# Absolute Quantitation of Serum Antibody Reactivity Using the Richards Growth Model for Antigen Microspot Titration

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Microarray Production and Measurements

#### 2.2. Sample Handling and Signal Detection

#### 2.3. Measurements with 14D5 Monoclonal Antibody

#### 2.4. Analysis of the Microarray Data

#### 2.5. Fitting of Binding Curves

_{i}is the inflection point and d is the asymmetry parameter (for details see Supplementary Text S1).

**z**) with growth rate 1, where z is the serum dilution factor. Due to the linear dependence of the logarithm fluorescent intensity on the logarithm printed Ag concentration, the upper limit of the logarithm fluorescent intensity is a Richards function of the negative logarithm serum dilution z with growth rate 1.

## 3. Results

#### 3.1. Experimental Setup and Properties of the Measurement System

#### 3.2. Curve Fitting

^{1}* ln[Ag]. Shape parameter d of the generalized logistic function allows for asymmetry in the binding curve. In our assay, d is an index, which characterizes the Ab composition of serum. The location of the fastest growth x

_{i}with respect to ln [Ag] provides a general measure of antibody affinity.

^{−5}to 10

^{−11}and molar concentrations from pM to nM, respectively. In order to better assess interactions over this range and to weigh curve fitting against signal-intensity-dependent variation, we use logarithmically transformed signals of binding. Thus, fitting the Richards curve to our measurements therefore requires the logarithmic form of the above equation:

_{i}to be in the measurement range.

#### 3.3. Characterization of Anti-Deamidated Gliadin Peptide Serum Antibodies

_{D}values expected to occur in serum and peptide dilutions that extended well beyond that range. Serum samples were diluted to span about two orders of magnitude and correspond to dilutions conventionally used in serological diagnostics. Binding data were fitted using the lnR

_{2}(x) function of Equation (7) introduced above and generated curves were overlain on the binding data (Figure 2).

_{i}and ${\mathsf{\gamma}}_{\mathrm{Ab}}$ are sufficient to quantitatively characterize the distribution of Ab thermodynamic activity in the tested serum sample as long as the Richards curve models binding events.

#### 3.4. Measurement of Reference Monoclonal Antibody Properties

## 4. Discussion

_{D}of interaction, then the concentration of the analyte does not change significantly (<1%) during the measurement. The signal from a detection antibody is therefore correlated only with analyte concentration when other variables are kept constant, which conditions are characteristic of ambient analyte immunoassay [19]. The analyte in our assay is serum antibody and the K

_{D}of interactions are heterogeneous. Nevertheless, the amount of antigen and therefore its concentration in the total measurement volume is still negligible. Our method takes further advantage of microspots by varying the surface density of antigen for antibody capture. Since the relative concentrations of antibodies with different affinities are not changed by serum dilution, the affinity profile of antibodies that make contact with antigen on the solid support does not change either. Identical saturation of the antigen spots would give a linear increase of fluorescent signals in Figure 3A,B. The observed non-linearity indicates changing degrees of antigen saturation at a given serum antibody concentration. In physical chemistry the thermodynamic activity coefficient adjusts concentrations to effective concentrations, accounting for non-ideality of binding. Therefore, instead of fractional occupancy we use thermodynamic activity coefficient. Dependence of the activity coefficient of serum antibodies on the density of antigen [Ag] relative to the average affinity of interactions is modelled by the Richards function. By varying [Ag] we obtain a distribution curve of activity coefficients suitable for estimating values of the parameters of fitted function.

_{i}is related to the apparent standard chemical potential of serum antibodies. The lower the x

_{i}, the stronger the binding, similar to Ab-titration-based affinity determination approaches [15]. We propose to use the short name “lnK

_{D}” for the natural logarithm of molar concentration of antigen, x

_{i}, required to reach maximal relative growth of antibody activity coefficient γ

_{Ab}(inflection point of sigmoid curve in the log-lin scale). While the unit of chemical potential is that of energy (Joule/mole) here we would retain the unitless number derived from antigen concentration titration.

_{i}= A/2 to y

_{i}= A/e. The slope of the lnR

_{2}(x) function s = k/(d − 1) at minus infinity, in the case k = 1, meanwhile increases from s = 1 to ∞. This slope characterizes interactions as the antigen is diluted out to infinity. Infinite dilution is a special thermodynamic state when antigen molecules are in contact with antibody molecules only [47]. In this state, binding is determined only by interactions between Ag and Ab, without the interference of homotypic interactions. This ideal state is characterized by the limiting activity coefficient ${\mathsf{\gamma}}_{\mathrm{i}}^{\infty}$. Using our parametrization d-1 changed between 0 and 1. We propose that d − 1 is related to the limiting activity coefficient of Ab, ${\mathsf{\gamma}}_{\mathrm{Ab}}^{\infty}$, which characterizes the composition of the Ab mixture. We expect that parameter d can be used to characterize disease activity when antibody diversity and affinity is related to disease pathogenesis, with lower d implying immunological activity and higher d indicating the approach of equilibrium concentrations and stability of the immune response.

