(427h) Solution for the Diffusive Interaction from a Spherical Source to an Internally Reactive Spherical Sink

Particle-particle interactions such as competition, mutualism and commensalism (J. E. Bailey and D. F. Ollis, Biochemical Fundamentals, 1986) are common among biological and non-biological systems. A mixed culture of microorganisms will exhibit all of these interactions; hence the culture's efficacy will be determined by the efficiency of each particle to utilize these interactions to their advantage. Because the interactive diffusion reaction phenomenon is routine, understanding the process is relevant in Biotechnology, Environmental Engineering, Physiology, Cellular Biology and Chemical Engineering. The method of a bispherical expansion is used to express the exact analytical solution for an infinite medium containing two spherical particles; a spherical source, and an internally reactive spherical sink. The source-penetrable sink model includes two spheres of arbitrary radii a₁ and a₂. Sphere 2 is a pseudo-steady state source generating an intermediate product at the cellular membrane with zeroth order kinetic rate constant, σ (K. E. Foster and D. A. Lauffenburger, Biophys. J. 61, 518 (1992)). For the substrate two Fickian diffusion transport parameters are required the extracellular (bulk phase) transport with extracellular diffusivity D and the intracellular diffusion in sphere 1, D₁, followed by volumetric reaction within sphere 1 using first order kinetic rate constant, k₁. The probability of an internal reaction in sphere 1 (P), i.e. consumption rate of the intermediate product inside cell 1 per generation rate at the surface of sphere 2, is determined using four dimensionless parameters: cellular size ratio (a₁/a₂), center-to-center separation distance (d/(a₁+a₂)), sink internal Thiele modulus (φ), and the external to internal Fickian diffusion ratio (δ=D/D₁). The two-cell model is the next logical step to solving the higher cell density problems common in colonies of microorganisms for two reasons, it can account for cellular interaction and it is also capable of resolving systems that are too dense for the Smoluchowski diffusion reaction treatment of a single cell. The exact solution of the source?penetrable sink problem is compared with a commonly used alternate, the source-impenetrable sink problem with an effective surface reaction. This substitution reduces by one the number of necessary parameters and is a good approximation of the source-permeable sink solution for large values of the Thiele modulus. However when the modulus is small, the range for most microorganisms (P. B. Weisz, Science 179, 433 (1973)) there is significant error. The results of the source-penetrable sink problem show an inverse relationship between the separation distance, d and the reaction probability, P, an unexpected effect of particle size and reactivity is also discussed. Finally the limitations of the source-impenetrable sink with an effective surface reaction are exposed for a range of Thiele modulus and diffusivity ratio.