Team:ETH Zurich/Model/In Vivo

Behavior of CATE inside Tumor

FIXME

We developed a model to gauge the behavior of our sensing circuit in the real life conditions of solid tumor colonization.

Benefits

We could simulate for a geometry of the system closer to the real-life tumor conditions

Since it was not practically feasible to conduct experiments of bacterial colonization inside tumors, we simulated the bacterial colonization in a thin spherical layer inside a solid tumor considering the simplifications and assumptions as mentioned in the system specifications.

Exact diffusion physics of AHL was included witout any simplifications

Our MATLAB model uses a simplified AHL diffusion model with the assumption of negligible degradation inside the layer and and not taking into consideration the diffusion of AHL far from the source. Extending the diffusion physics ordinary differential equations into partial differential equations using the COMSOL model helped us gauge and verify the behavior of our tumor-sensing circuit in more real-life conditions pertaining to the intended application context of a solid spherical tumor. Using the results obtained from our simulations, we could check the behavior of the AND Gate Switching in different conditions of dcell and lactate.

Diffusion physics of Azurin was included to simulate the effect of lysis

To simulate the effect of lysis, our COMSOL model stops the production of Azurin and starts its diffusion when temperature reaches 42°C. This simulates the effect of increase in temperature with FUS to cause cell lysis. Using data obtained from such a simulation, we could also find the temporal-maximum concentrations of Azurin at each point in the tumor, effectively helping us to estimate the killing area and the time-scale of the treatment.

Simulation of different colonization patterns

Using our model, we also tried a few other colonization patterns to show our system works as expected inside a tumor while stays dormant in healthy tissue. We simulated the following patterns:

  • Homogeneous distribution in a Single spherical-shell-shaped layer in Tumor

  • Heterogeneous distribution in a Single spherical-shell-shaped layer in Tumor

  • Heterogeneous distribution in Double spherical-shell-shaped layer in Tumor

  • Homogeneous distribution in Healthy tissue

Model Overview

This section presents a brief overview of the COMSOL model.

Geometry ... more details here

As mentioned in system specifications, the tumour has been chosen as a solid sphere of radius 20mm and the bactierial colonization pattern as a homogenous distribution in a spherical shell-shaped 0.5mm thick layer in the tumour at a distance of 10mm from the centre of the tumor, as shown in Figure 1. For more details go to the detailed description of the model.

Geometry of tumor and bacterial colony
Figure 1: Geometry of the tumor and bacteria colony (green area: colonized by E. coli Nissle)

Equations ... more details here

Transport of Diluted Species physics was used in COMSOL to integrate diffusion into our model. The partial differential equation for diffusion of a species C with reaction source rate RC is \[\frac{\partial \text{[C]}}{\partial t} + \nabla \cdot (-D_{\text{C}} \nabla \text{[C]})= R_{\text{C}}\]. The reaction rates of the species depends on the domain – tumor (no production and only extracellular degradation) or bacterial layer (production and intracellular degradation). Read here about the details of the domain-wise reaction rates for each species (AHL, LuxI and Azurin) and equations used.

Parameters ... more details here

The parameters that were used in the COMSOL model were obtained partly from literature, partly from characterizations of previous iGEM teams and finally the most important ones were estimated by fitting our experimental data and tuning the fitted-results in the context of the intended applciation, as explained in detail by the Functional Parameter Search. Check out details about the model to read more about the different parameter values used.

For more details about the model go to the detailed description and Functional Parameter Search.

Simulating The Treatment phases

Our model helped us to simulate the three main phases of the CATE treatment: Growth, AND-Gate Switching (Environment sensing) and finally Lysis & Azurin Diffusion. For details see CATE in Action.

Phase 1: Growth

The growth is modelled using the exponential growth rate \[\frac{\mathrm{d} d_{\text{cell}}}{\mathrm{d} t} = \frac{1}{\tau} d_{\text{cell}} \left ( 1-\frac{d_{\text{cell}}}{d_{\text{cell,ss}}} \right )\], where τ is the doubling time, that we obtained by fitting E. coli Nissle growth curves from our lab experiments. For more details on the growth equation check the detailed description.

Phase 2: AND-Gate Tumor Sensing Switch

The AND gate Switch senses the environment based on Quorum sensing (dcell) and Lactatae concentration ([Lac]), as explained in detail in the description of the Tumor Sensing circuit. Based on the different combinations of dcell and [Lac], as mentioned in the system specifications, the AND gate switches 'ON' or 'OFF'. For detailed equations see the model description details.

