Team:ETH Zurich/modeling/int

From 2014.igem.org

(Difference between revisions)
m (Data)
m (Data)
 
(16 intermediate revisions not shown)
Line 12: Line 12:
== Model ==
== Model ==
-
In our design, integrases compute the output of the logic gates. Integrases allow to flip one fragment of DNA. The model we developed 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.
<br/>
<br/>
<br/>
<br/>
Line 47: Line 47:
* SF<sub>IntegraseName</sub>: flipped DNA binding site. This site has been irreversibly flipped.
* 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 ===
Line 121: Line 121:
|}
|}
-
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.  
<br/>
<br/>
<br/>
<br/>
-
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 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.
<html></article></html>
<html></article></html>
Line 133: Line 133:
=== 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 simplified version of this phenomenon proposed by the [http://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 [http://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.
Line 182: Line 182:
=== 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 [http://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 probabilistic considerations.
+
:As switching needs two active sites to be effective (for more information on flipping, check the [http://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'''
Line 192: Line 192:
;'''Assumption D'''
;'''Assumption D'''
: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.
: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  
Line 203: Line 203:
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
Line 212: Line 212:
$$\lambda_1 =  1.82e-07  (1.649e-07, 1.992e-07)$$
$$\lambda_1 =  1.82e-07  (1.649e-07, 1.992e-07)$$
-
[[File:ETH Zurich Integrase SR1.png|center|800px|thumb|Parameter fitting of the dissociation 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:
Line 230: Line 230:
=== 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 [http://2014.igem.org/Team:ETH_Zurich/modeling/int#Characterization:_KSABxb1 previous characterization].Thus, we obtain the following set of differential equations:$$\begin{align*}
We do the same modeling steps, as for the [http://2014.igem.org/Team:ETH_Zurich/modeling/int#Characterization:_KSABxb1 previous characterization].Thus, we obtain the following set of differential equations:$$\begin{align*}
Line 241: Line 241:
=== 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 [http://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'''
Line 267: Line 261:
$$\lambda_2 =  8.211e-07  (7.421e-07, 9.001e-07))$$
$$\lambda_2 =  8.211e-07  (7.421e-07, 9.001e-07))$$
-
[[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:
Line 286: Line 280:
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 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.
-
[[File:ETHZ_Integrase_Bxb1.jpg|500px|center|thumb|Predicted transfer function for Bxb1 module.]]
+
[[File:ETHZ_Integrase_Bxb1.jpg|500px|center|thumb|'''Figure 6''' Predicted transfer function for Bxb1 module.]]
-
[[File:ETHZ_Integrase_Pc31.jpg|500px|center|thumb|Predicted transfer function for ΦC31 module.]]
+
[[File:ETHZ_Integrase_Pc31.jpg|500px|center|thumb|'''Figure 7''' Predicted transfer function for ΦC31 module.]]
<html></article></html>
<html></article></html>
{{:Team:ETH Zurich/tpl/foot}}
{{:Team:ETH Zurich/tpl/foot}}

Latest revision as of 01:09, 18 October 2014

iGEM ETH Zurich 2014