Team:ETH Zurich/modeling/xor

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== Model ==
== Model ==
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=== XOR Logic Gate ===
=== XOR Logic Gate ===
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The truth table of an XOR gate in binary
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We consider a binary exclusive or (XOR) logic gate, with two inputs and one output.
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[[File:ETH_Zurich_XOR_Logic_Gate.png|400px|center|thumb|'''Figure 1''' Truth table of the XOR logic gate.]]
=== Biological Principles ===
=== Biological Principles ===
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The terminator can be flipped once if either DBxb1 or ΦC31 is present. The state T<sub>off</sub> can be caused by two reasons. We further decompose it into two different states: T<sub>offBxb1</sub>(flipping due to presence of Bxb1) and T<sub>offΦC31</sub>(flipping due to presence of ΦC31).
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The terminator can be flipped once if either DBxb1 or ΦC31 is present. The state T<sub>off</sub> can be reached via two possible transitions. We further decompose it into two different states: T<sub>offBxb1</sub>(flipping due to presence of Bxb1) and T<sub>offΦC31</sub>(flipping due to presence of ΦC31).
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''Insert Image of two states''
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[[File:ETH Zurich XOR Toffs.png|400px|center|thumb|'''Figure 2''' Decomposition of the on output into two terminator states.]]
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Flipping by integrases is irreversible. The initial state, in which the terminator is on, is different from the state after two switches. From this last state, no further evolution of the system is possible. Therefore, we decompose the T<sub>on</sub> into two different states: T<sub>on,i</sub>(initial state of the system, no flipping) and T<sub>on,f</sub>(final state of the system after two flips).
Flipping by integrases is irreversible. The initial state, in which the terminator is on, is different from the state after two switches. From this last state, no further evolution of the system is possible. Therefore, we decompose the T<sub>on</sub> into two different states: T<sub>on,i</sub>(initial state of the system, no flipping) and T<sub>on,f</sub>(final state of the system after two flips).
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''Insert Image of two states''
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[[File:ETH Zurich XOR Tons.png|400px|center|thumb|'''Figure 3''' Decomposition of the off output into two terminator states.]]
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=== XOR bio''logic''  gate ===
=== XOR bio''logic''  gate ===
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[[File:ETHZ_XORmodule.png|center|600px|thumb|The behaviour of XOR module as a function of activated Bxb1 sites (SABxb1) and ΦC31 sites (SAΦC31). The XOR behaviour is continuous since we modelled it deterministically.]]
 
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[[File:ETH_Zurich_XOR_Biologic_Gate.png|center|600px|thumb|'''Figure 4''' Truth table of the XOR biologic gate: Summary of the model, coupled with the biological explanation.]]
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[[File:ETHZ_BiologicalTruthTable.png|center|600px]]
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=== Other Chemical Species ===
=== Other Chemical Species ===
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To have the equivalent for the strain producing LuxAHL as output, it suffices to remplace every occurence of LasI by LuxI.
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To have the equivalent for the strain producing LuxAHL as output, it suffices to replace every occurence of LasI by LuxI.
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== Transfer Function ==
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=== Determistic Differential Equations ===
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We modeled the XOR gate module deterministically. Thus, we considered a continuous approximation of variables, like the number of integrases-DNA binding sites or the number of terminator. We wrote them as concentration. We do not take the species LasI and mRNA<sub>LasI</sub> into account because they are only produced by the XOR biologic gate and GFP is another product of this biologic gate that can be read out.
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$$\begin{align*}
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\frac{d[T_{on,i}]}{dt}&= -k_{ToffBxb1}[T_{on,i}][SA_{Bxb1}]^2 -k_{Toff\phi C31} [T_{on,i}][SA_{\phi C31}]^2 \\
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\frac{d[T_{offBxb1}]}{dt}&=k_{ToffBxb1} [T_{on,i}][SA_{Bxb1}]^2 -  k_{-ToffBxb1} [T_{offBxb1}] [SA_{\phi C31}]^2\\
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\frac{d[T_{off\phi C31}]}{dt}&=k_{Toff\phi C31} [T_{on,i}][SA_{\phi C31}]^2 -  k_{-Toff\phi C31} [T_{offBxb1}] [SA_{Bxb1}]^2\\
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\frac{d[T_{on,f}]}{dt}&=  k_{-ToffBxb1} [T_{offBxb1}] [SA_{\phi C31}]^2 + k_{-Toff\phi C31} [T_{offBxb1}] [SA_{Bxb1}]^2\\
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\frac{d[mRNA_{GFP}]}{dt}&=  k_{mRNA_{GFP}} (\frac{[T_{offBxb1}]}{\theta + [T_{offBxb1}]} +  \frac{[T_{off\phi C31}]}{\theta + [T_{off\phi C31}]}) - d_{mRNA_{GFP}}[mRNA_{GFP}]\\
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\frac{d[GFP]}{dt}&=  k_{GFP} [mRNA_{GFP}] - d_{GFP}[GFP]\\
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\end{align*}$$
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As there is a strong codon bias in both integrases' seuquences <sup>[[Team:ETH_Zurich/project/references|[14]]]</sup>, the flipping parameters were assumed such that they would not be a rate limiting step. (see the [https://2014.igem.org/Team:ETH_Zurich/modeling/parameters parameter pages])
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=== Deterministic simulation ===
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At steady state, the transfer function obtained by simulation is shown on the next figure. It is remarkable to see that even a little amount of one integrase is sufficient to switch the XOR gate on.
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[[File:ETHZ_XORmodule.png|center|600px|thumb|'''Figure 5''' The behaviour of XOR module as a function of activated Bxb1 sites (SABxb1) and ΦC31 sites (SAΦC31). The XOR behaviour is continuous since we modelled it deterministically.]]
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Latest revision as of 23:33, 17 October 2014

iGEM ETH Zurich 2014