BACKGROUND
the functional antibody repertoire , the set of all antibodies produced by plasma cells at any one time, determines the immune system's perception of the antigen universe.
the fabr is shaped throughout the life of an individual by various stages and selection events during b cell development that take place in the fetal liver, in the bone marrow and in secondary lymphatic organs.
as the fabr is subject to constant change due to continuous antigen encounter and establishment of immunological memory  <cit> , it encompasses a variety of specificities and affinities for a wide range of antigens  <cit> .
the fabr's investigation thus provides the possibility to gather information about both past and on-going immune responses, and ultimately about the immune state of the body  <cit> .
since the fabr is highly diverse and the production of antibodies is a hallmark of many infectious and autoimmune diseases, high-throughput immunoblot and microarray technologies have been used intensively for large-scale profiling of serum antibody binding  <cit> .
antibody profiling data is widely used for serological diagnostics by exploiting the fact that sera of control and diseased individuals may differ substantially in their fabrs  <cit> .
currently, serum-antibody profiling is usually performed by incubating a serum sample with a peptide or protein microarray.
afterwards, the reactivity of antibodies is estimated by measuring the fluorescence from a fluorochrome-coupled secondary antibody that binds to the constant region of the subset of serum antibodies studied  <cit> .
the importance of peptide microarrays as a tool for serological diagnostics has strongly increased over the last decade.
however, interpretation of the binding signals is still hampered by our limited understanding of the technology  <cit> .
this is in particular true for arrays probed with antibody mixtures of unknown complexity, such as sera.
to gain insight into how signals depend on peptide amino acid sequences, we probed random-sequence peptide microarrays with sera of healthy and infected mice.
for prediction of antibody binding profiles, we use a multivariate regression model based exclusively on the peptide library's amino acid composition without taking into account amino acid positional information.
this approach is related to methods of linear b cell epitope prediction which rely on propensity scales for epitope prediction  <cit> .
our method contrasts, however, with previously reported quantitative structure-activity relationship  modeling which, in conjunction with physico-chemical properties, relates amino acid positions and amino acid compositions of peptides and monoclonal antibodies to various response variables  <cit> .
we propose to examine, in vitro and in silico, the extent to which the validity of our approach depends on the composition of antibody mixtures.
the regression model led to the definition of amino acid-associated weights  as predictors of antibody-peptide reactivity.
we found that the position-independent peptide amino acid composition accounts for up to 40-50% in variation of antibody-peptide binding for healthy mice.
we demonstrate with a mathematical model the ensemble properties of highly diverse, random antibody mixtures in which no antibody dominates.
we call these mixtures "unbiased" and show that the properties of unbiased mixtures are the foundation to a high predictive performance of aaws.
we hypothesize that serum antibodies of healthy individuals resemble an unbiased mixture, while during an acute immune response, specific antibodies dominate antibody-peptide binding thus lowering predictive performance.
based on in silico and in vitro evidence, our work thus suggests that the faithfulness of antibody-peptide binding prediction with propensity scales  <cit>  decreases with increasing antibody dominance in a mixture.
RESULTS
in order to investigate the binding of antibody mixtures to large random-sequence peptide libraries, we asked two main questions: i) what is the impact of the peptides' amino acid composition on the binding to serum antibodies, ii) and how does the serum-antibody composition influence binding prediction?
experimental setup
to study the impact of amino acid composition of random-sequence peptide libraries on measured signal intensity, serum samples from  <dig> balb/c mice bred under specific pathogen-free  conditions were collected.
these mice were infected with hb .
further serum samples were collected at  <dig> dpi , at  <dig> dpi  and at  <dig> dpi  totaling  <dig> serum samples.
microarrays of npep =  <dig> random-sequence peptide probes  were incubated with the serum samples.
the peptide arrays used have been shown to be suitable for serological diagnostics by bongartz et al.