_{D}in a diagnostically relevant region of affinity is actually identified. By using microspots with antigen densities spanning the whole range of relevant concentrations, a single measurement can provide a distribution of apparent chemical potentials.

## 5. Conclusions

## 6. Patents

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | relative thermodynamic activity |

A | antibody binding capacity [FI] |

Ab | antibody |

[Ab] | antibody concentration [mol/L] |

AbAg | antibody–antigen complex |

[AbAg] | antibody–antigen complex concentration [mol/L] |

Ag | antigen |

[Ag] | antigen concentration [mol/L] |

B | auxiliary parameter |

BSA | Bovine Serum Albumin |

c | molar concentration [mol/L] |

C | auxiliary parameter |

d | shape parameter that determines asymmetry |

D | positive constant [L/mol] |

ELISA | Enzyme-Linked Immunosorbent Assay |

γ | thermodynamic activity coefficient |

IgA | immunoglobulin A |

IgG | immunoglobulin G |

k | rate of exponential growth |

K_{D} | equilibrium dissociation constant |

m | auxiliary parameter |

μ | chemical potential [J/mol] |

μ° | standard chemical potential [J/mol] |

mAb | monoclonal antibody |

PBS | Phosphate-Buffered Saline |

R | ideal gas constant [~8.314 J/K·mol] |

R(x) | Richards function |

RFI | Relative Fluorescence Intensity |

RIA | Radioimmunoassay |

s | slope of lnR(x) at minus infinity |

T | temperature [K] |

x_{i} | value of x at inflection point |

y_{i} | value of y at inflection point |

z | serum dilution factor [1/dilution] |

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**Figure 1.**Simultaneous titration of antigen density and serum antibody. Steps of the technology, starting with microarray fabrication and measurement, through image analysis to curve fitting and visualization of results are shown. Serum dilution is indicated by red drops, antigen density differences are represented by shades of blue circles. Several binding curves are transformed into one by fitting data with generalized logistic curves, yielding two parameters, x

_{i}and d, which characterize thermodynamic activity distribution of serum antibodies. Affinity is related to x

_{i}, the point of inflection; antibody heterogeneity is related to d, the asymmetry parameter. The slope at infinity “s” of the curve in the lower left corner is given by the equation shown.

**Figure 2.**Fitting in two dimensions. Examples of parallel measurements of IgG and IgA binding to diagnostic celiac disease peptide epitope are shown, with serum dilutions in (

**A**) three-fold and (

**B**) two-fold steps, as shown beside the panels, in five representative serum samples. Dot symbols stand for measurement data, lines are fitted curves.

**Figure 3.**Comparative characterization of serum antibody binding. Using the values of the parameters obtained by curve fitting, calculated thermodynamic activity coefficient distributions are comparable not only for (

**A**) serum samples but also for (

**B**) distinct antibody classes. Results shown in Figure 2 were used for the generation of normalized comparisons. Bar charts (

**C**) show mean d and x

_{i}values, and 95% confidence intervals as whiskers.

**Figure 4.**Conventional titration of serum. Mid-point and end-point titers of IgA and IgG are calculated by logistic fitting of binding to distinct antigen densities and are displayed as a function of antigen density. Logarithm of the titers at 5 tested antigen densities are represented by filled circles, connecting lines are linear interpolations. Error bars represent standard deviation.

**Figure 5.**Characterization of a monoclonal Ab for reference. Binding data of monoclonal Ab 14D15 was fitted using the same algorithm as for serum antibodies. Binding data and fitted curves (

**A**), calculated distribution of activity coefficient (

**B**) and estimates of parameters x

_{i}and d (

**C**) are shown.

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**MDPI and ACS Style**

Papp, K.; Kovács, Á.; Orosz, A.; Hérincs, Z.; Randek, J.; Liliom, K.; Pfeil, T.; Prechl, J.
Absolute Quantitation of Serum Antibody Reactivity Using the Richards Growth Model for Antigen Microspot Titration. *Sensors* **2022**, *22*, 3962.
https://doi.org/10.3390/s22103962

**AMA Style**

Papp K, Kovács Á, Orosz A, Hérincs Z, Randek J, Liliom K, Pfeil T, Prechl J.
Absolute Quantitation of Serum Antibody Reactivity Using the Richards Growth Model for Antigen Microspot Titration. *Sensors*. 2022; 22(10):3962.
https://doi.org/10.3390/s22103962

**Chicago/Turabian Style**

Papp, Krisztián, Ágnes Kovács, Anita Orosz, Zoltán Hérincs, Judit Randek, Károly Liliom, Tamás Pfeil, and József Prechl.
2022. "Absolute Quantitation of Serum Antibody Reactivity Using the Richards Growth Model for Antigen Microspot Titration" *Sensors* 22, no. 10: 3962.
https://doi.org/10.3390/s22103962