 of Ludovico, and the floor is yours. All right, thank you. Okay, can you hear me? So yeah, I'm Ludo, hi everybody. I work at the Statistical Physics of Genomes and Cell Labs in Milan, at EFOM and University of Milan. It's a group of Marco Lagomarsino, who's also here. So in my research, I broadly work on different aspects of cell growth and protein synthesis. And today I'm going to talk about an interpretation, so I mainly do mathematical modeling. And today I'm gonna talk about how protein synthesis can limit growth with a focus on total mRNA. Since we believe that that's an interesting aspect, given some recent data that's been overlooked. So before going to the main topic of this talk, just some definitions. So growth is of course fundamental in cell biology and physiology. But there are some subtleties sometimes in, it's very definition, growth involves many mechanisms. You can grow mass, you can grow volume, and you can grow surface. So just wanna say that I'm gonna focus on mass here, so biomass addition, and specifically protein synthesis. But it's partially because most of the mass, most of the dry mass is really protein mass, less than 50% or above across most organisms. All right, so in the literature, there are these relationships that link the quantities of protein synthesis to growth. So you've seen these plots repeatedly during the last few days. So just to remind you perhaps, so here on the left, you see for instance, the relationship between the ribosomal content as proxied by the ribosomal fraction against the growth rate. And you see that the ribosomal fraction grows with the growth rate. And this is the case both for E. coli and budding, interestingly. And one typical interpretation of this is that to grow faster, you really need to increase your ribosomal content. So in some sense, ribosomes are limiting to grow faster, at least when you per tour, but when you modulate the growth rate with nutrients. So more recently, people in Teres Voslav have found another relationship between the total mRNA concentration and the growth rate. And interestingly, the total mRNA concentration increases with the growth rate as well. So even if this is perhaps a bit naive, but you might have a similar interpretation to the one I've outlined before, you could say that perhaps you also need to increase your total mRNA concentration to grow fast. Of course, this is not necessarily the case because this is just a correlation, so it's hard to infer a causation, but it's an interesting question to ask at least, I think. So for yeast, I'm not aware of such a curve. If you are, please let me know. But on the other hand, we have evidence from another line of research that mRNA may be interesting in setting some growth limitations. And this line of research is the so-called growth cost of our expression. So this is a pretty well-known concept in the physiological community, but not only that one. So to explain that briefly to somebody who perhaps hasn't seen this, you see this plot on the left. On the y-axis, you have the growth rate, and on the x-axis, you have the proxy of your overexpressed proteins. So this is a protein that for all purposes doesn't do anything, it's unneeded. And you see that the growth rate goes down as you overexpress more of these unneeded proteins. And the typical, the standard framework to understand this is to say that as you overexpress more of this protein, you are allocating less ribosomes to proteins that are actually useful for growth. Such as ribosome proteins themselves, which causes the growth rate to go down. So more recently, there's been an experiment in budding yeast, which is very similar. So they overexpress unneeded proteins as well. But they have another control parameters, which is the stability of your unneeded transcript. Okay, so you overexpress a certain gene, a certain unneeded gene, and you're also able to tune the stability of the transcript corresponding to this unneeded gene. And what they see is basically that you get two curves when you tune the stability. I don't think it's super clear what the intuition behind the shift is, but the question would be, can the standard framework explain this shift, basically? And if not, it does mRNA limitation play a role in here, basically. And this is part of what my talk is going to be about. So from a theoretical standpoint, people have tried to systematically categorize what limit-sporting synthesis and one well-known work have identified two regimes, which we termed one regime, we call the translation-limited regime. In this regime, biophysically, what happens is that there are a few ribosomes on the transcripts, as you see in this picture here. And therefore, the ribosome number is really what limits protein synthesis. On the other hand, you have another regime which you can characterize biophysically by the fact that the transcripts are really especially saturated by ribosomes, okay? So in this case, what matters for protein synthesis, what limits protein synthesis is rather the number of mRNAs, okay? This is because if you have another ribosome, you cannot just bind on the transcript in this case, but