Team:ETH Zurich/modeling/int

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{{:Team:ETH Zurich/tpl/head|Integrases}}
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{{:Team:ETH Zurich/tpl/fortables}}
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== Model ==
== Model ==
-
In our design, integrases compute the output of the logic gates. Integrases allow to flip one fragment of DNA. The model we developped is described here.
+
In our design, integrases compute the output of the logic gates. Integrases allow flipping one fragment of DNA. Therefore, they are of central importance to our design. However, their characterization in literature<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup> is incomplete. In particular, quantitative insight into dimerization rates and DNA-binding rates is lacking. Such data is however necessary in order to be able to set up a mathematical model to describe the overall Mosai''coli'' process. Thus, we decided to estimate the missing parameters from published from published experiments based on a model that we developed ourselves.
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{| class="wikitable"
{| class="wikitable"
|-
|-
-
! Name
+
! '''Name'''
! Description
! Description
|-
|-
|'''Bxb1'''
|'''Bxb1'''
-
|Serine integrase that can fold into two conformations called Bxb1a and Bxb1b. Even if those conformations have the same sequence of amino acids, their tertiary structure is different. As it is experimentally difficult to differentiate these two conformations, we chose to to use a common notation for both conformations.
+
|Serine integrase that can fold into two conformations - Bxb1a and Bxb1b. We chose to use a common connotation for both conformations - Bxb1.
|-
|-
|'''ΦC31'''
|'''ΦC31'''
-
|Serine integrase that can fold into two conformations called ΦC31a and ΦC31b. Similarly to Bxb1, we chose to use a common notation for both conformations.
+
|Serine integrase that can fold into two conformations - ΦC31a and ΦC31b. We chose to use a common connotation for both conformations - ΦC31.
|-
|-
|'''DBxb1'''
|'''DBxb1'''
-
|Dimer of Bxb1. Bxb1a (respectively Bxb1b) forms a dimer DBxb1a (respectively DBxb1b). To be coherent, dimers are not differentiated depending on their spatial configuration.
+
|Dimerized form of Bxb1. We chose to use a common connotation for both homodimers, DBxb1a and DBxb1b.
|-
|-
|'''DΦC31'''
|'''DΦC31'''
-
|Dimer of ΦC31. ΦC31a (respectively ΦC31b) forms a dimer DΦC31a (respectively DΦC31b). The different spatial configuration are not taken into account.
+
|Dimerized form of ΦC31. We chose to use a common connotation for both homodimers, DΦC31a and DΦC31b.
|}
|}
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=== Modeling DNA-binding sites ===
=== Modeling DNA-binding sites ===
-
Each dimer of integrases can specifically bind to a DNA binding site. As the flipping is irreversible, these DNA binding sites can be three possible states:  
+
Each dimer of integrases can specifically bind to a DNA binding site. As the flipping is irreversible, these DNA binding sites can be in three possible states:  
* SI<sub>IntegraseName</sub>: inactive DNA binding site. No dimer is bound to this site, which has never been flipped.
* SI<sub>IntegraseName</sub>: inactive DNA binding site. No dimer is bound to this site, which has never been flipped.
-
* SA<sub>IntegraseName</sub>: active DNA binding site. A dimer is to this site.
+
* SA<sub>IntegraseName</sub>: active DNA binding site. A dimer is bound to this site.
-
* SF<sub>IntegraseName</sub>: flipped DNA binding site. This DNA binding site has been used by a flipping.
+
* SF<sub>IntegraseName</sub>: flipped DNA binding site. This site has been irreversibly flipped.
-
[[File:ETH_Zurich_Integrases_sites.png|center|800px|thumb|The three different states of DBxb1-DNA binding sites.]]
+
[[File:ETH_Zurich_Integrases_sites.png|center|800px|thumb|'''Figure 1''' The three different states of DBxb1-DNA binding sites.]]
