Model off-lattice

In section 1.1.2 it was already mentioned that one may wish to simplify models for polymers such as polyethylene by replacing CH2 groups by united atoms. If we simplify the problem further, replacing n successive CH2 groups by an effective bond between some effective units, we may end up with a model of the type shown in Fig. 1.3. 

In the model of a freely jointed chain (Fig. 1.3[a]) each polymer hence is modeled by a succession of N rigid bonds of length i jointed together at arbitrary angles. The steps of the Monte Carlo procedure then consist in rotations of beads. For example, bead i is rotated by a randomly selected angle from its old position to its new position. If the chains are treated as completely noninteracting, the static and dynamic properties of such a 

The pearl necklace model (Fig. 1.3[b]) is a somewhat more useful model, although it is strictly athermal but, by a proper choice of the ration h/ between the radius h of the excluded volume sphere around each bead and the bond length I, one can ensure automatically that chains cannot cross each other if they respect excluded volume restrictions (no spheres are allowed to overlap apart, possibly, from subsequent ones if one chooses l l h 1). 

The most popular and efficient off-lattice models are of bead-spring type (Fig. 1.3[c]) and are not only used for MC but also for MD and Brownian dynamics (BD) simulations. It often is advantageous not to use a simple harmonic potential for the bond lengths as in eq. (1.2) but rather allow only a finite extensibility of the chains. In the MD work one works with the so-called FENE potential  

Choosing = 2 / (t the potential is purely repulsive and also the force is nonsingular at r = rc. Note that eq. (1.7) acts between all pairs of beads, including bonded ones. 

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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. 

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-... 

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 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]. 

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. 

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. 

Thirumalai, D., and Klimov, D. K. (1999). Deciphering the timescales and mechanisms of protein folding using minimal off-lattice models. Curr. Opin. Struct. Biol. 9, 197-207. 

In this review, we focus on the effect of anisotropic interactions, in particular parallel attractions, and demonstrate that the inclusion of such interactions in a model leads to a great richness in possible polymer phase behavior. From a practical point of view, the model that we describe has the advantage that it is computationally very cheap—although this advantage comes at the price of sacrificing the greater realism of an off-lattice model. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

Off-lattice models consider chains composed of interacting units in the free space. Single chains or simulation boxes containing many-chain systems can be investigated. Usually the solvent is only considered according to its quality effects in thermal systems. Therefore it is assumed to fill the remaining space act-... 

According to the results, it is determined that the asphericities can be described in terms of polynomials in Forni et al. [140] also used an off-lattice model and an MC Pivot algorithm to determine the star asphericity for ideal, theta, and EV 12-arm star chains. They also found that the EV stars chains are more spherical than the ideal and theta star chains. In these simulations the theta chains exhibit a remarkable variation of shape with arm length, so that short chains (where core effects are dominant for all chains with intramolecular interactions) have asphericities closer to those to those found with EV, while longer chains asymptotically approach the ideal chain value(see Fig. 10). 

The simulations and theoretical approaches reveal that the density profile is somewhat lost in the presence of polydispersity [193,202], and special features (including a kink in the profile) are observed for a bimodal chain distribution [195,203 ]. The profiles are flatter for the highest grafting densities, due to the influence of high-order terms. Laradji et al. [204] performed an MC simulation with an off-lattice model that uses the Hamiltonian employed in the SCF calculations (i.e., it only considers binary interactions). The parabolic profiles are... 

Mitchell and Koper [64] have involved the density-functional theory to determine the parameters necessary for the construction of an off-lattice model with no freely adjustable parameters for Br electrodeposition on Au(lOO). 

Abstract. Various ways of extracting information on the conformational structure, dynamics and correlations between them from single-molecule measurements of florescence resonance energy transfer are surveyed. The information obtained via those various ways is then analyzed in detail in the case of an off-lattice model of a two-stranded coiled-coil polypeptide that follows Langevin dynamics. The analysis includes a consideration of the cases of a freely diffusing and surface-immobilized polypeptide as well as the effect of different types of surface and denaturation conditions. 

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