Contents 1 Overview 2 Dissociation Constants 2.1 Introduction 2.2 Discontinuity 3 Solution 3.1 Our new estimations for Kd 4 References

Overview

The following section shows how we used our different modeling tools to analyse our original system, find its flaws, change it, and then optimise it to increase glue output, lysis timing and robustness. This process led to the eventual change of the quorum sensing system but with any investigation, it is important to begin with the basics- in this case, the "dissociation constants"

Dissociation Constants

Introduction

Our model to describe the transcriptional dynamics of the constructs uses Hill kinetics; we have three repression Hill functions of the form:

Moreovre, the model has one activation Hill function of the following form:

and one repression / induction Hill function of the following 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 promoter in question. Similarly, [S] is the concentration of the inducer S and K_s is the dissociation constant for the inducer to the repressor X. The dissociation constant K_d is defined as follows:

where k_off and k_on are the on and off rates in the following reaction

The dissociation constant K_d has more biologically meaningful definition; it is the concentration of X at which the promoter will be free 50% of the time.

Discontinuity

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 that we convert the 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).

Here, we face a major problem, since 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 conflicting information we found. This is an extract from the ETHZ Wiki [6] , where a new column with the values of K_d in units of molecule-per-cell has been added.

Parameter	Value	Molecules per cell	Description
KLacI	0.1 - 1 [pM] OR 800 [nM]	0.00004-0.0004 molecules OR 322 molecules	LacI repressor dissociation constant
KIPTG	1.3 [µM]	522 molecules	IPTG-LacI repressor dissociation constant
KtetR	179 [pM]	0.07 molecules	TetR repressor dissociation constant
KcI	8 [pM] OR 50 [nM]	0.003 molecules OR 20 molecules	cI repressor dissociation constant
KHSL	0.09 - 1 [µM]	402 molecules	HSL-LuxR activator dissociation constant

And here are other parameters that we found in the literature:

Parameter	Value	Molecules per cell	Description	Reference
KLacI	~1*10 -12 M OR ~1.8*10-12 M	0.0004 molecules OR 0.00072 molecules	Dissociation constant for LacI to LacO DNA site	[1][2]
KIPTG	1*10-6 M	402 molecules	Dissociation constant for IPTG to LacI	[3]
KtetR	(5.6 ± 2) × 10-9 M OR 1.53*10-8 M	2.25 molecules OR 6.1506 molecules	Dissociation constant for TetR to TetO	[4][5]
KcI	50 * 10-9 M	20 molecules	Dissocitation constant for cI to DNA site	[6]

Solution

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

1. Most K_d values are measured in-vitro, which yields a low measurement since the conditions of the reaction - most notably the 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 promoter. We know from Thermodynamics that pressure and temperature will change reaction kinetics and hence, the in-vitro experiments will have different reaction rate and hence, different K_d values than would be found 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 unfeasibly low, since they do not take into account non-specific DNA binding and cell pressure.

3. In our model we describe concentrations in terms of molecules-per-cell, instead of moles-per-litre. Upon converting the K_d values from moles-per-lire to molecules-per-cell we found that many of the K_d values were less than 1 molecule per cell. This implies that less than one protein (LacI, TetR, etc.) is required to half the overall production. This is physically misrepresentative for a number of reasons, including the fact that the probability that one protein molecule will collide with a single promoter 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 that the K_d values were unfeasibly low and introduced the idea of non-specific DNA binding to us. He showed us a PhD thesis from one of his students, Dr. Bryony Payne, from 2006. In this thesis a far more direct and accurate measurement of the number of LacI tetramers present in the cell was made. It stated that on average 340 tetramers of LacI were present in an e. coli cell. Since under normal conditions (when glucose is available and no lactose is present) the LacO operon is repressed. We can assume that 340 tetramers will fully repress the lacO operon. From this value, we estimated the K_d value for the LacO operon to be 700 molecules-per-cell.

Our new estimations for K_d

According to Dr. Payne's 2006 paper; 340 LacI tetramers completely repress a promoter. Hence, roughly 170 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 have a lower affinity to each other than LacI to LacO. However, the in-vitro values suggest that TetR still binds 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 of the same order of magnitude, but possibly smaller, than K_TetR. In V.fischeri, it is unlikely that HSL enters the cell before the amplifying loop starts. Assuming this, the complex HSL-LuxR (P) has to have a high affinity for the lux box same as LacI to lacO.

Now we have:

K_LacI = 700 molecules per cell

K_IPTG = 1200 molecules per cell

K_P = 700 molecules per cell

K_cI = 7000 molecules per cell

K_TetR = 7000 molecules per cell

Back to Chemotaxis

Continue to Sensitvity Simulations

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] https://2007.igem.org/title=ETHZ/Parameters

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