Better understanding of protein folding process can provide physical insights on the function of proteins and makes it possible to benefit from genetic information accumulated so far. Protein folding process normally takes place in less than seconds but even seconds are beyond reach of current computational power for simulations on a system of all-atom detail. Hence, to model and explore protein folding process it is crucial to construct a proper model that can adequately describe the physical process and mechanism for the relevant time scale. We discuss the reduced off-lattice model that can express <em>&alpha;</em>-helix and <em>&beta;</em>-hairpin conformations defined solely by a given sequence in order to investigate a protein folding mechanism of conformations such as a <em>&beta;</em>-hairpin and also to investigate conformational conversions in proteins. The first two chapters introduce and review essential concepts in protein folding modelling physical interaction in proteins, various simple models, and also review computational methods, in particular, the Metropolis Monte Carlo method, its dynamic interpretation and thermodynamic Monte Carlo algorithms. Chapter 3 describes the minimalist model that represents both <em>&alpha;</em>-helix and <em>&beta;</em>-sheet conformations using simple potentials. The native conformation can be specified by the sequence without particular conformational biases to a reference state. In Chapter 4, the model is used to investigate the folding mechanism of <em>&beta;</em>-hairpins exhaustively using the dynamic Monte Carlo and a thermodynamic Monte Carlo method an effcient combination of the multicanonical Monte Carlo and the weighted histogram analysis method. We show that the major folding pathways and folding rate depend on the location of a hydrophobic. The conformational conversions between <em>&alpha;</em>-helix and <em>&beta;</em>-sheet conformations are examined in Chapter 5 and 6. First, the conformational conversion due to mutation in a non-hydrophobic system and then the conformational conversion due to mutation with a hydrophobic pair at a different position at various temperatures are examined.

Better understanding of protein folding process can provide physical insights on the function of proteins and makes it possible to benefit from genetic information accumulated so far. Protein folding process normally takes place in less than seconds but even seconds are beyond reach of current computational power for simulations on a system of all-atom detail. Hence, to model and explore protein folding process it is crucial to construct a proper model that can adequately describe the physical process and mechanism for the relevant time scale. We discuss the reduced off-lattice model that can express <em>&alpha;</em>-helix and <em>&beta;</em>-hairpin conformations defined solely by a given sequence in order to investigate a protein folding mechanism of conformations such as a <em>&beta;</em>-hairpin and also to investigate conformational conversions in proteins. The first two chapters introduce and review essential concepts in protein folding modelling physical interaction in proteins, various simple models, and also review computational methods, in particular, the Metropolis Monte Carlo method, its dynamic interpretation and thermodynamic Monte Carlo algorithms. Chapter 3 describes the minimalist model that represents both <em>&alpha;</em>-helix and <em>&beta;</em>-sheet conformations using simple potentials. The native conformation can be specified by the sequence without particular conformational biases to a reference state. In Chapter 4, the model is used to investigate the folding mechanism of <em>&beta;</em>-hairpins exhaustively using the dynamic Monte Carlo and a thermodynamic Monte Carlo method an effcient combination of the multicanonical Monte Carlo and the weighted histogram analysis method. We show that the major folding pathways and folding rate depend on the location of a hydrophobic. The conformational conversions between <em>&alpha;</em>-helix and <em>&beta;</em>-sheet conformations are examined in Chapter 5 and 6. First, the conformational conversion due to mutation in a non-hydrophobic system and then the conformational conversion due to mutation with a hydrophobic pair at a different position at various temperatures are examined.

Deposition of protein fibers with a characteristic cross-β sheet structure is the molecular marker associated with human disorders ranging from Alzheimer's disease to type II diabetes and spongiform encephalopathy. Given the large number of non-disease related proteins and peptides that have been shown to form amyloid fibrils in vitro, it has been suggested that amyloid fibril formation represents a generic protein phase transition. In the last two decades it has become clear that the same protein/peptide can assemble into distinct morphologically and structurally amyloid aggregates depending on the solution conditions. Moreover, recent studies have shown that the early stage, oligomeric amyloid assemblies are the main culprit in vivo. We have investigated the amyloid assemblies formed under denaturing conditions for Hen Egg White Lysozyme (HewL) whose human homologue is directly implicated in hereditary non-neuropathic systemic amyloidosis. Our early investigations showed that HewL can aggregate via at least two distinct assembly pathways depending on solution ionic strength at fixed pH, temperature, and protein concentration. By combining Dynamic Light Scattering (DLS), Static Light Scattering (SLS) and Atomic Force Microscopy (AFM) we showed that at low ionic strength, the pathway is characterized by the nucleation and growth of long (several micron), rigid fibrils (RF) via monomers assembly. A second, high ionic strength pathway is characterized by the rapid assembly of monomers into globular oligomers that further polymerize into curvilinear fibrils (aO/CF). At NaCl concentrations above 400 mM, aggregation resulted in precipitate formation. Next, we used Foureir Transform Infrared spectroscopy (FTIR) and an amyloid-specific dye, Thioflavin T (ThT), to show that both RF and (a)O/CF are amyloidogenic species, but they have detectable structural differences. Moreover, we have determined that each assembly pathway has unique SLS, DLS, FTIR and ThT response signatures that help determine the assembly type prior to AFM imaging of aggregates. Taking advantage of the morphological, structural and kinetic signatures for the two distinct HewL amyloid aggregates I mapped out their amyloid aggregates phase diagram spanning over two orders of magnitude in protein concentration and from 50 to 800 mM NaCl in ionic strength. This is the most complete phase diagram for amyloid aggregates of a given protein up to date. The phase diagram has three distinct regions delineated by sharp boundaries. The RF- aO/CF was called Critical Oligomer Concentration, and we commonly refer to “above the COC” as the region were aO/CF are kinetically favored.. In the region of low salt/high protein concentrations, RF were the only amyloid species to nucleate and grow. As both salt and protein concentrations increase, aO/CF become the kinetically favored species, and RF nucleate and grow after several days of incubation. At high protein and high salt concentrations, aO/CF form very fast and eventually lose solubility forming a precipitate (Ppt). Cross-seeding experiments showed that RF is the thermodynamically stable aggregate phase, while the O/CF are the metastable species. Finally, we used the phase diagram to design experiments that would allow us to reveal the RF nucleation mechanism in presence of aO/CF. RF nucleation above the COC can undergo either via internal restructuring of aO/CF (NCC) or through a random coalescence of monomers into a nucleus (NP). The experimental results obtained so far strongly indicate that RF nucleate via NP mechanism both below and above the COC.