Team:Aberdeen Scotland/hillinput

Contents 1 Introduction 2 Repression of a Promoter 3 Activation of a Promoter 4 Input Function for an Inducer Molecule 5 References

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 k_on describes the collisions of X and D that occur per protein per unit time at a given concentration and k_off 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 K_d is the dissociation - or equilibrium - constant. K_d 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] = 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

 

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, [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]. 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

 

(IX.1)

Rearranging (IX.1), we can write

 

(IX.2)

where K_x is the dissociation constant - defined as the ratio of k_off to k_on 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 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)

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Continue to PDEs/Next Steps

References

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