Ensemble virtual

The VIGRAL approach represents the reflecting surface of a defect as an ensemble of virtual point sources. At every measurement point of the B-Scan, the detecting transducer responds... 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

The growing computahonal power available to researchers proves an invaluable tool to investigate the dynamic profile of molecules. Molecular dynamics (MD) and Monte Carlo (MC) simulahons have thus become pivotal techniques to explore the dynamic dimension of physicochemical properhes [1]. Furthermore, the powerful computational methods based in parhcular on MIFs [7-10] allow some physicochemical properhes to be computed for each conformer (e.g. virtual log P), suggesting that to the conformahonal space there must correspond a property space covering the ensemble of all possible conformer-dependent property values. 

In this section we introduce the model system on which virtually all approximate exchange-correlation functionals are based. At the center of this model is the idea of a hypothetical uniform electron gas. This is a system in which electrons move on a positive background charge distribution such that the total ensemble is electrically neutral. The number of elec-... 

The objects of statistical mechanics however, are not single systems, but ensembles of many, say v, physically similar systems, i.e. systems with Hamiltonians H(qi,pi,t). The distribution of such a virtual ensemble over the phase space will be described by means of the density function q(qi,Pi, t), called the phase density. It is a real non-negative quantity and will always be normalized to unity, i.e. 

Motion of the virtual ensemble in phase space may be likened to fluid flow by introducing a 2n-dimcnsional vector of phase velocity v with components qi,pi(i = 1,2,..., n). Since the systems of the virtual ensemble can neither be created nor destroyed in the flow, the equation of continuity... 

Fig. 15.25 Partial display of virtual screen of conformational ensemble of " 75 000 structure library based on multiple templates. A display of fitness scores to a three-point pharmacophore model provides initial signs of similarity of a library design to that model. Subsequent virtual screening against a more stringent four-point pharmacophore model further highlights potentially useful library designs.

Bradley and coworkers used the 3D pharmacophore ensemble model to filter a virtual combinatorial library of 3924 N-substituted glycine peptoids (30) containing three known a, actives down to a set of 639 products. Using a cut-down technique, a 160 compound combinatorial library was designed in which the number of compounds that passed the ensemble model filter was maximized. This library contained two of the three known actives present in the original 3924 compound virtual library. This represents a substantial enrichment [(2 actives/160 products) X 100 = 1.25% vs (3 actives/3924 products) x 100 = 0.076%]. 

SAP produces a set of virtual excitations from the fully occupied to the unoccupied Kohn-Sham orbitals thus producing a fictitious statistical ensemble. A thermodynamic interpretation of SAP is presented in [50] (see [51,52] as well), where two main observations are given. First, the redistribution of single particle states in the smoothing procedure leads to... 

Molecular dynamics (MD) should be performed for binding pockets defined mostly by side chains of flexible protein residues to generate an ensemble of binding sites. Such an ensemble can be used for subsequent docking or virtual screening in a parallel fashion. 

In protein structure prediction, potentials are used to assign an energy-like quantity to a conformation of a protein molecule. If this quantity enables us to distinguish the native state of a protein, the potential is regarded as a reasonable model for a protein-solvent system. The rationale behind this relies on two assumptions (a) a solved protein in its native state can be described by an ensemble of closely related conformations, and (b) in this state the system is in the global minimum of free energy. Virtually all techniques designed for structure prediction are based on these principles [3,4]. 

If all possible combinations were equally probably, we would observe stochastic behavior like primary nucleation, so that crystal growth kinetics would be virtually unpredictable. However, a few molecular paths for crystal growth are highly preferred over others, these paths combine in an ensemble to provide the macroscopic observations of crystal growth described in the next section. 

With no further approximations we can now consider an ensemble of molecules, which represents the components of the mixture under consideration as swimming around individually in the infinitely extended virtual conductor. Because the electric field of the molecules is perfectly screened off by the conductor, there is no interaction of the molecules. We know the total energy of this ensemble by just summing the total energies of the individual molecules in this virtual, but distinguished state, which we consider as the north pole of our globe (Fig. 1.1). 

The ESPS method draws on and synthesizes a number of ideas in the extensive free-energy literature, including the importance of representations and space transformations between them [63, 68, 69], the utility of expanded ensembles in turning virtual transitions into real ones [23], and the general power of multicanonical methods to seek out macrostates with any desired property [27],... 

For fuzzy virtual screening purposes, highly active molecules with different scaffolds are combined into an MTree model. By combining the information of remotely related actives into a single model, efficient database searches with molecule ensembles are possible. 

Bolstad ES, Anderson AC (2009) In pursuit of virtual lead optimization pruning ensembles of receptor structures for increased efficiency and accuracy during docking. Proteins 75(l) 62-74... 

Chemical species — is an ensemble of chemically identical molecular entities that can explore the same set of molecular energy levels on the time scale of the experiment. The term is applied equally to a set of chemically identical atomic or molecular structural units in a solid array. For example, two conformational isomers may be interconverted sufficiently slowly to be detectable by separate NMR spectra and hence to be considered to be separate chemical species on a time scale governed by the radiofrequency of the spectrometer used. On the other hand, in a slow chemical reaction the same mixture of conformers may behave as a single chemical species, i.e., there is virtually complete equilibrium population of the total set of molecular energy levels belonging to the two conformers. 

Pickett et al. [68] describe a program, DIVSEL, for selecting reactants while taking account of the pharmacophoric diversity that exists in the final products. They describe a 2-component library where the reactants in one pool are fixed and a subset of reactants is to be selected from the second pool. The virtual library is enumerated and a pharmacophore key is generated for each of the product molecules. Reactants are selected from the second pool using a dissimilarity-based compound selection process that represents a candidate reactant by a pharmacophore key that covers an ensemble of products. 

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