(29d) A Systematic Model-Based Approach for the Design and Control of Protein Crystallization

Because of its high resolution, packed-bed chromatography is the most widely used method for separation in bioprocesses. However, this purification method has an operating cost, roughly the cost of resin divided by the number of times that the resin can be reused, that scales linearly with throughput. As bioreactors have become more productive, the operating costs of chromatography have become a significant portion of the total costs of manufacturing. On the other hand, crystallization from liquid solution has cost that scales sub-linearly with throughput, which is why this method is heavily used for the purification of amino acids, active pharmaceutical ingredients, and intermediates. Although crystallization is used for the production-scale purification of some therapeutic proteins such as insulin, research and development are needed to develop technology effective for most therapeutic proteins [1].

Technology for the design and control of the crystallization of proteins is much less mature than for small molecules. Methods for inducing supersaturation are also more limited due to need of maintaining protein stability and quality. Many proteins are easily denatured by changes in temperature and pH, addition of precipitants, and agitation. Proteins have complex thermodynamics, slow kinetics, large uncertainties, and potential for protein aggregation that greatly restrict allowable paths through the phase diagram, which is equivalent to threading an unknown narrow winding path through an uncertain high-dimensional space.

Although numerous studies on protein crystallization have been published, most of them performed scale-up from micro-batch scale experiments to medium-scale batch crystallizers without modeling the crystallization kinetics. The presentation describes a systematic approach to the design and control of protein crystallization from droplet-scale to medium-scale by a combination of first-principles models and model-based optimization.

Preliminary well- or vial-based experimental data can provide lower bounds on nucleation rates. Most experimental studies apply screening methods like hanging-drop or sitting-drop vapor diffusion systems [2]. Nucleation within such drops is describable by a stochastic model in the form of a Master equation [3]:

dP₀(t)/dt = -B₀(t)V(t)P₀(t), P₀(0)=1,

where B₀ is the nucleation rate which is a function of states that change with time (more details below), P₀ is the time evolution of the probability that a droplet contains no crystals, and V is the volume of the droplet. A lower bound for the nucleation rate can be obtained by comparing mean induction time calculated from the model and experimental results.

The quantity of the protein for carrying out preliminary experiments for the estimation of crystallization kinetics and thermodynamics is very limited during the initial stages of process development. This observation motivated our development of a droplet-based evaporative crystallizer for the evaluation of candidate crystallization conditions obtained from publications and/or preliminary studies, and for the estimation of crystallization kinetics with minimum quantity of protein. Controlled evaporation and rehydration of the droplet are enabled by feeding air of controlled temperature and humidity into the chamber containing the droplet. Real-time image data are collected using a microscope through an optically visible chamber. This system is describable by a crystal population balance and solution mass balances:

∂f(L,t)/∂t = J(t)f(L,t)/V(t) + B₀(S,T,µ₂)δ(L) − G(S,T)∂f(L,t)/∂L,

dC_P(t)/dt = J(t)C_P(t)/V(t) − 3ρ_ck_vG(S,T)µ₂(t),

dC_B(t)/dt = J(t)C_B(t)/V(t),

dV(t)/dt = −J(t),

where f is the crystal size distribution, L is the characteristic length, G is the crystal growth rate, B₀ is the crystal nucleation rate (which can be a combination of primary and secondary nucleation), J is the evaporation rate, C_P is the protein concentration in solution, C_B is the concentration of buffer/precipitants, S is the supersaturation (which is a function of the solubility and the solution concentrations), T is the temperature, and µ₂ is the second-order moment of the crystal size distribution. Real-time data during evaporation and rehydration allow the estimation of crystallization kinetics and yield. Furthermore, controlled evaporation and rehydration allow exploration of the phase diagram and evaluation of the potential path in the scaled-up crystallization.

For scale-up, a mixed-tank design of a medium-scale continuous evaporative crystallizer was developed. The pressure in the crystallizer is lowed by a vacuum pump to induce evaporation, and real-time imaging and product collection are enabled through the tube leaving the crystallizer controlled by a valve. A model for this system, which is similar to the droplet-based system, is

∂f(L,t)/∂t = (J(t)−F_in(t))f(L,t)/V(t) + B₀(S,T,µ₂)δ(L) − G(S,T)∂f(L,t)/∂L,

dC_P(t)/dt = F_in(t)(C_P,in(t)−C_P(t))/V(t) + J(t)C_P(t)/V(t) − 3ρ_ck_vG(S,T)µ₂(t),

dC_B(t)/dt = F_in(t)(C_B,in(t)−C_B(t))/V(t) + J(t)C_B(t)/V(t),

dV(t)/dt = F_in(t) − F_out(t) − J(t),

where F_in and F_out are the volumetric flow rates of protein solution entering and leaving the crystallizer. Estimated model parameters from the droplet-based crystallization system are used to design optimal controlled operations for the medium-scale crystallizer.

The proposed overall systematic approach for the design and control of protein crystallization based on very small quantities of protein is evaluated in a detailed simulation case study using phase equilibria and crystallization kinetics experimentally reported in the literature for lysozyme [4]–[7], which is an antibacterial protein.

References:

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[2] A. McPherson, Introduction to protein crystallization, Methods, vol. 34, no. 4, pp. 254-265.

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[4] D. Rosenbaum, P. C. Zamora, and C. F. Zukoski, Phase Behavior of Small Attractive Colloidal Particles, Phys. Rev. Lett., vol. 76, no. 1, pp. 150-153, 1996.

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