Protein Crystal Shape and Size Control in Batch Crystallization: Comparing Model Predictive Control with Conventional Operating Policies

In this paper, we focus on a batch protein crystallization process used to produce tetragonal hen egg white lysozyme crystals and present a comparative study of the performance of a model predictive control (MPC) strategy formulated to account for crystal shape and size distribution with conventional operating strategies used in industry, namely, constant temperature control (CTC) and constant supersaturation control (CSC). Initially, a comprehensive, batch crystallizer model is presented involving a kinetic Monte Carlo (kMC) simulation model which describes the nucleation and crystal growth via adsorption, desorption, and migration mechanisms on the (110) and (101) faces and mass and energy balances for the continuous phase, which are developed to estimate the depletion in the protein solute concentration and the variation in the crystallizer temperature. Existing experimental data are used to calibrate the crystal growth rate and to develop an empirical expression for the nucleation rate. Simulation results demonstrate that the proposed MPC, adjusting the crystallizer jacket temperature, is able to drive the crystal shape to a desired set-point value with a low polydispersity for crystal size compared to CTC and CSC operating policies, respectively. The proposed MPC determines the optimal operating conditions needed to obtain protein crystals of a desired shape and size distribution as it helps avoid the small crystal fines at the end of the batch run. ■ INTRODUCTION Protein crystallization plays a crucial role in the $1 trillion pharmaceutical industry and has been a major contributor to both scientific advancement and economic growth. More than 100 therapeutic proteins have been licensed and a number of additional therapeutic proteins are currently under testing. For example, therapeutic proteins such as albumin which regulates the colloidal osmotic pressure of blood and globulin which boosts the immune system in our body against infectious diseases are the main proteins of human cells. Additionally, the therapeutic protein structure can be investigated via nuclear magnetic resonance for proteins of small molar mass (less than 30 000). Alternatively, X-ray crystallography is suggested for the study of the structure of proteins with molecular weight over 30 000. To use these methods, protein crystals, which are usually produced through a batch crystallization process, need to be of desired morphology with a low polydispersity. Here, we use tetragonal hen egg white (HEW) lysozyme which is a widely used model protein and relatively easily crystallizable with a molecular weight of 14 388. In the past few years, researchers have attempted to model protein nucleation and crystal growth, and significant advances have been made in the field of control of crystallization processes introducing new techniques such as the direct nucleation control and statistical control chart based switching. Their efforts make it possible to control the shape and size distributions of the protein crystals, however, no significant advance has been made associated with crystal growth initiated by nucleation in the consideration of mass and energy balances. To this end, we include mass and energy balances in order to estimate better the depletion in the protein solute concentration and the drop in the crystallizer temperature due to the heat of fusion by crystallization. Similar to our previous works, we assume the solid-on-solid lattice model which will cause the crystal to be very compact by avoiding voids and overhangs while depositing particles onto the crystal lattice, and only monomer units are considered in the attachment events. It is also assumed that the monomer is not aggregated with water, and it is a pure lysozyme molecule. The attachment rate is independent of the local surface microconfiguration, whereas detachment and migration events are highly dependent on it. To account for the dependence of detachment and migration rates on the surface configuration, kinetic Monte Carlo (kMC) simulations are needed to compute the net crystal steady-state growth rate. Kinetic Monte Carlo simulation methods represent a dynamic interpretation of the Master equation and have been widely used to simulate dynamic molecular processes. To implement our kMC methodology in the consideration of the entire lattice sites and account for mass and energy balances for the continuous phase, we extend the methodology of reference 29 to the rate equations originally developed by Durbin and Feher. The reader may also refer to our previous works for the details of the methodology for single crystal growth and for a set of crystals nucleated at different times, respectively. Our main contribution is a quantitative comparison of the performance of a novel model predictive control (MPC) strategy to that of two other conventional operating polices: constant Special Issue: David Himmelblau and Gary Powers Memorial Received: February 22, 2013 Revised: May 8, 2013 Accepted: May 24, 2013 Published: May 24, 2013 Article