Lattice models exhaustive enumeration

One way of limiting the conformational space available to a protein is to confine a model polypeptide to a lattice. In doing so, unrealistic distortions are imposed on protein structure. However, lattice models offer the possibility to enumerate the entire conformational space available to a polymer chain. A detailed atomic picture is not typically employed with lattice models. However, a variety of lattices of increasing complexity facilitate more detailed chain representations, A trade-off exists between the detail of the models and the ability to evaluate conformational alternatives exhaustively. 

In determining the density of states of protein models by MC methods, a number of other approaches are also useful. For systems for which it is feasible, exhaustive enumeration of all the conformations of the system by computer provides a viable and exact solution to the problem [61-66]. A number of powerful techniques have been developed, some for exhaustive enumeration of all conformations of relatively short lattice chains [61,63] others, for enumeration of protein conformations that are confined to a compact space [62,64-66]. However, in general protein models, the number of conformations is beyond the power of computer enumeration for these systems, MC methods are the main effective techniques. 

Lattice models are remarkably useful in answering the conceptual question posed above. To infer the sequence to structure mapping, we performed an exhaustive enumeration of all self-avoiding conformations for the sequences confined to cubic lattice with N = 15 [44]. The RB model has been used in the energy function with the parameters Bq = —0.1 and B = 1. Protein-like structures are not only compact but also have low energy. We first computed the... 

Hinds and Levitt developed an even coarser-grained model, which proved to emulate small proteins reasonably well, despite its simplicity. Multiple occupancies of lattice sites were permitted in the sense that zero to three residues were placed between the adjacent sites of a tetrahedral lattice. Exhaustive enumeration of all conformations for proteins occupying up to 40 vertices were possible by this approach. Optimal alignment of residues between vertices, elimination of extended conformations by a volume constraint, and evaluation of the resulting conformations on the basis of knowledge-based contact energies, led to folds that could capture the overall path of the native structures. 

In principle, an exhaustive enumeration of allowed transitions would generate not only the correct kinetics, but also the elapsed time associated with each step. This is often feasible for lattice models, but it presupposes some knowledge of the allowed transitions and a good kinetic model to describe the rates associated with each transition. On the other hand, full enumeration of allowed transitions in off-lattice models can be prohibitive, and the n-Fold Way KMC cannot strictly be applied. Importantly, the KMC method reduces to the conventional MC method for special cases where in Equation (6.7) and the denominator of Equation (6.8) are both constants, independent of configuration. Here again, appropriate selection of moves that satisfy these characteristics in an MC simulation is essential to obtain realistic dynamics. For two recent reviews of these methods, the reader is referred to References [48] and [49]. 

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