# Mechanisms of the IgA immune response in the gut

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## Log-likelihood maximization principle

The log-likelihood maximization principle  allows to get the best tting parameters that t a data set with an assumed probability distribution function.
Let us suppose we have at our disposal the data of a set of draws x1; :::; xN from a random variable X. We know the shape of its distribution { let us say for example that X is normally distributed { but the parameters of the distribution function (in our example, the variance and/or the mean which characterize the normal distribution) are unknown. The maximization of the log likelihood allows for an estimation of these parameters from the data of x1; :::; xN. Let us write f(x) the probability distribution of X, with the parameter we want to estimate. The likelihood can be dened as the quantity L(), with: L() = NY i=1 f(xi).
The best t of the data will be obtained for the value of maximizing the quantity L. Figure 1.5 provides a qualitative explication of this principle. Thus, in order to nd this best-tting parameter, one must solve the equation @L @ = 0 in . In many cases, the log-likelihood ln(L) is a quantity easier to manipulate; it does not change anything for the maximization, since a probability distribution function is always positive, and that the logarithm is a strictly monotonically increasing function.

### Biological grounds to the one-population model

In the experiments, bacteria are taken from a suspension at a certain concentration C0 to be orally inoculated to mice. Thus we will consider in our model that the sample of bacterial solution of volume V0 given to a mouse contains a number of bacteria k Poisson-distributed of average of N0 = C0V0. Once inside the organism, the bacterial population may undergo a rst bottleneck, best taken into account by an identical probability for each bacteria to settle in the cecum, instead of being killed by the acidity in the stomach before reaching it or being directly carried out of the niche . We will consider this phenomenon to be instantaneous and to happen at t = 0, while in practice, inoculated bacteria reach the cecum over a couple of hours. Writing P(n; t) the probability to nd a total number of bacteria n at time t in the cecum: P(n; t = 0+) = 1X k=n Nk 0 k! e􀀀N0 k n n(1 􀀀 )k􀀀n.

#### Establishment probability with the WITS loss

In the case where c = 0, no WITS can be lost during the Markov process, and all the WITS that are lost are lost during the Poissonian initial sampling. The probability to lose the tag i during the rst step writes as the probability to draw a zero in the Poisson distribution of mean ni: Probability to lose strain i = (ni)0 0! e􀀀ni = e􀀀ni Since we suppose all WITS are all inoculated in the same proportions, then the probability to lose a strain is the same for each strain, and equals the mean number of strains lost, which is available in our data set (it can a priori be averaged on several mice at once): Proportion of WITS lost = e􀀀n0.

Limit of the approximation c = 0

With rmax and the estimated from the WITS loss, we ran some preliminary simulations. By choosing arbitrary values of c remaining small (of the order of 0:1rmax) we noticed that the proportion of WITS lost after the Poissonian initialization was not a negligible portion of the totality of the WITS lost. We thus looked for a more accurate way of estimating the parameters and c when c is not zero. The way to estimate r (taken either equal to rmax or to rmean) remains unchanged.

The need for a new observable to disentangle and c

We have thereby seen that when c cannot be approximated to zero, the data of the proportion of WITS lost only gives a constraint linking and c. In order to separate the eects of and c on the WITS distribution, additional information must be extracted from the data. A possibility to do so is to look for an additional observable, to complete the data of the proportion of zeros in the distribution. Another possibility would simply be to apply numerically the loglikelihood maximization principle (as explained in section 1.2.1.5) on the whole WITS distribution. However, this method is just a way to obtain the best t possible by introducing a black box providing no actual understanding on the data. On the other hand, the use of another observable can provide a feedback on the model we choose: either it provides totally new information and allows to disentangle the roles of and c, or at least we can check if the informations provided by both observables on the parameters are coherent. This is the direction we will follow in the next section, by looking for an observable to characterize the variability of the WITS distribution.

