WTF Model

Assumption: negligible AHL degradation in the layer

The colony produces AHL which diffuses out of the layer. In steady state, we only have diffusion perpendicular to the colony, which is modelled with a diffusion constant \( D = \SI{4.9e-6}{cm^2 s^{-1}} \) (this is the diffusion of AHL in water —is it appropriate?) (citation needed). There is also extracellular degradation of AHL, modelled by \( k_{\text{deg}} = \SI{5e-4}{min^{-1}} \) (citation needed). The average time needed for AHL to diffuse out of the layer (i.e. the time interval after which the concentration of AHL drops by \( 50\% \)) is:

\[\mathrm{d} t = \frac{w^2}{D} = \frac{0.05^2}{4.9 \cdot 10^{-6}} \simeq \SI{510}{s}\]

During time \( \Delta t \) there is also extracellular degradation of AHL. The percentage that gets degraded before diffusing out of the layer is less than:

\[\begin{aligned} \Delta t \frac{k_{\text{deg}} [\text{AHL}]}{[\text{AHL}]} &= \Delta t \, k_{\text{deg}} \\ &= \frac{510}{60}\cdot 5 \cdot 10^{-14} \simeq 0.4 \%\end{aligned}\]

Therefore, there is negligible degradation in the layer in comparison to diffusion.

Assumption: Only care about local AHL

How far does AHL reach before begin degraded? Considering the half-life of the protein:

\[\begin{aligned} t_{1/2} &= \frac{\ln(2)}{k_{\text{deg}}} \simeq \SI{1400}{min} \\ \\ d &= \sqrt{D t_{1/2}} \simeq \SI{6}{mm}\end{aligned}\]

Since \( d < r_1 = \SI{10}{mm} \), AHL will be substantially degraded before it reaches a colonization area far from where it is produced. We will thus assume that each cell is affected by the protein produced locally, i.e. in a small area around the cell.

Simplified model of AHL diffusion

Given that \( w \ll r_1 \) we assume that the colony appears locally as an infinite sheet of width \( w \). According to the assumptions 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 parallepiped of surface \( \mathrm{d} S \) and width \( \mathrm{d} x \) perpendicular to \( x \) axis, we can use Fick’s law to model diffusion of AHL. The flux of the protein is:

\[\Phi(x) = - D \left. \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x} \right|_{x}\]

The small volume can be either inside (\( |w/2| < x \)) or outside (\( |w/2| > x \)) the layer. The change of protein amount in a small volume inside the layer within time \( \mathrm{d} t \) is equal to diffusion plus production:

\[\begin{aligned} \mathrm{d} n &= (( \Phi(x) - \Phi(x + \mathrm{d} x) ) \mathrm{d} S + P \mathrm{d} V) \, \mathrm{d} t \\ \frac{\mathrm{d} n}{\mathrm{d} V \mathrm{d} t} &= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P \\ \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P\end{aligned}\]

where \( P \) is the volumic production in \( \si{mol L^{-1} s^{-1}} \). Outside the layer there is diffusion and degradation:

\[\begin{aligned} \mathrm{d} n &= (( \Phi(x) - \Phi(x + \mathrm{d} x) ) \mathrm{d} S - k_{\text{deg}} \mathrm{d} m) \, \mathrm{d} t \\ \frac{\mathrm{d} n}{\mathrm{d} V \mathrm{d} t} &= D \frac{\mathrm{d}^{2} [\text{AHL}]}{\mathrm{d} x^2} - k_{\text{deg}} \frac{\mathrm{d} m}{\mathrm{d} V} \\ \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= D \frac{\mathrm{d}^{2} [\text{AHL}]}{\mathrm{d} x^2} - k_{\text{deg}} [\text{AHL}]\end{aligned}\]

Diffusion inside the layer

We are interested in the concentration profile of AHL in the steady state. We assume that diffusion happens faster than cellular growth, so \( P \) is considered constant, and we proceed as follows for \( x \in (-w/2, w/2) \):

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= 0 \\ \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} &= - \frac{P}{D} \\ [\text{AHL}](x) &= - \frac{P}{2D} x^2 + c_0 x + c_1 \\ [\text{AHL}](x) &= - \frac{P}{2D} x^2 + c_1 & \text{\( c_0 x = 0\) because of symmetry}\end{aligned}\]

The concentration has a distinct parabolic shape.

