Model Overview

This section presents a brief overview of the COMSOL model.

Geometry

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.

Equations ... how

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 R_C 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 ... get their values

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.

RESULTS

We could simulate for a geometry of the system closer to the real-life tumor conditions ... check out our results

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. This helped us to test our tumour sensing AND-Gate switch functionality in all the possible real-life scenarios that CATE might encounter in context of the intended application.

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 d_cell 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
  

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.

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.

In Figure 5, d_cell shows the growth of the cell density inside the layer and P_Lux-Lac shows the main function responsible for the switching functionality based on the environmental conditions of d_cell 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. d_cell and [Lac], there are 4 possible binary combinations that CATE can encounter in real-life scenario, viz.:

Tumor Colonization

High d_cell AND High [Lac]

Healthy tissue Colonization

High d_cell AND Low [Lac]

Tumor NOT colonized

Low d_cell AND High [Lac]

Healthy tissue NOT colonized

Low d_cell AND Low [Lac]

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

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

To further push the limits of our system, we tested the environment sensing AND-gate switching functioning in different colonization patterns. Although any possible colonization pattern can be simulated, we simulated the logically closest patterns (compared to a spherical shell layer), that we could think of, and introduced heterogenity through partitions.

Case 1: Homogeneous distribution in a Single spherical shell layer in tumour

already simulated as shown in previous results

Case 2: Heterogeneous distribution (Partitions) in a Single spherical shell layer in tumour

another possible feasible scenario

Case 3: Heterogeneous distribution (Partitions) in Double spherical shell layer in tumour

introducing some more heterogenity in the system

Case 4: Homogeneous distribution throughout Healthy tissue

with d_cell = 0.0005, as in this geometry the AND-gate is more easily activated as comparaed to other geometries, since difusion can not occur within the inner region where bacteria colonize, and thus this colonized inner region acts as a huge source, with no diffusion between the cells, as shown in the Figure 15.

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

The above figures show that our Tumor sensing circuit functions properly in the different colonization patterns simulated. According to Figure 15, in case of colonization of a healthy tissue, the switch is only partially triggered (slower rise and lesser final steady state [Azu]), when compared to Case 1 in Figure 12, that has a fully ON switch indicated by the rapid rise and approximately 3.5 times higher-fold increase in [Azu]

Killing Area

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

• COMSOL Multiphysics 5.2a by COMSOL Inc.

• MATLAB R2016b by MathWorks