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Modeling

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3 models are used to measure our project.

Bacteria scale

We use logistic equation to fit the mono-restrict factor growth of bacteria, and the bacteria scale itself is the factor.

dX/dt=Vmax(1-X/Xmax)X

In this equation, ‘X’ means the bacteria scale, it’s the function of‘t’, time. 'Vmax' represents the possible max relative growth speed of the bacteria . 'Xmax' is the upper limit of the growth. In this function, we can find out that the instantaneous growth speed is determined by the relative growth speed, upper limit of the growth, and the instantaneous scale. With time passing by, the scale grows larger, while the speed turns down.

Fig.1 The fitting curve of the growth of bacteria cells.

1. Fit Results:
(1)General model:
f(x) = X/(exp(-X*(C3 + V*x)) + 1)
(2)Coefficients (with 95% confidence bounds):
C3 = -1.248 (-1.962, -0.5334)
V = 0.07547 (0.02994, 0.121)
X = 3.055 (2.717, 3.393)

2. Goodness of fit:
SSE: 1.99
R-square: 0.9164
Adjusted R-square: 0.9045
RMSE: 0.3771

As the foundation of our modeling, the logistic model fits quite well with our data. It takes about 30 hours for the bacteria to reach the upper limit. According to the growth law, we can say that the highest growth speed is around 15h, when the scale is half of the maximum.

Product scale

We assume that our product comes from two parts, one is during the growth, and the other is through their livelihood.

dP/dt=αdX/dt+βX

In this equation, ‘X’ and ‘t’ are the bacteria scale and time. ‘P’ means the product. We can see the familiar ‘X’ and ‘t’, that is how we base the production on the bacteria growth. 'α' and 'β' indicates the proportion of the two processes. With these parameters, we can find out when is the most important part of time in the whole fermentation process. This is quite useful, because we can decide the length of fermentation and decide the timing to do some changes.

Fig.2 The original fitting curve of product formation

1. Fit Results:
(1)General model:
f(x) = C+(A*0.0255*3.055*exp(0.0754*x))/(3.055-0.0255+0.0255*exp(0.0754*x))+B*3.055/0.0754*log((3.055-0.0255+0.0255*exp(0.0754*x))/3.055)
(2)Coefficients (with 95% confidence bounds):
A = 8392 (4254, 1.253e+04)
B = -588.1 (-885.2, -291)
C = -201.6 (-312.1, -91.16)

2. Goodness of fit:
SSE: 344.8
R-square: 0.9165
Adjusted R-square: 0.898
RMSE: 6.19

Although this can reach a better R-square, our parameter β meet a negative number which should not happen. However, it can show that the product is mainly synthesized from the first part. Then we fit the data again with only the first part of our equation.

Substrate scale

Bacteria use the provided carbon source to propagate, maintain their livelihood and synthesize the product.

dS/dt=-γdX/dt-δX-ε*dP/dt

In this equation, 'S' is the substrate, 'γ', 'δ',and 'ε' are the ratio of the three outlets. Our genetic operation is aimed to make the bacteria put more energy into the product. In a way, these parameters can show whether our strategies are effective or not.

Fig.3 The altered fitting curve of product formation

1. Fit Results:
(1)General model:
f(x) = C+(A*0.0255*3.055*exp(0.0754*x))/(3.055-0.0255+0.0255*exp(0.0754*x))
(2)Coefficients (with 95% confidence bounds):
A = 204 (115.1, 292.9)
C = 16.45 (4.215, 28.68)