Ion exchange chromatography of proteins-prediction of elution curves and operating conditions. I. Theoretical considerations.

A mathematical model is proposed for the elution of proteins on ion exchange columns by a linear gradient increase and stepwise increase of ionic strength in order to predict relationships between the elution characteristics (the peak position, the peak width, etc.) and the operating conditions (the flow rate, the slope of gradient, etc). This model is in principle based on the continuous-flow plate theory, in which the protein concentration and ionic strength dependent distibution coefficient between proteins and ion exchangers and zone sperading effects are taken into consideration. The advantage of this model is its simplicity since it requires only two parameters: The distribution coefficient and the number of plates. Since the distribution coefficient of proteins depends on both the protein concentration and ionic strength of the elution buffer, the number of plates should vary with time. However, it is extremely difficult to take into consideration the time-dependent number of plates. Therefore, we assume that the number of plates is constant and related to that number derived from a mass balance model which includes longitudinal dispersion and gel phase diffusion. On the basis of these assumptions, a method for determining the number of plates by the moment method is presented. Although the dependencies of the peak position and peak width on the slope of linear gradient are predictable by numerical calculations of the present model, simpler methods for prediction of these dependencies are desirable. A graphical method is proposed for prediction of the peak position. For prediction of the peak width, an asymptotic solution is derived from a quasi-steady-state model.