Thermodynamic phase diagram of amyloid-ß (16-22) peptide.

The aggregation of monomeric amyloid ß protein (Aß) peptide into oligomers and amyloid fibrils in the mammalian brain is associated with Alzheimer's disease. Insight into the thermodynamic stability of the Aß peptide in different polymeric states is fundamental to defining and predicting the aggregation process. Experimental determination of Aß thermodynamic behavior is challenging due to the transient nature of Aß oligomers and the low peptide solubility. Furthermore, quantitative calculation of a thermodynamic phase diagram for a specific peptide requires extremely long computational times. Here, using a coarse-grained protein model, molecular dynamics (MD) simulations are performed to determine an equilibrium concentration and temperature phase diagram for the amyloidogenic peptide fragment Aß16-22 Our results reveal that the only thermodynamically stable phases are the solution phase and the macroscopic fibrillar phase, and that there also exists a hierarchy of metastable phases. The boundary line between the solution phase and fibril phase is found by calculating the temperature-dependent solubility of a macroscopic Aß16-22 fibril consisting of an infinite number of ß-sheet layers. This in silico determination of an equilibrium (solubility) phase diagram for a real amyloid-forming peptide, Aß16-22, over the temperature range of 277-330 K agrees well with fibrillation experiments and transmission electron microscopy (TEM) measurements of the fibril morphologies formed. This in silico approach of predicting peptide solubility is also potentially useful for optimizing biopharmaceutical production and manufacturing nanofiber scaffolds for tissue engineering.

A numerical study to determine the effect of ligament stiffness on kinematics of the lumbar spine during flexion.

BACKGROUND: There is a wide range of mechanical properties of spinal ligaments documented in literature. Due to the fact that ligaments contribute in stabilizing the spine by limiting excessive intersegmental motion, those properties are of particular interest for the implementation in musculoskeletal models. The aim of this study was to investigate the effect of varying ligament stiffness on the kinematic behaviour of the lumbar spine. METHODS: A musculoskeletal model with a detailed lumbar spine was modified according to fluoroscopic recordings and corresponding data files of three different subjects. For flexion, inverse dynamics analysis with a variation of the ligament stiffness matrix were conducted. The influence of several degrees of ligament stiffness on the lumbar spine model were investigated by tracking ligament forces, disc forces and resulting moments generated by the ligaments. Additionally, the kinematics of the motion segments were evaluated. RESULTS: An increase of ligament stiffness resulted in an increase of ligament and disc forces, whereas the relative change of disc force increased at a higher rate at the L4/L5 level (19 %) than at the L3/L4 (10 %) level in a fully flexed posture. The same behaviour applied to measured moments with 67 % and 45 %. As a consequence, the motion deflected to the lower levels of the lumbar spine and the lower discs had to resist an increase in loading. CONCLUSIONS: Higher values of ligament stiffness over all lumbar levels could lead to a shift of the loading and the motion between segments to the lower lumbar levels. This could lead to an increased risk for the lower lumbar parts.

Nucleation of polymorphic amyloid fibrils.

One and the same protein can self-assemble into amyloid fibrils with different morphologies. The phenomenon of fibril polymorphism is relevant biologically because different fibril polymorphs can have different toxicity, but there is no tool for predicting which polymorph forms and under what conditions. Here, we consider the nucleation of polymorphic amyloid fibrils occurring by direct polymerization of monomeric proteins into fibrils. We treat this process within the framework of our newly developed nonstandard nucleation theory, which allows prediction of the concentration dependence of the nucleation rate for different fibril polymorphs. The results highlight that the concentration dependence of the nucleation rate is closely linked with the protein solubility and a threshold monomer concentration below which fibril formation becomes biologically irrelevant. The relation between the nucleation rate, the fibril solubility, the threshold concentration, and the binding energies of the fibril building blocks within fibrils might prove a valuable tool for designing new experiments to control the formation of particular fibril polymorphs.

Confounding the paradigm: peculiarities of amyloid fibril nucleation.

Fibrils of amyloid proteins are currently of great interest because of their involvement in various amyloid-related diseases and nanotechnological products. In a recent kinetic Monte Carlo simulation study (Cabriolu, R.; Kashchiev, D.; Auer, S. J. Chem. Phys.2012, 137, 204903), we found that our simulation data for the rate of amyloid fibril nucleation occurring by direct polymerization of monomeric protein could not be described adequately by nucleation theory. It turned out that the process occurred in a peculiar way, thus confounding the nucleation paradigm and demanding a new theoretical treatment. In the present study, we reconsider the theoretical approach to nucleation of amyloid fibrils and derive new expressions for the stationary rate of the process. As these expressions provide a remarkably good description of the simulation data, by using them we propose a theoretical dependence of the amyloid-ß(40) fibril nucleation rate on the concentration of monomeric protein in the solution. This dependence reveals the existence of a threshold concentration 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. 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.

