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Simulations of Crystallization and Nucleation

Modelling crystal nucleation
Understanding crystal nucleation is a grand challenge for the future because of its wide implications ranging from climate to human health and welfare. For example the nucleation of ice clouds in the upper troposphere affects the energy flow into and out of the earth’s atmosphere and thereby our climate, and the aggregation of proteins into highly ordered amyloid fibrils is related to the onset of diseases such as Alzheimer’s and Parkinson’s. Of particular interest is the understanding of the early stages of crystallization as the smallest crystal nuclei often determine the structure and properties of the resulting solid. The experimental observation of the smallest crystal nuclei in a supersaturated solution is challenging because they form rarely, and when they form their appearance is fairly short (depending on the type of molecules and the solution conditions their lifetime can range from nano to picoseconds). In addition, it is difficult to probe experimentally crystal nuclei that are composed of only a few atoms, because it requires their visualization on an atomistic length scale (i.e. a few angstrom). 

Molecular simulations are in principle ideally suited to overcome these problems and have become a major tool to investigate crystal nucleation. There exist various different numerical techniques to calculate the shape and height of a nucleation barrier, to study the mechanism of formation and structure of the critical nucleus and to obtain quantitative predictions of crystal nucleation rates, which can provide a rigorous test of both experiments and existing theories. This has been demonstrated in a Monte Carlo simulation study of homogeneous crystal nucleation in hard sphere colloids, which is a simple model used to investigate general aspects of crystal nucleation. Figure 1 shows a configuration of a critical nucleus of about 100 hard spheres (shown in yellow) that form in a supersaturated liquid (blue spheres).

Figure 1: Taken from Ref. Nature, 409 1020 (2001)

With the rapid development of computer power and resources it is now possible to simulate crystal nucleation for more complex molecular systems such as water, calcium carbonate, sodium chloride, and polymers.  

Molecular simulations of crystal nucleation in biomolecular systems is much more challenging. The main reasons for this is that peptides and proteins are composed of many atoms that interact with complex potentials so that it becomes impossible to perform molecular simulations on a timescale necessary to follow the early stages of crystallisation with atomistic resolution. This limitation can be overcome by using a simplified model for peptides. In recent work we have been able to perform Molecular Dynamics simulation of a system of hundreds of peptides on a millisecond timescale to study the effect of nanoparticles on the peptide aggregation (Figure 2A). Our simulations reveal a generic condensation-ordering transition in which the peptides first condense on the nanoparticle surface (Figure 2B) before they reorder into highly ordered crystal-like protofilaments (Figure 2C).

Figure 2: Taken from Ref. PLoS Comput Biol, PLoS Comput Biol, 4 e1000458 (2009)

Alternatively, we approach the problem of nucleation of amyloid fibrils by treating the process in the framework of the general theories of nucleation of new phases. We have recently shown that the aggregation of proteins into amyloid fibrils is a peculiar type of nucleation that cannot entirely be described in the framework of classical nucleation theory. Therefore we have developed a new theory to describe amyloid fibril nucleation and derived new expressions for the stationary rate of this process. Using them we propose a theoretical dependence of the amyloid-β40 fibril nucleation rate on the concentration of monomeric protein in the solution. This dependence reveals the existence of a threshold concentration (see Figure 3) below which the fibril nucleation in small enough solution volumes is practically arrested, and above which the process occurs vigorously, because then each monomeric protein in the solution acts as fibril nucleus.

Figure 3: Taken from Ref. J. Am. Chem. Soc. 135 1531 (2013)

The presented expressions for the threshold concentration and for the dependence of the fibril nucleation rate on the concentration of monomeric protein can be a valuable guide in designing new therapeutic and/or technological strategies for prevention or stimulation of amyloid fibril formation.