Team:TU-Munich/Modeling/Kill Switch

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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

Figure 1: Example solution to our siRNA model with parameters k1=1, k2=2, k3=2, k4=1.
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.

Figure 1, created by our MatLab script #siRNA_model shown below, on the right shows a solution to this system for some example parameters. In this case the vitality of the cell decreases to somewhere around 0.3 while at the same time after an initial peak the siRNA concentration also drops upon which the vitality settles in at about 0.4.

So it appears, that the siRNA approach does not achieve the required death of the cells. In the following it will be shown, that this is the case for all parameter values.

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 using the results from above, we get: 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 our model

So for the siRNA there always is a stable point for the vitality between (but excluding) 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. Eventually a steady state at a lower vitality is reached, where the vitality stays constant.


Nuclease Model

Governing equations

Figure 2: Example solution to our nuclease model with parameters p1=1, p3=1, p4=0.5.
We determined the governing equations of this model to be the following:
TUM13 nuc formula.png

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

Figure 2, created by our MatLab script nuclease_model shown below, on the left shows a solution to this system for some example parameters. It clearly shows that in this case the vitality of the cell decreases to 0 and remains there, i.e. that the cell has died.

In the following we will verify, that this is the case for all parameter values.


Calculation of stable points and analysis

Any steady state given by V* and N* of this system must to satisfy
TUM13 nuc stable satisfy.png


The Hessian matrix of this system for the steady point is easy to solve
TUM13 nuc hessian ev.png


The eigenvector corresponding to the zero eigenvalue is TUM13 nuc eigenvec.png. It is obvious from the equations that any disturbance away from the steady state along this vector will decay back to the steady state, so this is a stable attractor.

Results of our model

The nuclease always reduces the vitality of the moss to 0, i.e. kills it off completely.


Interpretation

Again this result is very intuitive, since the moss cannot regenerate from the destruction of its genome.

Conclusions

For a functional kill-switch it is necessary, that the cells are actually killed and not just live on with reduced vitality. So based on our modeling 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.


MatLab Scripts

siRNA_model

% f = @(R,V,k) [k(3)* V  - k(4) * R , -k(1) * R * V + k(2) *(V  -1)* (R  -1)];
g = @(k,y)    [k(3)* y(2)- k(4)*y(1); -k(1)*y(1)*y(2)+k(2)*(y(2)-1)*(y(1)-1)];

%k= [  k1,  k2,  k3,  k4 ]; %insert the appropriate reaction rate constant
k = [   1,   2,   2,   1];

[TOUT, YOUT] = ode45(@(t,y) g(k,y) ,[0,10],[0;1]);

plot(TOUT, YOUT(:,1), 'markersize', 15, 'linewidth', 5)
hold on; plot(TOUT, YOUT(:,2), 'r', 'markersize', 15, 'linewidth', 5); hold off

legend('siRNA concentration','Vitality');
xlabel('time');
set(gca,'FontSize',24);
set(gcf,'position', [100 100 600 600]);

axis square
set(gcf,'Color','w');
export_fig siRNA_pic.png

nuclease model

% f = @(N,V,k) [p(3)* V  - p(4) * N , -p(1) * N * V  ];
g = @(p,y)    [p(3)* y(2)- p(4)*y(1); -p(1)*y(1)*y(2)];

%p= [  p1,  p2,  p3,  p4 ]; %insert the appropriate reaction rate constant
p = [   1, NaN,   1, 0.5 ];

[TOUT, YOUT] = ode45(@(t,y) g(p,y) ,[0,20],[0;1]);

plot(TOUT, YOUT(:,1), 'markersize', 15, 'linewidth', 5)
hold on; plot(TOUT, YOUT(:,2), 'r', 'markersize', 15, 'linewidth', 5); hold off

legend('Nuclease concentration','Vitality')
xlabel('time');
set(gca,'FontSize',24);
set(gcf,'position', [100 100 600 600]);

axis square
set(gcf,'Color','w')
export_fig nuc_pic.png


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