you need other mRNAs. So in this slide, I sort of want to claim that neither of these two regimes fully explains, seem to fully explain the observation that I've shown you so far. So in the translation-limited regime, it's rather intuitive, I think, to say that the ribosomal fraction would increase with the growth rate, but it's not really clear why the mRNA concentration would increase with the growth rate. And I'll try to show that, and you can make a similar argument for the growth cost of our expression. If you do the standard experiment, the theory based on resource location of ribosomes reproduces very nicely the data, but this recent data where people also tune the transcript stability is perhaps a bit harder to understand. So with this, I want to formulate what are our questions, given all I've said so far. So the first one is a bit more of a conceptual question, I guess. So how can, we want to try to understand if there are some limitation regimes of protein synthesis where both mRNA and ribosomes limit growth, okay? So the idea is that perhaps there are regimes on top of the two that I've outlined above that are known in the literature. And then we want to try to use this more theoretical framework to try to understand some or try to help understand some of these observations, including the fact that mRNA increases the cell square faster. And also these overexpression experiments where people tune also their transcription machinery. All right. So I'll go to the results section here, starting with the first question. And so yeah, I'm a theoretician and really this part of the work is mainly model driven, I would say. And to try to tackle this question we define a central dogma model with both transcriptional and the translation layer. The key feature of this model, it's the competition for ribosomes from mRNAs and the competition for RNA-Ps from genes. I'm not going to go into all the details, but I wanted to just catch a little bit in more detail the translation layer, which is key to try to answering our first question. So our picture, our simplified picture of translation is exemplified here in this cartoon. So ribosomes elongate on the transcript with a certain elongation rate and they bind to the transcript with a certain initiation rate. There's a pool of ribosomes that binds to the transcript. This ribosome density here, it's a parameter that characterizes basically the amount of, you can say that it's a proxy of the amount of spatial saturation on the transcript. So given this picture, we introduced two key assumptions for our model. And the first one is that translation is really initiation limited. So you can write some equations, this is the simplest one, where the protein production rate is proportional to the number of transcript times its initiation rate. And we say that initiation rate is proportional to the concentration of free ribosomes. The second assumption is that there is a steady ribosome density. This is, I think, pretty intuitive. So what this means here is really that initiation current, so to speak, must be equal to the current on the transcript. This is reflected in this equation here, where you have the initiation rate, that must be equal to the current on the transcript. There are some cadets here, if you like. Philip, the other day, talked about some of these things, but if you want to know more details, we can talk about this later. But within some approximation, this is all true. The important point here is that from these equations, you get the relationship between the free ribosomes and the bound ribosomes. And if you work out basically some algebra, you can re-express the protein production rate in terms of the total concentration of ribosomes and the total concentration of mRNA. So I'm gonna skip the steps for this presentation, but here you see kind of the result of that. And I want to quickly outline what are these terms here, but you see that the protein production rate is now expressed as a function of the total amount of mRNA and the number of ribosomes. So indeed, here you have the number of ribosomes, so the production rate is proportional to the number of ribosomes, but you have this kind of, you can say it's menten term here, which really can be interpreted as the fraction of bound ribosomes. And you see that there is a key parameter here that will determine how much the total mRNA concentration matters in this rate. Okay, this is a translation rate, and this can be interpreted as the fraction of ribosome translating this particular protein on this particular sector. This is not that important, though, for our purposes because we're really interested in the total protein production rate, right, which is a proxy of the total mass, so you just sum up all of these equations and you obtain that you have that the total protein production rate is proportional to the number of ribosomes times this kind of Michaelis-Menten factor that depends on the total mRNA concentration. So of course the key would be, the key is this parameter that in this model sets how much total mRNA is important as well, but what I wanna stress here is that both