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1 Extended Discussion: Mathematical modelling of the planktonic population kinetics in the presence of high-avidity IgA In this section we present a mathematical model of the interactions between planktonic bacteria that lead to the formation of bacterial clumps. In a first stage, we will consider both the effects of small-scale turbulent mixing in the gut and motility-induced encounters on clump formation by collision (i.e. classical agglutination). In a second stage, we will include the effect of growth and thereby develop the concept of enchained growth. General principles For two particle types, indexed by i and j (for example, bacteria and tracer particles), with respective densities C i and C j, the encounter rate between them is given by 11 E = βc i C j, (1) where E is the number of encounters between a particle of type i and a particle of type j per unit time and per unit volume. The encounter rate kernel β is determined by the process responsible for the encounter, and can include for example Brownian diffusion, sedimentation, motility and mixing. Each of these processes will have its own expression for β, but the encounter rate can still be predicted in the form of Equation (1). In what follows, we calculate the relevant encounter rate kernels for the system at hand. 1. Classical agglutination a. Motility-induced encounters We first consider encounters due to the swimming of bacteria in a quiescent fluid. For bacteria of two different types (for example, two different sizes), with radii a i and a j, diffusivities D i and D j, and densities C i and C j, respectively, the encounter rate kernel is β = 4π(D i + D j )(a i + a j ). The collision frequency between cells is therefore given by 11 : E mot = 4π(D i + D j )(a i + a j )C i C j. (2) We are interested in the encounter dynamics within a single bacterial population, so can therefore set i = j to find that E mot = 16πDaC 2, where D = D i = D j, a = a i = a j and C = C i = C j. Upon collision, two bacteria will stick to one another with a probability σ. We call this parameter σ the particle cross-linking strength (sometimes also called the adsorption efficiency), a quantity whose value falls between 0 and 1 and which is related in our case to the cross-linking strength of IgA. The diffusivity of a bacterium due to Brownian motion is of the order D B = m 2 s 1 (= 360 μm 2 h 1 ), and for clusters of bacteria it is even lower, because of their larger size. Brownian motion is negligible as an encounter process when bacteria are motile, because the encounter rate is then 1

2 dominated by motility. The diffusivity of bacteria due to swimming is much higher than that due to Brownian motion, and can be estimated using the relationship 51 D = v2 τ 3(1 α), (3) where v is the swimming speed, τ is the average run time (the time between reorientations in the bacterial trajectory), and α is the mean value of the cosine of the angle between successive runs. For v = 5 μm s 1, we estimate this diffusivity to be D~10 11 m 2 s 1, two orders of magnitude larger than D B. The overall aggregation dynamics in the system may be described by an infinite set of coupled ordinary differential equations 52,53 for the concentrations of different bacterial clump sizes. Single cells (henceforth referred to as planktonic ) can either stick to one another, or join an existing clump of size i 2. However, we ignore encounters with larger clumps of bacteria due to the fact that their diffusivity is negligible (clumps are non-motile) and their density is very low at the beginning of the experiment. Changes in the concentration of planktonic bacteria due to cell motility can be found simply by multiplying the cross-linking strength σ with the actual encounter rate due to motility E mot. That is, dc dt = 16πaσDC 2 = 16πaσv2 τ mot 3(1 α) C2. (4) Notably, high-affinity IgA actually leads to a loss of flagella-driven motility (v = 0) (Extended Data Fig. 4a and b), and therefore the contribution of motility to classical agglutination is equal to zero for the following calculations (with the exception of Extended Data Fig. 7h). b. Mixing-induced encounters The fluid motion in the system will also contribute to the encounter between bacteria. In this calculation, we neglect the motility of the cells, and consider encounters only through turbulent mixing. As above, we will focus on the dynamics of planktonic (individual) bacteria, neglecting interactions with larger clumps. Under these circumstances, the encounter rate is 11 E mix = 1.3γ 8a 3 C 2, (5) where the turbulent shear rate is estimated as γ = (ε/ν) 0.5, ε is the turbulent mixing dissipation rate (W kg 1 ), and ν is the kinematic viscosity of the fluid (m 2 s 1 ). Mixinginduced changes in the density of planktonic bacteria are therefore governed by the equation: dc dt = 10.4σa 3 ( ε 0.5 mix ν ) C 2. (6) Note we have again included the cross-linking strength coefficient σ. Changes in the planktonic cell density C arising either through motility-induced encounters or mixinginduced encounters, together fall under the umbrella of classical agglutination 1,2. 2

