Interdependence of behavioural variability and response to small stimuli in bacteria

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Original Entry: Peter Foster, AP 225, Fall 2011

General Information

Authors: Heungwon Park, William Pontius, Calin C. Guet, John F. Marko, Thierry Emonet & Philippe Cluzel

Publication: Park et al. Interdependence of behavioural variability and response to small stimuli in bacteria. Nature (2010) vol. 468 (7325) pp. 819-U114

Keywords: chemotaxis, fluctuation-response theorem, signalling network, Escherichia coli

Summary

The goal of this paper was to test whether or not a connection could be made between between natural variability and stimulus response in cells. This was examined using the E. coli chemotaxis network. The chemotaxis network regulates cellular movement by controlling whether flagella rotate in the clockwise (CW) or counterclockwise (CCW) direction. When the flagella rotate in the CCW direction, they form a bundle and propel the cell forward while rotation in the CW direction causes the bundle to unravel and randomly orients the cell. The signaling network serves to regulate the probability that the motors rotate in the CW vs. the CCW direction in response to external stimuli. Because of the stochastic nature of signaling events, there is a variability of the natural CW bias between different cells in the steady state (Figure 1a). Furthermore, for a given input stimulus, the time spent in response to the stimulus varies between cells. In this paper the connection is drawn between this variability and response.

Figure 1, taken from [1].


First, the output in response to a stimulus was measured. A small (10nM aspartate) stimulus was applied and the amount of time spent in CCW rotation immediately following the stimulus was measured (Figure 1b, black dots) and plotted against the initial pre-stimulus CW bias. The amount of time in the second successive CCW rotation immediately following the stimulus was also measured (Fig 1c). The black line in figures 1b and 1c shows a fit to the steady state behavior in absence of the stimulus. What's important is that the response is small and quickly relaxes back to steady state behavior. The grey triangles are from a simular experiment that used a much larger stimulus (1um aspartate). The response to this larger stimulus also relaxed back to near pre-stimulus behavior by the second CCW rotation. To quantify the response time, the lengths of time of CW and CCW rotation were measured after the stimulus was applied. The response time is defined as the sum of the post-stimulus CCW rotation times that were longer than the mean pre-stimulus response time + the time of the CW rotations in between - the mean pre-stimulus CCW time.


The behavioral variability can be defined in terms of a noise amplitude. A time series of switching times between CW and CCW rotation was used to compute the power spectral density which was integrated to obtain the amplitude of the noise. There is a bit of subtlety here, in that the noise carries contributions from two different sources, the spontaneous noise associated with signaling events and the stochastic switching behavior of the bacterial motor which needed to be (and was) decoupled. Figure 2 shows a plot of the response time vs. the pre-stimulus noise from signaling events. To generate the response, the small (10nM aspartate) stimulus was used.The letter on each point corresponds to cells from the bins in figure 1(a) and averaging was done in each bin. The response time was found to have a linear relationship with the pre-stimulus signaling noise with the linear fit (forced through the origin) having R2 = 0.8 . The inset of the plot shows the same thing, but for the larger (1um aspartate) stimulus. The linear fit for the larger stimulus was less appropriate, with the fit having R2 = 0.4 . Thus, in this system, the response is linear with the signaling noise only for the small stimulus.



Figure 2, taken from [1].


Discussion

Time and time again, the fluctuation-dissipation theorem has proven to be a useful tool to analyse systems. However, the fluctuation-dissipation theorem has the disadvantage that it only applied to systems in thermal equilibrium. From a biological perspective, this is bad because most systems in biology are very far from being in a state of thermal equilibrium. However, the fluctuation-dissipation theorem can be extended to the fluctuation-response theorem , which can deal with non-equilibrium states provided that they can quickly relax to some steady state, have Markovian dynamics, and the perturbation is small. This is the case for the experiment in this paper. To me, what's interesting about this experiment is that it uses a biological system as a testbed for an analysis from statistical mechanics. As a consequence, we not only gain some experimental validation for the fluctuation-response theorem but we gain interesting insight into the biological system itself.

References

[1] Park et al. Interdependence of behavioural variability and response to small stimuli in bacteria. Nature (2010) vol. 468 (7325) pp. 819-U114