Team:LMU-Munich/Project/Modeling

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=== Fitness W ===
=== Fitness W ===
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''For analysis of the impact of each parameters, we've used a function to determine the fitness '''W''' of the model. The fitness describes the integral of the surplus of toxic cannibalism proteins over the proprietary antidote. Therefore, a successful suicide equals a high fitness of the model. This means plotting W against one parameter, while statistically bruteforcing versatile value combinations of the remaining parameters, one can see how the parameter effects the quality of the model. That way we can figure out what options we have to manipulate our suicide system to adjust it.''  
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''For analysis of the impact of each parameters, we used a function to determine the fitness '''W''' of the model. The fitness describes the integral of the surplus of toxic cannibalism proteins over the proprietary antidote. Therefore, a successful suicide equals a high fitness of the model. As we do not know the parameters in for our model we sampled the whole parameter space using a uniform sampler. Successively we used a Nadaraya-Watson kernel densiy estimator to approximate the expected value of W given a single parameter value. This allows us to asses the expected change in W when changing a single parameter, whithout knowledge  of all other parameters.''  

Revision as of 14:34, 17 October 2014

 

Modeling

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Chemotaxis

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Suicide Switch

For safety issues we want BaKillus to autoinduce apoptosis after successful completion of his task. This can be achieved by shutting down the production of resistance providing anti-toxin proteins. (see: Suicide-Switch) Still, BaKillus must produce sufficients amounts of toxins before inducing apotosis. Therefor, the negative feedback, which eventually stops production of anti-toxin must be sufficiently delayed to provide a sufficiently long timeframe for toxin production. However, as simple as this idea might sound in theory, robust practical implementation of such a module requires precise finetuning of the kinetics of the respective signalling cascade. In this model we tried to assess whether the inclusion of a secondary sigma factor could theoretically enhance the efficiency of the killing mechanism.

In the following we will compare two different models, one including only one sigma factor and one including two sigma factors. The models will be compared based on a fitness function which describes the difference between the toxin and antitoxin production over time. This should provide a measure of the effective number of produced molecules toxin molecules.

Basic Model

The basic model only contains one sigma factor. The system of differential equations was derived following the standard paradigm of gene expression including mRNA aswell as protein dynamics. Inhibitory and Promoting activity were modeled via hill kinetics.

LMU14 Equations1.png


Variables

msdpABC:      Quantity of SdpABC mRNA
psdpABC: Quantity of SdpABC proteins
msdpI: Quantity of SdpI mRNA
psdpI: Quantity of SdpI proteins
mSiAlt1: Quantity of sigma factor mRNA
pSiAlt1: Quantity of sigma factor proteins
mfsrA: Quantity of FsrA mRNA
pfsrA: Quantity of FsrA proteins
Pqs: Quantity of QS proteins


Inquired Parameters

C0,Pqs: Basic concentration of QS proteins
C0,psdpI: Basic concentration of SdpI proteins
C0,fsrA: Basic concentration of FsrA proteins
k0,SiAlt1: Transcription Rate of sigma factor
k1,SiAlt1: Translation Rate of sigma factor
C0,SiAlt1:      Basic concentration of sigma factor proteins
k0,fsrA: Transcription Rate of FsrA
k1,fsrA: Translation Rate of FsrA


Fixed Parameters

In a first evaluation, the variation of following parameters were figured out to cause only insignificant variation of the model's fitness. Therefore they were keept at these specified values for all subsequent simulations:

Degradation rate (g1) of mRNA: 12 molecules per hour
Degradation rate (g2) of proteins: 1 molecule per hour
Transcription rate (k0) of Sdp: 7.2 per hour
Translation rate (k1) of SdpI & SdpABC:      10 per hour


Fitness W

For analysis of the impact of each parameters, we used a function to determine the fitness W of the model. The fitness describes the integral of the surplus of toxic cannibalism proteins over the proprietary antidote. Therefore, a successful suicide equals a high fitness of the model. As we do not know the parameters in for our model we sampled the whole parameter space using a uniform sampler. Successively we used a Nadaraya-Watson kernel densiy estimator to approximate the expected value of W given a single parameter value. This allows us to asses the expected change in W when changing a single parameter, whithout knowledge of all other parameters. 


% par0 = lhsdesign(10000,12); 
par0 = rand(100000,12);
par = 10.^(4*par0 - 2); % Coveres range from 0.01 to 100

par(:,3) =12; % g1
par(:,5) =1; % g2
par(:,2) =0.12*60; % k0_sdp
par(:,4) =10; % k1_sdp

tout = linspace(0,10,100);
tic 
for j = 1:size(par,1)
   [~,~,x,y] = simulate_RRE_cannibalism(tout,par(j,:));
    %figure
    %plot(tout,y)
    %legend('p_{p_sdpABC}','p_{p_sdpI}')
   