=== Reactions ===
=== Reactions ===
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{| class="wikitable"
{| class="wikitable"
|-
|-
-
! Name
+
! '''Name'''
! Description
! Description
! Status from literature
! Status from literature
! Status after characterization
! Status after characterization
|-
|-
-
|k<sub>DBxb1</sub>
+
|'''k<sub>DBxb1</sub>'''
|Dimerization rate of Bxb1
|Dimerization rate of Bxb1
|Unknown
|Unknown
-
|Fitted
+
|Fitted with Bonnet's data<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>
|-
|-
-
|k<sub>-DBxb1</sub>
+
|'''k<sub>-DBxb1</sub>'''
|Dissociation rate of DBxb1
|Dissociation rate of DBxb1
|Unknown
|Unknown
-
|Fitted
+
|Fitted with Bonnet's data<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>
|-
|-
-
|k<sub>SABxb1</sub>
+
|'''k<sub>SABxb1</sub>'''
|Rate of formation of SA<sub>Bxb1</sub> from DBxb1 and SI<sub>Bxb1</sub>
|Rate of formation of SA<sub>Bxb1</sub> from DBxb1 and SI<sub>Bxb1</sub>
|Unknown
|Unknown
-
|Fitted
+
|Fitted with Bonnet's data<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>
|-
|-
-
|k<sub>-SABxb1</sub>
+
|'''k<sub>-SABxb1</sub>'''
|Dissociation rate of SA<sub>Bxb1</sub>
|Dissociation rate of SA<sub>Bxb1</sub>
|Unknown
|Unknown
-
|Fitted
+
|Fitted with Bonnet's data<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>
|-
|-
-
|d<sub>Bxb1</sub>
+
|'''d<sub>Bxb1</sub>'''
|Degradation rate of Bxb1
|Degradation rate of Bxb1
|Unknown
|Unknown
|Assumed
|Assumed
|-
|-
-
|d<sub>DBxb1</sub>
+
|'''d<sub>DBxb1</sub>'''
|Degradation rate of DBxb1
|Degradation rate of DBxb1
|Unknown
|Unknown
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|}
|}
-
Replacing every occurence of Bxb1 by ΦC31 gives the set of parameters for ΦC31. The same status can be applied to those parameters.  
+
Substituting ΦC31 for Bxb1 gives the set of parameters for ΦC31. The same status can be applied to those parameters.  
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-
Even if degradation rates were not determined specifically for the serine integrases and their dimerized form, degradation rates of proteins in ''E. coli'' are available. To characterize integrases behavior, we focus on finding the parameters for dimerization and DNA-binding.
+
Even if degradation rates were not determined specifically for the serine integrases and their dimerized form, degradation rates of proteins in ''E. coli'' are available. We assume that the degradation rates of dimerized forms are two times higher than the degradation rates of monomers. Typically, d<sub>DBxb1</sub> = 2*d<sub>Bxb1</sub>. To characterize integrases behavior, we focused on estimating the parameters for dimerization and DNA-binding.
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<html><article id="Parameter"></html>
<html><article id="Parameter"></html>
-
== Characterization: K<sub>SABxb1</sub>==
+
== Characterization of the integrase DNA-binding reaction==
=== Data ===
=== Data ===
-
The parameter fitting is based on Bonnet's paper<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>. Their experimental setup is different from ours. However, they experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.
+
The parameter fitting is based on the data available from Bonnet's paper<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>. Their experimental setup is different from ours. They induce the production of the integrase Bxb1 with aTc. Bxb1 can then flip a terminator put between a constitutive promoter and the gene for a reporter protein. They experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.
-
[[File:ETH Zurich Bonnet S4.jpg|center|400px|thumb|Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper ''Amplifying Genetic Logic Gates''<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>]]
+
[[File:ETH Zurich Bonnet S4.jpg|center|400px|thumb|'''Figure 2''' Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper ''Amplifying Genetic Logic Gates''<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>.]]
-
To use their experimental data, we have to model their induction mechanism using aTc. We use a simplify version of it proposed by the [https://2013.igem.org/Team:UCSF 2013 iGEM team UCSF]. They modeled the induction with aTc as a leaky Hill function. In the following table are the set of parameters introduced to model induction.
+
To use their experimental data, we have to model their induction mechanism using aTc. We use a simplified version of this phenomenon proposed by the [https://2013.igem.org/Team:UCSF 2013 iGEM team UCSF]. They modeled the induction with aTc as a leaky Hill function. In the following table are the set of parameters introduced to model induction.
