Modelling effects of internalized antibody: a simple comparative study

Abstract

Background: The modelling framework is proposed to study protection properties of antibodies to neutralize the effects of the plant toxin (ricin). The present study extends our previous work by including (i) the model of intracellular transport of toxin to the Endoplasmic Reticulum and (ii) the model of the internalised antibodies (when antibody is delivered directly into the cytosol).

Method: Simulation of the receptor-toxin-antibody interaction is implemented by solving the systems of PDEs (advection-diffusion models) or ODEs (rate models) for the underlying transport coupled with mass-action kinetics.

Results: As the main application of the enhanced framework we present a comparative study of two kinds (external and internalised) of antibodies. This comparison is based on calculation of the non-dimensional protection factor using the same set of parameters (geometry, binding constants, initial concentrations of species, and total initial amount of the antibody).

Conclusion: This research will provide a framework for consistent evaluation and comparison of different types of antibodies for toxicological applications.

Figure 1

The schematic diagram of receptor-toxin-antibody system: (a) - Scenario I and (b) - Scenario II. △ – toxin, ◇ – antibody, conglutinated △ and ◇ in circle – toxin-antibody complex, external sphere – cell membrane, internal sphere – ER envelope.

Figure 2

Influence of toxin diffusivity and absorption constant on antibody protection factor. (a) Scenario I. Solid and dashed lines correspond to value of u_T determined for γ=0.1 and γ=0.05. Toxin diffusivity κ_T: 10^-2 (1), 10^-3 (2), 5·10^-4 (3). (b) Scenario II. Absorption constant γ=0.1, toxin diffusivity κ_T: 10^-2 (1), 5·10^-4 (2). $u_A^{0,e}=u_A^{0,i}=1$ in cases (a) and (b).

Figure 3

Effect of variation of toxin diffusivity on parameter δ in case${\mathbf{u}}_{\mathbf{A}}^{\mathbf{0}\mathbf{,}\mathbf{e}}\mathbf{=}\mathbf{1}$, ${\mathbf{u}}_{\mathbf{A}}^{\mathbf{0}\mathbf{,}\mathbf{i}}\mathbf{=}\mathbf{4.87}$. Solid and dashed lines correspond to value of δ determined for γ=0.1 and γ=0.01. Toxin diffusivity κ_T: 10^-2 (1), 10^-3 (2), 5·10^-4 (3).

Figure 4

Long-time asymptotic behavior of metric parameter δ(t). Solid and dashed lines correspond to δ determined for γ=0.1 and γ=0.01 in case $u_A^{0,e}=u_A^{0,i}=1$, κ_T=0.01.

Figure 5

Effect of variation of toxin diffusivity and absorption constant on parameter δ. Toxin diffusivity: (a)κ_T=10^-4, (b)κ_T=10^-3. Solid ($u_A^{0,e}=u_A^{0,i}=1$) and dashed ($u_A^{0,e}=1$, $u_A^{0,i}=4.87$) lines correspond to δ determined by advection-diffusion models, and bullets ($u_A^{0,e}=u_A^{0,i}=1$) and circles ($u_A^{0,e}=1$, $u_A^{0,i}=4.87$) to δ determined by well-mixed models for γ: 0.1 (1), 0.01 (2).

Figure 6

Effect of variation of toxin advection velocity v on parameterδ. Advection-diffusion models (solid lines): v=0.001, γ=0.01 (1), v=0.001, γ=0.1 (2), v=0.05, γ=0.01 (3), v=0.05, γ=0.1 (4); toxin diffusivity: (a)κ_T=10^-3, (b)κ_T=10^-4. Well-mixed models (dashed lines): γ=0.01 (1), γ=0.1 (2).