Presentation of the experimental platform
In this chapter, we describe the microfluidic chip used throughout this thesis for microbiology applications. The emphasis will be on the technical questions related to the chip: what is measured, how, and what are the lim-itations when we run an experiment. But we will also try to conceptualize what it means to conduct experiments in a device like this one. We will de-scribe as well the typical growth experiments that will be analyzed in more depth in the following Chapters.
An original microfluidic chip for parallel mea-surements of bacterial growth
Design of the chip
The breaking process
The microfluidic chip that is used in this thesis was designed in our lab and is described in-depth in . We are just going to give here its main charac-teristics for the reader to understand the experiments presented later on.
To be concise, the chip comes down to a big chamber, with two inlets and one outlet, that contains a static array of about 1500 nano-liter sized water-in-oil droplets immobilized in anchors, or traps, in which cells can be encap-sulated (Fig. 2.1(A),(B)). Its originality lies in the way by which the droplets are created. Indeed, the droplets are created directly on the anchors thanks to the so-called « breaking » technique , which is based on a surface-tension effect. In fact, the anchors are holes that are patterned in the PDMS ceiling of the chamber, and they act as wells of surface energy .
This principle was at first designed to keep a droplet trapped in an exter-nal oil flow . Let us imagine a big microfluidic chamber, with a small height difference between the glass floor and the PDMS ceiling, and infinite dimensions in the other directions. If a big water-in-oil droplet is introduced in such a chamber, it will be squeezed between the floor and the ceiling and adopt a pancake-like shape. However, the most energetically favorable shape for a droplet is a sphere, which minimizes its surface and thus its surface en-ergy. If now a hole is patterned in the ceiling of the chamber, the the droplet can enter the hole. By doing so, it releases some of its surface energy thanks to the height difference: if the dimensions of the trap are correctly chosen, the surface of the trapped droplet can be smaller than its surface when squeezed in the chamber. Thus it is more energetically favorable for the droplet to remain in the trap, and this trapping force can be quantified.
The same principle is extended in the chip to create droplets in the traps. In fact, let us go back to the big droplet trapped in the anchor that we have described in the last paragraph. If the external oil flow is increased above the trapping force, under certain conditions the droplet will break into two parts. Most of it is flushed away by the oil flow, and a small part remains trapped in the anchor, creating a smaller droplet with a predictable volume .
This is then an efficient and robust way of creating droplets directly on the anchors, and the chip used here is the generalization of this simple principle to a an array of 1500 anchors in a big chamber. Hence, an array of 1500 water-in-oil droplets can be created in less than ten minutes.
The method to create the droplets is described in Fig. 2.1(C),(D). First, the chamber is completely filled with fluorinated oil through inlet 1. Then, it is filled with a suspension of bacteria at the desired concentration in bacterial growth medium through inlet 2. This creates a big plug of aqueous solu-tion in the chamber and the anchors, surrounded by an oil lubrication film (thanks to the first step). Finally, an oil flow is applied again through inlet 1. It pushes the aqueous plug and « breaks » it, leaving a droplet in each anchor, encapsulating at the same time the bacteria in these droplets.
The multiple possibilities of this microfluidic device are fully described in , we will just give the reader a quick summary here. First of all, this whole « breaking » process can be realized with liquid agarose instead of pure culture medium. After the loading, the agarose can be gelified, and the exter-nal oil can be removed by replacing it with an aqueous solution. Hence, the droplets containing bacteria can be perfused with any solution: the micro-environment of the bacteria can be controlled over time. For instance, an-tibiotics can be added to study the stress response of bacteria. We can also perfuse the chip with a gradient of any compound, with a different concen-tration for each row of the droplet array. Finally, with the agarose droplets, a particular droplet can be selectively retrieved from the chip: we can precisely melt it with a laser and retrieve its content by pushing it out of the chip with an external oil flow. Its content is then viable and available for any off-chip analysis (sequencing, susceptibility testing..).
