Team:Imperial College/Motility

As part of our chasis characterisation process, we have decided to model B. subtilis motility. In order to do this, the approach illustrated below was taken. The first phase of modelling involved data collection using microscopy techniques and cell tracking. Collected data was then analysed using algorithms which enabled us to extract distributions of parameters as defined in our model.

Materials

We used the Zeiss Axiovert 200 inverted microscope and Improvision Volocity acquisition software. This system offers a full incubation chamber with temperature control and a highly sensitive 1300x1000 pixel camera for fast low-light imaging. Video images are captured into memory by the system at a basal video frame rate of 16.3Hz. This can be further increased to 27.9Hz by performing x4 binning. A short video of swimming B. subtilis is shown:

Method

In order to choose suitable tracking software, we generated a synthetic video and applied tracking algorithms to the data. We then assessed the reliability, validity and errors associated with the various tracking methods. We chose manual tracking as our method of tracking due to its high reliability and acceptable error.

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We manually tracked motile B. subtilis, obtaining two-dimensional coordinate data points which describes by the trajectory of the cells. The open source tracking software can be found [http://rsbweb.nih.gov/ij/plugins/track/track.html here].

Data Extraction

The coordinate data obtained was then fed into algorithms to model cell trajectory and motility. Algorithms used to extract motility data and fit cell trajectory data to models can be found in the appendices

A simple mechanical model was developed by taking an analogy of B. subtilis propelled by its flagellum with that of a boat propelled by its motor. As the bacteria swims in the medium, it experiences two opposing forces: the flagellum force and drag force provided by the liquid medium. The drag force is analagous to friction which opposes the forward motion of objects on solid surfaces. The mechnical model is illustrated below:

From the figure above, we equate the drag force and flagellum force to obtain:

Solving this first order ODE, we derive an expression for cell velocity:

, where

Solving the first order ODE for displacement, we derive an expression for cell trajectory:

Using our fitted data, we are able to determine parameters:

In this model, we attempt to obtain a distribution for the flagellar force, which is represented by parameter A. We assume that the medium is homogenous and the drag coefficient is constant throughout the medium, hence the distribution of flagellar force will be sufficiently be described by parameter A.

During the model fitting process, we modelled the motility of B. subtilis to have a maximum of two runs. We keep the value of parameter alpha constant since it is a ratio of the drag coefficient to the mass. We then let the final velocity of the cell's first run to be equivalent to parameter B the initial velocity of the cell's second run. Hence the parameter which is allowed to change is parameter A, the ratio of flagellum force to drag coefficient of the medium.

The following figure shows the results of our model fitting. We have introduced a change in flagellar force at certain points of the cell trajectory so as to achieve a better fit. A maximum of two runs were allowed for each cell trajectory.

The MATLAB Distribution Fitting Tool was used to model the distribution of parameter A. Parameter A was found to be exponentially distributed. The following figures describe the probability density function and cumulative density function.

From our model fitting process, we can see that flagellum force is exponentially distributed. Our mechanical model though simple, fits the cell trajectory data extremely well as shown in the figure above. Further work which can be done would be to utilise a movable stage to track the movement of B. subtilis over its entire run so as to obtain a distribution of other motility parameters associated running and tumbling events.