components, right, both ribosomes and the total mRNA, here appear in the total protein production rate. So the two components are co-limiting in a sense, there isn't just a single limitation in the system. All right, so next I'm gonna be a bit more sloppy, but this is kind of the picture that we have for translation. So we've identified three limitation regimes. Two of them are similar to what are really identical to what has been proposed in the literature before. So the translation-limited regime only, in the translation-limited regime only ribosomes determine the total protein production rate, and this has the same biological interpretation, few ribosomes are on the transcript. In what we've called the transcription-limited regime, all the mRNA matters for the protein production rate, and you have the similar biological interpretation. What we also find basically an intermediate regime, something that you might call an intermediate regime, which you have termed this complex formation limitation regime. And in this case, both ribosomes and the total mRNA levels matter for setting the total protein production rate. So I want to talk a little bit more about this one in terms of what's the intuition behind it, and then we're gonna try to see how we might connect to the data. So to give you an intuition about this, I'm gonna show you a toy model of this toy model basically. And in this simple model of reaction kinetics, we imagine that you have two species, A and B, and they form a complex AB with a certain binding constant. And once you have a complex, you produce a product P, and you also dissociate, so you can do all of this again basically. So this is highly reversible, of course, but you can still write down a few equations and use mass action kinetics to get the total production rate of your product P. And we really get a kind of pseudo phase diagram that really looks like the one that I've shown you for the more biological model. Basically in two regimes, you have that the total product rate is proportional to just one of the two components, either A or B, depending on which one is present in the least amount. But you also find the third regime here, where the total product rate is proportional to the concentration of A times the concentration of B. So in this case, you can say that the species are co-limiting for your product rate. So we call this the complex formation limitation regime because you can show that the important step here, the limiting step here, it's basically this part of this diagram here. So if the step that is limiting in the reaction is the formation of the complex, then you will get a production rate that is dependent on both species. So you can imagine to make kind of drawing an analogy between these abstract species and the more concrete ones that we are considering. So A would be the mRNA, B would be the ribosomes, and P might be the proteins. And the question is just to give you a hand with the intuition, but you can see that you can interpret things in a similar way. And these two limitation regimes, and we join analogy between these two limitation regimes in the toy model and in the more biological model. All right. So just to take perhaps a little break, I just wanted to mention that you can make a similar kind of argument and you can make a similar kind of diagram for transcription as well. So we have similar interpretation with DNA and RNA polymerases instead. For the purposes of this talk and most of our work for the transcription layer of the model, we focus on the situation where mRNA synthesis is limited by RNA polymerases, but we're also exploring all the combinatorics here. And yeah, Philip was talking about some of this the other day. Okay. So we try to go to the next two questions. Now, and so the next question that we wanted to try to address is why should mRNA increase the cells grow faster? So to do that, we use our model to derive how mRNA changes with the growth rate by optimizing the cellular composition across nutrient conditions. So of course, optimization may be very naive. People have talked about this quite a lot in these few days. So we still did it to try to get a feel of what happens in that case at least. So we tried to get the mRNA against the growth rate by doing this optimization. We also added another ingredient that's based on data from E. coli. And this ingredient is something that is basically an increasing transcription efficiency. So what this means is that as growth conditions improve, people have shown that RNA polymerases become more active with the growth rate. And when we mix this, when we consider our model in the complex formation limited regime on top of this ingredient, we really find that the mRNA concentration is linear with the growth rate. And importantly though, the crucial point is really that you have both growth laws, right? You have both the mRNA growth law and the ribosomal growth law. So next, I just wanted to mention what happens if you remove the second ingredient. Okay, so this increase in transcription efficiency. So this is the curve that I've just shown. So with this axon ingredient motivated by