where we are. And then people can come in as they come in. And so, again, we left behind the 1960s and we've jumped ahead to the 2010s, basically. And what we're looking at is this night-heart-magasinic relationship between the ribosome abundance and the protein abundance as exemplified by a straight-line relationship between the two when we change the nutrient that we're feeding our bacterium. So we're feeding the quality, or changing the quality of what we feed them that toggles this growth rate. And we see this positive correlation between the ribosome abundance measured as a mass fraction and the growth rate. And again, that was rationalized by saying that ribosomes are catalysts of protein synthesis. But irrespective of the interpretation, we end up with empirically this straight-line relationship that correlates positively the growth rate with the ribosome abundance so much so that we can describe it by a straight line where the slope is written somewhat idiosyncratically only because we want to interpret this as, we suppose that we can interpret this as a translation rate and then we have some offset. Contrast that now, so now this is not night-heart-magasinic, but we have a similar linear relationship between the ribosome abundance and the growth rate when we start to fiddle with the translation rate. So when we go in and keep the nutrient the same but inhibit the rate of protein synthesis by, in this example, chlorenphenicol, we take something like, say, a circle and we start to march the ribosome abundance up at the same time marching the growth rate down. And so we have a negative correlation between growth rate now and ribosome abundance. And again, that correlation is so strong that it's more or less linear which allows us then to qualify it or quantify it, sorry, with some slope again written idiosyncratically and some intercept. Here we haven't imposed any interpretation yet on the intercept or on the slope except to say that it looks like the slope is steeper, sorry, steeper if the cells were originally growing more slowly, i.e. this kappa is smaller for slow-growing cells and bigger for fast-growing cells before you had the antibiotic. And so that motivated me to call it the nutrient quality or to correlate with the nutrient quality. We'll talk more in detail about where this comes from or how we could possibly interpret this in a moment, but I want to show you one more set of experimental data before we do that, before I do even that. Are there any questions from this morning? Did anything sort of come up over lunch? Does anyone have any questions either about the data or about the, I don't know what you call this, the compression of the data into these two linear expressions? It's okay? Okay. So again to remind you this is the protein fraction of ribosomal proteins, the total proteins and virtually no one cares about that. Very often when people are dealing with bacteria what they're thinking of is all of this going on in the background. The ribosomes do what they do fine. What I'm interested in is my own particular protein whether it's this enzyme or that enzyme or it's insulin or it's whatever. It's certainly not ribosomes. The only people who care about this are bacterial physiologists. And so then the question is how does this growth dependence in the abundance of ribosomes trickle down and impose growth dependence on various proteins of interest? Okay and that's what I want to talk about. What those proteins might be is what we'll get to as we go but at least initially we want to turn our attention away from ribosomal proteins to other proteins in the cell whatever those might be. And the place to start would be a protein that's not regulated in any way. And so what I want to talk about here is what that looks like at the genetic level and where regulation comes in typically and then what happens when we measure the amount of these or the abundance of these proteins under these different growth perturbations. So that's where we're headed probably the next 20 minutes or so. But any questions before we start that out? Alright so a more general interest is the growth dependence or even whether there is growth dependence in non-ribosomal proteins. Okay in the place to start that analysis is to look at what's called an unregulated or constitutively expressed protein. And I'll tell you what that means. Place to begin is with an unregulated also called constitutive. Okay and I'll tell you genetically what that means in a second. So what we're going to do now is look in more detail at the DNA. So I have been up till now more concerned with what happens after the DNA is turned into RNA transcribed into RNA and what happens to that RNA when it's then translated into protein. But if we look and we zoom in functionally on a strip of DNA in a bacterium and in broad strokes this story is similar in eukaryotes and higher organisms. If we zoom in on this DNA it has a particular architecture. So we have what's called a promoter region. This is called the promoter and this is where that RNA polymerase binds. So this is or at least recognizes that it should start initiating transcription of the DNA. So it may bind non-specifically here and there but if it happens to bind in this promoter region it recognizes that the DNA after this region is important. And in bacteria or in E. coli there's a particular set of binding motifs that are called the well the just binding motifs. That will depend to some extent there are different variations but predominantly there are two patches that it recognizes. And then there's the transcriptional start. So transcriptional start site which is right here and then it transcribes in that direction. And then there's over here the gene of interest of the protein of interest. I'll say yeah let's say protein of interest is encoded in the DNA that's downstream of this promoter. Also encoded in this DNA is a ribosome binding site. So this is a ribosome binding site. So what happens is that this polymerase binds so the polymerase looks like sort of like a catcher's mitt like a baseball mitt. Binds onto this thing and it starts turning the DNA or transcribing the DNA into this RNA. And that RNA comes off with a ribosome binding site on it which is a motif that the ribosome recognizes. And then the instructions for your protein of interest which the ribosome then translates. And then there's a stop signal here. This is a transcriptional and when it hits that typically the RNA polymerase falls off. RNA polymerase falls off and finds somewhere else to go. Okay so that that's in some detail that the layout of the DNA strip. In these details like the promoters the promoter of the binding sites and the ribosome binding site were all worked out during the 60s and the 70s some of the early 80s. And in eukaryotic system there's still a lot of active research devoted to what precisely happens in the promoter regions and the transcriptional start and stop sites. Okay so any questions about this? I don't expect you to maintain this taxonomy in your head but I want to show you what it looks like when we say that a gene or the transcription of a gene or protein is regulated. So this is the layout in some detail. So remember this is just a bunch of DNA bases that gets transcribed into an RNA and then gets translated into a protein. Let me pause though. Does anybody have any questions about any of those things? Basically nomenclature but does the nomenclature make sense? All right and so this is what Minode one is Nobel Prize for. He discovered that in addition to these promoter binding sites, so that your RNA polymerase binding sites, you can sometimes have what he called operator regions. And these are motifs or strings of DNA bases that are recognized by some other protein in the cell. And so the protein binds down and occludes the binding of the RNA polymerase. This is what you call a repressor. So proteins can bind here and regulate transcription. And that can be a positive or a negative interaction with the polymerase. Negative is obviously way easier to engineer. You just put something that blocks it and so they also tend to be much more common. Positive would require some cooperativity. So the protein would bind somewhere, help recruit a polymerase and then activate transcription in that way. Those are more rare. But hopefully the negative interaction is not too unclear.  