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Rowland Institute at Harvard, Harvard University, Cambridge, Massachusetts
Correspondence: Address reprint requests to Howard C. Berg, Rowland Institute at Harvard, Cambridge, MA 02142. E-mail: hberg{at}mcb.harvard.edu
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ABSTRACT |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). Peritrichously flagellated bacteria, such as Escherichia coli and Salmonella, have several such flagella, each attached to a rotary motor that is embedded in the cell wall. Although the complete flagellum/motor complex contains 
25 proteins, the bulk of the flagellum itself is composed of a single protein, flagellin (2,3). The flagellin homopolymer, called the flagellar filament, comprises more than 99% of the length of the flagellum and provides the structural stiffness necessary to generate thrust during swimming.
A Salmonella filament is normally a left-handed helix, but environmental perturbations can trigger a sudden, discrete change to a new shape. All of these shapes, called polymorphic forms, are helices (4,5); some are left-handed and some right-handed (Fig. 1). The most extreme forms are straight left- or right-twisted rods. A polymorphic transformation from one shape to another can be caused by changing pH, salinity, or temperature (6–8), by adding alcohols (9) or sugars (10), or by applying forces or torques to the filament (11,12). All of these transformations are reversible provided the conditions do not depolymerize the filament (6,7). In addition, mutations in flagellin can change the basic polymorphic form (13). The structure of the right-type straight form has been determined by x-ray fiber diffraction at 9-Å resolution (14) and by electron cryomicroscopy and image reconstruction at 4-Å resolution (15); a truncated right-handed flagellin can be crystallized, yielding an x-ray crystallographic structure at 2-Å resolution (5). The left-type straight form is less well characterized, but based on electron cryomicroscopy (16) and x-ray fiber diffraction measurements (2), it is believed to be slightly longer than the right-type form.
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subunits per turn, with two turns required to step from one subunit to the next along a protofilament.
A simple geometric model by Calladine (4,17,18), based on earlier work by Asakura (19), explains the observed spectrum of flagellar polymorphic forms, in accord with physical data. The model assumes that 1), each individual flagellin monomer can switch between two states, "L" and "R," that have slightly different inherent twist and length (20), and 2), the equilibrium pattern of monomer states minimizes the elastic energy of the filament.
The trivial cases of 100% L and 100% R states correspond to L-type and R-type straight filaments. Between these extremes, elastic energy is minimized when like states self-segregate along protofilaments, so that one can meaningfully refer to the "state" of a protofilament, and like protofilaments cluster together. Implicitly, there is some three-dimensional geometric incompatibility between monomer shape and the flagellar symmetry: homogeneous L and R wild-type flagellins must not fit neatly into the filament's 11-fold symmetry because the pH-neutral, room-temperature "normal" filament form contains 9/11 L and 2/11 R. The two-state model is buttressed by experiments in which chimeric filaments, composed of a mixture of mutant flagellins locked in the L and R states, yielded intermediate polymorphic forms (19,21,22). We use a modified version of the Calladine model containing three free parameters: the inherent twists of the L and R protofilaments (
L and R) and the maximum curvature of the filament (
max). The twist and curvature of the filament are
 | (1) |
 | (2) |
max with n (23). The parameter max is directly related to the L and R protofilament geometries, and typical values in the literature are consistent with measurements of protofilament structure (23). A computer simulation of filament extension identified a conformational change in a ß-hairpin as the physical switch between the L and R states (5). Although the Calladine model predicts the shapes of various forms, it says nothing about the relative stability of each form or the forces required to transform from one polymorphic form to the next. In this work we use optical tweezers to stretch isolated flagellar filaments and measure the force associated with polymorphic changes.
