To be able to understand the behavior of our model, and later how to tune it via genetic engineering, we first need to establish the equations that govern its responses. Here is a detailed description of our model.
The tumor sensing circuit is composed of several proteins that interact with small molecules (AHL and lactate) and DNA (at the transcription factors binding sites). To establish a model describing the behavior of our circuit, we first had to understand the way these interactions are happening inside of the cell. Building on the description of the Tumor Sensor circuit, here is a more detailed overview:
Simplification of the lactate sensing
Let us first focus on the lactate sensing part of the circuit. In the cell, two proteins are produced:
LldP: a transmembrane protein enabling the transport of extracellular lactate into E. coli.
LldR: a transcription factor, repressing the activity of the hybrid promoter when not bound to lactate. Lldr releases repression once it binds to lactate.
To model precisely the regulation of the hybrid promoter by lactate, it would be necessary to take into account all the following points:
How the intracellular lactate concentration behaves in regard to the expression level of LldP and the extracellular lactate concentration
What is the binding constant between LldR and the lactate
What is the binding dynamics of Lldr to the operon, and how it affects the transcription rate downstream
In an effort to simplify our model to reduce it to the most meaningful parameters, and because it has already extensively been studied and characterized by previous iGEM teams, we have chosen not to take into account the complexity of the lactate sensing pathway and rather use a phenomenological model to describe its influence. We rely on the characterization of the lactate sensor using several expression regulation sequences done by the ETH 2015 iGEM team. We consider therefore that lactate sensing follows a Hill function as following:
As a result, the schematics of the circuit can be simplified this way:
Quorum Sensing sensor modelization
The sensing of the bacterial cell density is done via a quorum sensing circuit. The principles behind quorum sensing is that, via the expression of the enzyme LuxI, each bacteria produces a basal amount of a small molecule (here N-acyl homoserine lactone, AHL) that diffuses in the environment and into neighboring cells. When AHL is present in sufficient quantity, it binds to the intracellular LuxR and induces the production of more LuxI, which in turn results in the production of more AHL. This positive feedback loop results in the activation of the operon containing the LuxI gene when the cell density reaches a critical threshold.
Concerning LuxR
LuxR is under a constitutive promoter of strength \( a_{\text{luxR}} \) and its degradation rate is \( d_{\text{luxR}} \). AHL binds and stabilizes LuxR; LuxR-AHL molecules can only act as transcription factors when they form a tetramer (2*AHL+2*LuxR). Since we are modeling the steady state, the following simplifications apply:
We consider that the total amount of LuxR present in the cell is constant, and only depends on its constitutive expression and degradation rate.
We consider the global binding equilibrium between LuxR and AHL without taking into account the intermediary dimers.
The expression of the main operon containing the LuxI, Bfr and Azurin genes is regulated by a hybrid promoter activated by the quorum sensing and repressed by the lactate sensing (the repression being released in presence of lactate). This hybrid promoter should behave as a AND-gate: mathematically, this corresponds to multiplying the Hill functions describing their behavior.
Along with the lactate concentration, the intracellular levels of LuxR-AHL complexes affect LuxI expression. With \( a_{\text{LuxI}} \) being the maximal production rate of LuxI, \( d_{\text{LuxI}} \) the degradation rate, \( k_{\text{LuxI}} \) the leakiness of the promoter and \( P_{\text{Lux-Lac}} \) the combined effect of the \( P_{\text{Lux}} \) and \( P_{\text{Lac}} \) regulating sequences behavior, the ODE governing the production of \( [\text{LuxI}] \) can be written as following:
AHL is produced intracellularly by LuxI and diffuses then freely through the membrane [1]. Modeling the production of AHL is quite straightforward: it is proportional to the amount of LuxI present intracellularly. To describe the production per unit of volume though, we have to take into account the bacterial cell density present locally and take it as a dilution coefficient (for instance, if the cells occupy locally half of the volume, then the intracellularly produced AHL would be instantly diluted two times as it diffuses into the surrounding environment).
