Resonance in the response of the bacterial flagellar motor to thermal oscillations

Mahmut Demir and Hanna Salman

Phys. Rev. E 95, 022419 – Published 28 February 2017

Abstract

We have studied the dynamics of the Escherichia coli flagellar motor's angular velocity in response to thermal oscillations. We find that the oscillations’ amplitude of the motor's angular velocity exhibits resonance when the temperature is oscillated at frequencies around 4 Hz. This resonance appears to be due to the existence of a natural mode of oscillation in the state of the motor, specifically in the torque generated by the motor. Natural modes of oscillation in torque generation cannot result from random fluctuations in the state of the motor. Their presence points to the existence of a coupling mechanism between the magnitude of the torque generated by the motor and the rates of transition between the different states of the motor components responsible for torque generation. The results presented here show resonance response in torque generation to external perturbations. They are explained with a simple phenomenological model, which can help future studies identify the source of the feedback mechanism between the torque and the interactions responsible for its generation. It can also help us to quantitatively estimate the strength of these interactions and how they are affected by the magnitude of the torque they generate.

Received 8 June 2016
Revised 2 November 2016

DOI:https://doi.org/10.1103/PhysRevE.95.022419

Physics Subject Headings (PhySH)

Flagella Motor proteins

Physics of Living Systems

Authors & Affiliations

Mahmut Demir^1,* and Hanna Salman^1,2,†

¹Department of Physics and Astronomy, The Dietrich School of Arts and Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
²Department of Computational and Systems Biology, School of Medicine, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

^*Present address: Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven 06511, CT.
^†Corresponding author: hsalman@pitt.edu

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Vol. 95, Iss. 2 — February 2017

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Figure 1
Experimental procedure. The experimental setup used for temperature modulation and recording of tethered cells’ rotation is presented in (a). The infrared laser was focused into the sample and its power was controlled via a custom written labview program. The temperature around the tethered bacteria [presented in the top frame of (b)] was measured using the temperature sensitive dye BCECF as can be seen in the bottom frame of (b) and the time series presented in (c). Examples of the temperature measurements as a function of time using this method for linear and oscillatory change in temperature are presented in (d,e), respectively. The different lines in (e) represent the actual temperature measurement, which is noisy, and the best fit to a sinusoidal curve as indicated in the figure legend. Observations were made on an inverted microscope in fluorescence mode with a 100× objective. The tethered cells’ rotation was recorded using a CCD camera. For more information, see the Supplemental Material [18].
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Figure 2
The effect of temperature oscillations’ amplitude on the resonance frequency. The position of the resonance (resonance frequency) and the relative change in the amplitude of the angular velocity oscillations do not depend on the amplitude of the temperature oscillations. We have changed the amplitude of the temperature oscillations around 31 °C, from 0.7 °C to 6.5 °C. The data points in the main figure depict the relative change in the angular speed oscillations’ amplitude as a function of the temperature oscillations’ frequency. The different shapes depict different temperature oscillations’ amplitude as indicated in the legends. Each data point is the average of at least 30 measurements. Resonance frequencies for all amplitudes of temperature oscillations were obtained by fitting the data to the resonance equation [Eq. (1) in the text]. The fits are depicted by the lines in the figure. Resonance frequencies as a function of the temperature oscillations’ amplitude are presented in the inset. The error bars depict the fit errors. Our findings also found the relative change in the oscillations’ amplitude of the angular velocity to be similar.
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Figure 3
The oscillations’ amplitudes of the motor's angular velocity and torque in response to temperature oscillations. (a) The relative amplitude $[\frac{ω_{amp} (f)}{ω_{0}} / \frac{ω_{amp} (0.2 Hz)}{ω_{0}}]$ of the angular velocity oscillations was calculated as described in Materials and Methods. Relative amplitudes of a single experiment set were normalized with the relative amplitude at 0.2 Hz, and the normalized relative amplitudes are averaged for all experiments. The average normalized relative amplitude increases as the oscillation frequency increases up to 3–4 Hz and decreases sharply above resonance frequency (circles). The line is the best fit to the equation in the text. Comparison with the speed histogram (bars) confirms that bacterium-surface hydrodynamic interactions are not the cause of the observed resonance. (b) The relative maximal torque magnitude calculated from the ratio of the relative speed as described in the main text, which eliminates the effect of viscosity changes due to temperature as explained in the text. Each data point in the figure was obtained by averaging at least 30 different measurements from several different bacteria. The error bars depict the standard deviation of the different measurements.
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Figure 4
The response of the bacterial flagellar motor's speed to linear change in temperature. The motor's relative rotational speed as a function of time in response to linear temperature change applied at different rates is presented in (a) (the laser was applied as a pulse) and (b) (the laser power was increased over a period of 30 s). Gray dashed lines depict the rotational speed of different bacteria. Black thick lines depict the averages of each set of experiments. The speed was normalized by its value at room temperature. The number of bacteria recorded for temperature increases as a pulse and over the period of 30 s are 46 and 54, respectively. (c) The average relative speed is presented as a function of temperature for both modes of temperature increase. Note that the change in speed is almost linear with temperature in both cases and no deference is detected within our error estimate.
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Figure 5
Spectral analysis of the bacterial rotational speed at steady state. The power spectrum density functions of a tethered bacterium recorded for ∼3 min at a 30 fps are presented. In (a) the measurement was done at 31 °C, and as clearly evident, the rotational speed exhibits a natural mode of oscillation at the same frequency where the resonance was observed in Fig. 3, even though the rotational speed of this bacterium was ∼2 Hz. The power spectrum density function was measured for several bacteria at different temperatures as well, as indicated in each panel of (b). Note that the position of the resonance does not seem to depend on temperature or steady-state angular velocity, and appears to fluctuate around 4 Hz. It is, though, more pronounced for higher temperature. (c) The resonance frequency obtained by fitting the spectra in (b) to the resonance equation [Eq. (1) in the text] is plotted as a function of temperature (left) and average steady-state angular frequency (right). The error bars depict the fit error.
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Figure 6
A phenomenological model. The torque generating units in the motor can be in one of two states, active and inactive, or engaged and unengaged. The transition between the two states depends on the torque being generated by the motor at the time of transition.
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