Saddle Point Search Research Articles

A theoretical search for folding/unfolding nuclei in three-dimensional protein structures.

When a protein folds or unfolds, it has to pass through many half-folded microstates. Only a few of them can be seen experimentally. In a two-state transition proceeding with no accumulation of metastable intermediates [Fersht, A. R. (1995) Curr. Opin. Struct. Biol. 5, 79-84], only the semifolded microstates corresponding to the transition state can be outlined; they influence the folding/unfolding kinetics. Our aim is to calculate them, provided the three-dimensional protein structure is given. The presented approach follows from the capillarity theory of protein folding and unfolding [Wolynes, P. G. (1997) Proc. Natl. Acad. Sci. USA 94, 6170-6175]. The approach is based on a search for free-energy saddle point(s) on a network of protein unfolding pathways. Under some approximations, this search is rapidly performed by dynamic programming and, despite its relative simplicity, gives a good correlation with experiment. The computed folding nuclei look like ensembles of those compact and closely packed parts of the three-dimensional native folds that contain a small number of disordered protruding loops. Their estimated free energy is consistent with the rapid (within seconds) folding and unfolding of small proteins at the point of thermodynamic equilibrium between the native fold and the coil.

Saddle Points in Random Matrices: Analysis of Knuth Search Algorithms

We present an analysis of algorithms for finding saddle points in a random matrix, presented by Donald E. Knuth as Exercise 1.3.2-12 in The Art of Computer Programming . We estimate the average computing costs of three saddle point search algorithms. Amusingly, the asymptotic results in this analysis about matrix saddle points uses the same approach that leads to the celebrated saddle point method in complex analysis.

Formation and [formula omitted] isomerization of macrocyclic β-methoxyenones

Semiempirical MO and molecular mechanics calculations were carried out in order to rationalize the formation and isomerization of isomeric β-methoxyenones obtained by methylation of the precursors of the natural macrocyclic diarylheptanoid constituents of Garuga species. The tentative torsional parameters of CCCO in the standard MM2 ∗ force field were altered in order to reproduce the relative stability of the E and Z diastereomers. The mechanism of isomerization was studied by a computational procedure previously developed by one of the authors based on the saddle point search on the potential energy surface. The calculation suggests a dipolar mechanism for isomerization.

Viewpoint 11 — approaches to the global minimum problem

Algorithms of conformational space search, namely, methods of ranking conformational isomers of flexible molecules, are discussed with emphasis on the search of low-energy regions. Perspectives on future developments in the saddle point search and the conformational interconversion paths are also mentioned.

AMI and PM3 study of a low molecular weight structural mimic of hydrogen exchange within the catalytic center of aspartic proteases

Based on recent X-ray studies, a low molecular weight model of the active center of aspartic proteases is proposed. The model is small enough to enable unattended geometry optimizations (including search for saddle-points) by molecular orbital methods. It consists of two malonic acid molecules and a water molecule; there is a carboxylic dimer at one end and the water molecule is located between the carboxylate and the carboxyl group at the other. The latter structure reproduces the geometry of the catalytic center of the native enzyme penicillopepsin with a root-mean-square deviation of 0.46 A for five O--O distances. The AMI and PM3 molecular orbital methods were used to study the H-bond exchange within the model. Both methods lead consistently to the following conclusions: Among 2 pairs of symmetry-equivalent stationary states of the catalytic center there are at least 4 symmetry-independent hydrogen-exchange pathways, and many more when including symmetry of the center. Energetics and geometry of all identified pathways are presented. In summary, they result in “juggling” all three active center protons (COON and HOH) among all five active center oxygens (COO−, COOH and H2O) providing the center with a high delocalisation with respect to the actual position of its anionic site and/or its protonation status. The relevance of the delocalisation of the acidic proton to the mechanism of enzymatic action is briefly discussed.

Analysis of the topological features of the conformational potential energy surfaces of spiro [2.5] octanes by molecular mechanics calculations

Allinger's force field MM2 is expanded to the calculation of the static and dynamic properties of 1-hetero spiro[2.5]octanes by additional parameters. This model is shown to reproduce satisfactorily trends in conformational energies of these compounds. A detailed map of the chair →- chair inversion process is constructed and the calculated rates are compared to experimental data. For the exploration of the interconversion paths, a new method including a quadratic procedure is used which makes it possible to avoid the pitfalls of the exhaustive point-by-point mapping technique and provides high accuracy in the search for saddle points. Finally, the conformational potential energy surface is analyzed in terms of the relationships which exist between the number of critical points of various orders.

Search for saddle points of energy hypersurfaces using a multi-dimensional space of ?guiding? coordinates

For the determination of saddle points (SPs) of adiabatic potential surfaces a novel method denning a “pseudo reaction path” (PRP) is presented. The PRP consists of two components, the one is being the function of some selected “guiding” coordinates and the other is depending on the remaining ones. The tangent components of the PRP are parallel and antiparallel to the normals of the tangential planes of the equipotential surfaces defined by the two groups of coordinates. PRPs starting from points in an appropriately chosen domain of the configurational space arrive at the SP.

Stability problem in electromagnetic systems and possible solutions

The letter is concerned with dispersion relations of active electromagnetic systems which are polynomial functions of either frequency or wavenumber. In this case, the search for saddle and branch points needed for stability analysis can be performed by 2-dimensional rather than 4-dimensional search.