 last lecture, and so they won't miss anything if they're slow to come. And so to remind you guys, well maybe I'll let you get several for a second. Thanks. Alright, so to remind you, last time we were looking at the consequences of this proteome partitioning picture. And so we spent a lot of time in the early parts of these lectures looking at some of our first historical works connecting different observables to the physiology of the cell. And then we finally arrived at these two empirical relationships. One for the ribosome abundance either under nutrient change or translation inhibition. And then its companion set of data for the expression of say what we call the constitutive protein, which is an unregulated protein. So the surprising mirror symmetry of these two sets of data suggested that we have some type of proteome constraint. That is to say that the entirety of the protein profile of the cell if you like can usefully be divided into a minimal number of subsets. And then necessarily regulation of one of these subsets implies indirect regulation on all others. So if you make one piece of the pie bigger, you must necessarily make all other pieces smaller. Well, redistribute them in some way. The totality of the remaining pieces must get smaller. And so we looked at the simplest possible pie, which was two pieces, that wasn't sufficient, so we introduced a third piece of the pie which is inaccessible. So we had some growth rate independent part, some part that's related to protein synthesis proteins. And another which is related with or is composed of proteins that are responsible for supplying amino acids and energy and other non-protein synthesis roles in the cell. And so we weren't interested or I haven't spoken at length about the microscopic makeup of these sectors. And one point that was made last lecture is that we may know that the fractions are bound by these constraints, but their composition can vary very substantially depending upon what you're growing in. And that's a biological question. So what precisely are the proteins that are in this p-sector? You could ask that. That's mechanistic. That's if you like the statistical mechanics. What we talked about last lecture is the thermodynamics. Very much in the spirit of the mid-1800s thermodynamics, asking what's the very best we can do under a given scenario? What's the maximum efficiency of our heat engine, for example? So we're not interested in the engineering details that go into that, but rather what's the best we can expect? And so the simplest scenario that I proposed last lecture is suppose we want to make a protein that's useless, useless for the cell. It provides no advantage to growth and it's probably, I mean the only reason we would do this is either to to interrogate our picture of the cell or to make bioproducts, for example insulin or some other bio commodity that's not useful for the bacterial growth. And the argument was that if we take this proteome partitioning seriously, then we know without any parameter fitting that the growth rate at best is going to linearly decrease with the protein mass fraction of this unnecessary protein. And this was the idea that if we have a piece of the pine and we start wedging in useless sectors, we are necessarily going to constrain those sectors which we rely on to drive the growth of the bacterium. And then the data that I presented is this one, where these lines then are our best possible scenarios where we only have linear decrease in growth rate upon expression of a useless protein. And that these best estimates or best guesses or maximum efficiencies, if you like, are consistent irrespective of how you choose to make that protein. You might, however you, whatever the engineering details, you can expect at best a linear decrease in the growth rate and possibly, you know, depending upon the toxicity or other effects, less than that. And so that's what we were about three quarters of the way through last lecture. And then the suggestion was that we can go even further. Imagine this is not a useless protein, but rather it changes the growth rate in some way. The way that we looked last time was an antibiotic resistant protein. Well, then we can get indirect regulation through this proteome constraint that gives us feedback loops that's not evident in, say, the genomic material of the bacterium. I call these indirect because there's no molecules that are mediating this event. It's the growth of the bacterium itself that's mediating this feedback. And so we had this scenario where we had a resistance protein that inhibits an antibiotic. The antibiotic in its turn inhibits the translational or protein synthesis rate of the bacterium. And the translational rate is, translation rate is related to the growth rate of the bacterium in a positive way. The faster you can translate, the faster you can grow up to a point. And so what we looked at then, let me fade this to black so we don't need them anymore. So what we looked at then was this open loop scenario, which is our two weeks ago vision of what was going on. This chemical inhibits this machine, which inhibits growth. And we end up with a positive relationship between this resistance protein and the growth rate. But now when we take into account the physiology of the organism, this, if you like, fizzy or growth mediated feedback link. And now we still have a positive relationship, but it's a loop. It's a closed loop relationship. And as we said last time, these types of loops can exhibit by stability, which is very close to happening with that speaker in this microphone. It's this positive feedback. Okay, so that's what we were at the end of last lecture. What I want to do today is talk a little bit more about different scenarios of this type where we can expect positive feedback and by stability under different conditions. And then I want to talk about antibiotics without resistance protein and how physiology colors the response of the organism to these antibiotics. But before I do that, let me pause. Are there any questions about anything that we've done? Either today or, I mean yesterday. It's okay? All right. So let's look at a couple of, this feedback loop's okay. The logical connections make sense. Blonde arrows mean negative correlation, positive means, or spiky arrows mean positive correlations. So we can construct these types of, I don't know what to call it, interaction maps, different scenarios. So I'm going to talk about two in addition to this one. So for example, suppose we have a protein, I'll call it Y. This is a toxic protein that inhibits metabolism somehow. Okay, and the question would arise, why would the cell make its own poison? Why would it poison itself? It has many of these systems. E. coli has probably