Computer-generated models of a ring and their Fourier transforms. (A) Top view of a ring of 34 spherical subunits. (B) Fourier transform of A. The inner part of the transform consists of narrowly spaced, uniform rings of intensity. This pattern corresponds to the J₀ term of the transform. At the edge, a stronger, periodic ring appears, which corresponds to the J₃₄ term of the transform (∗). (C) Side views of the ring. The upper three images correspond to side projections of the ring in A taken at different orientations. The bottom view is the average of a set of 100 side projections taken at different angles. Note that the periodicity seen in the first three is effectively lost in the averaging process. (D) Power spectra derived from the Fourier transforms of the side views. The dashed curves correspond to the average of the power spectra of the 100 side views taken at random orientations and to the power spectrum of the average of the 100 side views. The solid line is the difference of the two, which has been scaled up by a factor of 10. ∗ marks the start of the J₃₄ term seen in B.

Gallery of basal-body images. (A–C) The three good top views. B′ and C′ are copies of B and C with ellipsoids drawn to correspond to the circumference of the rings. (D) A comparison of the two ellipsoids from B′ and C′ showing that the ring in B is smaller than that in C. (E) Fourier transforms of two top views spliced together (the top half corresponds to the ring in A; the bottom half corresponds to the ring in B). ∗ marks the positions of the J₃₄ and J₃₃ terms. The peaks in both transforms occur at a radius of 1/39 Å⁻¹. (F) A plot of local cross-correlation peaks vs. rotation angle for the three top views shown in A–C. The peaks provide a measure of the number of subunits in each C ring. There should be one peak for each subunit. There are four gaps in the plot for the ring in A, which presumably correspond to four missing peaks. Allowing for these missing peaks, we determine that n = 34 count for the rings in A and C, and n = 33 for the ring in B. (Bar = 100 Å in A.)

(A–C) Three side views of the C ring. Note that the rings are of different diameter. The left sides are aligned, and one of the units in A is marked with an arrowhead. The arrowhead is repeated at the same spatial position in B and C to make the difference in diameter clearer. (Bar = 100 Å.) (D) Fourier transforms of side views. The side views are projected onto a line perpendicular to the axis. The projection of the average of these views (Fig. 4 A or B) is subtracted. The amplitude distribution of the Fourier transforms of these differences projections are averaged. The two curves display the amplitudes corresponding to the wild-type and to the deletion-fusion mutant. There are peaks at a radius of 1/39 Å⁻¹ (see arrows), the same position as the peaks seen in Fig. 2E.

Averaged images of basal bodies. (A) Average of wild-type basal bodies embedded in vitreous ice. The M and C rings are labeled MR and CR, respectively. (B) Average of basal bodies from the deletion-fusion mutant. (C) Average of basal bodies from the full-length fusion mutant. The bar in each image is drawn to the same length to demonstrate the change in the diameter of the C ring. (Bar = 100 Å.)

Schematic of the model for the mechanism of motor rotation. (A) A contour plot of the density of the averaged image in Fig. 4A, except that the L and P rings and part of the hook are included. The stud and its connection to the peptidoglycan layer are shown in cartoon form. The stud is not attached to the C or M ring here, but it is in B. The bar connecting the C ring and the M ring is intended to represent the C ring levers and their binding sites on the M ring. The zig-zag lines here and in Fig. 3F depict possible elastic elements. The box indicates the part of the motor shown in cartoon form in B–F. (B) The model represents a short segment of the full motor. It shows five of the C ring (CR) subunits, four of the M (MR) ring subunits, and one stud (MotA–MotB complex). The difference in the number of subunits in the M and C rings in the model represents the differences in the number of subunits in these rings in the basal body. In particular, not all 34 levers from the C ring can mate with the bonding sites on the M ring, which has only 26 sites. The extra levers in our model form a complex with the MotA–MotB stud(s). In the model, we have shown only one extra lever, which is in a complex with a stud. We assume that all of the forces from the elastic elements are initially in balance. (C) The stud and unpaired C ring lever move, perhaps by Brownian motion, to the left and bind to the M ring–lever complex. On binding, the elastic elements generate a force, and the M and C rings move. (D) Motion stops when the forces are again balanced. (E) The unpaired lever from the C ring is now swapped so that it is bound to the M ring, and the stud and its new partner in the C ring are freed from the complex. Force is generated, and the M and C rings again move. (F) When the motion stops, the M ring and C ring have each moved by one subunit, and hence by different angles. The M ring has undergone 1/26 of a revolution, whereas the C ring has only undergone 1/34 of a revolution.