So you're trying to get in the door and transcribe this and he says no, go away. So that's what I mean by a regulated gene. And by gene I mean the DNA that encodes the protein you're interested in. All right, now for real stop. Any questions about any of this? Okay, so what I want to talk about is the case where we have none of these sites. We don't have any, no other proteins bind to regulate the protein that I'm going to be looking at. And how do I make sure that that's true? I make this up. I design this myself. Okay, I mean I don't personally, but one designs it oneself. And so what we're going to talk about is an unregulated protein, protein, and we want to be able to measure it in some sensible way. And so usually what we make is either something that fluoresces, which is what people do nowadays, or in the olden days what they would do is make an enzyme that the cell doesn't need for growth, but which catalyzes a reaction we can do in a test tube. And so we need some kind of reporter. And so this is either a fluorescent protein, or typically a fluorescent, fluorescent protein, that's a protein that glows. Or it's an enzyme. So depending upon how much of that enzyme we have, our reaction in the test tube is going to go faster, slow. Bless you. And so the data that I'm going to show you is with a particular enzyme that the cell doesn't need for growth. That's key. It's not going to affect the growth rate. But we can measure its abundance very easily with chemistry. And then I want to ask, what happens here with this picture, if instead of the horizontal or the vertical axis being the ribosomal abundance, I have the mass fraction of this enzyme. So I'm going to call this Phi Z, because the enzyme is encoded by some, anyway, I'm going to call it P. Phi P, so then I don't have to explain where the P comes from for protein. And this is going to be, I do need to put a Z here. Dang it. Okay, I'm going to put a Z here, because it comes from a piece of DNA that's been named LacZ. But that matters not one bit. So that Z corresponds to this enzyme. And so I'm going to look at milligrams of Z protein per milligrams of total protein. So in contrast to the ribosomal protein, now I have some other something that's not driving protein synthesis. It's in some sense a passive spectator. Does the experimental scenario make sense? And then, well, I'll show you what the, what the ordnance of the graph will look like, and then let's talk about the experiment. So here, again, I'm going to plot this mass fraction. Here's a growth rate. And I'm going to use these same set of media, triangle, square, circle. I'm going to see what I get. And then I'm going to add cornfinical and see what I get. So that's the scenario. All right. And this is what I get. Dried, and then let me get out of the way. Okay, so this is nutrient. And in this direction, there's less space to dried, but this is antibiotic. Okay, does everybody see what I've drawn? I mean, not, well, yeah, operationally, do you see what I drew? And then, second of all, does it make sense? So this drawing is meant to correspond with that right-hand side picture, which is the complement, if you like, of this ribosome abundance picture. So all the colors are the same, all the numbers are the same. They're different experiments, but they correspond to the same experimental conditions. All right. So the table set. Does everybody understand, though, what's going on in both graphs? So one, the one on the left is ribosome mass fraction. The one on the right is unregulated protein mass fraction. And now my question is, what's, what's going on? First of all, what's the relationship between the two pictures? I mean, qualitatively. And then how would you quantify that? How would you rationalize it? All right. Yeah. All right. So, so here's the idea that this bi Z plus bi R is some constant. Why do you say that? Exactly. They're like mirror images. When one goes up, the other one goes down. And so it looks, and both of them are linear. And so it looks like there's a linear constraint operating between the two. So her suggestion is that if you had only the plots on the left, plus this linear constraint, you would be able to explain the, or you'd be able to redraw the plot on the right, at least up to some, some scaling. Yeah. Yeah. Yeah. But, but this is just one small, this is just one small enzyme. And then, and the knees are ribosomal proteins. And so, so it's a big leap possibly to say that there are only two kinds of proteins. One kind of behaves like this and one kind of behaves like this. I agree with you as a, as a scientist that would be, that's a, what would you call it parsinum? It's a sort of a good hypothesis, right? It's a simplifying hypothesis. But it's not, it's not obvious that that's true. Let's, let's see if it is. So what she's suggesting essentially is, as he suggests, that we have two zero order approximation, two types of proteins. And the reason that we write it like this, and well, let's, let's talk a bit more about that, but let's pause there. Does everybody see that if this, this is what those two plots are suggesting? Yeah. Okay. We can go even further. I mean, I'm asking you to look at it by eye, but if we plot it now with the vertical axis on the right and the vertical axis on the left as two sides of the same graph, we'll get something that looks like that. So with the same data, we can plot phi r and phi z on the same set of axes and use the growth rate between them. And you get something like this. So you get, where's, it's going to be a mess. I mean, all, all kinds of things are going to fall on top of each other. But the point is that for every, every say pink two, pink two is going to be on this line. Green 12 is going to be on this line. All of the data collapses onto a single line, which suggests that this idea is a sensible one. That these two fractions are linearly related to one another. Does that make sense? I'll write it down here so that we can. And so now taking this idea a little farther, which is further, is his suggestion, but which is what if we have two types of proteins, please, or sectors of proteins. So we have this phi p, let's say, which, of which, of which phi z is a subset. So we have phi p, phi z finds itself inside there and phi r. So the, and suppose the probably the simplest assumption is to suppose that these are, these are sorry, exhaustive categories. So then we would have the entire protein divided into two pieces. You'd have p proteins and r proteins and this fraction would be phi r. Does everybody understand the image? Then that would take care of both of these pictures. All you really need to regulate is phi r and then phi p passively gets squished or expanded. Because this picture is not quite right, but we'll interrogate why it's not right in a second. Does it, does anybody not see what would motivate a person to propose this? It's okay. All right. Okay, this is not quite the, because we know that as we go to zero growth rate, this protein fraction, this phi r only goes up to about 0.6 maybe at the most, 0.5. So as growth rate goes to zero and this translation rate goes to zero, phi r does not go to one. It goes to about 0.55 or something like that. Sorry, I should write it like this. Let me get out of the way in a second. It goes to phi r, goes to this phi r max, which is about equal to 0.55. All right. So there's got to be at least one other fraction in there that's, say, growth rate independent. So the next, next simplest picture would be something like this.  