Team:Aberdeen Scotland/hillinput

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) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/2/2b/Picture5.gif"> <p style="float: left; width: 32%; text-align: right;">(II.3) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/6/60/Picture6.gif"> <p style="float: left; width: 32%; text-align: right;">(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: <p style="float: left; width: 25%; text-align: left;"> <p style="float: left; width: 46%; text-align: center;"><img src="http://2009.igem.org/wiki/images/3/36/Picture7.gif">or<img src="http://2009.igem.org/wiki/images/3/3a/Picture8.gif"> <p style="float: left; width: 25%; text-align: right;">(III) Substituting (III) in (II.4) we find <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/e/e1/Picture9.gif"> <p style="float: left; width: 32%; text-align: right;">(IV) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/a/ad/Picture10.gif"> <p style="float: left; width: 32%; text-align: right;">(IV.1) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/e/e9/Picture11.gif"> <p style="float: left; width: 32%; text-align: right;">(IV.2) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/c/c4/Picture12.gif"> <p style="float: left; width: 32%; text-align: right;">(IV.3) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/3/3a/Picture13.gif"> <p style="float: left; width: 32%; text-align: right;">(IV.4) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/c/c9/Picture14.gif"> <p style="float: left; width: 32%; text-align: right;">(IV.5) <p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/5/5f/Picture15.gif"> <p style="float: left; width: 32%; text-align: right;">(IV.6) Consequently, the probability <img src="http://2009.igem.org/wiki/images/3/3d/Picture16.gif"> that the site D is free is dependent on [X]. Similarly, the promoter activity, p, is defined by <p style="float: left; width: 25%; text-align: left;"> <p style="float: left; width: 46%; text-align: center;"><img src="http://2009.igem.org/wiki/images/5/5f/Picture17.gif"> or <img src="http://2009.igem.org/wiki/images/e/e9/Picture18.gif"> <p style="float: left; width: 25%; text-align: right;">(V) where β is the maximal transcription rate of the promoter. If <img src="http://2009.igem.org/wiki/images/2/2d/Picture19.gif"> then [X] = K d 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 <p style="float: left; width: 25%; text-align: left;"> <p style="float: left; width: 46%; text-align: center;"><img src="http://2009.igem.org/wiki/images/5/5f/Picture17.gif"> or <img src="http://2009.igem.org/wiki/images/a/a6/Picture41.gif"> <p style="float: left; width: 25%; text-align: right;">(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 <p style="float: left; width: 25%; text-align: left;"> <p style="float: left; width: 46%; text-align: center;"><img src=" http://2009.igem.org/wiki/images/c/c0/Picture20.gif"> or <img src="http://2009.igem.org/wiki/images/e/e5/Picture42.gif"> <p style="float: left; width: 25%; text-align: right;">(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. <p style="float: left; width: 15%; text-align: left;"> <p style="float: left; width: 66%; text-align: center;"><img src="http://2009.igem.org/wiki/images/a/a9/Picture21.gif"> or <img src="http://2009.igem.org/wiki/images/0/07/Picture22.gif"> or <img src="http://2009.igem.org/wiki/images/f/fb/Picture23.gif"> <p style="float: left; width: 15%; text-align: right;">(IIX) Writing the mass-action kinetic equation for S binding with X yields

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/e/e5/Picture24.gif"> <p style="float: left; width: 32%; text-align: right;">(IX) In the steady state http://2009.igem.org/wiki/images/8/8d/Picture25.gif

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/e/ef/Picture26.gif"> <p style="float: left; width: 32%; text-align: right;">(IX.1) Rearranging (IX.1), we can write

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/d/dc/Picture27.gif"> <p style="float: left; width: 32%; text-align: right;">(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

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/0/00/Picture28.gif"> <p style="float: left; width: 32%; text-align: right;">(X) Rearranging equation (IX) we obtain the Michaelis–Menten equation

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/9/90/Picture29.gif"> <p style="float: left; width: 32%; text-align: right;">(X) Substituting the third form of equation (IIX) in (IX.2)

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/f/f3/Picture30.gif"> <p style="float: left; width: 32%; text-align: right;">(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: <p style="float: left; width: 25%; text-align: left;"> <p style="float: left; width: 46%; text-align: center;"><img src="http://2009.igem.org/wiki/images/8/85/Picture31.gif">or<img src="http://2009.igem.org/wiki/images/b/bf/Picture32.gif"> <p style="float: left; width: 25%; text-align: right;">(XII) S binding on X is similarly described by the mass-action kinetic equation:

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/9/9e/Picture33.gif"> <p style="float: left; width: 32%; text-align: right;">(XIII) In steady state

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/8/84/Picture34.gif"> <p style="float: left; width: 32%; text-align: right;">(XIV) Substituting (XII) in (XIV)

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/1/15/Picture35.gif"> <p style="float: left; width: 32%; text-align: right;">(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.

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/6/60/Picture36.gif"> <p style="float: left; width: 32%; text-align: right;">(XVI) where http://2009.igem.org/wiki/images/8/84/Picture38.gif The concentration of unbound X to XT is given by

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/8/8b/Picture39.gif"> <p style="float: left; width: 32%; text-align: right;">(XVII) Combining (V) with (XVII) we obtain the input function of a gene regulated by a repressor

<p style="float: left; width: 32%; text-align: left;"> <p style="float: left; width: 32%; text-align: center;"><img src="http://2009.igem.org/wiki/images/f/ff/Picture40.gif"> <p style="float: left; width: 32%; text-align: right;">(IIXX)