 Thank you very much for the introduction. Yeah, it's all correct. I'm a PhD student with Hildegard Oeke here in Ploon and I'm very glad for the opportunity and I would like to thank the organizers for this very nice workshop. So let me start. Okay, I hope you see my slide. So as it was already mentioned a few times in the workshop, I think I do not have to say that there is many plasmids with multiple copies. So experimental studies from the recent years have shown that multi-copy plasmids potentially the evolution of antibiotic resistance. So so Averro Semilan and colleagues, for example, showed in a paper from 2016 for E. coli with a plasmid that has a plasmid copy number of 19 that it evolves resistance against high antibiotic concentrations. So he is shown on the right side where this is more likely if the resistance gene that is involved is on the multi-copy plasmid here shown in black, then if the involved resistance gene is on the chromosome where no antibiotic resistance could be seen for those large concentrations in this antibiotic ramp experiment where the population was transferred to increasing concentrations. So there are many reasons why this should be the case. So first, there is an increased mutation target for genes on the multi-copy plasmid. Second, there are gene dosage effects that can lead to a higher degree of resistance if the resistance gene is on the plasmid. And more general speaking, that's why I also put this other reference here, there's advantages of possible coexistence of several plasmid variants within a single cell to, for example, escape fitness trade-offs as shown in this paper by Rodriguez Beltran and others. So before I start, let me just give you a quick outline of my talk. So in the first part, I will show you some results of a mathematical study that we did that exactly deals with this scenario I just depicted on the evolution of antibiotic resistance on multi-copy plasmids. And in the second part of my talk, I'm going to elaborate a little bit on the fixation dynamics on multi-copy plasmids. So the question that interested me most when I read this paper by Ivor Samilan was, how does the probability of evolution of antibiotic resistance actually depend on the plasmid copy number? And to answer this, we built a mathematical model where we assumed that there is a susceptible population of bacteria declined due to an antibiotic stress. And by some chance, a de novo mutation, so here depicted by this plasmid copy that has a different color, arises just on a single copy. And in the model, plasmids replicate. And at the same time, the plasmid copies replicate. So before cell division, plasmid copies are distributed to the daughter cells. And for this case, there's two outcomes. So either there's an equal distribution of the plasmids with the two variants, or there's a segregation of the mutant plasmid, the mutant resistant plasmid. So basically, all our analysis came back to the one question. And that was like, can a single cell or what is the probability that a single cell can rescue the population? And we found that there is two factors that I want to kind of have as a general picture. So the first factor is due to a large copy number, the cell will likely or more likely produce progeny that has resistance mutations. And the other factor is stochastic loss, which means that mutations arising on a multi copy plasmid, so a plasmid with a greater number is more likely. So the establishment probability, so this is P, the establishment probability is actually exactly the inverse of the probability of stochastic loss. And this has a certain dependence on the plasmid copy number and it often decreases. While the other factors, so what I call the mutational supply in the following, is increasing. And with this, we use the framework of evolutionary rescue and formulate this basically as a single equation that has a dependence on the copy number n. So let me show you two examples. So in the first example, the resistant mutant allele is dominant. And what does this mean? So I have here on the x-axis also depicted by those small bacteria. In the example of copy number three, the number of mutant plasmids within the cell. And for the case of what I call a dominant mutation, the cell fitness increases with the first mutant copy. And no matter how many mutant copies there are within one cell, the reproductive fitness is the same. So as I already said, mutational supply is increasing with a copy number. But as our analysis show that we do with a multi-type branching process, the establishment probability, so the inverse of the stochastic loss, the establishment probability that a cell leaves a lineage that rescues the population in the end declines with a plasmid copy number. But in total, so the probability of rescue of antibiotic resistance evolving is an increasing function, no matter of how large the effect of this resistance mutation is. In the second example, I'm going to show you the same result for recessive mutation. What means that only a cell that has only mutant copies is resistant and has a positive Malthusian fitness. So for this we see here shown the establishment probability over the plasmid copy number, a strong decline. And this leads basically to a general trend of decreasing rescue probability with the plasmid copy number. So I hope that this kind of convinces, of course, with many assumptions that we make in our medical model that is only boils down to the single equation that the dominance relationship is actually the size of factor for resistance evolution on multi-copy plasmid. And the follow-up question we asked was how does the dominance of resistant alleles on a multi-copy plasmid is actually shaped? And I'm not going too much into detail here, we have built a small model of antibiotic degradation and we look here on two scenarios. In the first scenario, the production of antibiotic degrading enzymes is costly and we model this by a degradation rate that depends on the relative number of mutant compared to wild-type plasmid copies. And for this here show rescue probability over the antibiotic concentration, we see that greater copy numbers here shown in red are optimal only for the low concentrations. So in the other scenario we looked at plasmids are costly where the plasmid cost decreasing reproductive fitness is an increasing function with the plasmid copy number. And here the exact opposite shows where the greater copy numbers again shown in red are optimal only for the higher concentrations. And this is although we cannot really compare exactly with the data I've shown in the beginning, a similar trend to what we see here where for the medium concentrations we see that actually the high copy number plasmid is not optimal compared to the chromosome and only for