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 transcription rate varies depending on the polymerase affinity to the promoter.

First we split the different constructs according to the different mRNA they produce. We have 5 different constructs, each of which is assigned an arbitrary name. The first construct, which produces mRNA for glue production. The two proteins X and Y mix together to form the glue. Currently we are using GFP as a proof of concept - whereby Lambda cI is 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. We name mRNA coding for Antiholin mRNA_antilysis. Finally, the quorum sensing proteins, LuxI and LuxR are transcribed by the last mRNA, which is termed mRNA_qs.

Figure 1: Circuit of our Pico Plumber

As shown in figure 1, three repressors and one activator are built into the circuit. We use Hill input function to describe the behaviour of the activator and the repressors. 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 the concentration of Holin is 1000 higher then the concentration of Antiholin. This number is an assumption; which had to be made due to a lack of information available. [7-8]

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 the cell is high enough, the HSL concentration inside the cell increases as well 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 [2-4]. The other trigger, IPTG, is released from the hole in the pipe. Since we wouldn’t have LacY in the cell, that is a lactose permease protein actively helping IPTG diffusing inside the cell, we just need to consider IPTG diffusion through the cell membrane [5][6]. Beside the natural degradation of the proteins, we also take into account the dilution due to cell growth. All this information is summarized in the following differential equations:

Equations for mRNA transcription

The differential equations for the mRNA transcription include a production and a degradation term. The DNA sequence is placed on p plasmids and transcript at a rate β. The mRNA degrades with a rate α, and a dilution term including cell reproduction. We use Hill input functions to model the behaviour of the repressors and the activator.

Equations for Protein translation

Ribosomes translate proteins at a rate from the mRNA code. The second term describes the degradation a rate α including dilution.

Equations for Protein translation

HSL and IPTG diffuse through the cell membrane at a rate theta. If the outside concentration of the molecules is bigger then the concentration inside, HSL and IPTG diffuse inside the cell otherwise the cell releases HSL and IPTG. HSL collide with LuxR to form the complex P at a rate kP, the complex P breaks into HSL and LuxR at a rate k-P.

Parameter description

Parameter	Description
X	Degradation of X
Y	Degradation of Y
λ-CI	Degradation of lambda CI
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
P	Rate of formation of the HSL-LuxI complex
-P	Rate of dissociation of the HSL-LuxI complex

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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

[7] Young, Ry, Ing-Nang Wang and William D. Roof. “Phages will out: strategies of host cell lysis.” Trends in Microbiology 2000; 8(3):120-8.

[8] Christos G. Savva, Jill S. Dewey, John Deaton, Rebecca L. White, Douglas K. Struck, Andreas Holzenburg and Ry Young. The holin of bacteriophage lambda forms rings with large diameter. Molecular Microbiology 69(4), 784–793. 2008.