The quest for another observable

We have seen in the previous section that the data of the number of WITS lost provided an information linking the parameters and c, but was not sucient to disentangle their eects. In this section, I look for a second observable on the WITS distribution that could provide complementary information on the parameters, and that could be measured in the experiments. I thus present dierent possible observables to characterize the variability in the WITS distribution. The evenness index was rst chosen for continuity reasons with previous work of our collaborators, and proves to be convenient for simulations, but is hard to handle analytically. To overcome this diculty, two other observables will be considered in the following: the variance on the nal numbers of WITS, and the variance restrained on surviving WITS. Then the variance over the growth rate will be introduced in order to take into account the initial variability in the inoculum.

Variance over the growth factor

Among all the indicators of variability for the WITS distribution seen in the previous sections, the simple variance over the absolute nal numbers is the only quantity that can easily be analytically predicted. However, until now, we have always made the assumption that the initial numbers of WITS were all equal, which meant that the h nal numbers could be seen as h independent draws from the same random variable. But in the experiments, there is always some initial variability in the concentrations of the solutions of the dierent WITS prepared to make the inoculum. This translates into dierent ni values, and this variability is sometimes non-negligible compared to the nal variability. It is thus essential that this initial variability can be taken into account, and we will see in this section how to do so. The solution we propose is to look at the variance over the ratio between the nal (mi) and initial (ni) numbers of WITS mi=ni. These expressions might have been derived in the literature, but we rederive them, building on the previous section.

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Strategy for parameters estimation

Now that we have settled for the choice of our second observable for the variability in the WITS distribution, with the variance over the growth factors, let us go back to the question of parameters estimation. Using the dierent observables, let us look at the constraints imposed on the parameters by the data.

Constraint on and c from the mean growth rate

In addition to the data on the WITS distribution, we can also exploit the initial and nal total count of bacteria measured experimentally combined to the data of the plasmids dilution for another observable, the one of the mean growth rate. As seen in equation 2.9, it expresses as: Bf N0 = exp((r 􀀀 c)t).

