Nebraska Wesleyan University, April 2000

permission granted to use and distribute free in an academic setting.

model and data from

*Competition models in Population Biology,* Paul Waltman (SIAM:1983)

325 Eep Street

Lonlinc, SK 04685

17 April 2000

Rigorous Mathematical Contractors, Inc.

Suite 3, Strawmarket Business Plaza

Lonlinc, SK 04685

Dear Rimac:

As you are undoubtedly aware, MedRes, Inc., is a leader in the research into and manufacture of a wide variety of medical products. Most recently, we are experimenting with the development of a line of carefully (genetically) engineered microorganisms. These are to be cultivated in a chemostat, a laboratory apparatus which provides a constant stream of nutrient to the microorganisms in a culture vessel. As the nutrient is pumped in the culture is removed at the same rate, thereby maintaining a constant volume in the culture vessel and allowing a constant harvest of the microorganisms from the outflow. After a number of abortive experiments, we have with some trepidation concluded that we need a good, mathematical, understanding of what is going on in the culture vessel to guide our experimental work. It is to undertake this analysis that we are contacting you.

However, we did succeed in obtaining some preliminary results from the
multifaceted Dr. P. Gavin LaRose, who provided the following
suggestions for the mathematical analysis. In the simplest case we
need to track the concentration of the nutrient (substrate) in the
chemostat, *S*, and the number of microorganisms, *x*. If the input
concentration of nutrient is *S _{0}*, then

The number of microorganisms increases as they grow, and decreases as
a result of washout from the culture vessel. The growth rate of the
organisms is empirically determined to be growth = *m x S* /
(*a* + *S*), where *m* is the maximal growth rate and
*a* the Michaelis-Menton, or half-saturation constant.
The organism
growth rate is related to the consumption of nutrient by a constant
*G*, which is the ratio of organism formed to nutrient consumed;
this with the volume *V* of the culture vessel allows the use of this
growth term to model the nutrient consumption in the substrate
equation. Finally, the washout term is just proportional to the
number of organisms in the culture vessel, being washout = *D
x*.

The estimable Dr. LaRose then proposed that these equations could be
scaled appropriately, defining *S* = *S _{0} u*,

where

Table 1: System parameters |
||

parameter | min value | max
value |

S _{0} |
5x10^{-6} g/L |
1x10^{-4} g/L |

D |
6x10^{-2} per hr |
7.5x10^{-2} per hr |

G |
2.5x10^{10} cells/g |
6.2x10^{10} cells/g |

a |
1x10^{-6} g/L |
3x10^{-6} g/L |

m |
0.41 per hr | 1 per hr |

V |
200 ml | 200 ml |

Given this information, we need a confirmation that the equations given above are correct, and what the possible ranges of values for the parameters in these equations are. We are then curious to know what we can tell about the quantity of microorganisms we will see emerging from the chemostat.

The second part of this project, which may be more interesting, is to
determine the behavior of the system when there are two microorganisms
in the chemostat competing for the substrate. We are willing to
assume that there is no interaction between the organisms other than
indirectly, through depletion of the nutrient stream. Thus, the
addition of the second organism adds a second depletion term to the
equation in (1) for the substrate and contributes a second equation
for the microorganism population. This results in there being two
values for *M*, *M _{1}* and

Finally, we will unfortunately be needing your final report in fairly short order, as the chemostat will be sitting around unused until such time as we hear from you. We expect your final report no later than the 5th of May. We have arranged with the ubiquitous Dr. P. Gavin to provide mathematical advice if you should find it necessary in the course of your work, and he has indicated that you must contact him with the other member(s) of your research team on or before the 24th of April to confirm your initial progress on the project. Failure to meet any of these deadlines will, of course, result in a significant penalty.

Sincerely,

Cever Etkoop, M.D.

President, Medres, Inc.

ce:glr

Gavin's DiffEq Project 3, Spring 2000

Last Modified: Wed Apr 19 11:50:14 CDT 2000

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