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.Comment: Accepted to IJM

A Statistical Mechanical Approach to Protein Aggregation

We develop a theory of aggregation using statistical mechanical methods. An example of a complicated aggregation system with several levels of structures is peptide/protein self-assembly. The problem of protein aggregation is important for the understanding and treatment of neurodegenerative diseases and also for the development of bio-macromolecules as new materials. We write the effective Hamiltonian in terms of interaction energies between protein monomers, protein and solvent, as well as between protein filaments. The grand partition function can be expressed in terms of a Zimm-Bragg-like transfer matrix, which is calculated exactly and all thermodynamic properties can be obtained. We start with two-state and three-state descriptions of protein monomers using Potts models that can be generalized to include q-states, for which the exactly solvable feature of the model remains. We focus on n X N lattice systems, corresponding to the ordered structures observed in some real fibrils. We have obtained results on nucleation processes and phase diagrams, in which a protein property such as the sheet content of aggregates is expressed as a function of the number of proteins on the lattice and inter-protein or interfacial interaction energies. We have applied our methods to A{\beta}(1-40) and Curli fibrils and obtained results in good agreement with experiments.Comment: 13 pages, 8 figures, accepted to J. Chem. Phy

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.Comment: Updated version with correction

Quantum coherence of the molecular states and their corresponding currents in nanoscale Aharonov-Bohm interferometers

By considering a nanoscale Aharonov-Bohm (AB) interferometer containing a parrallel-coupled double dot coupled to the source and drain electrodes, we investigate the AB phase oscillations of transport current via the bonding and antibonding state channels. The results we obtained justify the experimental analysis given in [Phys. Rev. Lett. \textbf{106}, 076801 (2011)] that bonding state currents in different energy configurations are almost the same. On the other hand, we extend the analysis to the transient transport current components flowing through different channels, to explore the effect of the parity of bonding and antibonding states on the AB phase dependence of the corresponding current components in the transient regime. The relations of the AB phase dependence between the quantum states and the associated current components are analyzed in details, which provides useful information for the reconstruction of quantum states through the measurement of the transport current in such systems. With the coherent properties in the quantum dot states as well as in the transport currents, we also provide a way to manipulate the bonding and antibonding states by the AB magnetic flux.Comment: 10 pages, 7 figure