And then these two sort of are dynamically allocated depending upon the growth conditions. So either you've got high nutrients, you know, high translation rate or whatever it might be. This other half of the pie chart changes. So this slider is where the regulation is. And this, this piece here then would be phi r max minus phi r min, say. So the maximum that this guy can take is going to be equal to one minus this phi q. So that's how we would get an estimate for this, this growth rate independent piece. And we don't know yet what that is, but with that pie chart in mind, then we can reconstruct quantitatively this plot from this plot. Okay. Let me pause. There's a lot of hypotheses, a lot of what ifs here. But I want to pause. How does that seem? Yeah. They're their own guys. Okay. So if we, if we go to these data points, let me show you a couple of what the, what this pie chart would look like for a couple of data choices. Okay. So suppose we took, it's probably easier here because I have discrete symbols and then I don't need to color anything. So suppose I take circle, which is low growth. So here's some examples. So I grow in circle, which is slow growth. So poor nutrient, no antibiotic. So I can't, I can't do anything with this q, but I know that I have low r and high p. So that would be red circle over there or what I just called blank circle here. r is low, p is high. I can look at another example where I have what I called triangle, which on this colored figure would be light green. In that case, this is now rich nutrient. I would have the opposite. So I'd have something like lots of r, not much p. So this is a fraction. So not, so it won't necessarily be the same number of proteins constant. So don't, don't think of this as say a homeostatic mechanism in time or something like that or in concentration. But think of it as a, as a impossible constraint to escape that if you have only 100% protein, you can allocate it, you know, only in fractions from zero to one. And so then that, what this allows us to do is get away from questions of regulation. We say I don't know how it goes from here to here. But depending upon the question you ask, we don't care how it does this. We know that it does though. Because manifestly, I mean, we have the evidence that when one goes up, one goes down. And we can look at what the consequences of this is without having to worry about whether our interpretation is right. And this gives us a lot of solid ground to work on. It's, it's not just verbal. I mean, it's mentally solid ground. So, and then I'll show you one more, which is going to be what this one with chloramphenicol with antibiotic. We will then look like this, but it will grow very slowly. So this now has low lambda, high growth rate, low growth rate. Okay. These are different ways that, that we can explore the space of ribosome abundance and growth rate. Okay. And as you said, there's got to be some regulation that gets you from one pie chart to another as you change the conditions. And that's absolutely true. And in some cases we know what that is. And in some cases we don't. But I think this will maybe be more satisfying. The primary point of regulation, at least at this level of resolution, is that ribosome abundance. The cell controls how many ribosomes it's making. And I'll talk about how it does that in, you know, maybe this lecture, but maybe next week. But then the point is that as you grow faster or you grow slower, if say you grow faster, you dilute out proteins. So you don't need to actively degrade, which is a nice thing about bacteria. And they usually do not degrade things. They just dilute them away. So as you grow, if you stop making something, it eventually disappears. And so it's enough, it's enough to turn up ribosome regulation to guarantee that the P is going to shrink. This P fraction is going to shrink. That's maybe not convincing at this point. So at this point, let's just leave it as a conjecture, a thought experiment. Hypothesis is better. Because I want to leverage this view to give us some interpretation of this nutrient parameter. Okay, and again, everything that we're going to talk about next week is sort of higher level than this. We don't actually need to know what this means. We don't need to have an interpretation for it, which is good. But on the other hand, it's useful as humans to have some justification or rationalization for this so that we know how we might change it when we design our experiments. Okay, and so what I want to do is take this hypothesis and ask what can it tell us about this kappa n parameter in the same way that we used the protein synthesis hypothesis to give us an interpretation for this parameter. But before I do that, at the level of hypothesis, does everyone see that this coarse graining of the total protein in the cell gives us one picture from the other? Bless you. So if you knew the picture on the left, you could draw the picture on the right and vice versa. Is that okay? Okay, so so to this set of growth laws then, all I need to add is these are empirical constraints. And then what I add to them is this hypothesis. You have some growth rate, some growth rate, independent part of the proteome that you just can't touch. And we haven't said what that might be, because at this level, we don't know yet. I mean, in this course, we don't know yet, we'll get to what it is in a second or next week, are an MP. Okay, and these guys together, occupy a fraction of this fire max minus fire min. So that that's, that's where I want to pause for a second. Any, any questions? Any frustrations? Yes, good. Okay, so Phi Z is just an unregulated protein. So it's got no regulation. It's simply made. So it's not in fire. So is that okay? Exactly exactly. Exactly. So what we had then is that this Phi P was going to be not, I don't know how we do it in set theoretic notation, but Phi P are not Phi R. I don't know. It's got better way to say it. But so we had this initial proposal that the Phi P's are one minus Phi R. We only had two types. And we said that Phi Z is going to be like that. Yeah, a sub protein of those Phi P's because it's certainly not a sub protein of these. But then, as was said, even before we did this, this one doesn't need to need to be a one, it could be a constant lesson one. And in fact, it is. It's about 0.5. So then we modified this. So we have Phi Z is something with 0.5 minus Phi R. So it's not R. On the other hand, it's got growth dependence. So it can't be part of these guys which have no growth dependence. So it's still a logical compliment, except that we have another class that it can't be a part of. I don't know if that was was that helpful? It's okay. Okay. Any other questions? Yeah. Oh, so let's see. Yeah, you're right. See Phi R max would be one minus Phi Q. This is so yeah, it depends if you want to include this into Phi Q. You could do you could have a little slice. Sorry, is that does that make it less clear? What I wanted to do is tie that to the vertical axis on this graph. If that black line solid black line pass through the origin, then we would be fine. But I'm taking this offset as being included into Phi Q. I hope that doesn't drive you crazy. If it does, then don't worry about this. Phi Z is an unregulated protein. So think of Phi P as not ribosome. It's any protein is not a ribosome protein, which includes these unregulated proteins that we made up. Okay. Is that okay? But we haven't given it any functional significance yet. And so that's what I want to do now. But so far, we're saying this constraint appears to be operating between these two groups. But what could be the origin of that constraint? Is that sensible? Any other questions? How do we go pretty quick? Okay. Okay, so we use this this idea to then try to rationalize this this parameter that appears in the growth law. Now these two again, I'll say it again, these two empirical constraints operate independently of how we want to interpret these slopes. On the other hand, I think it's helpful to have at least a working hypothesis for the interpretation. So of this kappa n parameter. And so I'll write this out and you know, let's write it out and then talk about it. Okay, so remember that the the way that protein is made here is with a ribosome summarizing amino acids at some rate, kappa t that we had before. So this is protein. And this is an amino acid. Okay, for now, I'm going to ignore say these fire mints, not this guy, but I'll ignore this because it conceptually it's going to make it a little bit easier, I think. So that's our core process of protein synthesis. And so we know that amino acids are being consumed. So amino acids are consumed by protein synthesis. That's just a restatement of how proteins are made. So now what we're going to ask is how are they supplied? And so let's look at the flux of amino acids. Let me call a the the mass if you like a free amino acids. I've got some consumption that looks like this kappa t times by R. But what about supply? Because the