Colloids and Surfaces A: Physicochemical and Engineering Aspects 143 (1998) 117–131 Predicting equilibrium constants for ion exchange of proteins — a colloid science approach W. Richard Bowen *, Li-Chun Pan, Adel O. Sharif Biochemical Engineering Group, Centre for Complex Fluids Processing, Department of Chemical and Biological Process Engineering, University of Wales Swansea, Swansea, SA2 8PP, UK Received 2 November 1997; accepted 8 May 1998 Abstract A mathematical model for predicting the equilibrium constants ( K eq ) of ion exchange of proteins has been developed. The model is based on a description of the colloidal interactions between a protein molecule and a charged surface within an electrolyte solution. The electrostatic interactions are quantified using a solution of the non-linear Poisson–Boltzmann equation obtained by a finite element technique combined with a Newton sequence and an automatic adaptive mesh refinement incorporating error estimation. London–van der Waals’ interactions are calculated using an unretarded Hamaker constant. The approach enables a priori prediction of K eq for protein ion exchange in terms of protein size, protein zeta potential (and hence pH ), ion-exchanger zeta potential and electrolyte concentration. All of these parameters are readily quantified. The distance of closest approach (z 0 ) between protein and ion exchanger must also be specified. For ion exchange of bovine serum albumin (BSA), there was good agreement between theory and experiment for the variation of K eq with pH with a constant value of z 0 . This confirms the predictive capability of the approach developed. Good agreement between theory and experiment for the variation of K eq with ionic strength could be obtained if z 0 was allowed to vary with ionic strength. Overall, this fundamental approach has promise to become a general method of predicting K eq for protein ion exchange. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Ion exchange; Protein; Adsorption equilibrium constant; Colloidal interactions; Finite element method C f final concentration of protein solution in bulk volume Nomenclature (mol m-3) a particle radius (m) C i initial concentration of protein solution in bulk volume A 132 Hamaker constant (J ) (mol m-3) A IX effective surface area of ion exchanger (m2) C 0 bulk concentration of protein (mol m-3) b i mean number per unit volume of ions in the double C s surface concentration of protein (mol m-2) layer (m-3) C UV Cauchy constant ( =n2 0 -1) b i (2) mean number per unit volume of ions in the bulk fluid d distance to OHP from the particle surface (m) (m-3) D sphere–planar surface separation (m) c i concentration of ion species i (kmol m-3) e elementary charge (1.602×10-19) (C) C i relative concentration of ion species (c i /I ) G total electrostatic free energy around a colloidal particle (J) * Corresponding author. Tel./fax: +44-1792-295862; G* s dimensionless electrostatic free energy for a single isolated sphere e-mail: r.bowen@swansea.ac.uk 0927-7757/98/$ – see front matter © 1998 Elsevier Science B.V. All rights reserved. PII S0927-7757(98)00512-3