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METHODS |
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,24). Three liters of QM medium (10 g Difco bacto peptone + 10 g Difco yeast extract + 10 ml 30% glucose + 10 ml 40 mM pH 8 potassium phosphate buffer per liter of water) were inoculated with 300 ml of a saturated culture of Salmonella typhimurium SJW1103 (a phase-1 stable derivative with normal filaments of serotype i (25)) and grown overnight at 37°C with shaking at 200 RPM to aerate. Cells were pelleted by centrifuging 15 min at 8000 x g and then resuspended in 42 ml polymerization buffer (5 mM potassium phosphate buffer pH 6.5 + 150 mM NaCl). Flagella were sheared from the cells in this suspension using a modified Waring blender, and the cell bodies were pelleted out (15 min at 8000 x g). The supernatant fraction was further cleaned of cell debris by centrifuging 15 min at 15,000 x g. The resulting suspension was purified by three rounds of repolymerization. To perform a round of repolymerization: 1), pellet filaments 1 h at 78,000 x g and 4°C and discard supernatant; 2), resuspend filaments in 4–8 ml polymerization buffer; 3), reduce filament length by sonicating suspension 5 min at 50% power with a clean immersion sonicator (Heat Systems–Ultrasonics, Farmingdale, NY, model W225); 4), depolymerize filaments 5 min at 65°C; 5), clean monomer by centrifuging 1 h at 100,000 x g and 4°C and discarding precipitate; 6), make polymerization seeds: harvest a small fraction of supersaturated monomer solution, mix with an equal volume of 2M Mg2SO4 + 10 mM potassium phosphate (pH 6.5), polymerize 1 h at room temperature, spin down seeds 1 h at 78,000 x g, discard supernatant, and resuspend in original volume of polymerization buffer; 7), combine monomer and seeds and homogenize mixture by sonicating 5 min at 50% power; and 8), polymerize overnight at room temperature.
The three rounds of repolymerization used progressively smaller seed fractions of 20%, 10%, and 5% of total monomer volume. The resuspension volume in step 2 was decreased from 8 to 4 ml to keep the total monomer concentration (measured after depolymerization) around 1.5 OD280 (nominally 5 mg/ml) because 25% of flagellin was lost in each round of purification.
Repolymerized filaments were labeled with an amine-reactive Cy3 dye (Amersham Biosciences, Piscataway, NJ, Cat. No. PA23001) for 1.5 h in PBS (10 mM pH 7.0 phosphate buffer + 67 mM NaCl + 100 µM EDTA) in a variation of the method of Turner, Ryu, and Berg (26). To avoid breaking filaments, excess dye was removed by gently filtering with a 0.2-µm filter and flushing with 100 times the reaction volume of PBS. The majority of filaments were 10–25 µm in length, with a small population of extremely long (up to 70 µm) filaments. Labeled filaments were refrigerated in polymerization buffer until use.
Phase diagram
The phase diagram for Salmonella filaments (Fig. 2) was mapped using combinations of HCl (pH 2–4), 10 mM potassium phosphate buffer (pH 4–10), and NaOH (pH 10–12). Dilute samples of unlabeled repolymerized filaments in the appropriate buffer/salt combination were equilibrated 30–60 min at room temperature and observed with dark-field illumination.