several dozen of what are called toxin-antitoxin pairs, which are poisons that it makes for itself. And let me talk about this and we'll talk about the interaction and we'll ask why would it do this. And so we have growth rate over here and we have some kind of linkage here, which I'll talk about in a second. But the type of protein that I want to focus on today as I say inhibits metabolism, so in a phenomenological picture that we've been discussing that would be a change in this nutrient assimilation rate. So I would have a negative linkage here because it's inhibiting it. I have a positive linkage here that the higher that nutrient assimilation rate is, the faster these cells can grow. And then I have an unknown linkage between the growth rate change and the abundance of this unregulated protein. And so suppose I have an unregulated protein, let's call it Y, and I have the growth rate. As I change the nutrient quality, we had this data where we had, say, circle, square, triangle. Now suppose I'm here at triangle and I start to inhibit the nutrient environment. What happens to that triangle? So I'll tell you the growth rate goes down, so the triangle moves to the left. But now the question is, does it go up or does it go down toward the origin? Is that a sensible question? Does everybody see what I'm saying here? So this is, we'll call it a protein mass fraction. Why don't I call it Phi Y just so you can map it onto what we had before? And remember this guy is unregulated, so it's going to do what the P proteins do. So suppose I start inhibiting this Kappa N, this nutrient. What do you expect the triangle to do? Is that okay? I mean, is the question sensible is what I mean? Not does the answer present itself, but is the question sensible? Okay, so let me contrast this. This is an aside. I had that X protein before. I had growth rate. I had the same scenario here for nutrient. And now when I added my antibiotic, I had this. So that was an inhibited Kappa T. That is to say I expected a positive correlation between the protein fraction of my inhibitory or my resistance protein and the growth rate under this condition of inhibiting Kappa T. Now, focus on this guy again, and now I want to inhibit Kappa N. Ignore for the moment that it's the protein itself that's inhibiting Kappa N. If say I was using a chemical to inhibit Kappa N, what would happen to this triangle? So Kappa T is the same. Kappa N now is getting smaller. I mean, just guess. There are really only two choices. Well, there are many, but two are sensible. Oh, that was not the right thing to say. If you guess, you'll probably be right. You just shout it out. You want to guess? No, nobody does. I think if I tell you though that it's not going to be, it's not as pedagogically satisfying. You just guess. Oh, I don't want to bully you guess. So we have choices probably this way is one choice, maybe this way is the other choice. And what I'm suggesting is that this is indeed what will happen. You'll go up like this, and so you'll get the triangle, which is in some particular growth environment. So say the triangle is in glucose or something. You don't change the chemistry of the test tube, but by inhibiting Kappa N, you move this triangle like this. So in this scenario, the movement that you make is exactly the same as what you make if you grow on poor nutrient conditions. That's the scenario I have in mind. Here we're moving orthogonal to that, because we're inhibiting the translation of the nutrient quality of the medium. Let me pause. Do you guys follow me when I'm saying that there's a contrast between the way we inhibit the growth and the way that these proteins are expressed? So here when we inhibited translation protein synthesis, the cell reacts by making more lobosomal proteins, which means that it necessarily makes less of these other proteins. Here we do the opposite. We start to inhibit the nutrient quality, and so the cell responds by making more proteins to try and keep the rate at which nutrients are coming into the cell high. So you fiddle the per-protein rate. It just makes more proteins to try and compensate, but necessarily when it does that, it makes less ribosomes and it grows more slowly. So there's always a price to pay. So this comes from this feedback picture, but also the meaning of this parameter, kappa n. Let me pause. First thing is, is this picture sensible? And then second question is, what's the missing linkage in this feedback loop? So first question is, is this picture sensible? Does everybody see what I'm drawing here, basically? So this would be kappa n regular. Then this is kappa n a little bit smaller. And this would be kappa n a little bit smaller, where this kappa n naught is bigger than kappa n1, which is bigger than kappa n2. And the way that I'd been doing that previously is by growing it in different sugars or something. In this case, the way I'm doing it is by making this poisonous protein X. And X acts very much like an antibiotic. It just jams in and includes the making of different nutrients in the cell. I see furrowed brows, and I feel like I've made this more complicated than it needed to be. Are there any questions about that? So the way that this is going, do you mean like what's the linkage between y and lambda, or why is this up? Why is this up? So the way, remember for this one we had that the growth rate is equal to this kappa t phi r, or delta phi r if you like or whatever it is. I mean there's an offset, but just ignore the offset. And so what we did is make this smaller. The way that the cell responds to that is to make this bigger. But then if it makes this bigger, it necessarily makes this smaller. Oh, is that a p? That should be an r. And so it's trying to make this, restore this back to its original state, but then it necessarily is going to reduce the growth rate. And then it's going to meet a balance because these two lines need to be the same. Did that help? Was that helpful? So that's here. Over here it's the opposite. So now you start to fiddle with this and the cell compensates by making more of this. It actually compensates by making less of this, but it's easier to think of it. Makes more of this, which necessarily makes less of that. And so you cut your supply rate and that necessarily cuts also your consumption rate. Or you try to increase your supply rate, which necessarily closes off your consumption rate. So there's always this give and take. Whatever you do on one side, you have to do the opposite on the other side. Is that better? Any other questions? So I could have, instead of making this protein, what you can do is add inhibitors. So suppose I'm growing in glucose here, I can add a chemical that looks an awful lot like glucose, but the cell can't eat. And all it does is