Brownian aggregation rate of colloid particles with several active sites

Abstract

We theoretically analyze the aggregation kinetics of colloid particles with several active sites. Such particles (so-called "patchy particles") are well known as chemically anisotropic reactants, but the corresponding rate constant of their aggregation has not yet been established in a convenient analytical form. Using kinematic approximation for the diffusion problem, we derived an analytical formula for the diffusion-controlled reaction rate constant between two colloid particles (or clusters) with several small active sites under the following assumptions: the relative translational motion is Brownian diffusion, and the isotropic stochastic reorientation of each particle is Markovian and arbitrarily correlated. This formula was shown to produce accurate results in comparison with more sophisticated approaches. Also, to account for the case of a low number of active sites per particle we used Monte Carlo stochastic algorithm based on Gillespie method. Simulations showed that such discrete model is required when this number is less than 10. Finally, we applied the developed approach to the simulation of immunoagglutination, assuming that the formed clusters have fractal structure.

Dynamic quantification of antigen molecules with flow cytometry

“…First, the rate constant is independent of the specific instrument and can be recalculated for different fluorochrome labels (using the theory published in [12]).We emphasize that no experiment-specific calibration would be needed. Second, the rate constant is determined only by the interacting antibody-antigen pair (assuming that the temperature and pH are controlled).…”

“…The model is applicable for independent binding sites, i.e., when the size of the binding site is much less than the distance between binding sites [12]. We extended the model by accounting for time shift t 0 (Sec.…”

Dynamic quantification of antigen molecules with flow cytometry

“…First, the rate constant is independent of the specific instrument and can be recalculated for different fluorochrome labels (using the theory published in [12]). We emphasize that no experiment-specific calibration would be needed.…”

“…From the binding rate constant k + , it was possible to estimate the radius b of the binding site (a circular approximation of the shape of the site placed on a spherical reagent) using following expression [12]: (3) where η is the viscosity of the media, k B is the Boltzmann constant, T is the temperature; R 1 and R 2 are radii, N 1 and N 2 are valences of the first and second reactants, correspondingly. It should be noted that equation (3) is applied if bond formation is diffusion limited, -this is not necessary the general case, but this is true for many particular antibodies.…”

“…The radius of the beads in our experiments was 3 µm (approximately the same as the radius of the cells). The radius of antibody molecules can be estimated from the diffusion coefficient using Stokes-Einstein equation [12]: (4) Since the shape of the antibody molecule is not spherical (the diffusion coefficient depends also on the shape), equation (4) is an approximation that can be used to estimate the radius of the equivalent spherical particle (which has the same diffusion coefficient as the antibody molecule). On the other hand, the diffusion coefficient of the molecule can be estimated using the known relationship [17] between the diffusion coefficient (in cm 2 s −1 ) and the molar mass (in Da), M, of a protein (in water at room temperature) (5) It is known that anti-CD8a molecule is the 150 kDa type IgG immunoglobulin with the valence N=2, and PE is the protein with approximate molecular weight of 240 kDa.…”