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¶ ¶
* Department of Physics, Beckman Institute for Advanced Science and Technology, Center for Biophysics and Computational Biology, University of Illinois at Urbana-Champaign, Urbana, Illinois; Graduate School of Frontier Biosciences, Osaka University, Osaka, Japan; and ¶ Dynamic Nano Machine Project, International Cooperative Research Project, Japan Science and Technology Agency, Tokyo, Japan
Correspondence: Address reprint requests to K. Schulten, E-mail: kschulte{at}ks.uiuc.edu.
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ABSTRACT |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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). The flagellar filament behaves differently depending on the direction of the torque applied by the basal body. Under counter-clockwise rotation (as viewed from the exterior of the cell), several flagella form a single helical bundle (composed of left-handed helices) that acts to propel the cell along a straight line; this is known as the running-mode of motion. Under clockwise rotation, the individual filaments dissociate from the bundle and form separate right handed helices, causing the cell to rotate in place; this is known as the tumbling mode (2). The filament itself forms one of several supercoils, depending on the rotation of the motor; these supercoils have been extensively classified, and generally fall into the normal form for the running mode of motion, and one of the semicoil, curly I, and curly II forms in the tumbling mode (3,4). Varying the duration of running and tumbling modes, bacteria can move up or down a stimulus gradient by a biased random walk (2).
In this study, we concentrate on the flagellar filament of Salmonella typhimurium, which has been well characterized in experiments and for which a combined x-ray crystal/cryo-EM structure is available (5–7). This structure is composed of multiple copies (monomers) of a single protein, flagellin. The filament is formed by the helical arrangement of these monomers along its axis, with 11 monomers per turn. For the purposes of understanding filament behavior, it is generally much more useful to consider the filament to be composed of 11 intertwined protofilaments; each protofilament is a stack of monomers that forms a long-pitch coil along the length of the filament. Note that these two representations are identical; each protofilament is simply the subset of monomers occupying identical positions in their respective turns of the overall filament helix.
Early theoretical work on the flagellum suggested that the observed supercoiling forms of flagella arise from the 11 protofilaments individually assuming a so-called short or long conformation. These are defined as such since the interface of subsequent monomers in a protofilament (and adjacent monomers in other protofilaments) can take on two conformations that lead to slightly different spacings between these monomers along the length of the protofilament, without significant internal changes to each monomer. This rearrangement of protofilaments can also give rise to supercoiling of the filament, in which case the location of the short protofilaments becomes the inner face of the supercoil and the location of the longer protofilaments becomes the outer face (8,9). X-ray fiber diffraction studies on straight filaments confirmed the existence of two protofilament forms in the cases where all protofilaments in a filament took on either the long or short conformation; the structure with long-type interfaces formed left-handed helices (and thus this protofilament conformation was denoted L) and the shorter type formed right-handed helices (with a protofilament conformation denoted R) (10). Theoretical models of filaments containing varying numbers of protofilaments with each of these interface types resulted in supercoils very close to those observed experimentally; the normal supercoil, for instance, corresponds to a filament with two R and nine L protofilaments (denoted 2R/9L) (10). This correspondence gave rise to the so-called bistable-protofilament model for flagellar supercoiling, in which different supercoiling forms are thought to arise due to the presence of different ratios of L- and R-type protofilaments. In this model, the interface type is thought to remain uniform through any given protofilament (10,11). Recent molecular dynamics (MD) simulations (12) have suggested an atomic level mechanism for the L-R distinction, based on the presence of a set of "switch" interactions formed in L, but not R, protofilaments.
Despite extensive theoretical and experimental studies, the mechanisms responsible for inducing polymorphic transitions in protofilament arrangement between the running and tumbling modes remain unclear. The chiral nature of the flagellin monomers, which show a scooplike curvature (see Fig. 1) on the section interacting with solvent, offers a tantalizing clue on how the filament may interact differently with solvent under different rotational directions. One expects that the friction experienced by the filament differs substantially between the opposite rotation directions, which could then lead to a rearrangement of protofilaments and, hence, to different supercoiling patterns. However, this effect has not been quantified, because the resolution in experiments with functioning flagella is not high enough, the structure is too large to do atomistic simulations easily, and it seems to be too complex for a simple mathematical model description. It is even unclear if the solvent does play an important role for the polymorphic transition. Indeed, recent studies (10,12,13) have focused only on the role of protein-protein interactions, and it is possible that only these internal interactions control filament supercoiling. Thus, there is a need to model the behavior of the flagellar filament on a scale where supercoiling may be observed, but the interactions between the single protein monomers are also resolved. To this end, we have developed a new coarse-grained (CG) MD approach for large biomolecular assemblies, and applied it to the bacterial flagellum.
