Motility of bacteria

Figure 1 : Shape of the chemotactic response function of E.coli to glucose, measured using our novel chemotactic inference method.
      Noise for bacteria takes the form of Brownian motion, deviating their course of motion in typical times of the order of a few seconds, and molecular noise, perturbing the measurement of local concentrations of chemoattractants (see HC Berg, E. coli in Motion, Springer 2003). This makes that chemotactic bacteria lack the sense of position, that they cannot keep a straight course and have a relatively short time to accumulate enough detections for a reliable measurement of local gradients. The advantage, as compared to macro-organisms, is that a local gradient of concentration does exist and it provides, if properly extracted from the noise, a reliable cue to climb to regions of higher concentration. Unicellular organisms have developed sophisticated strategies to infer strength and orientation of gradients of chemoattractants, inferred by comparing intensities of the stimulus, either at different locations or at different times. The size of bacteria is typically smaller than eukaryotic cells and biophysical constraints make that they tend to assay gradients in time rather than across the cell (Berg and Purcell, Biophys J 20, 93-219 1977). The ascent of gradients of chemoattractants is then realized by regulation of the tumbling frequency. Tumbling is a scattering event, with the bacterium briefly halting its run and restarting in a direction different than that of income. A tumble involves the change in direction of one or more of the flagella propelling the bacterium and a sequence of changes in their handedness and pitch (Turner et al., J. of Bacteriology 182 2793-2801 2000). The frequency of tumbling is reduced if detections make the bacterium estimate that its run is up-gradient. The consequence is that runs in “good” directions last longer and a net flow results. Quantitatively, the frequency of tumbling at time t is regulated by the convolution
 of past bacterial detections n(t-s) of the attractant with the response function (kernel) K(s). At the molecular level, the response function is shaped by the molecular processes of (de)phosphorylation and (de)methylation. The response to any chemotactic stimulus (not saturating chemoreceptors) is predicted from the function K(s), which justifies its fundamental importance.

     Experiments to measure the chemotactic response function are realized using the classical technique (Silverman & Simon, Nature, 249, 73-74, 1974) where flagella are tethered to a glass slide via an antibody to flagellin that sticks to glass. Flagella cannot rotate, whilst counter-rotation of the bacterium is visible at the microscope. This occurs when only a single flagellum is tethered and E. coli is pre-treated by different means to reduce it to a mono-flagellated state. The statistics of rotations clock-wise or anti-clockwise (corresponding to phases of run and tumble) are thus measured in response to different stimuli. The tethering technique is valuable and commonly used (Paster and Ryu, PNAS 105 5373-5377 2008). However, previous description makes it clear its intrusiveness and tethering of the bacteria is preventing any observation of their motion and correlation with the response function K(s). Furthermore, work by (Turner et al., J. of Bacteriology, 182, 2793-2801, 2000) shows that tumbling events are not in one-to-one correspondence with changes of directions of individual flagella, which is what the technique assays. An alternative technique escaping these limitations would be highly valuable. This is the project that we undertook in 2008 and preliminary results for the response K(s) of E. coli to glucose (a strong chemoattractant) are shown in Fig. 1 The biphasic shape with two lobes having roughly an equal area qualitatively agrees with classical results on the response to aspartate (Block et al., Cell, 31, 215-226, 1982; Segall et al., PNAS, 83, 8987, 1986). Bacteria weight positively detections in the second past the current time, while detections further backwards in the past are given negative weight. In view of its importance for the function of motility, one expects the response function K(s) to be under selective pressure. Some works have tried to rationalize the shape observed in Fig. 1(Schnitzer, Phys. Rev. E, 48, 2553, 1983; Strong et al., Phys. Rev. E, 57, 4604, 1998; Clark & Grant, PNAS, 102, 9150, 2005; Kafri & da Silveira, 2008 Phys. Rev. Lett.100 238101 2008). However, arguments are rather ad hoc, e.g. they postulate two lobes of equal area. This is unsatisfactory as, e.g., no such property holds for the response to temperature (Paster and Ryu, PNAS 105 5373-5377 2008). It has also been pointed out (de Gennes, Europ. Biophys. J., 33, 691, 2004) that a mean flux up-gradient does not require two lobes and it is in fact maximized by a single positive lobe. Conversely, the density of bacteria at the steady state is maximal at locations of highest chemoattractant concentrations for a response with a single negative lobe. A principled way to compromise between the two previous opposite tendencies is clearly lacking and cannot be reduced to a fitting issue. Similar conflicting requirements arise in fact quite often in optimization arguments about the architecture of signal transduction pathways.

