Team:ETH Zurich/modeling/qs

{{:Team:ETH Zurich/tpl/head|Quorum Sensing}}
{{:Team:ETH Zurich/tpl/fortables}}

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{{:Team:ETH Zurich/tpl/scrollbutton3|Model|green}}
{{:Team:ETH Zurich/tpl/scrollbutton2|Parameters|blue}}
{{:Team:ETH Zurich/tpl/scrollbutton3|Leakiness|red}}
{{:Team:ETH Zurich/tpl/scrollbutton|Cross-talk|blue}}
{{:Team:ETH Zurich/tpl/scrollbuttontworows|Alternate|Design|green}}
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<html><article id="Model"></html>
== Model ==

The Quorum sensing module is mainly involved in receiving signals from the sender cells. The sender cells produce some signaling molecules (inducers) which diffuse out of their membrane, then diffuse in receiver cells membrane, and bind to the regulator molecules in the receiver cells, thus activating the transcription of certain genes. In order to characterize the quorum sensing module with a transfer function, we consider different initial inputs of external AHL, and see how much output is produced, as it was done in the [https://2014.igem.org/Team:ETH_Zurich/expresults#Quorum_Sensing quorum sensing experiments].


As diffusion through the membrane is very fast<sup>[[Team:ETH_Zurich/project/references|[27]]]</sup>, according to Fick's law of diffusion, internal and external concentration of AHL can always be considered as equal. This can also be observed in the [https://2014.igem.org/Team:ETH_Zurich/modeling/diffmodel#Results diffusion model results]. When an initial external AHL concentration is given, AHL diffuses into the cells very quickly (less than 20 seconds)<sup>[[Team:ETH_Zurich/project/references|[27]]]</sup> until internal AHL concentration equals external concentration. Then as soon as some internal AHL is consumed in the cell, it is taken up again without affecting external concentration, because external volume is very high compared to internal volume. Therefore we can consider in this module that external AHL (which is equal to internal AHL) only degrades, with the rate of extracellular decay.
 
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<br/>
 
=== Chemical Species ===
{| class="wikitable"
|-
! '''Name'''
! Description
|-
|'''LuxAHL'''
|30C6-HSL is an acyl homoserine lactone which diffuses into the cell and mainly binds to LuxR. Here we consider internal concentrations of LuxAHL.
|-
|'''LuxR'''
|Constitutively expressed regulator protein that can bind LuxAHL and stimulate transcription of Bxb1.
|-
|'''RLux'''
|LuxR and LuxAHL complex which can dimerize.
|-
|'''DRLux '''
|Dimerized form of RLux.
|-
|'''mRNA<sub>Bxb1</sub>'''
|mRNA of the Bxb1 integrase being transcribed by the Lux promoter.
|-
|'''Bxb1'''
|Serine integrase that can fold into two conformations - Bxb1a and Bxb1b. We chose to use a common connotation for both conformations - Bxb1.
|-
|'''LasAHL'''
|30C12-HSL is an acyl homoserine lactone which diffuses into the cell and mainly binds to LasR. Here we consider internal concentrations of LasAHL.
|-
|'''LasR'''
|Constitutively expressed regulator protein that can bind LasAHL and stimulate transcription of ΦC31.
|-
|'''RLas'''
|LasR and LasAHL complex which can dimerize.
|-
|'''DRLas'''
|Dimerized form of RLas.
|-
|'''mRNA<sub>ΦC31</sub>'''
|mRNA of the ΦC31 integrase being transcribed by the Lux promoter.
|-
|'''ΦC31'''
|Serine integrase that can fold into two conformations - ΦC31a and ΦC31b. We chose to use a common connotation for both conformations - ΦC31.
|}

<br/>

=== Reactions ===

For the Lux system:
$$ \begin{align}
&\rightarrow LuxR \\
LuxAHL+LuxR & \leftrightarrow RLux\\
RLux+RLux &\leftrightarrow DRLux\\
DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\
P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\
mRNA_{Bxb1}&\rightarrow Bxb1\\
LuxAHL &\rightarrow \\
LuxR &\rightarrow  \\
RLux &\rightarrow\\
DRLux &\rightarrow\\
mRNA_{Bxb1} &\rightarrow\\
Bxb1 &\rightarrow
\end{align}$$

