Chemotaxis

Motivation

To stay true to its title 'pathogen-hunting microbe', BaKillus should ideally be able to swim towards its prey via chemotaxis. However, the ability to react to an AIP- or CSP-gradient requires the introduction of a chimeric receptor into Bacillus subtilis. Chemotaxis receptors are quite complex and can, for example, take different methylation states to allow for attenuation of the chemotactic response. This complexity puts the construction of a custom receptor beyond the scope of an iGEM project.

To evaluate how a hypothetical chemotactic BaKillus would act in an experimental setting, we developed a model to simulate the movement of chemotactic hunters in response to an attractant gradient produced by a growing colony of prey bacteria.

Overview

Our multi-scale hybrid model essentially consists of 3 components: growth of prey bacteria (algebraic model), diffusion of attractant molecules (spatial partial differential equation) and chemotactic behavior of the hunter bacteria (stochastic spatial agent model coupled to differential equations for internal state with a stochastic driving force). All these processes take place in a defined geometric space (representing, for example, a microfluidic device). The experimental scenario we wish to capture in our model is the following: a colony of prey bacteria is spotted onto the arena, growing in place and producing attractant molecules. A fixed population of hunter bacteria is placed into the arena as well. Their reaction (biased movement) to the building attractant gradient is what we seek to observe.

Growth of prey

The prey bacteria will grow on a population level, according to a logistic growth curve with explicit solutions (ref!):

\[ y = ln \frac{N}{N_0} = \frac{A}{1 + e^{\frac{4\mu_m}{A} (\lambda - t) + 2}}\]

Here, \(N_0\) is the population size at \(t = 0\), \(A = ln \frac{N_\infty}{N_0}\) is the asymptote, \(\mu_m\) is the growth rate and \(\lambda\) the duration of the lag-phase.

Diffusion of attractants

The concentration of attractant molecules (\(c\)) in space changes according to a partial differential equation comprising diffusion, degradation and production by the prey bacteria.

\[ \frac{\partial c}{\partial t} = D\Delta c + p c_{prey} - a c\]

\(D\) is the diffusion coefficient, \(a\) is the degradation rate, \(p\) the rate of production and \(c_{prey}\) the concentration of prey/producer bacteria.

Chemotaxis

The chemotaxis of the hunter bacteria (BaKillus) is represented by an agent-based model adapted from [Matthäus ref]. Each bacterium has its own methylation and phosphorylation states of the chemotactic pathway which change according to a system of ordinary differential equations in response to attractant concentrations. In a classical 2-state model of bacterial chemotaxis, each bacterium will be in either running or tumbling state at each time point. In a running step, bacteria will swim forward in a straight line whereas in a tumbling step, they will change their direction by a random angle picked from a gamma distribution.

It should be noted that the model used here actually describes chemotaxis of E. coli. While the wiring of the chemotaxis pathway differs slightly, the same behavior is expected for B. subtilis. Therefore, we opted for the use of the better studied E. coli-model.

Implementation

Suicide Switch

Motivation

For safety issues we want BaKillus to autoinduce cell lysis after successful completion of his task. This can be achieved by shutting down the production of resistance proteins. (see: Suicide-Switch)

Still, BaKillus must produce sufficient amounts of toxins before inducing cell lysis. Therefore, the negative feedback, which eventually stops production of the immunity protein must be delayed to provide a sufficiently long time frame for toxin production. However, as simple as this idea might sound in theory, robust practical implementation of such a module requires precise fine tuning of the kinetics of the respective signalling cascade. In this model we tried to assess whether the inclusion of an alternative sigma factor could theoretically delay 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 immunity production over time. This should provide a measure of the effective number of produced 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 as well as protein dynamics. Inhibitory and promoting activity were modeled via hill kinetics.

Variables

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

MATLAB-Script

% 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

Output

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 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.

The black line marks the averages of the stochastic model.

Results

As we can see, C_0,psdpI has no effect on the dynamics of the model. While high basic concentrations of P_qs, FsrA and the sigma factor have reciprocal impact on the fitness of the model, the transcription and translation rates of FsrA and the sigma factor affect the model's fitness positively.
Therefore, the best results could be achieved with low basic concentrations and high translation and transcription rates.