Off-lattice

Off-lattice models enjoy a growing popularity. Again, a particle corresponds to a small number of atomistic repeat units... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

For structures with a high curvature (e.g., small micelles) or situations where orientational interactions become important (e.g., the gel phase of a membrane) lattice-based models might be inappropriate. Off-lattice models for amphiphiles, which are quite similar to their counterparts in polymeric systems, have been used to study the self-assembly into micelles [ ], or to explore the phase behaviour of Langmuir monolayers [ ] and bilayers. In those systems, various phases with a nematic ordering of the hydrophobic tails occur. 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

Polymer simulations can be mapped onto the Flory-Huggins lattice model. For this purpose, DPD can be considered an off-lattice version of the Flory-Huggins simulation. It uses a Flory-Huggins x (chi) parameter. The best way to obtain % is from vapor pressure data. Molecular modeling can be used to determine x, but it is less reliable. In order to run a simulation, a bead size for each bead type and a x parameter for each pair of beads must be known. 

Despite their contribution to the understanding of protein folding, the correspondence between lattice models and real proteins is still very limited. The first step toward making such models more realistic is to remove the lattice and study off-lattice minimalist models. Simple off-lattice models of proteins can have proteinlike shapes with well-defined sec-... 

An off-lattice minimalist model that has been extensively studied is the 46-mer (3-barrel model, which has a native state characterized by a four-stranded (3-barrel. The first to introduce this model were Honeycutt and Thirumalai [38], who used a three-letter code to describe the residues. In this model monomers are labeled hydrophobic (H), hydrophilic (P), or neutral (N) and the sequence that was studied is (H)9(N)3(PH)4(N)3(H)9(N)3(PH)5P. That is, two strands are hydrophobic (residues 1-9 and 24-32) and the other two strands contain alternating H and P beads (residues 12-20 and 36-46). The four strands are connected by neutral three-residue bends. Figure 3 depicts the global minimum confonnation of the 46-mer (3-barrel model. This (3-barrel model was studied by several researchers [38-41], and additional off-lattice minimalist models of a-helical [42] and (3-sheet proteins [43] were also investigated. 

Figure 3 The mimmum energy conformation of the off-lattice 46-mer P-baiTel model. Hydrophobic residues are in gray, hydrophilic residues in black, and neutral residues are white. (Adapted from Ref. 44.)...

H Nymeyer, AE Garcia, JN Onuchic. Folding funnels and fnrstration in off-lattice mimmalistic protein landscapes. Proc Natl Acad Sci USA 95 5921-5928, 1998. 

The data structure organization described above has been implemented in the BFM as well as in a very efficient off-lattice Monte Carlo algorithm, discussed in detail in the next chapter, which was modified to handle EP and used to study shear rate effects on GM [57]. 

We use the off-lattice MC model described in Sec. IIB 2 with a square-well attractive potential at the wall, Eq. (10), and try to clarify the dynamic properties of the chains in this regime as a function of chain length and the strength of wall-monomer interaction. 

A. Milchev, K. Binder. Polymer solutions confined in slit-hke pores with attractive walls An off-lattice Monte Carlo study of static properties and chain dynamics. J Computer-Aided Mater Des 2 167-181, 1995. 

A. Milchev, W. Paul, K. Binder. Off-lattice Monte Carlo simulation of dilute and concentrated polymer solutions under theta conditions. J Chem Phys 99 4786-4798, 1993. 

S. K. Kumar, M. Vacatello, D. Y. Yoon. Off-lattice Monte Carlo simulations of polymer melts confined between two plates. J Chem Phys 59 5206-5215, 1988. 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

The usual structure of off-lattice chain models is reminiscent of the Larson models the water and oil particles are represented by spheres (beads), and the amphiphiles by chains of spheres which are joined together by harmonic springs... 

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. 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

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. 

Off-Lattice Models with Coarse-Grained Side Chains... 

A step closer toward realism is taken by off-lattice models in which the backbone is specified in some detail, while side chains, if they are represented at all, are taken to be single, unified spheres [44-50]. One indication that this approach is too simplistic was given in [51], which proved that for a backbone representation in which only Ca carbons were modeled, no contact potential could stabilize the native conformation of a single protein against its nonnative ( decoy ) conformations. However, Irback and co-workers were able to fold real protein sequences, albeit short ones, using a detailed backbone representation, with coarse-grained side chains modeled as spheres [49, 52-54]. 

Although their medium-resolution model was successful for a-helical proteins, folding P-hairpin structures have been difficult. In general, many off-lattice approaches have been tested, and although definitive proof does not exist in most cases, there appears to be a growing consensus that such off-lattice models are not sufficient. 

Irback A, Peterson C, Potthast F, Sommelius O. Local interactions and protein folding A three-dimensional off-lattice approach. J Chem Phys 1997 107 273-... 

In contrast to the lattice models discussed below, off-lattice models allow the chemical species under consideration to occupy in principle any position in space, so that important information concerning the relaxation and space distribution of the constituents of the system can be obtained. We discuss next some applications of these models to electrochemical problems. 

D. Y. (1988) Off-Lattice Monte-Carlo Simulations of Polymer Melts Confined between2 Plates./. Chem. Phys., 89, 5206-5215. 

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