Team:Aberdeen Scotland/chemotaxis

Chemotaxis Model

Introduction

When bacteria move in reaction to a chemical this is chemotaxis. A model of this is important to us, as the Picoplumber will consist of a colony of bacteria dispersed in water that swim towards potential leaks marked by chemicals. The aim of our chemotaxis model is to simulate how bacteria respond to a source of chemoattractant diffusing through a virtual pipe from a breach in the pipe inner wall. Using this simulation we can ascertain roughly how long it takes bacteria to swim towards a leak in the pipe, and from this the required density of bacteria in the pipe to seal a hole of known diameter in a given time. A model of thousands of bacteria, each independently moving, is also useful as part of modelling how populations of interdependent gene regulatory networks behave. It is our hope to integrate a chemotaxis model and multiple gene network simulation together in order to check for any emergent behaviour and to see if the system will work.

E.Coli is a motile strain of bacteria, which is to say it can swim. It is able to do so by rotating its flagella, which are rotating tentacle like structures on the outside of the cell. Chemotaxis is the movement up gradients of concentration of chemical attractants (usually a food source) and away from poisons. E. Coli is too small to be able to detect any chemical gradient between the two ends of itself, and so must head in any random direction and then compare the new concentration at a new point to the previous one. Their motion is described by ‘runs’ and ‘tumbles’. If the concentration of food molecule has increased after a run in a certain direction the bacteria will be unlikely to tumble (which is to change direction), and if it has decreased then the likelihood of trying a new direction is very high. The way in which this motion occurs is by altering the direction of flagella rotation. When the bacterium is on a run it is rotating its flagella counter clockwise, and when it is tumbling it is rotating them clockwise. When bacteria respond to a chemoattractant it is always within a range of values, so that there is a minimum and maximum concentration between which bacteria chemotax.

Designing a Model

Chemotaxis in Escherichia Coli, being well studied, has a multitude of papers about the subject, each with different data and parameters. In order to construct a useful model we need reliable, and if possible, accurate data. The paper ‘Motility of Escherichia coli cells in clusters formed by chemotactic aggregation’ [2] is where most of the data used in the model comes from. Another older paper, ‘Chemotaxis in Bacteria’ by Julius Adler [1], examined how the speed of bacteria during each ‘run’ varies randomly around an average value. The paper did not offer a mathematical approximation of this random variation; it simply reported their experimental results and discussed them. Therefore to mimic this variation a random number at each ‘tumble’ is added to a minimal speed such that the average value of this random speed is the average speed of the bacteria. For example, the average swimming speed according to my available data is 27 micrometres per second [3], which in the program is represented by a speed of 25 micrometres per second plus a random number between zero and four. Modelling E. Coli’s behaviour as it swims up a chemotactic gradient is of course limited by inaccurate and unreliable data. The tumbling frequency for example, according to some papers in the seventies is reported to be as high as several hertz, whereas more recent papers have one hertz being the maximum tumbling frequency. One paper, ‘temporal comparisons in bacterial chemotaxis’, informs of the fact that bacteria actually look as far back in time when swimming as 4 seconds, but usually take the previous second as their basis on deciding whether or not concentration has increased or not. It is for this reason I have chosen one hertz to be the maximum rate at which bacteria tumble when moving down a chemoattractant gradient.

Although for one specific strain of E. Coli the motion could be very well studied, the actual strain used for our project may behave differently, as they vary in motility. An example of this would be that some strains of E. Coli swim as fast as 50 micrometres per second. Another example would also be that some mutant strains of E. Coli tumble very infrequently and can check concentrations as infrequently as every 30 seconds. The only way to ensure accuracy and reliability in simulating how our bacteria behave is to actually experiment with our own strain and compare this to the model. The model can then be altered to fit the data, or we could attempt a more in depth analysis to try and find the tumbling frequency behaviour and swimming speed of our own strain.

In order to design a simulation, some simplifications have to be made. In fact, our model has many more simplifications than most. This chemotaxis model for instance neglects the fact that E. Coli do not travel in straight lines, they often take curved paths due to unequal firing of the flagella and almost always there is a slight curvature due to Brownian motion in the fluid. Usually in a static fluid there are minute convection currents due to local temperature variations, these can transport bacteria against their own will, as we have actually seen for ourselves under the microscope. The model also does not consider the fact that E. Coli change in size, and for collisions the bacteria are simply modelled as spheres of 2 micrometres diameter. The tendency of bacteria to congregate into small groups by quorum sensing is also ignored.

Modelling Behaviour

How the tumbling frequency varies as the bacteria move up and down a chemical gradient is hard to define and model exactly. One paper suggests that the frequency of tumbling is roughly one hertz, and decreases to almost zero when moving up a chemotactic gradient. To attempt to model this we have simplified greatly the behaviour, and have tried several different types of behaviour. At every third iteration, which is one second, the bacteria compare the current concentration at that point to the concentration at the previous point, and if the concentration has decreased then the bacteria tumbles:

   if 1>c(t)/c(t-1)    
       Then tumble      
   end

The first behaviour model was to simply program the bacteria to have a probability of 60% (or any other percent) of not tumbling when moving up a gradient and 100% when moving down a gradient described by the statement where rand is a random number:

   if 1>c(t)/c(t-1)    
       Then tumble      
   elseif 0.6<rand    
       Then tumble
   end      

However it is known that behaviour will vary depending on how much the difference in concentration is between current and previous points. If the ratio of current to previous is greater than one then the probability of tumbling is between roughly 20% and 40%, expressed by:

   elseif (0.6+(p1(t)/10))<rand    
       Then tumble      

Where p1(t) = c(t)/c(t-1), the ratio of current to previous concentration. If the statement to the left is less than rand then the bacteria tumbles. The denominator of the fraction p1(t)/10 and the base probability of 0.6 have both been chosen with no scientific grounding. If actual data was obtained about our strain of bacteria, the parameters could be tuned to fit experimental findings.