<cit> .
each probe consisted of l =  <dig> out of  <dig> 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  <dig> 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 s → for each igm and igg serum sample and for each of the  <dig> monoclonal antibodies.
we use the terms signal intensity or antibody binding profile interchangeably to denote s →.
each signal intensity vector 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 .
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
we built a linear statistical model to relate the amino acid composition of our peptide library to measured signal intensities
s →=xw →+ε→,
where s → is the signal intensity vector and x the amino acid composition matrix  of the peptide library.
the x matrix is formed by counting the occurrences of each of the  <dig> amino acids in each peptide which results in a matrix with  <dig> columns and  <dig> rows.
importantly, x does not contain information about the position of an amino acid in a given peptide sequence.
the aaws vector w → indicates the contribution of every amino acid to the measured signal intensity.
furthermore, the residual of the regression model, ε→, captures the part of s → which cannot be explained by x alone.
aaws and residuals were estimated by partial least squares regression  .
once the vector w → has been estimated, we use the regression model to predict measured signal intensities given the peptides' amino acid composition.
figure  <dig> illustrates that the predicted signal intensities s → ^=xw → are in good agreement with signal intensities 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, q <dig>  which was determined by 10-fold cross-validation .
the predictive performance equals  <dig> for perfect predictions and is close to zero for poor predictions.
all  <dig> balb/c serum samples resulted in a median predictive performance of  <dig>  .
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 .
the model we propose is similar to bit string models  <cit>  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 h →, the twenty components of which were drawn from a uniform distribution on the closed interval  <dig> <cit> .
a peptide pi→ of l amino acids is thus represented by a vector of l numbers drawn from h →.
an antibody binding site is represented by a vector ak→ of length l. the binding strength of each position is given by a number between - <dig> and  <dig> that is drawn randomly from a uniform distribution and is scaled such that t→ak→= <dig>  the binding association between peptide pi→ and antibody ak→ is computed as the dot product of the two vectors, yi,k=→tpi→.
thus, the binding association yi, k depends explicitly on an amino acid's position in a given peptide sequence.
an expression for the simulated signal intensity, based on the law of mass action, can be obtained from classical langmuir adsorption theory  <cit> :
si=∑k=1nabkki,k1+ ∑k=1nabkki,k
where k is the concentration of antibody k with ∑k=1nabk= <dig>  the thermodynamic equilibrium association constant for antibody k binding peptide i is defined as ki,k=exp-Δrgort with Δrg=expβ0+β1yi,krt.
logarithmizing the results of equation  <dig>  and centering them to zero and unit variance, we obtained a vector of normalized simulated signal intensities s →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 nab =  <dig> binding to a simulated peptide library of  <dig> 14-mers.
the peptide library used in the simulation determines the amino acid composition matrix xsim.
we estimated simulated intensities s →sim  and respective weights w →sim  using the linear regression model s → ^sim=xsimwsim→.
prediction of simulated signal intensities yielded a predictive performance  of  <dig> , and the correlation between h → and w →sim was found to be r =  <dig>  , which indicates a very good recovery of h →.
recall that signal intensities were simulated in an amino acid position-dependent manner, while the composition-based regression model  relies on the amino acid position-independent matrix xsim.
further, our simulation framework enabled us to show in silico that the predictive performance increases with growing antibody diversity .
the same is true for the pairwise correlation of computed aaws , which nears perfection  with increasing antibody diversity , as does the correlation of aaws with h → .
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  <dig> balb/c mouse serum samples  with that of the  <dig> human monoclonal igg antibodies .
we found both a significantly higher predictive performance  and significantly higher pairwise correlations between aaws for serum antibodies  than for monoclonal antibodies, which confirms the predictions of our mathematical model .
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 , and early chronic phase   <cit> .
we found that predictive performance  and pairwise correlation of aaws decrease significantly in the course of the immune response .
in order to compare the experimental results with the mathematical model, we simulated signal intensities for  <dig> random mixtures of  <dig> different antibodies  and found that, when multiplicative gaussian noise is introduced into the simulated signal intensities, both predictive performance and pairwise correlation of aaws decrease .
by increasing the concentration of one monoclonal antibody  to a sufficiently high level , 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  <dig> 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 .
