 Lula transposition events from plasmids to other replicons. Also, no, yes, no, the title is to be announced. So he will announce himself the title. Sorry about this. No problem. Let me share. So I will present very briefly a model on the evolution of plasmids size. I'd like to thank the organizers for the wonderful place where the conference should have been too bad that it didn't work out, but still, this is a wonderful place. So the model I will briefly present is about the evolution of plasmid content and especially plasmid length in terms of number of genes, or if you want, or total length of the genome. Of course, plasmids have very dynamic gene content, and it's quite natural to expect that there will be many different phenomena acting depending on the type of plasmid and so on. So here, I will oversimplify things a lot. And most of my reasoning won't apply to most of the plasmids, most likely, but I think it's fair to say that there are too many forces that will dominate, in any case, the dynamic of the plasmid. So one is the mutation, gene loss and acquisition, and one is the selective forces acting on the specific gene content and how it determines fitness. Now, if we look at the mutational pressure, there are several cases that we can consider. The symmetric case where gain and loss or insertion deletion pressures are the same is probably unlikely. The case of a deletion bias or a loss bias is possible. So a situation where plasmids tend to lose genes over time and tend to shrink. However, it's unlikely to imply that in the long term, plasmid would be stable. It's not impossible. You can have selective forces that are strong enough, but it's some way a real scenario. The other scenario is to have insertion bias or gain bias in the sense that plasmids tend to grow with time, at least mutationally, in terms of acquiring more and more genes. This is a case where we can have, for example, a mutation selection bias in this context, and we can have a stable plasmid population. Then there could be more complicated size-dependent biases. So for example, having a constant insertion pressure, so a constant rate of acquisition of genes independently the size of the plasmid, but increasing the pressure for deletion. This may be an interesting scenario. It's quite complicated to assess really what's the trends towards acquisition of lots of genes at different lengths. I will focus to get some understanding on how plasmids may evolve on the case of insertion bias. And I will focus on another simplification of that case, just to get a more of the glimpse of the fundamental phenomena here. So the case where there is a very strong bias towards gene gain, so a very strong insertion pressure, essentially neglecting gene loss. Of course, gene losses will be there. We simply assume that it's essentially not comparable to the rate of gene acquisition. And of course, there will be selection of gene content, both in terms of fitness gains by acquiring more genes and in terms of fitness costs. This is a situation where we have essentially mutation and selection acting in different directions. And this is a classic of evolutionary biology. In most of evolutionary biology, what we see in terms of diversity is determined by a balance between the entropic pressure, the mutation pressure that generates entropy and selective pressures that keep it at bay. And this is another of these kind of models where we can assess in a very simple way what may be the fitness of these plasmids. So we assume simplifying a lot that essentially we have a fixed fitness advantage for a gene. Of course, we will have fluctuating selection. We will have environmental dependence on these advantages. But we assume that overall across all different environments that a plasmid could define itself, they will have a fitness advantage S per gene. We also assume that there will be redundancy in gene content. So that acquiring new genes doesn't necessarily means acquiring new actual functions. Genes could be simply copied or copies of each other or genes could anyway cover more or less the same function such a way that they are redundant. And therefore there's a diminishing return in the acquisition of more and more genes. And we moderate with the classical coupon where to approach sort of exponential situation in fitness. We assume a constant fitness cost per gene and we can add some more complication for the advantage in terms of conjugation rate and the cost of the conjugation machine or mobility-related genes and so on. So the fitness has the shape that you can see here in terms of the length or the size of the plasmid. The case that is the most interesting is the first one where essentially there is some gain for the plasmid to grow a little bit at least when it's small. But then you see that the cost of additional genes and the diminishing return brings the fitness down. Now, this simple model, one could write the equation solving them with the method of characteristics and so on. But I go towards the main intuitions from the model because they are actually the thing that is really important. So the first thing is that if we compare what is the most likely plasmid size in this model versus what is the size that will maximize the fitness of the plasmid, we find them as match. The fitness of the, the size that maximize the fitness of the plasmid with increasing selection, of course, both of them increase but the peak is always quite far. So the most likely plasmid length is quite, quite longer than the fitness peak. So we have a sort of insertion burden of insertion load in the sense that this mutational pressure to acquire genes is essentially causing the plasmid to contain more genes that would be needed. And selection is trying, actively trying to get rid essentially of these plasmids that have excessive insertion load. And curiously enough, this insertion load in this model implies that plasmids are marginally persistent. So the fitness peak as you see is here but the most likely length is precisely when the fitness of the plasmid is around zero because essentially under stronger insertion pressure under strong pressure for acquisition new genes, the point at which it's quite likely to find a plasmid is at the end of its acquisition process when the sector pressure is really strong against it and the fitness is precisely zero. So in this model, a bit paradoxically, plasmids find themselves on the border of evolution and stability. Of course, one can include the machinery in the communication machinery in this model. And the idea is that the more effective is the communication machinery in terms of the effectiveness of conjugation versus the cost of the machinery, the more the plasmid can carry more no-mobility-related genes. So we expect the genes that plasmid that have a more effective machinery will also carry more genes on average, more accessory genes on the... And therefore, they will be longer. Now, one can ask in this model what happens when we increase environmental stress. So for example, antimicrobial concentration. If this increases gradual, what happens is simply the size of the plasmids increases gradually because essentially these plasmids tend to acquire gradually genes that antimicrobial resistance or resistance or stress resistance genes. However, if these increases is very rapid, what happens that there is a mismatch between what will be the evolution equilibrium and the state where the plasmids finds itself. So the exertion load is lower than what one would expect for that plasmid. And essentially, the plasmid has a transient stress-induced fitness boost. So that if I suddenly increase the environmental stress, paradoxically, I find that plasmids tend to be at a higher fitness, at a relative fitness in the environment and therefore tend to spread more widely. Now, this model is oversimplified. There are several more realistic extensions. One of the obvious ones is include the gene loss, at least some components of gene loss and different constituents of the genome. So different parts with different functions and so on. We have done some of that. For example, here there's a two component model with two different sets of genes that can have different fitness and fitness cost. More in general, these models are of course oversimplified, but they offer some glimpse about some interesting phenomena that may happen in actual plasmids and may help explaining them. And with that, I would like to thank you for your attention. Thank you, Luca. Maybe we have one question from Simone Pompeii. Ciao, Luca. I have a question here. So which units are you using in your definition of the fitness? It's like per generation because this S is the selection coefficient. It looked quite big. It was about one or two. So I was using an arbitrary unit. So I was not specifying the time scale over which that happens. I mean, you can't... Okay, so it's not a generation in general. No, no, no. I mean, you can play a lot with the model. What is general are the consequences that I listed that don't depend on the parameters. So the most likely length is always the length at which you get your fitness no matter what is your absolute scale and so on. Thank you. Nothing else? Okay, so thank you, Luca. We have now the talk by Adam Roberts. He couldn't be here today because of important personal reasons. And he will try to have him somewhere in the next days. So now we have a short time for a break, Alice. Or you want a general discussion.