3 Notably, both classical agglutination due to mixing-induced encounters, and classical agglutination due to motility-induced encounters, depend upon the square of the bacterial density. For lower bacterial densities, this encounter rate will be extremely small, and therefore potentially allows for other mechanisms (e.g. enchained growth) to more strongly influence the concentration of planktonic bacteria. We determined that for bacterial densities below approximately 10 8 CFU g 1, the frequencies of interaction between bacteria are simply too low for significant clumping to occur by classical agglutination (based on empirical estimates of the gut environment, and confirmed by in vivo data. See section 4 "Estimation of parameters", Fig. 1k, pink symbols). 2. Enchained growth Thus far we have ignored in the model the influence of cell growth on the density of planktonic bacteria. We assume now that each cell grows at a constant rate λ (h 1 ). In the absence of IgA, cell growth almost always results in the production of distinct, planktonic daughter cells. However, high avidity IgA tends to enchain the progeny cells to one another, inhibiting the production of planktonic daughter cells. This inhibitory effect is called enchainment. We introduce the parameter δ (a number between 0 and 1) to quantify the probability that a dividing cell escapes enchainment and separates into two planktonic cells (δ = 0 is complete enchainment). The contribution of enchained growth to the density of individual planktonic bacteria C is then given by dc dt = (2δ 1)λC. (7) grow In this component of the model, we neither account for growth and division of cells within clumps, nor for flow-induced fragmentation of clumps, and furthermore the only way in which planktonic (individual) cells are produced is by division from other individual cells. Therefore, if δ < 1/2, then cell growth leads to a decline in the density of planktonic cells C, because a dividing cell is more likely to become a clump of size 2 than separate completely. However, if δ > 1/2, growth leads to an increase in C over time. 3. Bacterial population dynamics Changes in the density of planktonic cells occur through both classical agglutination, in the form of motility-induced and mixing-induced encounters (Eqs. (4) and (6)), and through enchained growth (Eq. (7)). The overall dynamics are then given by dc dt = dc dt + dc mot dt + dc mix dt, (8) grow or, equivalently, by dc dt = 16πaσDC2 10.4σa 3 ( ε ν ) 0.5 C 2 + (2δ 1)λC, (9) 3

