Team:Aberdeen Scotland/hillinput

From 2009.igem.org

(Difference between revisions)
 
(32 intermediate revisions not shown)
Line 1: Line 1:
{{:Team:Aberdeen_Scotland/css}}
{{:Team:Aberdeen_Scotland/css}}
{{:Team:Aberdeen_Scotland/header2}}
{{:Team:Aberdeen_Scotland/header2}}
 +
 +
== Introduction ==
 +
This section details the derivation of the input functions for repressors and activators that we are using for our differential equations describing the Pico Plumber. For further reading we recommend Uri Alon “An Introduction to Systems Biology”.
 +
{{:Team:Aberdeen_Scotland/break}}
== Repression of a Promoter ==
== Repression of a Promoter ==
-
During repression of a promoter a repressor protein, X, binds to a DNA site of the promoter, D.  The product of this binding process is [XD]. [XD] can also fall apart into [X] and [D] again:
+
During repression of a promoter a repressor protein; X, binds to a DNA site of the promoter, D.  The product of this binding process is [XD]. [XD] can also dissociate into [X] and [D] again:
<html><br>
<html><br>
Line 14: Line 18:
</html>
</html>
-
where k<sub>on</sub> describes the collisions of X and D that occur per time per protein at a given concentration and k<sub>off</sub> determines the strength of the chemical binding X and D. In form of a differential equation, the rate of change of [XD] is described by
+
Where k<sub>on</sub> describes the collisions of X and D that occur per protein per unit time at a given concentration and k<sub>off</sub> determines the strength of the chemical bond between X and D. In the form of a differential equation, the rate of change of [XD] is described by
<html><br>
<html><br>
Line 59: Line 63:
<div class="space"></div>
<div class="space"></div>
</html>
</html>
-
Equation (II.4) is called the chemical equilibrium constant equation, where Kd is the dissociation or equilibrium constant. Kd has units of concentration.  
+
Equation (II.4) is called the Chemical Equilibrium Constant equation, where K<sub>d</sub> is the dissociation or equilibrium - constant. K<sub>d</sub> has units of concentration.  
-
Therefore, transcription of a gene only happens whenever the repressor is not bound. That is when D is free. The total concentration of the DNA sites [DT] can be written with the help of the conservation law:
+
Therefore, transcription of a gene only happens whenever the repressor is not bound. That is to say, when D is free. The total concentration of the DNA sites [DT] can be written with in terms of the conservation law:
<html><br>
<html><br>
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
Line 125: Line 129:
<div class="space"></div>
<div class="space"></div>
</html>
</html>
-
As a result, the probability <html><img src="https://static.igem.org/mediawiki/2009/3/3d/Picture16.gif"></html>describing that the site D is free is dependent on [X]. The promoter activity, p, is defined by  
+
Consequently, the probability <html><img src="https://static.igem.org/mediawiki/2009/3/3d/Picture16.gif"></html> that the site D is free is dependent on [X]. Similarly, the promoter activity, p, is defined by  
<html><br>
<html><br>
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
Line 134: Line 138:
<div class="space"></div>
<div class="space"></div>
</html>
</html>
-
where β is the maximal transcription rate of the promoter. If <html><img src="https://static.igem.org/mediawiki/2009/2/2d/Picture19.gif"></html> then [X] = K<html><sub>d</sub></html> and the promoter activity is reduced by 50%. This [X] needed to repress the promoter activity by a half is called the repression coefficient.  
+
where β is the maximal transcription rate of the promoter. If <html><img src="https://static.igem.org/mediawiki/2009/2/2d/Picture19.gif"></html> then [X] = K<html><sub>d</sub></html> and the promoter activity is reduced by 50%. This particular [X] needed to repress the promoter activity by a half is called the Repression Coefficient.
 +
If several protein units of X are binding on D - in a dimeric or tetrameric fashion, for example - then we can apply the Hill function for repression of a promoter that is
 +
<html><br>
 +
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 46%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/5/5f/Picture17.gif"> or <img src="https://static.igem.org/mediawiki/2009/a/a6/Picture41.gif"></p>
 +
<p style="float: left; width: 25%; text-align: right;">(VI)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
{{:Team:Aberdeen_Scotland/break}}
{{:Team:Aberdeen_Scotland/break}}
== Activation of a Promoter ==
== Activation of a Promoter ==
-
In the case of an activation of a promoter, an activator protein, X, binds to its DNA site of the promoter and increases the rate of transcription of the promoter.
+
In activation of a promoter, an activator protein, X, binds to a DNA site of the promoter and increases the rate of transcription of the promoter.
-
Similarly to the repression of the promoter the promoter activity, p, in the case of activation can be derived as
+
Similarly, in the case of repression of the promoter - the activity, p, can be derived as
<html><br>
<html><br>
-
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
+
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
-
<p style="float: left; width: 32%; text-align: center;"><img src="
+
<p style="float: left; width: 46%; text-align: center;"><img src="
-
https://static.igem.org/mediawiki/2009/c/c0/Picture20.gif"></p>
+
https://static.igem.org/mediawiki/2009/c/c0/Picture20.gif"> or <img src="https://static.igem.org/mediawiki/2009/e/e5/Picture42.gif"></p>
-
<p style="float: left; width: 32%; text-align: right;">(VI)</p>
+
<p style="float: left; width: 25%; text-align: right;">(VII)</p>
<br>
<br>
<br>  
<br>  
Line 151: Line 164:
{{:Team:Aberdeen_Scotland/break}}
{{:Team:Aberdeen_Scotland/break}}
-
== Michaelis –Menten Equation for a Repressor Protein ==
+
== Input Function for an Inducer Molecule ==
-
