PMC

Piezoelectric stator model: (a) Stator front; (b) Stator back.

a An unbound stator unit is in a closed state (D) in which it freely diffuses within the inner membrane and does not translocate protons. Upon interacting with a rotor of radius R, the stator unit transitions into a loosely bound state (L). In this state, the protons start flowing through the stator unit, resulting in the production of a torque Γ on the rotor. As a reaction, a force F = Γ/R acts on the stator unit, which displaces it in the direction of the force. This displacement away from the point of tether lowers the off rate and results in a tightly bound state (T). A bound stator unit can occasionally detach from the rotor, resulting in a transiently unbound “hidden” state (H). b A simplified 4-state model of stator dynamics. A stator unit can go between the diffusive state (D) and the loosely bound state (L) with an on rate, k_on and an off rate k_off,l. From the L state, it can go to the tightly bound state (T) with a rate k_t and back with a rate k_l, both of which depend on the torque (Γ). From the T state, it can transition to the D state at a much lower off rate k_off,t. This model also includes the transiently unbound hidden state (H). The H state couples with either the T or the L state with rates k_h and k_−h.

Model of the flagellar rotor with 24 radially arranged copies of FliG (segments) surrounded by 16 stator elements (H⁺ channels) arranged as 8 S-T pairs (stator complexes). The two elements of a stator complex undergo brief rotational movements in a coordinate mode (curved arrows), as protons pass through the channels. At any given moment, one set (S in this scheme) interacts with the apposed rotor segment by antipodal charges and turns the rotor 1/48th of its circumference (half a segment) as shown in . Then the stator complex swings back (as in ) without contacting antipodal charges at the rotor face, and brings the T set of elements into position ready to carry out the next step. Each of the 16 stator elements takes 24 productive and 24 unproductive turns, before one revolution of the rotor is completed.

(a) A typical sequence of events that advance the rotor by one step of 2π/12. Consider the initial position of the rotor shown at ➊. The third site from the left is held in the electrostatic well of the stator charge. Step ➊ → ➋: the rotor fluctuates until the third (empty) site thermally jumps out of the potential well of the stator charge. This jump is biased by the transmembrane potential and is helped by the dielectric barrier preventing the first (empty) rotor site from entering the low-dielectric medium of the stator from the right. ➋ → ➌: once the third rotor site is out of the potential well of the stator charge, it quickly binds a sodium ion from the periplasmic (acidic) reservoir. ➌ → ➍: the positive stator charge pulls the empty fourth rotor site into its potential well. Because the second rotor site is neutralized, it can pass through the dielectric barrier. ➍ → ➎: once the second rotor site passes out of the stator its sodium ion quickly dissociates into the cytoplasmic reservoir. Once empty, it cannot go back into the low dielectric rotor–stator interface. ➎ is exactly the same state as ➊, but shifted to the left by one rotor step. (b) Free energy diagram of one rotor site as it passes through the rotor–stator interface. The chemical reactions of ion binding and dissociation to the rotor site switch the potentials seen by the rotor site between that corresponding to an empty site (solid line) and that corresponding to a neutralized site with an ion bound (broken line). Step A → B: the rotor diffuses to the left, bringing the empty (negatively charged) site into the attractive field of the positive stator charge (R227). B → C: once the site is captured, the membrane potential biases the thermal escape of the site to the left (by tilting the potential and lowering the left edge). C → D: the site quickly picks up an ion from the periplasmic channel, which drops the site to the neutralized site potential (broken line). D → E: this allows the occupied site to pass through the dielectric barrier. (If the site diffuses to the right, the ion quickly dissociates from the site as it approaches the stator charge.) E → F: upon exiting the stator the site loses its sodium ion. Now charged, the site sees the stator dielectric barrier, which prevents back-diffusion. The cycle decreases the free energy of the system by an amount equal to the electromotive force: Δμ = Δψ −2.3(RT/F)ΔpNa, where R is the gas constant, F is the Faraday constant, and pNa is the negative log of the Na concentration. The free energy changes accompanying ion binding from the periplasm (P) and dissociation to the cytoplasm (C) are ΔG_P = −(2.3RT/F)(pK_a − pNa_P) and ΔG_C = −(2.3RT/F)(pNa_C − pK_a), respectively.