consumption rate of amino acids is going to be proportional to the protein synthesis rate. And this is the protein synthesis rate. If we believe that night hard, a magasinic interpretation that we had before. I'm going to pause. Is that okay? So this is protein synthesis rate. Is that okay? So far so good. All right. Now the supply flux. So imagine that this, let me call this Jn maybe Jn. So J. And so what I think that this is probably conceptually the easiest scenario to imagine it's of course not exhaustive. But imagine that the amino acids don't need to be made by the cell. Imagine that we have them growing in a test tube and we're supplying all the amino acids. How do we get amino acids into the protein? We need to transport them across the wall of the of the bacteria. Okay, so here I'm going to imagine a very simple scenario. But then I'll suggest that we can generalize this to a more complex situation. So for simplicity, or for now, let's say for simplicity now, imagine the amino acids are supplied in the growth medium. So they're in the test tube. And so the only the only rate that we need to consider is a transport rate. So their supply rate will equal the transport rate. Is that scenario clear ish? I haven't said anything about the transport rate, but then we can go back to what Manot did and imagine a potential transport rate. Okay, so the scenario is that the cell doesn't need to chemically synthesize amino acids, it can do that if it's called upon to do so. But it grows much faster if all it has to do is transport them in. Okay, and so let's look at this transport rate. And so what I'm going to imagine is that there's some maximum transport. So this would be amino acids per second. And then it's mediated by some transporter protein. So I'm going to call that Phi t. So this is the transporter protein, which is proportional to the protein concentration to if you like. And so the more of this transporter you have the faster you're able to transport these amino acids. If you like, if you if you're comfortable with enzyme kinetics, this transporter is playing the role of an enzyme that converts external amino acids to internal amino acids. And there should be, if we're going to be sort of careful about this, there should be some concentration dependence. So that if we have high external concentrations of amino acids, we can saturate this rate. Otherwise, depending upon the concentration scale that's set by this, this affinity constant, we might have slower transport. So we can fiddle with this transport rate either by changing the concentration of the amino acids, but by changing the concentration of this transport. Okay, so this is a very, sorry, generic model of transport. And it really doesn't matter what the details are, as I'll show you in a second. What's most important is that we have a protein that's mediating this transaction. Okay, and what we're assuming is that the transport rate is linearly proportional or directly proportional to that protein abundance or that protein fraction or that protein concentration depending upon your units. And for all the transporters that we know, that's a reasonable assumption. It's a very simple process. It goes, you know, through basically through a pipe, the more pipes you have, the faster you can transport. Let me pause. Any questions about the setup, the scenario? Yeah. Yeah. No, so this is outside the cell. This is outside. And this guy is inside. Well, that's an important distinction. So this is something I can change with my experiment. That a is what the biology determines with the bacterium itself determines. Yeah. It's a good point. Any other questions? Other mechanics here? Yeah. Oh, sorry. I'll go back to him just because it didn't stop. It's not fast enough. Yeah, it's not fast enough. There are some things that are just passively diffused. But if they're important, they're always actively transported. Transport means sort of exactly what we mean casually when we say transport. It means you take it from here and move it to here. And so this is the amino acid outside the cell. It needs to get physically brought into the cell. Sometimes depending on the chemical, it can passively do that by diffusion. But either because of the structure of the chemical or because it's very important, we have or the cells have dedicated proteins that do that action. And they often consume energy. So they are like the general laborers that pull nutrients out of the environment and into the cellular environment. Is that okay? So they can do that more rapidly than diffusion. They can do it more specifically than diffusion. They can also do it in cases where the diffusion would be inhibited by chemical reasons. So it's like there's a fatty layer around the cell. So if it's something that's soluble in water, it's not going to pass through the fat. That makes sense. Right? And then the other thing is they can also work against a concentration gradient. So you could have more amino acid or higher concentration inside the cell than you have outside. You don't but there are some things that you have a higher concentration of inside the cell. So all of these necessitate an active protein member to catalyze. Does that make sense? Just from a physicist's point of view, not necessarily the structure of these transports or anything like that, just a necessity from a physical point of view. It's okay? Yeah. This guy? Oh, KD. This is a concentration scale that tells you how much amino acid you need to get the maximum rate. So if you had an external concentration that was about the same as KD, your transport rate would be half of its maximum. So this is just telling you how good the transporter is, how specific it is. If this number is very, very small, then it is always, no matter what the concentration is outside, it's always working at maximum rate. If KD is very big, you need a lot of amino acid outside in order to get this transporter to do its job. So this is telling you how specific it is, if you like. The point the point here is that this rate can saturate with amino acid concentration, usually. Yeah, this will depend on the chemical details of the transporter. But you'll see in a second that all of this is going to wash out. But just thinking that that's what this is. This is telling you how sensitive the transporter is to changes of concentration of the amino acid. So hopefully you can see that if there was no amino acid outside, the transport rate would be zero necessarily, right? And that takes care of this. Also, if there's a huge amount of amino acid and it's all trying to get in, it's very easy for the transporter to work at maximum rate. And so this also saturates the one if amino acid is much, much bigger than this KD. So this KD just sets a constant, the characteristic concentration scale. Does that make sense? Is that sensible? Yeah. Could you, could you say one more time? Sorry. Yeah, here we're thinking of only active channels. Yeah, ignore, for simplicity, just imagine transport is only active. And it's only outward actively transporting as amino acid. Okay, you're right, there's passive, or there can be passively transported molecules. Any other questions? Yeah. Could you say one more time there from the so? Yeah, so there are two scenarios that I want to contrast. One is, if there's no amino acid outside the cell, the cell needs to make amino acids. And so it's got dedicated enzymes that will take bits and pieces from here and there to make its own amino acid. In that case, the supply flux is, is from other pro from proteins within the cell. What I want is a, I want to consider first of all a simpler scenario where the amino acids chemically are present outside the cell. And the only protein you need is the protein that pulls them from outside into inside. The reason I do that is because in the other scenario, you need a network of proteins. And the, and the simplification that I'm about to make is less clear. You can make the same simplification. And it's in the lecture notes. But as a first step, it's incomprehensible. So I'm going to do this simplification here. And then we'll talk about how you would do it to more complicated examples, like, like what you would call biosynthesis of amino acids. Is that okay? Any other questions? Okay. Okay. So again, the thing that, so I