Why does bacteria cluster

These mice are nevertheless protected against pathogen spread from the gut lumen to systemic sites like lymph nodes, liver or spleen. A classic idea in immunology is that, as one antibody has several binding sites, antibodies aggregate bacteria when they collide into each other. But this effect would be negligible at realistic densities of a given bacterium in the digestive system, simply due to very long typical encounter times between bacteria recognized by the same sIgA see appendix A in S1 Text.

We have shown that actually, the main effect is that upon replication, daughter bacteria remain attached to one another by sIgA, driving the formation of clusters derived from a single infecting bacterium [ 12 ]. Clustering has physical consequences: the produced clusters do not come physically close to the epithelial cells.

And as interaction with the epithelial cells is essential for S. Typhimurium virulence, this is sufficient to explain the observed protective effect. If sIgA was perfectly sticky, we would expect all bacteria to be in clusters of ever increasing size.

In these experiments, despite observing S. Typhimurium clusters in the presence of sIgA, there are still free S. Typhimurium, and small clusters. One possibility would be that not all bacteria are coated with sIgA. But in these experiments, it has been demonstrated that they are extended figure 2c of [ 12 ]. Indeed, a gram of digestive content contains at most 10 11 bacteria, and typically 50 micrograms or more of sIgA [ 13 ], of molecular mass of about kD. This leads to about sIgA per bacteria.

Nevertheless, most bacteria already encountered by the organism will be coated with many sIgA, and thus the cluster size is not limited by the number of available sIgA.

Another possibility is that the sIgA-mediated links break. Such breaking has been demonstrated to be dependent on the applied forces in related systems [ 14 , 15 ]. As there is shear in the digestive system, because mixing is needed for efficient nutrients absorption, it is plausible that links break over time. Small clusters are linear chains of bacteria, bound by sIgA, with these links being broken over time by the forces induced by the flow.

As bacteria are similar to each other, it is, at another scale, analogous to other physical systems [ 16 ], such as polymers breaking under flow [ 17 ]. The main difference is that these chains grow by bacterial replication. Growth and fragmentation are competing effects, and the modelling of these chains can be viewed as statistical physics, to predict their length distribution, whether there is a typical chain length, or if large chains of ever-increasing length dominate the distribution, and how the growth in number of free bacteria depends on the bacterial replication rate.

This could have very important biological consequences. In this model, the fast-growing bacteria are selectively targeted by the action of the immune system. That could be a simple physical mechanism to target the action of the immune system to the fast-growing bacteria which are destabilizing the microbiota, and thus to preserve microbiota homeostasis.

In the following, we present different plausible models of bacteria clusters dynamics, and the methods to study them. Then we give, for each model, the resulting dynamics and chain length distribution, before putting these results in perspective with experimental data. Eventually, we discuss the results. As some biological details are unknown, studying different models enables to show which key results are robust; and differences confronted to experimental data give some indications about which are the most likely.

We perform a new analysis on microscopy images that were produced for [ 12 ]. We analyzed images of cecal content in vaccinated mice for the early data points 4 and 5 hours of experiments starting from a low inoculum 10 5 , to minimize the clustering from random encounters. Further details on our analysis can be found in appendix G in S1 Text , as well as a brief description of the experiments from which the images were produced.

We consider low bacterial densities, so encounters between unrelated bacteria are negligible. Thus, we consider each free bacteria and each cluster of bacteria independently of the others. Salmonella are rod-shaped bacteria, which replicate by dividing in two daughter bacteria at the middle of the longitudinal axis. Thus if the daughter bacteria remain enchained, they are linked to each other by their poles.

With further bacterial replications, the cluster will then be a linear chain. This is consistent with experimental observations, in which clusters are either linear chains, with bacteria attached to one or two neighbors by their poles, or larger clusters which seem to be formed as bundles of such linear clusters pannel A Fig 1.

Our aim is to model the dynamics of these chains. Representative experimental images of bacterial clusters in cecal content of vaccinated mouse at 5h post infection with isogenic GFP and mCherry expressing S.

Top images: complex clusters made from bundles of linear clusters, which could be re-linked single chains left or formed from at least two independent clones indicated by fluorescence, right. Bottom images: linear clusters which dynamics we aim to model. Fixed replication time or fixed replication rate the latter is chosen for the base model. Consequences of link breaking. A first element is the bacterial replication see Fig 1C. Another way, that we will generally use, less realistic but easier for calculations, is to assume that there is a fixed replication rate r.

A second element is that when bacteria replicate, they may be able to escape enchainment see Fig 1B , but likely with low probability see discussion in appendix B in S1 Text. A crucial element is the possibility for the links between bacteria to break. We will also explore the case when the link breaking rate is force-dependent, in which case not all the links have the same breaking rate. Another crucial element, is to model what happens when the chain breaks see Fig 1D. If the subparts come in contact again at the same poles and get linked again, then this could simply be modeled by an effectively lower breaking rate.

