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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 flotation 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:

Figure 1: Force Diagram (vertical only) of a bacteria in a liquid.

Normally, a bacterium's density is greater than water (E.Coli is about 1100kg/m³, compared to water's 998kg/m³
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-aquae¹
• 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 from Stokes Law²:

Flotation Only

Figure 2: Effect of gas vesicles on the upwards speed of E. Coli, over 24 hours. From left to right, each bacterium has +10% filling (beginning from 0% and ending at 50%) - we've started them off at different distances along the "flask".

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. Taking the x-axis of the giff graph to be in increments of +10% filling, 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, or the vacoule density could be different for our particular organism.

Flotation plus Random Walk

With a behaviour now outlined for out bacteria's flotation, we combined this small model with our previous random walk model. The upwards movement was incorporated by including an additional, simple y component of (0.2um/s), and plotting the path as before. Below is a comparison of the walk, before and after the addition.

Figure 3: Comparison of Random Walk with and Without Flotation Component

By a quick visual comparison, we can see that the addition of the flotation component has made no real change to the pattern of bacterial movement. Upwards movement is not favored. Because of this, there is a real case for the removal of the bacteria's natural propulsion.

Given the low terminal velocity, the gas vesicles as they are here have very little effect on the natural movement. This was to be expected – in nature, bacteria rarely have both locomotion systems, they use on or the other. Thus, it would benefit the project and the aims we had in mind, for the flagellar propulsion system to be removed from our bacteria strains – more on this later.

What this means for the project

Removal of flagella
As mentioned above, the results from this modelling supports the removal of the our bacteria's existing propulsion system, to ensure a solely upward movement is obtained. If the average swimming speed is 20μm/s, and the upwards speed is about ~0.2μm, the bacteria will basically swim as normal. By knocking out the genes, we can ensure all movement is due to the gas vesicles – and can also free up some more of the cells resources. It will take much longer to travel a similar distance, but all the movement will be upwards, as we desire. However, this is not the final solution.

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 0.25μm a second. This is in the very low end of the literature estimate, which is quoted at <2μm/s⁴, and is at a much higher % filling.
For the system to be industrially viable, this speed would have to be drastically increased. As a possible solution, we explored the addition of Ag43.

Ag43
Ag43 is an antigen that, when expressed, causes bacteria expressing it to stick together/clump. 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 20μms^-1).

(Credit to the Aberdeen iGEM Team for bringing this antigen to our attention at the Scottish iGEM meetup.) It was, however, decided that the modelling for this system would be incredibly complex. In our 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. The terminal velocity equation only holds for smooth, spherical objects.

Figure 4: Theoretical and Realistic surface areas of a bacterial sphere

To provide a simple comparison, and to illustrate the effect of radius/diameter on the terminal velocity, we calculated the terminal velocity of 4 theoretical bacterial spheres i.e perfect spheres: 1,10,100 and 1000μm in diameter. The %fill of gas vesicles was kept at a constant 15%, and bacterial size increase due to this was not considered. This left radius as the only changing variable (equation uses radius rather than diameter). The medium was water, and the gas vacoule density 252kg/m³ as before.

Figure 5: Effect of diameter on the terminal velocity of "perfect" bacterial spheres in water medium.

Summary

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.

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 behavior of the bacterium and produce data for others using our system – numerical, graphical and pictorial

References
1. Values for bacteria density
2,4. Walsby, AE. Gas Vesicles. Microbiological Reviews. 1994. Volume 58. p 121&138.http://www.ncbi.nlm.nih.gov/pmc/articles/PMC372955/.(accessed June 2014)
3. Stoke's Law.http://en.wikipedia.org/wiki/Stokes%27_law.(accesses June 2014)