get in the way. So the cells grabbing molecules that thinks it's glucose, it's really just garbage, and then its growth rate will be smaller. And to compensate, it tries to make more, well, it tries to make less ribosomal proteins and necessarily more non-ribosomal proteins. But then the growth rate goes down. So that would be what's called a metabolic inhibiting, or metabolism inhibiting antibiotic. You would get the same type of response. So is that it? Is that all right? Okay, so think about this. If you find a puzzling, let's talk about it. So tomorrow I'll have a tutorial here at 4.30. That's Thursday? No, wait, yeah, Thursday, here, at 4.30. We can talk about it then. For now, even if you don't like it, it's not comfortable. Assume that it's true. Then what would be the consequence here? Exactly. And so now again, we have this possibility of a positive feedback loop. We have two negatives make a positive. So don't not understand that, for example. And so again, we have this possibility of a feedback which then presents this possibility of bistability. And so now without even knowing, say, the details of how this is happening, we can suggest that under some conditions we might have two populations. One that has lots of this Y and is growing very slowly, or not at all. And ones that have none of this Y and are growing very quickly. And the reason that the cell makes these types of proteins is because when it's not growing, it's very impervious to many types of threats. And so it uses these types of proteins to survive, for example, antibiotics. So you have some of the cells are not growing at all. Most of them are growing happily. And these guys are like a secret cache of bacterial spores, if you like, so that when antibiotics are not the rest of their friends, a few generations or hours later they can wake up and start growing again. So this is a stochastic switch from, what would you call it, a deep sleep back to life. And when they're in deep sleep they're resistant to antibiotic stresses. And so these are sometimes called persistor cells. And if you're interested, I'll post some papers on the course webpage. But the point here is that the bi-stability is just implicit in the interconnections between how this protein works and how it affects growth rate. And then this indirect loop coming from the physiology. Let me pause. So the ones that are particularly, so different antibiotics target different parts of the cell, lifestyle basically. So the ones, the antibiotics that we've talked about so far target protein synthesis. And as we'll see in 10 minutes or so, whether the cell's growing faster, growing slow, it will change its response to the antibiotic depending upon chemical properties of the antibiotic if it targets ribosomes. But the ones that you're talking about are usually ones that are called like penicillin and ampicillin which attack the cell wall. So they inhibit cell wall synthesis. So if you've stopped growing, you're synthesizing cell wall components. And so their effect is, you know, it's not there. So it's like, you know, you've finished your meal, you're just sitting there and I tell you, you can't have any more to eat. You say, oh, all right. I mean, it doesn't matter to me. I'm not growing. Oh, they're actively synthesizing cell wall to grow. And so this antibiotic is inhibiting that. And so then it makes tears in their cell wall when they burst. Yeah, everything is growing happily. Yeah, except that this antibiotic is tearing, basically tearing holes in the wall. Yeah. Well, no, they're under tremendous osmotic trigger pressure. So it's like a water balloon. You see, it just explodes. Yeah. So this then brings to mind questions about how growth rate couples to antibiotic susceptibility, which is what we'll talk about in five minutes. Okay, we'll come back to this. But the point here is that for many classes of antibiotics, if you're not growing, the antibiotic doesn't touch you. And so your hope as a, I mean, I'm putting this in anthropomorphic terms, but the hope of the bacterium is that they can stay asleep long enough that the antibiotic threat will wash away. And for the most part, that tends to be true in real life. All right? Okay, so this is the why. This is the how. And the point that I want to draw your attention to is this bi-stability. You don't need a lot of cooperativity between any of these actions in order to get bi-stability because you have this feedback loop. All right, one more example, and then let's step back. The last feedback example I want to talk about is for bioproduction. So, and not necessarily just for bioproduction. So suppose I have some protein here, which is now a regulator, that turns off over expression. Okay, and so the experiments that I showed you last lecture where we were making this useless protein, the way that we controlled the making of that useless protein was that there was one protein that kept that off, and then we added a chemical to relieve that. So we relieve that repression. So this guy turns off over expression by u. By u, of course, inhibits growth rate. So this is now growth rate. And then the question is, again, this regulator itself is going to be unregulated. What's the missing link? And so if I have now, let's go back to our triangle here. So this is growth rate. This is this abundance of this regulator z. And I start to make an overexpressed protein. What happens to unregulated proteins of which this is one? So this guy then falls into this p-class of proteins. What happens to p as you start to overexpress? Is that a sensible question? So we had three different plots here. This is phi x. We had triangle. And then we had this. This is growth lambda. And so this was inhibiting kappa t. This was inhibiting kappa n. And now we're here, and we're increasing phi u, this useless protein. Where's the triangle going to go? Again, we're going to move to the left because the growth rate is going to go down. The most important part is whether we're going to do that with a positive slope or a negative slope. Because that's going to determine the branch of this feedback loop. You could do a meta-analysis of what I've been doing so far and ask yourself, wait, he's done two positive feedback loops. Probably he's going to do a third positive feedback loop. What polarity would you need the interaction to be in order for that to happen? Okay, so what do you want here? Do you want a positive correlation? Like this? Is that okay? Okay, so she suggests that ignoring the vector of physiology, just going from my past actions, this would give you a feedback loop that was positive. Does everybody believe that? Two negatives make a positive. It's got the same topology as this. Just sort of rotate it. Rotate it. All right. I would submit that