Estimating the Directional Flexibility of Proteins from Equilibrium Thermal Fluctuations

The directional flexibility of proteins is an equilibrium molecular property which is accessible to both experiment and computation. Single molecule force spectroscopy (SMFS) experiments report effective directional spring constants to describe the collective anisotropic response of a protein structure to mechanical pulling forces applied along selected axes. On the other hand, computational methods have thus far employed either indirect force based nonequilibrium simulations or coarse-grained elastic network models (ENM) to predict protein directional spring constants. Here, we examine the ability of equilibrium atomistic Molecular Dynamics (MD) simulations to estimate the directional flexibility and mechanical anisotropy of proteins. MD-derived effective directional spring constants are found to correlate well with SMFS spring constants (ρ² = 0.97-0.99; Adj R² = 0.92-0.99) and unfolding forces (ρ² = 0.85-0.97; Adj R² = 0.63-0.91) for five different globular proteins. Specifically, the computed spring constants reproduce the mechanical anisotropy reported by SMFS along five different directions of green fluorescence protein (GFP) and six directions of the immunoglobulin-binding B1 domain of streptococcal protein G (GB1). Further, protein dynamics as captured in MD can be translated into spring constants which can distinguish the N-C directional flexibility of ubiquitin (Ub) from two structurally homologous small ubiquitin-like modifier (SUMO1 and SUMO2) isoforms. We apply our computational framework to study the mechanical anisotropy of Ub along the seven lysine-C-term directions which are functionally relevant. We show that Ub possesses two distinct flexibility scales along these directions which roughly differ by an order of magnitude. Further, our studies reveal that the mechanical anisotropy of Ub is modified in contrasting ways by the binding of two partner proteins (UBCH5A and UEV) which attach and recognize these biomolecular tag proteins. On the basis of equilibrium MD benchmarks for flexibility along 2485 bond vectors in Ub, we propose and validate a new covariance-propagation scheme to extract spring constants from ENM normal modes. We also critically examine the ability of ENM to predict directional flexibility of proteins and suggest modifications to improve these intuitive and scalable descriptions.