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,15), viral capsids and their elements (16,17), the elasticity of actin filaments (18), lipid bilayers (19,20–22), and the aggregation of lipoprotein particles (23,24). The flagellar filament seems to be an ideal system to apply a CG technique to (12). Its structure is known; it consists of multiple copies of a single protein, the flagellin monomer (making it relatively easy to parameterize); and reliable experimental data are available for comparison. Addressing many aspects of flagellar filament dynamics using traditional all-atom MD would require simulations of tens of millions of atoms for timescales on the order of micro- to milliseconds; both the time and size scales are several orders of magnitude beyond what is currently feasible.
Our CG model (presented below) uses 15 point masses to represent each monomer (flagellin), and the assembly is then simulated with a simple implicit solvent model that accounts for protein-solvent interaction through a frictional term added to the MD description. The number of considered degrees of freedom is 
4000 times smaller than for the all-atom model with explicit solvent, and the integration time step can be raised substantially leading to faster simulation. In fact, we were able to simulate 0.5-µm segments of flagellar filament for tens of microseconds to study the effects of varying torque and solvent conditions on flagellar dynamics. Although quantitative agreement with experimental results is still relatively poor due to the very high speed at which we needed to rotate the model filament, the flagellar motion we observe is qualitatively consistent with the experimentally observed behavior of the filament and the bistable protofilament model. In particular, we find that solvent viscosity should play a key role in inducing transitions between different coiling and supercoiling states of the filament.
In this article, we define coiling as the twist observed within the individual protofilaments of the structure about the center of the filament, and supercoiling as twist of the entire filament.
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METHODS |
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The developed CG model is designed primarily for simulations of large-scale motions of macromolecular assemblies for which shapes of constituent molecules do not change significantly; for example, refolding of proteins is not accounted for in our CG model. A single protein in the CG representation should behave as an elastic object of definite form, the spatial distribution of CG beads reproducing the shape of the protein. Proteins appear in various shapes, and in many cases, this shape is not compact, but instead consists of several domains connected by relatively thin and disordered links or involves long tails. When reproducing the shape by a number of CG beads, it is important to reconstruct the bulk of the protein's compact regions as well as links and tails. Accordingly, a method which assigns each CG bead to represent the same number of atoms would be inefficient, since either the tails would be misrepresented or the number of CG beads representing compact regions would be too large. Instead, one should use a representation that adapts the beads to the topology of proteins, i.e., three-dimensional bulky bodies, planar sheets, or linear links and tails. In addition, one may want to use a multiresolution method that permits coarser and finer representations in different parts of the macromolecule (protein). Such methods were developed in the field of neural computation (25), and we chose to use one of them, the so-called topology-generating network (26,27).
The algorithm, in its simplest uniform resolution mode, works as follows. For an all-atom model of the protein (a crystal structure) consisting of Na atoms (total mass M), with coordinates ri and masses mi (i = 1, 2, ..., Na), the mass distribution is used as a target probability distribution, 
, for the neural network. The number NCG (NCG << Na) of neurons representing p(r) is specified from the start, the neurons being subsequently identified with the beads of the coarse-grained representation. One chooses NCG according to the desired level of coarse-graining (we used a ratio Na/NCG = 500). Initial positions Rk (k = 1, 2, ..., NCG) of the beads are assigned randomly. Then Nstep adaptation steps are carried out in which representative positions for the beads are determined. During each step, three operations are performed. First, a protein atom is chosen randomly (according to the probability p(r)); its position r serves as an input for the next operations. Second, for each CG bead k, one determines the number nk of CG beads j with |r – Rj| < |r – Rk|. Third, positions of the beads are updated (k = 1, 2, ..., NCG) according to the rule
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and 
at each step u (u = 0, 2, ... , Nstep) are represented by the functional form 
, with 0 = 0.2 NCG, 
, 0 = 0.3, and 
. Usually, Nstep = 200 NCG is enough to obtain a well-converged distribution of CG beads. A typical run of this algorithm for flagellin (500 residues, 15 CG beads) takes less than a second of computer time. The CG and all-atom models of the flagellum filament and of a single unit protein are compared in Fig. 1. Once a single protein is coarse-grained, the distribution of CG beads can be mapped on the other orientations of the same protein; in this way, all unit proteins building up the flagellum filament are coarse-grained identically. Both the coarse-graining and mapping programs were implemented in C++ and can be applied to any other molecular system without changes.
