# Model

Model A:

A mathematical approach to the toxins production in E. Coli.

All of our toxin production was controlled under the pTac system, utilized the DNA-binding characteristics of LacI transcription factor (BBa_C0012) to regulate downstream gene expression. We use IPTG (Isopropyl ß-D-1-thiogalactopyranoside) as an inhibitor of LacI, to induce the production of toxic proteins.

The lethality of toxins is proportional to the amount they were produced. Under ideal situation, the production rate is uniform while no proteins were lost, hence the concentration would constantly increase inside the cytoplasm. However, the protease and other factors contribute to a decent degradation rate which reduce the net amount of toxic proteins.

Contents:

Variables:

The relationship between them have been illustrated in the diagram below.

We want to visualize this relationship in a mathematical equation. Based on the kinetics diagram above, several mathematical equations could be derived:

In Chen, Ting, Hongyu L. He, and George M. Church’s model, the overall rate of protein synthesis was determined by two steps, rate of mRNA production and rate of protein production. Each step determined by the two factors: synthesis rate and degradation rate. For example: the net rate of LacI mRNA (r) production could be modeled as the rate of transcription on LacI gene minus the rate of degradation. 1083 is the length of gene and the transcription has been assumed to occur at 15 pET28a circuits.

Then, mRNA act as a template for translation. We could apply this equation to find the overall change in abundancy of LacI proteins, from translation and degradation rates. Where 360 is the number of amino acids of LacI.

According to Tomer Kalisky et al, the amount of active LacI (Ra) yet inhibited by IPTG after induction could be calculated using the dissociation constant and the total number of LacI proteins (Rt). [IPTG] is the concentration of IPTG in cell, assumption has been made that the concentration of IPTG is same in and out of cytoplasm (= 10-3M).

However, the abundance of active LacI should be converted into concentration (M) to find the coefficient respect to the rate of transcription under IPTG induction. In our model, we borrow the ideas from team Aberdeen_Scotland’s:

Where L is the Avogadro’s constant: ~6.02×1023;
V is the average volume of bacterium: 6.7×10-16 dm3

The equations for overall rate of mRNA synthesis for both toxins have been derived:

Solve the differential equations to hence find the abundancy formulae for each peptide respectively.

We use the function ode45 in MATLAB® to solve these ODEs, the coding of our programs is here.

Fig1 the variation of abundance of each molecule (LacI mRNA, LacI, Cry7Ca1 mRNA and Cry7Ca1) with respect to time. The graph suggested that the induction of IPTG at 6th hour would result in a significant increase in number of Cry7Ca1(Bt toxins) and the maximum number of this toxin could achieve is slightly below 500 per cell.

Fig2 the variation of abundance of each molecule (LacI mRNA, LacI, xncht mRNA and xncht) with respect to time. Which the induction of IPTG at 6th hour is expected to result in a significant increase in number of xncht(chitinase) till plateaued at a value just above 1400 molecules per cell.

Reference:

[1]: Munro, P. D., Ackers, G. K., & Shearwin, K. E. (2016). Aspects of protein–DNA interactions: a review of quantitative thermodynamic theory for modelling synthetic circuits utilising LacI and CI repressors, IPTG and the reporter gene lacZ. Biophysical reviews, 8(4), 331-345.
[2]: Kalisky, T., Dekel, E., & Alon, U. (2007). Cost–benefit theory and optimal design of gene regulation functions. Physical Biology, 4(4), 229.
[3]: García LR, Molineux IJ. Rate of translocation of bacteriophage T7 DNA across the membranes of Escherichia coli.J Bacteriol. 1995 Jul177(14):4066-76 p.4069 left column 2nd paragraph.
[4]: Mosteller RD, Goldstein RV, Nishimoto KR. Metabolism of individual proteins in exponentially growing Escherichia coli. J Biol Chem. 1980 Mar 25 255(6):2524-32.
[5]: Guet CC et al., Minimally invasive determination of mRNA concentration in single living bacteria. Nucleic Acids Res. 2008 Jul36(12):e73. doi: 10.1093/nar/gkn329. p.4 right column bottom paragraph.
[6]: Kennell D, Riezman H (1977) Transcription and translation initiation frequencies of the Escherichia coli lac operon. Journal of molecular biology 114: 1-21. AND Neidhardt FC (1987) Escherichia Coli and Salmonella Typhimurium: Cellular and Molecular Biology. American Society for Microbiology.