;For the Las system:
\begin{align}
&\rightarrow LasR \\
LasAHL+LasR & \leftrightarrow RLas \\
RLas+RLas & \leftrightarrow DRLas\\
DRLas+P_{LasOFF} & \leftrightarrow P_{LasON}\\
P_{LasON}&\rightarrow P_{LasON}+mRNA_{\phi C31}\\
mRNA_{\phi C31}&\rightarrow \phi C31\\
Las-AHL &\rightarrow \\
LasR &\rightarrow \\
RLas &\rightarrow\\
DRLas &\rightarrow\\
mRNA_{\phi C31} &\rightarrow \\
\phi C31 &\rightarrow \\
\end{align}

=== Differential Equations ===

Applying mass action kinetic laws, we obtain the following set of differential equations.
$$\begin{align*}
\frac{d[LuxAHL]}{dt} &= -d_{LuxAHL}[LuxAHL]\\
\frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[LuxAHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\
\frac{d[RLux]}{dt} &=  k_{RLux}[LuxAHL][LuxR] - k_{-RLux}[RLux]  - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ 
\frac{d[DRLux]}{dt} &= k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\
\frac{d[P_{LuxON}]}{dt} &= k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\
\frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + k_{mRNA_{Bxb1}} [P_{LuxON}] - d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\
\frac{d[Bxb1]}{dt} &=  k_{Bxb1} [mRNA_{Bxb1}] - d_{Bxb1}[Bxb1]\\
\end{align*}$$


The same holds true for the Las system.

'''From the original set of reactions, we reduce the rate of production of mRNA<sub>Bxb1</sub> to a Hill function of RLux instead of Mass action kinetics in terms of P<sub>LuxON</sub>  and P<sub>LuxOFF</sub>. For more information please check the [https://2014.igem.org/Team:ETH_Zurich/expresults characterization section].'''



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<html><article id="Parameters"></html>

== Characterization: K<sub>mLux</sub> and K<sub>mLas</sub> ==

=== Data ===

For the Quorum sensing module we used established experimentally determined parameters for the rate of formation of RLux (reference). Since, in the literature the other parameters were estimated or fitted to their data, we decided to determine the parameters specific to our system. Hence, we used our data for the remaining parameters. Our data was mainly a transfer function of normalized GFP concentration as a function of input LuxAHL concentrations. (link to data)

=== Assumptions ===

'''Assumption A'''

We assumed that the dimerization of RLux to DRLux is quick. Quasi steady state approximation (QSSA)  as follows

$$\frac{d[DRLux]}{dt} = k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \approx 0\\$$

'''Assumption B''' 

Further, from literature, we found that DRLux is specific to DNA and the dissociation constant is low (k<sub>m</sub> = 0.1nM) {Reference}.  Therefore, we using QSSA again,

$$\frac{d[P_{LuxON}]}{dt} = k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}] \approx 0\\$$

Solving, we get the rate of production of mRNA<sub>Bxb1</sub> as

$$\frac{d[mRNA_{Bxb1}]}{dt} = L_{P_{Lux}} + \frac{k_{mRNA_{Bxb1}}[RLux]^2}{K_{mLux}^2 + [RLux]^2 }- d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\$$

where 

$$K_{mLux} = \sqrt{\frac {k_{-P_{LuxON}}}{k_{P_{LuxON}}}.\frac {k_{-DRLux} + d_{DRLux}}{k_{DRLux}}}$$

is a lumped parameter which we fitted to our data.

Similarly, lumped parameter K<sub>mLas</sub> was derived for the las system and fitted to a transfer function of normalized GFP concentration as a function of input Las-AHL.

<!--Further,
$$K_{mRNA_{Bxb1}} = k_{mRNA_{Bxb1}} P_{LuxTOT}$$ which we assumed. -->

=== Parameter fitting ===

We used MEIGO Toolbox to fit the parameters to the experimental data. We used the concentrations at the end of five hours from each simulation and fit it to the experimental concentrations at the same time.


[[File:ETHZ_LuxParameterFitting.png|center|500 px|thumb|'''Figure 1''' Lux QS Module fitted to experimental data from riboregulated Lux system.]]


Using the 'DHC' local-solver (Direct search method) in MEIGO, we found the lumped parameters 
$$K_{mLux} = 0.45 \pm 0.00051 nM$$ 

and 

$$K_{mLas} = 10 \pm 0.0082 nM$$ 

respectively.

[[File:ETHZ_LasParameterFitting.png|center|500 px|thumb|'''Figure 2''' Las QS Module fitted to experimental data from riboregulated Las system.]]

