The thermodynamics of adsorption
Until now, only the physical processes that take place within the fluid phase of the chromatography column have been mentioned. We need to incorporate the adsorption of components inside the adsorber beads to complete the chromatography model.
Prevalent models describe the concentration of adsorbed species as a function of the species’ concentrations in the mobile phase and the concentration bound to the adsorber. In fact, most of the models employed today consider adsorption as a reaction, where one mol of proteins and one mol of ligands react to one mol of a protein-ligand complex. In the case of large proteins, multi-point binding often occurs so that the needed amount of ligands is increased by a stoichiometric factor of n.
In the next step, the law of mass action is applied to this chemical equation. It states that the molar concentrations of the reactants taken to the power of their stoichiometric factors is equal to an equilibrium constant K. This results in the equilibrium isotherm equation, a mathematical model that can be coupled directly to the equation system describing fluid dynamics.
Usually, both adsorption and desorption are very fast processes. In many models, the instantaneous formation of the equilibrium is assumed. In general, the isotherm equation is expressed by the rate of change of the concentration of the adsorbed species over time.
Historically, the Langmuir isotherm is the most extensively studied model. This model considers single-point binding only. It is isocratic, which means the mobile phase does not change its composition during a run. In particular, a change in the buffer concentration will not have any effect on adsorption. Due to lacking practical relevance in preparative chromatography, this model, in its basic form, is predominantly considered in academic research.
The typical Langmuir-shaped relation between the protein concentration in the stationary phase and the protein concentration in the mobile phase is shown. At low protein concentrations in the mobile phase, the ratio is linear and described with the slope k. At higher loading concentrations, the concentration in the stationary phase reaches a maximum binding capacity called qmax. The ratio between q and c results in the characteristic isotherm curve.
The Steric Mass Action (SMA) isotherm is a common model for ion exchange chromatography (IEC). It models multi-point binding, the influence of counter-ions and displacement. The basic idea of this isotherm model is a stoichiometric exchange of ions on an oppositely charged surface. Hereby, the adsorbing protein is assumed to free v moles of salt ions which reside on the ligands involved in binding. The reaction equation becomes
At constant buffer composition, the SMA model has a very similar shape compared to the Langmuir model. With an increasing mobile phase concentration, a state of saturation is reached. In contrast to Langmuir, this saturation concentration is not a protein-specific qmax but expressed as a function of the total ionic capacity. This allows the study of the influence of raw material variability once the model is built. Further effects included in the model are steric hindrance and hydrophobic repulsion. When approaching saturation, a certain number of binding sites cannot be accessed by proteins in the solution anymore. This is because they are sterically shielded or unusable because of stronger repulsive forces.
The separation principle of Hydrophobic Interaction Chromatography (HIC) is based on the reversible interaction between hydrophobic patches on the protein surface and the hydrophobic ligands of the stationary phase. Advances in the fundamental understanding of HIC have been achieved by modeling work based on Mollerup’s thermodynamic framework. It extends the simple equations derived from the law of mass actions with so-called asymmetric activity coefficients. This allows the simulation of salt-controlled chromatography processes.
Another approach by GoSilico and the Karlsruhe Institute of Technology (KIT) considers an equilibrium between well-ordered and bulk-like ordered water molecules on the hydrophobic surface. The challenge of incorporating salt effects in HIC modeling was solved by formulating the hydration number of ions as a function of the salt concentration. This model was later extended to include the effect of pH changes as well.
The basic assumption behind current Mixed Mode Chromatography (MMC) isotherms is that both adsorption modes described above, ion exchange and hydrophobic interaction, take place at the same time. The reaction equation can be extended accordingly so that binding requires both charged and hydrophobic binding sites. The resulting equation is a combination of the Steric Mass Action (SMA) model and Mollerup’s HIC model with asymmetric activity coefficients. It was first demonstrated by Nfor et al. in 2010. A similar combination of SMA with GoSilico’s HIC model has produced equally good results using fewer protein-specific parameters. In the practical industrial use of chromatography, an understanding and model of both interaction modes is rarely needed. Often, mixed mode adsorbers are used as a replacement for process steps for pure ion exchange or hydrophobic interaction that were used in the past. If the process is run in a similar fashion, it is sufficient to assume that one mode dominates and model the MMC as IEC or HIC.
Other modes of chromatography which have been successfully modeled in the past are reverse phase chromatography (RPC) and affinity chromatography (AC). Especially for mAb processes, Protein A chromatography is of interest here. However, the model derivation quickly leaves the path of formulating chemical equations and applying the law of mass action. Please contact us to learn more about possibilities, limitations and use cases.