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Section 2: Bacteria and Buoyancy With the main focus of our project being the gas vesicles, we decided to model the effect that they would have on the
bacterium's path.

The net force on the bacterium should be positive for floatation to occur. This force is composed of three components:
The buoyancy force, the force due to gravity and Stokes drag force. They are applied thus:

Normally, a bacterium's density is greater than water (E.Coli is about 1100kg/m^3, compared to water's 998kg/m^3
at 20 degrees Celcius). The presence of gas vesicles replaces some of the mass of the cell with lighter gas,
thus reducing the density.
As with the random walk model, we'll state our assumptions up here.

• The density of a gas vacoule would be 250kg/m3, a value obtained by Walsby in his study of the cyanobacteria anabaea flos-aquae
• It is believed that the bacteria will increase in volume as a result of gas vesicle production. We'll roughly estimate this to be half the % of total volume the vesicles will fill.
• Again, we assume the bacteria is a sphere.

With some preliminary calculations, we decided that the movement would be essentially linear – while there is an acceleration phase at the beginning while the forces are unbalanced, the time it takes to reach a constant terminal velocity is negligible compared to the time for which we're observing the system.

One of the problems with this model is finding the E. Coli mass that will be replaced by the vesicles – we need this in order to find the final density. This was estimated by replacing a given volume (5, 10, 15 etc), with an equivalent (lower density) volume of gas vesicle.

We would use the density and the assumptions above to calculate a new density for the bacteria. This is put into the terminal velocity equation:
The following giff shows the effect of gas vesicles on the upwards speed of E. Coli, over 24 hours.

The gif also compares salt water to fresh. Saltwater has a higher density than fresh, and so the bacteria experience a greater buoyancy force. As we can see, they don't move very fast! This is to be expected, as they
exist on a much smaller scale than us, who are used to seeing things moving in meters per second.

It's generally well known that, for positive buoyancy to occur,
the maximum space the gas vesicles can fill is 10% - any more than this,and they begin to have a negative effect on the cell's protein resources. We see that,in our model, upwards movement was attained at ~12% filling. This is a bit larger than normal, but again could be due to the simplification of the model and density calculations – more of the bacterial mass could have been replaced than we are estimating.

The main problem we see is our speed – it really is very slow! Based on this data, it would take a 50% filled bacteria 24 hours to travel 2cm – a speed of 0.25um a second. This is in the very low end of the literature estimate, which is quoted at <2um/s, and is at a much higher % filling. We are at least confident that we're not out by a large amount.

Based on this, we can probably say our model is reasonably accurate. Inaccuracies are due to the assumptions we've made in the course of the calculations. Each of these assumptions on their own would probably not change the final answer by too much. However, combined, they could very easily account for our discrepancies.

What this means for the project
The main issue we've found with this modelling is the lack of data on E.Coli and how it responds to gas vesicles. Any values for density of gas vacuoles has all been taken from cyanobacteria species, which have been extensively studied. This has definitely given us some direction for the next section of the project, the measuring. It was hoped we would be able to characterise the behaviour of the bacterium and produce data for others using our system – numerical, graphical and pictorial.

The addition of Ag43
This was a possibility that arose very early in the project's timeline. Ag43 is an antigen that, when expressed, causes bacteria expressing it to stick together.
We know on principal that, the larger the sphere, the more buoyancy (and the faster it will go), and it is also well documented that large colonies will travel faster (Walsby gives a value of 20ums-1) For the system to be industrially viable, the bacteria would have to move faster.
It was decided that the modelling for this system would be incredibly complex because in this case, a simple smooth sphere model is far too simplistic. In reality, while the overall shape may be roughly spherical, the surface area will be much larger, due to gaps between the bacteria.
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