Modeling of Mass Transfer Limitation in Biomolecular Assays

Many biomolecular assays involve the capture of an analyte by ligands that are attached or immobilized upon a solid surface. Often the binding kinetics of the ligand and analyte are fast enough that the capture step is limited by diffusion or mass transfer of the analyte from the bulk fluid phase onto the surface. In this contribution, after a brief survey of various mathematical models for mass transfer–limited analyte capture, we analyze one model problem. The model involves the capture of analytes by suspended solid spherical beads on whose surface many ligands are attached. The rate of association of the ligand and analyte molecules is taken to be high enough that the overall rate of the capture process is limited by diffusion. Two distinct limits are examined analytically. In early times, when the analytes near a bead are being captured and the depletion layers in the neighborhood of the individual beads are nonoverlapping, the problem is modeled as a diffusion/surface reaction problem about a single bead, and analytical results are obtained that describe the amount of analyte captured as a function of time. At later times, when the depletion layers from neighboring beads begin to overlap, the problem is modeled by means of generalized Taylor dispersion or macrotransport theory. In each case, scaling laws are derived that characterize the capture efficiency as a function of number density, volume fraction, and radius of beads.