# Hidden stochastic nature of a single bacterial motor

Probability density functions of CCW (gray) and CW (black) time intervals as a function of CW bias, from experimental data. The white lines represent the fit that the authors found between the measured data and the Gamma distribution function.

The bacterium E. coli has a rotary flagellar motor that switches between clockwise (CW) and counter-clockwise (CCW) motion in a stochastic manner. This study found that the intervals for the CW and CCW stages could be described by a Gamma distribution, and the authors suggest that this indicates there may be "hidden Markov steps" involved in the process of motor direction switching. The number of hidden steps seems to be a key dynamical parameter that determines the motor switching behavior in a bacterium cell as well as in large cooperative molecular systems more generally.

## General Information

Keywords: soft machine, molecular motor, nucleation, Poisson step, cell

Authors: Ekaterina A. Korobkova, Thierry Emonet, Heungwon Park, and Philippe Cluzel.

Date: February 10, 2006.

The Institute for Biophysical Dynamics, The James Franck Institute, The University of Chicago, 56540 South Ellis AvenueChicago, Illinois 60637, USA

Physical Review Letters 96, 058105. [1]

## Summary

It has been found that the CW bias, or the probability that the motor will rotate clockwise, increases nonlinearly with the concentration of the signaling molecule CheY-P. In this study the authors analyzed binary time series of motor direction switching events to quantitatively characterize the switching behavior. They found that the CW and CCW time intervals could be modeled with the Gamma distribution function

$G_r(\tau) = \frac{\nu^r \tau^{r-1} e^{- \nu \tau}}{\Gamma(r)}$

where $r$ and $\nu$ are the the shape and scale parameters respectively. The $r$ parameter represents the number of hidden steps in the Poisson process that repeats at the rate $\nu$. The Gamma distribution analysis conducted by the authors suggests that each motor switch is preceded by hidden Poisson steps. Previous studies had assumed an exponential behavior, or the necessity of only one Poisson step.

(A) Energy diagram for motor subunits: white and black circles represent inactive and active conformations of a subunit, respectively. (B) On the left, birth and growth of active domains on an idealized ring of the motor. On the right, a switch occurs when domains comprise the entire ring.

Duke et. al. suggested that the motor switching behavior depends on energy changes. In this model, "the conformational change of one subunit spreads through an idealized ring of closely packed subunits to mediate a switch," according to the authors. This is inconsistent with the experimental findings of this study, however, unless the model is modified such that the conformational changes of multiple subunits are necessary to mediate a switch. The authors propose that the number of conformational changes needed is equal to the number of hidden Poisson steps required for the mathematical fit to the data.

## Connection to soft matter

First of all, this study is concerned with the mechanism governing a bacterial motor, a soft machine. The authors suggest that the mechanism lies in conformational changes to subunits along an idealized ring within the motor. The conformational changes are due to small energy changes, and therefore statistical mechanics should be applied.

"The probability that the binding of one CheY* molecule induces one nucleation is a fixed characteristic of the motor and behaves like ~ $exp(\Delta E / k T)$ ," according to the authors. The concentration of CheY* is proportional to the probability that one CheY* will bind with a subunit of the motor, and therefore increasing the concentration will increase the rate of motor switching in a way consistent with the Gamma distribution function found from the experimental results.