To turn a repressor system from Off state to On state, we need an input signal (for example a molecule called inducer,S) such that the repressor protein, X, binds off the promoter side DNA. The inducer forms a complex with X varying X’s affinity to D.  
+
To turn a repressor system from the off-state to the on-state, we need an input signal - for example a molecule termed an inducer, S.  The repressor protein, X, dissociates from the promoter side DNA. The inducer forms a complex with X - changing X’s affinity to D.  
-
The total amount of concentration of the repressor, [X_T], can be considered as a product of the repressor protein forming a complex with the inducer, [XS] and the repressor protein in its free form [X], whereby free does not differ between bound to the promoter’s DNA site or not.
+
The total concentration of the repressor, [X<sub>T</sub>], can be considered as a product of the repressor protein forming a complex with the inducer, [XS] and the repressor protein in its free form [X]. By "free", we do not differentiate between the repressor being bound to the promoter’s DNA site or not.
<html><br>
<html><br>
<p style="float: left; width: 15%; text-align: left;">&nbsp;</p>
<p style="float: left; width: 15%; text-align: left;">&nbsp;</p>
<p style="float: left; width: 66%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/a/a9/Picture21.gif"> or <img src="https://static.igem.org/mediawiki/2009/0/07/Picture22.gif"> or <img src="https://static.igem.org/mediawiki/2009/f/fb/Picture23.gif"></p>
<p style="float: left; width: 66%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/a/a9/Picture21.gif"> or <img src="https://static.igem.org/mediawiki/2009/0/07/Picture22.gif"> or <img src="https://static.igem.org/mediawiki/2009/f/fb/Picture23.gif"></p>
-
<p style="float: left; width: 15%; text-align: right;">(VII)</p>
+
<p style="float: left; width: 15%; text-align: right;">(IIX)</p>
<br>
<br>
<br>  
<br>  
<div class="space"></div>
<div class="space"></div>
</html>
</html>
-
Writing the mass-action kinetic equation for the S binding with the X leaves us with
+
Writing the mass-action kinetic equation for S binding with X yields
<html><br>
<html><br>
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/e/e5/Picture24.gif"></p>
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/e/e5/Picture24.gif"></p>
-
<p style="float: left; width: 32%; text-align: right;">(II.X)</p>
+
<p style="float: left; width: 32%; text-align: right;">(IX)</p>
<br>
<br>
<br>  
<br>  
<div class="space"></div>
<div class="space"></div>
</html>
</html>
-
In steady state https://static.igem.org/mediawiki/2009/8/8d/Picture25.gif
+
In the steady state https://static.igem.org/mediawiki/2009/8/8d/Picture25.gif
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/e/ef/Picture26.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(IX.1)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
Rearranging (IX.1), we can write
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/d/dc/Picture27.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(IX.2)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
where K<sub>x</sub> is the dissociation constant - defined as the ratio of k<sub>off</sub> to k<sub>on</sub> when S binds to X. Substituting (VII) in (IX.2) yields
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/0/00/Picture28.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(X)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
Rearranging equation (IX) we obtain the Michaelis–Menten equation
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/9/90/Picture29.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(X)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
Substituting the third form of equation (IIX) in (IX.2)
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/f/f3/Picture30.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(XI)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
That is to say, the concentration of X not bound to S. Thus, as in the case of LacI, only X not bound to S can bind to D - thereby repressing production.
 +
However, repressor proteins are binding on D as protein-subunits - and activation is fully achieved if the inducer is attached to these subunits.  To describe this binding process we consider n molecules of S binding to X and by the conservation law:
 +
<html><br>
 +
<p style="float: left; width: 25%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 46%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/8/85/Picture31.gif">or<img src="https://static.igem.org/mediawiki/2009/b/bf/Picture32.gif"></p>
 +
<p style="float: left; width: 25%; text-align: right;">(XII)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
S binding on X is similarly described by the mass-action kinetic equation:
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/9/9e/Picture33.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(XIII)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
In steady state
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/8/84/Picture34.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(XIV)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
Substituting (XII) in (XIV)
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/1/15/Picture35.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(XV)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
Rearranging (XV) leaves us with the Hill equation describing the probability that the DNA site is bound compared to an average over-binding and dissociation of S.
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/6/60/Picture36.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(XVI)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
where https://static.igem.org/mediawiki/2009/8/84/Picture38.gif
 +
The concentration of unbound X to XT is given by
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/8/8b/Picture39.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(XVII)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
Combining (V) with (XVII) we obtain the input function of a gene regulated by a repressor
 +
<html><br>
 +
 