Breakdown of nucleation theory for crystals with strongly anisotropic interactions between molecules.

We study the nucleation of model two-dimensional crystals in order to gain insight into the effect of anisotropic interactions between molecules on the stationary nucleation rate J. With the aid of kinetic Monte Carlo simulations, we determine J as a function of the supersaturation s. It turns out that with increasing degree of interaction anisotropy the dependence of ln J on s becomes step-like, with jumps at certain s values. We show that this J(s) dependence cannot be described by the classical and atomistic nucleation theories. A formula that predicts the identified J(s) behavior is yet to be derived and verified, and the present study provides the necessary data and understanding for doing that.

Size distribution of amyloid nanofibrils.

We consider the size distribution of amyloid nanofibrils (protofilaments) in nucleating protein solutions when the nucleation process occurs by the mechanism of direct polymerization of ß-strands (extended peptides or protein segments) into ß-sheets. Employing the atomistic nucleation theory, we derive a general expression for the stationary size distribution of amyloid nanofibrils constituted of successively layered ß-sheets. The application of this expression to amyloid ß(1-40) (Aß(40)) fibrils allows us to determine the nanofibril size distribution as a function of the protein concentration and temperature. The distribution is most remarkable with its exhibiting a series of peaks positioned at "magic" nanofibril sizes (or lengths), which are due to deep local minima in the work for fibril formation. This finding of magic sizes or lengths is consistent with experimental results for the size distribution of aggregates in solutions of Aß(40) proteins. Also, our approach makes it possible to gain insight into the effect of point mutations on the nanofibril size distribution, an effect that may play a role in experimentally observed substantial differences in the fibrillation lag-time of wild-type and point-mutated amyloid-ß proteins.

Phase diagram of polypeptide chains.

We use a coarse grained protein model that enables us to determine the equilibrium phase diagram of natively folded α-helical and unfolded ß-sheet forming peptides. The phase diagram shows that there are only two thermodynamically stable peptide phases, the peptide solution and the bulk fibrillar phase. In addition, it reveals the existence of various metastable peptide phases. The liquidlike oligomeric phases are metastable with respect to the fibrillar phases, and there is a hierarchy of metastability. The presented phase diagram provides a solid basis for understanding the assembly of polypeptide chains into the phases formed in their natively folded and unfolded conformations.

Communication: Conformation state diagram of polypeptides: a chain length induced α-ß transition.

By using a generic coarse grained polypeptide model, we perform multicanonical molecular dynamics simulations for determining the equilibrium conformation state diagram of a single homopolypeptide chain as a function of the chain length and temperature. The state diagram highlights the thermal regimes of stability for various conformational patterns in polypeptides, including swollen, random and collapsed coils, globular structures, extended and bended α helices, and compact ß bundles. Remarkably, at low temperatures we observe a sharp transition from extended α helix to compact ß bundles as the chain length increases. This finding indicates that the chain length is one of the intrisic factors that can trigger α-ß transformations in a broad class of polypeptides.

Insight into the correlation between lag time and aggregation rate in the kinetics of protein aggregation.

Under favorable conditions, many proteins can assemble into macroscopically large aggregates such as the amyloid fibrils that are associated with Alzheimer's, Parkinson's, and other neurological and systemic diseases. The overall process of protein aggregation is characterized by initial lag time during which no detectable aggregation occurs in the solution and by maximal aggregation rate at which the dissolved protein converts into aggregates. In this study, the correlation between the lag time and the maximal rate of protein aggregation is analyzed. It is found that the product of these two quantities depends on a single numerical parameter, the kinetic index of the curve quantifying the time evolution of the fraction of protein aggregated. As this index depends relatively little on the conditions and/or system studied, our finding provides insight into why for many experiments the values of the product of the lag time and the maximal aggregation rate are often equal or quite close to each other. It is shown how the kinetic index is related to a basic kinetic parameter of a recently proposed theory of protein aggregation.

Phase diagram of alpha-helical and beta-sheet forming peptides.

The intrinsic property of proteins to form structural motifs such as alpha helices and beta sheets leads to a complex phase behavior in which proteins can assemble into various types of aggregates including crystals, liquidlike phases of unfolded or natively folded proteins, and amyloid fibrils. Here we use a coarse-grained protein model that enables us to perform Monte Carlo simulations for determining the phase diagram of natively folded alpha-helical and unfolded beta-sheet forming peptides. The simulations reveal the existence of various metastable peptide phases. The liquidlike phases are metastable with respect to the fibrillar phases, and there is a hierarchy of metastability.

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.

Nucleation of amyloid fibrils.