Discuss: The symmetry of the boundaries probably no longer applies if we depart from the infinite sheet assumption; the concentration of AHL inside of the colony layer should be higher than that on the outside, due to different rates of diffusion.

Diffusion outside the layer

Applying a similar QSSA, we solve the the differential equation as follows for \( x \in (-\infty, -w/2) \cup (w/2, +\infty) \):

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= 0 \\ \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} - \frac{k_{\text{deg}}}{D} [\text{AHL}] &= 0 \\ \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} - \kappa^2 [\text{AHL}] &= 0 & \kappa = \sqrt{k_{\text{deg}}/D} \\ [\text{AHL}](x) &= c_2 \exp{(-\kappa x)} + c_3 \exp{(\kappa x)} + c_4 \end{aligned}\]

The constants \( c_* \) are in principle different for the two regions; due to symemtry, they are the same. We calculate the constants \( \kappa, 1/\kappa \):

\[\begin{aligned} \kappa &= \sqrt{k_{\text{deg}} / D} \\ &= \sqrt{\frac{\SI{5e-4}{min^{-1}}}{\SI{4.9e-6}{cm^2 s^{-1}}}} \\ &\simeq \SI{0.13}{\milli\metre^{-1}}\end{aligned}\]

\[1/\kappa \simeq \SI{7.67}{\milli\metre} \\\]

TODO: Interpret \( \kappa, 1/\kappa \).

Boundary conditions

AHL concentration should be \( 0 \) at \( \pm \infty \) (in our MATlab model infinity occurs at a large finite distance from the tumor). Thus, we need to prune the parts of the solutions that grow without bounds as \( |x| \to + \infty \) and set \( c_4 = 0 \):

\[\begin{aligned} [\text{AHL}] = \begin{cases} c_3 \exp{(\kappa x)} & x < -w/2 \\ c_2 \exp{(-\kappa x)} & x > w/2 \\ \end{cases}\end{aligned}\]

The flux and concentration of AHL should be continuous at the boundary of the small volume:

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x}(w^{+}/2) &= \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x}(w^{-}/2) \\ -\kappa c_2 \exp{(-\kappa w/2)} &= -\frac{P}{D} \frac{w}{2} \\ c_2 &= \frac{Pw}{2D\kappa} \exp{(\kappa w/2)}\end{aligned}\]

\[\begin{aligned} [\text{AHL}](w^{+}/2) &= [\text{AHL}](w^{-}/2) \\ c_2 \exp{(-\kappa w/2)} &= -\frac{P}{2D} \frac{w^2}{4} + c_0 \\ c_0 &= c_2 \exp{(-\kappa w/2)} + \frac{P}{2D} \frac{w^2}{4} \\ c_0 &= \frac{Pw}{2D\kappa} + \frac{Pw^2}{8D} \\ c_0 &= \frac{Pw}{2D}(1/\kappa + w/4)\end{aligned}\]

We can simplify this result since \( 1/\kappa \simeq \SI{7.67}{\milli\metre} \) and \( w/4 \simeq \SI{0.125}{\milli\metre} \):

\[c_0 \simeq \frac{Pw}{2D\kappa} = \frac{Pw}{2 \sqrt{D k_{\text{deg}}}}\]

We thus effectively disregard the intra-layer variation of concentration, as negligible compared to the gradient produced by diffusion outside the layer.

Full solution

The final concentration of AHL is:

\[[\text{AHL}] = \frac{Pw}{2 D \kappa} \cdot \begin{cases} \exp{(-\kappa (w/2 + x))} & x < -w/2 \\ \kappa/w \left( \frac{w^2}{4} - x^2 + \frac{w}{\kappa} \right) & -w/2 < x < w/2 \\ \exp{(\kappa (w/2 - x))} & x > w/2 \end{cases}\]

Relating \( [\text{AHL}] \) and \( [\text{luxI}] \)

The concentration of AHL at the boundary is \( \frac{Pw}{2 D \kappa} \). This concentration should be such that Quorum Sensing is active precisely when the density of bacteria is high enough. As can be seen, the only parametre that can be readily tuned is the volumic production rate \( P \). This rate is proportional to the production rate of AHL inside a single cell, and the constant of proportionality is the surface coverage of cells, \( d_{\text{cell}} \). We assume that the intracellular degradation of AHL as well as the intracellular sequestering of AHL inside the cells by other molecules is negligible compared to diffusion (is that reasonable?). Thus the production of AHL is proportional to the concentration of the luxI enzyme:

\[\begin{aligned} P &= d_{\text{cell}} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} \\ &\simeq d_{\text{cell}} a_{\text{AHL}} [\text{luxI}] \\ \\ [\text{AHL}](w/2) &= d_{\text{cell}} \, a_{\text{AHL}} [\text{luxI}] \frac{w}{2 D \kappa}\end{aligned}\]

