A. R. Leach. Molecular Modelling: Principles and Applications. Prentice Hall, 2001.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the actual particles of such a system are atoms or molecules. Such systems are most accurately described by quantum theory. Fortunately, for massive particles at sufficient high temperature, quantal effects are negligible small and classical mechanics can be used [84] Molecular dynamics (MD) [2, 75, 97] describing a classical particle molecular system as a function of time has been used for several decades. MD has been successfully applied to understand and explain macro phenomena from micro structures, since it is in many respects similar to real experiments. For example, transport and ....

....current angle, and m the reference angle for constraint m. Some systems also include the so called Urey Bradley term, a harmonic spring between atoms i and k; K UB (jj x ik jj r UB ) where K UB and r UB are Urey Bradley parameters. Dihedral and improper (also known as torsion) bonded forces [75] define the interaction (torsion) between four bonded atoms. They are modeled by an angular spring between planes formed by the first three atoms and the second three atoms. The potential for a dihedral or improper force between atoms i, j, k, and l is given by ; U K t (1 cos(n ) if ....

Andrew R. Leach. Molecular Modelling: Principles and Applications. Addison-Wesley Longman, Reading, Massachusetts, July 1996.

....simulating molecular motion. It samples the conformation space of a system of molecules in order to study how they relax to or fluctuate around the equilibrium state. A key property of MC simulation is that, in the limit, the conformation space is sampled according to the Boltzmann distribution [18]. MC simulation starts at some initial conformation and performs a random walk in . Let be the conformation at the current step of this random walk. To obtain the next conformation, a conformation is chosen from a small neighborhood of , with a uniform or Gaussian distribution centered ....

.... for each conformational parameter 2600 uniformly at random from its allowable range (see Section 7 for a discussion of more efficient sampling strategies) Next, for each node , the algorithm finds its nearest neighbors according to a suitable metric such as the root mean squared distance [18]. It then creates 6 M.S. Apaydn et al. an edge between and every neighboring node and attaches to it the transition probability ( 0 (3) are the Boltzmann factors at and , and and are the numbers ....

A.R. Leach. Molecular Modelling: Principles and Applications. Longman, Essex, England, 1996.

....conformation space of a system of molecules in order to compute quantities such as average energy and heat capacity, or the distribution of molecules in a system. A key property of MC simulation is that, in the limit, the conformation space is sampled according to the Boltzmann distribution [Lea96] MC simulation starts at some initial conformation and performs a random walk in . Let be the conformation at the current step of this random walk. To obtain the next conformation, a conformation sampled from a small neighborhood of , using a uniform or Gaussian distribution centered at ....

.... a value for each conformational 8 parameter 7 uniformly at random from its allowable range (see Section 7 for a discussion of more efficient sampling strategies) Next, for each node , the algorithm finds its nearest neighbors using a suitable metric such as the RMS distance [Lea96] It then creates an edge between and every neighboring and attaches to it the transition probability defined by 1 2 2 2 3 2 2 24 . 7 , 8; A7BDCE FDG 8JID (3) are the Boltzmann factors at and are the number of neighbors ....

A.R. Leach. Molecular Modelling: Principles and Applications. Longman, Essex, England, 1996.

....trial steps, also as a function of k. In (b) the largest value is obtained for k = 1. Hence, it is common practice in MCS to change few DOFs (picked at random) at each trial step [33, 35, 40, 50, 51, 63, 64] 1. 3 Computing the energy Various energy functions have been proposed for proteins [20, 33, 37, 38, 57]. For all of them, the dominant computation is the evaluation of non bonded terms, namely energy terms that depend on distances between pairs of non bonded atoms. These may be physical terms (e.g. van der Waals and electrostatic potentials [38] heuristic terms (e.g. potentials between atoms ....

....have been proposed for proteins [20, 33, 37, 38, 57] For all of them, the dominant computation is the evaluation of non bonded terms, namely energy terms that depend on distances between pairs of non bonded atoms. These may be physical terms (e.g. van der Waals and electrostatic potentials [38]) heuristic terms (e.g. potentials between atoms that should end up in proximity to each other [20] and or statistical potentials derived from a structural database (e.g. 33] To avoid the quadratic cost of computing and summing up the contributions from all pairs, cuto# distances are ....

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A. Leach. Molecular Modelling: Principles and Applications. Longman, Essex, England, 1996.

....conformation space of a system of molecules in order to compute quantities such as average energy and heat capacity, or the distribution of molecules in a system. A key property of MC simulation is that, in the limit, the conformation space is sampled according to the Boltzmann distribution [Lea96] MC simulation starts at some initial conformation and performs a random walk in C. Let q be the conformation at the current step of this random walk. To obtain the next conformation, a conformation q # is sampled from a small neighborhood of q, using a uniform or Gaussian distribution ....

