Team:TU-Munich/Modeling/Kill Switch

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(→siRNA Model)
(→siRNA Model)
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====siRNA Model====
====siRNA Model====
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=====Governing equations=====
We determined the governing equations of this model to be the following:[[File:TUM13_siRNA_formula.png|center]]
We determined the governing equations of this model to be the following:[[File:TUM13_siRNA_formula.png|center]]
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with initial conditions V(0) = 1 and R(0) = 0, where at the time t=0 the trigger is activated.
with initial conditions V(0) = 1 and R(0) = 0, where at the time t=0 the trigger is activated.
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The stable points V* and R* of this system have to satisfy [[File:TUM13_siRNA_stable_satisfy.png|center]]
 
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Defining [[File:TUM13_siRNA_alpha_beta_def.png]] we get the following quadratic equation for the stable point of V [[File:TUM13_siRNA_stable_quadratic.png|center]]
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=====Calculation of stable points and analysis=====
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The steady points V* and R* of this system have to satisfy [[File:TUM13_siRNA_stable_satisfy.png|center]]
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If '''α = 1''', the unique stable point is [[File:TUM13_siRNA_alphaIS1_stable.png]].
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Defining [[File:TUM13_siRNA_alpha_beta_def.png]] we get the following quadratic equation for the steady point of V [[File:TUM13_siRNA_stable_quadratic.png|center]]
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If '''α = 1''', the unique steady point is [[File:TUM13_siRNA_alphaIS1_stable.png]].
To analyze the stability of these the eigenvalues of the Hessian matrix  H [[File:TUM13_siRNA_alphaIS1_Hessian.png|center]]
To analyze the stability of these the eigenvalues of the Hessian matrix  H [[File:TUM13_siRNA_alphaIS1_Hessian.png|center]]
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must be computed. The eigenvalues are [[File:TUM13_siRNA_alphaIS1_EV.png|center]]
must be computed. The eigenvalues are [[File:TUM13_siRNA_alphaIS1_EV.png|center]]
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These are both negative, so this is a '''stable point'''.
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These are both negative, so this is a '''stable point''', i.e. an attractor.
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If '''α ≠ 1''', the stable points are [[File:TUM13_siRNA_stable_points.png|center]]
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If '''α ≠ 1''', the steady points are [[File:TUM13_siRNA_stable_points.png|center]]
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So there is only one stable point in this range, namely:
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So there is only one steady point in this range, namely:
[[File:TUM13_siRNA_stable_realistic_point.png|center]]
[[File:TUM13_siRNA_stable_realistic_point.png|center]]
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[[File:TUM13_siRNA_EigVals.png|center]]
[[File:TUM13_siRNA_EigVals.png|center]]
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So [[File:TUM13_siRNA_END.png]], which means that this is a stable attractor.
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So [[File:TUM13_siRNA_END.png]], which means that this is a '''stable attractor'''.
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=====Result of the modelling=====
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'''So for the siRNA there always is a stable point for the vitality between 0 and 1, so the moss is not killed-off completely, just impeded in its growth.'''
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=====Interpretation=====
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This result makes intuitive sense, because as the function of the cell is repressed the cell produces less of the inhibiting siRNA, which leads to a regeneration of the cell.
====Nuclease Modell====
====Nuclease Modell====

Revision as of 15:22, 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:

  1. siRNA method: When some trigger is activated, siRNA is expressed inhibiting the expression of a vital gene
  2. 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

Governing equations
We determined the governing equations of this model to be the following:
TUM13 siRNA formula.png

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


Calculation of stable points and analysis
The steady points V* and R* of this system have to satisfy
TUM13 siRNA stable satisfy.png
Defining TUM13 siRNA alpha beta def.png we get the following quadratic equation for the steady point of V
TUM13 siRNA stable quadratic.png


If α = 1, the unique steady point is TUM13 siRNA alphaIS1 stable.png.

To analyze the stability of these the eigenvalues of the Hessian matrix H
TUM13 siRNA alphaIS1 Hessian.png
must be computed. The eigenvalues are
TUM13 siRNA alphaIS1 EV.png

These are both negative, so this is a stable point, i.e. an attractor.


If α ≠ 1, the steady points are
TUM13 siRNA stable points.png


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

  • for α > 1: TUM13 siRNA alphaGT1 lowerbound.png
  • for α < 1: TUM13 siRNA alphaLT1 lowerbound.png


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

TUM13 siRNA stable realistic point.png


By expanding the fraction by TUM13 siRNA expandby.png we can rewrite V* as

TUM13 siRNA V rewritten.png


So TUM13 siRNA V in01.png.


Now look at the eigenvalues of the Hessian matrix to analyze the stability
TUM13 siRNA Hessian.png


Defining TUM13 siRNA EigVals b.png, the eigenvalues are given by

TUM13 siRNA EigVals.png

So TUM13 siRNA END.png, which means that this is a stable attractor.


Result of the modelling

So for the siRNA there always is a stable point for the vitality between 0 and 1, so the moss is not killed-off completely, just impeded in its growth.

Interpretation

This result makes intuitive sense, because as the function of the cell is repressed the cell produces less of the inhibiting siRNA, which leads to a regeneration of the cell.

Nuclease Modell

TUM13 nuc formula.png
TUM13 nuc initial.png
TUM13 nuc stable satisfy.png
TUM13 nuc hessian ev.png

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.

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