We note the following in the above equation:

\( a_{\text{AHL}}, D, \kappa, w \) are constant parametres
\( d_{\text{cell}} \) varies bewteen tumor and healthy tissue, but is otherwise fixed
\( [\text{luxI}] \) is the only readily tunable parametre

Thus, in order to calibrate the Quorum Sensing activation, we have to tune the intracellular concentration of luxI. A plot of AHL as a function of distance from the colony layer is given below. The value of (extracellular) \( [\text{AHL}] \) has been normalized against intracellular \( [\text{luxI}] \).

We note that at a distance of \( 2 r_1 = \SI{20}{\milli\metre} \) from the layer, the relative concentration has dropped at \( 0.21 / 2.7 < 8\% \) of that near the colony layer.

Concerning LuxR

LuxR is under a constitutive promoter of strength \( a_{\text{luxR}} \) and its degradation rate is \( d_{\text{luxR}} \) (to be specified). AHL binds and stabilizes LuxR; LuxR-AHL molecules can only act as transcription factors when they form polymers (citation needed). LuxR also forms polymers with itself, which are inactive. Since we are modelling the steady state, where the concentration of AHL is high enough, the following simplifications apply:

LuxR does not form dimers with itself; it exists either as single molecule or bound to AHL
LuxR-AHL exists as a single molecule or as a dimer. We assume that the formation and dissociation of dimers is very fast compared to other reactions (e.g. binding of LuxR to AHL), so that the two forms are at equilibrium.
Degradation of bound forms of LuxR is negligible compared to degradation of free LuxR.

We can now write the following equations:

\[\begin{aligned} [\text{LuxR}]_0 &= \frac{a_{\text{LuxR}}}{d_{\text{LuxR}}} & \text{steady state concentration} \\ [\text{LuxR-AHL}] &= K_2 [\text{LuxR}]^2 [\text{AHL}]^2 & \text{rapid equilibrium} \\ [\text{LuxR}] &= [\text{LuxR}]_0 - 2 [\text{luxR-AHL}] & \text{mass conservation}\end{aligned}\]

where the binding constant is \( K_2 = \SI{5e17}{\molar^{-2}} \) (citation needed). Substituting \( [\text{AHL}] \) and \( [\text{LuxR}] \), we get:

\[\begin{aligned} \label{eq:conserve} [\text{LuxR-AHL}] &= K_2 ([\text{LuxR}]_0 - 2 [\text{LuxR-AHL}])^2 \left( d_{\text{cell}} \, a_{\text{AHL}} [\text{LuxI}] \frac{w}{2 D \kappa} \right)^2 \end{aligned}\]

Influence of Lac concentration

Our hybrid promoter is affected by levels of Lac as follows: While Lac is not present, LldR dimerizes and binds to the \( O_2 \) sites, preventing RNApolymerase from binding and thus repressing transcription of the output operon. Once Lac is present, it quickly binds to LldR, and removes it from the promoter region; thus transcription occurs. In fact, the resulting Lldr-Lac molecule also promotes the said transcription, once it forms dimers with itself. Thus, presence of Lac has a significant impact on the transcription rate. The formal model for LldR and Lac interaction would be as follows:

\[\begin{aligned} [\text{LldR}]_0 &= \frac{a_{\text{LldR}}}{d_{\text{LldR}}} & \text{constitutive production} \\ [\text{LldR}]_0 &= [\text{LldR}] + 2 [\text{Lldr-Lac}] + 2 [\text{Lldr}]_2 & \text{mass conservation} \\ P_{\text{Lac}} &= \frac{1}{1 + \left(\frac{[\text{LldR}]_2}{K_{\text{LldR}_2}} \right)^{n_{\text{LldR}}}} \frac{\left(\frac{[\text{LldR-Lac}]}{K_{\text{LldR-Lac}}} \right)^{n_{\text{LldR-Lac}}}}{1 + \left(\frac{[\text{LldR-Lac}]}{K_{\text{LldR-Lac}}} \right)^{n_{\text{LldR-Lac}}}}\end{aligned}\]

This system is overly complex and we were not able to find values for the relative parametres in literature. We will thus condense our model as if Lac is directly acting as a promoter in our system:

\[\begin{aligned} P_{\text{Lac}} &\simeq \frac{\left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}}{1 + \left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}}\end{aligned}\]

We expect that \( n_{\text{Lac}} \) is fixed to sufficiently high value; we will try to tune \( K_{\text{Lac}} \) so that the construct behaves differently in healthy and tumor tissue.