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 .
the differences between weights in figure  <dig> 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  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  with the z-scale developed by sandberg and colleagues  <cit> .
the z-scale aggregates in matrix form  <dig> physico-chemical amino acid properties for every one of the  <dig> examined amino acids .
the average aaws yield an absolute correlation coefficient higher than  <dig>  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  <cit>  , kolaskar and tongaonkar  <cit>  , chou and fasman  <cit>   and by emini and colleagues  <cit>   and found the resemblance with them to be poor .
notably, the compared propensity scales also do not highly correlate .
discussion
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 .
furthermore, we find aaws to be comparable across healthy balb/c mouse serum samples .
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 .
therefore, we hypothesize that the average aaws for healthy mice, shown in figure  <dig>  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 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 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 h → .
our mathematical model thus predicts that high predictive performance and high correlation of estimated aaws and 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 .
in addition, we showed that noise reduces predictive performance .
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  <cit> , and the functional diversity is estimated to be of the order of  <dig>  <cit> .
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, vj recombination and--in later stages of an immune response--somatic hypermutation arrange and mutate these segments in a largely random fashion  <cit> .
our results suggest that randomness in fact prevails.
this is consistent with the hypothesis that antibody repertoires can potentially recognize the entire antigenic universe  <cit> .
the predictive performance of healthy balb/c mice is not perfect but amounts to a median of  <dig> .
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  <cit> .
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 .
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  <cit> .
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 , are not significantly correlated to widely used propensity scales.
blythe and flower tested  <dig> amino acid propensity scales on a set of  <dig> epitope-mapped proteins.
they found that even the best set of scales perform only marginally better than random  <cit> .
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  <cit> .
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 .
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  <dig> should be interpreted with caution.
in addition to being indicative of both amino acid antibody binding preferences and physico-chemical properties , signal intensity may also be influenced by two other factors:  the accessibility of peptides and  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  <dig> and  <dig> 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  <cit> .
in order to minimize this effect, we performed secondary antibody correction on the log-transformed signal intensities.
CONCLUSIONS
we show that due to ensemble properties of unbiased mixtures, the position of amino acids in a linear epitope is no longer determinative for binding prediction.
we found that prediction of peptide-binding as well as consistence of aaws was best in sera of healthy balb/c mice.
therefore, we defined a signature of health characterizing the binding behavior of serum of healthy individuals.
this finding has far-reaching significance for the field of serological diagnostics.
furthermore, our findings have also deep implications for the field of b cell epitope mapping as we have discovered an important special case which enables amino acid scale prediction of peptide binding.
we show that amino acid scale prediction of peptide binding is justified only for unbiased mixtures.
for other cases, alternative methods have to be sought.
we thus showed that a knowledge of the composition of the used polyclonal mixture is essential for both the choice of the prediction method as well as the interpretation of results.
in the future, it will be of great interest to investigate the effects of a more detailed representation of binding in the mathematical model, and to study the effect of non-uniform antibody concentration distributions on predictive performance.
indeed, it has recently been shown for healthy zebrafish that the b cell clone repertoire follows a power-law distribution  <cit> .
thanks to our minimal assumptions approach, the conclusions of our model are independent of species, genetical background and individual exposure history.
future studies have to verify these predictions.