{| class="wikitable"
{| class="wikitable"
|-
|-
-
!Name
+
!'''Name'''
!Value
!Value
!Description
!Description
!Reference
!Reference
|-
|-
-
|A<sub>L</sub>
+
|'''A<sub>L</sub>'''
|8.904 mRNA min<sup>-1</sup>
|8.904 mRNA min<sup>-1</sup>
|Basal expression level of tet promoter
|Basal expression level of tet promoter
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|-
|-
-
|B<sub>L</sub>
+
|'''B<sub>L</sub>'''
|443.7 mRNA min<sup>-1</sup>
|443.7 mRNA min<sup>-1</sup>
|Maximal expresion level of tet promoter
|Maximal expresion level of tet promoter
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|-
|-
-
|n
+
|'''n'''
|2.551 (no units)
|2.551 (no units)
|Hill exponent
|Hill exponent
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|-
|-
-
|K<sub>m</sub>
+
|'''K<sub>L</sub>'''
|11.45 µmoles
|11.45 µmoles
|Half-maximal effective concentration of aTc
|Half-maximal effective concentration of aTc
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|[[https://2013.igem.org/Team:UCSF/Modeling 2013 iGEM team UCSF]]
|-
|-
-
|k<sub>mRNA<sub>Bxb1</sub></sub>
+
|'''k<sub>mRNA<sub>Bxb1</sub></sub>'''
|Unknown
|Unknown
|Translation rate of Bxb1
|Translation rate of Bxb1
-
|
+
|Assumed
|}
|}
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=== Assumptions ===
=== Assumptions ===
;'''Assumption A'''
;'''Assumption A'''
-
:The back-reaction from DBxb1 binding to site inactive is considered to be negligible, compared to the flipping rate. That is to say that once a site is active, it can only be flipped. Thus, an active site would only be a transitional state in our whole cell model. As the flipping is not modeled in our integrase subsystem, active sites are not transformed into flipped sites at the end of the information pipeline. Thus, we consider that we can express the switching rate given active site concentration.
+
:The back-reaction from DBxb1 binding to the inactive site state is considered to be negligible compared to the flipping rate. That is to say that once a site is active, it can only be flipped. Thus, an active site would only be a transitional state in our whole cell model. As the flipping is not modeled in our integrase subsystem, active sites are not transformed into flipped sites at the end of the information pipeline. Thus, we consider that we can express the switching rate given active site concentration.
;'''Assumption B'''
;'''Assumption B'''
-
:As switching needs two active sites to be effective (for more information on flipping, check the [https://2014.igem.org/Team:ETH_Zurich/modeling/xor XOR gate page]), the switching rate is approximated to: $${(\frac{SA_{Bxb1}}{S_{TOT}})}^2$$ This approximation is understated by statistical considerations.
+
:As switching needs two active sites to be effective (for more information on flipping, check the [https://2014.igem.org/Team:ETH_Zurich/modeling/xor XOR gate page]), the switching rate is approximated to: $${\left(\frac{SA_{Bxb1}}{S_{TOT}}\right)}^2$$ This approximation is understated by probabilistic considerations.
;'''Assumption C'''
;'''Assumption C'''
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;'''Assumption D'''
;'''Assumption D'''
-
:The activation by aTc is assumed to be dominant over degradation of Bxb1 and dimerization of Bxb1. It is supposed to be valid on the range of aTc concentration considered.
+
:The activation by aTc is assumed to be dominant over degradation and dimerization of Bxb1. It is supposed to be valid on the range of aTc concentration considered.
