descent into the native state, and ‘‘glass-forming’’ substances, such as Ar19 and Ar55 clusters, that are characterized by sawtooth-like landscapes [40,41]. Similar to topological mapping this analysis method characterizes the surface by its basin structure, highlighting the connectivity between the basins and following the basin-to-basin transitions. However, whereas topological mapping focuses on global connectivity, the staircase analysis tries to highlight pathways toward the native state.

C. Mapping Atomistic Free Energy Landscapes

The thermodynamics of protein folding, like those of other chemical reactions, are governed not by energy but by free energy, which is a combination of energy (enthalpy) and entropy. The foregoing mapping approaches focus only on the energy component of free energy, mapping energy as a function of conformation space. Although free energy can be inferred from these ‘‘energy landscapes’’ by evaluating conformation volumes and relating them to entropy [62], this is not a very accurate estimate of the free energy. Because free energy is not a function of any single conformation but rather of the whole conformation ensemble, ‘‘free energy landscapes’’ should be charted as a function of effective reaction coordinates, unlike ‘‘energy landscapes,’’ which are a function of conformational coordinates. In lattice studies a convenient reaction coordinate was the discrete enumeration of ‘‘native contacts’’ Q. An equivalent, though continuous, reaction coordinate appropriate for an all-atom model must be defined before a detailed free energy map of a protein can be obtained. Once a reaction coordinate is defined, statistical sampling methods can be used to estimate energy and entropy along the reaction coordinate, resulting in the desired chart.

An example of such an effort is the study by Sheinerman and Brooks [70] of the free energy landscape of a small α/β protein, the 56-residue B1 segment of streptococcal protein G. As discussed above, the first step in such an endeavor is to define an appropriate reaction coordinate. To this end the native state of the protein was first characterized through nanosecond time scale molecular dynamics simulations. From these simulations a set of 54 ‘‘native nonadjacent side chain contacts,’’ similar to those used in lattice simulations, were identified. These contacts were then employed to define a continuous reaction coordinate ρ, which measures similarity to the native state based on actual distances between the components of these 54 ‘‘native contacts.’’ Once the reaction coordinate was defined, high temperature MD simulations were used to sample a large number of protein conformations, which were then divided into groups according on their ρ values. For each group of conformations with a common ρ value, the center of the group was picked and subjected to an importance sampling, using a harmonic biasing potential along the reaction coordinate ρ. The slices of the potential of mean force that were generated in this way were combined to give a free energy map for this protein as a function of two reaction coordinates, the native contacts coordinate ρ and the radius of gyration Rg. The results indicated that this α/β protein undergoes a two-step folding. Folding commences with an overall collapse that is accompanied by the formation of 35% of native contacts that are not spatially adjacent. Only later do the rest of the contacts gradually form, starting with the α-helix and only later continuing to the β-sheet. Water was present in the protein core up to a late stage of the folding process. A few similar studies have been performed for other proteins, such as the three-helix bundle protein (a 46-residue segment from fragment B of staphylococcal protein A) [71].

VI. SUMMARY

In this chapter we reviewed the main computational approaches that have been used to study the basic biophysical process of protein folding. The current theoretical framework for understanding protein folding is based on understanding the underlying energy and free energy landscapes that govern both the folding kinetics and its thermodynamics. A variety of computational models are being used to study protein folding, ranging from simple theoretical models, through simple lattice and off-lattice models, to atomic level descriptions of the protein and its environment. In recent years the focus of these studies has been gradually shifting toward the more detailed atomic level description of the process, with new computational methods and analytical techniques helping to gain additional insight into this fundamental process.

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