Elbers

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

Elber R and Karplus M 1987 A method for determining reaction paths in large molecules application to myoglobin Chem. Phys. Lett. 139 375... 

Garel T, Orland H and Thirumalai D 1996 Analytical theories of protein folding New Developments in Theoretical Studies of Protein Folding e6 R Elber (Singapore World Scientific) pp 197-268... 

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

Elber, 1996] Elber, R. Reaction path studies of biological molecules. In Recent developments in theoretical studies of proteins (Advanced series in physical chemistry, Vol. 7). R. Elber, editor. World Scientific, Singapore, 1996. 

Olender and Elber, 1996] Olender, R., and Elber, R. Calculation of classical trajectories with a very large time step Formalism and numerical examples. J. Chem. Phys. 105 (1996) 9299-9315... 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

R. Elber. Novel methods for molecular dynamics simulations. Curr. Opin. Struc. Biol., 6 232-235, 1996. 

R. Olender and R. Elber, Yet another look at the steepest descent path , J. Mol. Struct. Theochem and the proceeding of the WATOC symposium, 398-399, 63-72 (1997)... 

Czerminski R and R Elber 1990. Self-A voiding Walk Between 2 Fixed-Points as a Tool to Calculate Reaction Paths in Large Molecular Systems. International Journal of Quantum Chemistry 824 167-186. 

Elber R and M Karplus 1987. A Method for Determining Reaction Paths in Large Molecules Application to Myoglobin. Chemical Physics Letters 139 375-380. 

Simmerling C and R Elber 1995. Computer Determination of Peptide Conformations in Water Differei Roads to Stracture. Proceedings of the National Academy of Sciences USA 92 3190-3193. 

The highest probability paths will make the argument of the exponential small. That will be true for paths that follow Newtonian dynamics where mr = F(r). Olender and Elber [45] demonstrated how large values of the time step ht can be used in a way that projects out high frequency motions of the system and allows for the simulation of long-time molecular dynamics trajectories for macromolecular systems. 

Elber et al. [48] applied this method to explore the dynamics of the C-peptide in water with impressive results. More than 30 trajectories of C-peptide were generated, and the process of helix fonnation in water was examined. Remarkably, a time step of 500 ps was used, which allowed for the study of peptide folding on extended time scales. 

Eollowing the computational protocol of Czerminski and Elber [39,40] a number of restraints and constraints are added to (1) encourage the mean-square distances between adjacent structures to be approximately constant. 

I am grateful to Vio Buchete, Ron Elber, Stefan Fischer, and the editors for helpful comments. 

R Elber, ed. Recent Developments in Theoretical Studies of Proteins. Singapore World Scientific, 1997. 

Czerminski and Elber [64], who generated an almost complete map of the minima and barriers of an alanine tetrapeptide in vacuum. Using the master equation approach they were able to smdy aspects of this system s kinetics, which involve the crossing of barriers of different heights. 

R Czermmski, R Elber. Reaction path study of conformational transitions m flexible systems Application to peptides. I Chem Phys 92 5580-5601, 1990. 

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