 the functional antibody repertoire, the set of all antibodies produced by plasma cells at any one time, determines the immune system's perception of the antigen universe. the fabr is shaped throughout the life of an individual by various stages and selection events during b cell development that take place in the fetal liver, in the bone marrow and in secondary lymphatic organs. as the fabr is subject to constant change due to continuous antigen encounter and establishment of immunological memory, it encompasses a variety of specificities and affinities for a wide range of antigens. the fabr's investigation thus provides the possibility to gather information about both past and on-going immune responses, and ultimately about the immune state of the body. since the fabr is highly diverse and the production of antibodies is a hallmark of many infectious and autoimmune diseases, high-throughput immunoblot and microarray technologies have been used intensively for large-scale profiling of serum antibody binding. antibody profiling data is widely used for serological diagnostics by exploiting the fact that sera of control and diseased individuals may differ substantially in their fabrs. currently, serum-antibody profiling is usually performed by incubating a serum sample with a peptide or protein microarray. afterwards, the reactivity of antibodies is estimated by measuring the fluorescence from a fluorochrome-coupled secondary antibody that binds to the constant region of the subset of serum antibodies studied. the importance of peptide microarrays as a tool for serological diagnostics has strongly increased over the last decade. however, interpretation of the binding signals is still hampered by our limited understanding of the technology. this is in particular true for arrays probed with antibody mixtures of unknown complexity, such as sera. to gain insight into how signals depend on peptide amino acid sequences, we probed random-sequence peptide microarrays with sera of healthy and infected mice. for prediction of antibody binding profiles, we use a multivariate regression model based exclusively on the peptide library's amino acid composition without taking into account amino acid positional information. this approach is related to methods of linear b cell epitope prediction which rely on propensity scales for epitope prediction. our method contrasts, however, with previously reported quantitative structure-activity relationship modeling which, in conjunction with physico-chemical properties, relates amino acid positions and amino acid compositions of peptides and monoclonal antibodies to various response variables. we propose to examine, in vitro and in silico, the extent to which the validity of our approach depends on the composition of antibody mixtures. the regression model led to the definition of amino acid-associated weights as predictors of antibody-peptide reactivity. we found that the position-independent peptide amino acid composition accounts for up to 40-50% in variation of antibody-peptide binding for healthy mice. we demonstrate with a mathematical model the ensemble properties of highly diverse, random antibody mixtures in which no antibody dominates. we call these mixtures "unbiased" and show that the properties of unbiased mixtures are the foundation to a high predictive performance of aaws. we hypothesize that serum antibodies of healthy individuals resemble an unbiased mixture, while during an acute immune response, specific antibodies dominate antibody-peptide binding thus lowering predictive performance. based on in silico and in vitro evidence, our work thus suggests that the faithfulness of antibody-peptide binding prediction with propensity scales decreases with increasing antibody dominance in a mixture.  in order to investigate the binding of antibody mixtures to large random-sequence peptide libraries, we asked two main questions: i) what is the impact of the peptides' amino acid composition on the binding to serum antibodies, ii) and how does the serum-antibody composition influence binding prediction? experimental setup to study the impact of amino acid composition of random-sequence peptide libraries on measured signal intensity, serum samples from balb/c mice bred under specific pathogen-free conditions were collected. these mice were infected with hb. further serum samples were collected at dpi, at dpi and at dpi totaling serum samples. microarrays of npep = random-sequence peptide probes were incubated with the serum samples. the peptide arrays used have been shown to be suitable for serological diagnostics by bongartz et al.. each probe consisted of l = out of 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 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 s → for each igm and igg serum sample and for each of the monoclonal antibodies. we use the terms signal intensity or antibody binding profile interchangeably to denote s →. each signal intensity vector 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. 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 we built a linear statistical model to relate the amino acid composition of our peptide library to measured signal intensities s →=xw →+ε→, where s → is the signal intensity vector and x the amino acid composition matrix of the peptide library. the x matrix is formed by counting the occurrences of each of the amino acids in each peptide which results in a matrix with columns and rows. importantly, x does not contain information about the position of an amino acid in a given peptide sequence. the aaws vector w → indicates the contribution of every amino acid to the measured signal intensity. furthermore, the residual of the regression model, ε→, captures the part of s → which cannot be explained by x alone. aaws and residuals were estimated by partial least squares regression. once the vector w → has been estimated, we use the regression model to predict measured signal intensities given the peptides' amino acid composition. figure illustrates that the predicted signal intensities s → ^=xw → are in good agreement with signal intensities 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, q which was determined by 10-fold cross-validation. the predictive performance equals for perfect predictions and is close to zero for poor predictions. all balb/c serum samples resulted in a median predictive performance of. 