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

Abstract

In spite of its pivotal role in the characterization of humoral immunity, there is no accepted method for the absolute quantitation of antigen-specific serum antibodies. We devised a novel method to quantify polyclonal antibody reactivity, which exploits protein microspot assays and employs a novel analytical approach. Microarrays with a density series of disease-specific antigens were treated with different serum dilutions and developed for IgG and IgA binding. By fitting the binding data of both dilution series to a product of two generalized logistic functions, we obtained estimates of antibody reactivity of two immunoglobulin classes simultaneously. These estimates are the antigen concentrations required for reaching the inflection point of thermodynamic activity coefficient of antibodies and the limiting activity coefficient of antigen. By providing universal chemical units, this approach may improve the standardization of serological testing, the quality control of antibodies and the quantitative mapping of the antibody–antigen interaction space.

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“…where Ab b is bound antibody concentration, Ab t is total antibody, and Ag is total antigen concentration. We have previously described that Ab b is a logistic function of log(Ag) (see Supplementary Text S1 of [37]), which is the solution of this logistic differential equation. This means that the rate of change of generation of bound antibody while increasing Ag is determined by the actual bound antibody concentration, its relationship to the total antibody concentration, and the rate parameter.…”

“…The rate of change in this quantity, while changing reaction conditions by titrating antibody or antigen, is the derivative of the function that relates the amount of complex formed to the logarithm of the titrated component. We proposed the use of the generalized logistic model (GL) or Richards growth model [40,41] instead of 5PL [37], because the Richards growth function, like the 5PL, is the solution of the differential equation…”

“…The binding of antibodies to antigens and the clearance of the formed complexes should be harmonized, otherwise either immune complexes would accumulate (high affinity, low clearance, as it happens under some pathological conditions) or would unnecessarily remove antibodies and antigen (low affinity, By fitting the Richards curve to experimental binding data from Ag and Ab titration experiments we can obtain the parameters that are suitable for the quantitative characterization of serum Ab. One of these is the antigen concentration at the point of inflection [Ag] i and the other is the limiting activity coefficient (Figure 4) [37]. The first is an estimate of the apparent equilibrium dissociation constant (determined by average standard chemical potential) of the antibodies bound to the Ag.…”

“…For a reaction where we increase Ag concentration and follow the concentration of bound Ab (equivalent to measuring Ab-Ag complexes), we can rewrite the logistic differential equation where N is the number of entities and K is the capacity of the system for such entities, and r is the rate of exponential growth, as where is bound antibody concentration, is total antibody, and is total antigen concentration. We have previously described that is a logistic function of (see Supplementary Text S1 of [ 37 ]), which is the solution of this logistic differential equation. This means that the rate of change of generation of bound antibody while increasing Ag is determined by the actual bound antibody concentration, its relationship to the total antibody concentration, and the rate parameter.…”

“…The rate of change in this quantity, while changing reaction conditions by titrating antibody or antigen, is the derivative of the function that relates the amount of complex formed to the logarithm of the titrated component. We proposed the use of the generalized logistic model (GL) or Richards growth model [ 40 , 41 ] instead of 5PL [ 37 ], because the Richards growth function, like the 5PL, is the solution of the differential equation but is parameterized in a way that the logarithm of the inflexion point [ Ag ] i is one of the parameters. This differential equation implies that besides the activity coefficient of antibody and the rate parameter, the rate of generation of bound antibody while increasing log ( Ag ), is determined by a power of the activity coefficient.…”