the data, but if you just assume a constant transcription efficiency, you, in the model, we get a curve of the total mRNA concentration. The ceiling increases with the growth rate, but you get a kind of a square root growth law rather than a linear growth law. So qualitatively it still makes sense, perhaps, but you get a striking difference at the quantity level. Okay, so I'll go to my final questions, which is basically whether we can interpret these over expression experiments when people also modulate transcription quantities. So I'll just remind you that our inspiration here, it's this experiment where people have over express certain protein in budding yeast and then they also change the stability of the unneeded protein. Okay, so you have an expression in unneeded protein and you change the mRNA stability of this unneeded protein as well. You get these two curves basically. So the idea is what happens when you do this, what happens in our model in the different limitation regime when you try to apply this perturbation. All right, and to do that, we first looked at the translation limited regime so where all the ribosomes are really important for the protein production rate. And you see on this plot on the y-axis, you see the relative growth rate. So here there is no unneeded protein, there is no over expression. And here this is a proxy basically of the level of unneeded proteins. Okay, so this is kind of the standard curve that I've seen before. And the different colors here are a proxy of the relative stability of the unneeded transcript. And so the darker the color, the more unstable and the lighter the color, the more stable. And what you see here indeed is that all of these curves are basically on top of one another. So the shape of this curve doesn't really change when you change the stability of the transcript. This is not really what you see, what you saw in those data in budding yeast. So when you look instead at what happens in the complex formation limitation regime, this is exactly the same plot, okay? So same legend here. And you see that the curve as you change the stability of the transcript do not sit on top of one another. But instead you get a steeper slope as you make the transcript more unstable. This is an interesting qualitative difference between this limitation regime for this cost of protein over expression. So the kind of explanation for this or a very rough explanation for this is that within our model, as you produce a certain unneeded proteins, you always get a decrease in the ribosomal fraction but also in the total mRNA concentration. But if you decrease the total mRNA concentration in this ribosomal limited regime, really the growth rate isn't affected while in the complex formation limited regime, the decrease in the mRNA also affects the growth rate. So you have an extra effect on the growth cost and this is really what causes the differences in the slope here. All right, then to basically conclude, I'll just show you that we've tried to use our model for reproducing this real data. We are able to use the over expression experiment in the stable mRNA condition to fit the parameters of the model. And therefore we make a real prediction under mRNA stability change. And you see that under the transition limited regime, of course the slope remains the same. You cannot get this kind of shift while in the complex formation limited regime, you will get a nice fit on the curve when you make the transcript unstable. All right, so with this I'm basically done. I'll just summarize briefly what I said. So our first question was more of a conceptual one. So how can both mRNA and ribosomal limited growth? And our answer is this complex formation limiting regime where it's the formation of the mRNA ribosomal complex that is limited. And this kind of emphasizes is, I think points to perhaps exploring also, there's perhaps exploring also cases where you don't have just a single limiting components, right? But you have co-limiting components. In our second question was basically try to try to, somehow help understanding this kind of mRNA workflow, right? And we showed it, within our model, you would have such a law if cells leave in the complex formation limiting regime. This isn't necessarily the case that they do, right? But still, it's somehow a way to validate the complex formation limiting regime. Finally, we tried to ask, what is the overexpression broadcast due to transcription? And the idea is that the simple idea is that we use this complex formation limiting regime. If total mRNA goes down, during the overexpression experiment, growth is affecting this regime. And this explains the pattern that you saw in the overexpression experiment in biting yeast. All right, so with that, I'm finished and I'd be happy to take any questions. So this is our work on Biarchive if you want to know more. Thank you. So you said you optimized this model. What are the costs to having too many mRNAs? How is this entering your model? Right, so we basically are able to obtain an expression of the growth rate in terms of the protein fractions, okay? So the ribosomal fraction, for