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Team:HZAU-China/Modeling/Gray logistic
From 2013.igem.org
Aim:
To know the growth curve of Bacillus subtilis in the dog’s blood.
Steps:
1. Experimentally measure the number of bacteria;
2. Establish the gray logistic model to simulate the growth of bacteria;
3. Determine the parameters through experiments;
4. Test the predicted results.
Results:
The gray logistic model gives good prediction and the model precision is excellent.
Background:
The color of blood is so deep that it is not fit to measure the OD value to determine the growth of bacteria in the blood. So we chose dilution-plate method to detect the number of total bacteria. We coated a large number of plates. If you want to know the details of the experiment, please click [[here]]. The logistic model of population can well predict the increase of population.
Establishing the logistic model:
In the blood environment, the number of bacteria has a maximum value K. When the bacteria number approaches K, the growth rate approaches zero. Then the population growth equation is as follows:
.The solution of the equation is :
,
where N0 is the size of bacterial population and r is population growth rate. For convenience, we rewrite the above equation as
where A=K, 
and r are unknown parameters. N is the logarithm of the colony-forming unit (CFU) of Bacillus subtilis.
Determining the parameters using the gray system theory:
To determine the parameters of the equation,we used the gray system theory. The equation can be rewritten as:
,
,
;Using the matrix equation in linear algebra we could determine the parameters α and β .
,
,
From the results, we know the value of posterior-variance is 0.1931, lower than 0.35, so that the model precision is excellent.
In conclution, the growth curve of our engineered bacterium in dog's blood is given by
; where N(t) is the logarithm of the CFU of Bacillus subtilis.
Reference:
1.Shiqiang Zhang, China's Population Growth Model Based on Grey System Theory and Logisitic Model[C]. 2010:4. (In Chinese)
2.Xiaoyin Wang, Baoping Zhou 2010. Mathematical modeling and mathematical experiment. Beijing : Science press. (In Chinese)
