (Xolair) is used for the Fc-binding.
Binding curve fittings, which are performed with the model to values measured by flow cytometry as input data, yield affinity estimates that are verified with an ideal binding curve fitting ( Figure 1D). SF370 is added to a serial dilution of either IdeScleaved αM, or Xolair, and thereafter stained with Fab or Fc specific fluorescently conjugated F(ab') 2 fragments, respectively. The determined binding site locations are used to calculate the best fit with the model to the measured binding curve with the K D value as the unknown variable. The affinity estimates with the model are very similar to those with an ideal binding curve fitting and the calculated Fc affinity is in agreement with previously reported data (5).

Simulations of Fab and Fc Binding Capability of M and M-Like Proteins
To investigate what factors affect the IgG binding of M protein, the effect of the IgGFc affinity on binding is computed at different total IgG concentrations (Figures 2A,B). This was done for two different antibody samples; one with Fab affinities corresponding to that of pooled IgG and one with negligible Fab affinities. According to the first simulation (Figure 2A), the Fc affinity of protein H is precisely at the level to reach maximum Fc binding and minimum Fab binding for the range of physiologically relevant IgG concentrations. However, M protein does not seem to bind Fc at any of these concentrations for the pooled IgG sample. For the antibody sample with no Fab binding ( Figure 2B), M protein only binds Fc at serum-like concentrations. This simulation of the IgGFc affinity indicates that protein H has an impact on IgG binding in most host environments, whereas M protein only has an effect in plasma for hosts with very low M protein specificity. This illustrates how calculations with our model can shed light on the implications of IgG binding by bacteria and help determine circumstances under which an effective antibody binding is prominent.
Simulations were performed to study the clonotype-specific binding of a polyclonal IgG sample ( Figure 2C). These calculations indicate that the Fc binding mainly depends on the concentration of the clonotype present whereas Fab binding depends on the clonotype affinity and, at higher concentrations, the epitope location. As some clonotypes may have overlapping epitopes, these antibodies, despite having a higher affinity, may be outcompeted by other clones with slightly higher concentrations. Moreover, according to this simulation a few antibody clones constitute a large proportion of the total Fab binding at plasma-like concentrations. The clonotype-specific simulations illustrate that epitope location and competition between clones are important for the M protein-IgG interaction.

Estimation of Fab Binding for Polyclonal IgG Samples
We wanted to apply our model to situations that are relevant for both mechanistic understanding as well as for potential treatment scenarios, meaning we had to expand it to polyclonal IgG. Calculations are presented here for IVIgG and IgG in serum from healthy donors. The effect of adding specified amounts of specific or pooled IgG to human serum is computed with the model. IVIgG is a complex polyclonal IgG sample that presumably contains a large number of antibody clones with a broad spectrum of affinities to M protein. Fab binding of IVIgG to M protein is estimated by measuring binding curves to SF370 wild type as well as an M protein mutant ( Figure 3A). The difference between these binding curves should correspond to Fab-binding of IVIgG to M protein. The model fitting yields a mean affinity of 10 3.0±0.1 nM −1 and a range in log 10 of 0.55 ± 0.8. Figure 3A shows a heatmap over the error of the fit for the mean  shows normalized binding as a function of total IgG concentration for the measured binding of IVIgG to M protein together with the calculated best fit to the model output. Data is from nine experimental repeats and the error bars indicate standard deviation (n = 9, mean ± SD). The heatmap shows the error of the fit for the mean and range of the affinity distribution for a broad range of values. (B) Binding curves of donor serum to SF370. The behavior of IgG in human serum from healthy donors is studied with the model. As with IVIgG, the Fab binding of serum IgG to M protein is estimated to be the difference between binding of serum IgG to SF370 wild type and the M protein mutant. A model fitting is performed for one of the serum binding curves (donor 4). The right plot shows normalized binding as a function of total IgG concentration for the measured binding of serum IgG to M protein together with the calculated best fit to the model output.
and range of the affinity distribution, illustrating the sensitivity of the model to these parameters. We have thus measured binding of IVIgG to M protein and shown how we can estimate affinity values of this polyclonal sample using the model.

