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<title>Project - Gaëlle Thurre</title>
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<h1>How antibiotics affect population size</h1>
<p>
</p>
<h2>Effects of killing rate and division rate</h2>
<p>
There are two classes of antibiotics: bacteriostatic (inhibit bacterial growth and/or division) and bactericidal (kill the bacteria).
Bactericidal antibiotics show reduced bacterial survival in batch experiments. Bacteriostatic antibiotics show a constant the number of surviving
bacteria over time. In those batch experiments, the survival of bacteria is assessed counting the colony forming units (CFU) per mililiter of culture.
Different concentrations of antibiotics have different effects on bacteria. It is believed that at low concentrations,
bactericidal antibiotics become bacteriostatic.
Recently, single cell experiments using <I>Mycobacterium smegmatis</I> (Wakamoto & al 2013) have shown that the constant CFU observed under low concentration of antibiotics
is in fact caused by balanced division and killing and not because of a static population of non-dividing cells.
This finding has a big impact because the higher the division rate, the higher the mutation rate! Doses of antibiotics that were thought
to maintain population size by inhibiting cell division could in fact promote resistance because cells keep dividing.
<br/><br/>
In that context, our lab and in particular Katrin Schneider is interested in the study of division and killing rates at the single cell level during the
exposure of different concentrations of antibiotics.
To quantify division and killing at the single cell level, Katrin performs time-lapse experiments using a microfluidic device.
She grows cells until micro-colonies of 10-15 cells are formed. She then applies different concentrations of antibiotics and is able to count
the number of division and killing events over time.
<br/><br/>
In preliminary, unpublished results, using bactericidal antibiotics, she noticed that an increase of division rates at low antibiotic concentrations is mainly the cause of the increase of survival.
At high antibiotic concentrations the division rate is lower. On the other hand, the killing rate seems to remain more or less constant at all concentrations.
Frustratingly, she observes a lot of variability in the results, especially at low concentrations.
Indeed, micro-colonies exposed to the same drug concentration can display completely different outcomes, ranging from a lot of survival to none at all.
We would like to try to better understand the reason for this variability by analysing three different phenomena.
<br/><br/>
- <b>"Drift" </b>- all cells are virtually identical (chapter I)
<br/><br/>
- <b>Phenotypic variations</b> - all cells are genetically identical but vary in phenotype which is not inherited (chapter II)
<br/><br/>
- <b>Mutations</b> - a positive mutation occurred during the timelapse and is inherited. (chapter III)
<br/><br/>
In the last part, we will show a 2D spatial model of bacteria that divide and are killed following the phenotypic variation model.
<h2>I Drift</h>
<h3>I.1 When Division rate > Killing rate</h3>
Let's assume that the division rate (twice every 150 min) is bigger than the killing rate (1 every 150 minutes), reflecting a case of very low concentration of antibiotics.
In this first chapter, I simulate cells that are all equal, and have all equal probability to divide or be killed. This is implemented in the simulation by giving every minute,
every cell 2 chances over 150 to divide and 1 chance over 150 to die. This might give rise to inconsistencies
at the single cell level (a cell could possibly divide twice in two minutes) but the probabilities are correct at the population level.
<br/><br/>
Even if all cells are equal, we already observe just by chance, a quite big variation in the fate of the starting micro-colonies.
After 17 hours some colonies are composed of over 10'000 cells where as some are only of less than 2'000.
</p>
<iframe src="1_Project_random_division_death.html?N=10&killing_rate=0.00667&division_rate=0.013" class="simulation" id="simulation1" marginwidth="0" marginheight="0" scrolling="no" ></iframe>
<p>
<a class="button button-primary" href="javascript:document.getElementById('simulation1').contentWindow.location.reload();">RELOAD SIMULATION</a>
</p>
<h3>I.2 When Division rate = Killing rate</h3>
<p>
In the case where the division rate is the same than the killing rate we can observe an interesting phenomenon when starting with a small population size (10 individuals).
After simulating 1'000 minutes of growth, in some micro-colonies all the individuals died whereas in others the population size slightly increased. Again all individuals are equal and have no prior advantage.
This effect could to some level be compared to the drift of two neutral alleles, that just by chance either go to fixation or are lost over the generations.
<br/><br/>
This specific behavior is dependent on the initial population size which is small in this example. The bigger the starting population, the slower the loss of entire micro-colonies would be.