she's right. Why do you think this is positive? So again, this is that indirect loop that comes from the physiology. If I started making this useless protein, what would happen to the triangle then? Exactly. So come down here. So both R proteins and P proteins have the same polarity when you start making an overexpressed protein. They both go down to zero. Along with the growth rate. They both have positive correlation with the growth rate. P goes down. Growth rate goes down. R goes down. Everybody goes down. And that's what gives you this positive correlation. As growth rate goes down, this guy goes down. But wait a second. Then that means this guy goes up, which means that this guy goes down, which means that this guy goes up. Did I do that right? No, I didn't. This guy goes up, which means that this guy goes down, which means that this guy goes up, which means that this guy goes down. The point here is, again, we can get by stability. So here's lambda over no growth rate. And depending upon the details of this regulator, we can end up with a situation where we have low amounts of this regulator, which means high overexpression, great and low growth rate, great. But then the cell might switch to a case where this guy is very high and this is very low and the growth rate is very high, in which case you have two populations. One is growing slowly. So this is say Z, abundance of Z or something. It's basically efficacy of this Z binding. So you would have a situation where what you want are these guys on the low that are making lots of this useless protein for you but the cells growing slowly. What you don't want are these guys that are growing fast because they'll overtake your tank and they're not making what you want. If you like, they're parasites from your point of view. They're eating up all the food that's in the test tube and they're not making any bioproduct for you. So from a biomanufacturing point of view, this type of topology is problematic because you often get that while you're making this overproduced protein, the cells will thwart you either through evolution or through this type of switching where you'll end up with parasitic cells that overgrow your population. Let me pause. Let me pause actually. Any questions about either of these three positive feedback loops? The important thing that I want to draw your attention to is not their various applications but the fact that in all cases it's a missing connector that gives us the weird bystability and positive feedback loopiness that closes the loop if you like, that links the growth rate to this protein that in turn changes the growth rate. And that these dotted lines, this one I didn't dot, these dotted lines again are not in the genomic material of the bacterium. You could sequence it a thousand times. You would never see this. This comes from the idea that these bacteria are growing and they need to deal with their internal resources in such a way to maintain that growth. All right, let me pause. Any questions? Yeah. Then I think a natural question is if these are bad from our point of view, can we find a way to break one of these or flip the polarity so that we don't have this anymore? So the experiments that I showed you, no, they were negative going to a positive. But if this one were positive, then suddenly you would have a homeostatic system so it would try to stay at that level. So the growth rate would go up. Okay, it would make more of that to bring it down. The growth rate would go down. It would make more to make it go up or less to make it go up. So this would be desirable. And then another question here would be can we find antibiotics that don't have this type of dependence? Well, if you had an antibiotic that did this, so it inhabited Kappa N, then you would have a negative here and much better off. And so children's diaper rash cream is of this type. It inhibits metabolism, not protein synthesis and many types of antibiotics are of that type. And so you can start thinking now totally differently of not just chemical details, but whole mode of action details that would thwart this kind of positive feedback loop. I think that's a really productive strategy. Is that okay? Any other questions? This type of feedback, I mean of course in bioproduction it's bad, but there may be other huge systems in the cell that the cell wants to use in a bistable way. And one is flagella assembly. So it's got this whole suite of motors if you like that allow it to move and it costs a huge amount of cellular resource to make them and they make them only under conditions of acute starvation and scavenging. And very often, depending on how severe the starvation is, only a subset of the bacteria will commit to that program. And it does it with a topology very similar to the one that I have here. Like this. And so in that case the bistability is again like in the persister cell, a bat hedging strategy. Have some guys doing one strategy, most doing another one just in case the whole casino burns down and you want somebody to win. Alright, let me pause, yeah. Sorry, can you say one more time? In real life, we're with antibiotics. In this scenario? Yeah, so in real life you say you've got some topical infection. And so it's... What am I doing? What am I doing here? You've got an antibiotic, yeah. So what you want is that this antibiotic is going to decrease the growth rate. So you want this to work well. What the bacterium wants is that not to happen. So it's making this resistance protein. But by making the antibiotic attack the metabolism you've made it so that the loop, the physiological feedback doesn't help the bacterium, it only helps you. Does that make sense? So you can play these engineering games of flipping the topology around to help you rather than to help the bacterium. I don't think I answered your question, though. Was that okay? Any other questions? And so you were playing a game outside of the physiology of the organism. We're looking at these coarse-grained partitionings of the proteome. We have a couple of sort of rule-based ideas that, okay, if you're in this protein group you'll do this, in this protein group you'll do that. But we're not asking how is that regulation implemented and we frankly don't care, right? For questions of this type, we're at a high enough level that these rule-based scenarios are adequate, okay? And as such, we don't need any fitting parameters. It's just sort of, you know, if the, you know, you're at a constant temperature, you increase the pressure or then the volume's going to go down or something like this, right? Very loosey-goose macroscopics. All right, let me pause. Any questions, though? Okay, so let's go to the opposite, in the opposite direction. So now what I want to do is look at a