4 with initial condition C(t = 0) = C 0. The motility-dependent diffusivity is D = v2 τ. From 3(1 α) Eq. (9) it follows that if δ 1/2, the bacterial density will monotonically decrease towards zero (dc/dt < 0 for all t) if there is any form of classical agglutination (i. e., σ 0 and at least one of D or ε is not zero). We therefore focus on the case in which δ > 1/2. 4. Estimation of parameters We now turn our attention to estimating the parameters of the model. The kinematic viscosity of pure water at 30 is ν = m 2 s 1 = μm 2 h 1. However, for the intestinal contents of the common brushtail possum, which is expected to be similar to the present system, this value is at least three orders of magnitude larger 54. We therefore adopt the value ν = m 2 s 1 = μm 2 h 1. Larger values would further diminish the effect of fluid mixing on classical agglutination. The cell growth rate is measured directly from experiments to be λ = 2 h 1 (Extended Data Fig. 1d). The average run time τ = 1 s and reorientation cosine α = 0.33 for the bacteria are taken as published values for E. coli 51. The cell radius (Extended Data Fig. 7b) and swimming speed (Extended Data Fig. 4a,b) are quantified for the present system. The radius is taken to be the major axis length, thereby including the contribution of the flagella to the effective cell size (a = 2.15 μm and a = 1.00 μm for growing and nongrowing cells respectively). As a conservative upper bound on the turbulent mixing dissipation rate, we take ε = 10 2 W kg 1. This is considerably higher than even the strongest turbulent mixing in the ocean, but is intended to provide a robust upper bound on the effect of fluid mixing in this system. The full set of parameters used is displayed in Extended Data Fig. 7c. 5. Solution curves. For δ > 1/2, Eq. (9) has one stable equilibrium point (dc/dt = 0) at C (2δ 1)λ = 16πaσD σa 3 ( ε. (10) )0.5 ν Moreover, Eq. (9) can be solved analytically, yielding A C(t) = (A/C 0 B)e At + B, (11) where A = (2δ 1)λ and B = 16πaσD σa 3 (ε/ν) 0.5. Extended Data Figures 7d and 7e show families of solution curves for C(t) from Eq. (11) for different values of σ and δ, respectively, for a swimming speed v = 0 μm s 1 and an initial density of cells C 0 = 10 3 CFU g 1. The planktonic bacterial population grows exponentially for early times and then plateaus at an equilibrium density C. The existence of an equilibrium density of planktonic bacteria is supported by experimental data demonstrating that the size of the planktonic population remains relatively constant during exponential growth in the gut lumen (Extended Data Fig. 7f). 4

5 The main parameters that can be tuned by oral vaccination are δ and σ, as these characterize the physical interactions between the bacteria. Both of these parameters affect the predicted equilibrium density of planktonic bacteria C (Eq. (10)). This reflects the fact that the stable equilibrium is attained through a balance between growth and clumping. Modifying the cross-linking strength parameter σ only influences the dynamics noticeably once the bacterial population grows to a point where significant encounters occur (i.e., it does not affect the initial increase in the density of planktonic bacteria; Extended Data Fig. 7d). Increasing the value of σ results in a lower value of C (Extended Data Fig. 7d). The effects of increasing δ are twofold, as evident in Extended Data Fig. 7e. As δ is increased (i.e., cells are more likely to separate successfully), both the initial rate of increase in the density of planktonic bacteria (the exponential increase in Extended Data Fig. 7e for t 5h) and the equilibrium density C (the plateau in Extended Data Fig. 7e) increase. It is worth noting that the swimming speed v also affects the value of C, since the cell diffusivity is proportional to v 2. Highavidity IgA-binding results in loss of flagella-driven motility in the intestinal lumen (Extended Data Fig. 4a). Interestingly, using the model to simulate the effect of reintroducing motility in the vaccinated intestine actually increases the agglutination efficiency (Extended Data Fig. 7h) suggesting that IgA-driven loss of motility may even benefit S.Tm. The two parameters δ and σ have been treated as independent so far. However, their values will both depend on the presence, density and avidity of specific IgA, and will therefore be related. A high density of specific IgA will give rise to a low value of δ (low probability of successful separation) and a high value of σ (high probability of sticky collisions). The model allows us to calculate the number of clustered and planktonic bacteria as a function of time for any inoculum size. We can also assess the percentage of clustered bacteria at a given time point (t = 3 h) after inoculation, as a function of the number of cells at that time. We first consider a system in which bacteria are fixed (not growing) and are therefore not subject to enchained growth. This eliminates the parameter δ and allows the cross-linking strength, σ = 0.19, to be directly extracted by fitting the model to experimental data (magenta curve in Fig. 1k). The population dynamics of growing bacteria (blue curve in Fig. 1k) are subject to both enchained growth and classical agglutination, and with the previously measured value of σ, we find that δ = The fact that δ + σ 1 is consistent with the expected parameter interdependence, since escaping enchained growth and being trapped by classical agglutination represent almost opposite processes, driven by identical IgA cross-linking. For most calculations, we explicitly use the extracted values of δ and σ. However, when assessing the effects of varying σ and δ (Extended Data Fig. 7g), we will set δ = 1 σ. Since we require δ > 1/2 in order to achieve a non-trivial steady state density of planktonic bacteria, we require σ < 1/2 in our new model formulation. Introduction of IgA to the system will have a two-fold effect on the density of planktonic bacteria: it will reduce the rate of production of planktonic cells by enchaining their growth, and 5

