Definition of system specifications

Goal

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

Overview

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 operating 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 operating 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.

Density of bacterial cells

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.

Geometry of tumor and bacterial colony — Geometry of the tumor and bacteria colony (green area: colonized by E. coli Nissle)

Volume of E. coli cell: 1 µm³ [3]
Radius of tumor Tumor radius: 20 mm
Radius of colony shell (concentric to tumor): 10 mm
Width of colony shell: 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]. The proportions were kept and adapted to a model tumor of 4 cm in diameter, to be closer to a situation in the human body (the colonization was studied only in mice). In addition to this data, we took into account the measured overall concentration of E. coli in the tumor, 1e9 CFU/g 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, as follows:

We model the bacteria as inhabiting a layer inside a spherical tumor of radius \( r_2 = 20 \text{mm} \). According to [4], the bacteria colonize the interface between live and necrotic tumor tissue, which we model occurring at radius \( r_1 = 10 \text{mm} \). The width of the colony is set to \( w = 0.5 \text{mm} \).

We expect that the concentration of bacteria in the tumor will be approximately 1e9 CFU/g (citation needed). The volume of a single bacterium is \( V_{\text{cell}} \approx 1 \mu m^3 \). We assume the density of the tumor to be the same as water: \( \rho_{\text{tumor}} = 1 \text{g/mL} \). We now calculate the effective volumetric occupancy of the cells (\( d_{\text{cell}} \) —percentage) as follows:

\[\begin{aligned} V_{\text{bacteria}} &= d_{\text{cell}} V_{\text{layer}} & \text{assuming homogeneity} \\ \frac{V_{\text{bacteria}}}{V_{\text{tumor}}} &= d_{\text{cell}} \frac{V_{\text{layer}}}{V_{\text{tumor}}} \\ \frac{V_{\text{cell}}\, N_{\text{cell}}}{m_{\text{tumor}}\, \rho_{\text{tumor}}} &\simeq d_{\text{cell}} \frac{4 \pi w r_{1}^{2}}{4/3 \pi r_{1}^{3}} & \text{assuming } w \ll r_{1} \\ \\ d_{\text{cell}} &\simeq \frac{N_{\text{cell}}}{m_{\text{tumor}}} V_{\text{cell}} \frac{1}{\rho_{\text{tumor}}} \frac{r_{1}}{3 w} \\ &\simeq 1e9 \text{g}^{-1} \cdot 1 \mu m^3 \cdot 1 \text{mL.g}^{-1} \cdot \frac{10 \text{mm}}{1.5 \text{mm}} \\ &\simeq 5\%\end{aligned}\]

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%

Criteria on the azurin production

Now that we know at what points to evaluate our system, we can set criteria that have to be met to validate that the activation of the production of azurin is specific enough for our needs (priority 1), as well as the highest possible (priority 2) to be able to treat the biggest part of the tumor possible.

Relative criteria

We only want azurin to be produced in the situation of a high lactate level (5 mM) and high bacterial cell density (5% density). This means that we want to maximize the ratio of azurin production in (high lactate, high density) state compared to the production in (low lactate, low density) state, or (low lactate, high density) and (high lactate, low density) states. To assess whether our system is good enough, we will set these quantified criteria:

Criteria for a functional circuit: percentage of maximum Azurin expression allowed in each condition

We have to keep in mind that these criteria are very arbitrary, and that their primary goal is to guide us in the engineering of our bacterial strain. They are not based on measured toxicity level of azurin, as azurin is already quite specific for tumor cells, therefore relying on such toxicity data could lead us to be allowed to express more azurin in healthy tissues, because it would not be less toxic there either way.

Absolute azurin expression

As the colonizing bacterial population is concentrated in a small shell inside of the tumor (which represent only a fraction of the tumor volume) and is at most occupying 5% of the space there, we can already anticipate that it will have to synthetize the highest amount of azurin possible to be able to kill the most tumor cells possible. However, as we have no idea about the effective expression of azurin in E. coli cells, we can only relate it to the expression of another protein in our circuit. As the azurin will be expressed on the same operon than luxI, an enzyme taking part into the bacterial cell density sensing, we can try to get a sense of how stronger must be the expression of azurin in regards to luxI to reach a given arbitrary azurin concentration.

This way, we'll know whether our circuit leads intrinsically to high expressions of the luxI-azurin operon (the expression level of azurin compared to luxI will not be that high in this case), or on the contrary weak expressions of this operon (in this case the expression of azurin will have to be much higher than the one of luxI to reach the target concentration).

We take as a reference value for azurin concentration 1 mM (absolute maximum of concentration of a given protein that can be achieved in E.coli [5]). This will be the goal to reach when our circuit is fully induced, in the (high lactate, high density) state.

Team:ETH Zurich/Model/Environment Sensing/system specifications