We choose the interaction potentials between the CG beads in the form of the CHARMM force field (28), i.e., bonded interactions are described by harmonic bond and angle potentials (we did not use dihedral potentials), and the nonbonded potentials include only 6–12 Lennard-Jones (LJ) and Coulomb terms. After the positions of CG beads are defined, we find for each atom the CG bead closest to it; the atom then belongs to the domain of this bead, the so-called Voronoi cell. The total mass and charge of each domain are assigned to the corresponding CG bead. Two CG beads are bonded if the distance between them is <25 Å (with this choice, all CG beads representing the flagellin are in one network, but regions that are not connected in the all-atom model remain disconnected in the CG model). Separate monomers interact only through nonbonded forces.
To determine the parameters for bonded interactions, we ran a 7-ns-long, all-atom MD simulation of a single flagellin monomer. Assuming that each bond and angle is an independent harmonic oscillator, and following the distances between the centers of mass of the domains corresponding to certain CG beads, we extracted the equilibrium bond lengths (or angles) and spring constants from the mean distances (angles) and respective root mean-square deviations (RMSD). This procedure can be illustrated by the simple example of a one-dimensional harmonic oscillator, with a particle moving along the x coordinate in the potential V(x) = f(x – x0)2/2. With the system in equilibrium at temperature T, the average position 
x
is equal to x0, and the RMSD is given by kBT/f (kB is the Boltzmann constant). Using an MD simulation, one can compute x and the RMSD, thus obtaining x0 and f. These formulas change slightly for bonds and angles, but the principle that the two parameters of a harmonic oscillator can be derived from the mean and RMSD of the coordinates, holds. This procedure allows one to systematically obtain suitable values for the interaction parameters.
To parameterize the nonbonded potentials, we chose several segments of the flagellin monomer, 500 atoms each (roughly representing a single CG bead), and carried out 10 all-atom simulations of various pairs of these segments, for 10 ns each. The potential of mean force (PMF) between the two segments was obtained for every pair using the Boltzmann inversion method (17,24,29). The PMFs (see Fig. 2) were found to be similar to an LJ potential in shape, and for each pair the well-depth was 4 kcal/mol. Therefore, the LJ well-depth for each CG bead is set to 4 kcal/mol. The all-atom simulations mentioned here were performed in vacuum, with the total charge of the protein neutralized, because the Coulomb interactions between large charged domains of the protein should be included in the Coulomb interactions between the charged beads, and not in the bonded or LJ interactions. The LJ radius for each CG bead is calculated as the radius of gyration of the respective all-atom domain, increased by 2 Å (approximately an average LJ radius of an atom in the CHARMM force field). This choice was verified by comparison with the PMFs extracted from the all-atom simulations: for each pair of protein segments in those simulations, the position of the PMF minimum was close to the sum of the gyration radii of each segment, increased by 4 Å.
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The MD program NAMD 2.5 (30) was used to simulate the CG system, with an implicit solvent model implemented. Three basic features of water are chosen to be represented by the implicit solvent, namely, viscosity, fluctuations due to Brownian motion, and dielectric permittivity. The relative dielectric constant is set to 80 everywhere. The effect of the liquid environment in our simulation is accounted for by the presence of frictional and fluctuating forces. This is done by solving for each bead the Langevin equation
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is a damping coefficient controlling frictional forces (for bead velocity v), and 
(t) is a temperature-dependent random force related to the frictional forces through the fluctuation-dissipation theorem. The scaling of the friction term with velocity, –v, is key to our modeling, particularly given that, as we apply a rotation to the filament, beads along the outer edges of the filament will experience the strongest force opposing motion. In effect, this means that the frictional effects of the solvent will apply a torque to the monomers in our system, the frictional torque pointing in the opposite direction to the torque that induces filament rotation. This frictional torque, arising mainly through the beads at the periphery since they move fastest on average, induces a rearrangement of the flagellum monomers as described in Results.
The motions of CG beads obey the Langevin equation, so aside from the potential energy-based forces between the beads, viscous damping and random forces from the solvent are introduced, with a single parameter defining the strength of both (see (30)); this parameter, the damping (or friction) constant , is directly related to the diffusion constant D, D = kBT/(m), where kB is the Boltzmann constant, T is the temperature, and m is the mass of the diffusing object. To select a proper -value, five all-atom MD simulations of single segments (500 atoms each) of a monomer in a periodic water box were performed. The diffusion rate for every segment was then computed, resulting in -values in the range 12–15 ps–1. The value = 14 ps–1 was accordingly used in our CG simulations (use of the Langevin equation, according to the fluctuation-dissipation theorem, allows one also to maintain a constant temperature, which we set to 300 K). Ions (CG beads carrying a charge of 25|e| and representing 25 Na+ ions each) were added to neutralize the total negative charge of the flagellum.
