In order to model the chemotactic motion of the engineered bacteria, a simulation program was designed and produced. The simulation models the motion of a population of bacteria in a small environment composed of two chemoattractants. This page provides an overview of the simulator followed by an in-depth discussion of the mathematical model used [1,2]. The inputs to the model and the possible outputs are highlighted. An analysis of the assumptions made in development, and resulting shortcomings of the tool is also conducted.

Overview

The simulation tool, as shown in the alongside figure, models the chemotactic motion of a population of bacteria in a small environment (a petri dish). As in the project setup, two chemoattractants (shown in red and green) are present in the environment and influence the motile action of the bacteria. The representative bacteria take the colour of the chemical towards which they are attracted, and move about the environment according to a movement algorithm. Various options are available to the user of the tool to specify the experiment setup (see later) and the panel on the left allows for this.

The tool may be viewed at this link. It is necessary to install the Unity Web Player to run the application.

Development Tools Used

The simulator was developed in the C# language and utilises the Unity3D graphics engine. This engine was chosen as a development tool above other platforms such as MATLAB or Java for a variety of reasons. Firstly, the engine is free to use and supports different development environments including easy web deployment. Secondly, the ease of implementation allowed the developer to prototype and deploy iteratively for the team. Functionality such as graphical user interface integration was provided and allowed more time to be dedicated to the modelling of the bacterial motility.

The Movement Algorithm

Chemotaxis in E. coli bacteria is achieved by the altering of a signalling pathway to the bacterial flagellar motors [1]. The flagella of the bacterium are capable of both clockwise and counter-clockwise motion. The former leads to a reorientation of the bacterium: a tumble [1,2]. The latter leads to a fairly straight propulsion of the bacterium through the medium: a run [1,2]. By altering the frequency of tumbles and runs, the bacterium effectively undergoes selective locomotion towards more favourable regions. This so-called biased random walk [2] is simulated in the program with a two-state model.

Effectively, each bacterium is associated with the particular chemoattractant that dictates its motion (specified by the functioning of the riboswitch). The diffusion of this chemical is simulated by the inverse square law; the point concentration decreases with time, whilst the spread (or radius) of the chemical increases. The first state of the model occurs when a bacterium is outside this chemical radius. In this instance, a bacterium’s movement is dominated by tumbles. The second instance is when the bacterium lies within the chemical concentration gradient. In this case, the frequency of runs is higher and dictated by the distance between the bacterium and the chemical point-source. Again, an inverse square relationship was used.

Inputs and Outputs

The simulator allows users to specify the experimental setup. Alterable variables include:

• The number of bacteria to simulate
• The bacterial speed
• The bacterial sensitivity to the chemoattractants
• The initial concentration of attractants
• The diffusion rate constant of the attractants
• The distribution grid resolution and size (see below)

The simulation program models the bacterial motion and creates useful measurements of the scenario. These include:

• A discreet probability density function of bacterial location (as in the figure below)
• The maximum and average times of flight it took a bacterium to reach each attractant

Verification

In order to verify the model, it was necessary to determine the extent to which the directed chemotaxis varied the bacteria’s motion from that of random movement. To this end, the number of bacteria in the toggled state (after coming into contact with the chemoattractant) over time, as a function of the initial chemical concentration was determined. A number of runs of the simulation tool under the same initial chemical concentration were performed. The simulator output the number of toggled bacteria sampled every 2 seconds. These data were then averaged to obtain a statistical mean of the particular experiment. This process was repeated over a range of concentration values, and plotted as below:

As may be seen from the above figure, the initial chemical concentration does indeed have an effect on the rate at which bacteria successfully navigate to the attractant. The chosen arbitrary reference concentration sees an almost 10 fold increase of the number of bacteria in the toggled state over the chosen 2 minutes of simulation time. Even at half of the reference concentration, a six fold increase over pure random motion was obtained. Concentration increases above 10 times the reference concentration had increasingly minimal impact on the chemotaxis of the bacteria. This is due to the overall underlying stochastic nature of the movement and physical distance between where the bacteria are initially located and the chemoattractant.

Collaboration

The modelling software was discussed with the Imperial College team. Specifically, the use of a non steady-state model was discussed and ultimately decided upon due to the team's implementation of the environment. A non steady-state model is used since the chemoattractant is not replenished with time, and so will diffuse uniformly over the entire environment.

