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Molecular model
From 2009.igem.org
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 greatly simplified the behaviour. Our simplifications however, are not as extensive as some models which simply have a probability of 0% of tumbling when moving up a chemotactic gradient.
Firstly, as an introduction to how the model works, we must understand how the bacteria decide to tumble when moving away from a source of chemoattractant. At every second, which is every third iteration, the bacteria compare the current concentration at that point to the concentration at the previous point, and if it has decreased then the bacteria tumbles:
In the previous described motion the bacteria will seek out the source of chemoattractant very well. However bacteria do still tumble when moving up a concentration gradient, and so this must be accounted for. The most simple way of describing this is giving the bacteria a probability of X% (with X < 100) of not tumbling when moving up a gradient and 100% when moving down a gradient described by the statement:
However, it is known that the behaviour varies depending on the difference in concentration between the current and the previous locations. If the ratio of these concentrations is greater than one, then the probability of not tumbling is roughly between 60% and 80%, expressed by either the following possible statements:
In order to experiment and to look at how various functions cause varying behaviour, we used many different functions like above for the probability of not tumbling. To illustrate these different behaviours the embedded video below shows bacteria that are given one hundred seconds to travel 8000 micrometers towards a source of chemoattractant - although the speed with which each bacterium moves is multiplied by six for the sake of making it easier to see them move:
In the above video, the colours correspond to the probability of not tumbling as follows (where p(t) = c(t)/c(t-1) and a probability of 1 = 100% and 0 = 0%): Green = 0.6, Cyan = (0.6+(p(t)/10)), Red = 0.6*p(t), Black = 0.6*p(t) + (p1(t)/10), Blue = 0.6*(p(t)^3).
As it can be seen in the video, the speed at which the bacteria travel is very similar. The red and green colonies, however, do not congregate as tightly around the source of food as the others do.
All the data and probabilities of tumbling when moving up a chemotactic gradient have been chosen without scientific grounding except educated guess-work after reading through the literature. If actual data could be obtained about our strain of bacteria, the parameters could be tuned to fit experimental findings. Therefore, another useful comparison is to look at how the bacteria move depending on the basic probability of not tumbling, chosen originally as 60% in our model. In the embedded video below the following colours have the respective probabilities of not tumbling: Green = 40%, Cyan = 50%, Red = 60%, Black = 75%, Blue = 90%.
As it can be seen, 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. In our experience of watching E.coli under the microscope, however, 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. Last year's University of Bristol team 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 embedded video below 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 diffuse from a source into the surrounding water - modelling the 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:
Where x is the radial distance from the source, t is the time since diffusion began and erfc is the complementary error function, D is the coefficient of diffusion and C(0) is the concentration at the source at time zero. The diffusion coefficient of the aspartate ion, or aspartic acid, was calculated from [5] as the diffusion coefficients of the amino acids correlate strongly to their molecular mass. The concentration at the source is assumed to be the saturation point of aspatic acid, as the boundary layer of water closest to the dissolving solid would be saturated.
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’ [1]. This has been accounted for in the model and can be shown by the video below in which aspartate is diffusing out (without eddy diffusion and with data from [5]) from a source in the middle of the bacteria. The black circle represents the point where the concentration is 10-8 M, and everything inside it is greater than 10-8 M:
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 our 2-d model tumbling of bacteria due to collisions and stopping of overlap are included. However if we implement this effect and count how many bacterial collisions occur, the frequency is surprisingly low. The model which detects collisions and changes the behaviour is extremely computationally intensive for MATLAB when dealing with thousands of bacteria and has little impact on the behaviour. The model is mostly concerned with 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.
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