Revision as of 15:18, 26 October 2017

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Model of our circuit

The tumor sensing circuit is composed of several proteins interacting 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. BLABLA references DAVID. Here is a detailed overview of the tumor sensing circuit:

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 expression of the hybride transcription factor when unbound to lactate, and deactivated when bound to it.

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 regards 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:

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

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 each bacteria produces via the expression of the enzyme LuxI a basal amount of a small chemical (here AHL) that diffuses in the environment and into neighboring cells. When AHL is 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.

We can therefore write the following equations:

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

Hybrid Lux-Lac promoter

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:

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

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, we get:

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

Production of AHL

AHL is produced intracellularly by LuxI and diffuses then freely through the membrane [1]. Modeling the production of AHL 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 bacteria 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).

\[\begin{aligned} P_{\text{AHL}} &= d_{\text{cell}} a_{\text{AHL}} [\text{luxI}] \end{aligned}\]

The missing link between AHL production and its concentration: DIFFUSION

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 into 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 hypothesis which will be detailed below, get the relationship between AHL production and its local concentration around bacterial cells.

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) (http://jb.asm.org/content/185/5/1485.long). We also take into account extracellular degradation of AHL, described by the degradation constant \( k_{\text{deg}} = \SI{5e-4}{min^{-1}} \) (http://jb.asm.org/content/185/5/1485.long). 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:

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

Therefore, we can consider for further work that the degradation of AHL happening in the layer is negligible.

Assumption: AHL doesn't diffuses 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 reaches before begin degraded. Considering the half-life of AHL and the mean distance covered in this given time by diffusion:

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

Since \( d < r_1 = \SI{10}{mm} \), AHL will be substantially degraded before it reaches a 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 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:

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

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 the diffusion inside the layer

We are interested in the concentration profile of AHL in the steady state. Indeed, we assume that diffusion happens faster than the colonization of the bacteria (happening over 2 days REF), so \( P_{\text{AHL}} \) is considered constant in this quasi steady state assumption (QSSA), and we proceed as follows for \( x \in (-w/2, w/2) \):

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

The concentration has a parabolic shape.

Solving the diffusion outside the layer

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

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= 0 \\ \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} - \frac{k_{\text{deg}}}{D} [\text{AHL}] &= 0 \\ \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} - 1/\kappa^2 [\text{AHL}] &= 0 & \kappa = \sqrt{D/k_{\text{deg}}} \simeq \SI{7.7}{\milli\metre} \space \text{is the characteristic length of diffusion}\\ [\text{AHL}](x) &= c_2 \exp{(-x/\kappa)} + c_3 \exp{(x/\kappa)} + c_4 \end{aligned}\]

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:

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x}(w^{+}/2) &= \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x}(w^{-}/2) \\ -(1/\kappa) c \exp{(- w/2\kappa)} &= -\frac{P_{\text{AHL}}}{D} \frac{w}{2} \\ c &= \frac{P_{\text{AHL}}w}{2D\kappa} \exp{(w/2\kappa)}\end{aligned}\]

We also have thanks to the continuity of the concentration of AHL:

\[\begin{aligned} [\text{AHL}](w^{+}/2) &= [\text{AHL}](w^{-}/2) \\ c \exp{(-\kappa w/2)} &= -\frac{P_{\text{AHL}}}{2D} \frac{w^2}{4} + c_1 \\ c_1 &= c \exp{(-w/2\kappa)} + \frac{P_{\text{AHL}}}{2D} \frac{w^2}{4} \\ c_1 &= \frac{P_{\text{AHL}}w\kappa}{2D} + \frac{P_{\text{AHL}}w^2}{8D} \\ c_1 &= \frac{P_{\text{AHL}}w}{2D}(\kappa + w/4)\end{aligned}\]

If we assess the order of magnitude of each term, we notice that 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:

\[[\text{AHL}] = \frac{P_{\text{AHL}} w \kappa}{2 D} \cdot \begin{cases} \exp{(\frac{1}{\kappa} (x + w/2))} & for \space x < -w/2 \\ \ 1+\frac{1}{w \kappa}\left(\frac{w^2}{4} - x^2 \right) & for \space -w/2 < x < w/2 \\ \exp{(\frac{1}{\kappa} (w/2 - x))} & for \space x > w/2 \end{cases}\]

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, where bacteria are. This is the AHL concentration they will be exposed to and to which they will react. If we try to calculate the concentration of AHL at x=0, which is the maximum concentration, 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 lengths numerical values 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. 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 intuitively in a high homogenization of AHL concentration into the layer.

