# E coli growth curve

Throughout the project, we have observed that colonies with larger construct correctly transform into the e coli results in a slower growth. To investigate whether the growth of e coli in neb5-alpha will decrease due to the large plasmid contained, we hope to model the growth of different transformed e coli using different growth models and curve fitting methods. Unless specified, in all equations below, a is the upper limit of the growth curve, b and k sets the horizontal displacement and growth rate respectively.

## Method

A single clone is picked from a perti dish, then incubated in 2 mL of LB with ampicillin for 16 hours, then diluted such that OD600 is 0.04-0.06. The cultures are then shaken at 250 r.m.p in 37 C. OD600 is taken at 15-minutes to 1-hour intervals. E coli with inserts of pET-Blue-2 (size: 3600 bp) and our construct FP 1.2.2 16 (size: 14471bp) are compared. Experiments are duplicated.

## Assumptions

s
 Assumptions Justification OD600 = 0.1 corresponds to 2*10^7 cells/mL. It is taken for convenience as calibration is very time consuming, and it is given by [1]. The OD600 of the overnight culture is the maximum OD600. OD600 remains constant staring from the stationary phase [2] and 16 hours of incubation takes the e coli to stationary phase [3].

## Results

 Time (h) 16.1 (OD600) 16.2 (OD600) pET-Blue-2.1 (OD600) pET-Blue-2.2 (OD600) 0.00 0.059 0.052 0.045 0.048 0.52 0.038 0.035 0.059 0.066 1.02 0.064 0.056 0.115 0.188 2.00 0.183 0.136 0.349 0.355 2.55 0.293 0.295 0.599 0.645 3.20 0.443 0.438 0.813 0.883 3.47 0.574 0.538 1.113 1.248 3.75 0.658 0.626 1.326 1.352 4.02 0.768 0.715 1.439 1.63 4.67 1.101 1.02 2.128 2.273 5.08 1.352 1.65 2.355 2.597 16.00 2.871 3.045 3.195 3.279

## Modelling

### Gompertzfunction

Gompertz model is the most frequently used sigmoid model to fit growth data in biology [4].

The form is given by [5]:

### Logistic function

The Logistic function is proposed Pierre François Verhulst, which takes a common “S” shape.

The form is given by [5]:

### Curve Fitting

Curve fitting is done by the curve fitting app in Matlab.

Coefficients of the Gompertz function when f(x) represents OD600 are as follows:

 a b k 16.1 2.877 7.258 0.4341 16.2 3.059 10.43 0.5145 pET-Blue-2.1 3.24 9.755 0.6461 pET-Blue-2.2 3.345 10.89 0.6933

Coefficients of the Logistic function when g(x) represents OD600 are as follows:

 a b k 16.1 2.871 74.28 0.8241 16.2 3.054 153.8 0.9832 pET-Blue-2.1 3.205 73.74 1.047 pET-Blue-2.2 3.307 81 1.099

Coefficients of the Gompertz function when f(x) represents log10(cells/mL) are as follows:

 a b k 16.1 8.834 0.262 0.2928 16.2 8.874 0.2766 0.2886 pET-Blue-2.1 8.898 0.2669 0.4016 pET-Blue-2.2 8.901 0.2566 0.4121

Coefficients of the Logistic function when f(x) represents log10(cells/mL) are as follows:

 a b k 16.1 8.829 0.2979 0.3184 16.2 8.869 0.3169 0.3153 pET-Blue-2.1 8.89 0.3031 0.4341 pET-Blue-2.2 8.895 0.2898 0.4431

## Conclusion

By comparing the growth rate of each model, we can see that a e coli containing a larger plasmid grows slower. Therefore, when overexpressing modified nif genes, it will take longer time to reach the designated phase compared to smaller constructs like PETase.

# References

 [1] R. Milo, P. Jorgensen, U. Moran, G. Weber and M. Springer, "BioNumbers—the database of key numbers in molecular and cell biology," Nucleic Acids Research, vol. 38, no. Database , p. D750–D753, 2009. [2] R. M. Maier, "Bacterial Growth," in Environmental Microbiology, Elsevier Inc, 2009, pp. 37-54. [3] B. G. Hall, H. Acar, A. Nandipat and M. Barlow, "Growth Rates Made Easy," Molecular Biology and Evolution, vol. 31, no. 1, p. 232–238, 2014. [4] K. M. C. Tjørve and E. Tjørve, "The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family," PLoS ONE, vol. 12, no. 6, p. e0178691, 2017. [5] P. R. Koya and A. T. Goshu, "Solutions of Rate-state Equation Describing Biological Growths," American Journal of Mathematics and Statistics, vol. 3, no. 6, pp. 305-311, 2013.