The velocities arising in the simulations and giving rise to frictional forces need to be compared with the average speed of free diffusion, to ensure that the employed frictional force model holds. In our simulations, the flagellar filament (20 nm in diameter) was rotated at a rate of 100 turns per millisecond, i.e., CG beads at the outer edge of the filament move by 6 nm in one µs. With = 14 ps–1 and average CG bead masses of 2000–4000 Da, the diffusion coefficient D is 50 nm2/µs. According to the Einstein relation, the average displacement due to diffusion is then 7 nm in 1 µs. Thus, the velocities arising due to the rotation are on the same order of magnitude as those arising due to the diffusion at equilibrium, and the viscous force model employed is appropriate.
All-atom MD simulations performed to parameterize the CG model were carried out using the CHARMM22 force field (28) and NAMD 2.5. A cutoff of 12 Å was used for the nonbonded interactions, and the particle-mesh Ewald summation (grid step <1 Å) was employed for long-range electrostatics. Temperature (300 K) and pressure (1 atm) were controlled using Langevin dynamics with a damping constant of 5 ps–1 and a Nosé-Hoover Langevin piston with period of 100 fs and decay time of 50 fs. The integration timesteps were 1 fs, 2 fs, and 4 fs for bonded, nonbonded, and long-range electrostatic interactions, respectively.
The CG simulations (Table 1) were run with a single time step of 400 fs and 30 Å cutoff for nonbonded interactions. In most of the simulations, the flagellum filament (whole or a basal segment consisting of 30 unit proteins, see Table 1) was rotated with constant angular velocity; for this purpose, a harmonic force with spring constant of 5 kcal/(Å2 mol) was applied to the bottom part (relative to the orientation in Fig. 1 a) in each affected unit protein. Analysis and visualization were performed using VMD (31). All simulations used CG filament models created from an R-type straight filament (13) employing the methods described in this section.
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RESULTS |
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500 times faster than that seen for S. typhimurium in nature (32). This difference in rotation speed must be kept in mind throughout the analysis of our results, and makes it difficult to draw any quantitative comparison between our results and experiments. Nevertheless, the qualitative behavior of the filament seen in our simulations already reveals several important insights.
Simulation of the coarse-grained filament in solvent (simulation F1100) in the absence of applied torque showed that the overall superstructure of the flagellar filament is stable in our model. However, the overall height of the filament decreased from 528 nm at the beginning to 495 nm and the average maximal radius (measured for the outermost bead of each turn) of the filament dropped from 9.6 nm to 8.9 nm. Both of these figures represent a roughly 7% decrease in dimension, indicating that our model may slightly underestimate the size of the filament components. The shrinking in length is due to the relatively small number of beads per monomer; increasing the number of beads should repair it. Despite this shrinking, the overall flagellum structure remained stable. For example, the RMSD value for a single ring of 11 monomers is 0.89 nm per bead and the protofilament twist per monomer increased from 4° to 4.6°, smaller than the 5° change (and switch in handedness) involved in a supercoiling transition (11). The RMSD value of the individual monomers is 0.85 nm per bead; the close RMSD values of monomers and rings indicates a uniformity of the simulated structure. The height, angle, and RMSD values all change rapidly at the beginning of the simulation, but have stabilized completely after 15 µs.
Coiling of protofilaments and supercoiling of the flagellum
The structures of the filament in simulations F1100swim and F1100tumble, after 25 µs of dynamics, are shown in Fig. 3 a. As in the case of the unrotated filament, we do observe a slight shortening in both cases, with the height dropping to 511 nm in F1100tumble and 509 nm in F1100swim. The overall structure of the filament is preserved in each case, and the average internal RMSD of the monomers is 0.93 nm in F1100swim and 0.94 nm in F1100tumble. It must be noted that in both of these simulations, the rotated section of the filament begins to tear away from the rest of the assembly after roughly 20 µs; after this point the rotated region is able to slip along the interface with the rest of the filament and, thus, fails to fully transmit the rotational force. This tearing is almost certainly caused by the fact that the filament in these simulations is rotated 500 times faster than physiological flagella (32), and thus is exposed to much greater drag than a normal flagellum would be. Because of this, for the purposes of the following analysis all snapshots and data are taken up through 20 µs of simulation unless otherwise noted.