We consider nucleation of amyloid fibrils in the case when the process occurs by the mechanism of direct polymerization of practically fully extended protein segments, i.e., beta-strands, into beta-sheets. Applying the classical nucleation theory, we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) constituted of successively layered beta-sheets. Analysis of this expression reveals that with increasing its size, the fibril transforms from one-dimensional to two-dimensional aggregate in order to preserve the equilibrium shape corresponding to minimal formation work. We determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as explicit functions of the concentration and temperature of the protein solution. The results obtained are applicable to homogeneous nucleation, which occurs when the solution is sufficiently pure and/or strongly supersaturated.

Atomistic theory of amyloid fibril nucleation.

We consider the nucleation of amyloid fibrils at the molecular level when the process takes place by a direct polymerization of peptides or protein segments into ß-sheets. Employing the atomistic nucleation theory (ANT), we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) composed of successively layered ß-sheets. The application of this expression to a recently studied peptide system allows us to determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as functions of the supersaturation of the protein solution. Our analysis illustrates the unique feature of ANT that the size of the fibril nucleus is a constant integer in a given supersaturation range. We obtain the ANT nucleation rate and compare it with the rates determined previously in the scope of the classical nucleation theory (CNT) and the corrected classical nucleation theory (CCNT). We find that while the CNT nucleation rate is orders of magnitude greater than the ANT one, the CCNT and ANT nucleation rates are in very good quantitative agreement. The results obtained are applicable to homogeneous nucleation, which occurs when the protein solution is sufficiently pure and/or strongly supersaturated.

Efficient high-quality volume rendering of SPH data.

High quality volume rendering of SPH data requires a complex order-dependent resampling of particle quantities along the view rays. In this paper we present an efficient approach to perform this task using a novel view-space discretization of the simulation domain. Our method draws upon recent work on GPU-based particle voxelization for the efficient resampling of particles into uniform grids. We propose a new technique that leverages a perspective grid to adaptively discretize the view-volume, giving rise to a continuous level-of-detail sampling structure and reducing memory requirements compared to a uniform grid. In combination with a level-of-detail representation of the particle set, the perspective grid allows effectively reducing the amount of primitives to be processed at run-time. We demonstrate the quality and performance of our method for the rendering of fluid and gas dynamics SPH simulations consisting of many millions of particles.

Impacts of multiple stressors on freshwater biota across spatial scales and ecosystems.

Climate and land-use change drive a suite of stressors that shape ecosystems and interact to yield complex ecological responses (that is, additive, antagonistic and synergistic effects). We know little about the spatial scales relevant for the outcomes of such interactions and little about effect sizes. These knowledge gaps need to be filled to underpin future land management decisions or climate mitigation interventions for protecting and restoring freshwater ecosystems. This study combines data across scales from 33 mesocosm experiments with those from 14 river basins and 22 cross-basin studies in Europe, producing 174 combinations of paired-stressor effects on a biological response variable. Generalized linear models showed that only one of the two stressors had a significant effect in 39% of the analysed cases, 28% of the paired-stressor combinations resulted in additive effects and 33% resulted in interactive (antagonistic, synergistic, opposing or reversal) effects. For lakes, the frequencies of additive and interactive effects were similar for all spatial scales addressed, while for rivers these frequencies increased with scale. Nutrient enrichment was the overriding stressor for lakes, with effects generally exceeding those of secondary stressors. For rivers, the effects of nutrient enrichment were dependent on the specific stressor combination and biological response variable. These results vindicate the traditional focus of lake restoration and management on nutrient stress, while highlighting that river management requires more bespoke management solutions.

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 Alzheimer's and Parkinson's 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 beta-sheets within the disordered oligomeric aggregates that form early under the effect of hydrophobic forces. Individual beta-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-beta structures emerge preferentially from disordered compact dynamical intermediate assemblies.

Simple Model of the Effect of Solution Conditions on the Nucleation of Amyloid Fibrils.

It is well known that peptide and protein fibrillation is strongly affected by the solution conditions, but a fundamental understanding of how amyloid fibril nucleation depends on solution pH, salt concentration, and solvent is absent. Here, we use expressions from Debye-Hückel theory to describe the interactions between charged amino acids in combination with our recently developed nonstandard nucleation theory to predict the concentration dependence of the fibril nucleation rate under different solvent conditions. The general rule that emerges from these considerations is that changes in solution pH, salt concentration, and solvent that increase the bonding energy between the fibril building blocks decrease the fibril solubility and promote fibril nucleation, in line with experimental observations. The simple analytical relations among the nucleation rate, fibril solubility, and binding energies provide a tool to controlling and understanding amyloid fibril formation by changing the solution conditions.

Effects of river bank heterogeneity and time of day on drift and stranding of juvenile European grayling (Thymallus thymallus L.) caused by hydropeaking.

