table of contents Largescale motions of biomolecules involve linear elastic deformations along lowfrequency normal modes, … 
Biology Articles » Biophysics » Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins » Methods
Methods

where r_{a} denotes the coordinates of atom a, and r_{a}_{, b} = r_{a} – r_{b}. 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 cutoff 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 atoms from xray 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 lowfrequency 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 (23–27). These studies suggest that largescale conformational transitions are dominated by lowfrequency 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_{n} = d·a_{n}/da_{n}, where a_{n} 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 S_{0} = S_{I} and final state S_{F}. For a structure, S_{k}, we redefine a Tirion potential, whose original conformation is S_{k}, and perform normal mode analysis, which defines a new normal mode coordinate, {q_{n}^{k}}, with vectors, {a_{n}^{k}}. The conformational difference between S_{k} and S is recalculated, d^{k}. Structure S_{k} is displaced along a chosen number of highest overlap normal modes (one or three in this study) toward the final state leading to the S_{k}_{+1} structure, which is defined as a point on the current coordinate, {l_{n}^{k}}, by l_{n}^{k} = d^{k}·a_{n}^{k}L, 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 S_{k}, 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(R^{open}, R^{closed}), where R^{open} is the rmsd to the open form from each structures on the conformational change path and R^{closed} is to the closed form. The mapping between two reaction coordinates, R^{closed} = L(R^{open}), is defined as the path L between open and closed form that minimizes the function ∫_{L}f(R^{open}, R^{closed})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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