Sampling‐based exploration of folded state of a protein under kinematic and geometric constraints

Abstract

Flexibility is critical for a folded protein to bind to other molecules (ligands) and achieve its functions. The conformational selection theory suggests that a folded protein deforms continuously and its ligand selects the most favorable conformations to bind to. Therefore, one of the best options to study protein-ligand binding is to sample conformations broadly distributed over the protein-folded state. This article presents a new sampler, called kino-geometric sampler (KGS). This sampler encodes dominant energy terms implicitly by simple kinematic and geometric constraints. Two key technical contributions of KGS are (1) a robotics-inspired Jacobian-based method to simultaneously deform a large number of interdependent kinematic cycles without any significant break-up of the closure constraints, and (2) a diffusive strategy to generate conformation distributions that diffuse quickly throughout the protein folded state. Experiments on four very different test proteins demonstrate that KGS can efficiently compute distributions containing conformations close to target (e.g., functional) conformations. These targets are not given to KGS, hence are not used to bias the sampling process. In particular, for a lysine-binding protein, KGS was able to sample conformations in both the intermediate and functional states without the ligand, while previous work using molecular dynamics simulation had required the ligand to be taken into account in the potential function. Overall, KGS demonstrates that kino-geometric constraints characterize the folded subset of a protein conformation space and that this subset is small enough to be approximated by a relatively small distribution of conformations.

Geometric analysis characterizes molecular rigidity in generic and non-generic protein configurations

“…velocities that are consistent with the velocity constraint equations (9), for any u̇ ∈ ℝ n – r . Perturbing a molecular conformation with a vector selected from a sufficiently small neighborhood of the origin in the nullspace of J , i.e., {Δ q N ∈ ℝ n | |Δ q N | ≪ 1, J Δ q N = 0 } maintains hydrogen-bond distances in linear approximation and can be used to efficiently probe conformational space [41, 9, 29]. …”

“…Hydrogen bonds and other non-covalent interactions are encoded as holonomic constraints, resulting in nested, interdependent cycles that require coordinated changes of the degrees of freedom, effectively reducing the dimensionality of configuration space. The remaining motions are known as floppy modes and yield collective motion of the degrees of freedom in a lower-dimensional constraint manifold Q [5, 34, 36, 41]. The constraints reduce conformational flexibility or can even completely rigidify larger substructures of biomolecules by merging rigid bodies through rotationally locked degrees of freedom or hydrogen bonds.…”

Modeling Structures and Motions of Loops in Protein Molecules

“…The methods described in [32,70,136,140], also referred to as sampling-based methods, present promising first steps in this direction. Moreover, the employment of IK techniques to address kinematic constraints in sampling-based methods is allowing new applications on modeling the equilibrium flexibility not only of specific protein fragments, such as loops, but of entire protein chains [71,154,159]. IK techniques are efficient and allow sampling-based methods to spend computational resources on sampling a large number of closed conformations.…”

Exploring the energy landscapes of flexible molecular loops using higher‐dimensional continuation

“…The atlas is initialized with just one chart and new charts are added as the tree branches escape from the balls associated with the charts. Note that recently, a related technique also using the tangent space to generate samples has proved useful in studying the local flexibility of proteins with several loops and hundreds of degrees of freedom [63,64], but in this approach, only one tangent space is used and the projection from the tangent space to the variety is not considered.…”