577 Sticking coefficients of adsorbing proteins Daniel R. Weaver and William G. Pitt Department of Chemical Engineering, Brigham Young University, Provo, Utah 84602, USA The protein sticking coefficient, 4, the fraction of collisions that result in adsorption, is a function of the molecular interactions between the protein and the surface. A random walk and diffusion- to-capture model was used to describe the kinetics of protein adsorption. The assumption of a constant sticking coefficient leads to a first-order model of the kinetics. A solution of the problem of adsorption from a semi-infinite medium with first-order kinetics at the boundary was obtained by numerical simulation on the computer. The results of the computer simulations match the time dependence observed experimentally. A correlation was developed to estimate C#J from experimental data. I#J has been found to be in the range 10-5-10-* for several protein adsorption kinetic studies reported in the literature. Keywords: Protein adsorption, diffusion, sticking coefficients, Received 11 November 1991; revised 8 January 1992; accepted 9 January 1992 The adsorption of proteins at interfaces has been shown’ to be a complex phenomena which includes the diffusion of the protein particle through the aqueous solution and the collision and interaction of the protein at the interface. One important goal in the study of protein adsorption has been the measurement of the affinity between the protein and the surface. The affinity has been broadly defined as a measure of the interaction between the protein and the surface. It has been suggested by Horbett and Brash’ that one measure of the affinity is the protein sticking coefficient. The sticking coefficient, denoted 4, is the fraction of the collisions between the protein and an available surface site that result in adsorption. It has been hypothesized by Horbett and Brash, and it is the hypothesis of this study, that the sticking coefficient can be deduced from kinetic measurements under conditions near the diffusion limit. Diffusion-controlled adsorption of protein from non- flowing solution has often been modelled by3 where cs is the concentration of adsorbed protein, cs is the initial (bulk) protein concentration, D is the diffusion coefficient, and t is time. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Equation 1 has several shortcomings which will be discussed. However, it has been used by several investi- gators of protein adsorption kinetics because experimental data match the functional form of Equation 1. It has been frequently observed3-’ that, initially, the amount adsorbed increases in proportion to t”‘. However, in many cases the initial observed slope dcsld(t”‘) has been found to be lower than 2cB(Dln)“’ predicted by Equation Z. It has Correspondence to Professor W.G. Pitt. 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1992 Butterworth-Heinemann Ltd 0142-9612/92/090577-06 also been frequently observed that after a period of time the adsorption rate decreases sharply due to surface filling. The departure of the initial slope from the diffusion limit has been attributed to a sticking coefficient that is not unity. Physical mechanisms such as energy barriers and reversible adsorption have been proposed to explain the departure of the sticking coefficient from unity. In the present study, the nature and consequences of the departure of the slope from the diffusion limit is investigated within the framework of a model for the sticking coefficient. To facilitate the investigation, a parameter Q is defined to be the ratio of the experimentally observed slope to the slope of the theoretical diffusion limit Q, the dimensionless observed slope, will be shown to be a measure of the departure of the experiment from completely diffusion-controlled conditions. The goal of this study included four main aspects: 1. To determine if kinetic limitations coupled with diffusion limitations could produce this functional t”* dependence with the accompanying decrease in the slope. 2. To determine how the sticking coefficient, @, correlates with the slope Q. 3. To develop a model or correlation to calculate the sticking coefficient from kinetic measurements. 4. To evaluate experimental data and determine an order of magnitude estimate of protein sticking coefficients. Biomaterials 1992, Vol. 13 No. 9