mechanistic, so a detailed chemical interaction model for antibiotics interacting with the protein synthesis machinery that I'm then going to embed in the physiology of the organism. So the model itself is going to be very simple because I want to draw your attention to the coupling with the physiology, not necessarily complications arising from detailed mathematical modeling. But I think it's a good, it's a good example to end with because it gives a good connection to biomathematical biophysics modeling that's been done in the past and hopefully bringing it into a physiological context. All right, so let me pause one more time. Does anybody want to talk about this anymore? Okay, let me erase it and then let's talk about antibiotics. Okay, so this was hinted at a little bit about how does the mode of action of an antibiotic interact with the growth of the organism, particularly with these persister cells. We talked about, just briefly, were antibiotics that target membrane synthesis, right? And if you're not growing, you're not making membrane, you're not susceptible to these antibiotics. So here what I want to talk about are antibiotics that target protein synthesis. Okay, and the reason that I want to focus on these is that we don't really need anything more than the magasanic night heart data that we started with at the beginning of this course. Need a little bit more, but not too much. So remember that we had these ribosomal proteins that were related to the growth rate by these two near linear relationships. So we had the, how do I do it? Triangle. So we had that if you change the nutrient conditions, you would get this positive correlation between ribosomal abundance and growth rate. So this is now nutrient quality, say. And then we had this opposite correlation if we started inhibiting translation. But now imagine that your antibiotic is targeting this ribosome. Well, your target abundance is changing with growth rate. It's changing depending on how virulent or how fast this infection is growing in your body. That's this line. And then when you start adding the antibiotic, you start to change the target abundance again. And so the question is, does that make these bacteria more or less susceptible to your antibiotic? So the target, let me write that out. So the target, which is the ribosome, growth rate dependence, how does that affect or even does that affect susceptibility? What I mean by that is, does it take more or less antibiotic to kill a fast growing bacterium than it does to kill a slow growing bacterium? So, i.e., does it take more or less antibiotic to kill a fast growing cell versus a slow growing cell? Question mark. Or does it even matter? And so we spoke briefly about this case of penicillins and ampicillins where there it's decidedly true that ampicillin preferentially kills things that are growing. If you're not growing, you're safe from ampicillin. It's the same true here. If I put a certain fixed amount of antibiotic into my test tube, will it only kill the cells if they're growing faster than a given growth rate or, conversely, slower than a given growth rate? If that makes sense. Or should it matter at all? So that's a question that I want to talk about today. It's okay? And so it does make a difference. It does make a difference if the cells are growing faster or slow, but it's more complicated than that. So I'll show you two tetracycline, sometimes used for acne, and streptomycin, almost never used anymore. So these are two antibiotics that target the ribosome. And in fact, chemically, or maybe what would you say, molecularly, if you look at the ribosome, their target sites are almost on top of one another. So they inhibit protein synthesis by messing with the machinery at almost precisely the same spot. And what I'm going to plot here is the growth rate normalized to the growth rate with no antibiotic. So this is what I'm going to call the drug-free growth rate. So no antibiotic, and this is the growth rate. And then along the horizontal is a concentration of these antibiotics. So concentration of tetracycline, concentration of streptomycin, and these are called growth inhibition curves. And so I can vary the concentration of these antibiotics and still get exponential growth, and that's what the vertical is going to be. And now if I have here circle is slow, square is medium, triangle is fast, what do I get? I get something like this. This guy doesn't plateau, he keeps going. And this one would be slow, medium, fast. That is to say that for tetracycline, I need less tetracycline to get a more dramatic relative change in growth rate. And then streptomycin is a different story altogether. So for streptomycin, the inhibition curves look like this, and the dependence is flipped. So here's slow, medium, fast. And so let me show you the data, and then let's talk about it, and then let's see if we can make sense of it. So the arrow is telling you how fast these cells are growing. So green is very fast doubling times without the antibiotic, red is very slow doubling times without the antibiotic. And what we'll talk about is that the concentration scale really doesn't matter. It's comparing apples to oranges. What I want to talk about is this flipping between the fast and the slow growth and the shape of the curves if we can get to it. Let me pause. Does everybody see the problem? Maybe not the resolution, but the problem? Yes, yes, yes. I mean these are, yes. But then the question is, can we take what we've learned so far and figure out why these are shaped like this and these are shaped like this? Or at least give a rationalization for it. Because on the paper there shouldn't be any difference, right? There are two chemicals that are attacking the same piece of machinery. Why such different behavior? So far so good. All right. Okay, and let me say one more thing. What we'll look at primarily is this half inhibition concentration. And we'll do that in a little while, but I want to bring it up now so there's not an unfamiliar concept. So the IC50 is the half inhibition concentration. And this is when this is, so this antibiotic concentration equals the IC50 when lambda over lambda naught equals one half. Okay, so what you do is look at the halfway point. This is the IC50 in fast growth. This is the IC50 in medium growth and so on. So the point here that I'm making is that this IC50, this half inhibition concentration is strongly growth rate dependent. Okay, and that's not the usual story. Usually we take it as a material property of the antibiotic, which it certainly is not. Not only is it growth dependent, it depends on the antibiotic as to what its growth dependence is. So let's see if we can get to the bottom of this. Let me pause though. Is the data, is what I'm showing you