.... a value for each conformational 8 parameter q 1 , q 2 , uniformly at random from its allowable range (see Section 7 for a discussion of more efficient sampling strategies) Next, for each node v i , the algorithm finds its nearest neighbors using a suitable metric such as the RMS distance [Lea96] It then creates an edge between v i and every neighboring node v j and attaches to it the transition probability P ij defined by # # # # # # # otherwise; 3) where # i and # j are the Boltzmann factors at v i and v j , and d i and d j are the number of neighbors of v i and v j ....

A.R. Leach. Molecular Modelling: Principles and Applications. Longman, Essex, England, 1996.

....as follows: First, an initial conformation of interest is selected. Then a new conformation is sampled around the original one according to a move set. The new conformation is accepted or rejected based on the energy difference between the pair of conformations, according to Metropolis criterion [Lea96] The simulation is performed for enough number of steps so as to compute relevant properties of the system under study. The stochastic nature of the molecular motion process requires one to gather many simulation paths to make such study thorough and precise. MC Simulation is limited to ....

A. Leach. Molecular modelling: principles and applications. Prentice Hall, New York, 1996.

....material is gradually reduced until the material crystallises. If the cooling is performed slowly enough, large, perfect crystals form, corresponding to the global minimum of the free energy for the system. Simulated annealing (SA) is a computational procedure that mimics this physical process [34]. Generally, the internal energy (rather than the free energy) of the initially molten system is minimized as a function of the temperature. If the cooling rate is infinitely slow, the SA will always (in principle) find the global minimum. However, since infinitely slow cooling is obviously not ....

....energy (metastable) isomers are frozen out. This is analogous to the formation of glassy structures from rapidly cooled (quenched) liquids. At each temperature, the system is equilibrated by performing a certain number (N MC ) of Monte Carlo cycles. The normal Metropolis Monte Carlo procedure [34] has been adopted. Each MC cycle involves going through the N atoms of the cluster in sequence (i.e. each MC cycle consists of N MC operations or steps) At each MC step, the chosen atom is displaced randomly and the change in potential energy accompanying this move (DV clus ) is calculated. All ....

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A. R. Leach, Molecular Modelling: Principles and Applications (Longman, Harlow, 1996).

....Hamiltonians to detect instability. The test problem is a set of 125 water molecules harmonically restrained to a 10 A radius sphere. The water is based on the TIP3P model [25] without cutoffs and with flexibility incorporated by adding bond stretching and angle bending harmonic terms (cf. Ref. [12]) 4 10.256 10.258 10.260 10.262 10.264 10.266 10.268 10.270 0 20000 40000 60000 80000 100000 time (fs) 8th order 6th order 4th order Figure 2: Closer look at higher order truncations for decalanine. 440 430 420 410 400 390 380 0 200 400 600 800 1000 time (fs) energy Figure 3: Energy ....

A. R. Leach. Molecular Modelling: Principles and Applications. Addison-Wesley Longman, Reading, Mass., July 1996.

....list of references is given in Example 2 below. Models of dynamic nature have also been used in various occasions, such as updating and learning in graphical models or the probabilistic expert systems (Spiegelhalter and Lauritzen 1990, Kong, Liu and Wong 1994) simulating protein structures (Leach 1996; Vasquez and Scherago 1985) genetics (Irwing, Cox and Kong 1994) and combinatorial optimizations (Wong and Liang 1997) An example of expert system updating can be found in Berzuini et al. 1997) In this article, we study Monte Carlo computation methods for real time analysis of dynamic ....

....a MCMC scheme to produce a more efficient transition proposal chain. The Hastings rejection procedure described in Section 8.1 can be used in combination. This type of Monte Carlo methods (sometimes called configuration biased Monte Carlo ) have been tested effective for simulating biopolymers (Leach, 1996). See Irwing et al. 1994) Kong et al. 1994) Wong and Liang (1997) and others for more examples. We hope that the results reported here can stimulate more interest and effort from other researchers on this type of problems. 8 APPENDIX 8.1 The Invariant Distributions of the Hastings ....

Leach, A.R. (1996), Molecular Modelling: Principles and Applications. Addison Wesley Longman: Singapore.

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A. R. Leach. Molecular Modelling: Principles and Applications. Prentice Hall, 2001.

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A. R. Leach. Molecular Modelling: Principles and Applications. Addison-Wesley Longman, Reading, Mass., July 1996.

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Leach, A.R.: Molecular Modelling: Principles and Applications. Addison-Wesley, Reading, Massachusetts (1996)

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A. R. Leach. Molecular Modelling: Principles and Applications. Addison-Wesley Longman, Reading, Massachusetts, July 1996. 14

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A. R. Leach. Molecular Modelling: Principles and Applications. Addison-Wesley Longman, Reading, Massachusetts, 1996.

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A. R. Leach, Molecular Modelling: Principles and Applications, AddisonWesley, Reading, Massachusetts, 1996.

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A. R. Leach. Molecular Modelling: Principles and Applications. Addison-Wesley Longman, Reading, Massachusetts, July 1996.

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A.R. Leach. Molecular Modelling: Principles and Applications. Longman, Essex, England, 1996.

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A. Leach. Molecular modelling: principles and applications. Prentice Hall, New York, 1996.

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