Competitive interaction between two different spherical sinks

Abstract

Competitive interactions within diverse mixed populations of chemically active sites are prevalent throughout nature, science, and engineering. Their effects are readily seen in the distribution of dead and surviving aerobic cells within a thick biofilm and particle shape changes during the growth and coarsening of crystals. Even in the most dilute case, competition for a reactant requires at least two spheres/cells, and the solution of the two-spherical sink problem is of interest for several reasons. The solution accurately describes lower cell concentration behavior (10(8) cells/l), and like the Smoluchowski diffusion-reaction treatment for a single sphere, the analysis is extremely helpful in understanding the fundamental phenomena of the effect on the first spherical sink of the presence of a second different spherical sink. In addition these exact solutions are required for the systematic extension to higher density behavior by rigorous expansions in the spherical sink densities. The method of the twin spherical expansion is used with a formal matrix elimination scheme to generate an exact solution for two distinct spherical sinks of differing sizes and kinetics. The two sinks exist in a medium, which supplies a reactant to the sinks via Fickian diffusion. The two sinks compete for the same reactant with different first-order reactions occurring at the surface of each sink. Earlier work focused on two spherical sinks of the same size with identical surface reaction kinetics. This work has been advanced to allow for diversity in the theory of cellular or reactive sink competition. A number of interesting higher order interactive phenomena are observed in this paper when the different reactive sinks are in close proximity. (c) 2004 American Institute of Physics.

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

“…In its modern form, the GMSV was thoroughly developed for studying diffraction of electromagnetic waves on surfaces of several bodies [17] and then applied in various fields. It is striking how many names were given to the method under consideration by different authors: "the method of addition theorems" [18], "the method of reduction to the ISLEA" [19], "the method of irreducible Cartesian tensors" [20], "the method based on the theory of multipole expansions" [21], "the generalized Fourier method" [22], "the Rayleigh multipole method" [23], "the method of twin multipole expansions" [24], "the direct method of re-expansion" [25], "a twin spherical expansions method" (or just "twin expansions technique") [26], "the method of a bispherical expansion" [27], "the multipole re-expansion method" [28], "the multipole expansion method" [29], and a particular case of "the method of series" [30]. The GMSV has been successfully applied in the elasticity theory [31], heat transfer [32], diffraction theory [30,33] and other branches of mathematical physics [18].…”

“…Traytak and Tachiya investigated diffusive interaction between two spherical sinks in an electric field by means of the GMSV [41]. Tsao, Strieder et al and Traytak et al used the GMSV to calculate rigorously the electric field effects and to study reactions on two different spherical sinks and on spherical source and sink [24,26,27,42,43,44,45]. A more general form of the GMSV was elaborated to compute the steady-state reaction rate for an irreversible bulk diffusion-influenced chemical reaction between a mobile point-like particle and static finite three-dimensional configurations of spherical active particles [25,46,47,48].…”

Size effects in reactive circular site interactions

“…Indeed similar conclusions were obtained when the WFW steady state reaction solutions for two spheres by Uhm et al 8 were compared with the corresponding exact solutions of McDonald and Strieder. 9 For the case of two equal sites, Ivanov and Lukzen 10 do not present numerical results for R 1 . Instead, for γ = 1, they suggest that the simpler analytical formula (see Eq.…”

“…This type of significant error for the WFW method, found for small γ when two very active sites are near contact, is also similar to that found in the case of very reactive spherical sites. 8,9 Kang et al 5 discussed the site reaction for two diffusioncontrolled circular sites of the same size (γ = 1) located an arbitrary distance apart on a larger inert, impenetrable sphere. As their modified Brownian dynamic (BD) simulation to calculate R 1 included an estimated "numerical cutoff," and assumed "long time asymptotic" forms, a comparison against an exact site reaction rate value would be useful.…”

“…This can be accomplished by the use of the second shift formula (8) to eliminate cos (mφ)J m (ru) from the concentration series (6) in a manner similar to the derivation of the site 1 concentration series form (9). To generate the site 2 {A n m , A n m } coefficient equation, this site 2 concentration series must be evaluated over all points on site 2, i.e., z = 0, −π ≤ φ ≤ π , and 0 ≤ γ r ≤ 1.…”