More likely, if the subparts come in contact again, they do so laterally, forming larger clusters of more complex shapes. Because in these clusters, most bacteria have more than two neighbors, and more contact surface, they are much less likely to escape.

To simplify, we will consider that these clusters do not contribute anymore to releasing either free bacteria or linear chains. We assume no death. Bacterial death would break chains. Crucially, in this latter case, free bacteria, and all chains are lost at the same rate. The c value has a complex effect on stochastic quantities, such as the probability to have at least one chain of a given length.

We consider the beginning of the process, early enough so that the carrying capacity is far from reached, and thus the replication rate is constant. We do not consider generation of escape mutants which are not bound by IgA. We consider only the average numbers of free bacteria and linear chains of different lengths, and we do not count more complex clusters, as they do not contribute to free bacteria dynamics in our model.

For each model, we write the equations for the derivative of these numbers with respect to time. With N the vector of the mean number of free bacteria, linear chains of length 2, 3, etc. The results are obtained in part via analytical derivations and in part via numerical studies.

The latter are obtained in Mathematica by numerically solving the eigensystem written for chains up to length n max , chosen large enough not to impact the results. Besides, we obtain chain length distributions the components p i of P , which could be compared to experimentally observed distributions. For longer chains, there are two outermost links on each side of the chain which breaking releases one free bacterium and one one-bacterium shorter chain.

Even for this simple version, the system of equations is hard to solve in the general case. The base model is represented in black in all the left panels to ease comparison between models. Open circles linked by solid lines: numerical results. C, D: Model with bacterial escape. E, F: Fixed time between replications. G, H: Model with linear chains independent after breaking. The colored dotted lines are the analytical approximation The colours represent the same q values than for the left panel.

All curves are almost overlaid for small r. I,J: Model with force-dependent breaking rates. The colored dotted lines the analytical approximation 32 , and the black dashed line the numerical result for the base model.

Eq 2 simplifies to: 3 Assuming that i is large, 4 is required. Part of the discrepancy is that Eq 4 is an approximation for large i , and thus does not hold at small chain length. This is similar to the base model presented before, except that we take into account that upon replication, bacteria may not be perfectly bound, and may escape Fig 1B.

Similarly to the base model, we study numerically the growth rate as a function of the replication rate Fig 2C. The larger the replication rate, the more the deviation between the growth rate and the replication rate, which would be its value in the absence of clusters. Consequently, the total rate of production of free bacteria at time t is. The shape of the relation between free bacteria growth rate and effective replication rate Fig 2E is very similar in the fixed replication time vs.

When the replication is at fixed time intervals instead of a fixed replication rate, the maximum growth rate is higher, and it dips faster at increasing effective replication rate. Indeed, in the case of fixed replication rate, the distribution of durations between two replications is exponential, thus more spread.

Close to the maximum, the presence of short replication intervals makes that there can be more cluster formation, and conversely, at higher replication rates, the presence of longer replication intervals results in more production of free bacteria. We show here the main steps to calculate analytically an approximation for the chain length distribution, and more details are given in appendix D in S1 Text. We define n i t the number of chains of length i at t with t taken just before a replication.

Then previous equation leads to: 15 We make the assumption that the first term of the sum is large compared to the rest of the sum assumption discussed in appendix D in S1 Text.

Consequently, the growth will be close to its value in the absence of clustering, i. Compared to the case with fixed replication rate, the distribution is much narrower Fig 3. The base model is represented in black. All intermediate q values are between the black and the red curves. In this model, when a chain breaks, the two resulting chains remain independent and can thus continue to participate in the dynamics of the system: 19 We recognize here the equation studied in [ 18 ], where they described chains of growing unicellular algae.

As it has been shown, the steady state solution of the system is: 20 with C a constant dependent on the initial state of the system. In the steady state, the growth rate is equal to the replication rate. Note that the resulting chain length distribution is then exactly equal to its approximation in the base model 5. More realistically, after breaking, chains will have some probability to either encounter each other and remain trapped in more complex clusters, or to escape and become independent.

The proportion of free bacteria thus decreases over time, and decreases with increasing replication rate r. What drives link breakage? The links could break if there was some process degrading the sIgA, but the sIgA are thought to be very stable [ 19 ]. Another possible explanation for link breaking is that the bound antigen can be extracted from the bacterial membrane, at a rate which may vary exponentially with the force [ 15 , 20 ].

The forces applied on the links are likely mostly due to the hydrodynamic forces exerted by the digesta flow on the bacterial chain. Taking the linear chain as a string of beads, as done for polymer chains, and in a flow with a constant shear rate, the force is predicted to be larger as the chain grows longer, and the largest at the center of the chain [ 17 ].

A more detailed discussion and the calculations can be found in appendix E. The growth rate as a function of the replication rate has a qualitatively similar shape as for the base model Fig 2I , with a finite replication rate maximizing the growth rate.