The missing link between AHL production and its concentration: DIFFUSION MODEL
To be able to close mathematically the feedback loop, we still miss an equation: we need to know how the intracellular production of AHL translates into the AHL concentration in the environment. For that, we have to consider the diffusion of AHL around the colonized area of the tumor. By solving the equation governing AHL transport, we can, under certain assumptions which will be detailed below, get the relationship between AHL production and its local concentration around bacterial cells.
Assumptions
Assumption: negligible AHL degradation in the layer
The colony produces AHL which diffuses out of the layer. Because of the symmetry of the problem, we only consider diffusion in the radial direction, being characterized by the diffusion constant \( D = \SI{4.9e-6}{cm^2 s^{-1}} \) (diffusion constant of AHL in water) [4]. We also take into account extracellular degradation of AHL, described by the degradation constant \( k_{\text{deg}} = \SI{5e-4}{min^{-1}} \)[4]. Using the relationship between the mean square distance and time in a Brownian movement, we can estimate the average time needed for AHL to diffuse out of the layer:
\[\Delta t = \frac{w^2}{D} = \frac{0.05^2}{4.9 \cdot 10^{-6}} \simeq \SI{510}{s}\]
During this time \( \Delta t \), we can estimate the magnitude of the degradation of AHL. The proportion of AHL that gets degraded before diffusing out of the layer can be estimated as:
Therefore, we can consider for further work that the degradation of AHL happening in the layer is negligible.
Assumption: AHL doesn't diffuse far from where it is produced
To assess whether we would have to consider in a given point of the tumor the AHL coming from every part of the tumor or only the closest area, we have to estimate how far AHL diffuses before being degraded. Considering the half-life of AHL and the mean distance covered in this given time by diffusion:
Since \( d < r_1 = \SI{10}{mm} \), AHL will be substantially degraded before it reaches a colonized area far from where it is produced. We will thus assume that we can restrict the study of the diffusion of AHL to a local one, without considering the AHL that could come from the other side of the tumor.
Model of AHL diffusion
Given the latter assumption, and given \( w \ll r_1 \), we assume that the colonized area appears locally as an infinite sheet of width \( w \). According to the first assumption stated above, we consider only production and diffusion to be significant within the sheet, and diffusion and degradation to be significant outside the sheet. We model the sheet as perpendicular to the \( x \) axis, centered at \( 0 \), covering the interval \( [-w/2, w/2 ] \).
For a small parallelepiped of surface \( \mathrm{d} S \) and width \( \mathrm{d} x \) perpendicular to \( x \) axis (see figure below), we can use Fick’s law to model diffusion of AHL.
The flux of the protein is:
\[\Phi(x) = - D \left. \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x} \right|_{x}\]
When inside (\( x < |w/2| \)) of the layer, the change of protein amount within time length \( \mathrm{d} t \) is equal to diffusive transports plus production:
\[\begin{aligned}
\mathrm{d} n &= (( \Phi(x) - \Phi(x + \mathrm{d} x) ) \mathrm{d} S + P_{\text{AHL}} \mathrm{d} V) \, \mathrm{d} t \\
\frac{\mathrm{d} n}{\mathrm{d} V \mathrm{d} t} &= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P_{\text{AHL}} \\ \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P_{\text{AHL}}\end{aligned}\]
where \( P_{\text{AHL}} \) is the volumic production of AHL in \( \si{mol L^{-1} s^{-1}} \).
