Coarse-Grained Protein Dynamics Studies Using Elastic Network Models

Abstract

Elastic networks have been used as simple models of proteins to study their slow structural dynamics. They consist of point-like particles connected by linear Hookean springs and hence are convenient for linear normal mode analysis around a given reference structure. Furthermore, dynamic simulations using these models can provide new insights. As the computational cost associated with these models is considerably lower compared to that of all-atom models, they are also convenient for comparative studies between multiple protein structures. In this review, we introduce examples of coarse-grained molecular dynamics studies using elastic network models and their derivatives, focusing on the nonlinear phenomena, and discuss their applicability to large-scale macromolecular assemblies.

Keywords: allostery; coarse-grained model; elastic network; molecular dynamics; molecular machine; nonlinearity; normal mode analysis; protein.

Figure A1

An illustrative example of the nonlinearity. Two springs are initially aligned straight without strains; each spring obeys Hooke’s law (equilibrium length: $l_0)$. If we pull the midpoint perpendicularly to the initial spring direction, the strength F of the resultant force $\mathbf{F}$ depends nonlinearly on the displacement x. Adapted from [62].

Figure 1

Schematic diagram of the elastic network model (ENM). Each spring $(i,j)$ obeys Hooke’s law; i.e., the strength of its elastic force ${\mathbf{F}}_{ij}$ is proportional to its deformation from the equilibrium length, and the force direction coincides with the spring direction. The resultant force ${\mathbf{F}}_i$ on particle i is the sum of forces ${\mathbf{F}}_{ij}$ exerted by all the springs connected to the particle. In dynamic simulations in the overdamped limit (Section 2.4), the velocity $\dot{{\mathbf{R}}_i}$ of particle i is directly calculated from ${\mathbf{F}}_i$. Adapted from [62].

Figure 2

Relaxation paths of ENMs. Two molecular motors were compared: (A,C) Myosin V and (B,D) KIF1A; (A,B) Structure of the motors; positions of the markers 1,2,3 are shown. The ATP-bound structure of myosin V and ADP-bound structure of KIF1A were used as the reference structures to construct the ENMs; (C,D) Trajectories of conformational changes. Horizontal and vertical axes show the distances between markers 1 and 2, and markers 1 and 3, respectively. Gray lines display trajectories starting from 100 different initial deformations, prepared by applying static force toward random direction to each particle. In most cases, these trajectories quickly converge to an energetic valley (represented by a bundle of trajectories in (C,D)), and slowly return to the reference structure (stable state). This behavior is common for these two motors, and the direction of the valley around the reference structure agrees with the slowest normal mode. Red lines represent transitions between two states of the motors, which start from the nucleotide free state of myosin V (i.e., corresponding to the conformational transition upon ATP binding), and from the ATP-bound state of KIF1A (i.e., corresponding to the transition from the ATP-bound state to the ADP-bound state); label 0 shows the initial condition and the relaxation proceeded 1, 2, …. In myosin V, the trajectory was similar to those from random deformations, and the normal mode approximation holds well except for initial changes. In KIF1A, in contrast, the relaxation progressed with multiple steps, and the trajectory converged to the slowest normal mode direction only at the final stage of relaxation (<1 Å of distance change). Reproduced from [61] with modification.

Figure 3

Ligand-induced conformational changes simulated by ENMs. (A) structure of G-actin and (B) its ENM representation; (C) In the nucleotide binding pocket (NBP), two additional particles (ADP and P${}_i$) were introduced to mimic ligand binding. The natural lengths of springs connected to P${}_i$ (solid lines in the schematic) was set shorter than the distances in the reference structure, so that they introduce attractive forces (arrows), mimicking the shrinkage of the NBP upon ATP binding; (D) The motion introduced by the ligand, together with additional interactions (breakable links shown in purple lines), suggested ATP-induced transition to the closed conformation (left to center), which may explain the acceleration of actin polymerization by ATP. A metastable closed state after ATP hydrolysis and P${}_i$ release (right) was also observed. Reproduced from [71] with modification.

Figure 4

Entire operation cycles from ENM. (A) Table showing the main experimental efforts and observed/proposed mechanisms for the operation of the hepatitis C virus (HCV) helicase molecular motor. The references for this table are [70,77,78,79,80,81,82,83]; (B–D) Summary of results of structurally resolved dynamical simulation for HCV helicase based on ENM modeling of this motor; (B) Structures obtained from our simulations are compared with crystal structures from [81]; (C) Ratcheting inchworm translocation observed in the simulations is shown as snapshots from a single cycle. Employing the alternating hand-on hand-off mechanism of grip control on the nucleic-acid strand, the motor domains can translate their nucleotide-related internal opening and closing motion into their transport by 1 base-pair (bp) per consumed nucleotide; In (D), coupling of inchworm translocation to mechanical separation of duplex strands is shown as snapshots [70,75].