Protein lattice

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. 

Each lattice protein is compact and of a single type (i.e., a homopolymer) and features a binding site along one face (Fig. 14.1a). Complementary lattice... 

A database of lattice protein-ligand complexes is now constructed with the following steps ... 

Liang, F., Annealing contour Monte Carlo algorithm for structure optimization in an off-lattice protein model, J. Chem. Phys. 2004,120, 6756-6763... 

There are also strong hints that protein structures fulfil relations very similar to those reported for the RNA structures. In a recent paper [25] it was shown that the first regularity, the existence of a few common and many rare structures, is indeed observed with lattice protein models as well. Extensive neutrality seems to be present with proteins, as the data on the molecular clock of evolution derived lfom sequence comparisons show [1]. 

Recent computational studies of thermophilic adaptation described in this article make use of genomic/proteomic data (32, 43, 53, 62), simulations of model lattice proteins (62), and off-lattice all-atom simulations of natural proteins (43, 53). High-throughput analysis reveals signals of novel mechanisms of protein [entropic mechanism (53)] and DNA [purine-purine base stacking (32)] thermostability and urges us to consider... 

Liwo, A., Odziej, S., Pincus, M. R., Wawak, R. J., Rackovsky, S. and Scheraga, H. A. (1997). A united-residue force field for off-lattice protein-structure simulations. I. Functional forms and parameters of long-range side-chain interaction potentials from protein crystal data. J. Comp. Chem., 18, 849-873. 

Kolinski and Skolnick [42,147] developed a series of high-coordination lattice protein models that are capable of describing protein structures at various levels of resolution. The simplest of these, the chess-knight model... 

Figure 4. Folding of a 38-mer (210) lattice protein model from the coiled state to the native state. The left side is a snapshot of the unfolded conformation the right is the lowest-energy conformation. (Adapted from Hao and Scheraga [18].)...

Figure 5. Relative entropy of a 38-mer (210) lattice protein model determined by the ESMC procedure with enhanced sampling procedures (including both the CBMC and jump-walking techniques). The inset shows the standard deviations of the computed entropy function.

For further details regarding the development and interpretation of the order parameters, Refs. 20 and 26-28 consider on-lattice protein models, while Refs. 21, 23 and 29 consider field theory approaches. References 24, 25, and 30 provide overviews of and general arguments for the spin-glass approach to the protein folding problem. 

The two-dimensional square lattice protein folding model discussed earlier provides a simple basis for probing this issue. The model has the advantage of allowing one to carry out many exact calculations to check the predictions from first-order sensitivity theory. Unlike molecular dynamics or Monte Carlo simulations, there are no statistical errors or convergence problems associated with the calculations of the properties, and their parametric derivatives, of a model polypeptide on a two-dimensional square lattice. 

NMR residual dipolar couplings (RDCs), in the form of the projection angles between the respective internuclear bond vectors, have been used by Haliloglu et as structural restraints in the ab initio structure prediction of a test set of six proteins. The restraints have been applied using a recently developed SICHO (5/de CHain Only) lattice protein model that employs a replica exchange Monte Carlo algorithm to search conformational space. The proteins studied were ACP (77 residues), rubredoxin (53 residues), NSl, a 73 residue RNA binding/dimeriz-ation domain, Nodf (35 residues), Crd (137 residues) and Bret domain with 92 residues. 

The reader should of course notice that this lattice model is very closely related to the coarse grained view of proteins that we discussed earlier in this Chapter. Physicists often call lattice models toys — but this is actually just a joke of restraint humor is the best way to avoid pompous seriousness which is incompatible with science. Lattice proteins is a very serious business. They are used to test the theories, to get the hints on how to improve the theories. For instance, we mentioned that only properly selected sequences are foldable this idea was tested with lattice models and beautifully proved right. 

Knots can of course be addressed also for toy lattice proteins discussed in section 10.10. The shortest cubic lattice knot has 24 monomers (Figure C11.7 a), but it is not space filling. The shortest space filling (open ended) lattice polymer to have knot is 36-mer. Its conformation with a trefoil knot is shown in the Figure C11.7. By the way, if the sequence of monomer species in it is properly selected (see Section 10.6), it folds in virtuo, of course) quite successfully, and not much slower that the corresponding chain without a knot — which opens even wider the question as to why real proteins are statistically less likely to have knots than random. 

In addition to the HP model, several other low-resolution protein models have been developed to provide insight into the nature of protein folding. The AB model, used in the studies mentioned by Shakhnovich et al. and studied extensively by Socci and Onuchic, is similar in spirit to the HP model. The AB model is a three-dimensional lattice protein with 27 monomers. Each monomer is either of type A or type B, and the interaction energy between two nearest-neighbor, nonbonded monomers is Ec for an AA or BB pair, and E for an AB pair, where Ei < E < 0. In this scheme, like contacts are favored over unlike contacts, and there is an overall driving energy toward... 

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