 +
<p style="float: left; width: 32%; text-align: left;">&nbsp;</p>
 +
<p style="float: left; width: 32%; text-align: center;"><img src="https://static.igem.org/mediawiki/2009/f/ff/Picture40.gif"></p>
 +
<p style="float: left; width: 32%; text-align: right;">(IIXX)</p>
 +
<br>
 +
<br>
 +
<div class="space"></div>
 +
</html>
 +
 
 +
 
 +
<html>
 +
<table class="nav">
 +
<tr>
 +
<td>
 +
<a href="https://2009.igem.org/Team:Aberdeen_Scotland/parameters"><img src="https://static.igem.org/mediawiki/2009/e/ed/Aberdeen_Left_arrow.png">&nbsp;&nbsp;Back to Parameters</a>
 +
</td>
 +
<td align="right">
 +
<a href="https://2009.igem.org/Team:Aberdeen_Scotland/modeling/pde">Continue to PDEs/Next Steps&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2009/4/4c/Aberdeen_Right_arrow.png"></a>
 +
</td>
 +
</tr>
 +
</table>
 +
</html>
 +
 
 +
{{:Team:Aberdeen_Scotland/break}}
 +
== References ==
 +
Alon, Uri. “An Introduction to Systems Biology Design Principles of Biological Circiuts.” London: Chapman & Hall/CRC, 2007.
{{:Team:Aberdeen_Scotland/footer}}
{{:Team:Aberdeen_Scotland/footer}}

Latest revision as of 09:46, 18 August 2009

University of Aberdeen iGEM 2009

Contents

Introduction

This section details the derivation of the input functions for repressors and activators that we are using for our differential equations describing the Pico Plumber. For further reading we recommend Uri Alon “An Introduction to Systems Biology”.