Model Fitting and Titration Simulations of Monoclonal and Pooled IgG
To test this model in host-environment like conditions, binding of IgG in serum from individual donors was studied. As with IVIgG, Fab binding of IgG to M protein in serum from healthy individual donors is measured and estimated with the model (Figure 3B). Interestingly, the Fab binding curves of the serum samples show large variations in binding to M protein. A model fitting performed for one of the serum binding curves is shown in Figure 3B together with its affinity estimates. We next calculate how this binding curve is altered when specified amounts of αM or IVIgG is added to the serum, corresponding to different routes of administration (Figure 4). We show simulations of how the binding curve for donor 4 is altered as the present IgG is increased by a range of percentages of either IVIgG or αM (Figure 4). A proportional increase in present IgG illustrates the case of intravenous IgG administration with an equally proportional dispersion to tissue environments. For the clinical use of IVIgG as treatment for invasive infections, about 200% of serum IgG (∼10 mg/ml) is typically administered (12). Additionally, we show simulations of how this binding curve is altered as specific concentrations of antibody are added at each IgG concentration. An increase of present IgG by a specified concentration (Figure 4) can represent administration by injection to an infected area or a non-proportional delivered local dose. These simulations illustrate how we can use this model to represent various pathophysiological scenarios of delivered local IgG dose.

Phagocytosis in Serum With Monoclonal or Pooled IgG
To evaluate how added monoclonal or pooled IgG affects phagocytosis of S. pyogenes in serum, 20 µg/ml of either αM or IVIgG is added to 5% of serum ( Figure 5). This corresponds to a 4% increase of antibody in full serum. According to the simulations, at 5% serum 20 µg/ml of αM gives higher binding than the same amount of IVIgG (Figure 4). The phagocytosis experiments are performed according to a standardized assay based on normalization of persistent association of prey and phagocyte (26). Association measured with Xolair-coated bacteria serves as a baseline of non-Fcmediated association. An increase in association is seen as αM or IVIgG is added to the serum (Figure 5A, see full curve in Supplementary Figure 5A). There is a shift in the Hill coefficient (from 1.06 > 1.20) with the addition of αM, showing that the association behavior has changed to a more cooperative effect of association (26). The association is slightly higher for αM than for IVIgG (MOP 50 = 29.65 ± 7.6 (mean ± SD) for αM and MOP 50 = 39.75 ± 11.8 for IVIgG) (Figure 5B). This difference is however not statistically significant (p = 0.3).
When considering internalization, the proportion of phagocytes that internalize seem to be increased at higher MOP when αM or IVIgG is added, with αM having the strongest effect ( Figure 5C, see full curve in Supplementary Figure 5B). Considering the area under curve (AUC) as a cumulative measure of internalization, we see that serum with αM has the highest total internalization [αM AUC = 20,641 ± 660 (mean ± SD)], and that adding IVIgG to serum does not considerably increase total internalization (IVIgG AUC = 19,439 ± 525 and serum AUC = 19,299 ± 498). The difference in internalization between αM and IVIgG is however not statistically significant (p = 0.15). From the simulations performed in Figure 4, we expect an increase in Fab binding at the αM condition tested with the phagocytosis experiments. Overall, we find that the presented increase in association and proportional internalization agrees well with the predicted results from the model.