FIGURE 2.1: Taken from  Description of the microfluidic device and protocol for droplet production: (a) design of the microdroplet multiwell device. The central chamber has di-mensions 0.5 4.8 cm and contains a 2D array of 115 13 surface-tension anchors. Square anchors have side dimension d = 120 mm, spaced by d = 240 mm. The chamber height is h1 = 35 mm and the anchor height h2 = 135 mm. (b) The device, which fits on a microscope slide, is connected to two inlets and one outlet. (c) Time-lapse of the drop formation process. At t = 0, the cell sample fills the microfluidic chamber entirely, and is being pushed by fluorinated oil (FC40 + 0.5% surfactant) using a hand-pushed syringe. The arrow indicates the oil flow direc-tion. When the interface penetrates between two anchors, it de-forms and then breaks, which leaves a well-calibrated droplet in the anchor. The cell sample is colored in red for better visu-alization. Scale bar: 200 mm. (d) Cross-sectional schematic of the breaking process on anchors. The aqueous sample initially fills large regions and then gets divided into isolated droplets that fill each of the anchors. (e) Experimental histogram of the normalized droplet volumes on one chip. The orange line is the best Gaussian fit to the data, leading a standard deviation
s = 0.02.
Encapsulation of the bacteria
When bacteria are diluted in suspension, and then encapsulated in small droplets, the number of bacteria per droplet follows a Poisson distribution . This is true as long as there is no interaction between the bacteria.
Theoretically, this Poisson distribution comes from the simple fact that the probability for a given bacterium to be encapsulated in a given droplet of volume v, taken from a bigger volume of medium V, is just the ratio of the volumes v/V. In maths, this is known as a Bernoulli trial: a random experiment with exactly two possible outcomes, « success » and « failure ». The probability of getting a success (the bacterium is encapsulated in the droplet) is the ratio of the volumes v/V.
If there are N bacteria passing through our microfluidic chip during load-ing, then the number of bacteria in a given droplet is the sum of N inde-pendent Bernoulli experiments of parameters v/V: it follows a binomial dis-tribution of parameters (N,v/V), and the product of the two parameters is l = c0v where c0 is the concentration of bacteria in the medium. Then, if the bacteria are independent (which we believe to be true as they are diluted), since N 1 and v/V 10 3, we can apply Le Cam’s theorem , which states that the number of bacteria per droplet converges in law to a Poisson distribution of parameter l.
The probability of finding k bacteria in a droplet is then:
P(N0 = k) = . (2.1)
Amselem et al.  have checked that this was verified in our chip by counting the final number of colonies in each droplet.
One important thing to note about the Poisson distribution is that the probability of having an empty droplet is:
P(N0 = 0) = e l. (2.2)
As we can see, there is a direct relationship between this probability and the Poisson parameter l. This is very useful for us, as then the number of empty droplets on the chip can be used as an estimator for finding the Pois-son parameter used in a experiment:
lˆ = ln (pˆ0) = ln Ntot , (2.3)
where Nempty and Ntot are the number of empty droplets and the total num-ber of droplets, respectively, and pˆ0 represents the estimated probability of having an empty droplet.
The 95% confidence interval on the estimated value of pˆ0 is then :
pˆ0 2 pˆ0+, pˆ0 where pˆ0 pˆ0 1.96 s (2.4)
pˆ 0 Ntot pˆ 0 ,
which we can translate into a confidence interval on l: lˆ = ln pˆ0 . (2.5)
In our case, we have approximately 1000 droplets, and we can plot the ratio of the width of the confidence interval over ˆ (see Fig. 2.2). The width l of the confidence interval is smaller than 20% of ˆ only if we are between l l 0.5 and l 4. This gives us a range of concentration that can be used if we want to be able to estimate l, as the droplet size is approximately 2 nL: the range of concentration goes from 2.5 105 to 2 106 cells/mL. Of course, it is not always of interest to be able to evaluate l, as we will discuss later on.
FIGURE 2.2: Relative confidence interval when estimating l by counting the number of empty droplets
A conceptual approach to microfluidics experiments
Many different experiments can be conducted with this microfluidic plat-form. However, the interpretation of the results is often delicate because of this Poisson distribution and the fact that the number of cells per droplet is not known exactly but as a statistics. Experiments have to be well designed to avoid mistakes. We will try to explore here the different possibilities.
In the limit of very low l, the chip allows us to encapsulate almost only single cells. But as l is very low, the number of empty droplets will be large. For instance, if one wants to statistically achieve that 95% of the droplets contain only one bacteria, she has to choose l 0.1. In this case, the expected number of droplets containing bacteria is only 100. The statistics of such an experiment are then not the same at all as when the whole 1500 droplets are filled with bacteria. This relatively low number of droplets can be sufficient for a lot of applications, such as isolating unknown bacterial species in a sample for further analysis . But it seems to be too low if one wants to test the systematic response of single cells to antibiotics, for instance, where the heterogeneity plays a central role  and makes it often preferable to have a lot of data. Studies using other microfluidic technologies, like the well known « mother-machine »  seem to be preferable in this case .
