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Recently, Angel Ortiz and coworkers conducted a study on several superfamilies and showed that most structural variation of the structurally conserved protein core lies in the subspace spanned by only a few collective vibrational normal modes of one representative protein of the superfamily under study: evolutionary deformations follow closely dynamical deformations [1].
The slowest most collective vibrational normal modes are important for protein function. This suggests that natural selection might be behind the coincidence between evolutionary and dynamical deformation spaces. However, if we think of a mutation as a perturbation of the wild-type protein, according to the Linear Response Theory we expect such a perturbation to be relaxed by inducing in the system a response similar to the dynamical response of the unperturbed system to a natural fluctuation of the perturbed site. Thus, both mutation and selection are possible candidates to explain the observations of Ortiz and coworkers. The collective vibrational dynamics of proteins can be modeled by Elastic Network Models (ENM), in which a protein is modeled as a network of alpha carbons with springs connecting sites that are in contact in the 3D structure [2]. The model works because the collective motions, which are the slowest and of largest amplitude, depend on many sites, averaging out details of the local interactions. Thus, these motions depend basically on protein topology, rather than the details of the potential. As far as we know, there is no model of protein structure divergence that can be used to study the relationship between evolutionary and dynamical deformations. To study this relationship, and the relative roles of mutation and selection on structural divergence, we developed the Linearly Forced Elastic Network Model (LF-ENM). Just as ENM models can be derived by a quadratic Taylor expansion of the protein¢s potential around its equilibrium state [2], the LF-ENM can be obtained by a quadratic expansion of the potential energy potential with respect to coordinates and parameters. The LF-ENM Hamiltonian has a term identical to that of the ENM and a perturbative term, linear in the coordinates, which models mutation. Selection, in turn, is naturally modeled by the probability that the mutant will adopt the wild-type structure. This probability depends on a single parameter that represents selection pressure. The LF-ENM accounts for the observation that evolutionary deformations occur within the subspace of collective normal modes, thus providing a mechanism to explain such observation. For a test-case of a globin superfamily, we show that the main reason is mutational deformation: any given mutation produces a structural stress that is relaxed following the network of oscillators, just as dynamical fluctuations are relaxed. In contrast, selection has a limited effect of 10%-20% of the overall structural deformation. In conclusion, we show that, in contrast to naive selectionist interpretations, structural variation along collective coordinates is mainly mutational: that is just the way proteins are expected to vary under random substitutions of their amino acids. References [1] A. Leo-Macias, P. Lopez-Romero, D. Lupyan, D. Zerbino, and A. R. Ortiz, An analysis of core deformations in protein-superfamilies, Biophys. J. 88(2005)1291. [2] I. Bahar and A. J. Rader, Coarse-grained normal mode analysis in structural biology, Curr. Opin. Struct. Biol. 15(2005)586. |
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