Global and local indices for characterizing biomolecular flexibility and rigidity

Abstract

Understanding flexibility and rigidity characteristics of biomolecules is a prerequisite for understanding biomolecular structural stability and function. Computational methods have been implemented that directly characterize biomolecular flexibility and rigidity by constraint network analysis. For deriving maximal advantage from these analyses, their results need to be linked to biologically relevant characteristics of a structure. Such links are provided by global and local measures ("indices") of biomolecular flexibility and rigidity. To date, more than 14 indices are available with sometimes overlapping or only vague definitions. We present concise definitions of these indices, analyze the relation between, and the scope and limitations of them, and compare their informative value. For this, we probe the structural stability of the calcium binding protein α-lactalbumin as a showcase, both in the "ground state" and after perturbing the system by changing the network topology. In addition, we introduce three indices for the first time that extend the application domain of flexibility and rigidity analyses. The results allow us to provide guidelines for future studies suggesting which of these indices could best be used for analyzing, understanding, and quantifying structural features that are important for biomolecular stability and function. Finally, we make suggestions for proper index notations in future studies to prevent the misinterpretation and to facilitate the comparison of results obtained from flexibility and rigidity analyses.

CNA web server: rigidity theory-based thermal unfolding simulations of proteins for linking structure, (thermo-)stability, and function

“…Detailed definitions of the global and local flexibility indices are given elsewhere (17). In short, the global indices represent the macroscopic network flexibility (and rigidity) of each network state s : (i) The floppy mode density Φ refers to the number of internal independent degrees of freedom that are associated with dihedral rotations, normalized by the number of overall internal degrees of freedom associated with the total number of bodies in the network; (ii) the mean rigid cluster size S is computed with the size of the largest rigid cluster always being excluded; (iii) & (iv) the rigidity order parameter P ∞ denotes the fraction of the network belonging to the giant percolating cluster (type 1) or the actual largest rigid cluster (type 2).…”

“…Building on the ideas of Rader et al (15) on diluting non-covalent constraints in a constraint network, our group has developed the Constraint Network Analysis (CNA) approach that performs thermal unfolding simulations of proteins (16). CNA goes beyond the mere identification of rigid clusters and flexible regions in a biomolecule in that it (i) provides a refined modeling of thermal unfolding simulations that also considers the temperature dependence of hydrophobic tethers; (ii) allows performing rigidity analyses on ensembles of network topologies, either generated from structural ensembles or by using the concept of fuzzy non-covalent constraints; and (iii) computes a set of global and local indices for quantifying biomacromolecular stability (17). Furthermore, CNA robustly handles small-molecule ligands in general.…”

Kinematic Flexibility Analysis: Hydrogen Bonding Patterns Impart a Spatial Hierarchy of Protein Motion

“…Therefore, strict rigidity theory based methods rely on exceedingly local motions to estimate conformational flexibility, 20,27,28 ligand binding, 31 entropy, 33,34 or thermo-stability 38–40 compared to ENM or NMA. Nonetheless, they show convincing agreement with experimental data, if energy cutoffs to determine constraints are carefully chosen.…”

Role of Cys73 in the thermostability of farnesyl diphosphate synthase from Geobacillus stearothermophilus

“…Global indices are used to identify phase transition points T p , where the network shifts from rigid to flexible, while local indices are used to characterize the flexibility of each bond in the network. In the case of global indices, here we only consider the cluster configuration entropy H type2 , because T p which are calculated using this modified H have been found to be related to the thermostability of proteins, i.e., the melting temperatures ( T m ) of proteins (Radestock and Gohlke 2008, 2011; Pfleger et al 2013b). …”