=== Range of validity of the assumptions ===
These assumptions hold true for all input LuxAHL and LasAHL concentrations.

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<html><article id='Degradation'></html>
== Retrieving degradation rates==

We are considering quorum sensing experiments with riboregulator, where Plux is induced by LuxAHL or Plas is induced by LasAHL and GFP is produced instead of Bxb1, and this time we look at dynamic curves. 


[[File:ETH_Zurich_dGFP_Dynamic.png|500px|center|thumb| '''Figure 3''' Dynamic response of the promoter Plux to a dose entry at time t=0.]]


By adding an additional quasi steady state assumptions on R<sub>Lux</sub>, and neglecting degradation of R<sub>Lux</sub> compared to its unbinding rate, we can find :

$$\frac{d[GFP]}{dt}=LeakyLux+\frac{k_{mRNAGFP} k_{GFP} \alpha_{LuxR}^2}{d_{LuxR}^2(Km_{Lux}^2+\alpha_{LuxR})} \frac{[AHL]^2}{K_{mAHL}^2 + [AHL]^2}-d_{GFP}[GFP]$$

$$\text{with} K_{mAHL}=\frac{K_{mLux}^2 k_{-RLux}}{k_{RLux}(K_{mLux}^2+\alpha_{LuxR}^2/d_{LuxR})}$$

We have dynamic curves for different initial AHL concentrations. 

We can see in the equation above that for initial AHL concentrations much higher than K<sub>mAHL</sub>=0.3 nM, GFP is only produced and degraded and thus :

$$\frac{d[GFP]}{dt}=LeakyLux+\frac{k_{mRNAGFP} k_{GFP} \alpha_{LuxR}^2}{d_{LuxR}^2(Km_{Lux}^2+\alpha_{LuxR})}-d_{GFP}[GFP]$$

so by taking 
$$t_{1/2}=\frac{ln(2)}{d_{GFP}}$$
 from experimental curves, we find 
$$d_{GFP} = 4.9 . 10^{-3} min^{-1}$$


For initial AHL concentrations much lower than K<sub>mAHL</sub>=0.3 nM, we find
$$\frac{d[GFP]}{dt}=LeakyLux+\frac{k_{mRNAGFP} k_{GFP} \alpha_{LuxR}^2}{d_{LuxR}^2(Km_{Lux}^2+\alpha_{LuxR})} \frac{[AHL]^2}{K_{mAHL}^2}-d_{GFP}[GFP]$$

and thus a steady state
$$[GFP]=\frac{Constant}{(d_{GFP}-2d_{AHL})}(e^{-2d_{AHL}t}-e^{-d_{GFP}t})$$

 This curve has a maximum at 
$$t_{max}=\frac{1}{d_{GFP}-2d_{AHL}}ln\big(\frac{d_{GFP}}{2d_{AHL}}\big)$$
 
This way we can find from experimental curves 
$$d_{LuxAHL}=4,0.10^{-3} min^{-1}$$

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<html><article id="Leakiness"></html>

== Leakiness ==

The leakiness of promoters is a major issue in our system. As the signal propagates row-wise, error diffusion could lead to a totally different pattern. The goal is then to control the leakiness. This issue was particularly observed and adressed in the case of the Lux promoter during our [https://2014.igem.org/Team:ETH_Zurich/expresults experiments set]. This leakiness is dependent on LuxR concentration in the cell(see our biobrick characterization in the [http://parts.igem.org/Part:BBa_R0062:Experience Registry] for more information).
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Leakiness was modeled as an offset in the classical Hill function. 

$$rFluo = a + b \frac{[AHL]^n}{K_m^n + [AHL]^n}$$
$$\text{where rFluo is the relative fluorescence (absolute measured fluorescence value over OD),}$$
$$\text{a the basal expression rate (Leakiness),}$$
$$\text{b the maximum fold expression rate,}$$
$$\text{n the Hill coefficient,}$$
$$K_m\text{ the activation concentration of AHL.}$$


Given this offset and the maximal expression, the signal over noise ratio can be derived. This ratio, which can then be compared amongst all curves, characterizes the impact of leakiness on the behavior of a system. The leakier a construct in its native form is, the more impact the riboregulator will have, and the more likely it is for the riboregulator to increase the signal over noise ratio. Our final constructs (Promoters with a [https://2014.igem.org/Team:ETH_Zurich/expresults riboregulating system]) have the following parameters:

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{| class="wikitable"
|-
! '''Promoter used'''
! '''Signal over noise ratio'''
|-
|P<sub>Lux</sub> without riboregulating system
|23
|-
|P<sub>Lux</sub> with riboregulating system
|79
|-
|P<sub>Las</sub> without riboregulating system
|84
|-
|P<sub>Las</sub> with riboregulating system
|55
|}
<br/>
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We took the leakiness coefficients into account in our [https://2014.igem.org/Team:ETH_Zurich/modeling/whole whole cell model].

