# Serological Number for Characterization of Circulating Antibodies

^{*}

## Abstract

**:**

## 1. Introduction

_{D}, or the fractional concentration N

_{0}, of antibodies seem to be less relevant for the systemic analysis without knowing an immunogen. In addition, both parameters cannot be obtained in a high throughput manner in comparison with the binding motifs. However, it is interesting that both parameters have the same unit “mol/L” [9] and can build the dimensionless serological number, or S-number:

_{D}/N

_{0}

_{D}and such binding motifs. In this paper, we investigate the potential of the S-number as an additional parameter for studying the immune response and demonstrate the possibility of its determination via peptide arrays.

## 2. Results

#### 2.1. Analytical Estimation of the S-Number of Different Binding Motifs

_{bm}is the number of different binding motifs, S

_{k}is the serological number of the antibody with a number k, and k is the index of addition over all circulating antibodies. It is worth noticing that no special assumptions are needed to perform such additions. If the S-number relates to the entire humoral system, i.e., S = S

_{1}= S

_{k}, Formula (2) can be simplified to

_{k}/b

_{k}is constant for all k. After the addition over the fractional concentrations of all antibodies and averaging of the dissociation constants over Z

_{bm}, we obtain Formula (4), which states the relation between the S-number and the number of different binding motifs Z

_{bm}:

_{tot}is the total concentration of the antibodies in the serum. Applying in (4) the average $\overline{{K}_{D}}$ = 10

^{−8}mol/L and N

_{tot}= 10 g/L for the IgG antibodies, as well Z

_{bm}= 42 reported for the serum used in this paper, S = 0.007 is obtained.

#### 2.2. Dilution Measurements of the S-Number

## 3. Discussion

_{D}smaller by a factor ~3, i.e., has better affinity to the pathogen than the single peptide on a solid support.

_{c}= K

_{Dc}/ΔN. The cellular S-number Sc characterizes the generation of new plasma cells with a fractional concentration ΔN from the B cells with an improved cellular dissociation constant K

_{Dc}towards the antigen. The generated plasma cells in the bone marrow will, in turn, build a fractional S-number S

_{k}= K

_{Dc}/ΔN = K

_{Dk}/N

_{0k}. Here, the equations K

_{Dc}= β⋅K

_{Dk}and N

_{0k}= β⋅ΔN were taken into account. Thus, we see that the bone marrow transfers the serologic proportions from the B cells, with the improved affinity, to the circulating antibodies. The reason for a larger fractional concentration of the plasmablasts and their corresponding precursors—memory cells, with a smaller affinity to the antigen—may be due to the fact that B cells with smaller affinities reenter additional rounds of mutative replication and, therefore, they have a higher probability to be developed in larger amounts at the plasmablast stage.

## 4. Materials and Methods

#### 4.1. Equation for Measurement of the S-Number in Array Format

_{sat}is the maximum fluorescent signal at saturation, N the concentration of the analyte (antibody in our case) in the solution, and K

_{D}is the equilibrium dissociation constant [17]. K

_{D}can be measured in parallel for many molecules if one of the binding partners is arrayed in spots on a solid support and the other carries a fluorescent label, and if its concentration, N, is known [18].

_{0}, Formula (6) can be transformed to

_{obs}= I

_{sat}/2. This means that if the patient’s serum will be constantly diluted, the dilution coefficient will be equal to the S-number at the signal I

_{obs}= I

_{sat}/2. Actually, the S-number can be measured by other I

_{obs}and dilution values. The general formula has the form S = α∙k, by I

_{obs}= I

_{sat}/(α + 1), where α is an arbitrary positive number and can be defined from the fluorescent signal saturation curve. We chose I

_{obs}= I

_{sat}/2, because this value refers to IC50, known as half maximal inhibitory concentration and can be easily measured by the available analytic techniques. Please note, to determine the S-number, it is not necessary to know either the concentration of antibodies or coefficients that converts the fluorescent signal to concentration units. Such a simple approach is especially useful for measuring S-numbers by studying a large number of antibody interactions with combinatorically synthesized peptide arrays.

#### 4.2. Methods to Measure the S-number

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Substitutional analysis of peptide MVPEFSGSFPMR reveals the binding motif X–A/I/P/V–P–E–F–X–G–A/S–X–P–X–X [8]. The double spot array was used. Reproduced with permission, Copyright 2017, ELSEVIER.

**Figure 2.**Schematic of the long-term humoral memory via elimination of the long-lived plasma cells for “old” antigens (red and green nucleus) with plasmablasts with a new specificity (lilac nucleus).

N | Amino Acid Sequence | Characteristics of the Motif | S-number | Standard Deviation in % |
---|---|---|---|---|

1 | GGQVRSIHSGPT | heterogeneous motif | 0.00267 | 37 |

2 | KEVPALTAVETGAT | LXAXETX motif group, poliovirus motif | 0.00386 | 31 |

3 | MVPEFSGSFPMR | Staphylococcus aureus motif | 0.00620 | 56 |

4 | LIADLNAESTSR | heterogeneous motif | 0.00775 | 22 |

5 | VLSSTAIKVDSV | heterogeneous motif | 0.00865 | 49 |

6 | VMSVNASTTAAN | heterogeneous motif | 0.01183 | 41 |

7 | QMKAWFPQTTYD | KXXFPQXT motif | 0.01219 | 48 |

8 | LRPNAVQTDTLA | heterogeneous motif | 0.01302 | 39 |

9 | SWVLTATETGSS | LXAXETX motif group, poliovirus motif | 0.01427 | 34 |

10 | NPVEDYLDYSVI | NPVEXXX motif | 0.01490 | 60 |

11 | ETKSDDMLLSNV | heterogeneous motif | 0.01537 | 31 |

12 | AKIRMFLDTDYK | heterogeneous motif | 0.01841 | 58 |

13 | VDTINLPQNTIQ | heterogeneous motif | 0.02059 | 49 |

14 | TALDAVSTGFSW | heterogeneous motif | 0.02597 | 42 |

15 | QHWPTNVDSVTV | heterogeneous motif | 0.04362 | 24 |

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Palermo, A.; Nesterov-Mueller, A.
Serological Number for Characterization of Circulating Antibodies. *Int. J. Mol. Sci.* **2019**, *20*, 604.
https://doi.org/10.3390/ijms20030604

**AMA Style**

Palermo A, Nesterov-Mueller A.
Serological Number for Characterization of Circulating Antibodies. *International Journal of Molecular Sciences*. 2019; 20(3):604.
https://doi.org/10.3390/ijms20030604

**Chicago/Turabian Style**

Palermo, Andrea, and Alexander Nesterov-Mueller.
2019. "Serological Number for Characterization of Circulating Antibodies" *International Journal of Molecular Sciences* 20, no. 3: 604.
https://doi.org/10.3390/ijms20030604