Team:Aberdeen Scotland/parameters/invest 1

Dissociation Constants

Introduction

Our model uses hill kinetics; we have three repression hill functions of the form:

It also has one activation hill function of the form:

And one repression / induction hill function of the form

Where β is the maximal transcription rate, [X] is the concentration of protein X and K_d is the dissociation constant for molecule X to the operator in question, [S] is the concentration of the inducer, S and K_s is the dissociation constant for the inducer to the repressor, X. K_d is defined as follows:

Where k_off and k_on are the on and off rates in the equation

K_d has a more biologically meaningful definition however, it is the concentration of X at which the operator will be repressed 50% of the time.

The issue

The units of K_d are usually given in M, the molarity, or moles per litre. Our model works with the exact number of molecules so we convert our K_d values into molecules per cell. This is achieved as follows:

Where the volume of the cytoplasm of the cell is 6.7×10^-16 litres

This conversion constant of Avogadro’s number multiplied by the cytoplasm volume is ~ 402000000 (402 million).

The problem with this is that most dissociation constants found in the literature equate to a value of molecules per cell that is less than 1. Clearly in a cell with 10 plasmids and therefore 10 operators 1 molecule could not repress all of them.

Below is a table of the conflicting information we found. This is an extract from the EHTZ wiki [6] with the new column of the value in molecules per cell added.

Dissociation Constants Continued

Upon further investigation we have concluded that the vast majority of K_d values found in papers were inaccurately low for the following reasons:

1. Most K_d values are measured in vitro, which gives a falsely low measurement since the conditions in the reaction (most notable being salt concentration and pressure) are completely different than in an e. coli cell. The salt concentration affects the reaction significantly since it lowers the electrostatic affinity of the protein to the operon. We know from thermodynamics that pressure and temperature will change reaction kinetics and hence the in vitro experiments will different reaction rates and hence different K_d values than the true behaviour in the cell.

2. We have found measurements of K_d values which have been done in conditions which try to replicate in vivo conditions. These K_d values are better, but also far too low, since they do not take into account non-specific DNA binding and cell pressure.

3. In our model, we are working in molecules per cell, instead of molarity. Upon converting the K_d values from molarity to molecules per cell we found that a great deal the K_d values we found were less than 1 molecule per cell. This implies that less than one protein (lacI, tetR etc) is needed to half the production. This is physically infeasible, the probability that one protein molecule will collide with a single operon in the cell at the correct angle is close to zero.

We consulted with Prof. Peter McGlynn of the Institute of Medical Sciences in Aberdeen, who agreed with our analysis of the K_d values and introduced the idea of non-specific DNA binding. He showed us a PhD thesis from one of his students Bryony Payne from 2006. In this thesis a far more direct and accurate measurement of the lacO repression was made. It stated that 340 tetremers of LacI was required to fully repress the lacO operon. From this value, we estimated the Kd value for the lacO operon to be 700 molecules per cell.

Our new estimations for K_d

Starting our estimation from Prof. Mcglynn’s PhD student Bryony Payne; 2006, we are given that 340 LacI tetramers completely repress a promoter. Hence roughly 120 tetramers will give half repression. Assuming the tetramers are stable this gives a value of K_LacI , K_d for LacI to LacO, of 4×170 or K_LacI ~ 700.

For the LacI IPTG complex formation, we estimated K_IPTG ~1200 using [7] and our value of K_LacI above. TetR to TetO seems to have a lower affinity to each other than LacI to LacO. However, the in vitro values suggest that TetR binds still with a strong affinity to TetR. Thus the K_TetR value was roughly estimated to be up to 10 times K_LacI. The in vitro values for cI to its operon seem to suggest that the in vivo K_cI value is in the same order of magnitude, but possibily smaller, than K_TetR.

So we now have:

K_LacI = 700 molecules per cell

K_cI = 7000 molecules per cell

K_TetR = 7000 molecules per cell

References

[1] Mitchel Lewis (2005) The Lac repressor. C. R. Biologies 328 (2005) 521–548

[2] Falcon C.M and Matthews K.S. (2000) Operator DNA sequence Variation Enhances High Affinity Binding by Hinge Helix Mutants of Lactose Repressor Protein. Biochemistry. 39, 11074-11084

[3] Uri Alon, An introduction to systems Biology, p244

[4] Nucleic Acids Res. 2004; 32(2): 842–847. Two mutations in the tetracycline repressor change the inducer anhydrotetracycline to a corepressor Annette Kamionka, Joanna Bogdanska-Urbaniak, Oliver Scholz, and Wolfgang Hillen*

[5] Volume 272, Number 11, Issue of March 14, 1997 pp. 6936-6942, The Role of the Variable Region in Tet Repressor for Inducibility by Tetracycline, Christian Berens , Dirk Schnappinger and Wolfgang Hillen

[6] http://parts.mit.edu/igem07/index.php?title=ETHZ/Parameters

[7] Detailed map of a cis-regulatory input function – Y. Setty*,†, A. E. Mayo*,†, M. G. Surette‡, and U. Alon*,†,§

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