This is an IPTG-inducible model for the expression of the fusion protein CBD cipA-BMP2 under the control of Plac promoter and cloned in psb1c3. The vector does not contain the ORF of LacI and thus the LacI repressor protein concentration is that in E. coli strains (10nM per cell).1
The model consists of systems of differential equations that were solved in Python using Spyder 3.6 (Anaconda).

Entry of IPTG into E. coli
Mimicking lactose, IPTG is a chemical widely used in scientific research because it cannot be metabolized. IPGT induces the expression of genes under the control of Plac promoter.

IPTG enters into E. coli cells in a concentration dependent manner. This uptake is mediated by lactose permease and passive diffusion at low and high concentrations respectively.2 The IPTG uptake through passive diffusion can be described with the law of mass action expressing the balance between IPTG concentration outside (IPTGext) and inside (IPTGint) of E. coli :

Eq 1. The first term on the right side of Eq 1 represents the forward rate where kIPTGupt is the rate constant of IPTG uptake (0.92 min-1)3 while the second term has negative sign because it represents the backward rate where kIPTGout is the rate constant of IPTG output (0.05 min-1).3 Because of [IPTG]ext = [IPTG]TOTAL - [IPTG]int, it can be replaced in equation 1 in order to have IPTGint as the unique independent variable:

Eq 2. The LacI-IPTG interaction is assumed to be so fast and when the IPTG concentration largely exceeds that of lacI. The LacI active ([LacI]act), i.e that unbounded to IPTG, can be described with the Hill repression function.4,2


Eq 3. Equation 3 (Eq 3) describes the amount of unbounded LacI that actively repress the expression of a protein driven by the Plac promoter; the expression of CBD cipA-BMP2 in the present study. [LacI]TOTAL is the concentration of LacI inside E. coli, KIPTG is the Michaelis constant of IPTG-LacI binding (6000 M) 5 and hIPTG is the associated Hill coefficient (2).3,6

 

The CBD cipA-BMP2 mRNA concentration ([CB]mRNA) at time t can be described as follows:

The first term on the right side of the equation 2.6 represents the synthesis of mRNA. kPlac is the maximum transcriptional rate corresponding to the Plac promoter (0.5 nM min-1).3 αCEB+(1-αCEB) is the leakiness factor where αCEB is the transcriptional leakage of Plac promoter. is the repressing Hill function of Plac promoter where KLacI is the Michaelis constant of LacI-Plac promoter binding (6000 M), 3 and hLacI is the associated Hill coefficient(2).3 The second term represents the degradation of the [CB]mRNA where δ[CB] is the degradation rate.3,6,7
Because there is not constitutive expression of LacI protein from the plasmid cloned in E. coli carrying the CBD cipA-BMP2 construct and based on our preliminary measurements of total protein concentration using BCA method, the transcriptional leakage can be taken to be 0.8 and thus the balance for [CB]mRNA becomes:

Eq 5. Figure 1. Rate balance of IPTG diffusion through E. coli membrane. Forward and backward rates are plotted versus normalized [IPTG]int

IPTG-LacI repressor interaction
Because of IPTG is the inducer, it is important to evaluate its interaction with LacI repressor at different initial concentrations added to the system ([IPTG]TOTAL). For these values the [LacI]act can be calculated with Equation 3 (Table 1). From table 1 and figure 4, it can be deduced that a concentration of IPTG = 0.1 mM is saturating because the solutions derived from higher IPTG values converge.

Table 1. [LacI]act for different initial IPTG concentrations.

The mathematical model for the expression of CBD cipA-BMP2 is:

Eq 7.
The linearization of the mathematical model revealed an stable equilibrium point for the expression of CBD cipA-BMP2 because two negative igenvalues,

 

 

1. Oehler, S. (2009). Feedback Regulation of Lac Repressor Expression in Escherichia coli. Journal of Bacteriology, 191(16), 5301–5303. http://doi.org/10.1128/JB.00427-09
2. Ukkonen, K. Vasala, A. (2015). Protein expression by IPTG autoinduction in EnPresso B:Protocol with minimal manual work and superior yields compared to other media.BioSilta. Retrieved on june 24th, 2018 from https://bioscience.co.uk/userles/pdf/Application%20Note%20-%20Protein%20expression%20by%20IPTG%20autoinduction%20in%20EnPresso%20B.pdf.
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4. Semsey, S., Jaured, L., Csiszovszki, Z., Erdssy, J., Stger, V., Hansen, S. Krishna, S. (2014).The effect of LacI autoregulation on the performanceof the lactose utilization system in Escherichia coli. Nucleic Acids Research,41 (13),6381?6390.https://academic.oup.com/nar/article/41/13/6381/1120421.

5. University of California Berkeley.(2006).Berkeley. International Genetically Enginereed Machines.Retrieved on july 14th, 2018 from https://2006.igem.org/wiki/index.php/Berkeley
6. Ozbudak, E., Thattai, M., Lim, H., Shraiman, B. Van Oudenaarden, A. (2004).Multistability in the lactose utilization network of Escherichia coli. Nature, 427 (1476-4687), 737?740.https://www.nature.com/articles/nature02298.

7. Michel, D. (2013). Kinetic approaches to lactose operon induction and bimodality. Journal of Theoretical Biology 325, 6275.https://www.sciencedirect.com/science/article/pii/S0022519313000659?via

8. Moulton, G. (2014). Fed-Batch Fermentation A Practical Guide to Scalable Recombinant Protein Production in Escherichia coli. Retrieved from Google Books.

9. Galli, E., Midonet, C., Paly, E. Barre, F. (2017).Pressure and Temperature Dependence of Growth and Morphology of Escherichia coli:Experiments and Stochastic Model. Plos Genetics,13(3):e1006702.https://doi.org/10.1371/journal.pgen.1006702. ://jour-nals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1006702

10. Hovgaard, L., Frokjaer, S. van de Weert, M. (2013).Pharmaceutical Formulation Development of Peptides and Proteins. Second Edition. Retrieved from Google Books.

11. Bernstein, J., Khodursky, A., Lin, P., Chao, S. Cohen, S. (2002). Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two color fluorescent DNA microarrays. Proceedings of the National Academy of Sciences of the United States of America. 99 (15) 9697-9702. ://www.pnas.org/content/99/15/9697

12. Bremer, H., Dennis, P. P. (1996) Modulation of chemical composition and other parameters of the cell by growth rate. Neidhardt, et al. eds. Escherichia coli and Salmonella typhimurium: Cellular and Molecular Biology, 2nd ed. chapter 97, pp. 1559, Table 3

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The main goal of our work is to develop a functionalized cellulose-based biomaterial for bone and cartilage regeneration in order to allow a faster injuries recovery time and overcome remaining challenges in repairing these connective tissues. In biomedicine, emergent strategies have focused the use biomaterials loaded with drugs.1-3 We aimed to use bacterial cellulose, a biocompatible biomaterial, as scaffold for the delivery of the human bone morphogenetic protein 2 (BMP2). It is worth to note that the diffusion of a drug far of the action site can become ectopic tissue formation, representing a serious problem. Therefore, we focused to attach the BMP2 by the use of a cellulose binding domain (CBD). We choose the CBD of the scaffolding protein of C. thermocellum (CBD cipA) because the higher affinity compared to other CBDs, reported by the Team Imperial_2014. Moreover, in an effort for a longer broad application of a functionalized bacterial cellulose, such as wound healing, we also fused the CBD cipA with an elastin like polypeptide (ELP_C5) derived from the ELP- [V-150].