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. the model we propose is similar to bit string models 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 h →, the twenty components of which were drawn from a uniform distribution on the closed interval. a peptide pi→ of l amino acids is thus represented by a vector of l numbers drawn from h →. an antibody binding site is represented by a vector ak→ of length l. the binding strength of each position is given by a number between - and that is drawn randomly from a uniform distribution and is scaled such that t→ak→= the binding association between peptide pi→ and antibody ak→ is computed as the dot product of the two vectors, yi,k=→tpi→. thus, the binding association yi, k depends explicitly on an amino acid's position in a given peptide sequence. an expression for the simulated signal intensity, based on the law of mass action, can be obtained from classical langmuir adsorption theory: si=∑k=1nabkki,k1+ ∑k=1nabkki,k where k is the concentration of antibody k with ∑k=1nabk= the thermodynamic equilibrium association constant for antibody k binding peptide i is defined as ki,k=exp-Δrgort with Δrg=expβ0+β1yi,krt. logarithmizing the results of equation and centering them to zero and unit variance, we obtained a vector of normalized simulated signal intensities s →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 nab = binding to a simulated peptide library of 14-mers. the peptide library used in the simulation determines the amino acid composition matrix xsim. we estimated simulated intensities s →sim and respective weights w →sim using the linear regression model s → ^sim=xsimwsim→. prediction of simulated signal intensities yielded a predictive performance of, and the correlation between h → and w →sim was found to be r =, which indicates a very good recovery of h →. recall that signal intensities were simulated in an amino acid position-dependent manner, while the composition-based regression model relies on the amino acid position-independent matrix xsim. further, our simulation framework enabled us to show in silico that the predictive performance increases with growing antibody diversity. the same is true for the pairwise correlation of computed aaws, which nears perfection with increasing antibody diversity, as does the correlation of aaws with h →. 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 balb/c mouse serum samples with that of the human monoclonal igg antibodies. we found both a significantly higher predictive performance and significantly higher pairwise correlations between aaws for serum antibodies than for monoclonal antibodies, which confirms the predictions of our mathematical model. 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, and early chronic phase. we found that predictive performance and pairwise correlation of aaws decrease significantly in the course of the immune response. in order to compare the experimental results with the mathematical model, we simulated signal intensities for random mixtures of different antibodies and found that, when multiplicative gaussian noise is introduced into the simulated signal intensities, both predictive performance and pairwise correlation of aaws decrease. by increasing the concentration of one monoclonal antibody to a sufficiently high level, 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 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. 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. the differences between weights in figure 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 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 with the z-scale developed by sandberg and colleagues. the z-scale aggregates in matrix form physico-chemical amino acid properties for every one of the examined amino acids. the average aaws yield an absolute correlation coefficient higher than 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, kolaskar and tongaonkar, chou and fasman and by emini and colleagues and found the resemblance with them to be poor. notably, the compared propensity scales also do not highly correlate. discussion 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. furthermore, we find aaws to be comparable across healthy balb/c mouse serum samples. 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. therefore, we hypothesize that the average aaws for healthy mice, shown in figure 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 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 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 h →. our mathematical model thus predicts that high predictive performance and high correlation of estimated aaws and 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. in addition, we showed that noise reduces predictive performance. 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, and the functional diversity is estimated to be of the order of. 