put these pieces of information up here or these, these, these sort of this model for the transport in case you see an enzyme kinetics and you want to maybe more detail. But the point that's important here is that it's proportional to the transporter protein. And more importantly, if I, if I now have a situation with a concentration outside of my cell of amino acids doesn't change, which is what I need in order to guarantee balanced exponential growth. So in balanced growth, this concentration is constant. If it wasn't, then the supply rate would change. So then necessarily the growth rate would change. And so then necessarily I would not be in balanced growth. So everybody see the chain backward. And it's possible to do this to keep the concentration constant for a finite amount of time, which is the length of your experiment. And then you dilute them again and then you dilute them again. You keep diluting them. But always in such a way that this concentration remains more or less constant. The amount of amino acid that the cell consumes is so minuscule compared to the pool of amino acids available that you could not see it over the length of the experiment, which means the following that this transport rate looks like this. This is a constant. The concentration is fixed by me, the experimentalist. The KD is fixed by the chemical details of your transporter and a chemical out there. The K cat, which is how quickly this transporter will work again, is fixed by the nature of the transporter. So for a given growth scenario, this thing is just some number five, for example. Does that make sense? And I'm going to call it, I'll call it nu. So nu phi t, where this is a constant. And so that's why I say the details are not that important. Because I know that, so long as I've got this linear relationship between the transporter and the transport rate, that everything else needs to be constant, otherwise I won't have balanced growth. All right, so some magic's about to happen, but it needs to, this needs to be sensible, otherwise it's not an exciting magical event. It's more like, more muddy water. Is that all right? Is this, what would you call it, this proposition sensible? All right. Now the way that I'm going to rationalize this second growth law is the following. Now imagine that this phi t is also, I can't remember how you do this, is also, you do it like, he's contained in phi p. There we go. Is that right? Yeah. Phi t is contained within phi p. Imagine this transporter is not regulated the way that ribosomal proteins are regulated. I.e., this phi t is going to be some epsilon times phi p. If phi p increases, phi t also increases. Then, what do we have? Then we have this Jn is going to be epsilon times this nu times phi p. Okay, these are all what ifs. I'm saying what if, and then let's see what happens. So then, what do we have? We have dA dt is equal to this epsilon nu by p minus kappa t phi r. Is that okay? It's okay so far. So what if? And now we know two things about each of these. So first of all, these supply and consumption reactions operate much, much faster than growth rate. So on the time scale of the cellular growth, this change is about zero. These guys almost immediately equilibrate. That's an extra piece of information, but it's nonetheless true. Okay, so this pool of amino acids basically is constant inside the cell. Okay, and so then what do we have? We have that this equals that. Now, this one, we know from the night heart of magasinic that lambda is equal to this kT phi r. That was night heart of magasinic, and our work earlier today with that solid line. How about the right-hand side? Well, the right-hand side, is that okay, this part? This part, we know the phi p is equal to this, let's call it phi max minus phi r. That's a proteome partitioning constraint. And so then what do we end up with? When we put all this together then, we end up with lambda on this side is equal to epsilon nu times this thing. Does that make sense? Or r is equal to negative lambda over epsilon nu plus phi max. And so this looks remarkably like this second growth law with the interpretation of this nutrient capacity then is a measure chemically, so let me write that and then put a question mark here. The idea then is that it's a measure of how well the transporters transport what they need to transport. How rapidly can we make this transport rate by minimizing this protein cost? And so for some cells in some environments that might be, you might be able to do that very efficiently. So that would lead to a large KN. If it's an inefficient transport mechanism, then that's a low KN, right? Now this is the simplest possible scenario and doesn't have anything to do with biosynthesis. And so let's maybe talk about the more complicated case after a break. Let's take a five minute break and just unwind. Let me pause before we do that. And so the interpretation then is a nutrient quality just tells you how efficiently the organism can produce amino acids or supply amino acids to feed protein synthesis. Whether that's done through transport or whether that's done through some complex metabolic pathways, we should get the same answer in the end, okay? All right, let's talk more about it in five minutes. Okay, maybe I'll start, that's okay. All right, so let me come back to where we were and let's talk about it a little bit more. Okay, and so again, at the risk of being tedious and repeating this over and over again, I'll remind you that these two constraints are sort of empirical. They're like Ohm's Law or something like that. And in fact, I'll make that analogy even more strict in a second. So these are something that we see. There's no interpretation. On the other hand, they're almost impossible to use unless you tie some interpretation to these parameters. But just like Boyle's Law or something, unless you know what pressure is or temperature is, at least with a working hypothesis, these types of relationships are more or less look up tables. You know what, you give me this and that and I tell you these parameters or something like this. And so what we looked at initially was the interpretation of this parameter or this slope as somehow quantifying the protein synthesis rate per ribosome. Then we noticed that these two laws were mirror reflected in the protein, good, thanks. The protein fraction of non-ribosome related proteins which motivated this coarse-graining, if you like, of the proteome or the protein composition of the cell into some growth rate independent fraction and then two fractions that are dynamically allocated depending upon the growth conditions. And from that, so taking this and this, then we're in a position to rationalize the slope of the second growth law by appealing to amino acid flux balance. So here we appeal to protein synthesis balance. Here we're looking at flux of amino acids. And the idea was the following, that we have ribosomes making all the proteins in the cell. Some of these are ribosomal proteins which just facilitate this whole process. And so we have an intrinsic feedback loop here. And that gives us this growth law. On the other hand, we need to feed this process somehow. The substrates for protein synthesis have to come from somewhere. They either come from inside the cell through biosynthesis or they come from outside the cell through transport. Either way, there's a protein cost associated with bringing those nutrients into this reaction. Okay, and so what I wanna talk about today, which we're in the twilight of today, but at least next week, is this protein allocation constraint. Sure, you wanna make proteins as fast as possible to grow as fast as possible, but in order to do that, you need to supply nutrients as fast as possible. Specifically, you need to supply amino acids as rapidly as possible. And so the interpretation that we're using here, this working hypothesis, is that this parameter kappa n corresponds to this rate of amino acid or nutrient assimilation. So let me write that up. So protein synthesis and then nutrient assimilation. And here assimilation is meant to be a catch-all that means either transport or biosynthesis. And the one that we, the example that we looked at was a very simple transport mechanism. But what I wanna do now is look at an analogous system that you've seen