sensible? I mean, not, certainly not the explanation yet. It's going to take us a bunch of steps, but does everybody see the question? And so then the question, let me go even a step further. If somebody comes to you and they've got a chronic infection, then would it be better to give this antibiotic or this? And what I mean by chronic is that it's been there for a long time, but it's growing very, very slowly. Would you give this or this? And then wait. Next person comes in and they've got a very acute infection. So like an hour ago, they were fine, but now suddenly their lungs are full of bacteria. Would you give this antibiotic or this antibiotic? These are really, these are truly clinically relevant questions to ask. And just by looking at these curves, we can make a reasonable assessment. That you would need, oh, maybe you can tell me. So with a chronic, so slow burning, long lasting infection, which would you give? Streptomycin or tetracycline, right? So strep to mycin because you'd need less of it to kill these bacteria. And then the converse, right? If you've got a virulent infection, you need less of the antibiotic to really make a difference to slow this thing down. And clinically, typically these are the guys that are prescribed your respective of the virulence. Because the thought is, well, look how low these are here. You can really kill them. Here it's just gentle. But that's completely missing the point that there's some growth dependence in the susceptibility. All right, so let's look at a simple model of this system. So can we rationalize this? This growth dependence. And so let's take a look at a simple model. I'm going to put the model up in a cartoon. And then we'll put it up in mathematics. And you'll see that the mathematics is no more complicated than the cartoon. And the cartoon, I promise you, is simple. OK, so here is my cell. And here is my ribosome. Here is my antibiotic. I'm going to distinguish the antibiotic that's inside the cell from the antibiotic that's outside the cell. So I'll have some transport in or out. And I'll call this A out. And the only antibiotic, so I'm able to control this, but this is what does the damage. And the way it does the damage is by binding or unbinding to my ribosomes. And so I want to distinguish between these two cases. This guy I'm going to call unbound. And this guy I'm going to call bound. And what I want is an expression for the growth rate that depends on the external concentration of antibiotic. Because that's what these growth curves are. OK, does everybody see the scenario? So I have two processes here. I have transport. And I have binding and unbinding. And that's all I want to consider. I want it to be as simple as possible. It's OK? All right, so now let's go through the mathematics. In contrast to everything that we've done up to now, I want to talk in concentrations just because chemically that's the more natural framework. But the concentrations are related to these mass fractions by proportionality. So we'll deal with, so our state variables, we'll write it like that, are this internal antibiotic, this unbound, and this bound, which are all in units of concentration. And then I want to ask, how are their dynamics related to one another? And so it's probably best if I make a big plate of these. So we have, all right, these are all of my guys, bound, unbound, and antibiotic. Now the first part of the dynamics that's common to all of them is dilution through growth. And so each of these is going to have a dilution term. And the reason that we have that term in there is because we're not looking now at the total abundance in the test tube, we're looking at it per cell. And if you shut off the synthesis of any of these, like for example you don't make any of this unbound target, well then the cell is going to double and your concentration is going to half, then those are going to double and your concentration is going to quarter in all the cells and so on. You're going to have bi-nomial partitioning of the contents at every division. And in the end you'll get a rate of dilution that's exactly your growth rate, the exponential growth rate. Okay, that part is, sort of takes the least modeling. That's what we mean by exponentially growing cells. But let me pause though. Does anybody have a, is that okay? So we haven't talked about that, but I hope that that's okay. Alright, then the next process that we have is going to be binding and unbinding. So binding, unbinding. And for that I'm going to assume very simple binding and unbinding. So here what I'll have is some on rate, which is constant, multiplied by the amount of A time, or the density of A times the density of the unbound. Okay, so A is decreasing by these binding events and these binding events depend upon the density of these two. So it's like a Latke-Volterra model with predator prey, where this guy's a predator and this guy's a prey, if you like. And that's going to come all the way down, except that here the bound target is going to increase every time these guys decrease. And then, you know, goes the other way, this guy's going to decrease every time there's an unbinding event and these guys are going to then be replenished. So this is binding, this is unbinding, and that's how these state variables are related to one another at this level. I mean, there's going to be other relationships in a second. So I came for now, I'm assuming that those binding constants are constant and there's some fixed numbers. I don't know what they are, but we can figure it out later. What I'm going to do, I mean, we'll take a break in about five minutes, but what I'm going to do just to assuage any ill feelings you might have is non-dimensionalize this whole system. And what you'll see is it will end up with two parameters, one that sets your time scale, one that sets your concentration scale. And so all of these details are going to get washed away in that non-dimensionalization. But I'll show you that in a second. All right, any questions about either of these? Yeah, sure. These are rates at which the concentrations change depending upon the density of these guys. You can think of them as collision rates, if you like. So if the binding event depends upon a successful collision, then you've got the probability that a collision occurs times the probability that it's successful. And that's what this k on would be. And the probability that a collision occurs is going to depend on the density of these two species. Is that okay? So it's, again, it's a lot. If you've seen logcavoltaire in your ODE's courses, your differential equation courses, it's the same idea. If you have lots of predators and lots of prey, then you're going to get eating events very often. If you have too many predators, not