In the course of our project, we mainly focused on obtaining the flagellar force, which is associated with the parameter A.
During the model fitting process, we modelled the motility of B. subtilis to have a maximum of two runs. Modelling the a cell with only a single run is relatively straightforward as it only generates a single set of parameters. When modelling a cell with two runs, we keep the value of parameter alpha constant since it is the ratio of viscosity to mass. We then let the final velocity of the cell's first run to be equal to parameter B, the initial velocity of the cell's second run. The parameter which is allowed to change is parameter A, the ratio of flagellar force to viscosity of the medium.
The following figure shows the results of our model fitting for few trajectories. We have introduced a change in flagellar force at certain points of the cell trajectory so as to achieve a better fit. The data fits our model very well, sometimes surprisingly so, given the simplicity of our model. However, some results also suggest that a third or a fourth change of flagellar force would contribute to a better fit.
In general, the choice of the change of flagellar force is a very complex problem. Automatic detection can certainly be developed but we lacked time to look into such a solution. Instead we opted for the highly subjective manual detection of these changes.

Fitted Models with Experimental Data from 4 Cells

The MATLAB Distribution Fitting Tool was used to model the distribution of the amplitude of parameter A.
The amplitude of parameter A is very interesting since it represents both the flagellar force and the velocity of the cell after a sufficiently long time has elapsed. The following figures describe the probability density function and cumulative density function.
The distribution was found to be close to an exponential distribution. This is rather surprising as we would have expected a Maxwell distribution instead, similar to how the velocities of gas molecules are distributed. This could be due to the practical difficulty in determining with accuracy the changes of flagellar push and the consequent loss of accuracy in the extraction of the model parameters.


Probability Distribution for Parameter A
 Cumulative Distribution for Parameter A