Model B:

Mathematical approach to UV inducible promoter and T4 lytic protein production in vivo:

SOS response is a negative-feedback loop in many microbes in response to an environmental stress, typically ultraviolet (UV) radiation [1]. In E. coli, UV triggers the activation of RecA co-protease which cleaves the LexA transcription factor residues on certain regions of genome hence allow downstream gene to be expressed and repair the DNA damaged by UV light. This regulation has been utilized to design the UV-dependent promoter SulAp (BBa_K518010), where we apply this promoter to control the cell lysis in our project.

However, the transformation efficiency of our recombined E. coli with pSS2 is relatively low. We hypothesized that the high leaky (basal) expression of the T4 lysis proteins results in early cell death and slow the proliferation rate. That would be a critical information for industrial production if our design was applied in agriculture. Moreover, we want to deeply evaluate the activation of SulAp and synthesis of lytic proteins using math formulae, hence can be used to deduce the efficiency of cell disruption and protein release. Which will benefit the industrial production as well.

Similar to model A derived, we use the gene expression model suggested by Chen[7], Ting, Hongyu L. He, and George M. Church in 1999 as a template to design the mathematical approach of T4 lytic expression. Aiming to characterize the production of cell lytic proteins respect to time before and after UV inductions at 6th hour.

Constant:

Variables:

Assume that the initial value for all variables are 0.
The rate of change in the number of LexA mRNA can be represented as:

Then derive the instantaneous change in the number of LexA proteins:

When induced by UV, RecA cleavage took place. Therefore, the coefficient representing the decrease in LexA number = degradation rate (u) + cleavage rate (uRecA).
Apply the formula used in team Aberdeen_Scotland’s model to find the respective concentration of LexA:

Where L is the Avogadro’s constant: ~6.02×1023;
Then V is the volume of bacterium: ~6.7×10-16 dm3.

The amount of SulAp inhibited by LexA and the transcription rate of downstream T4 lysis genes could be calculated using the fraction, a like the one we borrow from Tomer Kalisky et al[8] to model the pTac system in model A:

Where 15 is the copy number of pSB1C3 in cytoplasm, 1800 is the length of T4 lysis gene.
Which the equation above was substituted to find the rate which lytic proteins are being synthesized.

The equations have been solved using MATLAB, (see our algorithm).

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Starting with very few LexA proteins, the expression level of lytic proteins was high since few SulAp regions were being inhibited. As LexA being synthesized and maintain at a high level for hours, the expression of T4 lysozyme maintain at a lower level. Until a steep drop in LexA number was resulted by the UV-activation of SOS-response in cytosol.
Moreover, the figure also illustrated a high basal expression of lytic proteins when not irradiated by UV. Which potentially support our hypothesis about cell death.

Reference:

[1]: Little, J. W., & Gellert, M. (1983). The SOS regulatory system: control of its state by the level of RecA protease. Journal of molecular biology, 167(4), 791-808.
[2]: Proshkin S, Rahmouni AR, Mironov A, Nudler E. Cooperation between translating ribosomes and RNA polymerase in transcription elongation. Science. 2010 Apr 23 328(5977):504-8. p.505 left column 2nd paragraph & table 1.
[3]: Mosteller RD, Goldstein RV, Nishimoto KR. Metabolism of individual proteins in exponentially growing Escherichia coli. J Biol Chem. 1980 Mar 25 255(6):2524-32.
[4]: Maurizi MR. Proteases and protein degradation in Escherichia coli. Experientia. 1992 Feb 15 48(2):178-201. p.181 table 2.
[5]: Guet CC et al., Minimally invasive determination of mRNA concentration in single living bacteria. Nucleic Acids Res. 2008 Jul36(12):e73. doi: 10.1093/nar/gkn329. p.4 right column bottom paragraph.
[6]: Lewis, L. K., Harlow, G. R., Gregg-Jolly, L. A., & Mount, D. W. (1994). Identification of high affinity binding sites for LexA which define new DNA damage-inducible genes in Escherichia coli. Journal of molecular biology, 241(4), 507-523.
[7]: Chen, T., He, H. L., & Church, G. M. (1999). Modeling gene expression with differential equations. In Biocomputing'99 (pp. 29-40).
[8]: Munro, P. D., Ackers, G. K., & Shearwin, K. E. (2016). Aspects of protein–DNA interactions: a review of quantitative thermodynamic theory for modelling synthetic circuits utilising LacI and CI repressors, IPTG and the reporter gene lacZ. Biophysical reviews, 8(4), 331-345.