Microscopic Theory of Protein Folding Rates.I: Fine Structure of the Free Energy Profile and Folding Routes from a Variational Approach

A microscopic theory of the free energy barriers and folding routes for minimally frustrated proteins is presented, greatly expanding on the presentation of the variational approach outlined previously [J. J. Portman, S. Takada, P. G. Wolynes, Phys. Rev. Lett. {\bf 81}, 5237 (1998)]. We choose the $\lambda$-repressor protein as an illustrative example and focus on how the polymer chain statistics influence free energy profiles and partially ordered ensembles of structures. In particular, we investigate the role of chain stiffness on the free energy profile and folding routes. We evaluate the applicability of simpler approximations in which the conformations of the protein molecule along the folding route are restricted to have residues that are either entirely folded or unfolded in contiguous stretches. We find that the folding routes obtained from only one contiguous folded region corresponds to a chain with a much greater persistence length than appropriate for natural protein chains, while the folding route obtained from two contiguous folded regions is able to capture the relatively folded regions calculated within the variational approach. The free energy profiles obtained from the contiguous sequence approximations have larger barriers than the more microscopic variational theory which is understood as a consequence of partial ordering.Comment: 16 pages, 11 figure

Non-Gaussian dynamics from a simulation of a short peptide: Loop closure rates and effective diffusion coefficients

Intrachain contact formation rates, fundamental to the dynamics of biopolymer self-organization such as protein folding, can be monitored in the laboratory through fluorescence quenching measurements. The common approximations for the intrachain contact rate given by the theory of Szabo, Schulten, and Schulten (SSS) [J. Chem. Phys. {\bf 72}, 4350 (1980)] and Wilemski--Fixman (WF) [J.\ Chem. Phys. {\bf 60},878 (1973)] are shown to be complementary variational bounds: the SSS and WF approximations are lower and upper bounds, respectively, on the mean first contact times. As reported in the literature, the SSS approximation requires an effective diffusion coefficient 10 to 100 times smaller than expected to fit experimentally measured quenching rates. An all atom molecular dynamics simulation of an eleven residue peptide sequence in explicit water is analyzed to investigate the source of this surprising parameter value. In analytical work, the polymer is typically modeled by a Gaussian chain of effective monomers. Compared to Gaussian dynamics, the simulated end-to-end distance autocorrelation has a much slower relaxation. The long time behavior of the distance autocorrelation function can be approximated by a Gaussian model in which the monomer diffusion coefficient D_0 is reduced to D_0/6. This value of the diffusion coefficient brings the mean end-to-end contact time from analytical approximations and simulation into agreement in the sense that the SSS and WF approximations bracket the simulated mean first contact time.Comment: 12 pages, 7 eps figures, to appear in J. Chem. Phy

Microscopic Theory of Protein Folding Rates.II: Local Reaction Coordinates and Chain Dynamics

The motion involved in barrier crossing for protein folding are investigated in terms of the chain dynamics of the polymer backbone, completing the microscopic description of protein folding presented in the previous paper. Local reaction coordinates are identified as collective growth modes of the unstable fluctuations about the saddle-points in the free energy surface. The description of the chain dynamics incorporates internal friction (independent of the solvent viscosity) arising from the elementary isomerizations of the backbone dihedral angles. We find that the folding rate depends linearly on the solvent friction for high viscosity, but saturates at low viscosity because of internal friction. For $\lambda$-repressor, the calculated folding rate prefactor, along with the free energy barrier from the variational theory, gives a folding rate that agrees well with the experimentally determined rate under highly stabilizing conditions, but the theory predicts too large a folding rate at the transition midpoint. This discrepancy obtained using a fairly complete quantitative theory inspires a new set of questions about chain dynamics, specifically detailed motions in individual contact formation.Comment: 18 pages, 8 figure

Plaza Central Peachtree. Atlanta-(EE.UU.)

This 70-storey hotel has been constructed to meet the requirements of the city of Atlanta which needed a building with a sufficient room capacity and adequate premises for Conventions. On a structure of reinforced concrete which serves as a base and in which the common areas are situated rises a big cylindric tower, covered with coloured glass and which contains the 1.100 rooms. 230 m above ground level, the construction is crowned with a roof top cocktail lounge and a revolving restaurant with a splendid view of the city. Among the most noteworthy characteristics of this hotel is the elegantly decorated entrance hall —atrium shaped and 7 storeys high— with a pond in the centre. Further premises worth mentioning in view of their design and dimensions are the great ball room, coffee-shops and luxurious restaurants, one of which is planned in different levels and in which the most impressive feature is a 30 m high waterfall.Este edificio de setenta plantas se construyó para responder a las necesidades hoteleras de la ciudad de Atlanta, que precisaba de una instalación con suficiente capacidad de habitaciones y preparación para albergar Convenciones. Sobre una estructura de hormigón armado, que sirve de base y en la que se sitúan las zonas comunes, se eleva una gran torre cilíndrica, recubierta de vidrio coloreado reflectante, destinada a distribuir las 1.100 habitaciones con las que cuenta el edificio. La construcción se corona, a 230 m de la cimentación, con una sala para cócteles y un restaurante giratorio desde el que se domina una espléndida vista del contorno. El edificio dispone de importantes servicios comunes, entre los que cabe destacar el hall de entrada —a modo de atrio y con una altura equivalente a siete plantas—, que está dotado de un gran estanque y variados elementos de gran efecto decorativo. Otros servicios notables por su diseño y dimensiones son la gran sala de baile, y las cafeterías, comedores y restaurantes de lujo, uno de los cuales, organizado en varios niveles, domina una cascada de agua de 30 m de altura