   W(j) = sum(y(:,1)-y(:,2));
end
toc

parname={'C0_Pqs','k0_sdp','g1','k1_sdp','g2','C0_p_sdpI', ...
'C0_fsrA','k0_SiAlt1','k1_SiAlt1','C0_SiAlt1','k0_fsrA','k1_fsrA'};
 
for k = [1,6,7,8,9,10,11,12]
  figure
  scatter(par(:,k),W',10,'filled')
  hold on

  [xx,ind]=sort(par(:,k));
  r = ksr(log10(xx),W(ind));
 
  plot(10.^r.x,smooth(r.f,10),'k-','LineWidth',5)
  set(gca,'Yscale','lin')
  set(gca,'Xscale','log')
  xlabel(parname{k})
  ylabel('W')
  saveas(gcf,['parameter_' parname{k}]);
  print('-depsc2','-r300',['parameter_' parname{k}])
end


The graphs show the statistical fitness W of the model plotted against each considered parameter. The black line resembles the average.


LMU14 model 1.jpg
LMU14 model 2.jpg
LMU14 model 3.jpg
LMU14 model 4.jpg
LMU14 model 5.jpg
LMU14 model 6.jpg
LMU14 model 7.jpg
LMU14 model 8.jpg


As we can see..... some go up. some go down. one doesn't matter at all. we now know what parameter effect the system in what way.

Delayed Model

We also considered a delayed SdpI inhibition by integrating a second sigma factor. The mathematical model for this system contains two additional equations for the new gene:


LMU14 Equations2.png


% par0 = lhsdesign(10000,12); 
par0 = rand(100000,15);
par = 10.^(4*par0 - 2); % Coveres range from 0.01 to 100

par(:,3) =12; % g1
par(:,5) =1; % g2
par(:,2) =0.12*60; % k0_sdp
par(:,4) =10; % k1_sdp

tout = linspace(0,10,1000);
tic 
for j = 1:size(par,1)
    [~,~,x,y] = simulate_RRE_cannibalism_twosigma(tout,par(j,:));
%      figure
%      plot(tout,y)
%      legend('p_{p_sdpABC}','p_{p_sdpI}')
%     
    W(j) = sum(y(:,1)-y(:,2));
end
toc

parname={'C0_Pqs','k0_sdp','g1','k1_sdp','g2','C0_p_sdpI','C0_fsrA', 'k0_SiAlt1', ...
 'k1_SiAlt1','k0_SiAlt2','k1_SiAlt2','C0_SiAlt2','C0_SiAlt1','k0_fsrA','k1_fsrA'};


for k = [1,6,7,8,9,10,11,12,13,14,15]
   figure
   scatter(par(:,k),W',10,'filled')
   hold on
 
   [xx,ind]=sort(par(:,k));
   r = ksr(log10(xx),W(ind));

   plot(10.^r.x,smooth(r.f,10),'k-','LineWidth',5)
   set(gca,'Yscale','lin')
   set(gca,'Xscale','log')
   xlabel(parname{k})
   ylabel('W')
   saveas(gcf,['red_parameter_' parname{k}]);
   print('-depsc2','-r300',['red_parameter_' parname{k}])
end


As above, the graphs show the statistical fitness W of the model plotted against each considered parameter. The black line resembles the average.


LMU14 model a.jpg
LMU14 model b.jpg
LMU14 model c.jpg
LMU14 model d.jpg
LMU14 model e.jpg
LMU14 model f.jpg
LMU14 model g.jpg
LMU14 model h.jpg
LMU14 model i.jpg
LMU14 model j.jpg
LMU14 model k.jpg


As we can see.....

Conclusion

Hi there!

Welcome to our Wiki! I'm BaKillus, the pathogen-hunting microbe, and I'll guide you on this tour through our project. If you want to learn more about a specific step, you can simply close the tour and come back to it anytime you like. So let's start!

What's the problem?

First of all, what am I doing here? The problem is, pathogenic bacteria all around the world are becoming more and more resistant against antimicrobial drugs. One major reason for the trend is the inappropriate use of drugs. With my BaKillus super powers, I want to reduce this misuse and thus do my part to save global health.

Sensing of pathogens

To combat the pathogenic bacteria, I simply eavesdrop on their communication. Bacteria talk with each other via quorum sensing systems, which I use to detect them and trigger my responses.

Adhesion

The more specific and effective I can use my powers, the lower the danger is of provoking new resistance development. So I catch pathogens whenever I get hold of them and stick to them until my work is done.

Killing

Talking about my work - killing pathogens is finally what I am made for. In response to quorum sensing molecules of the pathogens, I export a range of antimicrobial substances leading to dissipation of biofilms and the killing of the targeted bacteria.

Suicide switch

When the job is done and all the bad guys are finished, you don't need a super hero anymore. So after fulfilling my work I say goodbye to the world by activating my suicide switch.

Application

Of course I'm not only a fictional hero, but a very real one. In two different prototypes, I could be used for diagnosis or treatment of pathogen-caused diseases. However, there is still a whole lot of regulational and economical questions that have to be answered before.

See you!

So now you know my short story - and it is time for me to return to my fight for a safer world. Feel free to take a closer look on my super powers, the process of my development or the plans for a medical application.

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