Simplified exactly solvable model for β-amyloid aggregation

Invited review: Modeling milk stability

Novel insights into the stability of milk and milk products during storage and processing result from describing caseins near neutral pH as hydrophilic, intrinsically disordered, proteins. Casein solubility is strongly influenced by pH and multivalent ion binding. Solubility is high at a neutral pH or above, but decreases as the casein net charge approaches zero, allowing a condensed casein phase or gel to form, then increases at lower pH. Of particular importance for casein micelle stability near neutral pH is the proportion of free caseins in the micelle (i.e., caseins not bound directly to nanoclusters of calcium phosphate). Free caseins are more soluble and better able to act as molecular chaperones (to prevent casein and whey protein aggregation) than bound caseins. Some free caseins are highly phosphorylated and can also act as mineral chaperones to inhibit the growth of calcium phosphate phases and prevent mineralized deposits from forming on membranes or heat exchangers. Thus, casein micelle stability is reduced when free caseins bind to amyloid fibrils, destabilized whey proteins or calcium phosphate. The multivalent-binding model of the casein micelle quantitatively describes these and other factors affecting the stability of milk and milk protein products during manufacture and storage.

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.

Multiphysics of Prionlike Diseases: Progression and Atrophy

Many neurodegenerative diseases are related to the propagation and accumulation of toxic proteins throughout the brain. The lesions created by aggregates of these toxic proteins further lead to cell death and accelerated tissue atrophy. A striking feature of some of these diseases is their characteristic pattern and evolution, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. Here, we simulate the anisotropic propagation and accumulation of toxic proteins in full brain geometry. We show that the same model with different initial seeding zones reproduces the characteristic evolution of different prionlike diseases. We also recover the expected evolution of the total toxic protein load. Finally, we couple our transport model to a mechanical atrophy model to obtain the typical degeneration patterns found in neurodegenerative diseases.

Physical mechanism for biopolymers to aggregate and maintain in non-equilibrium states

Many human or animal diseases are related to aggregation of proteins. A viable biological organism should maintain in non-equilibrium states. How protein aggregate and why biological organisms can maintain in non-equilibrium states are not well understood. As a first step to understand such complex systems problems, we consider simple model systems containing polymer chains and solvent particles. The strength of the spring to connect two neighboring monomers in a polymer chain is controlled by a parameter s with s → ∞ for rigid-bond. The strengths of bending and torsion angle dependent interactions are controlled by a parameter s A with s A  → -∞ corresponding to no bending and torsion angle dependent interactions. We find that for very small s A , polymer chains tend to aggregate spontaneously and the trend is independent of the strength of spring. For strong springs, the speed distribution of monomers in the parallel (along the direction of the spring to connect two neighboring monomers) and perpendicular directions have different effective temperatures and such systems are in non-equilibrium states.

Finite-size corrections and scaling for the dimer model on the checkerboard lattice

Lattice models are useful for understanding behaviors of interacting complex many-body systems. The lattice dimer model has been proposed to study the adsorption of diatomic molecules on a substrate. Here we analyze the partition function of the dimer model on a 2M×2N checkerboard lattice wrapped on a torus and derive the exact asymptotic expansion of the logarithm of the partition function. We find that the internal energy at the critical point is equal to zero. We also derive the exact finite-size corrections for the free energy, the internal energy, and the specific heat. Using the exact partition function and finite-size corrections for the dimer model on a finite checkerboard lattice, we obtain finite-size scaling functions for the free energy, the internal energy, and the specific heat of the dimer model. We investigate the properties of the specific heat near the critical point and find that the specific-heat pseudocritical point coincides with the critical point of the thermodynamic limit, which means that the specific-heat shift exponent λ is equal to ∞. We have also considered the limit N→∞ for which we obtain the expansion of the free energy for the dimer model on the infinitely long cylinder. From a finite-size analysis we have found that two conformal field theories with the central charges c=1 for the height function description and c=-2 for the construction using a mapping of spanning trees can be used to describe the dimer model on the checkerboard lattice.

Cooperativity and modularity in protein folding

A simple statistical mechanical model proposed by Wako and Saitô has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Muñoz and Eaton and has since been referred to as the Wako-Saitô-Muñoz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related models, Saitô emphasized the importance of the hierarchical pathway in protein folding; folding starts with the creation of contiguous segments having a native-like configuration and proceeds as growth and coalescence of these segments. The Φ-values calculated for barnase with the WSME model suggested that segments contributing to the folding nucleus are similar to the structural modules defined by the pattern of native atomic contacts. The WSME model was extended to explain folding of multi-domain proteins having a complex topology, which opened the way to comprehensively understanding the folding process of multi-domain proteins. The WSME model was also extended to describe allosteric transitions, indicating that the allosteric structural movement does not occur as a deterministic sequential change between two conformations but as a stochastic diffusive motion over the dynamically changing energy landscape. Statistical mechanical viewpoint on folding, as highlighted by the WSME model, has been renovated in the context of modern methods and ideas, and will continue to provide insights on equilibrium and dynamical features of proteins.

