# Math 224--Differential Equations: Project 3, Spring 2000

## A Little Biochemistry...

### by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, April 2000

©2000 Gavin LaRose (glarose@umich.edu)
permission granted to use and distribute free in an academic setting.

model and data from
Competition models in Population Biology, Paul Waltman (SIAM:1983)

### Medres, Inc.

Office 728 Ratherold Hall
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 S0, then S will increase at a rate proportional to the difference between S0 and S (with constant of proportionality D, the input rate). Substrate is lost as a result of consumption by the microorganisms, which offsets this input rate.

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 = S0 u, t = T / D and x = N v, to obtain a simpler set of equations. He suggested that, for a correctly chosen microorganism scale N, the governing equations would simplify to

u'(T) = (1 - u(T)) - M u(T) v(T) / (A + u(T)), v'(T) = M u(T) v(T) / (A + u(T)) - v(T),

where M and A are some combinations of the parameters in the problem. For the case we are considering, we expect the system parameters to be as shown in table 1, below.

 Table 1: System parameters parameter min value max value S0 5x10-6 g/L 1x10-4 g/L D 6x10-2 per hr 7.5x10-2 per hr G 2.5x1010 cells/g 6.2x1010 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, M1 and M2, the first corresponding to the nutrient consumption resulting from v(T) and the second due to the second organism, w(T). Similarly, there may be two values of A, A1 and A2. In this case, we need to know if we will get a culture of both microorganisms from the chemostat, or if one will dominate the other. We would like to have as convincing evidence supporting your conclusion as is possible.

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