DISCUSSION
In this article we have presented a framework for a competitive antibody binding model together with its potential applications. Our implementation of the model is computationally efficient and we provide an easy to use, publicly available MATLABbased software. We have experimentally verified the model and have shown how this model can be used to determine binding characteristics to M protein of both monoclonal and polyclonal IgG samples as well as IgG in a serum environment. Additionally, the model has been used to perform in silico computations to understand the inherent behavior of the IgG-M protein interaction, such as the effect of the IgGFc affinity on the overall binding. Ultimately, we have illustrated how this model can be used as a tool to predict the effect of antibody treatments by simulating how binding of IgG in serum is altered as specified amounts of monoclonal or pooled IgG is added.
The model that we have developed takes into account aspects of the protein-antibody interaction that hitherto have not been considered. The concentration dependency of the IgGFc-binding is in line with previous results (8) and the model behaves as an ideal binding curve for monoligand binding. Finally, we have shown that functional assays are consistent with simulations of antibody treatment. Due to the fact that the competitive binding model incorporates parameter values that could not be determined experimentally, simulations with the model require certain assumptions regarding these. Specifically, the number of clonotypes in pooled IgG or serum and how their affinities are distributed is generally unknown and has therefore been based on measurements of specific antibodies (23,27). While the model agrees well for these conditions, the values extracted from the model need to be understood in the context of the assumptions made regarding these parameters.
It is possible to generalize and implement this model for other ligands and surface proteins. The competitive binding model could be improved by incorporating other relevant ligands of M protein that are present in the host environment. The framework is designed as such that it can be extended to include other human proteins that bind M protein, such as fibronectin, fibrinogen, C4BP, and factor H. As long as ligand specific information is available the model is readily applicable to any linear protein. Another possible extension of this model would be to calculate IgG binding to other bacterial surface proteins. Protein G of group C and G streptococci and protein A of S. aureus are globular proteins with five and three IgGFc-binding regions, respectively. This would require detailed information on the protein structures and location of epitopes.
Prediction of the competitive binding of pooled IgG gives insight into the previously observed concentration dependency of IgGFc-binding (8). The Fc-binding affinity is constant and there exists a spectrum of Fab-binding affinities to multiple epitopes on M protein. Thus, with an increase of concentration of antibodies, the probability of high-affinity Fab-binding increases to an extent where the Fc-binding is outcompeted. This together with the previous findings indicates that IgGFc-binding by bacteria has evolved to execute its function most efficiently in IgG-poor environments such as saliva, and may explain why severe invasive infections in blood are so rare.
Our chosen approach for IgG affinity measurements is based on several considerations. We wanted to study a situation that was as close as possible to in vivo, in this case IgG binding to their antigens at the bacterial surface and with the bacterial surface proteins in their native conformation and molecular surroundings. We did not want our model to require purified protein, as this would limit the applicability of the model. Additionally, purified proteins might adopt conformations not present in vivo, potentially affecting the specific interactions that we are trying to measure. Specific to M protein, this protein has known issues with conformational stability in its purified form (28,29). Flow cytometry is a commonly used method for measuring the affinity of cell-bound proteins and their ligands (30,31). We find that our implementation of flow cytometry-based measurement yields resulting affinities in agreement with those published previously (5).
Simulations illustrate how we can use this model to represent various pathophysiological scenarios of delivered local IgG dose and give an insight into what these scenarios might implicate. According to these calculations, whether pooled IgG or a specific mAb gives rise to the highest Fab binding depends on both the amount and route of administration. Specifically, 4% antibody treatment with mAb gives more binding than IVIgG. However, the inverse is true at 48% administration and above. In fact, the binding increase of the mAb treatment saturates at relatively low percentages while the binding of IVIgG continues to increase at higher concentrations. A probable explanation for this is that there exists only one epitope for mAb whereas IVIgG has several clones with different epitopes (8), thus the mean binding on M protein can increase to a higher extent with IVIgG than with mAb. A specified local dose as treatment gives rise to higher binding with IVIgG than with mAb in low IgG environments. At high concentrations treatment with a specified dose of mAb increases binding more than with IVIgG. Even though the mean affinity of IVIgG is lower than that of mAb, as per our calculation, there exist antibody clones in IVIgG with higher affinity than mAb. These results suggest that high affinity IgG clones have the most effect in low IgG environments. The increase in Fab binding with the treatments should increase Fc-mediated association of cells and bacteria.
The results from the phagocytosis experiments are consistent with the simulation, albeit with no statistically significant differences. Internalization of bacteria by cells link the antibody binding to a physiological effect. Altogether these results indicate that there are several factors to consider when selecting an antibody treatment. Even though pooled IgG yields higher binding in certain scenarios, it is more difficult to characterize the effect of this binding than with mAbs. Furthermore, the delivered local dose to infected regions needs to be considered. In the future, this model could potentially be used to determine the most efficient antibody treatment and route of administration for a given infection, as well as assist in identifying the therapeutic range of an antibody treatment for a specific patient.

