Statistical mechanical model for helix-sheet-coil transitions in homopolypeptides

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.

Simple moment-closure model for the self-assembly of breakable amyloid filaments

In this work, we derive a simple mathematical model from mass-action equations for amyloid fiber formation that takes into account the primary nucleation, elongation, and length-dependent fragmentation. The derivation is based on the principle of minimum free energy under certain constraints and is mathematically related to the partial equilibrium approximation. Direct numerical comparisons confirm the usefulness of our simple model. We further explore its basic kinetic and equilibrium properties, and show that the current model is a straightforward generalization of that with constant fragmentation rates.

Effect of hydrogen bond interaction on protein phase transition

We derive the grand partition function of protein chain by restricting dihedral angles to exist only in five distinct states and assume that the dominant noncovalent potential is the hydrogen bond interaction. We investigate the phase transition of protein secondary structures and the order of the transition through analyzing its heat capacity. Our theory demonstrates the presence of α-β-coil structural phase transition in the protein polyalanine.

The toxicity of amyloid β oligomers

In this review, we elucidate the mechanisms of Aβ oligomer toxicity which may contribute to Alzheimer's disease (AD). In particular, we discuss on the interaction of Aβ oligomers with the membrane through the process of adsorption and insertion. Such interaction gives rises to phase transitions in the sub-structures of the Aβ peptide from α-helical to β-sheet structure. By means of a coarse-grained model, we exhibit the tendency of β-sheet structures to aggregate, thus providing further insights to the process of membrane induced aggregation. We show that the aggregated oligomer causes membrane invagination, which is a precursor to the formation of pore structures and ion channels. Other pathological progressions to AD due to Aβ oligomers are also covered, such as their interaction with the membrane receptors, and their direct versus indirect effects on oxidative stress and intraneuronal accumulation. We further illustrate that the molecule curcumin is a potential Aβ toxicity inhibitor as a β-sheet breaker by having a high propensity to interact with certain Aβ residues without binding to them. The comprehensive understanding gained from these current researches on the various toxicity mechanisms show promises in the provision of better therapeutics and treatment strategies in the near future.

Normal modes and phase transition of the protein chain based on the Hamiltonian formalism

We use the torsional angles of the protein chain as generalized coordinates in the canonical formalism, derive canonical equations of motion, and investigate the coordinate dependence of the kinetic energy expressed in terms of the canonical momenta. We use the formalism to compute the normal-frequency distributions of the α helix and the β sheet, under the assumption that they are stabilized purely through hydrogen bonding. In addition, we obtain the free-energy relations of the α helix, the β sheet, and the random coil of a 15-residue polyalanine. Interestingly, our results predict a phase transition from an α helix to a β sheet at a critical temperature.

Exactly solvable model for helix-coil-sheet transitions in protein systems

In view of the important role helix-sheet transitions play in protein aggregation, we introduce a simple model to study secondary structural transitions of helix-coil-sheet systems using a Potts model starting with an effective Hamiltonian. This energy function depends on four parameters that approximately describe entropic and enthalpic contributions to the stability of a polypeptide in helical and sheet conformations. The sheet structures involve long-range interactions between residues which are far in sequence, but are in contact in real space. Such contacts are included in the Hamiltonian. Using standard statistical mechanical techniques, the partition function is solved exactly using transfer matrices. Based on this model, we study thermodynamic properties of polypeptides, including phase transitions between helix, sheet, and coil structures.