Biological grounds to the one-population model

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Table of contents

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