Parameterizing elastic network models to capture the dynamics of proteins

Abstract

Coarse-grained normal mode analyses of protein dynamics rely on the idea that the geometry of a protein structure contains enough information for computing its fluctuations around its equilibrium conformation. This geometry is captured in the form of an elastic network (EN), namely a network of edges between its residues. The normal modes of a protein are then identified with the normal modes of its EN. Different approaches have been proposed to construct ENs, focusing on the choice of the edges that they are comprised of, and on their parameterizations by the force constants associated with those edges. Here we propose new tools to guide choices on these two facets of EN. We study first different geometric models for ENs. We compare cutoff-based ENs, whose edges have lengths that are smaller than a cutoff distance, with Delaunay-based ENs and find that the latter provide better representations of the geometry of protein structures. We then derive an analytical method for the parameterization of the EN such that its dynamics leads to atomic fluctuations that agree with experimental B-factors. To limit overfitting, we attach a parameter referred to as flexibility constant to each atom instead of to each edge in the EN. The parameterization is expressed as a non-linear optimization problem whose parameters describe both rigid-body and internal motions. We show that this parameterization leads to improved ENs, whose dynamics mimic MD simulations better than ENs with uniform force constants, and reduces the number of normal modes needed to reproduce functional conformational changes.

Low-Frequency Harmonic Perturbations Drive Protein Conformational Changes

“…The ANM was extensively used for the investigation of protein dynamics for three main reasons. Firstly, the computed fluctuations are found to exhibit a good agreement with the B-factors obtained from crystallographic experiments, thus providing good estimates of the protein flexibility [17][18][19][20][21][22][23]. Secondly, the ANM low-frequency motions are found to describe fairly accurately the directionality of the protein conformational change [20,[24][25][26][27][28][29][30][31][32][33][34].…”

Optimising Elastic Network Models for Protein Dynamics and Allostery: Spatial and Modal Cut-offs and Backbone Stiffness

“…It is clearly the discrete bimodality of spring constants in the BENM models that gives the two discrete regions in their dispersion curves, and the discontinuous step in frequency between them. Recent work that optimises ENM models to aaMD by permitting just such a continuous parameterisation of bond strengths supports such a conclusion [21]. We may also deduce the interesting result that all dynamic modes up to the critical mode number of the BENM discontinuity operate without significant stretching of the protein backbone.…”

“…It should be noted that at this point, the corresponding eigenfunctions are not generally of physical significance at the level of coarse-graining associated with the ENM. Furthermore, the resulting discontinuity in the frequency dispersion, not present in aaMD calculations, indicates that more quantitatively faithful ENM models need to respect a continuous range of inter-residue bond strengths, as found in [21].In no case, however, was the allosteric free energy and co-operativley found to be affected by the backbone stiffening. Notwithstanding, a backbone stiffness ratio of 100 does give a better match of the dispersion spectrum in the narrow range of modes clearly dependent in a dominant way on that property, close to the discontinuity in the frequency dispersion.…”

“…Even at this level of coarse-graining, which dispenses with the huge space of possible parameterisation of fully-atomistic models, there remain open questions in regard to the optimal parameterisation, the range of physical applicability of ENM models, and possible refinements of the ENM model itself. These have been the subjects of recent reviews and analyses of the ENM approach [19][20][21]. Potential extensions and suggested refinements include the breaking of some bonds during structural fluctuation [22], the relative importance of refining protein geometry over residue specificity [23], and the introduction of specific residue-residue bond strengths [24].…”

Sampling of Protein Conformational Space Using Hybrid Simulations: A Critical Assessment of Recent Methods

“…Several approaches have been developed aiming to increase the specificity of CG approaches while retaining the high resolution of full atomic simulations. Many ENM-based approaches have focused on the optimization of the basic parameters, spring constants and cutoff distances/functions for inter-residue interactions (Hinsen, 1998;Yang et al, 2009;Kaynak et al, 2018;Kaynak and Doruker, 2019;Koehl et al, 2021), but such studies fall short of providing atomic level information. Instead, another research line, the development of the so-called hybrid methods that combine MD and NMA (using either ENMs or full atomic models) proved to be useful in recent years.…”