-
$$\frac{k_{mRNA_{Bxb1}}*(A_L + B_L * \frac{[aTc]^n}{[aTc]^n+K_L^n})}{d_{Bxb1}K_{DBxb1}} >> 1
+
$$\frac{k_{mRNA_{Bxb1}}*\left(A_L + B_L * \frac{[aTc]^n}{[aTc]^n+K_L^n}\right)}{d_{Bxb1}K'_{DBxb1}} >> 1
$$  
$$  
with  
with  
-
$$K_{DBxb1} = \frac{k_{-DBxb1} + d_{DBxb1}}{k_{DBxb1}}
+
$$K'_{DBxb1} = \frac{k_{-DBxb1} + d_{DBxb1}}{k_{DBxb1}}
$$
$$
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We consider the system at steady-state. After derivation, the following explicit equation can be retrieved:
We consider the system at steady-state. After derivation, the following explicit equation can be retrieved:
-
$$[SR]_{qss} = {(\frac{B_{L} * [aTc]^{n}}{ \lambda_1 K_L^{n} + (B_{L} +\lambda_1) [aTc]^{n}})}^2 $$
+
$$[SR]_{qss} = {\left(\frac{B_{L} * [aTc]^{n}}{ \lambda_1 K_L^{n} + (B_{L} +\lambda_1) [aTc]^{n}}\right)}^2 $$
where
where
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$$\lambda_1 =  1.82e-07  (1.649e-07, 1.992e-07)$$
$$\lambda_1 =  1.82e-07  (1.649e-07, 1.992e-07)$$
-
[[File:ETH Zurich Integrase SR1.png|center|800px|thumb|Parameter fitting of the dissociate rate constant of K<sub>SABxb1</sub>]]
+
[[File:ETH Zurich Integrase SR1.png|center|800px|thumb|'''Figure 3''' Parameter fitting of the dissociation rate constant of K<sub>SABxb1</sub>]]
We assume that:
We assume that:
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* k<sub>mRNA<sub>Bxb1</sub></sub> is of the order of magnitude 10<sup>-1</sup> min<sup>-1</sup> mRNA<sup>-1</sup>. We estimated to be a low value because the starting codon of Bxb1 is GTG (and not ATG) and this parameter also takes into account folding time.
* k<sub>mRNA<sub>Bxb1</sub></sub> is of the order of magnitude 10<sup>-1</sup> min<sup>-1</sup> mRNA<sup>-1</sup>. We estimated to be a low value because the starting codon of Bxb1 is GTG (and not ATG) and this parameter also takes into account folding time.
-
Thus, '''K<sub>SABxb1</sub>'s order of magnitude is 10<sup>-6</sup> nM'''. The interpretation of this dissociation constant is that the DNA binding reaction is really specific, as it can be expected for integrases.
+
Thus, '''K<sub>SABxb1</sub>'s order of magnitude is 10<sup>-6</sup> nM'''. The interpretation of this dissociation constant, K<sub>SABxb1</sub>, is that the DNA binding reaction is really specific, as it can be expected for integrases.
 +
 
 +
By assuming that k<sub>SABxb1</sub>, the rate of formation of SA<sub>Bxb1</sub>, is not rate limiting and fixing it to 1, we find that k<sub>-SABxb1</sub>'s order of magnitude is 10<sup>-6</sup> nM.
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<html><article id="Parameters2"></html>
<html><article id="Parameters2"></html>
-
== Characterization: K<sub>DBxb1</sub>==
+
== Characterization of the dimerization reaction==
=== Data ===
=== Data ===
-
The parameter fitting is based on Bonnet's paper<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>. Their experimental setup is different from ours. However, they experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.
+
The parameter fitting is based on data from Bonnet's paper<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>. Their experimental setup is different from ours. However, they experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.
-
[[File:ETH Zurich Bonnet S4.jpg|center|400px|thumb|Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper ''Amplifying Genetic Logic Gates''<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>]]
+
[[File:ETH Zurich Bonnet S4.jpg|center|400px|thumb|'''Figure 4''' Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper ''Amplifying Genetic Logic Gates''<sup>[[Team:ETH_Zurich/project/references|[9]]]</sup>]]
We do the same modeling steps, as for the [https://2014.igem.org/Team:ETH_Zurich/modeling/int#Characterization:_KSABxb1 previous characterization].Thus, we obtain the following set of differential equations:$$\begin{align*}
We do the same modeling steps, as for the [https://2014.igem.org/Team:ETH_Zurich/modeling/int#Characterization:_KSABxb1 previous characterization].Thus, we obtain the following set of differential equations:$$\begin{align*}
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=== Assumptions ===
=== Assumptions ===
-
;'''Assumption A'''
 
-
:The back-reaction from DBxb1 binding to site inactive is considered to be negligible, compared to the flipping rate. That is to say that once a site is active, it can only be flipped. Thus, an active site would only be a transitional state in our whole cell model. As the flipping is not modeled in our integrase subsystem, active sites are not transformed into flipped sites at the end of the information pipeline. Thus, we consider that we can express the switching rate given active site concentration.