the high concentrations we see the optimality of the high copy number or the multi-copy plasmid. Okay, so this brings me directly to the second part of my talk and here we have the question how long does it take until a beneficial mutation reaches fixation? And this is interesting like in face of the maintenance of wild-type plasmid as we will see in the following. So let's jump right into the model. So we consider population that has a small fraction of cells with a single beneficial mutant plasmid copy for example due to a transfer event. And cells with a mutant replicant copy here have an increased fitness 1 plus s. So this is the analog to a dominant mutation. And in this scenario where there's multiple copies within a cell it is possible that there is a time point where all cells in the population have the beneficial phenotype but the wild type is still in the population. So whereas what we call allele fixation there is a time when the homozygous mutant is fixed and the wild type is lost from the population and we call this difference in the time the heterosegosity window. So let me show you just one example of how we analyze this. So if we have a copy number of one so there is only one plasmid copy in the cell we only have homozygous mutant cells no heterosegosity. And we see here that after a very short time mutant cells and those are homozygous cells they will rise to fixation. So phenotype fixation and allele fixation they coincide but if we look at the same result of this deterministic model where we basically have a proxy here for fixation for copy number 32 we see that fixation of the mutant cells is way earlier compared to the homozygous mutant cells due to also rise of the heterosegous cells in the meantime. And we can show that for a certain degree of selection that is s with an arbitrary unit this heterosegosity window rises if s is greater than the inverse of the copy number n. And for this we can show that this heterosegosity window is approximately scaling with n times the natural logarithm of n. So what one should definitely keep in mind and I have mentioned this in the first part of my talk we assume here some random segregation of the plasmid copies to its daughter cells. And the fixation dynamic is is depending strongly on the plasmid segregation mode. So we for example look at in our model at some clustering of sister replicons where the plasmid does not resolve before it goes to the daughter cells and here it shows that this for example reduces both fixation times at the two levels of phenotype and of the genotype but also the heterosegosity window whereas if we have a separation of sister replicons as for example often assumed for for low copy plasmids where plasmid copies are pushed to the daughter cells we only see a rise of heterosegous cells up to fixation. So let me come to a summary. So for the first part I have shown you that the dominance relationship plays a crucial role in the evolution of antibiotic resistance of multi-copy plasmids and we have seen that the optimal copy number is actually often depending on the antibiotic concentration which shapes the dominance relationship. So for the second part where we look at the fixation dynamics I've shown you that we see a rise of heterosegous cells causing a heterosegosity window as we term it and of course there should be always mentioned that the segregation mode of the plasmid is crucial for the fixation dynamics. So with this I would like to thank my supervisor Hildegard and Tal, Anna, Anna and for the plasmid illustrations I would like to thank Fenerstuka. Thank you. Okay thank you it was really interesting so let's see if I can see if there are questions. Yeah there is Olivia please Olivia gone. Hi yeah thanks Alice really great talk Mario really like the work it's very interesting. Thank you. I have a few questions so in your first vignette when you looked at dominance and recessive relationships did you all investigate the in-between architecture where it scales potentially your phenotype of your cell with number? Because that's more similar to the beta-lactamase work of alvaro where gene dosage is happening. So there was two relationships that are I would say mainly important that we call those intermediate where it's something between dominant and recessive and for this it actually gets messy if you go to copy numbers greater than two where you see also an optimal copy number for the intermediate so not for the high but not for the low copy numbers where rescue gets most likely if you if you choose an intermediate intermediate copy number. And the second is gene dosage effects that people often assume also when it comes to beta-lactamases as I also mentioned at the beginning where you see establishment probabilities that can also rise with a copy number. So in the two examples I've shown dominant and recessive we only see the establishment probability falling but for the gene dosage effects where for example you need two out of two copies that have the mutation to get a cell population growing of course it can be a benefit to have a greater copy number because then you can actually increase the reproductive fitness with greater copy number. Does this answer your question? That was my first question but I'll hold off for my second just in case other people there's no other hands raised. Okay well then I selfishly I'm gonna ask um so beta-lactamase is a interesting gene dosage slash kind of more dominant genetic architecture but that's a type of like specific resistance mechanism so do you know of any genetic architecture for plasmid genes that have shown to be recessive when it comes to mutations? So um there are genes but I um so I cannot um I do not have any examples for the plasmids basically um or or where it was shown for the plasmid that the dominant is that case but uh let me quickly look if I can okay um maybe maybe I can um send you just references um because that'd be great I should I should not name any genes because that's um really um where can like start making mistakes here. Yeah thanks Oliver I was just I'm just selfishly interested in how genetic architecture of mutations might be related to the function of the gene itself um and beta-lactamase being like a free floating monomeric enzyme if you increase the activity of it and had some you know wild type enzyme around and some enzyme that is like better that obviously like leads to kind of like the intuition of dominance anyway so it's just curious about what. Yeah thanks for the question. Yeah thank you really great. Thanks. Okay so I had a question but I will send you an email because it's uh too complicated and I think I need to write it um and we should be on break until uh what's the time until 4 p.m so you have seven minutes well we will wait it's italy