METHOD Competitive Antibody Binding Model
Protein-protein binding is in essence an interaction resulting in a stable complex with lower free energy than when the proteins are unbound. The binding is formed through cumulative attractive forces between the atoms in the proteins and the complexity of these large molecular energy systems requires them to be described statistically.
By treating different binding possibilities on M protein as statistical weights and using statistical physics, a formula for the probability of a type of binding to occur at a specific site could be derived. The straight and rigid structure of M protein enables us to model it as a one-dimensional lattice. This lattice is divided into N number of sites, each denoted i. Consider The type of binding l for antibody clone S is characterized by a site-dependent binding constant K s,l (i) that relates to its binding affinity. λ is the number of sites i an antibody covers when bound to a specific site on the lattice (Supplementary Figure 1). Each possible binding state has a statistical weight which is given by the binding constant as well as the present free concentration of the antibody clone c s , as c s K s,l (i) (Supplementary Figure 1). We aim to calculate the probability p s,l (i) of a type of antibody binding to occur at a specific site on M protein. The probability p s,l (i) is where Z is the partition function and Z s,l (i) is the sum over all allowed Boltzmann weighted states for site i.
To numerically evaluate 1 we implemented the multi-ligand transfer matrix method for competitive binding as described by Nilsson et al. (19). The number of configurations that need to be considered in order to evaluate 1 increases exponentially with N. Using the transfer matrix method the probability resulting from all these configurations can be calculated efficiently. The transfer matrix method is applied to calculate Z and Z(i) for this system. There exists M = 1 + lλS number of possible states for each site i. An M × M transfer matrix T(i) is defined for each site i. T(i; m, m ′ ) gives the statistical weight of i being in state m provided i + 1 is in state m ′ . A specific order that is chosen for the possible states together with their respective statistical weights is illustrated in Supplementary Figure 2. The centermost region of the antibody is assigned the statistical weight for binding. Explicitly, the elements of the transfer matrices can be described as follows: • Site i and i + 1 are both empty: T(i; 1, 1) = 1 • Site i is empty while i + 1 is covered by the leftmost segment of antibody s: T(i; 1, 1 + λs) = 1, If i + 1 is covered by the leftmost segment of any antibody, it is only possible for i to be either empty or covered by the rightmost segment of any other antibody. • Site i is covered by the rightmost segment of antibody s and site i + 1 is empty T(i; 2 + λ(s − 1), 1) = 1 • Both site i and i + 1 are covered by antibody s; excluding all cases in which the centermost region of s is bound to site i: T(i, m + 1, m) = 1. • Site i is covered by the rightmost segment of antibody s and site i + 1 is covered by the leftmost segment of another antibody s ′ : T(i; 2 + λs − 1, 1 + λs ′ ) = 1 • Site i is bound to the centermost region of antibody s, i + 1 is hence covered by the following part of antibody s. T(i; 1 + λs, λs) = c s K s,l (i) The cases described above cover all possible states and therefore the remaining elements in the transfer matrix equal 0. To illustrate a simple case for the following part of this section, we show the transfer matrix for one type of antibody S = 1, l = 3 and λ = 3. M is thus 10 and each transfer matrix takes the form Z s,l (i) and thereby p s,l (i) can be calculated using the transfer matrices. The partition function Z is where v(1) and v(N + 1) are column vectors of length M, and consist of all allowed states of the first and last site on M protein, considering the geometry of the antibody and bacterial surface.
where the projection operator O s,l has been introduced. O s,l selects states wherein a particular type of binding occurs. For instance, the projection operator for all states associated with Fab binding, i.e., states for which l = 1, 2 and s = 1, 2..S, O Fab is, as per the previously defined choice of enumeration In the same manner O Fc is 0 for all elements except for those corresponding to Fc-binding, i.e., the states for which l = 3 and s = 1, 2..S. The probabilities p Fab (i) and p Fc (i) can thus be calculated for all sites on the M protein. The mean probability that sites are occupied by Fab or Fc bound antibodies for the bacterial protein as a whole is then where X = Fab or Fc.

