Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.     

Probing polar solvation dynamics in proteins: a molecular dynamics simulation analysis

Measurements of time-resolved Stokes shifts on picosecond to nanosecond time scales have been used to probe the polar solvation dynamics of biological systems. Since it is difficult to decompose the measurements into protein and solvent contributions, computer simulations are useful to aid in understanding the details of the molecular behavior. Here we report the analysis of simulations of the electrostatic interactions of the rest of the protein and the solvent with 11 residues of the immunoglobulin binding domain B1 of protein G. It is shown that the polar solvation dynamics are position-dependent and highly heterogeneous. The contributions due to interactions with the protein and with the solvent are determined. The solvent contributions are found to vary from negligible after a few picoseconds to dominant on a scale of hundreds of picoseconds. The origin for the latter is found to involve coupled hydration and protein conformational dynamics. The resulting microscopic picture demonstrates that a wide range of possibilities have to be considered in the interpretation of time-resolved Stokes shift measurements.     

New formulation for derivatives of torsion angles and improper torsion angles in molecular mechanics: Elimination of singularities

A new set of formulae is developed for the derivatives of torsion angle energy terms and is introduced into the program CHARMM. These formulae, which are based on derivatives of the torsion angle itself, avoid the singularities introduced by use of the derivatives of the torsion angle cosine. The potential energy can include any differentiable function of the torsion angle and there is no need for a special treatment for cases where planar conformations are not extrema. The resulting code is simpler than the original version and yields correct derivatives in all practical situations. Because the minimum of the torsion energy can be at any angle, the functionality of the existing energy routines is generalized. © 1996 by John Wiley & Sons, Inc.     

MCSS-based predictions of RNA binding sites

The diversity of RNA tertiary structures provides the basis for specific recognition by proteins or small molecules. To investigate the structural basis and the energetics which control RNA-ligand interactions, favorable RNA binding sites are identified using the MCSS method, which has been employed previously only for protein receptors. Two different RNAs for which the structures have been determined by NMR spectroscopy were examined: two structures of the TAR RNA which contains an arginine binding site, and the structure of the 16S rRNA which contains an aminoglycoside binding site (paromomycin). In accord with the MCSS methodology, the functional groups representing the entire ligand or only part of it (one residue in the case of the aminoglycosides) are first replicated and distributed with random positions and orientations around the target and then energy minimized in the force field of the target RNA. The Coulombic term and the dielectric constant of the force field are adjusted to approximate the effects of solvent-screening and counterions. Optimal force field parameters are determined to reproduce the binding mode of arginine to the TAR RNA. The more favorable binding sites for each residue of the aminoglycoside ligands are then calculated and compared with the binding sites observed experimentally. The predictability of the method is evaluated and refinements are proposed to improve its accuracy. Received: 24 April 1998 / Accepted: 4 August 1998 / Published online: 7 December 1998     

Optimal estimates of free energies from multistate nonequilibrium work data

We derive the optimal estimates of the free energies of an arbitrary number of thermodynamic states from nonequilibrium work measurements; the work data are collected from forward and reverse switching processes and obey a fluctuation theorem. The maximum likelihood formulation properly reweights all pathways contributing to a free energy difference and is directly applicable to simulations and experiments. We demonstrate dramatic gains in efficiency by combining the analysis with parallel tempering simulations for alchemical mutations of model amino acids.     

Folding of ubiquitin: a simple model describes the strange kinetics

The ubiquitin mutant UbG folding experiments of Sabelko et al., in which "strange kinetics" were observed, are interpreted in terms of a simple kinetic model. A minimal set of states consisting of a semicompact globule, two off-pathway traps, and the native state are included; the fully unfolded state is not considered because folding to the semicompact globule is fast. Both the low- and the high-temperature experiments of Sabelko et al. are fitted by a system of kinetic equations determining the transitions between these states. It is possible that cold- and heat-denaturated states of UbG are the basis of the off-pathway traps. The fits of the kinetic model to the experimental results provides an estimate of the rate constants for the various reaction channels and show how their contributions vary with temperature. Introduction of an on-pathway intermediate instead of one of the off-pathway traps does not lead to agreement with the experiments.     

