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From 2012.igem.org

The Human Gut Microbiota Regulation by E.coslim and Mathematical Modeling

　　As it has been mentioned, manipulating the number of Firmicutes and Bacteriodetes might be an effective method to achieve 'a slim glutton'. So the following content is about to provide a theoretical base for the whole project.

　　We assume that resources (glucose) and (fatty acid) are perfectly substitutable for both populations (Firmicutes) and (Bacteriodetes). We simplify this situation as an exploitative competition in a chemostat, and the ODEs are:

　　1. S(t) and R(t): concentrations of glucose and fatty acid

　　2. xi: biomass of the competing populations at time t

　　3. S0 and R0: concentrations of resource S and R in the feed bottle

　　4. D: dilution rate

　　[The specific death rates of the microorganisms are assumed to be insignificant compared to this dilution rate D]

　　5.Si and Ri: the rate of conversion of nutrient S to biomass of population xi

　　[if the conversion of nutrient to biomass is proportional to the amount of nutrient consumed, the consumption rate of resource S per unit of　　competitor xi is denoted

where ξi is the respective growth yield constant]

　　6. Gi: the rate of conversion of nutrient to biomass of population xi

　　[Since perfectly substitutable resources are alternate sources of the same essential nutrient, the rate of conversion of nutrient to biomass of population xi is made up of a contribution from the consumption of resource S as well as R: ]

　　Here we choose

; 　　And let

　　They denote the maximal growth rate of population xi on resource S(R) when none of the other resource is available.

PART I

　　A model is built to describe a quantitative relationship between Firmicutes and Bacteriodetes in people's intestines who are overweight.

　　Parameters mSi(mSi) can be assigned values to Firmicutes and Bacteriodetes so as to simulate the ability to utilize glucose and fatty acid. If their ability to use nutrient are given as follow:

	Firmicutes	Bacteriodetes
Glucose	+++ (mS1)	++(mS2)
Fatty acid	+ (mR1)	++(mR2)

　　Then we can set mS1=2.25, mR1=0.5, mS2=2.1, mR2=2.1. In order to make easier ODEs, we set S0=R0=D=1 and .

　　In this situation, the ratio N(Firmicutes)/N(Bacteriodetes) is rather high, usually achieving a value 8.0, and each of their absolute number, or concentration, is stable.

PART II

　　We try to add a Genetic Engineered Bacterium(GEB) into system. This ideal type of bacterium consumes glucose as well as fatty acid, thus makes itself a competitor to Firmicutes and Bacteriodetes. While it is reproducing in intestines, the competition among these three types of bacteria makes the number change gradually. At last, we hope to achieve a lower ratio of Firmicutes/Bacteriodetes.

　　The key point is to find out how competitive our GEB should be. In other words, we have to point out its ability to consume glucose and fatty acid---to study new parameters . For example, if mS3＞mS1, then we conclude that GEB has stronger ability to consume glucose than Firmicutes. We try to find out an appropriate pair of .

1. (2.5, 2.1)

	Firmicutes	Bacteriodetes	GEB
Glucose	+++ (mS1)	++(mS2)	++++(mS3)
Fatty acid	+ (mR1)	++(mR2)	++(mR3)

1'. (2.1,2.5)

	Firmicutes	Bacteriodetes	GEB
Glucose	+++ (mS1)	++(mS2)	++(mS3)
Fatty acid	+ (mR1)	++(mR2)	+++(mR3)

　　1 and 1' show that GEB are so competitive that others die out.

2. (2.1,0.4)

	Firmicutes	Bacteriodetes	GEB
Glucose	+++ (mS1)	++(mS2)	++(mS3)
Fatty acid	+ (mR1)	++(mR2)	＜+(mR3)

　　GEB is too weak to survive.

3. (2.1,2.11)

	Firmicutes	Bacteriodetes	GEB
Glucose	+++ (mS1)	++(mS2)	++(mS3)
Fatty acid	+ (mR1)	++(mR2)	++(mR3)

4. (2.25,0.5)

	Firmicutes	Bacteriodetes	GEB
Glucose	+++ (mS1)	++(mS2)	+++(mS3)
Fatty acid	+ (mR1)	++(mR2)	+(mR3)

　　In this situation N(firmicutes)/N(bacteriodetes)<8.0 is obvious. We successfully finished our task of finding a pair of =(2.25,0.5), and this kind of GEB is exactly what we hope to build.