Our objectives and projects for bacterial chemotaxis

Inferring the chemotactic response from images of swimming bacteria

     The set-up of the chemotactic inference method is as follows. We prepare a well-controlled profile of attractant by linking two large reservoirs, only one of them containing attractant, with a canal (a few mm long) and letting the attractant diffuse so as to reach a steady profile, monotonously increasing from the entrance towards the end of the channel. Bacteria are then injected at the entrance of the channel and will swim up the gradient. We record their trajectories in a field of about 650μm2 by a fast camera placed in the middle of the channel. Parameters are chosen so as to ensure that many events of tumbling are recorded in the field of vision and a sizable gradient of concentration is measured along bacterial runs. It is then possible to infer parameters of motion writing down the expression of the likelihood of the trajectories and inverting it to obtain the posterior distribution of the parameters of the model given the data, i.e. the trajectories. We have recently successfully applied a very similar scheme [1] for the inference of maps of forces in microdomains of cell membranes. In the absence of any attractant, the unknown parameters are the tumbling frequency and the angular diffusivity of bacteria. In the presence of the aforementioned controlled field of attractant, we can extract the parameters of the response function from the biased trajectories of the bacteria. Posterior distributions for the set-up that we have described are well peaked and allow making sharp estimates of the various parameters. To give a concrete sense of the value of the method, we show in Fig. 1 preliminary data for the chemotactic response to glucose of E. coli, obtained using our method. The major advantage of our method is that it is non-intrusive and directly assays the tumbling of bacteria. The project was supported in its initial stage by the CNRS program "Interface physique, biologie et chimie" and has reached an advanced stage. The methodology just described will be employed to measure the chemotactic response of E. coli and S. typhimurium to different chemoattractants, in particular comparing those that have different metabolic and uptake rates. The non-intrusive character of the method opens up the possibility of correlating the shape of the response function to swimming properties of the bacteria, which could not be done thus far. We shall also realize selective speed race experiments to analyze the genetic/epigenetic nature of the chemotactic response trait. Finally, we shall measure the diversity of the chemotactic response function in a bacterial population.

Measuring the chemotactic response at different densities of the surrounding colony

     We shall analyze the issue if and how E. coli regulates its runs as a function of the density of its surrounding colony. This is natural to ask as anybody who observed a typical colony entering in stationary phase (when swimming is mostly active), certainly remarked its high density. Since the density of a colony is not easy to control, we prefer to consider an artificially constrained microfluidic environment. The idea is to construct sort of a porous medium, measure the flux of bacteria (equivalent of Darcy law) and investigate the dependency of the permeability on the density of obstacles in the medium. The dependency should inform us about the runs that the bacterium is making and whether they are regulated to better zigzag. In practice, obstacles are micropillars of suitable aspect ratio, the construction of which will require going beyond typical soft lithography and using ICP-RIE or wet etching, possibilities that we are currently exploring. The first option appears to be viable. Hydrodynamic attraction to surfaces, deformation of the chemoattractant field due to pillars and the chemotactic response of E. coli will also be combined to model and predict the bacterial flux observed in the experiments.

Theoretical models for the bacterial chemotactic response

     As we stated previously, arguments proposed up to now for the functional and evolutionary logic of the bacterial chemotactic response functions in Fig. 1 are rather ad hoc, e.g. they postulate two lobes of equal area. This is unsatisfactory for at least two reasons. First, a single lobe is sufficient (and even more effective) to produce a mean flux of bacteria climbing a concentration gradient. Second, two lobes of different areas are experimentally measured for the chemotactic response to temperature. The reasons to evolve a response function with two lobes of roughly equal area (zero integral) are not understood and this is one of our goals. The model that we are developing provides an answer to this question, gives an excellent fit to the curves in Fig. 1 and, most importantly, provides further predictions to be tested in experiments. The model is based on the idea that the process of nutrient acquisition can be assimilated to a game against the rest of the colony. The choice of a chemotactic response function corresponds to a strategy played by the bacterium in this game. The experimentally observed chemotactic response is reproduced extremely well by the strategy that maximizes the minimum uptake of chemoattractant by the bacterium in any chemoattractant profile. This leads naturally to the condition of zero integral, without any ad hoc hypotheses. In other words, single-lobe responses are adapted to particular profiles of chemoattractants but perform poorly in others. Conversely, two lobes with equal integral ensure the best generic response, without any prior knowledge of the chemoattractant profile. In the language of game theory, the conflict specialists vs generalists is recast as MiniMax vs MaxiMin strategies.