Response of Rotation–Translation Blocked Proteins Using Langevin Dynamics on a Locally Harmonic Landscape

Abstract

Langevin dynamics is used to compute the time evolution of the nonequilibrium motion of the atomic coordinates of a protein in response to ligand dissociation. The protein potential energy surface (PES) is approximated by a harmonic basin about the minimum of the unliganded state. Upon ligand dissociation, the protein undergoes relaxation from the bound to the unbound state. A coarse graining scheme based on rotation translation blocks (RTB) is applied to the relaxation of the two domain iron transport protein, ferric binding protein. This scheme provides a natural and efficient way to freeze out the small amplitude, high frequency motions within each rigid fragment, thereby allowing for the number of dynamical degrees of freedom to be reduced. The results obtained from all flexible atom (constraint free) dynamics are compared to those obtained using RTB-Langevin dynamics. To assess the impact of the assumed rigid fragment clustering on the temporal relaxation dynamics of the protein molecule, three distinct rigid block decompositions were generated and their responses compared. Each of the decompositions was a variant of the one-block-per-residue grouping, with their force and friction matrices being derived from their fully flexible counterpart. Monitoring the time evolution of the distance separating a selected pair of amino acids, the response curves of the blocked decompositions were similar in shape to each other and to the control system in which all atomic degrees of freedom are fully independent. The similar shape of the blocked responses showed that the variations in grouping had only a minor impact on the kinematics. Compared with the all atom responses, however, the blocked responses were faster as a result of the instantaneous transmission of force throughout each rigid block. This occurred because rigid blocking does not permit any intrablock deformation that could store or divert energy. It was found, however, that this accelerated response could be successfully corrected by scaling each eigenvalue in the appropriate propagation matrix by the least-squares fitted slope of the blocked vs nonblocked eigenvalue spectra. The RTB responses for each test system were dominated by small eigenvalue overdamped Langevin modes. The large eigenvalue members of each response dissipated within the first 5 ps, after which the long time response was dominated by a modest set of low energy, overdamped normal modes, that were characterized by highly cooperative, functionally relevant displacements. The response assuming that the system is in the overdamped limit was compared to the full phase space Langevin dynamics results. The responses after the first 5 ps were nearly identical, confirming that the inertial components were significant only in the initial stages of the relaxation. Since the propagator matrix in the overdamped formulation is real-symmetric and does not require the inertial component in the propagator, the computation time and memory footprint was reduced by 1 order of magnitude.

Ligand-Induced Protein Responses and Mechanical Signal Propagation Described by Linear Response Theories

“…Recently, there has been renewed interest in using linear response theory (LRT) (2,(6)(7)(8)(9) to describe such ligand-triggered motions. In the framework of LRT, observed conformational changes result from a delicate interplay between protein intrinsic dynamics and the external perturbations that excite the most relevant global modes (2).…”

“…The Coalson group also derived a time-dependent version of the theory to characterize fine-and coarse-grained presentations of the ligand binding process on timescales of hundreds of picoseconds or longer (6,7). The authors also showed that the exaggerated curvature might be overcome by quasiharmonic approximation of the roughness of the energy landscape caused by anharmonicity (7).…”

Overdamped Dynamics of Folded Protein Domains within a Locally Harmonic Basin Using Coarse Graining Based on a Partition of Compact Flexible Clusters

“…Even though this suggests that these structures are rigid, it does not mean that they will necessarily remain rigid under an arbitrary applied force. In a previous paper, 10 we attempted to model rigid units and to incorporate these units into the dynamics of the system. We found that even though the motions were qualitatively similar to those of the all-atom control (i.e., the same system with no rigidity constraints imposed), the potential energy surface (PES) was steeper than the all-atom control so the relaxations of the Langevin modes were faster.…”

“…In our previous study of rigid block dynamics, 10 which did not allow for internal distortions within a block, we suggested two ways to rectify the problems associated with purely rigid blocking. One way is to add back a subset of the suppressed internal (vibrational) degrees of freedom in each block to the six rotation-translation block (RTB) degrees of freedom (dof's).…”

“…20 (3) The application of rigid blocking can artificially steepen the PES. 10 To remedy these issues, we have developed a tractable coarse-graining method which decomposes a protein into intrinsically flexible clusters (fc's) that will accurately describe the slowest overdamped collective motions. The proposed protocol produces a down-sampled version of the all-atom Hessian matrix such that the overdamped Langevin dynamics of the system (to be denoted here as Brownian dynamics 21 ) is found in nontrivial test cases to correlate well with the dynamics and kinematics of the all-atom control (where the motion of all atoms is time-evolved subject to the Hessian of the corresponding all-atom force field).…”