Team:TU-Munich/Modeling/Kill Switch

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Revision as of 15:04, 3 October 2013

Kill Switch Modeling

Purpose

The idea of our kill switch is to kill off our moss, as soon as it leaves the filter system. For this purpose two methods were proposed:

siRNA method: When some trigger is activated, siRNA is expressed inhibiting the expression of a vital gene
nuclease method: When some trigger is activated, a nuclease is released destroying the DNA of the cell

To decide between these two methods we modelled the vitality V of the cell (a number between 0 and 1, so a perfectly functional cell has V=1, a dead cell V=0) and depending on the tested method the concentration of siRNA R and nuclease N as appropriate. Both concentrations are normalized to the unit interval [0,1].

siRNA Model

We determined the governing equations of this model to be the following:

with initial conditions V(0) = 1 and R(0) = 0, where at the time t=0 the trigger is activated.

The stable points V* and R* of this system have to satisfy

Defining

we get the following quadratic equation for the stable point of V

If α = 1, the unique stable point is .

To analyze the stability of these the eigenvalues of the Hessian matrix H

must be computed. The eigenvalues are

These are both negative, so this is a stable point.

If α ≠ 1, the stable points are

Now only one of these is in the sensible range, because

for α > 1:
for α < 1:

So there is only one stable point in this range, namely:

By expanding the fraction by we can rewrite V* as

So .

Now look at the eigenvalues of the Hessian matrix to analyze the stability

Defining , the eigenvalues are given by

So , which means that this is a stable attractor.

Nuclease Modell

Conclusion

For a functional kill-switch it is necessary, that the cells are actually killed completely and not just live on with reduced vitality. So based on our modelling results the siRNA approach is not satisfactory, while the nuclease satisfies the requirement. As a result the team pursued the nuclease approach leading to our final kill-switch.

@@ Line 18: / Line 18: @@
 ====siRNA Model====
-We determined the governing equations of this model to be the following:
+We determined the governing equations of this model to be the following:[[File:TUM13_siRNA_formula.png|center]]
-[[File:TUM13_siRNA_formula.png|center]]
 with initial conditions V(0) = 1 and R(0) = 0, where at the time t=0 the trigger is activated.
@@ Line 45: / Line 44: @@
+So there is only one stable point in this range, namely:
 [[File:TUM13_siRNA_stable_realistic_point.png|center]]
-[[File:TUM13_siRNA_expandby.png|center]]
+By expanding the fraction by [[File:TUM13_siRNA_expandby.png]] we can rewrite V* as
 [[File:TUM13_siRNA_V_rewritten.png|center]]
-[[File:TUM13_siRNA_V_in01.png|center]]
-[[File:TUM13_siRNA_Hessian.png|center]]
+So [[File:TUM13_siRNA_V_in01.png]].
+Now look at the eigenvalues of the Hessian matrix to analyze the stability [[File:TUM13_siRNA_Hessian.png|center]]
-[[File:TUM13_siRNA_EigVals_b.png|center]]
+Defining [[File:TUM13_siRNA_EigVals_b.png]], the eigenvalues are given by
 [[File:TUM13_siRNA_EigVals.png|center]]
-[[File:TUM13_siRNA_END.png|center]]
+So [[File:TUM13_siRNA_END.png]], which means that this is a stable attractor.
 ====Nuclease Modell====