</p>
<iframe src="1_Project_random_division_death.html?N=10&killing_rate=0.00667&division_rate=0.00667" class="simulation" id="simulation2" marginwidth="0" marginheight="0" scrolling="no" ></iframe>
<p>
<a class="button button-primary" href="javascript:document.getElementById('simulation2').contentWindow.location.reload();">RELOAD SIMULATION</a>
</p>
<h3>I.3 When Division rate < Killing rate</h3>
<p>
When the killing rate is bigger than the division rate (1/150 vs 1/300) we see that most of the micro-colonies die after 1000 minutes.
They don't die at the exact same speed and the ones that still have some individuals left alive would have been killed if the simulation was longer.</p>
<iframe src="1_Project_random_division_death.html?N=10&killing_rate=0.00667&division_rate=0.0033" class="simulation" id="simulation3" marginwidth="0" marginheight="0" scrolling="no" ></iframe>
<p>
<a class="button button-primary" href="javascript:document.getElementById('simulation3').contentWindow.location.reload();">RELOAD SIMULATION</a>
</p>
<p>
In those three examples, I kept the killing rate the same and varied only the division rate, consistently with what Katrin observed using different drug concentrations.
We see that even when all micro-colonies are composed of the same number of perfectly equal individuals, their fate can be very different over time.
Under low concentrations of antibiotics, when the division rate remains bigger than the killing rate, they don't grow at the same speed.
Under medium concentrations of antibiotics, when the division equals the killing rate, some colonies survive, even tripple their size in 16 hours, whereas some die.
Under high concentration of antibiotics, most of the micro-colonies see all their individuals die, but some quicker than others.
</p>
<h4>Code</h4>
<pre><code>
var division_rate = 1/150; // per minutes
var killing_rate = 1/150; // per minutes
var N = 10;
var time_lapse_in_minutes = 1000;
var simulations = 10;
var data = [];
// calculate the population size for each time step
function next_time_step(current_N) {
var next_N = current_N;
// out of current_N cells, how many die if they have a probability to die of "killing rate" every minutes?
var number_of_cells_that_die = binomial_draws(current_N ,killing_rate);
next_N = next_N - number_of_cells_that_die;
//How many cells divide?
var number_of_cells_that_divide = binomial_draws(current_N, division_rate);
next_N = next_N + number_of_cells_that_divide;
return next_N;
}
function binomial_draws(n,p){ // n = number of draws, p = probability of success
var number_of_success = 0;
for (var i=0; i &lt n; i++){
var random_number = Math.random();
if (random_number < p){
number_of_success = number_of_success + 1;
}
}
return number_of_success;
}
// calculate the final population size for each simulation
function simulation(simulation_counter) {
var population_sizes = [];
population_sizes.push(N);
for (var i = 0; i &lt time_lapse_in_minutes; i++){
next_N = next_time_step(population_sizes[i]);
population_sizes.push(next_N);
}
return population_sizes;
}
// run all simulations
for (var i = 0; i &lt number_of_simulations; i++) {
population_sizes = simulation(i);
data.push(population_sizes);
}
</code></pre>
<h2>II Phenotypic heterogeneity</h2>
In this chapter, we will implement very small differences between individuals that are non-inheritable, in order to represent the phenotypic
heterogeneity between individuals of an isogenic population. Even if individuals remain very similar to each other, some can divide slightly more
often than others and some are killed less often.
<h3>II.1 When the Division rate = Killing rate with a std of 5%</h3>
<p>
When the killing rates and the division rates are gaussian distributed with a 5% standard deviation around the mean, the
survival does not seem to change compared to deterministic values of killing and division rate as simulated in the first part.
Here is an example with gaussian distributed division and killing rates with equal mean (1/150) and standard deviation (5%).
<br/><br/>
</p>
<iframe src="2_Project_individual_division_death.html?N=10&division_rate=0.00667&killing_rate=0.00667&noise_division=0.05&noise_killing=0.05" class="simulation" id="simulation4" marginwidth="0" marginheight="0" scrolling="no" ></iframe>
<p>
<a class="button button-primary" href="javascript:document.getElementById('simulation4').contentWindow.location.reload();">RELOAD SIMULATION</a>
</p>
The trend seems to be the same, which is confirmed by the histogram of final population size from 2'000 simulations performed for both conditions:
all cells being exactly equal compared to cells having phenotypic heterogeneities that are not inherited.
<iframe src="2_Project_individual_division_death_hist.html" class="simulation" id="simulation5" marginwidth="0" marginheight="0" scrolling="yes" ></iframe>
<p>
We saw in the first chapter that the fate of micro-colonies was already quite different even if all individuals where exactly the same.
In this second chapter we implemented phenotypic heterogeneity between individuals of an isogenic population. The heterogeneity between cells did not change
the final populations sizes distribution.