6 increase the rate at which cells that are produced are lost through classical agglutination (Extended Data Fig. 7). Under the assumption σ = 1 δ, we find that C, given by C λk = 16πaD a 3, (12) (ε/ν) 0.5 is proportional to k = (2δ 1)/(1 δ), which then summarizes into a single parameter describing the influence of IgA. The parameter k is sensitive to changes in δ. For example, changing from δ = 0.95 to δ = 0.80 results in a six-fold decrease in k and therefore in C. Extended Data Fig. 7g shows three different solution curves for various values of σ (with δ = 1 σ). Small changes in either the cross-linking strength parameter, σ or the probability of successful separation, δ, give rise to considerable changes in the overall population dynamics. We are now in a position to investigate the relative importance of classical agglutination and enchained bacterial growth. From the stochastic model of mln colonization (Extended Data Fig. 5), we can derive a relationship between the percentage reduction in the luminal planktonic S.Tm and the probability of having less than 10 CFU of S.Tm in the mesenteric lymph nodes at 24 h post-infection (Extended Data Fig. 7i). A 100-fold reduction in the planktonic luminal S.Tm population size is expected to result in this level of protection in approximately 50% of cases (Extended Data Fig. 7i). Comparison to the model solutions (Extended Data Fig. 7g,h) clearly demonstrates that increasing the efficiency of enchained growth (by increasing IgA cross-linking strength) and of classical agglutination (by increasing diffusivity or IgA cross-linking strength) is highly beneficial to reach this threshold reduction. 6. Notes on the model While this model gives rise to an equilibrium density of planktonic bacteria, we emphasise that it is not due to a carrying capacity of the system in the traditional sense (i.e., nutrient or space limitation). Instead, the equilibrium density is a dynamic equilibrium in which the rate of production of individual planktonic cells from cell division is equal to the rate of loss of individual planktonic cells through encounterdriven aggregation (classical agglutination). In the context of this model, the only mechanisms by which a clump of two cells can form are (i) sticking together of two individual cells (by motility or mixing, i.e., agglutination), or (ii) incomplete separation of cells during growth (enchained growth). Both of these processes are irreversible in our framework. Mechanism (ii) would give rise to clumps that were entirely clonal, i.e., only red or only green in a mixed infection of red- and green-fluorescent bacteria, while mechanism (i) can generate mixed clumps. The predictions of the model are that clumping by growth is dominant at lower densities (i.e., early time), but aggregation by sticking becomes more significant as the number of planktonic cells increases. This would suggest that, at early times, observed clumps would be mainly clonal [mechanism (ii)], while at later times the proportion of mixed clumps would increase [mechanism 6