enough prey, they're less frequent and so on. That the rate of a predator meeting a prey is going to depend on some density of both of them times some successful capturing of the prey. This k on is a constant. This whole thing is not a constant. So this is the rate constant. But then the rate at which RB is increased by that event depends on the density of these two guys. No, I don't think, I don't think I answered your question right. So this is constant. This is constant. Sorry, of course that one is. This one is. These two guys are constants. They're given by the chemical makeup of the antibiotic. But the rate, so these are called rate constants or binding constants. That's probably a better name for them. But then the rate, which is concentration per time, is given by this product. Is that unfamiliar? Did I explain that poorly? Does anybody want clarification on that? Yeah. Exactly. That's my next step. Yeah. So I'm taking it piecewise. We've got two terms that are missing still. You're exactly right. Maybe it's better if I write that up or just. So the thing is, if you haven't seen chemical kinetics before, this is troubling maybe. I know it was for me, but so we can pause here. There are two terms, as I say, two terms are still missing. How about I put those up and then let's come back to this? You know what? How about I put them up and then we take a break and then we talk about after? Because then you can think about it. OK, and so as he said, there's going to be a transport term as well. That's going to dictate how this interior antibiotic comes in and out. And so there's going to be some transport rate that depends on these two concentrations. And I'm going to take the simplest possible one, which is something that looks very much like this, where I have a constant permeability. So I have that J A increases by some P in A out minus P out A. So this is transport of the antibiotic into the cell minus transport of the antibiotic out. And that transport could be active. It could be diffusive. But for now I'm assuming that it's got these permeability constants that are constant. They have no growth dependence. OK, it brings in which now? A out is actually our control variable. You're right though. So this is what I can vary. This is what I want to read out, but you hit the nail on the head, we're going to have. Yeah, so A out is going to be what I can move. It's going to be my independent variable. And then I want growth rate as my dependent variable, but I have no equation for it. Is that OK? We'll come back to that. So that's transport. Transport is very similar to this binding, except that it's moving something in and out. The rate at which it moves in is faster if you have lots of antibiotic outside. And then the last thing that we have is that we need to be able to synthesize unbound targets. Otherwise, this system can never reach a steady state. We've got dilution, but if we have no synthesis to replenish the pool of ribosomes, then we're finished. But of course we know the cell makes ribosomes. So we have some synthesis rate, which is probably going to depend on the growth rate. But we don't know what that is. So that's synthesis rate. OK, so let me now make two remarks and then let's take a break. So the remark is, what is this synthesis rate? OK, I mean it could be anything. So suppose I make it, I don't know, a cubic equation in lambda and I get some answer. So what? I mean if we don't know what that is, we don't know what's going on in the system. Number two is that what I want is a growth rate. I want the growth rate. It's not given. I mean it's going to change. If I add this antibiotic, I'm going to change the growth rate. I don't know what that new growth rate is going to be. In fact, that's what I want to know. Long growth rate, lambda, which is a fourth variable. So what I really want is I have A, unbound, bound, and growth rate. But I'll have three equations. How do I close the system? And both of these remarks then bring us back to the course that we've been talking about. So all of this is fairly, or no, it's totally standard biophysics chemical modeling. But then fixing this and closing the system is where this microscopic or mechanistic model, if you like, interfaces with the cell physiology. So we'll talk about that in maybe in one moment. Does anybody have a question? Anybody have any questions before we break? Why don't we take a five minute break, 10 minute break, whatever, and come back and talk about it. If you have any questions about the chemistry, maybe you can call them now. We can hash it out before people get back. And to bring it back here, so let me erase some of this stuff. To bring it back to where we were, the equations that I have on the board are basically all chemistry. So there's no biology here. There is growth rate. I mean this is inevitable. We're going to dilute out the contents. But everything else is a chemist's purview. Now we want to ask, how do we bring the biology into the system? Because we're going to need it in two ways. We're going to need it to close the system to find this growth rate. And we're going to need it to specify the synthesis rate. And in both cases, our answer is by making the way that we're going to fix these two problems that's closure and synthesis rate, is by forcing this chemical system to be consistent with our physiological system, with these empirical constraints. And so maybe what I'm going to do is put this up on the slide so then I can use that part of the board. Oh no, that's not right. Yeah, there we go. Okay, so this is the plot that I'm thinking of. All right, so let's talk first about the synthesis rate. Okay, synthesis rate. Suppose I take the last two equations. So look at RU plus RB, which is going to be our total ribosomes. So I'm going to look at the total ribosomal pool. What are the dynamics going to look like? So I'm going to have now d by dt of RU plus RB. I'm going to have some dilution term. I can't get away from that. But the binding kinetics disappear. This guy cancels this guy, this guy cancels this guy, as it should. Because I don't care if the ribosome is bound or unbound. I just want to count all ribosomes. And so the bindings disappear and all I'm left with is this synthesis rate. Or at steady state, lambda times this R total is equal to this S lambda. That is to say that the synthesis rate perfectly balances the dilution rate, as it should. So far so good. But I know empirically what this is. So what can you tell me about the total target abundance under conditions of antibiotic or translational inhibiting antibiotics? So this R total is the total abundance, mass fraction of concentration of ribosomes as you inhibit protein synthesis. What's it going to look like? I mean, yeses. Positively