</p>
<h2>III Mutations</h2>
<h3>III.1 Inheritance of division rate increases survival</h3>
<p>
In the case of inherited growth rate (but not killing rate) we see enhanced survival between the micro-colonies.
Again, the killing rates and the division rates are gaussian distributed with a 5% standard deviation around the mean (1/150).
<br/><br/>
In this case the starting population is composed of individuals that are genetically different, i.e the daughter cells inherit the
growth rate of the mother.
After 2'000 simulations, only 300 micro-colonies die compared to 600 when the growth rate was not inherited.
</p>
<iframe src="3_Project_inherited_division_death_hist.html" class="simulation" id="simulation6" marginwidth="0" marginheight="0" scrolling="yes" ></iframe>
<h3>III.2 Mutations occurring during the time-lapse </h3>
The last case that we are going to see is when the starting population is again composed of genetically identical but phenotypically different individuals.
But this time, we estimate the emergence of resistance to the antibiotic with a probability of once every 10^6 divisions as described in the literature.
The resistance is again illustrated with a slightly higher division rate (mean + 2*std).
<br/><br/>
The resistance mutation event is so rare that it is hardly observable: on average, one mutation arises every 10'000 simulations.
After 20 simulation of 1'000 minutes I found the average number of divisions to be 87.
Thus 10^6 divisions / 87 divisions per simulations = 10'000.
It is thus unlikely to happen in the context of the experiments where we follow 10 to 50 micro-colonies.
Even if such a mutation occurs, the effect on the final population size might not even be notable,
as it is likely to arises late in the simulation when there are more cells.
<iframe src="4_Project_deNovo_mutation.html?N=10&division_rate=0.00667&killing_rate=0.00667&noise_division=0.05&noise_killing=0.05&force_of_mutation=2" class="simulation" id="simulation8" marginwidth="0" marginheight="0" scrolling="no" ></iframe>
<p>
<a class="button button-primary" href="javascript:document.getElementById('simulation8').contentWindow.location.reload();">RELOAD SIMULATION</a>
<br/><br/>
<h2>2D Spatial model</h2>
In this last part, we show a 2D spatial model starting with 15 bacteria, that divide and are killed following the phenotypic variation model.
This time, division rates and killing rates were assigned to each cell following a normal distribution with shortened tails, ensuring that
a bacterium can not divide in less than 20 minutes. The division rates are equal to the killing rates with a standard deviation of 5%.
Cells that are alive are displayed in dark grey. Cells that died remain on the grid as light grey dots. Similarly, in time lapse movies
most of the dead cells remain visible under phase imaging but appear less dark than the living ones.
As described in section 2, we observe after reloading the spatial simulation, that sometimes all cells die and that sometimes the simulation plays a long time with more and more cells alive.
But on average, we see approximately the same number of cells alive along the simulation, than there was at the beginning. This holds with the
fact that the division rates are equal to the killing rates, as shown in the figure of chapter II.1.
Interestingly, we observe that some cells are instantly killed and will never form micro-colonies whereas some succeed to do so.
The advantage certains cell have in the division rates due to phenotypic variations are not inherited. This shows again that if some
cells are able to forme colonies and not others can be explained just because of "chance" and not necessary because of an underlying mutation.
<iframe src="6_Spatial_simulation.html" class="simulation" id="simulation9" marginwidth="0" marginheight="0" frameborder="0" scrolling="no" onload="resizeIframe(this)" ></iframe>
<p>
<a class="button button-primary" href="javascript:document.getElementById('simulation9').contentWindow.location.reload();">RELOAD SIMULATION</a>
</p>
<h2>Discussion</h2>
The experiments Katrin has performed and the heterogeneity of the final population sizes she observed could likely be explained by
a simple model of non inherited phenotypical heterogeneity of the individuals as simulated in chapter two.
As no differences in population size and survival were reported using clonal cells, the even simpler simulations of drift in chapter one
can be used to explain the heterogeneity.
<br/><br/>
It could also be that a mutation occurs during the time lapse of the experiment. According to the literature, such a mutant should be encountered every 10^6 divisions,
meaning it will occur once every 10'000 micro-colonies which is very unlikely as Katrin follows 10 to 50 micro-colonies.
Even if such a mutation occurs, it could not be detected as it is likely to appear at the end of a time lapse, when there is a lot of cells.
When simulating inheritance, we assumed that it was representing the evolution of 10 individuals
that are not genetically identical but another simulation that could be made is when the <i> phenotypic traits </i> are inherited.
Wakamoto & al. showed that phenotypic traits can be inherited to some point for 2-3 generations (unpublished).
In that case, the survival should increase compare to simulations where there is no inheritance, but should not reach the results obtained with full inheritance when the trait is genetically transmitted.
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