before and then argue at least verbally how we can generalize this picture to more complicated metabolic reactions that supply nutrients and amino acids. All right, but let me pause. Any questions? So this stuff, I don't think there's anything. We can talk about, this is the interpretation that we ended up with at the end of the lecture 15 minutes ago or so. Is that okay? Is anybody concerned about the rationalization? All right, does the picture make sense? It's fine. Magasamic and night heart, we're looking here, which is fine. But that's at the expense of remembering that or thinking that the cell needs to supply the other side of that reaction. And how it does that is connected to how it does this. And it's connected through that proteome partitioning. So far so good. Okay, we can make an analogy of this system to something that you've seen before, which is electrical circuit analysis. And I'm gonna make that analogy because I think it's useful for a couple of reasons. One is that you then can see the level of abstraction that we're talking about here. And also we can use network analysis tools like Kirchhoff's loop law, for example, and Ohm's law to generate more information from this picture than is currently in the picture, if you know what I mean. So let me talk first about the analogy. So there's an analogy here that's sometimes called omics, which is a joke because, well, I won't explain the joke because it's not funny. But the idea is that this is analogous to electrical circuit. So we can rewrite these, rewrite the growth laws like this. Like this, lambda is equal to kappa t times, I'm gonna use now a delta phi r. And I'll tell you what I mean by that. And kappa n delta phi p. I've been using the word case and I'll keep that. For this delta phi r, what I mean here is the phi r, the mass fraction of ribosomal proteins. But now subtract off the growth rate independent part, okay? I'm not, this is what I empirically measure, but then in this analogy, what I'm concerned with is a part that's growth rate dependent. And similarly, this phi p, this delta phi p is going to be equal to this phi max minus phi r. Sorry, delta phi r. Where this, sorry. I should, where this phi max is the difference between these. So I haven't yet got to the analogy. The analogy starts here, but I'm renaming some of my variables so that the analogy is a little bit more clear. The problem with that is that renaming these variables makes things less clear. So let's clear that up first. So this is what I measure. This is what I measure. But I want this, this growth rate independent part to be gone. So I subtract that off. This is that proteome constraint that I had before, but now I'm lumping together all the growth rate independent parts together, okay? And so I can, in E. coli, over the growth rates that I showed you, this is about 0.5. But that's neither here or there. Okay, and so these two equations, doop-doop, are these two equations. Doop-doop. And they're identical to one another. I just, you know, clean them up a little bit. Let me pause. Is that okay? Just to clean up the notation. Not the motivation yet. It's okay. With this secondary constraint, and the constraint and the proteome constraint, by proteome constraint, I mean, this is all the proteins being made in the cell, which is sometimes called the proteome. And the constraint is that the pie chart is exhaustive. So the proteome constraint is the following. We have that delta phi r plus delta phi p is this phi max. And again, this is just a relabeling of the constants. Okay, is that okay? Just in terms of the relabeling. Not the motivation. I haven't told you my motivation yet. So this scenario, this proteome constraint, and these growth laws, these empirical growth laws, are mathematically identical to two resistors in series with a battery. So if I have a battery here, put this, there's the resistor, there's one resistor. There's another resistor. Okay, and the voltage of this battery is gonna be phi max. The voltage drop across this resistor is gonna be delta phi r. And the voltage drop across this guy is going to be phi p. The potential drop. And the resistance of these resistors is one over kappa t, one over kappa n. And the current in this analogy is a growth rate. So let me write all this out. So lambda is, the growth rate is like the current. One over kappa t, one over kappa n is like the resistance. Delta phi p, delta phi r is like the potential drop. Does that make sense? I mean, not, first of all, not why a person would want to now complicate a simple set of rules with this analogous picture, but given this picture and given what you know about Kirchhoff's loop laws, that the sum of the potential drops has to equal the driving potential, and that the current through a resistor is equal to the, what? The equals IR. The voltage, the potential drop across the resistor is equal to the current times the resistance. That should give you these two equations and this equation. So this is Kirchhoff's loop law. These are Ohm's laws. And just like in electrical circuit design, Ohm's law is the empirical. Like you can try and derive it from Maxwell's equations, but you're gonna go crazy.  They're empirical relationships. That's how they began their life, and that's how we use them. And so at this level of abstraction, we don't care, so suppose we now change this battery. Well, that's not good. How about we change this resistor? I pull out a resistor and I stick in a new one. Well, then the potential drop across here and the potential drop across here is going to change, but then, I mean, how that changes is the quantum mechanics. I don't care. I know that it's gonna change and I know what's gonna happen at the end, but the transitory period or the transitioning period is outside of this theory. And that's the very same thing that's going on here with the bacterial growth laws. Okay, let me pause here. Are you guys comfortable with this analogy?  You don't need to be comfortable with it. We'll explore it hopefully until it becomes comfortable. So one thing you might be thinking of then is why when I change nutrient did I get translation and why when I change translation did I get this nutrient parameter? That is to say, this guy came out when I changed kappa n, this one came out when I changed kappa t, it seems like they're backward. Well, it's because what you do is you think of these as a variable resistor, better think of this as a variable resistor. You twist this one and you measure the potential drop across this one. And that gives you the resistor. Same thing with the translation mutants. You adjust the resistance here, you make it bigger or smaller. And then you measure the potential drop, either here or here. And that tells you the resistance of this guy. By fiddling with one, you get the other. By fiddling with one, you get the other. Does that make sense? The other point that I wanna make here is that we can parametrically solve for the growth rate here. So we can solve for the growth rate in terms of these two resistors. And so notice that the overall growth rate can be written like this, where we have the voltage is equal to the current times the sum of the resistors. And I add them because they're in series. Or that this lambda is equal to this phi r max minus phi r omega, let's just put it as phi max. Phi max over one over kappa t plus one over kappa n. And that's exactly what you would get here if you set these left-hand sides equal to one another and then solve for lambda. Then you get a parametric equation for the growth rate in terms of these phenomenological parameters, kappa t and kappa n. All right, let me pause. Any questions about this? Okay, where are we going with this? So one way that you can think about this is that this is the engine of protein synthesis. This is protein synthesis. And this is metabolism or nutrient supply or whatever you wanna call it. And you can try to make one of them super efficient by decreasing the resistance. So you can make this rate high, super high so that the resistor is small and the voltage drop here is small. That is to say that the protein cost is small. But you can't make it zero, first of all. And any changes you make here are going, any changes in the protein drop here is going to change over here. So if you use less