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Large-scale motions of biomolecules involve linear elastic deformations along low-frequency normal modes, …


Biology Articles » Biophysics » Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins » Methods

Methods
- Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins

Preparation of Structure. Two conformations, open (Protein Data Bank ID code 4AKE [PDB] ) (15) and closed (Protein Data Bank ID code 1AKE [PDB] ) (16), of adenylate kinase are used. In addition to atoms included in the crystallographic coordinates, polar hydrogen atoms are added based on the AMBER united atom model (17). The hydrogen atoms are relaxed with a relatively short minimization by using the AMBER6 program (18).

The Protein Elastic Model: Tirion Potential. In this study, we use a coarse-grained model, Tirion potential, which is defined as follows. An interaction between two atoms, a and b, is the Hookean pairwise potential:

where ra denotes the coordinates of atom a, and ra, b = rarb. The zero superscript indicates the coordinate at the original conformation. The strength of the potential C is assumed to be the same for all interacting pairs. The total potential is

where R is the cut-off parameter, so that the interactions are limited to pairs separated by R. We use R of 8 Å, which has already been shown to be appropriate for normal mode studies of protein (23).

Originally, the Tirion potential was proposed for normal mode analysis. We go beyond normal mode analysis since the Tirion potential is not entirely harmonic as a function of the atomic Cartesian coordinates. The spring constant of the Tirion potential, C, is optimized so that the average of B factors of C{alpha} atoms from x-ray crystallography and the normal mode analysis coincide (19, 20). The normal modes were found by using the RTB approach (21).

This potential is crude but adequate for low-frequency motions. In many respects it is the elastic counterpart of the Go model landscapes used in protein folding simulations (22). Others have used normal modes to model the path of conformational change (2327). These studies suggest that large-scale conformational transitions are dominated by low-frequency modes. Nevertheless activation barriers and therefore rates are not be obtained from a pure normal mode approach, as pointed out by Mouawad and Perahia (28) in their study of hemoglobin allostery.

Overlap Between Normal Mode and Conformational Change. The overlap, a measure of the relevance of a given normal mode to the conformational change, is defined as cos{theta}n = d·an/|d||an|, where an is the direction of the normal mode n and d is the conformational change (21).

Iteration Procedure for Generating Nonlinear Conformational Change Path. From our initial attempt for describing energy profile with normal modes, we found that a single normal mode does not accurately represent the conformational changes (see Results). Thus we needed to define the conformational change pathway iteratively by the following procedure. We define the initial state S0 = SI and final state SF. For a structure, Sk, we redefine a Tirion potential, whose original conformation is Sk, and perform normal mode analysis, which defines a new normal mode coordinate, {qnk}, with vectors, {ank}. The conformational difference between Sk and S is recalculated, dk. Structure Sk is displaced along a chosen number of highest overlap normal modes (one or three in this study) toward the final state leading to the Sk+1 structure, which is defined as a point on the current coordinate, {lnk}, by lnk = dk·ankL, where L is a parameter that determines how far the structure is to be deformed; In our calculation, we use L = 0.1. In addition, displacement is limited by an rms deviation (rmsd) of 0.5 Å to the current structure Sk, i.e., L has to satisfy the following inequality:

where N is the number of atoms. This iterative procedure is continued 100 steps.

Mapping Two Energy Profiles from Open and Closed Structures. When we need to superimpose two energy profiles from the open and closed forms (see Results), since the two profiles are actually defined with different coordinates, i.e., rmsd to open and closed forms, respectively, a mapping between the two coordinates is used. We calculate the pairwise rmsds between structures along the conformational change path from open or closed form, f(Ropen, Rclosed), where Ropen is the rmsd to the open form from each structures on the conformational change path and Rclosed is to the closed form. The mapping between two reaction coordinates, Rclosed = L(Ropen), is defined as the path L between open and closed form that minimizes the functionLf(Ropen, Rclosed)dl/l + kl, where l is the length of the path L, and k is a parameter, which enforces that the length of path is short enough, k = 1 was used for our calculation.


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