Lattice simulation

Fig. 8.7 Cubic and tetrahedral (diamond) lattices, which are commonly used for lattice simulations of polymers.

To conclude, although the models used in lattice simulations are very simplified, the results provide general information on possible protein folding scenarios, albeit not on the detailed behavior of specific proteins, which would require more complex models and more accurate potentials. The contribution made by these simulations is that they enable an analysis of the structures, energetics, and dynamics of folding reactions at a level of detail not accessible to experiment. 

A few groups replace the Lennard-Jones interactions by interactions of a different form, mostly ones with a much shorter interaction range [144,146]. Since most of the computation time in an off-lattice simulation is usually spent on the evaluation of interaction energies, such a measure can speed up the algorithm considerably. For example, Viduna et al. use a potential in which the interaction range can be tuned... 

The model has not been studied very intensely so far in particular, none of the features which are characteristic for amphiphilic systems have been recovered yet. However, it is close enough to the successful vector models and simple enough that it might be a promising candidate for off-lattice simulations of idealized amphiphilic systems in the future. 

Thus random interfaces on lattices can be investigated rather efficiently. On the other hand, much analytical work has concentrated on systems described by Hamiltonians of precisely type (21), and off-lattice simulations of models which mimic (21) as closely as possible are clearly of interest. In order to perform such simulations, one first needs a method to generate the surfaces 5, and second a way to discretize the Hamiltonian (21) in a suitable way. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

From that time the validity of such parameters was confirmed by theoretical variational calculations (D. Diakonov et.al., 1984) and recent lattice simulations of the QCD vacuum (see (T. De Grand et.al., 1998 M.C. Chu et.al., 1994 T. DeGrand, 2001 P. Faccioli et.al., 2003 J. Negele, 1999)). The following figure represent results of lattice calculations (J. Negele, 1999). 

The figure represents dynamical quark mass M(p) from a lattice simulation (P. Bowman et.al., 2004). Here solid curve from instantons vacuum model no fitting (D. Diakonov et.al., 1986). So, rescattering of massless quarks on an instanton vacuum leads to the dynamical quark mass M(p), which is perfectly confirmed by lattice calculations. 

Cifra, P. and Bleha, T. Conformer populations and the excluded volume effect in lattice simulations of flexible chains in solutions, Polymer, 34 (1993). 3716-3722... 

For importance sampling in the lattice simulation, one can use the leading part of the determinant, [real, positive]. This proposal provides a nontrivial check on analytic results at asymptotic density and can be used to extrapolate to intermediate density. Furthermore, it can be applied to condensed matter systems like High-Tc superconductors, which in general suffers from a sign problem. 

Shell, M.S., Debenedetti, P.G., Panagiotopoulos, A.Z. Generalization of the Wang-Landau method for off-lattice simulations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2002, 66, 056703. 

Calculations were performed with both rigid and flexible lattices. Because of increased computational demands of allowing the framework to relax, flexible lattice simulations were done with a smaller system, together with a shorter time step relative to the rigid framework simulations. The adsorbate atoms were distributed evenly throughout the channels at the start of each simulation, and a nominal temperature of 300 K was selected. The size... 

The diffusion coefficients calculated from a simulation employing a flexible framework were all between 5 and 10 times larger than those calculated from fixed lattice simulations. A comparison between flexible framework results and NMR measurements (57) illustrated the influence of the cations in the experimental sample calculated diffusion coefficients from the cation-free (flexible) framework were approximately 5 times higher than the experimental results. The increase in diffusion coefficient as a function of loading found in experimental studies was reproduced by the simulations. 

Label-free optical detection 199 Lactate dehydrogenase 465 -469, 472 A-repressor folding kinetics 551 Lattice simulations 597 Leaving group 81, 86, 88-93 Levinthal s paradox 575,576, 598, 599,600... 

Off-lattice simulation 597 180/160 tracer experiments 262-266, 488-490 Oligomer 24 Operon 35... 

Fig. 3.27 Schematic of the Monle Carlo moves in the lattice simulations of micellization and adsorption of block copolymers by Mattice and co-workers, (a) Brownian moton of a chain (b) end flip of an end bead (c) two types of kink jump (d) reptalion of a chain (Zhan et at. 1993d).

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