Modeling of a single nanoparticle interaction with the human blood plasma proteins.

When nanoparticles are introduced into a physiological environment, proteins and lipids immediately cover their surface, forming a protein "corona". It is well recognized that the corona structure influences the biological response of the body. Two deterministic models for corona formation of the human blood serum proteins around a single nanoparticle are presented and studied numerically in this paper. One of them is based on a coupled system of PDEs and involves diffusion of proteins toward the nanoparticle surface. The other one is described by ODEs and is a limit version of the first model as the protein diffusivity tends to infinity. The protein diffusivity influence on the temporal corona structure is studied in detail. Results are presented using figures and discussion.

Modelling toxin effects on protein biosynthesis in eukaryotic cells.

We present a rather generic model for toxin (ricin) inhibition of protein biosynthesis in eukaryotic cells. We also study reduction of the ricin toxic effects with application of antibodies against the RTB subunit of ricin molecules. Both species initially are delivered extracellularly. The model accounts for the pinocytotic and receptor-mediated toxin endocytosis and the intact toxin exocytotic removal out of the cell. The model also includes the lysosomal toxin destruction, the intact toxin motion to the endoplasmic reticulum (ER) for separation of its molecules into the RTA and RTB subunits, and the RTA chain translocation into the cytosol. In the cytosol, one portion of the RTA undergoes degradation via the ERAD. The other its portion can inactivate ribosomes at a large rate. The model is based on a system of deterministic ODEs. The influence of the kinetic parameters on the protein concentration and antibody protection factor is studied in detail.

Modeling neutralization of Shiga 2 toxin by A-and B-subunit-specific human monoclonal antibodies.

A mathematical model for Shiga 2 toxin neutralization by A-and B-subunit-specific human monoclonal antibodies initially delivered in the extracellular domain is presented, taking into account toxin and antibodies interaction in the extracellular domain, diffusion of toxin, antibodies, and their reaction products toward the cell, the receptor-mediated toxin and complex composed of toxin and antibody to A-subunit internalization from the extracellular into the intracellular medium and excretion of this complex back to the extracellular environment via recycling endosomal carriers. The retrograde transport of the intact toxin to the endoplasmic reticulum and its anterograde movement back to the vicinity of the plasma membrane with its subsequent exocytotic removal to the extracellular space via the secretory vesicle pathway is also taken into account. The model is composed of a set of coupled PDEs. A mathematical model based on a system of ODEs for Shiga 2 toxin neutralization by antibodies in the absence of cell is also studied. Both PDE and ODE systems are solved numerically. Numerical results are illustrated by figures and discussed.

Modeling of toxin-antibody interaction and toxin transport toward the endoplasmic reticulum.

A model for toxin-antibody interaction and toxin trafficking towards the endoplasmic-reticulum is presented. Antibody and toxin (ricin) initially are delivered outside the cell. The model involves: the pinocytotic (cellular drinking) and receptor-mediated toxin internalization modes from the extracellular into the intracellular domain, its exocytotic excretion from the cytosol back to the extracellular medium, the intact toxin retrograde transport to the endoplasmic reticulum, the anterograde toxin movement outward from the cell across the plasma membrane, the lysosomal toxin degradation, and the toxin clearance (removal from the system) flux. The model consists of a set of coupled PDEs. Using an averaging procedure, the model is reduced to a system of coupled ODEs. Both PDEs and ODEs systems are solved numerically. Numerical results are illustrated by figures and discussed.

Toxin effect on protein biosynthesis in eukaryotic cells: a simple kinetic model.

A model for toxin inhibition of protein synthesis inside eukaryotic cells is presented. Mitigation of this effect by introduction of an antibody is also studied. Antibody and toxin (ricin) initially are delivered outside the cell. The model describes toxin internalization from the extracellular into the intracellular domain, its transport to the endoplasmic reticulum (ER) and the cleavage inside the ER into the RTA and RTB chains, the release of RTA into the cytosol, inactivation (depurination) of ribosomes, and the effect on translation. The model consists of a set of ODEs which are solved numerically. Numerical results are illustrated by figures and discussed.

Modelling effects of internalized antibody: a simple comparative study.

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.

A reaction-diffusion model of the receptor-toxin-antibody interaction.

BACKGROUND: It was recently shown that the treatment effect of an antibody can be described by a consolidated parameter which includes the reaction rates of the receptor-toxin-antibody kinetics and the relative concentration of reacting species. As a result, any given value of this parameter determines an associated range of antibody kinetic properties and its relative concentration in order to achieve a desirable therapeutic effect. In the current study we generalize the existing kinetic model by explicitly taking into account the diffusion fluxes of the species. RESULTS: A refined model of receptor-toxin-antibody (RTA) interaction is studied numerically. The protective properties of an antibody against a given toxin are evaluated for a spherical cell placed into a toxin-antibody solution. The selection of parameters for numerical simulation approximately corresponds to the practically relevant values reported in the literature with the significant ranges in variation to allow demonstration of different regimes of intracellular transport. CONCLUSIONS: The proposed refinement of the RTA model may become important for the consistent evaluation of protective potential of an antibody and for the estimation of the time period during which the application of this antibody becomes the most effective. It can be a useful tool for in vitro selection of potential protective antibodies for progression to in vivo evaluation.

On the stability of separable solutions of a sexual age-structured population dynamics model.

The Sharpe-Lotka-McKendrick-von Foerster equations for non-dispersing age-sex-structured populations with a harmonic mean type mating law are considered and their separable solutions are analysed. For certain forms of the demographic rates the underlying evolution equations are reduced to systems of ODEs, the long time behavior of their solutions is studied, and the stability of separable solutions is discussed. It is found that for the constant death rates and constant sex ratio of newborns with stationary birth rates this model admits one one-parameter class of separable solutions, two such classes (repeated or different) or no such ones. In the case of special forms of age-dependent birth rates, solutions of one of these two different classes corresponding to the greater root of the characteristic equation are locally stable, solutions of the other one corresponding to the smaller root are unstable, and the population dies out if the model does not admit separable solutions or if initial densities of newborns are small enough in the case of the existence of separable solutions. In the case of constant vital rates, the model has no separable solutions or admits only one class of such ones that are globally stable.