Dynamic distance disorder in proteins is caused by trapping

Dynamic disorder in proteins, as demonstrated by variations in single-molecule electron transfer rates, is investigated by molecular dynamics simulations. The potential of mean force for the fluctuating donor-acceptor distance is calculated for the NAD(P)H:flavin oxidoreductase (Fre) complex with flavin adenine dinucleotide (FAD) and is found to be in agreement with that estimated from electron transfer experiments. The calculated autocorrelation function of the distance fluctuations has a simple exponential behavior at low temperatures and stretched exponential behavior at higher temperatures on femtosecond to nanosecond time scales. This indicates that the calculated dynamic disorder arises from a wide range of trapping times in potential wells on the protein energy landscape and suggests a corresponding origin for the stretched exponential behavior observed experimentally on longer time scales.     

Lagrangian formulation with dissipation of Born-Oppenheimer molecular dynamics using the density-functional tight-binding method

An important element determining the time requirements of Born-Oppenheimer molecular dynamics (BOMD) is the convergence rate of the self-consistent solution of Roothaan equations (SCF). We show here that improved convergence and dynamics stability can be achieved by use of a Lagrangian formalism of BOMD with dissipation (DXL-BOMD). In the DXL-BOMD algorithm, an auxiliary electronic variable (e.g., the electron density or Fock matrix) is propagated and a dissipative force is added in the propagation to maintain the stability of the dynamics. Implementation of the approach in the self-consistent charge density functional tight-binding method makes possible simulations that are several hundred picoseconds in lengths, in contrast to earlier DFT-based BOMD calculations, which have been limited to tens of picoseconds or less. The increase in the simulation time results in a more meaningful evaluation of the DXL-BOMD method. A comparison is made of the number of iterations (and time) required for convergence of the SCF with DXL-BOMD and a standard method (starting with a zero charge guess for all atoms at each step), which gives accurate propagation with reasonable SCF convergence criteria. From tests using NVE simulations of C(2)F(4) and 20 neutral amino acid molecules in the gas phase, it is found that DXL-BOMD can improve SCF convergence by up to a factor of two over the standard method. Corresponding results are obtained in simulations of 32 water molecules in a periodic box. Linear response theory is used to analyze the relationship between the energy drift and the correlation of geometry propagation errors.     

A lattice protein with an amyloidogenic latent state: stability and folding kinetics

We have designed a model lattice protein that has two stable folded states, the lower free energy native state and a latent state of somewhat higher energy. The two states have a sizable part of their structures in common (two "alpha-helices") and differ in the content of "alpha-helices" and "beta-strands" in the rest of their structures; i.e. for the native state, this part is alpha-helical, and for the latent state it is composed of beta-strands. Thus, the lattice protein free energy surface mimics that of amyloidogenic proteins that form well organized fibrils under appropriate conditions. A Go-like potential was used and the folding process was simulated with a Monte Carlo method. To gain insight into the equilibrium free energy surface and the folding kinetics, we have combined standard approaches (reduced free energy surfaces, contact maps, time-dependent populations of the characteristic states, and folding time distributions) with a new approach. The latter is based on a principal coordinate analysis of the entire set of contacts, which makes possible the introduction of unbiased reaction coordinates and the construction of a kinetic network for the folding process. The system is found to have four characteristic basins, namely a semicompact globule, an on-pathway intermediate (the bifurcation basin), and the native and latent states. The bifurcation basin is shallow and consists of the structure common to the native and latent states, with the rest disorganized. On the basis of the simulation results, a simple kinetic model describing the transitions between the characteristic states was developed, and the rate constants for the essential transitions were estimated. During the folding process the system dwells in the bifurcation basin for a relatively short time before it proceeds to the native or latent state. We suggest that such a bifurcation may occur generally for proteins in which native and latent states have a sizable part of their structures in common. Moreover, there is the possibility of introducing changes in the system (e.g., mutations), which guide the system toward the native or misfolded state.