High-head storage hydropower is deemed to be the ideal renewable energy source in Alpine regions to meet the increasing demand for daily peak electrical energy. However, this mode of operation - called hydropeaking - can imply severe hydrological and hydromorphological consequences for river ecosystems, affecting fish populations by e.g. drift and stranding of young life stages. Several fish-stranding experiments using physical models have been performed in the past, but until now very little is known about influences of time of day or gravel bank heterogeneity. We performed experiments during late summer 2013 with juvenile European grayling (Thymallus thymallus) (mean length: 53mm) in a nature-like experimental channel enabling hydropeaking simulations. In the first experiments (n=21) we observed relative drift and stranding rates for a single hydropeaking event focusing on the effect of time of day on a homogenous gravel bank. The second test series (n=15) focused on two dewatering potholes installed as potential traps. Additional experiments (n=6) were done with a reduced downramping rate to gain information about potential mitigation effects on stranding risk. During daytime and decreasing water level, we observed low drift rates of 15% and stranding rates below 5% in dewatering potholes and on homogenous gravel banks. However, in the presence of dewatering potholes, nighttime drift rates were about three times and stranding rates about ten times higher than on the homogenous gravel bank. A lowered downramping rate reduced drift to about a quarter and almost eliminated nocturnal stranding risk. These results might be used to effectively regulate water releases from high-head storage hydropower plants in a more suitable way for sensitive life stages of fish. Reducing the downramping rate or shifting peaks to daytime can reduce negative effects of hydropeaking in consideration of the morphological character of affected rivers.

Amyloid fibril nucleation: effect of amino acid hydrophobicity.

We consider the nucleation of amyloid fibrils when the process occurs by direct polymerization of fully extended peptides (i.e., ß-strands) into fibrils composed of successively layered ß-sheets with alternating weak and strong hydrophobic surfaces. We extend our recently developed nucleation model (Kashchiev, D.; Cabriolu, R.; Auer, S. J. Am. Chem. Soc. 2013, 135, 1531-1539) to derive general expressions for the work to form such fibrils, the fibril solubility, the nucleation work, the equilibrium concentration of nuclei, and the fibril nucleation rate as explicit functions of the supersaturation of the protein solution. Analysis of these expressions illustrates the effect of increased asymmetry between the weak and strong hydrophobic ß-sheet surfaces on the thermodynamics and kinetics of the polymerization process. In particular, the application of our theoretical framework to a simple model peptide system shows that lowering the hydrophobicity of one ß-sheet surface can hamper protein fibrillation because the threshold concentration below which the fibril nucleation is practically arrested, and above which the process occurs vigorously--because then each monomer in the solution acts as a fibril nucleus--is shifted to higher concentrations. This effect is entirely due to the effect of asymmetry of the two hydrophobic ß-sheet surfaces on the fibril solubility. In addition, with increasing asymmetry, the nucleation rate of one fibril polymorph becomes increasingly dominant, illustrating that there is a morphological selection between the two possible polymorphs.

Tracking polypeptide folds on the free energy surface: effects of the chain length and sequence.

Characterization of the folding transition in polypeptides and assessing the thermodynamic stability of their structured folds are of primary importance for approaching the problem of protein folding. We use molecular dynamics simulations for a coarse grained polypeptide model in order to (1) obtain the equilibrium conformation diagram of homopolypeptides in a broad range of the chain lengths, N = 10, ..., 100, and temperatures, T (in a multicanonical ensemble), and (2) determine free energy profiles (FEPs) projected onto an optimal, so-called "natural", reaction coordinate that preserves the height of barriers and the diffusion coefficients on the underlying free energy hyper-surface. We then address the following fundamental questions. (i) How well does a kinetically determined free energy landscape of a single chain represent the polypeptide equilibrium (ensemble) behavior? In particular, under which conditions might the correspondence be lost, and what are the possible implications for the folding processes? (ii) How does the free energy landscape depend on the chain length (homopolypeptides) and the monomer interaction sequence (heteropolypeptides)? Our data reveal that at low T values equilibrium structures adopted by relatively short homopolypeptides (N < 60) are dominated by α-helical folds which correspond to the primary and secondary minima of the FEP. In contrast, longer homopolypeptides (N > 70), upon quasi-equilibrium cooling, fold preferentially in ß-bundles with small helical portions, while the FEPs exhibit no distinct global minima. Moreover, subject to the choice of the initial configuration, at sufficiently low T, essentially metastable structures can be found and prevail far from the true thermodynamic equilibrium. We also show that, by sequence-enabling the polypeptide model, it is possible to restrict the chain to a very specific part of the configuration space, which results in substantial simplification and smoothing of the free energy landscape as compared to the case of the corresponding homopolypeptide.