Another behaviour that we used to model the decrease in tumbling frequency is having the probability of not tumbling linearly increase with the ratio of current to previous concentration:

   elseif 0.6*p1(t)<rand    
       Then tumble      

And another method to make the behaviour such that tumbling frequency is even less when travelling up a gradient is a combination of the two previous ones:

   elseif (0.6*p1(t) + (p1(t)/10))<rand    
       Then tumble     

And another method is to raise the ratio of current to previous concentrations to a power of two or three:

   elseif 0.6*(p1(t)^3)<rand    
       Then tumble      

The best way to illustrate how these different behaviours look is to show with the below animated gif where the bacteria are given one hundred seconds to travel 8000 micrometers towards a source of chemoattractant (although the speed with which each bacteria move is multiplied by six for the sake of making it easier to see them move):

In the above video, the colours correspond as follows to the probability of not tumbling as: Green = 0.6, Cyan = (0.6+(p1(t)/10)), Red = 0.6*p1(t), Black = 0.6*p1(t) + (p1(t)/10), Blue = 0.6*(p1(t)^3).

As you can see in the video, the speed at which they travel is very similar, the red and green colonies however do not congregate as tightly around the source of food as the others do.

Another useful comparison of behaviours is to look at how the bacteria respond to the parameters, namely the basic probability of not tumbling, chosen as 0.6 for our model. In the below animated gif the following colours have the following probabilities of not tumbling: Green = 0.4, Cyan = 0.5, Red = 0.6, Black = 0.75, Blue = 0.9.

As you can see, the blue colony finds the source far better than the others. The green and cyan colonies however find it hard to even locate the source in the given time. The red colony which corresponds to our choice does show some bacteria unable to find the source quickly; however in our experience of watching E. Coli under the microscope, some bacteria do take more time to find the source than others.

Exactly which probability to use here is not an easy choice, although through experiments in the lab we hope to be able to decide upon a value with confidence.

Another parameter is how often the bacteria decide to check the concentration gradient. Bristol last year used the figure of bacteria checking their concentration every three seconds, whereas after reading the literature and comparing papers we have decided to use a figure of once per second. This is because the tumbling rate in other papers seemed to be around one hertz, or sometimes higher, and if they only checked their concentration every three seconds this level of tumbling could never occur. In the below animated gif the following colours correspond to the following times for how often the concentration is checked: Green = 3 seconds, Cyan = 2 seconds, Red = 1 second, Black = 2/3 seconds, Blue = 1/3 seconds.

As the bacteria are attracted to a chemical, in our case aspartate ions, which diffuses from a source into the surrounding water, modelling this diffusion is also important. As the diffusion in all our cases is from what can be considered a point source and equal in all radial directions, the following solution to fick’s law can be used:

In the chemotaxis model this is used to calculate the concentration at any time in any location. Here however is where another series of simplifications and assumptions are made. The above solution of fick’s law is for a static fluid in which there exist no eddy currents. As there will be small temperature fluctuations there will also be convection currents mixing the fluid, and as the bacteria themselves swim around they will be further mixing the fluid. Estimating the eddy diffusion coefficient is very difficult to do accurately. For the purposes of the chemotaxis model in most cases eddy diffusion has been ignored as the fluid is for all intensive purposes static. However in the case where we were performing experiments, a greater level of turbulence and mixing exists as samples are taken from a channel of water every five minutes.

E. Coli is only capable of sensing our signalling chemical, aspartate, at levels above 10-8 M according to ‘Chemotaxis in Bacteria’. This has been accounted for in the model and can be shown by the below video in which aspartate is diffusing out from a source in the middle of the bacteria and the concentration of 10-8 M is represented by the black circle:

Our model, unlike many chemotaxis models is not agent based, and so collision behaviour of correctly shaped E. Coli cells has not been included. In some of the models several lines of code which cause the bacteria to tumble when a collision occurs has been included. However when running these programs we tried to count how many bacterial collisions occurred, and the frequency is surprisingly low. The code which detects collisions and changes the behaviour is extremely intensive computationally for MATLAB when dealing with thousands of bacteria and has little impact on the behaviour. The model is mostly interested in how long it takes E. Coli to travel through mostly empty space to our target source of chemoattractant, and not how inter-bacterial collisions affect a tight quorum of bacteria.

References

[1] Julius Adler. “Chemotaxis in Bacteria”. Annu. Rev. Biochem. 1975.44:341-356.

[2] Nikhil Mittal, Elena O. Budrene, Michael P. Brenner, and Alexander van Oudenaarden. “Motility of Escherichia coli cells in clusters formed by chemotactic aggregation”. PNAS. Vol 100. no 23. Nov 11, 2003.

[3] Kayo Maeda, Yasuo Imae, Jun-Ichi Shioi, and Fumio Oosawa. “Effect of Temperature on Motility and Chemotaxis of Escherichia coli”. JOURNAL OF BACTERIOLOGY, Vol. 127, No. 3 Sept. 1976, p. 1039-1046

[4] Jeffrey E. Segall, Steven M. Block, and Howard C. Berg. “Temporal comparisons in bacterial chemotaxis”. Proc. Nati. Acad. Sci. USA Biophysics Vol. 83, pp. 8987-8991, December 1986