Global minimization

Therefore, modeling a protein molecule amounts to deciding on the atoms considered to be essential and to specifying the contribution of the various interactions to the potential. Since the work to find the global minimizer increases drastically (and possibly exponentially) with the dimension of x, it is customary to use for larger proteins a reduced description that treats only very few atoms in each amino acid as essential. 

Unfortunately, the approach of determining empirical potentials from equilibrium data is intrinsically limited, even if we assume complete knowledge of all equilibrium geometries and their energies. It is obvious that statistical potentials cannot define an energy scale, since multiplication of a potential by a positive, constant factor does not alter its global minimizers. But for the purpose of tertiary structure prediction by global optimization, this does not not matter. 

C.D. Maranas, IP. Androulakis and C.A. Floudas, A deterministic global optimization approach for the protein folding problem, pp. 133-150 in Global minimization of nonconvex energy functions molecular conformation and protein folding (P. M. Pardalos et al., eds.), Amer. Math. Soc., Providence, RI, 1996. 

G. Ramachandran and T. Schlick. Beyond optimization Simulating the dynamics of supercoiled DNA by a macroscopic model. In P. M. Pardalos, D. Shal-loway, and G. Xue, editors. Global Minimization of Nonconvex Energy Functions Molecular Conformation and Protein Folding, volume 23 of DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, pages 215-231, Providence, Rhode Island, 1996. American Mathematical Society. 

Method Path minimization Path maximization Hyperplane minimization Hypersphere minimization Global minimization Points moved... 

B-spline coefficients that globally minimize the performance index is determined. 

Klepeis, J. L. I. P. Andronlakis M. G. Ierapetritou, et al. Predicting Solvated Peptide Configuration via Global Minimization of Energetic Atom-to-Atom Interactions. Comput Chem Eng 22 765-788 (1998). 

One can show (Ericksen, 1975) that the homogenous state w(x) = Wi corresponding to a deeper minimum of g(w Oo) is the global minimizer of the functional (2.1) another homogenous configuration w(x) = Wo (the metastable state), is only a weak local minimizer... 

In the trust region or restricted step method we determine in each iteration the global minimizer of the RSO model Eq. (3.10). In the global region a step is taken to the boundary of the trust region... 

When solving Eq. (5.8) the level shift parameter p must be chosen such that s(p) is the global minimizer on the boundary. From the discussion in Sec. Ill we know that p must be smaller than the lowest Hessian eigenvalue. Also, p must be negative since otherwise the step becomes longer than the Newton step. The exact value of p may be found by bisection or interpolation. 

Figure 4.35 Calculated structure for an HT 3-hexylthiophene tetramer obtained by using molecular mechanics modeling, where the globally minimized tetramers have been docked in an idealized manner to X-ray structural parameters, (a) Intermolecular p-stacking between the thiophene rings as inferred from a (90 °C) X-ray pattern of the film, (b) Lamellar stacking as inferred from X-ray scans of intensity versus 20 data. Reprinted from R.D. McCullough, S. Tristram-Nagle, S.P. Williams, R.D. Lowe and M. Jayaraman, ]. Am. Chem. Soc., 115,4910 (1993). Copyright (1993) American Chemical Society...

Figure 7 Structure of local and global minimization algorithms.

A.3 Search for the Correct Binding Mode (267-283). Just as there are many different approaches to the global minimization problem, most, if not all, have been applied to the docking problem. These include molecular dynamics, Monte Carlo sampling, systematic... 

H.L. Gordon, R.L. Somorjai, Applicability of the method of smoothed functionals as a global minimizer for model proteins, J. Phys. Chem. 96 (1992), 7116. 

C. Tsoo, C.L. Brooks, Cluster structure determination using Gaussian density distribution global minimization methods, J. Chem. Phys. 101 (1994), 6405... 

D. Shalloway, Application of the renormalization group to deterministic global minimization of molecular conformation energy functions. Journal of Global Optimization 2 (1992), 281. 

Kostrowicki, J. and Scheraga, H. A. (1996). Some approaches to the multiple-minima problem in protein folding, in Global Minimization of Nonconvex Energy Functions Molecular Conformation and Protein Folding.,. 

McDonald and Floudas (1994) and Michelsen (1994) also present methods to find the phase conditions for a given mixture. Both methods discover the number of phases and the compositions of these phases. McDonald and Floudas globally minimize the Gibbs free energy of the mixture using a computer pack-... 

Ultimately the aim of molecular modeling is the design (before synthesis) of novel materials. In molecular crystal chemistiy, this requires the ab initio prediction of solid-state structures. Although much progress has been made and the potential of such an ability is enormous, progress in this field has been hindered by problems with global minimization and force-field accuracy. 

An effective optimization of the linear parameters in can now be performed by globally minimizing a cost function 9 where f is either... 

The variational problem that the SCF method epitomizes can then be reduced to a global minimization of 9 i/( ), where [16]... 

The existence of multiple solutions (stationary paths of the action) suggests why in the discrete representation we need to use a global minimizer... 

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