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). Antibody-coated beads were prepared by adsorbing anti-Cy3 antibody (Abcam, Cambridge, MA, Cat. No. ab6902-1) onto 1.4-µm latex beads (Polysciences, Warrington, PA, Cat. No. 17133) in PBS + 1% bovine serum albumin (BSA). A dilute solution of Cy3-labeled filaments and beads was mixed with an appropriate buffer (either pH 7 PBS + 0.003% BSA or pH 4 100 mM acetate + 0.01% BSA) and an oxygen-scavenging solution (0.1 mg/ml glucose oxidase (Sigma-Aldrich, St. Louis, MO, Cat. No. G7016) + 0.018 mg/ml catalase (Sigma Cat. No. C100) + 3 mg/ml glucose) and loaded into a thin chamber composed of a microscope slide and coverslip assembled using double-sided tape (0.07 mm thick). The slide was placed in a custom-built open-loop x-y piezoelectric stage (27) on an inverted microscope (Nikon Diaphot 200). A temperature-controlled brass jacket around the objective lens, designed according to Khan and Berg (28), was thermally coupled to the sample slide by immersion oil. Epifluorescence images of filaments were captured by a framegrabber (model LG-3, Scion Corporation, Frederick, MD) using a black and white CCD camera (Marshall Electronics, Culver City, CA, model V-1070). Trap stiffness was calibrated by fitting the trapped bead power spectrum to a Lorentzian (29), and the quadrant photodiode (QPD) response was calibrated by scanning an immobilized bead through the beam focus. During data runs, noise from the Brownian motion of the bead in the trap was suppressed using a 10-Hz low-pass hardware filter (Wavetek Rockland, Rockleigh, NJ, model 852), and the filtered QPD signal was recorded using LabView (National Instruments, Austin, TX). The trap was focused just above the chamber surface (+2 µm) so that pulling forces would be perpendicular to the optical axis, and so that the stretched filament would lie in the focal plane.
Isolated filaments naturally settle to the bottom of a microscope slide, and their proximal ends (6) adhere to clean glass (cleaned for several minutes in 95% ethanol saturated with KOH, then copiously rinsed with water). After a suitable filament was identified, an antibody-coated bead was trapped and forced against the filament until it bound, and the filament was stretched to an approximately neutral length. Labview was used to drive the piezo stage with a periodic triangle wave; the true displacement of the stage was calculated by correcting the open-loop signal using the recorded positions of stuck beads. Before data acquisition, the piezo was cycled at least 10 times to overcome the piezo memory effect. In principle, the filament extension might be less than the stage displacement because the far end of the filament moves inside the optical trap. In practice, with the relatively stiff traps used (100–150 pN/µm), the correction for bead motion was negligible.
Force-extension curves were fit to Eq. 12 with n and n considered to be known parameters. The extension z is related to experimental displacements dz via z = zn + 
+ dz, where is the offset between the nominal origin (dz = 0) and the actual neutral filament position, and zn is the neutral axial length of the filament (given by Eq. 10 with = n and = n). To compute the fractional extension 
= z/L we need to know the filament contour length L; in practice, it is more convenient to identify the neutral filament position, measure zn (from the image of the filament), and calculate L using Eq. 10. Eliminating L leads to the expression
 | (3) |
n 
zn/L can be calculated from Eq. 10 to give 1 = 0.98, 2 = 0.85, and 3 = 0.30 for the hyperextended, normal, and coiled forms, respectively. The measured force was manually corrected by a constant offset to bring the slack region to zero force; this was required because of drift in the QPD amplifier-nulling electronics. In total, fitting of each data set used this manually fixed offset and two free parameters ( and EI). This offset was correlated with the fit parameter but did not affect the stiffness EI or the overall shape of the force-extension curves. |
RESULTS |
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). Regions labeled "coiled" generally contained some completely coiled filaments, some completely normal filaments, and some hybrid normal-coiled filaments composed of both forms, as in Fig. 3 A. Based on spot checks of a few of the regions, fluorescent labeling of filaments does not change the phase diagram.
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, pp. 311–315):
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
10% (23). The model predicts the forms' radii; measured values agree to within 0.05 µm. In hybrid filaments containing two different polymorphic forms, one can predict the angle between the two helical axes (31); our measurements agree with such a calculation to within 6° (data not shown).
Repeatedly pulling on filaments in the neutral-pH/moderate-salt normal phase at room temperature gives the series of curves shown in Fig. 4. Under rapid pulling, the filament usually traces a simple, hysteresis-free curve (upper trace). Occasionally, however, the measured force will jump suddenly to a lower curve. This corresponds to a sudden twist of the filament as a portion transforms from the normal (n = 2) to the hyperextended (n = 1) form. Under rapid cycling conditions, the filament often (9 of 19 cycles) completes a complete elastic extension-compression cycle without performing a polymorphic transformation, but when the filament is extended more slowly, it always transforms (10 of 10 cycles), and the transformation generally occurs at lower force levels (Fig. 4, inset). When we attempt to compress the filament (z 
–0.5 µm), the helix buckles (frames A and B) rather than sustain any negative force. The polymorphic transformation that occurred during extension is reversed during this buckling, but because it occurs during a uniform zero-force regime, we do not detect it.