 
-
;'''Assumption B'''
+
We keep the following assumptions from the previous characterization: '''assumption A''', '''assumption B''' and '''assumption C'''. A assumption is made:
-
:As switching needs two active sites to be effective (for more information on flipping, check the [https://2014.igem.org/Team:ETH_Zurich/modeling/xor XOR gate page]), the switching rate is approximated to: $${(\frac{SA_{Bxb1}}{S_{TOT}})}^2$$ This approximation is understated by statistical considerations.
+
-
 
+
-
;'''Assumption C'''
+
-
:Given the normalization of the paper, basal rate of production of Bxb1 is not taken into account. Thus, we consider that $$A_L = 0$$
+
;'''Assumption E'''
;'''Assumption E'''
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$$\lambda_2 =  8.211e-07  (7.421e-07, 9.001e-07))$$
$$\lambda_2 =  8.211e-07  (7.421e-07, 9.001e-07))$$
-
[[File:ETH Zurich Integrase SR2.png|center|800px|thumb|Parameter fitting of the dissociation rate constant of K<sub>DBxb1</sub>]]
+
[[File:ETH Zurich Integrase SR2.png|center|800px|thumb|'''Figure 5''' Parameter fitting of the dissociation rate constant of K<sub>DBxb1</sub>]]
We assume that:
We assume that:
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* k<sub>mRNA<sub>Bxb1</sub></sub> is of the order of magnitude 10<sup>-1</sup> min<sup>-1</sup> mRNA<sup>-1</sup>. We estimated to be a low value because the starting codon of Bxb1 is GTG (and not ATG) and this parameter also takes into account folding time.
* k<sub>mRNA<sub>Bxb1</sub></sub> is of the order of magnitude 10<sup>-1</sup> min<sup>-1</sup> mRNA<sup>-1</sup>. We estimated to be a low value because the starting codon of Bxb1 is GTG (and not ATG) and this parameter also takes into account folding time.
-
Thus, K<sub>DBxb1</sub>'s order of magnitude is 10<sup>-6</sup> nM. The interpretation of this dissociation rate constant is that the dimerization reaction is really specific, as it can be expected for integrases.
+
Thus, '''K<sub>DBxb1</sub>'s order of magnitude is 10<sup>-6</sup> nM'''. The interpretation of this dissociation rate constant is that the dimerization reaction is really specific, as it can be expected for integrases.
 +
By assuming that k<sub>DBxb1</sub>, the rate of formation of D<sub>Bxb1</sub>, is not rate limiting and fixing it to 1, we find that k<sub>-DBxb1</sub>'s order of magnitude is 10<sup>-6</sup> nM.
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<html><article id="Results"></html>
<html><article id="Results"></html>
 +
==Results==
==Results==
-
[[File:ETHZ_Integrase_Bxb1.jpg|300px]] [[File:ETHZ_Integrase_Pc31.jpg|300px]]
+
The figures show the predicted transfer function for the ratios of active sites of Bxb1 (SA<sub>Bxb1</sub>) to total number of sites (SBxb1<sub>TOT</sub>) and active sites of ΦC31 (SA<sub>ΦC31</sub>) to total number of sites (SΦC31<sub>TOT</sub>) as a function of their respective integrase concentrations based on the fitted parameters. With the fitted parameters, we found the Km value to be 10 nM.
-
The figures show the predicted transfer function for the SABxb1 and SAΦC31 as a function of their respective integrase concentrations based on the fitted parameters.
+
[[File:ETHZ_Integrase_Bxb1.jpg|500px|center|thumb|'''Figure 6''' Predicted transfer function for Bxb1 module.]]
 +
[[File:ETHZ_Integrase_Pc31.jpg|500px|center|thumb|'''Figure 7''' Predicted transfer function for ΦC31 module.]]
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{{:Team:ETH Zurich/tpl/foot}}
{{:Team:ETH Zurich/tpl/foot}}

Latest revision as of 01:09, 18 October 2014

iGEM ETH Zurich 2014