According to reasearch (citation needed), the concentration of lactate in blood, adipose tissue and muscle tissue is, respectively: \( \SI{0.6}{\milli M}, \SI{1.1}{\milli M}, \SI{1.9}{\milli M} \). We will consider a reference concentration of \( \SI{1.0}{\milli M} \). According to research (citation needed), the concentration of lactate in tumor tissue is \( \SIrange{5}{12}{\micro \mol g^{-1}} \). We will use \( \SI{7.1}{\micro \mol g^{-1}} \) as a reference value, as suggested in the paper. Assuming a tumor density of \( \SI{1e3}{g l^{-1}} \), we approximate lactate concentration in tumor tissue to be \( \SI{7.1}{\milli M} \).

To achieve differential response, we would like the value of \( K_{\text{Lac}} \) to be in about the middle of the range \( \SIrange{1}{7.1}{\milli M} \). For construct K1847009, we know that (ETH 2015) \( K_{\text{Lac}} \simeq \SI{2.361}{\milli M} \) and \( n_{\text{Lac}} \simeq 1.7 \). We obtain \( P_{\text{Lac}} \simeq 19\% \) in healthy tissue and \( P_{\text{Lac}} \simeq 87\% \) in tumor tissue, a 4-fold increase.

Incorporating positive feedback

The intracellular levels of AHL affect LuxI expression. Assumming \( a_{\text{LuxI}} \) is the maximal production rate of LuxI, \( d_{\text{LuxI}} \) is the degradation rate, \( k_{\text{LuxI}} \) is the leakiness of the promoter and \( P_{\text{Lux-Lac}} \) is the combined effect of the \( P_{\text{Lux}} \) promoter and \( P_{\text{Lac}} \) activator, the ODE governing the production of \( [\text{LuxI}] \) is:

\[\frac{\mathrm{d} [\text{LuxI}]}{\mathrm{d} t} = a_{\text{LuxI}} (k_{\text{LuxI}} + (1 - k_{\text{LuxI}}) P_{\text{Lux-Lac}}) - d_{\text{LuxI}} [\text{luxI}]\]

where

\[\begin{aligned} P_{\text{Lux-Lac}} &= P_{\text{Lux}} \, P_{\text{Lac}} \\ \\ P_{\text{Lux}} &= \frac{\left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR-AHL}}} \right)^{n_{\text{LuxR}}}}{1 + \left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR-AHL}}} \right)^{n_{\text{LuxR}}}}\end{aligned}\]

Solving the above at steady state:

\[\begin{aligned} \frac{\mathrm{d} [\text{luxI}]}{\mathrm{d} t} &= 0 \\ [\text{luxI}] &= \frac{a_{\text{luxI}}}{d_{\text{luxI}}} (k_{\text{luxI}} + (1 - k_{\text{luxI}}) P_{\text{Lux-Lac}})\end{aligned}\]

Substitutting this into \( \eqref{eq:conserve} \) and expanding:

\[\begin{aligned} [\text{luxR-AHL}] &= K_2 \left(\frac{a_{\text{LuxR}}}{d_{\text{LuxR}}} - 2 [\text{luxR-AHL}]\right)^2 \times \\ &\quad \times \left( d_{\text{cell}} \, a_{\text{AHL}} \frac{a_{\text{luxI}}}{d_{\text{luxI}}} \left(k_{\text{luxI}} + (1 - k_{\text{luxI}}) \frac{\left(\frac{[\text{luxR-AHL}]}{K_{\text{luxR-AHL}}} \right)^{n_{\text{luxR}}}}{1 + \left(\frac{[\text{luxR-AHL}]}{K_{\text{luxR-AHL}}} \right)^{n_{\text{luxR}}}} P_{\text{Lac}} \right) \frac{w}{2 D \kappa} \right)^2 \end{aligned}\]

We use fzero in MatLAB to find roots (\( [\text{luxR-AHL}] \)) for the above equation.

Team:ETH Zurich/WTF