Implementation of Model
In the following section we describe how the model has been implemented for the simulations in this article and present the reasoning behind the adopted values for parameters that could not be determined experimentally. The parameters for this model are N, λ, K s,l (i), c s and S.
To start with, for all simulations of polyclonal IgG each antibody s is assigned one binding site with an arbitrary location. Since the binding site locations of specific antibodies in polyclonal samples are generally unknown, these have been generated through randomization. Antibodies typically recognize a particular sequence of amino acids and thus have high affinity to a specific epitope. There may nevertheless occur binding to other epitopes on the bacterial protein with lower affinities. This aspect is incorporated in the formalism of the model. However we have made the simplification in the implementation that each antibody s only binds to a specific site i.
As the ratio between N and λ corresponds to the proportion of M protein that an antibody covers when bound, these parameters have been based on the width of the antibody relative to the length of the bacterial protein. λ was thus set to 7 and N set to 33 (8) and increasing N and λ proportionally from this did not affect the result (Supplementary Figure 4).
A polyclonal IgG sample is defined here by its number of clones S, a range of affinities to specific sites K s,l (i) and the concentration of each clone c s . Polyclonal samples such as pooled IgG presumably contain a large number of antibody clones with a broad spectrum of affinities to M protein. Previous studies suggest that serum from an immunized individual can have >100 specific IgG clones (27). Additionally, due to the arbitrary assignment of binding sites, computations with the model are more stable for an increasing amount of IgG clones (Supplementary Figure 3). Therefore, all calculations have been performed for 50-200 IgG clones. Affinity measurements of 1,450 monoclonal antibodies show that the exponentials of the K D values are normally distributed, suggesting that the dissociation constant for IgG is an inherently exponential measurement (23). We have therefore characterized antibody affinities in polyclonal samples as a normal distribution of the K D exponents. The range and mean affinities of these distributions are determined by fitting to experimentally measured values. All fittings to measured data have been done using a weighted least squares method. The free concentration of antibodies is attained through an iterative minimization with the total concentration as initial value. The implementation of the model has been written in MATLAB, and is available through GitHub (https://github.com/nordenfeltLab/ competitive-polyclonal-antibody-binding-model).

Bacterial Culturing Conditions
S. pyogenes strain SF370 wildtype and M mutant (32) were cultured overnight in THY medium (Todd Hewitt Broth; Bacto; BD, complemented with 0.2% (w/v) yeast) at 37 • C in an atmosphere supplemented with 5% CO 2 . Strain SF370 expresses M1 protein on its surface and is available through the American Type Culture Collection (ATCC 700294) (33). For the binding curve measurements SF370 expressing GFP was cultured with 1 µg/ml erythromycin. The bacteria were harvested at early log phase and washed twice with PBS. For the assays in which live bacteria were used 0.2% glucose was added to the PBS.

Antibody Preparation
Human serum collected from four healthy individuals was heat-inactivated at 56 • C for 30 min and stored in aliquotes at −20 • C. Intravenous immunoglobulin (IVIgG; Octagam, Octapharma) is IgG pooled from >3,500 healthy donors. Xolair (Omalizumab, Novartis) is a humanized monoclonal IgG that is IgE-specific, and thus only binds to M protein via Fc. αM is an M protein specific antibody (25). Based on reference values, the concentration of IgG in full serum was set to 10 mg/ml (34).

Opsonization of Bacteria
Serum and all antibody samples were incubated with IdeS, an enzyme that cleaves IgG at the hinge region, separating the F(ab') 2 from the Fc fragments (24) (1 µg/ml to 5 µg/ml depending on the maximum IgG concentration of the serial dilution) at 37 • C overnight. Bacteria were sonicated (VialTweeter; Hielscher) for 0.5 min to separate any aggregates and incubated with serial dilutions of serum, IVIgG, Xolair, and αM for 30 min on shake (400 rpm) at 4 • C to minimize metabolic activity of the bacteria. Unbound antibodies were washed away after incubation with PBS (3,220 g 3 min) 2-4 times, depending on the maximum IgG concentration of the serial dilution.

Flow Cytometry Acquisition
Flow cytometry acquisition was performed with CytoFLEX (Beckman-Coulter) with 488-and 638 nm lasers, and 525/40 FITC and 660/10 APC filters. The bacteria were gated by SSC-H and the GFP signal using control sample with only bacteria. The antibody signal background gating was performed using control with only bacteria and the antibody staining (Alexa Fluor 647-conjugated F(ab') 2 Fragment Goat Anti-Human IgGFab or IgGFc). Acquisition was set to run until at least 20,000 events were collected in the target population or 200 µl of sample had run through.