Outside (\( x > |w/2| \)) of the layer, where there is diffusion and degradation, we get:
\[\begin{aligned}
\mathrm{d} n &= (( \Phi(x) - \Phi(x + \mathrm{d} x) ) \mathrm{d} S - k_{\text{deg}} \mathrm{d} n) \, \mathrm{d} t \\
\frac{\mathrm{d} n}{\mathrm{d} V \mathrm{d} t} &= D \frac{\mathrm{d}^{2} [\text{AHL}]}{\mathrm{d} x^2} - k_{\text{deg}} \frac{\mathrm{d} n}{\mathrm{d} V} \\
\frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= D \frac{\mathrm{d}^{2} [\text{AHL}]}{\mathrm{d} x^2} - k_{\text{deg}} [\text{AHL}]\end{aligned}\]
Solving for diffusion inside the layer
We are interested in the concentration profile of AHL at steady state. Indeed, we assume that diffusion happens faster than the colonization of the bacteria (happening over 2 days [1]), so \( P_{\text{AHL}} \) is considered constant in this quasi steady state assumption (QSSA), and we proceed as follows for \( x \in (-w/2, w/2) \):
We have \( c_2 = c_3 := c \) due to symmetry of the problem.
Boundary conditions
AHL concentration is \( 0 \) at \( \pm \infty \). This implies \( c_4 = 0 \) and:
\[\begin{aligned}
[\text{AHL}] =
\begin{cases}
c \exp{(x/\kappa)} & for \space x < -w/2 \\
c \exp{(-x/\kappa)} & for \space x > w/2 \\
\end{cases}\end{aligned}\]
The flux of AHL should be continuous at the boundary of the colonized area:
If we assess the order of magnitude of each term, we notice that one can simplify this result since \( \kappa \simeq \SI{7.7}{\milli\metre} \) and \( w/4 \simeq \SI{0.125}{\milli\metre} \):
\[c_1 \simeq \frac{P_{\text{AHL}}w\kappa}{2D}\]
Full solution
The final concentration of AHL is:
\[[\text{AHL}] =
\frac{P_{\text{AHL}} w \kappa}{2 D} \cdot
\begin{cases}
\exp{(\frac{1}{\kappa} (x + w/2))} & for \space x < -w/2 \\
\ 1+\frac{1}{w \kappa}\left(\frac{w^2}{4} - x^2 \right) & for \space -w/2 < x < w/2 \\
\exp{(\frac{1}{\kappa} (w/2 - x))} & for \space x > w/2
\end{cases}\]
A dimensional analysis can confirm that [AHL] is a concentration in M, as \(P_{\text{AHL}}\) is a production rate in M.min-1, w and \(\kappa\) two lengths in m, and D a diffusion coefficient in m2.min-1 (the remaining expressions are dimensionless).
AHL concentration inside of the layer
To complete our model at the bacterial circuit level, we only need to know the AHL concentration inside the colonization layer. This is the AHL concentration the bacteria will be exposed to and to which they will react. If we compute the concentration of AHL at x=0, which is where the concentration is maximal, we get:
\[\begin{aligned}\text{AHL}(x=0) =
\frac{P_{\text{AHL}} w \kappa}{2 D}(1+\frac{w}{4 \kappa} )
\end{aligned}\]
With the numerical values of w and \(\kappa\) applying to our problem, since we have \(\frac{w}{4 \kappa} \simeq 0.02\), we can neglect this component of the equations, which amounts to ignore the intra-layer variation. The intuition is that this simplification is allowed to us because of the very fast diffusion compared to the width of the colonization layer (high \(\kappa\) and small w), which results in a high homogenization of AHL concentration into the layer.
We therefore get the following equation relating AHL concentration to its volumetric production:
\[[\text{AHL}] =
\frac{P_{\text{AHL}} w \kappa}{2 D}
\]
Initial test of our model
Our final in-vivo model comprises the following equations:
To simulate responses of our system under different values of lactate and bacterial cell density input, we solve this system using the fzero function of Matlab. Here is an example of the response of our system with typical values (see tables in the following part) to see how it behaves:
The white lines correspond to low levels of lactate and bacterial cell density (in healthy tissues) and the black lines represent the high levels (in tumor tissues). We can see that our system behaves well like an AND-gate as expected, but that the levels at which the transitions happen are not the right ones for our application. To find under which circumstances the system behaves as we need it to, we have to proceed to a parameter search.
