Definition of system specifications

GOAL

Determine the target specifications of our bacterial system regarding the detection levels and azurin production levels

In order to tune our bacteria so that it would behave correctly in vivo, we had to define precise, quantitative criteria that had to be met. This means that we first need to define the functioning points that our system will encounter (in other words, the different environmental conditions our bacteria will meet: tumoral or healthy tissues for instance). We will then need to determine the relative level of toxin production in our bacteria that we are willing to tolerate in area where there is no need for treatment. As a result, we will get objective quantitative criteria, which will help us build upon it a strategy to optimize the performance of our system, thanks to further modeling.

Determining functioning points

To know how to tune our system, we need to define the detection thresholds in term of bacterial cells density and lactate concentration for our AND gate. We will here establish for both variables a low value (in situations of healthy environment) and a high value(in situations of tumor environment).

Lactate concentration

A high local lactate concentration is a fairly reliable indicator of a tumorous environment. Tumorous cell exhibit indeed an erratic metabolic behavior leading to a lactate production way over the levels of healthy cells (this is part of the so called "Warburg effect" [1]). Basal concentration of lactate in healthy tissues has been found to be below 1 mM, whereas it was found to be larger than 4 mM and up to 40 mM in tumors. [2]

To build our model, we will take as reference value a lower lactate concentration of 1 mM for healthy tissues, and a higher one of 5 mM for tumors. As the activation of the synthesis of azurin is monotonous in regards to lactate concentration (the more lactate there is, the more it is activated), we believe that measuring the performance of our circuit with these two wisely chosen values will be suitable for a good assessment of its specificity.

Bacterial cells density

Estimating a precise distribution of cells into the tumor is not straightforward, and we had to apply assumptions that appeared reasonable to us to be able to estimate a plausible quantitative repartition of bacteria in solid tumors.

Volume of E. coli = 1 µm³ [3]
Diameter of the tumor = 20 mm
Diameter of E. coli colonization shell area = 10 mm
Width of E. coli colonization shell area = 0.5 mm

The three last values where obtained from an estimation made from the microscopy images of mouse tumors presented in a paper studying E. coli Nissle colonization of tumors [4]. In addition to this data, we took into account the measured overall concentration of E. coli in the tumor, 1x10⁹CFU.g^-1 to deduce an estimated more precise distribution of the bacteria in the tumor. Considering that the bacteria colonize preferentially the "shell area" at the border between the necrotic area of the tumor and the alive tissue, the effective concentration of bacteria in this area can be inferred from a simple proportionality equation:

We could therefore infer the bacterial density present in the tumor: 5% of the volume is locally filled with bacteria, in a shell shape. As for other tissues, to be on the safe side, we take as a reference value a concentration of bacteria corresponding to the maximum observed in any healthy organ, for any bacterial strain that as been tested in the paper. This amounts to 1x10⁷CFU.g^-1, and corresponds therefore to a bacterial density of around 0.05%

Steady state QS modelling

We model the bacteria as inhabiting a layer inside a spherical tumor of radius \( r_2 = \SI{20}{mm} \). According to (citation needed), the bacteria colonize the interface between live and necrotic tumor tissue, which we model occurring at radius \( r_1 = \SI{10}{mm} \). The width of the colony is set to \( w = \SI{0.5}{mm} \).

Calculating bacterial cell density

We expect that the concentration of bacteria in the tumor will be approximately \( \SI{1e9}{g^{-1}} \) (citation needed). The volume of a single stationary bacterium is \( V_{\text{cell}} \approx \SI{1}{\micro\metre^3} \); we assume the effective volume occupied by a live —moving— cell to be approximately the same. We assume the density of the tumor to be \( \rho_{\text{tumor}} = \SI{1}{g/mL} \) (citation needed). We now calculate the effective surface coverage of the cells (\( d_{\text{cell}} \) —percentage) as follows:

\[\begin{aligned} S_{\text{colony}} &= d_{\text{cell}} S_{\text{layer}} \\ w S_{\text{colony}} &= d_{\text{cell}} V_{\text{layer}} \\ V_{\text{colony}} &= d_{\text{cell}} V_{\text{layer}} & \text{assumming homogeneity} \\ \frac{V_{\text{colony}}}{V_{\text{tumor}}} &= d_{\text{cell}} \frac{V_{\text{layer}}}{V_{\text{tumor}}} \\ \frac{V_{\text{cell}}\, N_{\text{cell}}}{m_{\text{tumor}}\, \rho_{\text{tumor}}} &\simeq d_{\text{cell}} \frac{4 \pi w r_{1}^{2}}{4/3 \pi r_{1}^{3}} & \text{assumming } w \ll r_{1} \\ \\ d_{\text{cell}} &\simeq \frac{N_{\text{cell}}}{m_{\text{tumor}}} V_{\text{cell}} \frac{1}{\rho_{\text{tumor}}} \frac{r_{1}}{3 w} \\ &\simeq \SI{1e9}{g^{-1}} \cdot \SI{1}{\micro\metre^3} \cdot \SI{1}{mL g^{-1}} \cdot \frac{\SI{10}{\milli\metre}}{\SI{1.5}{\milli\metre}} \\ &\simeq 7\%\end{aligned}\]

According to (citation needed), the concentration of bacteria in healhty tissue 2–3 days after injection is at most \( \SI{1e6}{g^{-1}} \). We will use a concentration of \( \SI{1e7}{g^{-1}} \) as the highest concentration of bacteria in non-tumor tissue; we expect a \( d_{\text{cell}} \simeq 0.07\% \) in those areas.

References

^ Vander Heiden, Matthew G., Lewis C. Cantley, and Craig B. Thompson. “Understanding the Warburg Effect: The Metabolic Requirements of Cell Proliferation.” Science (New York, N.Y.) 324.5930 (2009): 1029–1033. PMC. Web. 18 Oct. 2017.
^ Yong Wu, Yunzhou Dong, Mohammad Atefi, Yanjun Liu, Yahya Elshimali, and Jaydutt V. Vadgama, “Lactate, a Neglected Factor for Diabetes and Cancer Interaction,” Mediators of Inflammation, vol. 2016, Article ID 6456018, 12 pages, 2016. doi:10.1155/2016/6456018
^ Bionumbers.org
^ Stritzker, Jochen, et al. "Tumor-specific colonization, tissue distribution, and gene induction by probiotic Escherichia coli Nissle 1917 in live mice." International journal of medical microbiology 297.3 (2007): 151-162.