7 (i)]. This prediction is qualitatively consistent with the experimental data (Fig. 1i-k, Extended Data Fig. 6). An important observation from the model is that loss of bacterial swimming due to IgAbinding actually decreases the efficiency of classical agglutination, as less movement translates into a lower probability of collision. This suggests that whilst genetic loss of motility slows down disease kinetics 38,55 (Extended Data Fig. 4c), IgA-mediated loss of motility may have opposing effects: on one hand, slightly decreasing the virulence of the attacking bacteria (Extended Data Fig. 4c), but on the other increasing the size of the planktonic population potentially still able to drive disease (Extended Data Fig. 7h). Notably, loss of motility on IgA-binding has also been observed for distantly related species such as Vibrio cholerae 10 and is expected to increase the reliance on enchained growth, rather than agglutination, for clumping. The planktonic bacterial population appears to be responsible for residual infection in the intestines of vaccinated mice. By considering the predominant processes leading to loss of planktonic bacteria, we can predict that IgA cross-linking strength is a critical determinant of the size of the population of planktonic bacteria. Of note, this parameter will be determined by the avidity of the IgA responses and the total IgA density, but also by the abundance of the recognized epitope on the bacterial surface, the anchoring of the epitope in the bacterial outer membrane and the ease with which the epitope can be modified by mutation/phase-shifting. Thus, antibodies against weakly expressed surface proteins, or against weakly anchored epitopes will induce enchained growth to a much lesser extent, regardless of antibody avidity. Further, phase-variation resulting in divergent structures, or highly variable epitope concentrations on progeny bacteria is expected to disrupt enchained growth. This consideration can inform the design of oral vaccines to induce IgA responses with optimal cross-linking ability. Supplementary Data References 55 Stecher, B. et al. Motility allows S. Typhimurium to benefit from the mucosal defence. Cell Microbiol 10, , doi: /j x (2008). 56 Suar, M. et al. Accelerated type III secretion system 2-dependent enteropathogenesis by a Salmonella enterica serovar enteritidis PT4/6 strain. Infect Immun 77, , doi: /iai (2009). 57 Suar, M. et al. Virulence of broad- and narrow-host-range Salmonella enterica serovars in the streptomycin-pretreated mouse model. Infect Immun 74, , doi: /iai (2006). 7

8 Supplementary Data: Table 1. Bacterial strains in this study Strain Description Resistances* Strain number, Source or Reference S.Tm wt S.Tm att S.Tm wt WITS S.Tm att WITS S.Tm ΔfliGHI S.Tm attδflighi S.Tm ΔwbaP P2cat Donor wt (SL1344) Recipient wt (14028) P2cat Donor att Recipient att SB300 S. enterica serovar Typhimurium SL1344 (wildtype) SB300 derivate, ΔinvG, ΔsseD Wildtype isogenic tagged versions of SB300 Wildtype isogenic tagged versions of S.Tm att P22-mediated transduction of flighi::tn10 into SB300 P22-mediated transduction of flighi::tn10 into S.Tm att SB300 derivate ssed::apht and ΔwbaP (lacking O- antigen) SB300 derivative carrying pcol1b9 cat (P2cat) 14028S derivative, lpfed::apht M2702 derivative carrying pcol1b9 cat (P2cat) 14028S derivative ΔinvG, ΔssaV lpfed::apht Sm SB Sm M Sm, Km M2608- M ,25 Sm, Km M3078-M Sm, Tet M Sm, Tet, Km M Sm, Km SKI10 39 Sm, Cm M Km M Sm, Cm M Km M P2cat Donor ΔfliGHI M913 derivative carrying pcol1b9 cat (P2cat) Sm, Tet M3184, This work Recipient ΔfliGHI 14028S derivative flighi::tn10 Sm, Tet M3185, This work S. Enteritidis S. enterica serovar Sm, Cm M1513 8,56 8

9 (M1513) Enteritidis ΔssaV::cat S. Choleraesuis S. enterica serovar Choleraesuis ATCC25957 (914/99) E. coli CFT073 Extraintestinal pathogenic E. coli that colonizes the mouse intestine as a commensal species. ATCC25957 (914/99) 41,57 CFT E. coli 8178 Mouse commensal E. coli IR::aphT Km E. coli 8178 KanR 28 *Sm: Streptomycin 50µg/ml, Km: Kanamycin 50µg/ml, Cm: Chloramphenicol 6µg/ml, Tet: Tetracycline 12.5µg/ml 9

10 Supplementary Data: Table 2. Plasmids used in this study Plasmids Description Resistances Ref pm975 PssaG::gfpmut2 Amp 20 pm965 PrpsM::gfpmut2 Amp 38 pfpv25.1 PrpsM::mCherry Amp 43 pam34 ColE type vector requiring IPTG for replication primer production Amp 44 pbadgfpmut2 pbadgfpmut2 Amp pz1603, This study pcol1b9 cat (P2cat) Conjugative plasmid from SL1344, marked with a Chloramphenicol resistance cassette Cm