Stefan Auer

Prediction of absolute crystal-nucleation rate in hard-sphere colloids

Crystal nucleation is a much-studied phenomenon, yet the rate at which it occurs remains difficult to predict. Small crystal nuclei form spontaneously in supersaturated solutions, but unless their size exceeds a critical value—the so-called critical nucleus—they will re-dissolve rather than grow. It is this rate-limiting step that has proved difficult to probe experimentally. The crystal nucleation rate depends on Pcrit, the (very small) probability that a critical nucleus forms spontaneously, and on a kinetic factor (κ) that measures the rate at which critical nuclei subsequently grow. Given the absence of a priori knowledge of either quantity, classical nucleation theory is commonly used to analyse crystal nucleation experiments, with the unconstrained parameters adjusted to fit the observations. This approach yields no ‘first principles’ prediction of absolute nucleation rates. Here we approach the problem from a different angle, simulating the nucleation process in a suspension of hard colloidal spheres, to obtain quantitative numerical predictions of the crystal nucleation rate. We find large discrepancies between the computed nucleation rates and those deduced from experiments: the best experimental estimates of Pcrit seem to be too large by several orders of magnitude.

Onset of heterogeneous crystal nucleation in colloidal suspensions

The addition of small ‘seed’ particles to a supersaturated solution can greatly increase the rate at which crystals nucleate. This process is understood, at least qualitatively, when the seed has the same structure as the crystal that it spawns. However, the microscopic mechanism of seeding by a ‘foreign’ substance is not well understood. Here we report numerical simulations of colloidal crystallization seeded by foreign objects. We perform Monte Carlo simulations to study how smooth spherical seeds of various sizes affect crystallization in a suspension of hard colloidal particles. We compute the free-energy barrier associated with crystal nucleation. A low barrier implies that nucleation is easy. We find that to be effective crystallization promoters, the seed particles need to exceed a well-defined minimum size. Just above this size, seed particles act as crystallization ‘catalysts’ as newly formed crystallites detach from the seed. In contrast, larger seed particles remain covered by the crystallites that they spawn. This phenomenon should be experimentally observable and can have important consequences for the control of the resulting crystal size distribution.

Suppression of crystal nucleation in polydisperse colloids due to increase of the surface free energy

The formation of small crystallites is governed by two competing factors: the free energy gained upon transferring constituent atoms, molecules or colloidal particles from the metastable liquid to the more stable solid, and the free energy needed to create the surface area of the crystallite. Because the ratio of surface area to bulk is large for small particles, small crystallites dissolve spontaneously under conditions where larger crystallites are stable and macroscopic crystal growth occurs only if spontaneously formed crystallites exceed a critical minimum size. On theoretical grounds, the probability of forming such critical crystal nuclei is expected to increase rapidly with supersaturation. However, experiments show that the rate of crystal nucleation in many systems goes through a maximum as the supersaturation is increased. It is commonly assumed that the nucleation rate peaks because, even though the probability of forming critical nuclei increases with increasing concentration, the rate of growth of such nuclei decreases. Here we report simulations of crystal nucleation in suspensions of colloidal spheres with varying size distributions that show that the probability that critical nuclei will form itself goes through a maximum as the supersaturation is increased. We find that this effect, which is strongest for systems with the broadest particle size distribution, results from an increase with supersaturation of the solid–liquid interfacial free energy. The magnitude of this effect suggests that vitrification at high supersaturations should yield colloidal glasses that are truly amorphous, rather than nano-crystalline.

Numerical prediction of absolute crystallization rates in hard-sphere colloids

Special computational techniques are required to compute absolute crystal nucleation rates of colloidal suspensions. Using crystal nucleation of hard-sphere colloids as an example, we describe in some detail the novel computational tools that are needed to perform such calculations. In particular, we focus on the definition of appropriate order parameters that distinguish liquid from crystal, and on techniques to compute the kinetic prefactor that enters in the expression for the nucleation rate. In addition, we discuss the relation between simulation results and theoretical predictions based on classical nucleation theory.

Kinetics and thermodynamics of amyloid formation from direct measurements of fluctuations in fibril mass.

Aggregation of proteins and peptides is a widespread and much-studied problem, with serious implications in contexts ranging from biotechnology to human disease. An understanding of the proliferation of such aggregates under specific conditions requires a quantitative knowledge of the kinetics and thermodynamics of their formation; measurements that to date have remained elusive. Here, we show that precise determination of the growth rates of ordered protein aggregates such as amyloid fibrils can be achieved through real–time monitoring, using a quartz crystal oscillator, of the changes in the numbers of molecules in the fibrils from variations in their masses. We show further that this approach allows the effect of other molecular species on fibril growth to be characterized quantitatively. This method is widely applicable, and we illustrate its power by exploring the free-energy landscape associated with the conversion of the protein insulin to its amyloid form and elucidate the role of a chemical chaperone and a small heat shock protein in inhibiting the aggregation reaction.