instance, and we also have an RNA polymerase fraction, okay? So the trade-off is basically between making ribosomes and between making RNA polymerases in this case. The RNA polymerase fraction is pretty small, actually, in E. coli, I suppose I know. But it is included in the model. Yeah, yeah, it is included in the model, yeah. Very nice model, thank you. I wonder if you or somebody else try to break down the mRNA into the mRNA responsible for production of ribosomal proteins and the rest of mRNAs. Would this linear trend persist, experimentally persist in both sectors? So if I understand your questions, in this Balakrishnan's paper, people have looked at the mRNA pi, if you like, which I think is what you're asking. Right, right, and I haven't read the paper, that's why maybe it's... And the mRNA pi and the protein pi is really roughly similar, okay, for most proteins. So you can argue that the ribosomal fraction, okay, it's basically the amount of mRNA corresponding to a certain sector divided by the total amount of mRNA, okay. And if the fraction of the biofibosomal grows and the other one grows linearly, what does happen to the rest? Yeah, maybe, yeah, I wanted to add that. The point is that the behavior of these fractions, though, really do not tell you anything about the total mRNA fraction, right, okay. So you have to do something else or understanding the total mRNA concentration, basically, okay. Actually, it's a very similar question. So a lot of plots we saw about growth laws were like total RNA to protein fraction that changes out with growth rate. But if the mRNA changes, not the same way as the rRNA, doesn't actually change a little bit. So the total mRNA to protein fraction maybe is not exactly like the ribosome fraction anymore. So you're saying that the total rRNA over protein? Like we need to only look at rRNA, I guess, to really know, right? To really know what? The ribosome fraction. Yeah, but it's interesting by the... Perhaps you're just saying that the mRNA is more compared to the rRNA? Yeah, maybe if one of them is much smaller, then it's easy. Yeah, that's right, but it's still interesting to ask what is the trend of the total mRNA, right, okay. No, of course, yeah. And I guess maybe it's a longer answer, but can you give a little intuition why you got like a square root solution for the non-changing version? I guess that... So perhaps the simple explanation mathematically is that if you imagine that your growth rate is proportional to one sector times another sector, okay? And there is something like a constraint, okay? So I'll just use things like that, okay? You can imagine that your maximum growth rate here, it's basically when they are equal, okay? And therefore you have, in the optimization solution, you have like lambda equal phi one to the power of two, okay? And if you invert it, you get a square root, okay? And there are similar arguments in some regimes, yeah? Thanks. I just want to add a few words. Your analysis is good, right? But then somehow the message is coming in, from the questions that are being asked here, I think the message came out a bit off, okay? And the thing is mRNA is not a limitation, right? So resources are tiny, right? It's compared to ribosomal RNA, so far they ask, it's tiny, okay? And the RNA polymerase is also tiny. But it's more, okay, but then you can easily, right? So the cell purposely touch it away RNA polymerase, so it's not saving any resource to reduce the mRNA, right? But that's what it's doing, that's fact, right? And so it's more at the level of, but it does that, of course you can describe it to me, yes, there are, it's a co-limiting, but it's a self-imposed co-limitation, right? And so then our view is more like that, okay? The cell knows that at the protein level, there's a constraint. And now you have to deal with that constraint, the constraint is a constraint, right? And it is pre-empting, so if you just make lots of mRNA, then the constraint will be implemented at the level of a ribosome need to choose which mRNA to make, right? But it's pre-empting that random choice by the polymerase by pre-limiting the mRNA, right? It's projecting the constraint onto the mRNA level, so that it's never in confusion, that's why we see the parallel pie chart for the mRNA and for the protein. So, yeah, but then if you think about it as optimization, oh, then it's a limitation, and I still have to deal with it, there's no resource problem at all. Well, I think I would agree with some of those points. Well, part of the work, though, is just exploring more theoretically some limitation regimes, okay? The what? Part of the work is just exploring, right, theoretically some limitation regimes, okay? Interpretation of the data, though, yeah, I agree that. I mean, at the end, you're right, it's code-limiting, right? But it's a self-imposed one, but so, well, this is our view that it's a self-imposed one because you know there's a limitation ultimately, but there's really no limitation mRNA level. Okay, okay. All right, any more questions? All right, that seems not the case. Let's give Ludovico a hand again, and we will reconvene at 40.