Team:ETH Zurich/Model/Environment Sensing/system specifications
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 includes on the one hand the detection thresholds in term of bacterial cells density and lactate concentration for our AND gate, and on the other hand the azurin output level that could be reasonably obtained in the tumor from the lysis of our bacteria. As this later criterium is tightly related to the bacterial cells density that can be achieved in the tumor (the more bacterial, the more azurin is released), let us begin with focusing on bacterial cells density.
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 [1]
- 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 [2]. 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, we take as a reference value a concentration of bacteria corresponding to the.
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
- ^ 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.