A Generic Mechanism of Emergence of Amyloid Protofilaments from Disordered Oligomeric Aggregates

The presence of oligomeric aggregates, which is often observed during the process of amyloid formation, has recently attracted much attention because it has been associated with a range of neurodegenerative conditions including Alzheimers and Parkinsons diseases. We provide a description of a sequence-indepedent mechanism by which polypeptide chains aggregate by forming metastable oligomeric intermediate states prior to converting into fibrillar structures. Our results illustrate that the formation of ordered arrays of hydrogen bonds drives the formation of β-sheets within the disordered oligomeric aggregates that form early under the effect of hydrophobic forces. Individual β-sheets initially form with random orientations and subsequently tend to align into protofilaments as their lengths increase. Our results suggest that amyloid aggregation represents an example of the Ostwald step rule of first-order phase transitions by showing that ordered cross-β structures emerge preferentially from disordered compact dynamical intermediate assemblies.

Characterization of the nucleation barriers for protein aggregation and amyloid formation

Despite the complexity and the specificity of the amino acid code, a variety of peptides and proteins unrelated in sequence and function exhibit a common behavior and assemble into highly organized amyloid fibrils. The formation of such aggregates is often described by a nucleation and growth mechanism, in which the proteins involved also form intermediate oligomeric aggregates before they reorganize and grow into ordered fibrils with a characteristic cross‐β structure. It is extremely difficult to experimentally obtain an accurate description of the early stages of this phenomenon due to the transient nature and structural heterogeneity of the oligomeric precursors. We investigate here the phenomenon of ordered aggregation by using the recently introduced tube model of polypeptide chains in conjunction with the generic hypothesis of amyloid formation. Under conditions where oligomer formation is a rare event—the most common conditions for forming amyloid fibrils by experiment—we calculate directly the nucleation barriers associated with oligomer formation and conversion into cross‐β structure in order to reveal the nature of these species, determine the critical nuclei, and characterize their dependence on the hydrophobicity of the peptides and the thermodynamic parameters associated with aggregation and amyloid formation.

Self-Templated Nucleation in Peptide and Protein Aggregation

Peptides and proteins exhibit a common tendency to assemble into highly ordered fibrillar aggregates, whose formation proceeds in a nucleation-dependent manner that is often preceded by the formation of oligomeric assemblies. This process has received much attention because disordered oligomeric aggregates have been associated with neurodegenerative disorders such as Alzheimers and Parkinsons disease. Here we describe a self-templated nucleation mechanism that determines the transition between the initial condensation of polypeptide chains into disordered assemblies and their reordering into fibrillar structures. The results that we present show that at the molecular level this transition is due to the ability of polypeptide chains to reorder within oligomers into fibrillar assemblies whose surfaces act as templates that stabilize the disordered assemblies.

Crystallization of weakly charged colloidal spheres: a numerical study

We report a numerical study of crystal nucleation in a system of weakly charged colloids. The interaction between the colloids is approximated by a repulsive hard-core Yukawa potential. We studied the dependence of the nucleation barrier and the nucleation rate on supersaturation as a function of both contact value and range of the interaction potential. We find that, at the same volume fraction, nucleation is much faster for these soft colloids than for hard spheres. This is partly because fluid–solid coexistence in charged colloids occurs at lower volume fractions than for hard spheres. But, in addition, the softness of the potential has a pronounced direct effect on the nucleation barrier through a lowering of the solid–liquid surface free energy. Moreover, the softness of the potential directly affects the pathway for crystal nucleation: even when the stable crystal phase has a face-centred cubic structure, we find that the initial crystal nuclei have a bcc structure.

A Condensation-Ordering Mechanism in Nanoparticle-Catalyzed Peptide Aggregation

Nanoparticles introduced in living cells are capable of strongly promoting the aggregation of peptides and proteins. We use here molecular dynamics simulations to characterise in detail the process by which nanoparticle surfaces catalyse the self-assembly of peptides into fibrillar structures. The simulation of a system of hundreds of peptides over the millisecond timescale enables us to show that the mechanism of aggregation involves a first phase in which small structurally disordered oligomers assemble onto the nanoparticle and a second phase in which they evolve into highly ordered as their size increases.