(385d) Investigation into the Applicability of “Conventional” Kinetic Models for the Process Modelling of Protein Crystallisation

The synthesis of protein and peptide therapeutics have seen orders-of-magnitude improvement in recent years, resulting in the economic bottleneck of these biopharmaceutics being shifted to downstream steps, which have not seen the same degree of development. Much of the cost associated with downstream processing is attributed to costly chromatography steps, which suffer from high solvent requirements and costly resins. As such, there is significant interest in the development of effective and cost-efficient purification workflows for proteins. Crystallisation is widely employed in the small molecule pharmaceutical industry, as these molecules generally have a high crystallisation propensity. As well as this, the crystalline form inherently presents favourable properties over amorphous precipitates, such as higher purity and stability. As well as this, the crystallisation process can be tuned to produce crystals of a desired size and shape for further processing. However, the usage of crystallisation for protein purification remains very limited, which is due to a combination of factors:

1. Proteins are highly complex structures compared to small molecules. The protein chain gives rise to significant degrees of flexibility and allows for secondary, tertiary, and quaternary structuring of the protein molecule in solution.
2. While small molecules can often be crystallised from simple solvents or solvent mixtures, protein crystallisation requires the usage of precise concentrations of buffers, salts, and precipitants, and a controlled solution pH to bring about crystallisation, in order to preserve the structure of the protein and prevent denaturation.
3. Protein crystals are highly hydrated structures due to the size of protein molecules, and as such exhibit noticeably different mechanical properties, such as hardness.
4. The fundamental mechanisms of protein crystal nucleation are an area of active research interest. It has been shown that the classical nucleation theory severely underpredicts the nucleation rate in many real protein crystallisation solutions. Proteins also exhibit distinct solution-dependent phase behaviour beyond crystallisation, such as amorphous precipitation, gelation, and liquid-liquid phase-separation. This presents additional complexities in the successful crystallisation of proteins.

Process modelling, which refers to the usage of numerical models to predict the behaviour of unit operations, is used extensively in tandem with small molecule crystallisation. For crystallisation, process modelling takes the form of a population balance model, which predicts the time evolution of the particle size distribution (PSD) of the crystallising species, using kinetic expressions to model crystal nucleation and growth, as well as more complex phenomena such as agglomeration and impurity inclusion. Process modelling is invaluable for model-based optimisation of factors such as crystalliser geometry and batch time for small molecule crystallisation. It also allows for model-based design of experiments, to minimise material usage during research and development of a purification strategy.

With the limited usage of protein crystallisation as a purification strategy, research into the applicability of process modelling of protein crystallisation is scarce, particularly for predicting protein nucleation. The research presented herein examines the applicability of “conventional” classical and empirical equations describing nucleation and growth, which are widely used for small molecules, on the crystallisation of model proteins. Initial studies focus on the crystallisation of lysozyme at the 100 mL and 1 L scale. Both a classical and empirical kinetic expression are tested for nucleation and growth, giving a total of four permutations (“cases”) of kinetics to be compared. The system is modelled using gPROMS FormulatedProducts, within which the necessary model equations (population balance, mass balance, and energy balance) are contained within the flowsheet shown in Figure 1. Experimental data (concentration and particle size) was gathered from two separate literature sources, in order to assess the transferability of results obtained. Using the built-in parameter estimation capabilities of gPROMS, the experimental data was used to produce parameter estimates for all four cases, the results of which are presented in Table 1.

Initial simulations using data at the 100 mL scale reveal the importance of including multiple particle size estimates alongside concentration data to extract sensible and statistically significant parameter estimates. From refined parameter estimation, it was shown that usage of the Classical Nucleation Theory is advantageous over empirical nucleation equations, as it results in a lower uncertainty in model parameters. The choice of growth equation had a smaller effect on the efficacy of the models, but when analysing correlation coefficients, it was found that using a classical two-step (diffusion and integration) growth model resulted in higher correlation between all parameters. As such, the combination of classical nucleation theory and power-law growth appeared to be the most suitable for modelling lysozyme crystallisation. Parameter estimates for the interfacial energy (0.528 and 0.766 mJ/m²) obtained here are also in line with literature results, which is surprising as it was expected that the classical nucleation theory would perform poorly compared to an empirical nucleation model. However, as shown in Figure 2, all models performed identically in terms of fit to experimental data for both particle size and concentration. These final estimates obtained at the 100 mL scale were then used to attempt to model separate experimental data for lysozyme crystallisation at the 1 L scale. Here, it was shown that the fit to experimental data was much poorer, but the model provided a much more reasonable estimate for the final particle size. By using the values obtained at the 100 mL scale for further parameter estimation, the model was able to be tuned to provide a much better fit to experimental data.

One explanation for the discrepancy between parameter estimates at the 100 mL and 1 L scale is the difference in buffer pH. It has been shown that the solution pH is a critical parameter in the crystallisation of proteins. To further investigate this, process modelling was also carried out the crystallisation of human insulin at the 1.5 mL scale at varying solution pHs. Using this data, it is shown that performing simple modifications to the pre-exponential factor for nucleation provides a superior fit to experimental data, as shown in Figure 3. This modification is justified as the pre-exponential factor is a kinetic factor, accounting for the frequency of collisions between protein molecules to bring about the formation of a nucleus. As the frequency of collisions is proportional to the concentration of protein in solution, the pre-exponential factor therefore has a concentration dependence.

It was also possible to extend the nucleation models to include a pH dependence, and experimental data at different pH values allowed for the estimation of the effects of pH on nucleation parameters – namely, the pre-exponential factor and the interfacial energy. It is shown that the interfacial energy decreases by around 20% and the pre-exponential factor increases by around seven orders of magnitude with increasing pH. Ultimately, this work provides a starting point for the application of population balance modelling and conventional kinetic nucleation and growth models to protein crystallisation, as well as insight into the effects of pH on crystallisation. Future work will hopefully research further into the effects of pH upon scale-up and in continuous protein crystallisation.