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Current address: Applied Information Systems Division, Cybermedia Center, Osaka University, Ibaraki, Osaka, Japan

Conceived and designed the experiments: YT ASM. Performed the experiments: YT. Analyzed the data: YT ASM. Wrote the paper: YT TY ASM.

The authors have declared that no competing interests exist.

The assumption of linear response of protein molecules to thermal noise or structural perturbations, such as ligand binding or detachment, is broadly used in the studies of protein dynamics. Conformational motions in proteins are traditionally analyzed in terms of normal modes and experimental data on thermal fluctuations in such macromolecules is also usually interpreted in terms of the excitation of normal modes. We have chosen two important protein motors — myosin V and kinesin KIF1A — and performed numerical investigations of their conformational relaxation properties within the coarse-grained elastic network approximation. We have found that the linearity assumption is deficient for ligand-induced conformational motions and can even be violated for characteristic thermal fluctuations. The deficiency is particularly pronounced in KIF1A where the normal mode description fails completely in describing functional mechanochemical motions. These results indicate that important assumptions of the theory of protein dynamics may need to be reconsidered. Neither a single normal mode nor a superposition of such modes yields an approximation of strongly nonlinear dynamics.

Biological cells use a variety of molecular machines representing enzymes, ion channels or pumps, and motors. Motor proteins are nanometer-size devices generating forces and actively moving or rotating under the supply of chemical energy through ATP hydrolysis. They are crucial for many cell functions and promising for nanotechnology of the future. Although such motors represent single molecules, their operation cycles cannot be followed in detail in simulations even on the best modern supercomputers and some approximations need to be employed. It is often assumed that conformational dynamics of motor proteins is well described within a linear response approximation and corresponds to excitation of normal modes. We have checked this assumption for two motor proteins, myosin V and kinesin KIF1A. Our results show that, while both these biomolecules respond by well-defined motions to energetic excitations, these motions are essentially nonlinear. The effect is particularly pronounced in KIF1A where relaxation proceeds through a sequence of qualitatively different conformational changes, which may facilitate complex functional motions without additional control mechanisms.

Protein machines, which may represent enzymes, ion pumps or molecular motors, play a fundamental role in biological cells and understanding of their activity is a major challenge. Operation of these machines is based on slow conformational motions powered by external energy supply, often with ligands (such as ATP). In molecular motors, binding of ATP and its subsequent hydrolysis induce functional mechanochemical motions, essential for their operation. These motions, which follow after an energetic activation, are conformational relaxation processes.

Large-scale conformational changes may take place in proteins as a result of ligand binding

It is known that relaxation processes in complex dynamical systems may be strongly nonlinear and deviate much from simple exponential relaxation. As an example borrowed from a distant field, we can mention the Belousov-Zhabotinsky reaction which exhibits a great variety of spatiotemporal patterns (pacemakers, rotating spiral waves) that are however only complicated transients accompanying relaxation to the equilibrium state

While partial unfolding and refolding, associated with ligand binding, are known for some protein machines, such as the enzyme adenylate kinase

Here, we provide detailed analysis of conformational relaxation processes, associated with ligand binding and hydrolysis, in two motor proteins — myosin V

Within the coarse-grained ENM approach, a protein is modeled as a network of point-like particles, corresponding to residues, which are connected by a set of elastic links

Despite a wide-spread misunderstanding, elastic dynamics is generally nonlinear. For example, macroscopic objects, such as ribbons or membranes, can still exhibit pronounced nonlinear effects of spontaneous twisting or buckling, while fully retaining their elastic behavior and not undergoing plastic deformations

Explicitly, relaxation dynamics of considered proteins is described by equations (3) in the

The reference conformation, used to construct the elastic network, was that of the ATP(analog) bound state (Protein Data Bank (PDB) ID code: 1W7J, with MgADP-BeFx as the ATP analog

The elastic network model is constructed for the ATP-bound structure as the reference state. The red line shows the trajectory in the plane of distances

The red trajectory in

Snapshots of the conformations of myosin V along the special relaxation path are shown. The essential light chain, displayed in green, is included into the model.

The attractive path corresponds to a deep energy valley in the energy landscape of myosin V. Once this valley is entered, the conformational relaxation motion becomes effectively one-dimensional and characterized by a single mechanical coordinate. The profile of the elastic energy along the bottom of such energy valley determines the dependence of the elastic energy on the collective mechanical coordinate (see

(A) Myosin V and (B) KIF1A. Elastic energy

The dotted blue line in

The reference conformation for KIF1A is the ADP-bound state (PDB ID: 1I5S, with MgADP

The ADP-bound structure has been used to construct the elastic network. The visualization labels are indicated in panel (A), and the relaxation paths are displayed in panels (B) to (D) in the same way as in

100 relaxation trajectories, starting from random initial conditions, are shown in

The red lines in

Thus, in KIF1A the deep energy valley leading to the reference ADP-bound state gets branched at some distance from it. The path corresponding to the functional mechanochemical motion from the ATP-bound state belongs to the side branch. Only at the final relaxation stage, in the immediate vicinity of the equilibrium, the valleys merge and the functional motion begins to coincide with the typical relaxation motion in this protein.