Repression of a Promoter

During repression of a promoter a repressor protein; X, binds to a DNA site of the promoter, D. The product of this binding process is [XD]. [XD] can also dissociate into [X] and [D] again:

 

(I)



Where kon describes the collisions of X and D that occur per protein per unit time at a given concentration and koff determines the strength of the chemical bond between X and D. In the form of a differential equation, the rate of change of [XD] is described by

 

(II)



At steady state the concentration [XD] does not change.

 

(II.1)




 

(II.2)




 

(II.3)




 

(II.4)



Equation (II.4) is called the Chemical Equilibrium Constant equation, where Kd is the dissociation - or equilibrium - constant. Kd has units of concentration. Therefore, transcription of a gene only happens whenever the repressor is not bound. That is to say, when D is free. The total concentration of the DNA sites [DT] can be written with in terms of the conservation law:

 

or

(III)



Substituting (III) in (II.4) we find

 

(IV)




 

(IV.1)




 

(IV.2)




 

(IV.3)




 

(IV.4)




 

(IV.5)




 

(IV.6)



Consequently, the probability that the site D is free is dependent on [X]. Similarly, the promoter activity, p, is defined by

 

or

(V)



where β is the maximal transcription rate of the promoter. If then [X] = Kd and the promoter activity is reduced by 50%. This particular [X] needed to repress the promoter activity by a half is called the Repression Coefficient. If several protein units of X are binding on D - in a dimeric or tetrameric fashion, for example - then we can apply the Hill function for repression of a promoter that is

 

or

(VI)



Activation of a Promoter

In activation of a promoter, an activator protein, X, binds to a DNA site of the promoter and increases the rate of transcription of the promoter. Similarly, in the case of repression of the promoter - the activity, p, can be derived as

 

or

(VII)



Input Function for an Inducer Molecule

To turn a repressor system from the off-state to the on-state, we need an input signal - for example a molecule termed an inducer, S. The repressor protein, X, dissociates from the promoter side DNA. The inducer forms a complex with X - changing X’s affinity to D. The total concentration of the repressor, [XT], can be considered as a product of the repressor protein forming a complex with the inducer, [XS] and the repressor protein in its free form [X]. By "free", we do not differentiate between the repressor being bound to the promoter’s DNA site or not.

 

or or

(IIX)



Writing the mass-action kinetic equation for S binding with X yields

 

(IX)



In the steady state Picture25.gif

 

(IX.1)



Rearranging (IX.1), we can write

 

(IX.2)



where Kx is the dissociation constant - defined as the ratio of koff to kon when S binds to X. Substituting (VII) in (IX.2) yields

 

(X)



Rearranging equation (IX) we obtain the Michaelis–Menten equation

 

(X)



Substituting the third form of equation (IIX) in (IX.2)

 

(XI)



That is to say, the concentration of X not bound to S. Thus, as in the case of LacI, only X not bound to S can bind to D - thereby repressing production. However, repressor proteins are binding on D as protein-subunits - and activation is fully achieved if the inducer is attached to these subunits. To describe this binding process we consider n molecules of S binding to X and by the conservation law:

 

or

(XII)



S binding on X is similarly described by the mass-action kinetic equation:

 

(XIII)



In steady state

 

(XIV)



Substituting (XII) in (XIV)

 

(XV)



Rearranging (XV) leaves us with the Hill equation describing the probability that the DNA site is bound compared to an average over-binding and dissociation of S.

 

(XVI)



where Picture38.gif The concentration of unbound X to XT is given by

 

(XVII)



Combining (V) with (XVII) we obtain the input function of a gene regulated by a repressor

 

(IIXX)




References

Alon, Uri. “An Introduction to Systems Biology Design Principles of Biological Circiuts.” London: Chapman & Hall/CRC, 2007.