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, vj recombination and--in later stages of an immune response--somatic hypermutation arrange and mutate these segments in a largely random fashion. our results suggest that randomness in fact prevails. this is consistent with the hypothesis that antibody repertoires can potentially recognize the entire antigenic universe. the predictive performance of healthy balb/c mice is not perfect but amounts to a median of. 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. 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. 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. 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, are not significantly correlated to widely used propensity scales. blythe and flower tested amino acid propensity scales on a set of epitope-mapped proteins. they found that even the best set of scales perform only marginally better than random. 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. 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. 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 should be interpreted with caution. in addition to being indicative of both amino acid antibody binding preferences and physico-chemical properties, signal intensity may also be influenced by two other factors: the accessibility of peptides and 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 and 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. in order to minimize this effect, we performed secondary antibody correction on the log-transformed signal intensities.  we show that due to ensemble properties of unbiased mixtures, the position of amino acids in a linear epitope is no longer determinative for binding prediction. we found that prediction of peptide-binding as well as consistence of aaws was best in sera of healthy balb/c mice. therefore, we defined a signature of health characterizing the binding behavior of serum of healthy individuals. this finding has far-reaching significance for the field of serological diagnostics. furthermore, our findings have also deep implications for the field of b cell epitope mapping as we have discovered an important special case which enables amino acid scale prediction of peptide binding. we show that amino acid scale prediction of peptide binding is justified only for unbiased mixtures. for other cases, alternative methods have to be sought. we thus showed that a knowledge of the composition of the used polyclonal mixture is essential for both the choice of the prediction method as well as the interpretation of results. in the future, it will be of great interest to investigate the effects of a more detailed representation of binding in the mathematical model, and to study the effect of non-uniform antibody concentration distributions on predictive performance. indeed, it has recently been shown for healthy zebrafish that the b cell clone repertoire follows a power-law distribution. thanks to our minimal assumptions approach, the conclusions of our model are independent of species, genetical background and individual exposure history. future studies have to verify these predictions. 
to gain insight into how signals depend on peptide amino acid sequences, we probed random-sequence peptide microarrays with sera of healthy and infected mice. this is in particular true for arrays probed with antibody mixtures of unknown complexity, such as sera. it has implications for both serological diagnostics and b cell epitope mapping. our results indicate that position-independent amino acid-associated weights predict linear epitope binding of antibody mixtures only if the mixture is random, highly diverse, and contains no dominant antibodies. the importance of peptide microarrays as a tool for serological diagnostics has strongly increased over the last decade. experimental results suggest that sera from healthy individuals correspond to that case, in contrast to sera of infected ones. mathematical modeling shows that this position-independent ansatz is only adequate for highly diverse random antibody mixtures which are not dominated by a few antibodies. multivariate regression analysis relating peptide sequence to measured signals led to the definition of amino acid-associated weights. we analyzed the resulting antibody binding profiles with regression methods and formulated a minimal model to explain our findings.
 the importance of peptide microarrays as a tool for serological diagnostics has strongly increased over the last decade. however, interpretation of the binding signals is still hampered by our limited understanding of the technology. this is in particular true for arrays probed with antibody mixtures of unknown complexity, such as sera. to gain insight into how signals depend on peptide amino acid sequences, we probed random-sequence peptide microarrays with sera of healthy and infected mice. we analyzed the resulting antibody binding profiles with regression methods and formulated a minimal model to explain our findings.  multivariate regression analysis relating peptide sequence to measured signals led to the definition of amino acid-associated weights. although these weights do not contain information on amino acid position, they predict up to 40-50% of the binding profiles' variation. mathematical modeling shows that this position-independent ansatz is only adequate for highly diverse random antibody mixtures which are not dominated by a few antibodies. experimental results suggest that sera from healthy individuals correspond to that case, in contrast to sera of infected ones.  our results indicate that position-independent amino acid-associated weights predict linear epitope binding of antibody mixtures only if the mixture is random, highly diverse, and contains no dominant antibodies. the discovered ensemble property is an important step towards an understanding of peptide-array serum-antibody binding profiles. it has implications for both serological diagnostics and b cell epitope mapping. 