Protein elastic network models and the ranges of cooperativity

Abstract

Elastic network models (ENMs) are entropic models that have demonstrated in many previous studies their abilities to capture overall the important internal motions, with comparisons having been made against crystallographic B-factors and NMR conformational variabilities. ENMs have become an increasingly important tool and have been widely used to comprehend protein dynamics, function, and even conformational changes. However, reliance upon an arbitrary cutoff distance to delimit the range of interactions has presented a drawback for these models, because the optimal cutoff values can differ somewhat from protein to protein and can lead to quirks such as some shuffling in the order of the normal modes when applied to structures that differ only slightly. Here, we have replaced the requirement for a cutoff distance and introduced the more physical concept of inverse power dependence for the interactions, with a set of elastic network models that are parameter-free, with the distance cutoff removed. For small fluctuations about the native forms, the power dependence is the inverse square, but for larger deformations, the power dependence may become inverse 6th or 7th power. These models maintain and enhance the simplicity and generality of the original ENMs, and at the same time yield better predictions of crystallographic B-factors (both isotropic and anisotropic) and of the directions of conformational transitions. Thus, these parameter-free ENMs can be models of choice whenever elastic network models are used.B factors ͉ conformational entropy P rotein dynamics can provide important insights into protein function. Most proteins carry out their functions through conformational changes of the structures. In X-ray structures, the information about thermal motions is provided by the Debye-Waller temperature factors or B-factors, which are proportional to the mean square fluctuations of atom positions in a crystal. Thus, accurate predictions of crystalline B-factors offer a good starting point for understanding the functional dynamics of proteins.A number of computational and statistical approaches has been proposed to predict protein B-factors from protein sequence (1-7), atomic coordinates (8-13), and electron density maps (14). The atomic coordinate-based methods such as molecular dynamics (MD) (15-18) and normal mode analysis (NMA) (19)(20)(21)(22) are computationally expensive, and thus, in the past decade, the elastic network model (ENM), with NMA, using a single parameter harmonic potential, usually coarse-grained, has been widely used for studying protein dynamics including B-factor predictions. The ENM for isotropic fluctuations is called the Gaussian network model (GNM) (23,24), where only the magnitudes of the fluctuations are considered. Its anisotropic counterpart, where the directions of the collective motions are also examined, is called the anisotropic network model (ANM) (25), and these can be compared with the experimental anisotropic temperature factors (26-29), but generally these comparisons are ...

Multiscale multiphysics and multidomain models—Flexibility and rigidity

“…4. This result is a close match to the findings of Yang et al 67 and their parameter free ENM (pfENM) model. In the pfENM, spring constants are scaled by an inverse power.…”

“…In the pfENM, spring constants are scaled by an inverse power. Yang et al 67 tested powers 1-10 and found second and third inverse power relationships were the most accurate for B-factor predictions. 67 In our study, we also test noninteger powers over the range 0.5-10.0 and come to a similar conclusion.…”

“…Yang et al 67 tested powers 1-10 and found second and third inverse power relationships were the most accurate for B-factor predictions. 67 In our study, we also test noninteger powers over the range 0.5-10.0 and come to a similar conclusion. The optimal value for υ is plotted against the optimal value for η and colored by the size of protein in Fig.…”

Predicting protein dynamics from structural ensembles

“…[6] This approach, which attempts to predict fluctuations and dynamics from a single protein structure, is not directly comparable to the LE4PD model we present here, where we model the dynamics and take the structural ensemble from experiment or by sampling an underlying atomistic model via MD simulation. Like other elastic network models [7][8][9][10][11] the coupled rotator model is capable of capturing the local variation in flexibility along the protein chain with no site-specific adjustable parameters, but because it begins from an empirical network description it requires a large amount of parameterization and specification of an overall rotational diffusion time τ 0 , a scaling factor k 0 , a cut-off distance R c , and a characteristic internal diffusion time t D . The model is explicitly limited to small displacements around a single conformational minima, and relaxation times centered upon a short characteristic internal diffusion time of ∼ 300ps.…”

Mode localization in the cooperative dynamics of protein recognition

“…[11] Other theoretical models based upon effective harmonic descriptions, which calculate sequence dependent protein flexibility, are Normal Mode Analysis (NMA), and general Elastic Network Models (ENM) including the Gaussian Network Model (GNM). [12][13][14][15][16] Differently than the LE4PD, those models were designed to study short-time vibrational fluctuations around the protein structure, which are dominated by the topology of native contacts. Because the LE4PD is a diffusive equation of motion which contains information about the extent of the intramolecular energy barriers, specific monomer friction coefficient, amino-acid specific local semiflexibility, degree of hydrophobicity, as well as hydrodynamics, it provides a realistic description of the motion of proteins in solution over a wide range of timescales, from the local vibrational fluctuations as measured by crystallographic B-factors, to the long-time dissi-pative dynamics.…”