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<html><article id="Cross-talk"></html>

== Cross-talk ==

We investigated the existence of cross-talk between three quorum sensing systems (Lux, Las and Rhl). Each quorum sensing system is based on three components: a signaling molecule, a regulatory protein and a promoter. Cross-talk implies non-orthogonality of the communication systems. It corresponds to the fact that LasAHL can activate the Lux promoter, even if it is not its native communicating pathway. There are 27 combinations possible and only 3 native combinations between signaling molecule, regulatory protein and promoter.
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[[File:ETH Zurich Crosstalk.png|1500px|center|thumb|'''Figure 4''' Each quorum sensing system is based on three components: a signaling molecule, a regulatory protein and a promoter. These elements are here ordered into three layers. Cross-talk evaluation can be done by comparing all combinations of those three elements. After collecting the [https://2014.igem.org/Team:ETH_Zurich/expresults experimental data] of all possible pathways, we modeled their influence.]]
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<br/>
From the [https://2014.igem.org/Team:ETH_Zurich/expresults exhaustive experimental data], the central role of regulatory proteins was identified, allowing the characterization of two-levels of cross-talk. The following parts in the Registry are presenting our results:
*[http://parts.igem.org/Part:BBa_R0062:Experience BBa_R0062]
*[http://parts.igem.org/Part:BBa_R0079:Experience BBa_R0079]
*[http://parts.igem.org/Part:BBa_I14017:Experience BBa_I14017]
*[http://parts.igem.org/Part:BBa_C0062:Experience BBa_C0062]
*[http://parts.igem.org/Part:BBa_C0179:Experience BBa_C0179]
*[http://parts.igem.org/Part:BBa_C0171:Experience BBa_C0171]

Each experimental data set was fitted to an Hill function using the [https://2014.igem.org/Team:ETH_Zurich/modeling/parameters Least Absolute Residual method]. 

$$rFluo = a + b \frac{[AHL]^n}{K_m^n + [AHL]^n}$$
$$\text{where rFluo is the relative fluorescence (absolute measured fluorescence value over OD),}$$
$$\text{a the basal expression rate,}$$
$$\text{b the maximum fold expression rate,}$$
$$\text{n the Hill coefficient,}$$
$$K_m\text{ the activation concentration of AHL.}$$

We took cross-talk into account in our [https://2014.igem.org/Team:ETH_Zurich/modeling/whole whole cell model].

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<html><article id="Alternate"></html>

== Alternate Design ==

As cross-talk is a burning issue in quorum sensing, we thought about a theoretical solution. [https://2014.igem.org/Team:Edinburgh Edinburgh iGEM team 2014] also worked on communication between ''E. coli''. They developed new communication channels via metabolic wiring. By assuming that quorum sensing molecules would not cross-talk with metabolites, we used their idea to develop a model on the molecular level. The idea is finally to combine our [https://2014.igem.org/Team:ETH_Zurich/modeling#Alternate_Design whole-cell model] and their idea on metabolites.

Metabolic wiring is based on the fact that when a resource is at disposal, a cell will produce an enzyme to break it down to smaller pieces. There is often a chain of metabolites. Using the fact that metabolites can diffuse through the membrane, this can allow communication (for more information, please see the [https://2014.igem.org/Team:Edinburgh Edinburgh iGEM team 2014] wiki).

=== Chemical Species ===

The species' names are generic because the implementation with particular metabolites implies biological considerations on cell growth and medium used. 

