Team:Aberdeen Scotland/internal/deterministic

Deterministic Model

Introduction

Our circuit is constructed on 5 different plasmids, with the aim of using no more than 2 different plasmids for the final construction. From each plasmid, mRNA is transcribed, followed by ribosome translation of these mRNA strings into proteins. For the different promoters the maximal production strength varies depending on the ribosome affinity for polymerase.

Firstly we split the different constructs according to the different mRNA they produced. We have 5 different constructs, each of which was assigned arbitrary names. The first construct, which produced mRNA for glue production - or in the current case GFP, which we are using as proof of concept - and Lambda cI was termed “mRNA_production”. The second construct which produces LacI and TetR is named “mRNA_latch”. Holin and Endolysin are transcribed from the mRNA produced by the “mRNA_lysis” construct. The name we gave for the mRNA with the code for Antiholin is “mRNA_antilysis”. Finally, the quorum sensing proteins, LuxI and LuxR are transcribed by the last mRNA, which was termed “mRNA_qs”.

Figure 1: Circuit of our Pico Plumber

As it is visible in figure 1, three repressors and one activator are built into the circuit. On the production plasmid, AND- gate behaviour, activated by the presence of LuxR-HSL and IPTG, triggers the production of the glue and cI. However, because the lux box is a transcriptional activator, the input of LuxR-HSL only increases the maximal transcription rate of the promoter. Thus, we have to assume a leakiness of the promoter on the production plasmid in the presence of IPTG [1]. Once cI is produced it represses mRNA_latch transcription, causing glue production to stay on even if there is no IPTG present. This is the latch behaviour we require. Another consequence of cI production will be repression of TetR. This leads to, after a natural degradation of TetR, the transcription of mRNA_lysis and hence production of Holin and Endolysin. The cell will then lyse when there are 1000 more Holin proteins than Antiholin proteins present in the cell. This number is an assumption; which had to be made due to a lack of information available. SOURCE

The two proteins, LuxI and LuxR, are constitutively produced. LuxI together with SAM, which is constantly present in the cell, forms - via an enzymatic process - HSL. HSL can freely diffuse in and out of the cell. If the concentration of HSL outside a cell is high enough, the HSL concentration inside the cell reaches a high enough level such that it can combine with LuxR and form a complex LuxR-HSL. In the future referred to as P, which activates the lux box on mRNA_production [2] [3] [4]. The other trigger, IPTG, is released from the hole in the pipe. Since we wouldn’t have LacY in the cell we just need to consider IPTG diffusion through the cell membrane [5][6]. Beside the natural degradation of the proteins, we also took into account the dilution due to cell growth. All this information was summarized in the following differential equations:

Equations for mRNA transcription

Equations for Protein translation

Equations for Complex formation

Parameter description

Parameter	Value	Description	Reference
X		Degradation of X
Y		Degradation of Y
λ-CI		Degradation of lambda CI
X		Degradation of X
LacI		Degradation of LacI
TetR		Degradation of TetR
Holin		Degradation of Holin
Endolysin		Degradation of Endolysin
Antiholin		Degradation of Antiholin
Protein		Translation rate of Protein
		Rate of production of HSL from LuxI
HSL		Rate of diffusion of HSL in/out of the cell
IPTG		Rate of diffusion of IPTG in/out of the cell
IPTG		Rate of diffusion of IPTG in/out of the cell

References

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

[2] Ward, J.P., J.R. King and A.J. Koerber. “Mathematical modelling of quorum sensing in bacteria.” IMA Journal of Mathematics Applied in Medicine and Biology 2001: 18, 263-292

[3] James, Sally et al. “Luminescence Control in the Marine Bacterium Vibro fischeri: An Analysis of the Dynamics of lux Regulation.” JMB 2000: 296, 1127-1137

[4] Goryachev, A.B., D.J. Toh and T. Lee. “System analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constant.” BioSystems 2006: 83, 178-187

[5] Chung, J. D. and Greogry Stephanopoulos. “On the Physiological Multiplicity and Population Heterogeneity of Biological Systems.” Chemical Engineering Science 1996: 51, 1509-1521

[6] Nichols, J.C. and K.S. Matthews. “Combinatorial Mutations of lac Repressor.” The Journal of Biological Chemistry 1997: 272, 18550 -18557