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Model for comS diffusion system

Summary

Design

Model

LacI

We use LacI as a repressor for the emitter gene construct. LacI repression can be cancelled by IPTG. This way we can induce production of RFP and T7' by puttig IPTG on the cells.

Inactivated LacI can not repress the pLAC promoter anymore. Note that we consider that the reaction between IPTG and LacI fires without any delay. This assumption is justified by the fact that this reaction is much faster than any other in our gene network.

Emitter gene construct - comS

The emitter gene construct is modeled by the following equations:

The reporter for the emitter gene construct (RFP) is modeled by the following equations:

Receiver and amplification gene construct - comK

The receiver and amplification gene construct is modeled by the following equations:

The reporter for the receiver and amplification gene construct (GFP) is modeled by the following equations:

Parameters

This design relies on comS as the signaling molecule going through the nanotubes.

The parameters used in this model are:

Parameter	Description	Value	Unit	Reference
	Active LacI concentration (LacI which is not inactivated by IPTG)	NA	molecules per cell	Notation convention
	IPTG concentration	NA	molecules per cell	Notation convention
	Inactived LacI concentration	NA	molecules per cell	Notation convention
	Total LacI concentration	TBD	molecules per cell	Steady state for equation
	T7 RNA polymerase (emitter, T7') concentration	NA	molecules per cell	Notation convention
	mRNA associated with T7' concentration	NA	molecules per cell	Notation convention
	T7 RNA polymerase (auto-amplification, T7'') concentration	NA	molecules per cell	Notation convention
	mRNA associated with T7'' concentration	NA	molecules per cell	Notation convention
	Maximal production rate of pVeg promoter (constitutive)	???	molecules.s-1 or pops	Estimated
	Maximal production rate of pLac promoter	0.02	molecules.s-1 or pops	Estimated
	Maximal production rate of pT7 promoter	0.02	molecules.s-1 or pops	Estimated
	Dissociation constant for IPTG to LacI	1200	molecules per cell	Aberdeen 2009 wiki
	Dissociation constant for LacI to LacO (pLac)	700	molecules per cell	Aberdeen 2009 wiki
	Dissociation constant for T7 RNA polymerase to pT7	3	molecules per cell	Estimated ADD EXPLANATION
	Translation rate of proteins	1	s-1	Estimated ADD EXPLANATION
	Dilution rate in exponential phase	2.88x10-4	s-1	Calculated with a 40 min generation time. See explanation
	Degradation rate of mRNA	2.88x10-3	s-1	Uri Alon (To Be Confirmed)
	Delay due tT7 RNA polymerase production and maturation	300	s	http://mol-biol4masters.masters.grkraj.org/html/Prokaryotic_DNA_Replication13-T7_Phage_DNA_Replication.htm
	Delay due to mRNA production	30	s	http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=104902&ver=5&hlid=58815 2kb/(50b/s) --> approximation: all our contructs are around 2kb

Results & discussions

Limits

[[File:TRNA_scheme.png|thumb|center|upright=3.0|tRNA Amber genetic design]] [[Team:Paris_Bettencourt/tRNA_diffusion|The amber suppressor tRNA diffusion.]] The idea of the system is to pass tRNA amber molecules through the nanotubes. At every moment of time in the receiver cell there is a certain amount of transcribed mRNA-T7 among the others mRNA. The behavior of tRNA amber that arrived in a receiver cell is random, so in order to describe its interaction with mRNA-T7 and its further translation we can reason in terms of probability. We can reason in two steps : first a tRNA amber molecule gets close to a mRNA molecule. Then, it binds it's anti-codon with a codon of the mRNA. This reasoning is similar to the problem of boxes and balls. There are two types of boxes: 'a' of the first type and 'b' of the second (which corresponds to the set of mRNA-T7 and mRNA-non-T7), and there are 't' balls(tRNA amber). All the balls are randomly distributed in the boxes. If there are two or more balls in some box of the first type (two or more tRNA amber per mRNA-T7) then a T7 molecule will be produced with a chance P_0. We have defined two models for this system which both rely on the following assumptions : * Each mRNA is defined as a 'box' * All the tRNA molecules are uniformly distributed in the boxes. * The number of tRNA_amber diffused through the nanotubes is much more smaller than the one of the mRNA. Thus the chance that three or more tRNA amber will "find" one mRNA-T7 is negligible comparing to the one of two tRNA amber (finding a mRNA-T7). In our model we will consider that at one moment of time each mRNA interacts with 0, 1 or 2 tRNA ambers. * The tRNA_amber placed in a correct box are always used We have defined two models for this system which both rely on the following assumptions : * Each mRNA is defined as a 'box' * All the tRNA molecules are uniformly distributed in the boxes. * The number of tRNA_amber diffused through the nanotubes is much more smaller than the one of the mRNA. Thus the chance that three or more tRNA amber will "find" one mRNA-T7 is negligible comparing to the one of two tRNA amber (finding a mRNA-T7). In our model we will consider that at one moment of time each mRNA interacts with 0, 1 or 2 tRNA ambers. * The tRNA_amber placed in a correct box are always used Let's define two types of boxes: mRNA_amber ('A') and other type of mRNA ('B'). We note the mRNA_amber producing T7 as mRNA*_amber. The latter appears if we have two tRNA_amber in one box. This model treats the repartition of tRNA_amber in the different boxes. [[File:Boxes_scheme.png|thumb|center|upright=3.0|A-box and b-box]] We note P(x) the probability of having x A-boxes containing two tRNA_amber. Thus P(x=1) corresponds to the probability of finding a couple of tRNA_amber in an A-box, thus to produce x T7 molecules. 't' is the number of tRNA_amber in the cell. We note:
* P(x=1)= (probability that 2 balls choose A-boxes) * (probability that these 2 balls choose the same A-box) + (probability that 3 balls choose A boxes) * (probability that 2 balls out of 3 choose the same A-box and the third doesn't) + ... + (probability that t balls choose an A-box) * (probability that 2 of these t balls choose the same A-box and the rest don't). * P(x=2)= (probability that 4 balls choose A-boxes) * (probability that these 4 balls choose 2 A-boxes, one A-box per pair of balls) + (probability that 5 balls choose A-boxes) * (probability that 4 balls out of 5 choose 2 A-boxes, one A-box per pair of balls and the fifth doesn't) + ... + (probability that t balls choose A-boxes) * (probability that 4 out of these t balls choose [t/2] A-boxes, one A-box per pair of balls and the rest don't). * ... * P(x=i)= (probability that 2i balls choose A-boxes) * (probability that these 2i balls choose i A-boxes, one A-box per pair of balls) + (probability that (2i + 1) balls choose A-boxes) * (probability that 2i balls out of (2i + 1) choose i A-boxes, one A-box per pair of balls and the rest don't) + ... + (probability that t balls choose A-boxes) * (probability that 2i out of these t balls choose [t/2] A-boxes, one A-box per pair of balls and the rest don't).
Hence: [[File:P(1).png|thumb|center|upright=3.0|A-box and b-box]]