Pseudo-one-dimensional nucleation in dilute polymer solutions

Pathogenic protein fibrils have been shown in vitro to have nucleation-dependent kinetics despite the fact that one-dimensional structures do not have the size-dependent surface energy responsible for the lag time in classical theory. We present a theory showing that the conformational entropy of the peptide chains creates a free-energy barrier that is analogous to the translational entropy barrier in higher dimensions. We find that the dynamics of polymer rearrangement make it very unlikely for nucleation to succeed along the lowest free-energy trajectory, meaning that most of the nucleation flux avoids the free-energy saddle point. We use these results to construct a three-dimensional model for amyloid nucleation that accounts for conformational entropy, backbone H bonds, and side-chain interactions to compute nucleation rates as a function of concentration.

Simplified lattice model for polypeptide fibrillar transitions

Polypeptide fibrillar transitions are studied using a simplified lattice model, modified from the three-state Potts model, where uniform residues as spins, placed on a cubic lattice, can interact with neighbors to form coil, helical, sheet, or fibrillar structure. Using the transfer matrix method and numerical calculations, we analyzed the partition function and construct phase diagrams. The model manifests phase transitions among coil, helix, sheet, and fibril through parameterizing bond coupling energy ɛh,ɛs,ɛf, structural entropies sh,ss,sf of helical, sheet, and fibrillar states, and number density ρ. The phase diagrams show the transition sequence is basically governed by ɛh, ɛs, and ɛf, while the transition temperature is determined by the competition among ɛh, ɛs, and ɛf, as well as sh, ss, sf, and ρ. Furthermore, the fibrillation is accompanied with an abrupt phase transition from coil, helix, or sheet to fibril even for short polypeptide length, resembling the feature of nucleation-growth process. The finite-size effect in specific heat at transitions for the nonfibrillation case can be described by the scaling form of lattice model. With rich phase-transition properties, our model provides a useful reference for protein aggregation experiments and modeling.

Frustration-induced protein intrinsic disorder

Spontaneous folding into a specific native structure is the most important property of protein to perform their biological functions within organisms. Spontaneous folding is understood on the basis of an energy landscape picture based on the minimum frustration principle. Therefore, frustration seemingly only leads to protein functional disorder. However, frustration has recently been suggested to have a function in allosteric regulation. Functional frustration has the possibility to be a key to our deeper understanding of protein function. To explore another functional frustration, we theoretically examined structural frustration, which is designed to induce intrinsic disorder of a protein and its function through the coupled folding and binding. We extended the Wako-Saitô-Muñoz-Eaton model to take into account a frustration effect. With the model, we analyzed the binding part of neuron-restrictive silencer factor and showed that designed structural frustration in it induces intrinsic disorder. Furthermore, we showed that the folding and the binding are cooperative in interacting with a target protein. The cooperativity enables an intrinsically disordered protein to exhibit a sharp switch-like folding response to binding chemical potential change. Through this switch-like response, the structural frustration may contribute to the regulation function of interprotein interaction of the intrinsically disordered protein.

Statistical mechanical treatments of protein amyloid formation

Protein aggregation is an important field of investigation because it is closely related to the problem of neurodegenerative diseases, to the development of biomaterials, and to the growth of cellular structures such as cyto-skeleton. Self-aggregation of protein amyloids, for example, is a complicated process involving many species and levels of structures. This complexity, however, can be dealt with using statistical mechanical tools, such as free energies, partition functions, and transfer matrices. In this article, we review general strategies for studying protein aggregation using statistical mechanical approaches and show that canonical and grand canonical ensembles can be used in such approaches. The grand canonical approach is particularly convenient since competing pathways of assembly and dis-assembly can be considered simultaneously. Another advantage of using statistical mechanics is that numerically exact solutions can be obtained for all of the thermodynamic properties of fibrils, such as the amount of fibrils formed, as a function of initial protein concentration. Furthermore, statistical mechanics models can be used to fit experimental data when they are available for comparison.

Inversion of the balance between hydrophobic and hydrogen bonding interactions in protein folding and aggregation

Identifying the forces that drive proteins to misfold and aggregate, rather than to fold into their functional states, is fundamental to our understanding of living systems and to our ability to combat protein deposition disorders such as Alzheimer's disease and the spongiform encephalopathies. We report here the finding that the balance between hydrophobic and hydrogen bonding interactions is different for proteins in the processes of folding to their native states and misfolding to the alternative amyloid structures. We find that the minima of the protein free energy landscape for folding and misfolding tend to be respectively dominated by hydrophobic and by hydrogen bonding interactions. These results characterise the nature of the interactions that determine the competition between folding and misfolding of proteins by revealing that the stability of native proteins is primarily determined by hydrophobic interactions between side-chains, while the stability of amyloid fibrils depends more on backbone intermolecular hydrogen bonding interactions.

In order to carry out their biological functions, most proteins fold into well-defined conformations known as native states. Failure to fold, or to remain folded correctly, may result in misfolding and aggregation, which are processes associated with a wide range of highly debilitating, and so far incurable, human conditions that include Alzheimer's and Parkinson's diseases and type II diabetes. In our work we investigate the nature of the fundamental interactions that are responsible for the folding and misfolding behaviour of proteins, finding that interactions between protein side-chains play a major role in stabilising native states, whilst backbone hydrogen bonding interactions are key in determining the stability of amyloid fibrils.