Shortcomings

The two-state model used to simulate the bacterial motion neglects the half-life and phosphorylation times of the signalling pathways. The three-state model of [2] could be used to provide a more accurate model. However, it is assumed that these times are negligible when compared to the time of flight of the bacteria. Larger chemical networks consisting of three or four attractants (or the inclusion of repellents) are also not catered for. As of yet, the variables in the simulator are also unitless. More detailed investigation into the underlying model is needed to fully characterise them.

[1] P. Spiro, J. Parkinson, H., Othmer, A model of excitation and adaptation in bacterial chemotaxis, Proceedings of the Nation Academy of Sciences, United States of America, Biochemistry, Vol 94, pp. 7263 – 7268, July 1997.

[2] C. Rao, J. Kirby, A. Arkin, Design and Diversity in Bacterial Chemotaxis: A Comparative Study in Escherichia coli and Bacillus subtilis, PloS Biology, Issue 2, Vol 2, pp. 239 – 252, February 2004.

Collaboration was also performed between the engineers to improve the modeling for both the teams. Several Skype conferences were held between the team from Imperial College London and the Wits CSIR team to discuss the design and theory behind our own chemotaxis models. The collaborative effort resulted in the Imperial College London kindly providing us with a model for our theophylline riboswitches. Their modelling results are as follows:

The Wits CSIR iGEM team intend to use a riboswitch to reprogram the chemotactic behavior of E.coli. The project includes engineering the attraction of the bacteria to theophylline[1]. CheZ is an important protein controlling the chemotaxis of bacteria. They used a theophylline riboswitch to control the expression of CheZ in CheZ deletion mutants in order to engineer the bacteria's movement towards theophylline[1]. We're using a riboswitch sensitive to theophylline to control the expression of CheZ. In the absence of theophylline, the start codon is covered so the translation of the strand cannot occur. In the presence of theophylline, the conformation of the riboswitch changes and the ribosome binding site is exposed[1]. Thus, the higher the concentration of the theophylline, the more will enter the cell resulting in the up regulation of CheZ expression. This will increase the frequency of directed movement[1].

The theophylline riboswitch can be modeled in three differential equations [2].

M ?= a?(T)-?M

(CheZ) ?= ßMT(P)-?CheZ

T ?= ?(P_ext )CheZ-dT

M, CheZ and T respectively stands for the concentration of CheZ mRNA, the concentration of protein CheZ and the concentration of theophylline. The constants a,ß,?,? and d are all positive, and respectively denote the CheZ-promoter transcription rate, the CheZ-mRNA translation rate and the mRNA, CheZ and theophylline degradation-plus-dilution-rates. ?(Text) is the theophylline transport rate per unit CheZ concentration. It is a function of number of theophylline receptors and external theophylline concentration. The function ?(T) and T(T) denote the theophylline-governed regulation ay the transcriptional and translation levels respectively (equation below [2]). KF is the equilibrium constant at transcriptional level and KT is the equilibrium constant at translation level.

F(T)= K_F/(K_F+T)

T(T)= K_T/(K_T+T)

Varying the parameter ?(Text) of above model could help us to understand how the number of receptors and external theophylline concentration effect the intracellular concentration of theothyline and hence the expression level of CheZ. The results are shown in Fig 1 and Fig 2 below. In addition, the response curve of CheZ against theophylline concentration with different theophylline transport rate was illustrated in Fig 3.

Both the expression of CheZ and intracellular concentration increases with increase of transport rate. The bacteria will produce more CheZ and therefore higher level of directed movement. The CheZ response curve shows that the transport rate can be used to tune the CheZ response, as the threshold intracellular concentration of theophylline required to trigger the CheZ response increases with transport rate.

Parameters:

[2] C. Rao, J. Kirby, A. Arkin, Design and Diversity in Bacterial Chemotaxis: A Comparative Study in Escherichia coli and Bacillus subtilis, PloS Biology, Issue 2, Vol 2, pp. 239 – 252, February 2004.

[1] P. Spiro, J. Parkinson, H., Othmer, A model of excitation and adaptation in bacterial chemotaxis, Proceedings of the Nation Academy of Sciences, United States of America, Biochemistry, Vol 94, pp. 7263 – 7268, July 1997.