  • ON state:

    Rapid and high-fold increase in [LuxI] or [Azu]

  • OFF state:

    Slow and negligible-fold increase in LuxI] or [Azu]

Phase 3: Lysis and Azurin Diffusion

The effect of lysis is simulated by a temperature controlled trigger of the diffusion of azurin that is produced in the cells. The temperature is increased from 0 to 42°C as a step function and when the temperature reaches 42°C, the production of Azurin and AHL stops, since the cells are lysed, and diffusion of Azurin begins which depletes Azurin out of the cell into the tumor, effectively inducing apoptosis of the tumor cells.

The model described above was simulated to test the working of the Tumor Sensing circuit. The three phases, as described above were simulated and the results for bacteria colonization of tumor are shown here.

Growth
Figure 2: Growth phase: 0 hr to 35 hr.
TurnON
Figure 3: AND-Gate Switch Sensing phase: 35 hr to 69 hr; The rapid increase in [Azu shows the switch ON of the AND-gate tumor sensor
Lysis
Figure 4: Cell Lysis and Azurin Diffusion phase: 69 hr to 100 hr
HighDHighL_norm
Figure 5: Normalized concentration of AHL and Azurin, Cell density (as a ratio of its steady state value), PLux-Lac and Temperature (as a ratio of its steady state value 42°C) probe plots at a point inside the layer of bacterial colonization in the tumor.

In Figure 5, dcell shows the growth of the cell density inside the layer and PLux-Lac shows the main function responsible for the switching functionality based on the environmental conditions of dcell and lactate.

During the growth phase, our sensing circuit is OFF (visible from Figure 1), and is triggered to turn ON once the desired cell density for quorum sensing is reached at around 40 hr, as shown in Figure 5 (and in Figure 2). Then once steady state is reached, the temperature step triggers the cell-lysis and stops the production of Azurin and AHL. Finally as is visible in Figure 3, all the Azurin diffuses out of the layer very rapidly, thus completing the treatment.

AND-Gate Tumor Sensor Characterization

To test our sensing circuit with the intended application in mind, we use the relative criteria set in the system specifications. Since our AND-Gate switch has 2 inputs for environment sensing viz. dcell and [Lac], there are 4 possible binary combinations that CATE can encounter in real-life scenario, viz.:

  • Tumor Colonization

    High dcell AND High [Lac]

  • Healthy tissue Colonization

    High dcell AND Low [Lac]

  • Tumor NOT colonized

    Low dcell AND High [Lac]

  • Tumor NOT colonized

    Low dcell AND High [Lac]

The simulation results between 35 hr and 75 hr for all the 4 possible scenarios that CATE can encounter, are shown below.

Case1_HighDHighL
Figure 6: Case: Tumor colonized - High dcell AND High [Lac]; AND-gate Switch is turned ON fully.
Case1_HighDLowL
Figure 7: Case: Healthy tissue colonized - High dcell AND Low [Lac]; AND-gate Switch is turned ON partially.
Case1_LowDHighL
Figure 8: Case: Tumor NOT colonized - Low dcell AND High [Lac]; AND-gate Switch is turned OFF fully.
Case1_LowDLowL
Figure 9: Case: Healthy tissue NOT colonized - Low dcell AND Low [Lac]; AND-gate Switch is turned OFF fully.
SemiLog_Azu_t
Figure 10: [Azu] vs Time with log scale on Y-axis; Shows the behavior of our tumor sensing circuit in all the different dcell and [Lac] conditions.
Azu_t
Figure 11: [Azu] vs Time with linear scale; Shows the behavior of our tumor sensing circuit in the different dcell and [Lac] conditions.

As is clear from the Figures 6-9, the AND-gate switch is fully turned ON for the case of tumor colonization with steady state Azurin concentration reaching 75000 nM (as shown in Figure 11). For the case of colonization of healthy tissue, the AND-gate switch is partially turned ON with steady state Azurin concentration at about 22000 nM. This is about 3.5 times lower than when the switch is fully ON. In the case of tumor and healthy tissue not colonized the steady state Azurin concentrations are around 20 nM and 17 nM, respectively, which represents a fully turned OFF AND-gate switch. This high-fold difference is better visible in the semi-log plot in Figure 10.

Bacterial colonization patterns

  • Homogeneous distribution in a Single spherical shell layer in tumour

  • Homogeneous distribution throughout Healthy tissue

  • Heterogeneous (Partitioned) distribution in a Single spherical shell layer in tumour

  • Heterogeneous (Partitioned) distribution in Double spherical shell layer in tumour

Limitations

Our model has some limitations. We do not model protein E production and cell lysis caused by it. Instead lysis is just simulated in effect as the end of production of AHL and Azurin and start of diffusion of Azurin. Moreover, a step signal is used as a trigger for the lysis. As mentioned in the parameters description, Azurin production is taken to be 10 times proportional to LuxI production. Also, killing mechanism of Azurin has not been modelled since that was not necessary to demonstrate the working of our project CATE in the scope of iGEM.

Tools used