The branching of the energy valley is already an indication of strong nonlinearity in the relaxation dynamics. We have also determined the profile of the elastic energy

(A) Conformation snapshots (seen from two different viewpoints). Switch I and switch II regions are indicated in green and orange, respectively. (B,C) Schematic representation of the relaxation motion, observed in the simulation from the ATP-bound state (red) to the ADP-bound state (blue). In switch I region, as shown in panel (B), reconfiguration of the structure (2), i.e., transformation from a loop to an

The normal mode description is broadly used in structural studies of proteins. The analysis of thermal fluctuations and the interpretation of the respective experimental structural data are traditionally performed assuming that fluctuations are linear and, hence, correspond to thermal excitation of various normal modes (see, e.g.,

Our numerical investigations of elastic conformational motions in two motor proteins (myosin V and KIF1A) have revealed however that in both of them the nonlinearities play an essential role. While slow conformational relaxation motions in myosin can still be qualitatively characterized in terms of the normal modes, the normal mode description breaks down

We want to emphasize that, when the dynamics is nonlinear, neither a single normal mode, nor a combination of many such modes can reproduce the motions. Thus, the normal mode description fails completely in this case and the problem is not that many normal modes must be taken into account. Actually, as we have shown, even for KIF1A, one normal mode would be sufficient to describe long-time relaxation within the harmonic domain — however, this domain is restricted to a tiny neighbourhood of the equilibrium state.

Thermal fluctuations have not been explicitly included into our dynamical ENM simulations. However, such fluctuations are effectively generating random conformational perturbations. In our study, relaxation processes starting from random conformational perturbations have indeed been considered.

In myosin V, one well-defined nonlinear conformational relaxation trajectory, leading to the equilibrium state, has been identified. Starting from an arbitrary initial conformation (but still without unfolding), rapid convergence to this special trajectory takes place. While the motion corresponding to the special attractive trajectory is initially nonlinear, it becomes harmonical later and a substantial part of the ordered conformational relaxation process is within the harmonic domain of the equilibrium state. Similar behavior has been previously noted by us

The situation is more complex for the monomeric kinesin KIF1A. Instead of a unique deep energy valley leading to the reference ADP-bound state, two such valleys, both leading to the equilibrium state, are present. These valleys correspond to two kinds of ordered conformational motions possible in the protein.

The first of them is relatively wide and, when thermal conformational fluctuations are excited, they would typically proceed along it. However, the conformational relaxation motion starting from the ATP-bound state follows a different path, which corresponds to the second energy valley branching from the typical fluctuation path already at very small deviations from the equilibrium state. Note that the branching takes place as the change in the distance between the molecular labels Glu233 and Ala286 is still less than an angstrom, which is much smaller than the intensity of typical thermal fluctuations for such a distance. Thus, the nonlinear effects in KIF1A are strong even for the typical thermal fluctuations.

Remarkably, such second relaxation path is also stable with respect to perturbations, i.e. structurally robust. Our numerical investigations reveal that motion along this path can be divided into two

Thus, in contrast to myosin, a single ATP binding event induces in KIF1A a complex, but ordered conformational motion characterized by two qualitatively different consequent phases. As we conjecture, this special dynamical property of KIF1A may be needed for the processive motion of this single-headed molecular motor

In myosin V, conformational motions driven by random thermal fluctuations are similar in their properties to the relaxation motion from the nucleotide-free state. This may facilitate exploitation of such fluctuation motions for the motor operation, as suggested by recent single-molecule experiments

Finally, we note that our study has been based on the elastic network approximation for proteins. More detailed descriptions, such as, e.g., G

In this study, we employ elastic network models where material points are connected by a set of elastic springs

The elastic forces obey the Hooke law and all springs have the same stiffness constant

Because slow conformational dynamics of proteins in the solvent is considered, the motions are overdamped (see

Explicitly, the relaxation dynamics is described by a set of differential equations:

To prepare random initial conditions, the following procedure has been employed. Random static forces

When relaxation from specific conformations has been considered, initial positions of all particles were allocated according to the respective PDB structures. When robustness of a relaxation path starting from a specific conformation was investigated, the initial condition was prepared by randomly shifting the positions of all particles with respect to their locations in that conformation with a certain root-mean-square displacement

To visualize conformational motions, three particles labeled as

The choice of the visualization labels is essentially arbitrary. In a simulation, motions of all residues were traced (see, e.g.,

The collective mechanical coordinate

For each point along the trajectory, the time

We provide a summary of the results on the normal mode description of conformational relaxation processes. If deviations

Equations (6) can be written in the matrix form as

The general solution of these linear differential equations is given by a superposition of

Generally, all normal modes are initially present. As time goes on, first the normal modes with the larger eigenvalues

Eigenvalues

Note that in both motor proteins a significant gap, separating the soft mode from the rest of the spectrum, is present. This means that, in the linear approximation, long-time relaxation in these proteins is effectively characterized by a single degree of freedom, representing the amplitude of the soft mode. The pattern of displacements of particles (i.e., residues) from the reference positions is determined by the eigenvector

In the plane

When relaxation is reduced to a single soft mode, the elastic potential is quadratic in terms of the mechanical coordinate, i.e.

Note that the representation of the relaxation process as a superposition (9) of normal relaxation modes holds only in the harmonic domain, i.e. when linearization (6) of full nonlinear relaxation dynamics equations (3) is valid. If dynamics is nonlinear and the linearization does not hold, relaxation dynamics cannot be viewed

As an extension, iterative normal mode analysis has been proposed

The motion of myosin V along the special relaxation path. t = 0 to 2000.

(1.25 MB MOV)

The motion of KIF1A along the special relaxation path. t = 0 to 50.

(0.76 MB MOV)

We thank M. Ueda, H. Takagi and T. Komori for helpful comments.

_{1}-ATPase.