Mathematical modelling and analysis of bacterial motility

In this post Gabriel Rosser, who completed his D.Phil. at the WCMB in early 2013, describes his research in bacterial motility.

Dr Gabriel Rosser

Bacteria have evolved myriad means of transporting themselves from one location to another. Twitching, gliding, floating and swimming are all methods seen in nature in some bacterium or other. Presumably the different mechanisms have evolved as different optimal solutions to the various needs of bacteria in their natural habitats, which include foraging for nutrients, escaping potentially harmful toxins and searching for new habitats to colonise. This is of more than simply academic interest: colonies of bacteria are responsible for life-threatening infections in humans, and cause billions of pounds of damage annually in industry by biofouling.

During my D.Phil., I focused on one particularly prevalent form of motility: swimming through a liquid medium mediated by one or more flagella. This is exemplified by many of the model bacterial species studied in recent years, including E. coli, R. spaeroides and P. aeruginosa. In many flagellate bacteria, motility proceeds by a series of approximately straight-line movements, interespersed by reorientation phases. Discussions with experimental collaborators from the Armitage lab (Department of Biochemistry) early on in my project indicated that experimental methods had progressed to the stage where it is possible to gain large amounts of microscopy video data from bacteria swimming in a glass capillary. Some example footage is shown below.

This process results in the availability of rich datasets containing tracks derived from bacteria swimming in a controlled medium, which is of great value in addressing many of the open questions about bacterial motility, such as the mechanism of reorientation, the effect of viscosity on swimming and how bacterial motility changes close to a surface. However, the limiting factor was the ability to analyse these tracking datasets meaningfully. I first turned my attention to this problem, developing novel Bayesian statistical analysis methods to automatically distinguish reorientation phases in the tracks [1]. This made use of a recently developed multitarget tracker based on a PHD filter [2]. The same example footage as above, this time overlaid with computed cell tracks, is shown below.

A second area of interest is in the optimisation of experimental methods. This is a particularly important area, since it may have a significant impact on the effectiveness of an experimental technique. This is an area of research that benefits from the application of mathematical models; it is neither practical nor economical to test too many variants of a given experimental protocol, yet the application of an appropriate model of the process allows us to consider the impact of a wide variety of parameters, with no restriction on the number of combinations we may test. I used a model of bacterial motion known as the correlated random walk to test the effect of the microscope camera sampling frequency on the observed process [3]. Tracks that are captured with a low sampling frequency will not be sufficiently detailed (see Figure 1). In this study, I showed that the apparent distributions of speeds and angle changes in bacterial tracks vary significantly with the sampling frequency. In particular, when finite duration stationary reorientation phases are included in the model of motion, the effects become more severe.

Figure 1: The effect of sampling on apparent motion. The lower track is simulated using a correlated random walk model of bacterial motion, in which bacteria are assumed to reorientate instantaneously. Circles indicate reorientation events, crosses indicate sampling points and two run lengths and a turning angle are shown. The upper panel shows the circled portion in greater detail. This is the observed track, with two apparent displacements and two apparent angle changes marked. Details of the notation used can be found in [3].

Finally, I considered the effect that Brownian motion has on bacterial motility. It has long been known that bacteria are sufficiently small that their movements are affected by the buffeting motion of the molecules in the surrounding liquid, characterised by translational and rotational diffusion of the bacterium (see Figure 2). However the exact nature of this effect is not fully understood.

Figure 2: Illustration of a single bacterium undergoing translational (left) and rotational (right) diff usion. Increasing transparency represents position and orientation in the more distant past. Dashed lines trace the trajectory of the cell centroid (left) or a point on the flagellum to show the angle changes (right).

Modelling a bacterium as a 'self-propelled particle' subjected to Brownian rotational buffeting, I interrogated the tracking data discussed previously. The model agrees well with the tracks of a bacterial mutant that does not exhibit reorientation phases. Considering tracks from a bacterium that does exhibit such reorientations, however, indicates that the reorientation mechanism cannot occur passively - the bacterium wouldn't change direction sufficiently if it simply waited for Brownian rotation to do the hard work. Based on these findings, I proposed several possible phenomenological explanations for the discrepancy, and demonstrated that a small change to the model of motion that incorporates an active reorientation mechanism agrees much better with the data [4].

References

1. G. Rosser, A.G. Fletcher, D.A. Wilkinson, J.A. de Beyer, C.A. Yates, J.P. Armitage, P.K. Maini and R.E. Baker (2013). Novel methods for analysing bacterial tracks reveal persistence in Rhodobacter sphaeroides. PLoS Comput. Biol. 9: e1003276.
2. T. Wood, C.A. Yates, D. Wilkinson and G. Rosser (2012). Simplified multitarget tracking using the PHD filter for microscopic video data. IEEE Trans. Circuits Syst. Video Technol. 22:702-713.
3. G. Rosser, A.G. Fletcher, P.K. Maini and R.E. Baker (2013). The effect of sampling rate on observed statistics in a correlated random walk. J. R. Soc. Interface 10:20130273.
4. G. Rosser, R.E. Baker, J.P. Armitage and A.G. Fletcher. Modelling and analysis of bacterial tracks suggest an active reorientation mechanism in Rhodobacter sphaeroides. Submitted.