Population-balance study of protein crystal growth from solution using a hyperbolic rate law

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Kinetic data for crystal size, recently reported for bulk protein crystal growth from solution, is shown to obey a hyperbolic rate law beginning at some critical time after the commencement of the growth run. Given this rate law, the linear growth rate is established and mass-balance methods are then used to derive analytic time dependent expressions for the nucleation rate, the supersaturation and the distribution function for crystal sizes. Our population-balance approach involves a truncated Fokker–Planck equation along with an integro-differential equation for mass conservation. This model is used to describe three cases of protein crystal growth from solution for which kinetic data has been reported: lysozyme, β–lactoglobulin and insulin. Additionally, we show that this empirical rate law for crystal size is approximately related to the Burton-Cabrera-Frank (BCF) and Kossel-Stranski (KS) growth rates. Further, insight into the so called two-step, process is obtained. For two of the growth cases studied here, two-step activity was experimentally confirmed. The hyperbolic tangent rate law most accurately describes growth beginning at some critical time when two-step activity begins to wane and the bulk crystal phase becomes the more dominate form in the solution. Therefore, there seems to be a transition between growth modes at the critical time. We propose here that two-step activity creates a diffusion limited growth situation in the solution for times less than the critical time. The kinetics of crystal size are estimated, during the early diffusion limited time period, using a known model for diffusion limited growth leading to the crystal radius being directly proportional to the square root of time. At the critical time, the two models meet and the kinetics of the crystal radius can thereafter be described using the hyperbolic rate law. Additionally, features of the hyperbolic growth rate model are used to develop a lever rule that can be used to predict the critical time when the transition between growth modes occurs.

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Journal of Crystal Growth



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