Introduction
I Population dynamics of a bacterial gut infection
Introduction
1 Experimental data and methods
1.1 Experimental data
1.1.1 Direct count of bacteria
1.1.2 Plasmids and mean number of generations
1.1.3 Wild type Isogenic Tagged Strains
1.1.4 Summary of experimental data
1.2 General methods
1.2.1 Analytical methods
1.2.1.1 Master equation
1.2.1.2 Generating function
1.2.1.3 Characteristics method
1.2.1.4 Moments of the probability distribution
1.2.1.5 Log-likelihood maximization principle
1.2.2 Computational and numerical methods
1.2.2.1 Gillespie algorithm
1.2.2.2 Tau-leaping procedure
1.2.2.3 Bessel’s correction
1.2.3 Summary of general methods
1.3 Symbols for Part I
2 One-population models
2.1 Biological grounds to the one-population model
2.2 Approximation c = 0
2.2.1 Replication rates
2.2.2 Establishment probability with the WITS loss
2.2.3 Limit of the approximation c = 0
2.2.3.1 WITS loss when c 6= 0
2.2.3.2 The need for a new observable to disentangle and c
2.3 The quest for another observable
2.3.1 Evenness
2.3.2 Variance
2.3.2.1 Mean expected variance
2.3.2.2 Comparison with the WITS loss
2.3.2.3 Variance of the variance
2.3.3 Variance conditioned on WITS survival
2.3.4 Variance over the growth factor
2.3.4.1 Mean growth rate
2.3.4.2 Variance
2.3.4.3 Variance on the variance
2.3.5 Summary of the quest for a new observable
2.4 Strategy for parameters estimation
2.4.1 Constraint on and c from the mean growth rate
2.4.2 Constraint on and c from the tags loss
2.4.3 Constraint on and c from the renormalized variance over the growth factor
2.4.4 Summary of the strategy for parameters estimation
2.5 Simulations and results
2.5.1 Determining the carrying capacity
2.5.2 Results
2.5.3 Discussion
3 Two-subpopulations models
3.1 Arguments for a two-subpopulations model
3.1.1 Biological arguments
3.1.2 Qualitative argument
3.1.3 Summary of the arguments for a two-subpopulations model
3.2 Analytical approach
3.2.1 Generating function
3.2.2 Observables calculations
3.2.2.1 Mean growth factor
3.2.2.2 WITS loss
3.2.2.3 Variance on the growth rate
3.3 Parameters search strategy
3.3.1 Carrying capacity and replication rates
3.3.2 and c with plasmid dilution, mean growth factor and WITS loss
3.4 Simulations and results
II Mechanisms of the IgA immune response in the gut
4 A new idea in immunology: enchained growth
4.1 Vaccination triggers sIgA production
4.2 Limit of the classical agglomeration idea
4.3 Modeling clonal loss with enchained growth
4.4 Conclusion
5 Enchained growth as a way to regulate microbiota homeostasis
5.1 Interplay between clusters growth and fragmentation
5.2 Models and methods
5.2.1 Elements of the various models
5.2.2 Methods
5.2.3 Argument for a low escape probability
5.2.4 Table of the symbols used
5.3 Clusters dynamics and distributions of sizes
5.3.1 Base model
5.3.1.1 Equations
5.3.1.2 Free bacteria growth rate as a function of the bacterial replication rate
5.3.1.3 Chain length distribution
5.3.2 Model with bacterial escape and dierential loss
5.3.2.1 Equations
5.3.2.2 Free bacteria growth rate as a function of the bacterial replication rate
5.3.2.3 Chain length distribution
5.3.3 Model with xed replication time
5.3.3.1 Equations
5.3.3.2 Free bacteria growth rate as a function of the bacterial replication rate
5.3.3.3 Chain length distribution
5.3.4 Model with linear chains independent after breaking (q > 0)
5.3.4.1 Limit case: subchains always independent after breaking
5.3.4.2 Intermediate case: chains independent or trapped after breaking
5.3.5 Model with force-dependent breaking rates
5.3.5.1 Equations
5.3.5.2 Free bacteria growth rate as a function of the bacterial replication rate
5.3.5.3 Chain length distribution
5.4 Comparison with experimental data
5.5 Summary of the results and discussion
6 Consequences of enchained growth on the evolution of antibiotic resistance
6.1 Introduction
6.2 Model
6.2.1 Within-host dynamics
6.2.1.1 Types of bacteria
6.2.1.2 Treatment
6.2.1.3 Within-host growth equations
6.2.2 Transmission
6.2.3 Between hosts
6.3 Methods and equations
6.3.1 General methods
6.3.2 Naive hosts
6.3.3 Immune hosts
6.3.3.1 Limit G N
6.3.3.2 Limit G N
6.3.4 Table of the symbols used
6.4 Results
6.4.1 Impact of clustering in the absence of mutations
6.4.2 Impact of clustering with mutations
6.4.2.1 Small number of generations
6.4.2.2 Large number of generations
6.4.2.3 Conclusion
6.4.3 But this eect can be countered by silent carrier eect
6.5 Discussion
Appendix
A Experimental data tables
B Variance of the variance
B.1 Variance of the simple variance
B.2 Variance of the variance on the growth factor
C Source code of the R simulations
D Constant division time instead of constant division rate?
D.1 Generating function
D.2 Mean population size
D.3 Fixed point and extinction probability
D.4 Variance and variance on the growth factor
D.5 Comparison with data
E Detailed derivation for the model with force-dependent breaking rate
F Proportion of mixed clusters and probability to transmit at least one mutant
G Approximations for the evolution of resistance model
G.1 Regime of a few generations within the host: both sG 1 and G N
G.1.1 First approximation: only states starting from 0, 1 and N resistant bacteria matter
G.1.2 When extinction is certain in the absence of mutations .
G.1.3 Ratio of spread of the bacteria in all immune vs. all naive host populations
G.2 Limit of a large number of generations, with both sG 1 and G N
G.2.1 Equations
G.2.2 Regime of sure extinction in the absence of mutations

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