In this article we present a quantitative evaluation of the convergence of the conformational coordinates of proteins, obtained by the Essential Dynamics method. Using a detailed analysis of long molecular dynamics trajectories in combination with a statistical assessment of the significance of the measured convergence, we obtained that simulations of a few hundreds of picoseconds are in general sufficient to provide a stable and statistically reliable definition of the essential and near constraints subspaces, at least within the nanoseconds time range. Proteins 1999;36:419-424.

From a specific definition of the roto-translational ͑external͒ and intramolecular ͑internal͒ coordinates, a constrained dynamics algorithm is derived for removing the roto-translational motions during molecular dynamics simulations, within the leap-frog integration scheme. In the paper the theoretical basis of this new method and its statistical mechanical consistency are reported, together with two applications.

The structural and dynamical behavior of the 41-56 b-hairpin from the protein G B1 domain~GB1! has been studied at different temperatures using molecular dynamics~MD! simulations in an aqueous environment. The purpose of these simulations is to establish the stability of this hairpin in view of its possible role as a nucleation site for protein folding. The conformation of the peptide in the crystallographic structure of the protein GB1~native conformation! was lost in all simulations. The new equilibrium conformations are stable for several nanoseconds at 300 K~Ͼ10 ns!, 350 K Ͼ6.5 ns!, and even at 450 K~up to 2.5 ns!. The new structures have very similar hairpin-like conformations with properties in agreement with available experimental nuclear Overhauser effect~NOE! data. The stability of the structure in the hydrophobic core region during the simulations is consistent with the experimental data and provides further evidence for the role played by hydrophobic interactions in hairpin structures. Essential dynamics analysis shows that the dynamics of the peptide at different temperatures spans basically the same essential subspace. The main equilibrium motions in this subspace involve large fluctuations of the residues in the turn and ends regions. Of the six interchain hydrogen bonds, the inner four remain stable during the simulations. The space spanned by the first two eigenvectors, as sampled at 450 K, includes almost all of the 47 different hairpin structures found in the database. Finally, analysis of the hydration of the 300 K average conformations shows that the hydration sites observed in the native conformation are still well hydrated in the equilibrium MD ensemble.

In this paper we present the quasi-Gaussian entropy theory in a comprehensive and consistent way, introducing a new derivation of the theory very suited for applications to molecular systems, and addressing its use in the case of multi-phase systems. A general derivation of the possible confinement of the system within a part of phase space is given, and for water it is shown that for this a hard sphere excluded volume model can be used. To obtain the temperature dependence of the pressure, a new differential equation is derived, and besides the previously introduced Gaussian and Gamma states, in this paper we also describe a new statistical state, the Inverse Gaussian state. We discuss the properties of these different statistical states and for water compare their thermodynamics with experimental data, finding that both the Gamma and Inverse Gaussian states are excellent descriptions. © 1997 American Institute of Physics. ͓S0021-9606͑97͒50305-9͔

A method is presented to mathematically extract concerted structural transitions in proteins from collections of crystal structures. The "essential dynamics" procedure is used to filter out small-amplitude fluctuations from such a set of structures; the remaining large conformational changes describe motions such as those important for the uptake/release of substrate/ligand and in catalytic reactions. The method is applied to sets of x-ray structures for a number of proteins, and the results are compared with the results from essential dynamics as applied to molecular dynamics simulations of those proteins. A significant degree of similarity is found, thereby providing a direct experimental basis for the application of such simulations to the description of large concerted motions in proteins.