References

^ Kaplan HB, Greenberg EP. Diffusion of autoinducer is involved in regulation of the Vibrio fischeri luminescence system. Journal of Bacteriology. 1985;163(3):1210-1214.

@@ Line 130: / Line 130: @@
      <p>When inside (<span class="math">\( x &lt; |w/2| \)</span>) of the layer, the change of protein amount within time length <span class="math">\( \mathrm{d} t \)</span> is equal to diffusive transports plus production:</p>
      <p><span class="math">\[\begin{aligned}
-         \mathrm{d} n &amp;= (( \Phi(x) - \Phi(x + \mathrm{d} x) ) \mathrm{d} S + P \mathrm{d} V) \, \mathrm{d} t \\
+         \mathrm{d} n &amp;= (( \Phi(x) - \Phi(x + \mathrm{d} x) ) \mathrm{d} S + P_{\text{AHL}} \mathrm{d} V) \, \mathrm{d} t \\
-         \frac{\mathrm{d} n}{\mathrm{d} V \mathrm{d} t} &amp;= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P \\     \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &amp;= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P\end{aligned}\]</span></p>
+         \frac{\mathrm{d} n}{\mathrm{d} V \mathrm{d} t} &amp;= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P_{\text{AHL}} \\     \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &amp;= D \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} + P_{\text{AHL}}\end{aligned}\]</span></p>
-     <p>where <span class="math">\( P \)</span> is the <em>volumic production</em> of <span style="font-variant: small-caps;">AHL</span> in <span class="math">\( \si{mol L^{-1} s^{-1}} \)</span>.</p>
+     <p>where <span class="math">\( P_{\text{AHL}} \)</span> is the <em>volumic production</em> of <span style="font-variant: small-caps;">AHL</span> in <span class="math">\( \si{mol L^{-1} s^{-1}} \)</span>.</p>
      <p>Outside (<span class="math">\( x &gt; |w/2| \)</span>) of the layer, where there is diffusion and degradation, we get:</p>
      <p><span class="math">\[\begin{aligned}
@@ Line 141: / Line 141: @@
 <section>
      <h2 id="diffusion-inside-the-layer">Solving for the diffusion inside the layer</h2>
-     <p>We are interested in the concentration profile of <span style="font-variant: small-caps;">AHL</span> in the steady state. Indeed, we assume that diffusion happens faster than the colonization of the bacteria (happening over 2 days REF), so <span class="math">\( P \)</span> is considered constant in this quasi steady state assumption (QSSA), and we proceed as follows for <span class="math">\( x \in (-w/2, w/2) \)</span>:</p>
+     <p>We are interested in the concentration profile of <span style="font-variant: small-caps;">AHL</span> in the steady state. Indeed, we assume that diffusion happens faster than the colonization of the bacteria (happening over 2 days REF), so <span class="math">\( P_{\text{AHL}} \)</span> is considered constant in this quasi steady state assumption (QSSA), and we proceed as follows for <span class="math">\( x \in (-w/2, w/2) \)</span>:</p>
      <p><span class="math">\[\begin{aligned}
          \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &amp;= 0 \\
-         \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} &amp;= - \frac{P}{D} \\
+         \frac{\mathrm{d}^2 [\text{AHL}]}{\mathrm{d} x^2} &amp;= - \frac{P_{\text{AHL}}}{D} \\
-         [\text{AHL}](x) &amp;= - \frac{P}{2D} x^2 + c_0 x + c_1  \\
+         [\text{AHL}](x) &amp;= - \frac{P_{\text{AHL}}}{2D} x^2 + c_0 x + c_1  \\
