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From 2010.igem.org

Mathematical models are very important in the design of an iGEM project. The Michigan modeling team is composed of Josh, Kevin, Candy and Jennifer. We are currently researching different models that have been previously done that can help us in our own modeling endeavors. To construct our models, we have taken advantage of MATLAB's Simbiology toolset. More information and tutorials on Matlab and Simbiology can be found here: [http://www.mathworks.com/academia/student_center/tutorials/launchpad.html?s_cid=0410_webg_igem10_294031 MATLAB Tutorial]

There are several components to a good mathematical model. These include assumptions, parameters, and equations.

Pili

We have decided to model the growth of a biofilm using pili. By modeling the flocculation of the algae, we hope to be able to predict how our reactions will proceed and determine quantities such as the optimal initial concentration of E. coli.

Assumptions
There is no growth or degradation of cells and algae.

• I made this assumption to simplify the model.

Floc formation only involves two particles. Collision efficiency is perfect.

• This assumption greatly simplifies the model.

Parameters
All rate constants set to 0.1.

• This is obviously unrealistic, but with experimental data, we can refine the numbers.

Initial quantities of E. coli and algae are 1 mol.

• We can perform assays to determine the ideal starting conditions.

The pili neural network has been characterized in several papers [1][2]. Essentially, the two recombinases FimB and FimE control an invertible DNA element that acts as a switch, known as FimS. When FimS is in the "on" position, the cell becomes fimbriated. It has been previously determined that the level of piliation depends on the ratio [FimE]/[FimB]. The goal of the modeling team is to determine the optimal ratio to promote flocculation.

Quorum Sensing

Quorum sensing has been modeled by other iGEM teams for previous competitions, including: [http://parts.mit.edu/igem07/index.php/Bangalore Bangalore] and Singapore

This model is based on the paper [http://jb.asm.org/cgi/content/short/189/16/6011 Quorum Sensing in Escherichia coli Is Signaled by AI-2/LsrR: Effects on Small RNA and Biofilm Architecture (doi:10.1128/JB.00014-07)] and was created with the help of Alex who is also on the quorum sensing team. More information about quorum sensing can be found here: Team:Michigan/Project and their notebook can be found here: Team:Michigan/Quorum_Sensing.

This model shows the movement of AI2 (Autoinducer-2) from outside the cell (AI2_o) to inside the cell (AI2_i). The final output after transcription and translation is GFP in the case of the Lux- mutants recieved; however, at the end of the project, this will hopefully cause flocculation of the microalgae.

This model is also specific to the Lux- mutants as it is not understood how LuxS works or is produced. [http://jb.asm.org/cgi/content/short/189/16/6011 (doi:10.1128/JB.00014-07)]

Assumptions

• Assume all massaction
  • This greatly simplifies the model and gives a good estimate; however, it is unrealistic.
• Initially, all LsrR bound to all operators to inhibit.
  • This also simplifies the model.
• Creation of LsrS is unknown so simply exists in the cell at a fixed concentration. Any output that occurs is due to the addition of AI2 from the person performing the experiment.
• Assume that the proteins that bring in AI2 (Lsr A, B, C, and D) exist in a protein complex and can be called one "species" in Matlab. Also assume that the protein complex is created and formed at the same time.
  • This simplifies the number of species in Matlab and also simplifies the model.
• Assuming no degradation for now... Molecule is "used" up in reaction.
  • This simplifies the model by removing degradation from the equations.

Parameters

• For now, all rate constants are 1.
  • This simplifies the model; however, it is not realistic.
• Assumptions with quantities
  • A lot of AI2_o in comparison with all other molecules.
    • This is saying that the experimenter will add a higher concentration of AI2 than the concentrations of various molecules that exist in the cell.
  • Initially low concentration of Lsr A,B,C,D transport proteins
    • Low basal expression of the transport proteins.
  • Low concentration of LsrK
    • Low basal expression of LsrK.
  • Low concentration of LsrR + Operator
    • In comparison with the protein concentration, the number of plasmids containing the Operator is low.

GFP Output

Since the Quorum Sensing Team is only measuring the output of GFP from the addition of AI2, this is the output that really matters. As can be seen, after a certain amount of time, the expression of GFP levels off.

--Jejihong 06:30, 18 August 2010 (UTC)

Surface Display

Oil Sands

We are developing a mathematical model which describes the degradation of naphthenic acids (NAs) by Pseudomonas putida and Pseudomonas fluorescens in a biofilm in a drip flow bioreactor under aerobic conditions. We will use a series of [http://controls.engin.umich.edu/wiki/index.php/Bacterial_Chemostat_Model chemostat] models to approximate the change in NA concentration along the length of the bioreactor and through time.

In developing the mathematical model, we made the following assumptions:

(1) It is known that P. putida and P. fluorescens synergistically degrade NAs [4]. The mechanism of the synergistic degradation and differentiation between the two species should have little effect in evaluating the efficiency of NA degradation, thus our models will assume a homogeneous mixture of P. putida and P. fluorescens and represent the two species as one organism.

(2) We will assume that the microbial population is in a steady state condition, with a balanced growth rate and death rate.

(3) As NAs are not known to be volatile, we assume that NAs remain in solution within the bioreactor.

(4) The biofilm is initially distributed homogeneously throughout the bioreactor, and diffusion of NAs into the biofilm occurs instantaneously.

(5) Both Pseudomonasspecies are distributed homogeneously throughout the biofilm

(6) Oxygen is not a limiting substrate.

(7) Cross sections remain unchanged through the length of bioreactor

Reactions

The system is represented by repeating units of the following reactions, with the units distinguished by a number n. All reactions are assumed to be mass action. For n = 0, NA_n-1 is null.

uOttawa used plug flow to describe nutrient flow in the intestines.

References

1. Kuwahara, H., Myers, C., Samoilov, M., Abstracted Stochastic Analysis of Type 1 Pili Expression in E. Coli.

2. [http://jb.asm.org/cgi/content/short/189/16/6011 Li, J., C. Attila, L. Wang, T. K. Wood, J. J. Valdes, and W. E. Bentley. 2007. Quorum sensing in Escherichia coli is signaled by AI-2/LsrR: effects on small RNA and biofilm architecture. J. Bacteriol. 189:6011-6020.]

3. Wolf, D., and Arkin, A., Fifteen Minutes of fim: Control of Type 1 Pili Expression in E. Coli. OMICS 6 2002

4. [http://www.ncbi.nlm.nih.gov/pubmed/17040229 Del Rio, L., Hadwin, A., Pinto, L., MacKinnon, M. and Moore, M. (2006), Degradation of naphthenic acids by sediment micro-organisms. Journal of Applied Microbiology, 101: 1049–1061.]