## Our Model

**The objective of the following modeling work is to develop a dynamic model that describes our recombinant strain and to establish practical methods for estimation of bioluminescence in the complex media.**

The diagram below demonstrates the data points of bioluminescence under 4L and 0.01M arabinose and are yet to be modeled.

Based on observation, we anticipated that our data points can be accurately modeled through a sigmoidal function, not only due to the S-shaped distribution of the data points, but also due to the natural properties of the data collected. For instance, according to the data points, the bioluminescence strength will reach a plateau from the third to forth hour. This is because as the bacteria becomes immersed in glucose, there is a maximum capacity in which the number of bacteria can grow to. Since bioluminescence strength is also dependent on the population of the bacteria in the glucose, it is able to be modeled by a sigmoidal function which is a function typically used model population trends in both biology and sociology.

The relationship between a sigmoidal function and arabinose-induced luminescence can be further extended. According to a previous study conducted by M. Nadri in 2006, the light production can be taken to be directly proportional to the protein concentration variations, as well as other cell activity and oxygen as limiting substrate, resulting in the function:

where X represents biomass concentration (g/l), P protein concentration in the reactor (g/l), φ_2 inducer effect, φ_3 oxygen limitation, μ specific growth reaction rate (1/h), and y_l yield coefficients.

In the study, the equation above is further simplified into:

where S is the substrate concentration (g/l), k_s is the saturation constant associated to the oxygen and k_I is the saturation constant associated to the inducer effect. The study successfully validates the choice of a sigmoidal model through comparing the mathematical model with the behavior of the system consisted of an *E. Coli *strain and plasmid PIT34 containing genes for bioluminescence and the pBAD promoter inducible by arabinose. Thus, we justified our use of a sigmoid function to model a similar system.

The following diagram illustrates the mathematical model we have computed:

Figure 2: 4L+0.01M Arabinose Luminescence over Time

The sigmoidal mathematical equation obtained from our data points using Graphpad Prism is:

The model predicts that as time approaches 5 hours, the luminescence strength will approach to the maximum capacity, which is estimated to be around 1.63×10^7 RLU.

Due to limitations in office hours, we were incapable of acquiring data points for a more extensive time period, hence decreasing the accuracy of the model for longer periods of time.

Nevertheless, observations in the following day indicate that the luminescence strength will decrease as the glucose the bacteria is immersed in is exhausted.

### Bibliography

Nadri, M., Trezzani, I., Hammouri, H. et al. Bioprocess Biosyst Eng (2006) 28: 217. https://doi.org/10.1007/s00449-005-0008-1