Compared to the base model, the number of chains decreases much faster with their length See Figs 2J and 3. Indeed, the breaking rates for each link increase importantly with the chain length, thus larger chains are much less stable than in the base model. We analyzed see appendix G in S1 Text microscopy images of cecal content from vaccinated mice infected with S.

Typhimurium, which were acquired for our previous study [ 12 ]. Most clusters are large, and of complex shape. But smaller clusters are linear, and we obtained the distribution shown by the black line and points in Fig 4. The model with fixed division time is a priori more realistic. It seems however, though there is not enough data to quantitatively ensure it, that there are less long clusters than expected see appendix G.

The data may be biased, as longer chains may not be fully in the focal plane. That the distribution is relatively narrow could also be compatible with force-dependent breaking rates.

The black dots and line are the experimental data. The horizontal dotted black line represents the case in which there is one chain of the given size. No chains longer than size 14 were detected in our experimental images. The red line and points are the numerical results for the model with fixed replication time.

We started from the recent finding [ 12 ] that the protection effect of sIgA, the main effector of the adaptive immune system in the gut, can be explained by enchained growth. Because sIgA are multivalent, they can stick identical bacteria together if they encounter each other. Early in infection, bacteria of the same type are at low density, thus typical encounter times are very long, but when a bacterium replicates, the daughter bacteria are in contact and thus can remain enchained to each other by IgA.

Bacteria in clusters are less motile than individual bacteria, and in particular, are not observed close to the epithelial cells. In the case of wild type S. Typhimurium, only free bacteria which can interact with the epithelial cells contribute to the next steps of the infection process.

Despite the presence of sIgA, some free bacteria are observed. It could be that they escape at the moment of replication. But, along with the observation that clusters do not grow indefinitely, it could also be a sign that the links between bacteria break. It is also physically expected that the links have some finite breaking rate. If the typical time between two bacterial divisions is much larger than the typical time for the link to break, then there would be no cluster.

Conversely, in the inverse case, bacteria will be very likely to be trapped in large clusters. Then, even if sIgA are produced against all bacterial types, the bacteria dividing faster will be disproportionately affected. We investigated if this qualitative idea holds with more realistic models. We started from a base model in which: bacteria replicate at a fixed rate; remain enchained upon replication; until the link between them breaks at a given fixed breaking rate, identical for all links; and considering that, because of the way bacteria such as Salmonella or E.

We studied this base model with a combination of analytical and numerical approaches. We also tested the robustness of our findings by studying separately several variations of the base model: a probability of escaping upon replication, loss rates, fixed replication time, non-zero probability for the subchains to escape, and force-dependent breaking-rates.

For each model, we studied how the growth rate of the free bacteria varies with the replication rate which would be equal if there were no clusters , and the distribution of chain lengths. And more spectacularly, in most of the models studied but not if more than half the subchains escape upon link breaking, or if there is a significant probability for bacteria to escape enchainement upon replication , the growth rate of the number of free bacteria is non-monotonic with the replication rate: there is a finite replication rate which maximizes the growth rate of non-clustered bacteria.

At very high replication rates, bacteria get trapped in more complex clusters and cannot contribute anymore to the free bacteria dynamics and thus to the next steps of the infection process.

The replication rate maximizing the growth rate is of the order of the breaking rate, though its specific value depends on the details of the model. The chain length distribution depends on the model see Fig 3. When replication occurs at fixed time, or when breaking rates are force-dependent, the proportion of longer chains decreases faster. There are models with different chain length distributions but qualitatively similar dependence of the growth rate on the replication rate, and the opposite is true too.

This shows that large clusters have little importance for free bacteria production, what matters most is the small chains dynamics. It is reassuring, as we did not consider buckling, which would make long linear chains fold on themselves and produce more complex clusters, and may bias the linear chain distribution for very large lengths.

It should also be noted that with fixed division time, not only the distribution is bumpy, as chains comprising a power of two number of bacteria are more frequent than others, but the distribution is also narrower.

We analyzed experimental data on clusters of S. Typhimurium in the cecum of vaccinated mice. The experimental chain length distribution is in line with the model of fixed replication time, which is indeed more realistic. There is however somewhat less large chains than expected. More data would be necessary to asses this more reliably. This could be because of possible bias in the data.

This could be also compatible with force-dependent breaking rates. Additional experiments, for instance to measure the breaking rate, could help by giving additional independent information and constrain the fitting. To test the dependence of the growth rate with the replication rate, an ideal experiment would be to compare similar bacterial strains, but with differing replication rates, and compete them in the same individual. It is however very challenging to obtain bacteria that differ only by their replication rate, particularly in vivo.

However, using in vitro results to draw conclusions on in vivo systems is limited. First, there could be chemical or enzymatic components of the lumen that could facilitate or hinder link breaking, and the non-Newtonian viscosity of the digesta could play a role in the mechanic forces felt by the links, thus a simple buffer may not mimic well the real conditions. More crucially, the exact forces felt by particles of the size of bacterial clusters are not well characterized.

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