However, the droplet technology that we have developed has an advan-tage over technologies resembling the « mother-machine »: the progeny of the initial cell is kept in a closed micro-reactor and not thrown away. In a sense, the lineage of the first cell encapsulated in the droplet can be tracked for a much longer time. However, we are in our droplets in a very peculiar situ-ation: this lineage can be enclosed, but the division tree is not accessible, as the bacteria don’t grow in 2D and cannot be individually tracked. The only observable that we have, if we want to take advantage of the relatively high throughput of our droplets, is the number of cells per droplet as a function of time, that is measured indirectly through the fluorescence signal (see next section). Our advantage, once again, is that the whole lineage is kept. The stochastic effects of the division at each generation accumulate in the droplet, even if we will see in the next chapter that the early stages dominate over the later generations. In a 2D system, because of the exponential growth, the bacteria very rapidly invade the whole chamber, and only a few generations (up to 7) can be tracked , or part of the descendants has to be thrown away . In our droplets, we can observe, for E. coli, we can estimate that more than 12 generations can be observed, simply because the growth is ex-ponential for 250 min, with a mean division time of 20 min.
In our experiments, we often choose the initial concentration of bacteria such that l 1. This ensures that only 30% of the droplets are empty, while keeping the number of bacteria per droplet quite low, to still have a trace of the cell-to-cell variability. Indeed, if the initial number of bacteria per droplet is too high, the population-size variability due to the stochasticity of the division process is hidden, as we will see in the next chapter.
As such, experiments in micro-droplets can be seen as an experimental Monte-Carlo process: it gives us the ability to repeat and follow in parallel the same stochastic experiment a large number of times. The distribution of the responses in the population can be obtained thanks to the statistics of this system, especially when compared to classical microbiology methods. We know through the Central Limit Theorem that the convergence to the solution in this kind of method goes as n, where n is the number of repetitions of the experiment. Going from a 96 well plate to a chip like ours with 1000 droplets increases then the precision on the final solution by a factor 3.
There are in fact two ways of applying this Monte-Carlo analogy to our droplets. The first one is to see the droplets as micro-reactors, a little bit like an expanded 96 well plate, with more and smaller wells, and to use them as growth reactors as we have explained above: the stochastic effects accumu-late in the droplet, and thanks to the big number of droplets, the distribution of growth responses can be obtained. This is the approach that we will de-velop in Chapters 3 to 4. The second one is to have a « digital biology » ap-proach , and to use the droplets as binary probes, 0 or 1, for instance to test the survival rate of the bacteria to a stress (antibiotic, oxidative..). This kind of approach is already widely used, for instance, for digital PCR . We will see in the last chapter how it can be applied in our case.
Time-lapse measurements of bacterial growth
We are going to focus now on the experiments of bacterial growth conducted with this microfluidic platform.
FIGURE 2.3: Example of the growth of E. coli in 7 droplets of our microfluidic chip, at 37 °C in LB medium. Fluorescence and bright field images are superposed, with artificial red color for fluorescence.
Principle of the experiment
The principle of the experiment is the following. The chip is loaded with bacteria as described above, with a Poisson parameter l close to 1. The bac-teria come from an overnight culture, and are re-diluted in fresh medium in the morning. They are loaded on the chip in exponential phase (OD 0.2). Then the chip is placed in a temperature-controlled motorized microscope at 37°C. The chip is scanned every 5 min, with a 10X magnification. The images are acquired with an EMCCD camera (ANDOR), with an exposure time of 40 ms and a numerical gain of 100. Those values are a compromise between the early stages and the final state. In the early stages, we want to increase the measurement sensitivity as there a few bacteria in the droplets, by increasing the exposure time and/or the gain. However, if we do not want to get com-pletely saturated images at the end, when there are a lot of bacteria in the droplets, we must keep those settings not too high.
By running this experiment, we wish to obtain growth curves of the bac-teria in each of the droplets that reflect, as least partially, the cell-to-cell vari-ability of the population, as we will study in the next chapter. We can see on Fig.2.3 that the growth is indeed variable in between the droplets. We can also observe that the droplet are shrinking throughout the experiment, with variability as well. We will come back to this point at the end of this chapter. Similar results can be obtained for B. subtilis, and with agarose gel instead of pure liquid medium inside the droplets, as displayed on supplementary movies 1 and 2 (see links in Appendix E).
