Analysis of functional motions in Brownian molecular machines with an efficient block normal mode approach: myosin-II and Ca2+ -ATPase

Abstract

The structural flexibilities of two molecular machines, myosin and Ca(2+)-ATPase, have been analyzed with normal mode analysis and discussed in the context of their energy conversion functions. The normal mode analysis with physical intermolecular interactions was made possible by an improved implementation of the block normal mode (BNM) approach. The BNM results clearly illustrated that the large-scale conformational transitions implicated in the functional cycles of the two motor systems can be largely captured with a small number of low-frequency normal modes. Therefore, the results support the idea that structural flexibility is an essential part of the construction principle of molecular motors through evolution. Such a feature is expected to be more prevalent in motor proteins than in simpler systems (e.g., signal transduction proteins) because in the former, large-scale conformational transitions often have to occur before the chemical events (e.g., ATP hydrolysis in myosin and ATP binding/phosphorylation in Ca(2+)-ATPase). This highlights the importance of Brownian motions associated with the protein domains that are involved in the functional transitions; in this sense, Brownian molecular machines is an appropriate description of molecular motors, although the normal mode results do not address the origin of the ratchet effect. The results also suggest that it might be more appropriate to describe functional transitions in some molecular motors as intrinsic elastic motions modulating local structural changes in the active site, which in turn gets stabilized by the subsequent chemical events, in contrast with the conventional idea of local changes somehow getting amplified into larger-scale motions. In the case of myosin, for example, we favor the idea that Brownian motions associated with the flexible converter propagates to the Switch I/II region, where the salt-bridge formation gets stabilized by ATP hydrolysis, in contrast with the textbook notion that ATP hydrolysis drives the converter motion. Another useful aspect of the BNM results is that selected low-frequency normal modes have been identified to form a set of collective coordinates that can be used to characterize the progress of a significant fraction of large-scale conformational transitions. Therefore, the present normal mode analysis has provided a stepping-stone toward more elaborate microscopic simulations for addressing critical issues in free energy conversions in molecular machines, such as the coupling and the causal relationship between collective motions and essential local changes at the catalytic active site where ATP hydrolysis occurs.

FIGURE 1

The difference between the prehydrolysis (1FMW) and hydrolyzing (1VOM) state of the Dictyostelium discoideum myosin-II motor domain. (Left) The superposition of the motor domain in the two states based on the backbone atoms in residues 1–650; red and blue corresponds to 1FMW and green and yellow indicates the 1VOM state. As illustrated by the arrow, the majority of the conformational difference occurs in the C-terminal converter region. (Right) The structural difference between the active sites of the two states, with the same color coding; the ATP molecule is shown in van der Waals representation, and the three important motifs for ATP hydrolysis are shown in ribbon-type representation.

FIGURE 2

The correspondence between structural difference and low-frequency normal modes in myosin-II motor domain. (a) The displacements of Cα atoms between the prehydrolysis (1FMW) and hydrolyzing (1VOM) structures, with the essential active site motifs highlighted; (b) the root-mean square fluctuations (RMSF) of Cα atoms in the prehydrolysis state calculated using the 100 lowest-frequency modes at 300 K (Eq. 1); (c) the individual involvement coefficients (Eq. 3) for the first 100 modes; and (d) the cumulative involvement coefficients (Eq. 4). In c and d, the overall translational and rotational modes are shown with negative mode indices; they contribute because the structures are superimposed with residues 1–650 (see text).

FIGURE 3

The RMSF of Cα atoms in selected normal modes of the prehydrolysis state myosin-II motor domain (1FMW) that have large individual involvement coefficients. For each mode, the three numbers above each plot are the vibrational frequency (in cm⁻¹), reciprocal participation ratio (Eq. 2), and the individual involvement coefficient (Eq. 3).

FIGURE 4

Superposition of end-structures (in cartoon format) in selected normal modes of the prehydrolysis state myosin-II motor domain (1FMW) that have large individual involvement coefficients. The end-structures are generated based on the corresponding equation of motion, and they correspond to a phase angle of 90° and 270°, respectively; to clearly illustrate the structural variations, a high temperature of 1500 K was used. In each mode, red and blue indicate one structure; green and yellow correspond to the other. The arrows indicate qualitative in-plane motion of the modes, whereas “⊕” indicates out-of-plane rocking motion. Similar to Fig. 3, the three numbers in each plot are the vibrational frequency (in cm⁻¹), reciprocal participation ratio (Eq. 2), and the individual involvement coefficient (Eq. 3). Also shown are the results (in Trace format) of dynamical domain analysis using the Dyndom software with these end-structures. Different domains are colored differently, with dark green indicating regions undergoing significant bending going from one end-structure to the other; for the hinge residues, see Table 2 and Supporting Materials. The arrows indicate the hinge-bending axis, where the colors of the axis and arrowhead indicate the two dynamical domains involved in the relative motion. In b, displacements in the ATP (in vander Waals representation) binding site are shown (see text).

