Figure 1.
Model for stepping of the flagellar motor.
A. Side view of the flagellar motor. B. Top view of the motor highlighting the model's essential ingredients. The passage of across the inner membrane causes the stretching of protein “springs” which link the peptidoglycan-anchored stator complexes (MotA/B) to the rotor (FliG, etc.). In the schematic, stretched springs are attached to a stator at one end, and to an attachment site (represented by blue dots) at the other end, and apply a torque to the rotor. Contact forces between the stators and the rotor also produce a potential of interaction, which is approximately 26-fold periodic due to the 26 FliG subunits. The 26-fold periodicity of FliG and the 11-fold periodicity of the hook and filament are represented.
represents the absolute angular position of the rotor. C. Left: Rotation of the rotor as a whole corresponds to a viscously damped random walk in a tilted corrugated potential
arising from the combined torque and contact potential. Right: Example of a trace generated by the model (blue) and the inferred steps (red) between local potential wells (shown with purple shading).
Figure 2.
Backward steps are smaller than forward steps.
A. Probability distribution of step sizes using a approximately 26-fold periodic potential (see main text), showing that backward steps are on average smaller than forward steps, in agreement with experiment. Inset: Backward steps rely on low barriers, which occur preferentially where angular steps sizes (e.g. ) are small. B–E. Forward steps immediately following or preceding backward steps are found to be smaller on average (
in the experiment,
in the model) than the mean of all forward steps (
in model and experiment, black dots). Crosses denote mean and standard deviation of backward and subsequent or previous forward steps, while black dots and horizontal lines give the mean (
) and standard deviations of all forward steps. Note change of scale between simulation and experiment. The experimental data are from [1].
Figure 3.
The height of barriers to backward steps is positively correlated with the backward step size.
For 100 randomly generated potentials , we plotted the size of each of the 26 steps versus the height of the corresponding barrier to backward steps (i.e. the energy difference between the minimum of
and its immediately preceding maximum). Each color corresponds to a particular potential. We used:
, with
, and
drawn uniformly at random with fixed total power
. The torque is
.
Figure 4.
Absolute position of the rotor matters.
A. A typical approximately 26-fold periodic tilted potential (red line) used in our simulations. For each of the 26 barriers in
, we show the frequencies of forward and backward steps across that barrier obtained analytically from first-passage theory (see Materials and Methods). Backward steps occur much more frequently at low barriers. B. Average backward and forward step sizes for each of the 26 barriers around the circle. Each color corresponds to a simulation with a different choice of the potential
. C. The ratio of backward over forward step counts for a given barrier decreases with the average forward step size (colors as in B). D and E. Same as B and C, but with experimental data [1]. Each color corresponds to a different cell. For each cell and each position around the circle, we show a data point only if there were at least 10 backward steps.
Figure 5.
The model predicts a sublinear torque-speed relation and a peak in rotor diffusion.
A. Rotation speed of the rotor as a function of torque for loads with different drag taken from [7]: solid curves, from top to bottom, ,
,
,
, and
. The potential was chosen to be perfectly 26-fold periodic:
, with
. As torque increases, the rotation speed asymptotes to the behavior expected in the absence of barriers,
, represented here for the case of the lowest load
(dashed line). At small torques, the rotation speed is limited by the rate of barrier crossing (left inset), while at high torques the tilt makes barriers easy to cross (right inset), and rotation is only limited by drag. B. Effective diffusion coefficient as a function of torque for a load with drag coefficient
: solid curves,
with
,
, and
. The real diffusion coefficient
is represented by a dashed line.
Figure 6.
The distribution of waiting times between steps (backward or forward).
A. Simulation with a perfectly 26-fold periodic contact potential . There is only one type of barrier, and the distribution of waiting times is roughly exponential. (The contact potential is
with
, and torque
.) B. Simulation with an approximately 26-fold periodic contact potential. There are 26 distinct barriers, each of them having a different characteristic waiting time. The overall waiting time distribution is therefore the sum of 26 exponentials. (The contact potential and torque are the same as in Fig. 2.) C. Experiment – the distribution is consistent with a sum of exponentials.