Note that the whole chip can be scanned, but it takes approximately 12 minutes to do so (for one fluorescent channel plus the bright field images). We were more interested about reducing the time interval between two ob-servations than by scanning the 1500 droplets at each point, which explains why only 900 droplets were scanned for this experiment.
Measuring the fluorescence
Once the data of our growth experiments have been acquired, we would like to analyze them. For each time point, two images of the chip are taken, one in fluorescence that we will use as a proxy for the number of cells in the drop, and one in bright field that we will use for the detection of the wells. Indeed, to gain time, we do not take one image per trap, but the whole chip is scanned with several traps on one image. We have written a home-built Matlab code to process the bright-field images, detect and track the wells over time, for a full description see . We will focus in this part on the fluorescence measurement.
As we have already said, we want to use the fluorescence measurement as a proxy for the number of cells in the droplets. As each cell is fluorescent, it is intuitive to think that the more cells there are in the droplet, the more in-tense the fluorescent signal. The fluorescence signal that we measure is then the sum of the contributions of the fluorescence signals of all the cells in the droplet. If we simplify things even more, all cells have the same fluorescent signal and then we obtain a proportionality relation between the number of cells per droplet and the number of cells in the droplet: Fluo = a f N, (2.6)
where N is the number of cells in the droplet, Fluo is the intensity of the fluorescence signal that we measure, averaged over the droplet, and a f is the proportionality coefficient between fluorescence and number of cells.
Of course, in real life things are not that simple. First of all, all cells do not have the same fluorescent signal, even in an isogenic population . This heterogeneity in the fluorescence signal of the cell will be discussed later on. Second, we have to deal with the existence of a measurement noise, and we are going to see how in the next paragraph.
Table of contents :
1.1 Celltocellheterogeneityinbacteria .
1.3 Exponentialgrowth .
1.4 Planofthethesis .
2.1 Anoriginalmicrofluidicchipforparallelmeasurementsofbacterial growth .
2.1.1 Designofthechip .
2.1.2 A conceptual approach to micro fluidics experiments .
2.2 Time-lapsemeasurementsofbacterialgrowth .
2.2.1 Principleoftheexperiment .
2.2.2 Measuringthefluorescence .
2.2.3 Analysisoftheresults .
2.2.4 Volumeandsizeofthecolony .
2.2.5 FluorescenceHeterogeneity .
2.2.6 SummaryoftheChapter .
3 Distributionofthenumberofcellswithtime:theBellman-Harris model
3.1 Introduction .
3.2 Theoreticalandnumericalresults .
3.2.1 Comparingthethreemodelsofcelldivision .
3.2.2 ClassicalcasestudiedbyBellmanandHarris .
3.2.3 Poissondistributedinitialnumberofcells .
3.2.4 Generation-dependentdivisiontime .
3.3 Experimentalresults .
3.3.1 GrowthRate,MeanandStandardDeviation .
3.3.2 Individualdivisiontimes .
3.3.3 Fullcomparison .
3.3.4 Towardsinference? .
3.3.5 Experimentalnoise .
3.4 Summaryandconclusion .
4.1 Introduction .
4.2 Evolutionoftheresidualswithtime .
4.2.1 Ideaanddefinition .
4.2.2 MathematicalProperties .
4.2.3 Choosing Dt .
4.3 Experimentalresiduals .
4.3.1 Aslopeproblem .
4.3.2 Residualsandnoise .
4.3.3 Experimentalsampling .
4.4 Binningtheresidualsbythenumberofcells .
4.4.2 Backtotheexperiments .
4.5 SummaryandConclusion .
5.1 Introduction .
5.2.2 Exposuretimetoantibiotics .
5.2.3 Growthunderantibioticstress .
5.3.1 Alittleintroduction .
5.3.2 Preliminaryresults .
5.3.3 Pursuingexperiments .
5.4 SummaryandConclusion .
A.1 Bacterialstrainsandmedia .
A.2 Microscopy .
B.1 Equivalence .
B.2 Momentsofadistribution .
B.3 Convergenceindistribution:definitionandlemma .
B.5 GammadivisiontimesinaBellman-Harrismodel .
D.1 Additivenoise .
D.2 Multiplicativenoise .
D.3 Heterogeneityofthefluorescence .