{| class="wikitable"
|-
! Name
! Description
|-
|'''A'''
|Metabolite
|-
|'''B'''
|Metabolite
|-
|'''Enz'''
|Enzyme that catalyzes the transformation from A to B
|-
|'''[A.Enz]'''
|Complex made of metabolite A and enzyme Enz
|-
|'''P_A'''
|Promoter induced by A. It can be either on or off (rescued or not)
|}

=== Reactions ===
$$\begin{align*}
&\rightarrow A \\
A + P_{Aoff} &\rightarrow P_{Aon} \\
P_{Aon} &\rightarrow P_{Aon} + Enz \\
A + Enz &\leftrightarrow [A.Enz] \\
[A.Enz] &\rightarrow B \\
A &\rightarrow \\
B &\rightarrow \\
[A.Enz] &\rightarrow \\
Enz &\rightarrow
\end{align*}$$

=== Parameters ===
{| class="wikitable"
|-
! Name
! Description
|-
|'''α<sub>A</sub>'''
|Production rate of metabolite A
|-
|'''d<sub>A</sub>'''
|Degradation rate of metabolite A
|-
|'''α<sub>Enz</sub>'''
|Production rate of enzyme Enz
|-
|'''d<sub>Enz</sub>'''
|Degradation rate of enzyme Enz
|-
|'''α<sub>B</sub>'''
|Production rate of metabolite B
|-
|'''d<sub>B</sub>'''
|Degradation rate of metabolite B
|-
|'''K<sub>d</sub>'''
|Parameter of the Michaelis-Menten function modeling the action of enzyme Enz on the substrate, metabolite A, in order to produce metabolite B
|-
|'''n'''
|Hill coefficient for the Hill function modeling the activation of the transcription of enzyme Enz with metabolite A as inducer
|-
|'''K<sub>A</sub>'''
|Activation concentration for the Hill function modeling the activation of the transcription of enzyme Enz with metabolite A as inducer
|}

Parameters are not well-known.

=== Deterministic Model ===
We derived this model doing the following assumptions:

;'''Assumption 1'''
:The induction of the promoter P<sub>A</sub> by A is supposed to follow an Hill function.

;'''Assumption 2'''
:The enzyme-based reaction from A to B is supposed to follow a Michaelis-Menten function.

$$\begin{align}
\frac{d[A]}{dt} &= \alpha_A - d_{A} [A] \\
\frac{d[Enz]}{dt} &= \alpha_{Enz} \frac{[A]^n}{K_{A}^n + [A]^n} - d_{Enz} [Enz] \\
\frac{d[B]}{dt} &= \alpha_B \frac{[Enz] [A]}{K_d + [A]} - d_{B} [B] 
\end{align}$$

=== Application to our project ===
Populations of bacteria will grow into a medium, which provides them metabolite A. The metabolite B will serve as communicating signal, like LasAHL or LuxAHL. It will be an input for the logic construct. As soon as metabolite B is being made available to the cell, the cell will produce the input for the logic construct.
<br/>
Moreover, one type of cell will produce B as output. Therefore, the output of the logic gate signal will correspond to the production (or absence of production) of the enzyme Enz, so that the A contained in the medium can be transformed to B. Enz will play the same role as LasI or LuxI in our original model.
<br/>
We separate the promoter activating the production of the enzyme with the production of the enzyme itself. Moreover, cells sense B and want to produce B. In our modules, we have to remplace equations in the sensing module and in the production module.
<br/>
<br/>
;'''Sensing Module'''
:For example, B would induce the production of the integrase, Bxb1.
$$
\begin{align*}
\frac{d[B]}{dt} &= \alpha_B - d_{B} [B] \\
\frac{d[Bxb1]}{dt} &= \alpha_{Enz} \frac{[B]^n}{K_{B}^n + [B]^n} - d_{Bxb1} [Bxb1]
\end{align*}
$$

;'''Production Module'''
:Here, output<sub>logic</sub> is the output of the XOR logic gate, factorized in one term for simplicty's sake. For more information on this function, see the [https://2014.igem.org/Team:ETH_Zurich/modeling/xor XOR gate modeling page].
$$\begin{align*}
\frac{d[Enz]}{dt} &= output_{logic} - d_{Enz} [Enz] \\
\frac{d[B]}{dt} &= \alpha_B \frac{[Enz] [A]}{K_d + [A]} - d_{B} [B]
\end{align*}
$$

=== Simulations ===

We implemented this solution in our [[Team:ETH_Zurich/modeling/whole#Alternate_Design|whole-cell model]]. As no parameters were known, we assumed their values to be in the range of standard rates. The results indicate that the system could work, the next step would be to test this prediction experimentally.

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{{:Team:ETH Zurich/tpl/foot}}