A theoretical and numerical multiscale framework for the analysis of pattern formation in protein crystal engineering

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The relevance of self-organization, pattern formation, nonlinear phenomena and non-equilibrium behavior in a wide range of problems related to macromolecular crystal engineering, calls for a concerted approach using the tools of statistical physics, thermodynamics, fluid-dynamics, nonlinear dynamics, mathematical modeling and numerical simulation, in synergy with experimentally oriented work. The reason behind such a need is that in many instances of relevance in this field one witnesses interplay between molecular and macroscopic-level entities and processes. Along these lines, two models are defined here and discussed in detail, one dealing with issues of complex behavior at the microscopic level and the second referring to the strong nonlinear nature of macroscopic evolution. Such models share a common fundamental feature, a group of equations, strictly related, from a mathematical point of view, to the ‘kinetic conditions’ used to model mass transfer at the crystal surface; model diversification then occurs on the basis of the desired scale length, i.e. according to the level of detail required by the analysis ('local' or 'global'). If the "local" evolution of the crystal surface is the subject of the investigation (distribution of the local growth rate along crystal face, shape instabilities, onset of surface depressions due to diffusive and/or convective effects, etc, i.e. all those factors dealing with the "local" history of the shape) the model is conceived to provide "microscopic" and "morphological" details. For this specific case a ‘kinetic-coefficient-based’ moving boundary numerical (CFD) strategy is carefully developed on the basis of Volume-of-Fluid methods (also known as Volume Tracking methods) and Level-set techniques, which have become popular in the last years as numerical techniques capable of modeling complex multi-phase problems as well as for their capability to undertake a fixed-grid solution without resorting to mathematical manipulations and transformations. On the contrary, if the size of the crystals is negligible with respect to the size of the reactor (i.e. if they are small and undergo only small dimensional changes with respect to the overall dimensions of the cell containing the feeding solution), the shape of the crystals is ignored and the proposed approach relies directly on an algebraic formulation of the nucleation events and on the application of an integral form of the mass balance kinetics for each protein crystal. The applicability and the suitability of the different sub-models are discussed according to some worked examples of practical interest. Pattern formation in these processes is described here with respect to crystal shapes, nuclei spatial discrete arrangements and the convective multicellular structures arising as a consequence of buoyancy forces, thus enriching the discussions with some interdisciplinary flavor.
Original languageEnglish
Pages (from-to)149–174
Number of pages26
JournalJournal for Multiscale Computational Engineering
Issue number2
Publication statusPublished - 2011


  • multiscale framework
  • protein crystal engineering
  • pattern formation


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