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The propagation speed of the motion can be estimated by checking the angular position (about the filament axis) of monomers throughout the length of the filament; monomers which have not yet been affected by the rotation, and all those in simulation F1100, show oscillations about a roughly constant value, whereas those affected by the rotation show a linear relationship between angular position and time. By this metric, motion in F1100swim had propagated 492 nm along the length of the filament in 20 µs, and that in F1100tumble propagated 494 nm during the same time period, giving a propagation rate of 2.5 x 104 µm/s (it is not surprising that our observation of the helical reversal in F1100swim lags behind this figure, since it will take some time after the rotation begins to reverse the handedness of a given area of protofilaments).
There is no experimentally measured quantity that should correspond directly to this propagation; for the sake of perspective, it can be compared to the overall transition rate of full flagellar filaments, but this should not be expected to match well. Turner and co-workers observed flagellar filaments 6 µm in length to undergo the complete transition from normal to curly I states in 0.1 s (33); this gives a propagation rate for the complete transition of 60 µm/s, which is much slower than the calculated rate.
The supercoiling states of the filaments from simulations F1100swim and F1100tumble are shown in Fig. 3 b. Pitch data from our simulations was extracted by plotting the position of a bead corresponding to the interface between the D0 and D1 domains (denoted the D0D1 bead; compare to Fig. 1) of greatest radius from the filament axis in 40 Å slices along this axis. In both cases, this supercoiled region appeared in an area several nanometers past the edge of the rotated monomers, and ended at or before the leading edge of rotational propagation. As seen in Fig. 3 b, the filament from simulation F1100swim forms a relatively long-pitch left-handed supercoil, whereas the filament from F1100tumble forms a shorter pitch right-handed supercoil. The difference in handedness between these supercoils matches that observed between the normal and curly I/II supercoils (8,9), but the high rotation speed applied in these simulations makes it unreasonable to compare simulations and experimental results in quantitative terms.
Protofilament contacts and heights
The polymorphic supercoiling of the flagellar filament is correlated with switching of several protofilaments between L- and R-forms, which have been found to have different repeat distances (52.7 Å and 51.9 Å, respectively) (7,10). A supercoiled filament then arises from the curvature generated by the presence of these repeat distances in the protofilaments; for example, a filament with two R-type protofilaments and nine L-type protofilaments (2R/9L) forms the normal configuration involved in the running mode, whereas the 5R/6L and 6R/5L combinations form curly states thought to be involved in the tumbling-mode (9,33). Recent MD studies have suggested the presence of switch-interactions between neighboring monomers, which arise between monomers involved in L-state (long) protofilaments, but are broken between R-state (short) protofilaments (7,12); using these switch-interactions as a criterion for identifying protofilaments as L- or R-state lead to the observation of the expected 2R/9L, 5R/6L, and 6R/5L states even in systems containing only 44 monomers (12).
The average rise per monomer for each of the 11 protofilaments for simulations F1100swim and F1100tumble is shown in Fig. 4 (calculated as the rise between D0D1 beads; see Fig. 1 a); in each case the data was calculated above the region of the filament being directly twisted, and <30 turns up the filament, to ensure that calculations were performed on a region consistently influenced by the turning. For simulation F1100swim beginning after 10 µs, one can recognize in Fig. 4 two distinct populations of protofilaments, one set of nine protofilaments with an average rise of 51.5 Å per monomer, and the other set of two protofilaments with an average rise of 49.5 Å. The protofilaments from simulation F1100tumble, in contrast, show a slightly broader distribution of heights, but do not have the same clear distinction between protofilament types; the average protofilament height after 20 µs is 50.8 Å in this case. It should be noted that the figures given here are for rise per monomer along the filament axis, not along the protofilament direction; it turns out that taking the measurement along the filament axis is more robust under the rotation conditions of our simulations. The difference of the stated experimental 52.7:51.9 Å (for L- and R-forms) and simulated 51.5:49.5 Å figures is due to the different measurement directions.
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). In the case of F1100swim, we observe nine L-type protofilaments and two R-type protofilaments (2R/9L); furthermore, as seen in the bottom panel of Fig. 4, the two short protofilaments are adjacent to each other. This should not be surprising, as it is sensible for the shorter protofilaments to be co-located on the interior face of the supercoil, an arrangement also found in the bistable protofilament model (11). In the case of simulation F1100tumble, there is no obvious distinction between L-type and R-type protofilaments; however, the protofilament wheel diagram in Fig. 4 does reveal that the four highest height protofilaments cluster along one face of the filament (along with a single moderate height protofilament), a configuration similar to the experimental curly II (6R/5L) state.