of this, you can use more of this and vice versa. If you use more of this, you use less of this. Okay, so there's an intrinsic constraint that's associated with the system. And what I wanna do for the rest of this course is explore the consequences of those types of intrinsic constraints. Now those types of constraints are independent of what mechanism you want to describe to these parameters. Once we have this empirical picture, we can look at what the consequences are. Let me pause, because that's important. Let me show you an example of a consequence in a second. So okay, okay, all right. So let's look at some consequences. The first set of consequences I wanna look at are sort of the first thing you would do with an electrical circuit, if you like. So maybe I'll start with that. So now, okay, this is the first example that I wanna talk about. Co-utilization of a carbon source. So this is a biological problem, but then it's got a physical solution that I think is quite elegant in this framework. And the idea is this, you grow your bacterium on two carbon sources, or two sugars, if you like. And what you sometimes see, what you sometimes see is that it uses one carbon source at the exclusion of the other, and then when it's exhausted that, it switches to the other carbon source. But that turns out to be a rarity. More often what you see is that it simultaneously uses both carbon sources and grows faster than on either carbon source alone. Okay, so what you see is, often, what you'll see is, so now this is your log base two of the number of cells, this is time. This is the doubling rate carbon source one. So sugar one, sugar two. And then this is on both together, and it's always less than mu one plus mu two. Okay, so you grow in two different types of carbon, or sugars. Sugar two gives you a slower growth rate than sugar one. So you mix them together, and you grow faster, but not linearly. You don't get M one plus M two. Okay, and now the question is, how do we understand this, or what's one way to understand this, and can we quantify what the doubling time is in the mixed sugar versus the doubling time in the single sugars? And I think this is worth discussing because it opens up two views. One is, how can we unpack this simple lumped circuit description of metabolism into more complicated units? And two, how can we use this type of phenomenological picture to get some rapid and deep insight into the biology? Okay, so two questions I have for you. One is the circuit diagram on the right hand side, sensible, does everybody see how I'm making the connection between that circuit and these growth laws? Anybody want to get clarified? Question number two, do you see this, is this scenario clear? What I'm drawing here? Growth in sugar two, growth in sugar one, growth in both sugars, always less than the sum of these two growth rates. Yeah, exactly, right? It's superb. So his suggestion is, is it like two resistors in parallel? Yes. All right, so let's unpack this with that. Do we have time to do all of it? Yeah, we do, all we need is five minutes. And so before I get to the parallel part, there's not, it's not necessarily going to be that this resistor itself is broken into two. This guy might be broken in series, and then one of those series connectors broken into two. And that's actually what happens, okay? And I'll talk about that in a second. So, how do we make sense of this? Can we explain this? Okay, here's a little bit of background before we can explain this. That in the years between this hypothesis and what I'm about to tell you, we've done, we've done some proteomics work to validate that idea, that in fact, you do get many proteins grouped, naturally cluster into sectors depending on their growth dependence. Okay, and two things come out of that, one of them is relevant to what I'm talking about. This part is not relevant, but I'll tell you, to you first, is that this Q sector is really just growth independent offsets in this P sector. So these are made of the same kinds of proteins, but then a given protein has some mass fraction that doesn't depend on growth, and then it's got some part that does. It's got like a DC offset. And then there's no way to get around it, it's just part of the regulations, part of the way the cells design. So that even if you're growing at zero growth, you're making a little bit of this protein. Or you're about to hit zero growth, anything. The other part is that these proteins that are non-ribosomal can be effectively partitioned further and deeper by looking at different types of growth inhibition. So here we've looked at inhibition of translation and nutrient if you like, but you can start limiting in different ways. You can say, wait, maybe I only want to limit by energy, or maybe I want to limit nitrogen or something like this. And that lets you refine what you mean by these non-ribosomal proteins. They too have different clusters within this cluster.  So this guy is untouched. And then the nutrient resistor, we can break up at least into three more functional units.  But this is at least the minimum that we need to answer the question at hand. Okay, so I've got a C, an A, and a U. So C, A, U, this guy I'll call R, or T, let's call him T. And these assignments have some correspondence to what the metabolism is doing. This one, these proteins are what are called catabolic, which means that they break things down. They scavenge carbon, they break things down to get energy. These are what are called anabolic. They build things up. So after catabolic, it smashes things down. I mean, the best, I think picture would be, so you take a big Lego set and you smash it to pieces and then the anabolic's build it up into whatever they want. So these are catabolic enzymes, anabolic enzymes, and these are unassigned. So these are whatever's left over. Okay, and the way that you get at these is by doing different limitations. And so you hold all of these fixed, their rate's fixed, their kappa's fixed, and then you fiddle with this by changing something. Okay, and so if we fiddle with this kappa C, what happens is so to get an estimate of this one over kappa A plus one over kappa U, sorry, short out one over kappa C. And what you do then is measure the proteins that are in this sector, i.e. the voltage drop across this resistor as a function of growth rate. And the limitation that you do is that you grow in different carbon sources, but you don't supply any amino acids or anything. And so this allows us to make everything from scratch, which means that this sector is as big as it gets. And what you see is something that looks very much like the constitutive protein that I showed you before, but that constitutive protein one has an intercept of about, so for unregulated, it has an intercept here of about 2.85 per hour, lambda, whereas here for this kappa C, I'm gonna call this thing lambda C, which is the limiting growth rate when you have none of these catabolic enzymes expressed. It's about, I think it's like 1.7 or something, heavily embedded. My point is that it's smaller, and that'll be useful, 1.2. Okay, so what am I looking at? Experimentally what I'm looking at is shorting out this resistor and measuring the current. And that's my phi C. And from that then I can estimate what these two resistors are. And so that gives me an estimate. So this lambda C provides an estimate one over kappa U, kappa T. So phi C is equal to this, phi max over one over kappa A plus one over kappa U plus one over kappa T. Let me stand away, okay? And I'm gonna use that in a second, but my point here is that experimentally, without co-utilizing carbon sources, I can independently assign values to these different members of my circuit. This is great, I mean, as a scientist, that's what you want. You wanna be able to do one experiment in some other room with some other set of, you know, scenarios, and then come into this other room and do another set of experiments that corroborates it. You want it to be as independent as possible. So let me pause though. I've got a lot on the board here, and it seems awfully far from here. Are there any questions on what I'm doing? What this lambda C is, for example? Pardon me, yeah, what lambda C is, sure. So lambda C in this scenario is, is you're measuring proteins that are in this