Analysis
The data was analyzed using MATLAB and files are available on GitHub. For serum and IVIgG samples, binding to M protein was derived as the difference between mean wildtype and M binding. The standard deviation is the square root of the sum of the variances of wildtype and M. Binding curve values were normalized to an interpolated saturation level prior to being evaluated with the model implementation. Measured binding curves are shown as mean and standard deviation of data points as described in the figure legends above. Affinity values were derived by minimizing the mean squared error of the model output and measured data using a MATLAB minimization function. K D values extracted from an ideal binding curve were derived in the same manner for a monoligand dose-response curve together with the measured data. The accuracy of predicted affinity estimates was calculated using the bootstrap method (35) and is the confidence interval calculated from 50 resamplings of the measured data.

Cell Lines
THP-1 cells (ATCC TIB-202, male) were maintained in Roswell Park Memorial Institute medium 1640 (Sigma-Aldrich) supplemented with 10% fetal bovine serum (Gibco) and 2 mM GlutaMAX (Life Technologies). Cells were grown at 37 • C in a 5% CO 2 atmosphere. The cell density was kept between (0.2 − 1.0)x10 6 cells/ml with a viability over 95% and split to 0.5x10 6 cells/ml the day prior to experiment.

Heat-Killing of Bacteria
After harvesting the bacteria, they were washed two times with 10 ml PBS (Gibco) at 4,000 × g for 10 min and resuspended in 1 ml PBS. Heat-killing was done at 80 • C for 5 min on a vigorously shaking heat block, followed by cooling the bacteria rapidly on ice.

Labeling of Bacteria
Heat-killed bacteria were stained with 5 µM Oregon Green 488-X succinimidyl ester (Invitrogen) at 37 • C with gentle rotation and protected from light for 30 min. This was followed by a change of buffer to sodium carbonate (0.1 M, pH 9.0) and staining with the pH-sensitive dye CypHer5E (General Electric) 20 µg/ml for 2 h in room temperature under gentle rotation in the dark. The samples were washed two times with Na-medium (5 min, 8,000 × g, swing-out rotor). The stock of labeled bacteria was stored up to 5 days in fridge protected from light and was used for all phagocytosis experiment. Prior to opsonization bacteria were sonicated until (VialTweeter; Hielscher) any large aggregates of bacteria were disperse, which was confirmed by microscopy. Staining was checked by flow cytometry (CytoFLEX, Beckman-Coulter), for CypHer5E, 1 µl of sodium acetate (3 M, pH 5.0) was added to confirm pH-sensitivity.

Phagocytosis Assay
The cell density was kept at 0.5 × 10 6 cells/ml on the day of experiment. Medium was changed to Na-medium (10 min, 4,696 × g swing out rotor). The concentration of THP-1 cells was measured prior to phagocytosis by flow cytometry and adjusted to 2,000 cells/µl, amounting to 100,000 cells per well. A 96-well plate was prepared with the bacteria with a multiplicity of prey (MOP) of 0-400. 50 µl of THP-1 cells were thereafter added on ice, resulting in a final volume of 150 µl. The plate was directly transferred to 37 • C on a heating-block with moderate shaking protected from light or kept on ice as control for internalization. Phagocytosis was haltered by putting the samples on ice for at least 15 min before data acquisition. Three independent experiments were performed.

Data Acquisition
Flow cytometric acquisition was performed using CytoFLEX (Beckman-Coulter) with 488 and 638 nm lasers and filters 525/40 FITC and 660/10 APC. Threshold was set at FSC-H 70,000 for phagocytosis and for bacteria FSC-H 2000 and SSC-H 2000. Gain was kept at 3 for FITC and 265 for APC. Acquisition was set to analyze at least 5,000 events of the target population with a velocity of 30 µl/min taking approximately 60 min for all samples. Throughout the data acquisition the plate was kept on an ice-cold insert.

Data Analysis
Flow cytometry data were analyzed using FlowJo version 10.2 (Tree Star). The THP-1 cells were gated on forward (FSC) and side scatter (SSC) height, then doublets were excluded by gating on FSC-H versus FSC-A. THP-1 cells positive for Oregon Greensignal were defined as associating cells, and of those THP-1 cells, cells positive for CypHer5E-signal were defined as internalizing cells. Associating and internalizing cells are expressed as the percentage of the whole phagocytic population. The data was further analyzed using the PAN-method (26).

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.