A simple model for assessment of anti-toxin antibodies

Abstract

The toxins associated with infectious diseases are potential targets for inhibitors which have the potential for prophylactic or therapeutic use. Many antibodies have been generated for this purpose, and the objective of this study was to develop a simple mathematical model that may be used to evaluate the potential protective effect of antibodies. This model was used to evaluate the contributions of antibody affinity and concentration to reducing antibody-receptor complex formation and internalization. The model also enables prediction of the antibody kinetic constants and concentration required to provide a specified degree of protection. We hope that this model, once validated experimentally, will be a useful tool for in vitro selection of potentially protective antibodies for progression to in vivo evaluation.

Figure 1

Schematic representation of the model for receptor-toxin-antibody interaction.

Figure 2

Simulated effect of antibody concentration on formation of toxin-receptor complexes C _R. Parameter λ = A ₀/C ₀, C ₀ = R ₀ + K ₁. The binding curves were created using the simulation package COPASI and the kinetic constants in Table 1. R ₀ = 5 nM,  T ₀ = 10 pM, C ₀ = 1.15 · 10⁻⁷.

Figure 3

Effect of antibody concentration on protection factor. Parameter Ψ (19) was determined from (20) (solid lines) and by using simulated values of C _R from Figure 2 at 2500 sec (△),  ϵ = 25.9.

Figure 4

Protection factor Ψ (19) as a function of parameter ϵ = K ₁/K ₂ and λ = A ₀/C ₀ ((20)): λ = 0.01 (⚪); 0.025 (△); 0.05 (□); 0.1 (▽); 0.25 (◊). The range of values for λ and ϵ below dashed line corresponds to 80% protection.

Figure 5

Different time scales for formation of receptor-toxin complex C _R (□) and associated toxin internalization T _i (solid lines). Results of COPASI simulation with kinetic constants from Table 1. λ = A ₀/C ₀, R ₀ = 5 nM, T ₀ = 10 pM, C ₀ = 1.15 · 10⁻⁷, ϵ = 25.9.

Figure 6

Comparison of parameters Ψ and Γ. Γ (△) was determined using values of T _i and T _i ⁰ at t = 10⁴ sec from toxin internalization time courses simulated using COPASI and the kinetic constants in Table 1. Parameter Ψ (solid line) was determined from (20). R ₀ = 5 nM, T ₀ = 10 pM, C ₀ = 1.15 · 10⁻⁷, ϵ = 25.9.

Figure 7

Establishment of the quasi-equilibrium state in the presence of antibody. C _R formation (△) was simulated using COPASI and the kinetic constants in Table 1. Γ (□) was determined using (25) and values T _i and T _i ⁰ at t = 10⁴ sec using simulated toxin internalization time courses. R ₀ = 5 nM, T ₀ = 10 pM, C ₀ = 1.15 · 10⁻⁷, λ = 0.05.

Figure 8

Relationship between toxin internalization time τ _i and protection factor Ψ (19). Solid line is formula (27) and (□) is simulation with COPASI. τ _i was determined as the time to internalize 5 · 10⁻¹⁴ M of ricin. All other parameters are the same as in Figure 7.