catabolic enzyme, or catabolic sector. And that, I haven't explained how you would, what do you call it, group these? But they have a distinct characteristic. One is that if you start limiting nitrogen, for example, these proteins change, but these don't, for example. It's a little like that. The point being that if you start to measure these enzymes that are in this group, their level drops with growth rate. As you start changing things, as you start changing the nutrient environment. Until they disappear at a growth rate of about 1.2. Which in this picture would correspond to shorting out this resistor. Just taking a wire and putting it like this and making this a zero resistance. Does that make sense? So what I'm doing with this line is parametrically ramping up this conductance, if you like, or decreasing this resistance to the point where it becomes a shorted out element. And then I'm looking at what the maximum current through this circuit would be if this resistor was not there. So I can't get here, but what I can do is sample along this line and extrapolate. And that's what you do experimentally. Set another way, I'll come back to you in one second. Set another way, we've got this metabolic resistor broken up into pieces and we can functionally tune, we can toggle the knobs of these pieces separately. And so what I'm doing is changing the knob that corresponds to carbon utilization. Yeah. This was that Phi P that I had. So this one is when the Phi P goes to zero. Okay, so that is when all of these would go to infinity. What's the fastest I can grow if all I'm doing is making rabbit zones? Is that, should we talk about that? Is that a better thing? That's okay. Is it okay? Okay, I do all this because, well, what I want to talk about is exactly what he said. If we have carbon co-utilization, well, it's like a modular switching in and out of two different types of resistors. One that's got a high resistance and one that's got a low resistance, depending on how happy the cell is to grow on that carbon source. And then co-utilization would be two independent resistors parallel to one another. This resistor eats that carbon source, this resistor eats the other. Why can you never get the sum of these two growth rates? Because you need to dump potential across both resistors. There comes a cost, I mean, there's a cost associated with making these proteins and these proteins simultaneously. Okay, and I'll show you that in a second. My purpose here is to suggest that that lumped circuit can be further expanded or magnified. And in fact, we can do so in a way that sort of tells us what those proteins are responsible for. Carbon utilization, amino acid biosynthesis largely, and then other stuff. And this is a guy that I want to focus on. So far, so good. Okay, so my other purpose was to say that independently we can figure out what the current through this system is or the maximum growth rate is if we throw this resistor right out. We short this out. Okay, and so then the idea is, so the simplest model of co-utilization following, so you've got this battery, you've got this resistor, you've got these one over kappa t, one over kappa a, one over kappa u, and then you have independent groups of enzymes that are chewing up carbon source one and carbon source two. This might not be true. Maybe they have shared enzymes like carbon source one also uses some of the proteins that carbon source two does. Well, then there would be another small resistor here that I'm ignoring. So the prediction that I get out of this model is assuming that these guys use completely orthogonal and separate non-overlapping sets of proteins to consume that carbon source. Okay, these catabolic enzymes are breakdown, they break down carbon sources. So this is now one over C one if you like and one over kappa C two. You know, how would I characterize those? I would look at the growth rate in carbon source two by itself and carbon source one by itself. Okay, and so I can characterize, this is the last thing we'll do and we'll break. So we can characterize these one over kappa C one and one over kappa C two by growing in carbon source one or carbon source two alone. If these two are really orthogonal, then that should give us all the information we need when we bring them together. Then if we use these Kirchhoff circuit laws, we should get for the growth rate that the growth rate on both carbon sources divided by this sort of this maximal growth rate is gonna look like this. And if you like, try and derive this. It's not too hard. It's applying Kirchhoff circuit loop law to this and the sum of parallel resistors where the resistors here are one over kappa C one and one over kappa C two. And what you will end up with is the following. Hat notation just means take the growth rate and divide it by that maximal phi C. Now this guy, it's not clear or it's not clear to me but this is always less than the sum of phi one plus phi two. And I'll write that. And then let me show you how it compares to experiments. This is always less than phi one plus phi two. Since co-utilization never gets you this sum but it always is bigger than phi one or phi two alone. And that's the same thing that we have in these electrical circuits. So here again, this is phi max. This guy is the growth rate. So if you don't see immediately where that formula comes from I don't blame you, it's a hassle. It's always a hassle to deal with parallel resistors but you can work through it and convince yourself that that's the formula that you do get. But the important part here is that this one is characterized, these two are characterized independently of this prediction. Something I do, one carbon source, one carbon source. And if these truly are orthogonal then I should get that. And what you see is something very nice. So if you have the predicted lambda one two and you have the measured lambda one two what would you, what you would hope is that they fall along a diagonal like this. And for the most part they do. So I forgot to put that on my slide set but I'll show you that on Monday or Tuesday, sorry. Except for a couple. So some choices of carbon don't do this. They group together and they're off diagonal. So most are here. But then some groups are off of this diagonal. And what's interesting is that those are ones that have something going on that's not like this. So what's going on for these carbon sources being co-utilized is some sort of passive regulation. Just like we would expect with a circuit. Here it's active. So one carbon source triggers regulation in the cell to cut off the transport of all other carbon sources for example, okay? And so these become the exceptions whereas this passive indirect regulation is the norm. And I think it's best understood in this context in the context of a circuit diagram where we take all the complexity of the cell and we abstract it away into these lumped phenomenological parameters that we can measure independently of one another. And that's key. And so again, it's got this flavor of 19th century physics and mathematics where we have the ideal gas law or we have the Ohm's law, Kirchhoff's laws, things like this. We're operating underneath, bubbling underneath is the statistical mechanics, is this huge sort of complicated chemical network. But at this phenomenological level, it's intelligible. We can work with it. Not only that, we can make quantitative predictions, okay? And all of this is contingent on the conservation of, in this case, electrical energy. So that's the Kirchhoff loop law. All of this stuff that I'm talking about is contingent on the conservation of total proteome fraction, if you like. That the fraction of proteins has to sum to one. And that's going to have implications for more complicated scenarios. This co-utilization was a complicated simple scenario if you like. Okay, let me pause though. Are there any questions? Does anyone want to, any part of this clarified? I feel like I snowballed you in the last, so I just steam-rollered over you. Which wasn't my intention. So think about it, maybe look at the lecture notes and then let's talk about it on Tuesday. So we've got a long weekend. We just relax. It's okay. If you do have a question, come and see me afterward. Otherwise, I'll talk to you guys on Tuesday. So enjoy the weekend.