-         [\text{AHL}](x) &amp;= - \frac{P}{2D} x^2 + c_1 &amp; \text{\( c_0 = 0\) because of symmetry}\end{aligned}\]</span></p>
+         [\text{AHL}](x) &amp;= - \frac{P_{\text{AHL}}}{2D} x^2 + c_1 &amp; \text{\( c_0 = 0\) because of symmetry}\end{aligned}\]</span></p>
      <p>The concentration has a parabolic shape.</p>
 </section>
@@ Line 171: / Line 171: @@
      <p><span class="math">\[\begin{aligned}
          \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x}(w^{+}/2) &amp;= \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} x}(w^{-}/2) \\
-         -(1/\kappa) c \exp{(- w/2\kappa)} &amp;= -\frac{P}{D} \frac{w}{2} \\
+         -(1/\kappa) c \exp{(- w/2\kappa)} &amp;= -\frac{P_{\text{AHL}}}{D} \frac{w}{2} \\
-         c &amp;= \frac{Pw}{2D\kappa} \exp{(w/2\kappa)}\end{aligned}\]</span></p>
+         c &amp;= \frac{P_{\text{AHL}}w}{2D\kappa} \exp{(w/2\kappa)}\end{aligned}\]</span></p>
 <p>We also have thanks to the continuity of the concentration of <span style="font-variant: small-caps;">AHL</span>:</p>
      <p><span class="math">\[\begin{aligned}
          [\text{AHL}](w^{+}/2) &amp;= [\text{AHL}](w^{-}/2) \\
-         c \exp{(-\kappa w/2)} &amp;= -\frac{P}{2D} \frac{w^2}{4} + c_1 \\
+         c \exp{(-\kappa w/2)} &amp;= -\frac{P_{\text{AHL}}}{2D} \frac{w^2}{4} + c_1 \\
-         c_1 &amp;= c \exp{(-w/2\kappa)} + \frac{P}{2D} \frac{w^2}{4} \\
+         c_1 &amp;= c \exp{(-w/2\kappa)} + \frac{P_{\text{AHL}}}{2D} \frac{w^2}{4} \\
-         c_1 &amp;= \frac{Pw\kappa}{2D} + \frac{Pw^2}{8D} \\
+         c_1 &amp;= \frac{P_{\text{AHL}}w\kappa}{2D} + \frac{P_{\text{AHL}}w^2}{8D} \\
-         c_1 &amp;= \frac{Pw}{2D}(\kappa + w/4)\end{aligned}\]</span></p>
+         c_1 &amp;= \frac{P_{\text{AHL}}w}{2D}(\kappa + w/4)\end{aligned}\]</span></p>
      <p>If we assess the order of magnitude of each term, we notice that can simplify this result since <span class="math">\( \kappa \simeq \SI{7.7}{\milli\metre} \)</span> and <span class="math">\( w/4 \simeq \SI{0.125}{\milli\metre} \)</span>:</p>
-     <p><span class="math">\[c_1 \simeq \frac{Pw\kappa}{2D}\]</span></p>
+     <p><span class="math">\[c_1 \simeq \frac{P_{\text{AHL}}w\kappa}{2D}\]</span></p>
 </section>
 <section>
@@ Line 187: / Line 187: @@
      <p>The final concentration of <span style="font-variant: small-caps;">AHL</span> is:</p>
      <p><span class="math">\[[\text{AHL}] =
-         \frac{P w \kappa}{2 D} \cdot
+         \frac{P_{\text{AHL}} w \kappa}{2 D} \cdot
          \begin{cases}
          \exp{(\frac{1}{\kappa} (x + w/2))} &amp; for \space x &lt; -w/2 \\
@@ Line 199: / Line 199: @@
 </p>
 <p><span class="math">\[\begin{aligned}\text{AHL}(x=0) =
-         \frac{P w \kappa}{2 D}(1+\frac{w}{4 \kappa} )
+         \frac{P_{\text{AHL}} w \kappa}{2 D}(1+\frac{w}{4 \kappa} )
          \end{aligned}\]</span></p>
 <p>With the lengths numerical values applying to our problem, since we have <span class="math">\(\frac{w}{4 \kappa} \simeq 0.02\)</span>, we can neglect this component of the equations, which amounts to ignore the intra-layer variation. This simplification is allowed to us because of the very fast diffusion compared to the width of the colonization layer (high <span class="math">\(\kappa\)</span> and small w), which results intuitively in a high homogenization of AHL concentration into the layer.

Difference between revisions of "Team:ETH Zurich/Model/Environment Sensing/parameter space"