Generalized Spring Tensor Models for Protein Fluctuation Dynamics and Conformation Changes

Abstract

In the last decade, various coarse-grained elastic network models have been developed to study the large-scale motions of proteins and protein complexes where computer simulations using detailed all-atom models are not feasible. Among these models, the Gaussian Network Model (GNM) and Anisotropic Network Model (ANM) have been widely used. Both models have strengths and limitations. GNM can predict the relative magnitudes of protein fluctuations well, but due to its isotropy assumption, it cannot be applied to predict the directions of the fluctuations. In contrast, ANM adds the ability to do the latter, but loses a significant amount of precision in the prediction of the magnitudes. In this book chapter, we present a single model, called generalized spring tensor model (STeM), that is able to predict well both the magnitudes and the directions of the fluctuations. Specifically, STeM performs equally well in B-factor predictions as GNM and has the ability to predict the directions of fluctuations as ANM. This is achieved by employing a physically more realistic potential, the Gō-like potential. The potential, which is more sophisticated than that of either GNM or ANM, though adds complexity to the derivation process of the Hessian matrix (which fortunately has been done once for all and the MATLAB code is freely available electronically at http://www.cs.iastate.edu/~gsong/STeM ), causes virtually no performance slowdown. In addition, we show that STeM can be further extended to an all-atom model and protein fluctuation dynamics computed by all-atom STeM matches closely with that by Normal Mode Analysis (NMA). Derived from a physically more realistic potential, STeM proves to be a natural solution in which advantages that used to exist in two separate models, namely GNM and ANM, are achieved in one single model. It thus lightens the burden to work with two separate models and to relate the modes of GNM with those of ANM at times. By examining the contributions of different interaction terms in the Gō potential to the fluctuation dynamics, STeM reveals, (i) a physical explanation for why the distance-dependent, inverse distance square (i.e., 1/r (2)) spring constants perform better than the uniform ones, and (ii), the importance of three-body and four-body interactions to properly modeling protein dynamics.STeM is not limited to coarse-grained protein models that use a single bead, usually the alpha carbon, to represent each residue. The core idea of STeM, deriving the Hessian matrix directly from a physically realistic potential, can be extended to all-atom models as well. We did this and discovered that all-atom STeM model represents a highly close approximation of NMA, yet without the need for energy minimization.