Number of native contacts

Let us consider a stochastic system described by a generic variable C. This variable may stand for the position of a bead in an optical trap, the velocity field of a fluid, the current passing through a resistance, of the number of native contacts in a protein. A trajectory or path V in configurational space is described by a discrete sequence of configurations in phase space. 

The number of native contacts between i and j (within i when i = j) in X... 

Fig. 5. Free energy landscape of a lattice model protein (see Sect. 2.2), as a function of two order parameters, the number of contacts C and the number of native contacts Qo (see Sect. 2.3). Unlike the energy landscape funnel picture, the free energy shows two stable states separated by a barrier (the transition state). Extended unfolded conformers quickly collapse to the molten globule, and have to overcome a barrier to folding to the native state. The funnel picture is thus reconciled with the two-state concept of a free energy barrier. Reprinted from Dinner et ah. Trends Biochem. Sci. 25, 331, (2000) with permission from Elsevier...

The number of native contacts is an important and often used parameter in protein folding descriptions. A contact is made when the a-carbons of non-adjacent residues are within a 6 A distance. A native contact is a contact that also occurs in a reference configuration representing the native state. This reference configuration can be taken from, for instance, the Protein Database (PDB) or from simulations (e.g. the most likely structure, minimum free energy, etc). The number of native contacts can be evaluated for arbitrary configurations and measures the similarity between the configuration of interest... 

Number of native contacts at the glycosylation site Number of native contacts at the conjugation site... 

Figure 3 The linkage between the position of the conjugation sites and the effect introduced by the conjugation. The glycosylation (a) and ubiquitination (b) sites are characterized by the number of native contacts the modification site is involved in. Both glycosylation and ubiquitination will show destabilization if the modification is made at a more structured position. Experimentally, it was shown that cross-linked dimers will be destabilized compared to the isolated monomers if the cross-linking is made through a structured residue (b, triangles).

One way of determining thermodynamic equilibrium is through the calculation of protein relaxation times. By choosing an appropriate variable to monitor folding progress (examples of which may be Rg, the radius of gyration, or Q, the number of native contacts formed) and monitoring its... 

Figure 11 Free energy surface as a function of the radius of gyration and the number of native contacts for protein A from an importance sampling molecular dynamic simulation. Adapted from Guo et al. ...

Figure 12 Free energy surfaces for protein A computed with REMD simulations, (a) shows the surface as a function of RMSD and the number of native contacts below the folding transition temperature. The native state shows two basins corresponding to a hydrated nearly folded state and to the dry folded state, (b) shows the surface as a function of RMSD and the number of native contacts at the folding transition temperature. The unfolded and folded states are equally populated at this temperature. Taken with permission from Garcia and Onuchic. ...

Reaction coordinate-based methods identify TSE structures as those conformations residing at the top of the free energy barrier projected onto a given reaction coordinate q (see Figure 13). Several different reaction coordinates have been introduced to extract TSE conformations from simula-tions/ " with the most common coordinate being the number of native contacts A serious concern with reaction coordinate-based... 

The quantity (N) denotes the ensemble average number of native contacts in a macroscopic state L (with L corresponding to the unfolded U, folded N or transition i states). A simplification of Eq. [28] results when the fraction of native contacts in the unfolded state can be neglected ... 

Figure 19 Numerical methods available for the evaluation of 4). The expressions rely on specific approximations, as discussed in detail in the main body of the text. Here A is the energy difference between the mutated and the wild-type protein, is the probability of contact formation for the ij pair in state L, and (N) is the ensemble average number of native contacts in state L (unfolded, native, or transition). The remaining terms have been introduced in the previous figures.

However, characterization of these multiple minima and maxima poses a serious problem because the order parameters are not always well defined. In the case of protein folding, the radius of the protein and the number of native contacts serve as two good order parameters. Further quantification can be accomplished by considering quantities such as the number of hydrophobic pair contacts. 

In connecting these ideas with earlier phenomenological models, it is not obvious how to reconcile the dependence of the rate on the structure with a nucleation mechanism, as in Ref. 50. The statistical relationship suggests that the transition state contains a considerable amount of native structure, while a nucleus, in the classic sense of the word, is a small part of the structure. However, it could be that a limited number of native contacts (i.e., those in the nucleus) are sufficient to confine the transition state ensemble to a native-like fold. This idea is supported by a recent analysis of the folding transition state of acylphosphatase in which key residues, as determined by a < ) value analysis, play a critical role [56]. 

For the study of the folding transition, the introduction of an effective parameter that uniquely describes the macrostate of the ensemble of heteropolymer conformations is useful. A typical measure is the contact number q( X), which for a given conformation X is simply defined as the fraction of the already formed native contacts n(X) in conformation X and the total number of native contacts tot in the final fold, i.e., (X) = (X)/ tot- Then, the statistical ensemble average of this quantity q X)) at a given temperature characterizes its macrostate. Roughly, for a two-state folder, if q X)) > 0.5, native-like conformations are dominating the statistical ensemble. If less than half the total number of contacts is formed, the heteropolymer tends to reside in the pseudophase of denatured conformations. 

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