Mathematical Modelling of Protein Precipitation Based on the Phase Equilibrium for an Antibody Fragment from E. coli Lysis

Abstract

Precipitation is an important operation in biopharmaceutical purification yet the mechanism of protein precipitation in multi-component solutions is not well understood. Existing models lack fundamental understanding of the process. In this paper, a new model describing how the protein solubility changes in the protein precipitation is proposed and is based on the phase equilibrium of the light liquid phase and dense solid phase. The model structure is generic and robust. It adequately reflects the non-linearity of protein precipitation kinetics and thus provides new fundamental insights into the protein precipitation in multi-component, complex protein solution. Two feed stocks of a pure fragment antigen-binding (Fab') solution obtained by chromatographic purification and a clarified Fab' homogenate solution from E. coli were used to examine the effect of ammonium sulphate concentrations and pH conditions on precipitation. It was found that the model can describe pure Fab' precipitation well, and identify the non-ideal behavior of Fab' precipitation in multi-component homogenates. Through statistical analysis, the model parameters have been further reduced from 8 to 4. The quality of the model is such that errors were within the acceptable statistical confidence limits, even when applied to multi-component impurity precipitation. The new model with fewer parameters is better than existing empirical models in reflecting the salting-in and salting-out effect of the protein precipitation. This demonstrated that the structure of the model is sound and over-fitting in the parameter estimation is avoided. The model can be applied directly to industrial processes for protein precipitation process design after appropriate calibration with the required operating conditions of pH and salt concentration. a 1 , b 1 , c 1 , d 1 constants in Equation (9) a 2 , b 2 , c 2 , d 2 constants in Equation (18) a 3 , b 3 , c 3 , d 3 , e 3 , f 3 constants in Equation (19) a 5 , b 5 , c 5 , d 5 , e 5 , f 5 constants in Equation (21) A, B constants in Equation (12) d C protein molar concentration in the dense phase (-1 L mol ⋅) i C other component molar concentration in the solution (-1 L mol ⋅) l C protein molar concentration in the light phase (-1 L mol ⋅) s C salt molar concentration (-1 L mol ⋅) C T the maximum protein concentration in the solution (-1 L mol ⋅) I ionic strength (-1 L mol ⋅) k s salt activity coefficient (-) k i components activity coefficient (-) m 3 the salt mole concentration (-1 L mol ⋅) l r protein activity coefficient in the light phase (-) d r protein activity coefficient in the dense phase (-) Q i molar concentration of ion i (-1 L mol ⋅) R g ideal gas constant (J·mol-1 ·K-1) R 2 coefficient of determination S Fab' concentration in the supernatant (-1 L mol ⋅) S 0 Fab' concentration in the feedstock (-1 L mol ⋅) T the absolute temperature (K) V l light liquid phase volume (L) V d dense phase volume (L)