For both the swimming and tumbling states found here, supercoiling parameters can be constructed from the protofilament heights and angular displacements using the procedure of Hasegawa and co-workers (11) and assuming the average height and angular displacement values for each L- or R-state cluster found in each of the simulations. Supercoils built using these parameters are shown in Fig. 3 c. The results indicate that the supercoiling observed in our simulations indeed arises from a mechanism similar to that seen in the bistable protofilament model, suggesting that solvent effects induce switching in the filament. Slower rotation is necessary to verify this.
Effect of the solvent
The results of our simulations of the 1100-unit filament demonstrate how coiling of the protofilaments and supercoiling of the flagellum arise along the filament's length. To investigate this behavior further, we studied a five-times shorter flagellum segment in simulations F220swim and F220tumble. We found that the speed of rotation propagation, switching of the protofilament states, and supercoiling of the segment are the same (within 10%) as in simulations F1100swim and F1100tumble, respectively. This demonstrates that the behavior of the flagellum is due to local interactions.
The function of the flagellum is to propel the bacterium, by pushing against the surrounding solvent, and thus, one may further probe how much of a role the solvent plays in driving local interactions to alter the quaternary structure of the flagellum. To address this question, we simulated a 220-unit flagellum segment in a nonviscous medium using a reduced -value of 2 ps–1 (simulations F220swim-ns and F220tumble-ns). We found that without solvent (i.e., with reduced friction) the flagellum rotates essentially as a rigid body, i.e., the relative positions of monomers are frozen.
The results of simulations F220swim-ns and F220tumble-ns are compared with those of F220swim and F220tumble in Fig. 5 a. Two representative protofilaments, highlighted in blue and red, maintain their initial helical conformation (to within ±0.5° per monomer) throughout simulations F220swim-ns and F220tumble-ns. In the presence of solvent (friction), the right-handed helix, which the corresponding protofilaments form, becomes shorter when the segment is rotated in the tumbling direction, and the helix becomes left-handed when rotated in the running direction. Enlarged snapshots from the simulations of the entire 1100-unit filament in solvent (Fig. 5 b) reveal protofilament arrangements very similar to those arising in simulations of the 220-unit segment in solvent. Thus, it is clear that the solvent (friction) plays a crucial role in the switching between the arrangements of protofilaments and, consequently, in producing supercoiling along the entire filament.
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The coarseness of our model and the overly fast rotation of the flagellum certainly introduce artifacts into the observed behavior, but nevertheless the role of filament-solvent interactions in protofilament rearrangement and filament geometry suggested by our simulations is likely significant. To our knowledge, the effect of solvent on flagellum supercoiling under conditions of running or tumbling has not been systematically investigated before. Our study suggests that both the structure of the flagellum and the effect of the liquid medium (viscous drag) are important for understanding the polymorphic transitions controlling bacterial swimming.
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CONCLUSION |
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While the results of our study are in qualitative agreement with experimental data, quantitative comparison is not yet successful due to the high rotation speed required because of our 10-µs simulation time-limit. In addition, the present model lacks a description of the detailed switching interactions between protofilaments. Overcoming this omission requires a multiscale coarse-graining methodology, in that different levels of detail need to be combined in a single CG description. Such multiscale resolution is possible for the CG scheme we used since the neural computation method employed here permits the resolution of a hierarchy of spatial properties (25).
The CG technique applied in our study has been developed in a general form, which allows one to employ it directly to investigate other macromolecular systems. The CG simulations can be carried out with the program NAMD, achieving a parallel computing performance in line with all-atom simulations. Therefore, concepts and methods developed for all-atom simulations can be transferred to CG simulations. With presently available computing resources, the new CG method is capable of simulating systems on a micrometer length-scale and soon on a microsecond timescale.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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This work was supported by National Institutes of Health grant No. PHS-5-P41-RR05969. The authors are grateful for supercomputer time provided by the Pittsburgh Supercomputer Center and the National Center for Supercomputing Applications via an National Science Foundation National Resources Allocation Committee grant No. MCA93S028. P.F. acknowledges support from the National Science Foundation Graduate Research Fellowship Program.
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FOOTNOTES |
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Submitted on July 17, 2006; accepted for publication September 6, 2006.
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REFERENCES |
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