# 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 *N _{0}* 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*is unknown parameter.

*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)