FIGURE 4

Superposition of end-structures (in cartoon format) in selected normal modes of the prehydrolysis state myosin-II motor domain (1FMW) that have large individual involvement coefficients. The end-structures are generated based on the corresponding equation of motion, and they correspond to a phase angle of 90° and 270°, respectively; to clearly illustrate the structural variations, a high temperature of 1500 K was used. In each mode, red and blue indicate one structure; green and yellow correspond to the other. The arrows indicate qualitative in-plane motion of the modes, whereas “⊕” indicates out-of-plane rocking motion. Similar to Fig. 3, the three numbers in each plot are the vibrational frequency (in cm⁻¹), reciprocal participation ratio (Eq. 2), and the individual involvement coefficient (Eq. 3). Also shown are the results (in Trace format) of dynamical domain analysis using the Dyndom software with these end-structures. Different domains are colored differently, with dark green indicating regions undergoing significant bending going from one end-structure to the other; for the hinge residues, see Table 2 and Supporting Materials. The arrows indicate the hinge-bending axis, where the colors of the axis and arrowhead indicate the two dynamical domains involved in the relative motion. In b, displacements in the ATP (in vander Waals representation) binding site are shown (see text).

FIGURE 5

The superposition of calcium-loaded and calcium-free Ca²⁺-ATPase structures when they are aligned based on (a) backbone atoms in the transmembrane domain (TMD) and (b) all backbone atoms. In c and d, the corresponding displacements in Cα are shown respectively. In a and b, brown-red-purple-dark blue corresponds to the E1·Ca²⁺ state, and gray-green-blue-yellow corresponds to E2. In c and d, pale blue indicates the A domain, yellow indicates the N domain, pink indicates the P domain, and the rest corresponds to the TMD.

FIGURE 6

Individual (a, b) and cumulative (c) involvement coefficients calculated when the calcium-loaded/-free forms are aligned based on all backbone atoms. In a and b, the eigenvectors associated with the calcium-loaded and calcium-free structures are used, respectively. The overall translational and rotational modes are shown with negative mode indices, although they do not contribute significantly (see text).

FIGURE 7

The RMSF of Cα atoms in selected normal modes of the calcium-loaded form (E1·Ca²⁺) of Ca²⁺-ATPase that have large individual involvement coefficients. For each mode, the three numbers above each plot are the vibrational frequency (in cm⁻¹), reciprocal participation ratio (Eq. 2), and the individual involvement coefficient (Eq. 3).

FIGURE 8

The RMSF of Cα atoms in selected normal modes of the calcium-free form (E2) of Ca²⁺-ATPase that have large individual involvement coefficients. For each mode, the three numbers above each plot are the vibrational frequency (in cm⁻¹), reciprocal participation ratio (Eq. 2), and the individual involvement coefficient (Eq. 3).

FIGURE 9

Similar to Fig. 8, but for a few modes of substantially higher frequencies that have large involvement coefficients for E2.

FIGURE 10

Superposition of end-structures in selected normal modes of the calcium-loaded form (E1·Ca²⁺) of Ca²⁺-ATPase that have large individual involvement coefficients. The end-structures are generated based on the corresponding equation of motion, and they correspond to a phase angle of 90° and 270°, respectively; to clearly illustrate the structural variations, a high temperature of 1500 K was used. The dotted circles indicate that the enclosed domains move almost like a rigid body. The brown-red-purple-dark blue corresponds to one end-structure, and gray-green-blue-yellow corresponds to the other. Also shown are the results (in Trace format) of dynamical domain analysis using the Dyndom software with these end-structures. Different domains are colored differently, with dark green indicating regions undergoing significant bending going from one end-structure to the other; for the hinge residues, see Table 3 and Supporting Materials. The arrows indicate the hinge-bending axis, where the colors of the axis and arrowhead indicate the two dynamical domains involved in the relative motion.

FIGURE 11

Similar to Fig. 10, but for the normal modes of the calcium-free form (E2) of Ca²⁺-ATPase.

SCHEME 1

The functional cycle of myosin-II-actin (only a myosin-monomer is shown).

SCHEME 2

The functional cycle of Ca²⁺-ATPase in the E1-E2 model.