### Model

In order to enhance the production of GABA in *Escherichia coli*, researchers have made well-established measurements on the production rate under various types of combinations of pH and temperature. However, the study done by *Pham et al.* focuses on the direct maximization of test results in experiments, but does not include a detailed, predictable model on the influence of pH and temperature. Our focus here is to go further to find values of pH and temperature that contribute to the maximum production of GABA in *Escherichia coli*, especially *E. coli* Nissle 1917, the subspecies that we are utilizing for GM GABA production. The result of this regression will be used to guide the setup of experimental groups of GABA producing circuits and to achieve high GABA yield.

##### Assumption

- The growth condition of bacteria in lab is reasonably far away from the extreme conditions for certain kind of bacteria.
- There is no effective variation within the time period that GABA is produced.
- To simplify the model, we assume that the growth of bacteria under this certain rage of time can be simplified into a polynomial function with a linear relationship instead of a typical logarithmic growth model.
- The function of relevant proteins (anti porter etc) is assumed stable throughout the experiment.
- The difference between individual subtype of bacteria is omitted.

##### Design

The production of GABA is influenced by two factors: the expression of GAD proteins in bacteria, and the amount of bacteria found in sample. This could be generalized as:

`R=NxxC`

**Where N is the number of bacteria, and C is the expression level of GAD operon.
**

The number of bacteria is influenced by its initial value and its grow condition. For every bacterium, there exist a theoretical maximum rate of growth and a corresponding condition, so the actual rate of growth can be found through finding the coefficient of the difference between the rate under actual conditions and under extreme conditions (when pH is really low, for example). Using a similar approach as outlined by Ross et al., we deduce that temperature has similar relationship with bacteria’s growth rate and therefore can be modeled in the same way. For this specific model, we focus on two specific factors, temperature and pH, to describe the bacteria growth:

`N=nxx[(rxx10^(-pH_min))xx(1-10^(pH_min)/10^(pH))]xx[r'xx(T-T_max)xx(T-T_min)]xxt`

**n is the original number of bacteria, r is the coefficient for pH changes, and pH is the actual pH value, pH min is the theoretical minimum value of pH that this bacterium can survive. r’ is the coefficient for temperature difference, and T is the actual temperature. Tmax and Tmin are the extreme conditions of temperature. Small t is time coefficient. Although the general growth model obeys the growth function, growth of bacteria in small time interval can be simulated as a linear relationship. **

The activity level of GAD, on the other hand, is also proportional to the concentration of hydrogen ion. Given experimental data, the group assumes that it obeys the following hyperbolic distribution [3]:

`C=c(pH-pH_min)(pH-pH_max)`

The overall expression can be obtained by combining these two equations. This gives an exponential equation with maximum power of two,

In this experiment, however, the above equation can be simplified. Given that the dataset the group has is far away from the extreme condition, and the coefficient obtained by researchers for r is negligible according to Ross et al., this formula may be reduced to a second-degree equation with two variables: T and pH.

`R=r'(T-T_max)(T-T_min)xxc(pH-pH_min)(pH-pH_max)xxtxxk`

Where: `k=nxx[(rxx10^(-pH_min))xx(1-10^(pH_min)/10^(pH))]`, constant

Then R becomes a polynomial equation that can be used to run regression through Wolfram Mathematica.

##### Data

`data = {{2.5, 25, 0.5}, {2.5, 30, 0.7}, {2.5, 37, 0.8}, {3.5, 25, 0.3}, {3.5, 30, 1.01},`

`{3.5, 37, 0.65}, {4.5, 25, 0.5}, {4.5, 30, 0.85}, {4.5, 37, 0.7}}`

Using the published data from Ross et al., a regression of degree 4 was run. The data point is production of GABA with respect to temperature and pH, and it is displayed as follow:

##### Results

As discussed in Assumptions, we assume time t and constant k are the same for a given set of experiment, they are omitted during the calculating process. Based on these assumptions, the following result was determined:

`125.504-78.0536x+10.8036x^2-8.18845y+5.10126xy-0.705095x^2y+0.131131y^2-`

`0.0814048xy^2+0.0112381x^2y^2`

This result is shown at the same graph with the original data points: where red dots are actual data and the yellow curve is the predicted model:

In order to find the conditions for lab condition of our own experiment, we determined the maximum value of that function within the domain of standard temperature and pH range. We found:

`{x->4.28, y->31.8}`

This theoretical optimal condition was used in our lab setting. See the test of these result at *demonstrate*.

##### References

[1]: Pham, V. D., Somasundaram, S., Lee, S. H., Park, S. J., & Hong, S. H. (2015). Efficient production of gamma-aminobutyric acid using Escherichia coli by co-localization of glutamate synthase, glutamate decarboxylase, and GABA transporter. Journal of Industrial Microbiology & Biotechnology, 43(1), 79-86. doi:10.1007/s10295-015-1712-8

[2]: Ross, T., Ratkowsky, D., Mellefont, L., & Mcmeekin, T. (2003). Modelling the effects of temperature, water activity, pH and lactic acid concentration on the growth rate of Escherichia coli. International Journal of Food Microbiology, 82(1), 33-43. doi:10.1016/s0168-1605(02)00252-0

[3]: Komatsuzaki, N., Shima, J., Kawamoto, S., Momose, H., & Kimura, T. (2005). Production of g-aminobutyric acid (GABA) by Lactobac paracasei isolated from traditional fermented foods. Food Microbiology 22 (2005) 497–504.