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DISCUSSION |
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 | (9) |
, Chapter 18). and are the curvature and twist of the rod, which, under the application of some combination of torque and tension, may differ from their intrinsic, unstressed values n and n. It is more convenient to describe the helix by its length z and winding angle 
(= 2
z/p):
 | (10) |
 | (11) |
U(z, )/z and 
= U(z, )/. Because one end of the filament is attached to a spherical bead, the filament end rotates freely to relieve any torque. Solving = 0 for the winding angle and substituting into the expression for F gives
 | (12) |
z/L is the normalized length. For an isotropic solid –1 < E/µ < , and most common substances lie in the narrow region 0 < E/µ < 
, whereas a rod of circular cross section has J = 2I = a4/2, where a is the radius. This suggests that 1 > EI/µJ > 
, and we will make the simplifying assumption that EI/µJ = 1. For right-handed helices a positive force produces a positive extension (
); for left-handed helices the convention is reversed, so a negative force produces an extension (
). In the following discussion, signs are reversed to make left-handed helices follow the more natural right-handed convention. Fig. 6 shows the data of Figs. 4 and 5 superimposed on a family of elastic force-extension curves given by Eq. 12 with n and n taken from Eqs. 6–8. For the normal:hyperextended and coiled:normal data, respectively, filament lengths are 19.5 and 7.6 µm, and fit parameters are = 0.12 µm and = 1.0 µm.
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–35). The slope of the elastic portion of the force-extension curve for the normal form is six times steeper than that for the coiled form, but that is entirely because of the difference in helical geometry between the two forms; the inherent stiffness of the filament itself is unchanged.
The derivation of Eq. 12 assumed that the bending and twisting stiffnesses (EI and µJ) are equal. If this were not the case, Eq. 12 would contain additional terms involving µJ and factors that depend on the helix geometry. A large difference between EI and µJ, as has been suggested (36), would have led to a different effective stiffness in our measurements. Because we do not observe any change in stiffness between forms, the approximation EI = µJ is probably quite good.
Equation 12 assumes that an axially aligned force produces a uniform extension or compression of the helix. Under compression, however, if the force is misaligned with the helical axis, then the filament will buckle. This problem is particularly severe with the normal form, which has a smaller radius, and as a result, the normal filament can sustain almost no compressive force before buckling into a flat, near-zero-force region.
Fig. 6 also shows intermediate curves corresponding to hybrid filaments, which are calculated by assuming that sections of different polymorphic forms act as springs connected in series. This formulation neglects the contact angle between the different forms, which varies from 20° (for normal:hyperextended) to 40° degrees (for coiled:normal). When a change in handedness occurs, the contact angle can be much larger. For instance, the coiled:semicoiled contact angle is 140° (Fig. 3 B), which is presumably why pulling on the coiled form triggers a transformation to the normal, rather than semicoiled, form, although both are more extended than the coiled form.
Polymorphic transformations occur in discrete, rapid steps, converting micrometer-long sections of filament at a time. In between transformations, the filament behaves as a linear elastic object that accumulates elastic strain energy, which is released during the next transformation. This phenomenon is consistent with some sort of activation energy or energy barrier because the transformation is not deterministic and is associated with a time scale, yielding force-extension curves that depend on strain rate as well as strain.
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ACKNOWLEDGEMENTS |
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This work was supported by National Institutes of Health grants AI016478 and AI065540.
Submitted on July 27, 2006; accepted for publication October 23, 2006.
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REFERENCES |
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3), and most, but not all, have been observed in the wild (see Yamashita et al. (14