or negatively correlated with growth rate? What's the slope? So here what I'm looking at is under translational inhibition. So I mean to think of tetracycline, streptomyosin, chlorinphenicol, for example. So what I'm looking at then is these colored lines up in the left-hand plot. So I start with my cells growing like light green circle. And then I start to add the antibiotic, chlorinphenicol in this case. It could be tetracycline, it could be streptomyosin, whatever. And I start going to where those numbers are, 2, 4, 8, 12. What am I measuring? I'm measuring the total ribosome abundance. I've got no way at the level of this experiment of knowing whether those ribosomes are bound or unbound by antibiotic. And so what I end up with is, exactly, I end up with some empirical relationship for this total ribosome abundance. Now I need to convert into concentrations rather than mass fractions. But I can do that. I mean it's just a proportionality constant. The point is that these are given, these are known to us from that experiment which has nothing to do with this scenario. These are independently determined. Actually, it has everything to do with this scenario. But does everybody see that? So then that means this synthesis rate is going to be equal to lambda times negative lambda over kappa n plus r max. That is to say we've specified the synthesis rate so that it's consistent with this empirical relationship that I show you at the top. There's no fitting. We've already fit that. I mean it's not even fit. It's just an empirical constraint. So that's one piece of information that we've used to make this chemical model consistent with the biological facts. Let me pause. Does everybody see that this is what's going to happen? First of all, that the vertical axis in this plot is the total ribosome abundance. I've got no way of knowing whether it's bound or unbound to antibiotic. And then second of all, that when I'm inhibiting it with antibiotic, I have this negative relationship. So under conditions of antibiotic inhibition, I travel along this line over there. And in order to make this differential equation consistent with that constraint, I need the synthesis rate to be lambda times that empirical constraint. Now the color of my circle here is going to depend on my initial growth rate. And so I can rewrite this kappa n parameter in terms of my drug-free growth rate. And that's going to move me then along the solid black line, depending on how virulent my infection is before I start treating it with antibiotic. So this is given by the growth rate before I add antibiotics. This is just an empirical constant. And then this growth rate dependence of the synthesis rate is fixed by the empirical constraints. Let me pause. Are you guys with me here? So here we've taken a chemistry and we've married it with the biology. We still don't have closure, but we can do that in a second. Is that okay? Okay. So it is a spirit of the exercise sensible. So what I'm doing is saying almost miraculously, if I add these two lines together, I can isolate for the synthesis rate in a way that I can force it to be consistent with that empirical constraint. That the total ribosome abundance goes up like a straight line with negative slope. And in fact, I know what that slope is depending on what my drug-free growth rate was. And I certainly know the intercept. All right. So that's question number one. Then for the closure argument, let's talk about the closure. Yeah, so exactly so. So the problem here is that these are concentrations and then this guy is not in units of concentration. And so the connection between mass fraction and concentration is just a proportionality constant. So if you like, this is like a cappella hat. But in fact, I'm not going to use this. I'm going to convert this into units of drug-free growth rate. So I'm going to convert this guy into a lambda knot. Is that okay? This guy? Or the lambda knot, yeah. So here I have that lambda knot is going to be equal to r max over 1 over kappa t plus 1 over kappa n. That's what I had from two days ago. And so I don't actually care about this. I really care about this. And so what I'll do is solve this equation for that in terms of lambda knot. And that's what I can measure. That's what comes from that solid black line. Okay, but more important is the, I mean, that's really important, but also the equal importance perhaps is this functional form. This is a line. Is that okay? It can't be just anything. It can't be constant. It can't be a cubic. It has to be a line. Otherwise it won't match what we know. We're constrained. Yeah. Yes, this is just that steady state. Exactly, exactly. So what I'm going to do is not talk about dynamics because I don't know what the dynamics are. The only constraints I have are at steady state and that's where I'm going to use this system. Okay. And I'm going to look at all those inhibition curves that I showed you were at steady state. So I let them adapt and then I let them grow at an exponential growth rate. But I didn't talk about transitions between concentrations because I don't know what the dynamics would be. Are there any other questions? That's an important point. All right. The last one is closure. So point number two is closure. And what I'm going to use to close this system is the other empirical relationship that we have. So here we have that in the absence of antibiotic, the growth rate is linearly proportional to this unbound ribosome abundance. So here we use the other constraint, other empirical constraint that looks like this, that R at steady state is some lambda over kT, again with units of concentration, plus some R min. And now my point is that in the absence of antibiotics, the only targets we have are unbound targets. And so now I have, so now I'm going to assume that even if some of my targets are bound, the only ones that are contributing to growth are these unbound. So I've had this empirical relationship, which is a solid black line, in the absence of antibiotics. And my assumption is that it's still going to hold under translational inhibition. And so then that allows me to close the system because now I have a fourth equation with this lambda, where these two guys are known to me. So this then becomes my fourth equation that allows me to relate one of my state variables RU to the variable of interest, lambda. Let me pause now. So I've used both of those lines, both of those lines, colored and black, to constrain this chemical system to be consistent with the empirical constraints. So I know that these targets change with growth rate. I know actually more than that, I know exactly how they change with growth rate along these straight lines. And so I can jam that into this chemistry to make it consistent with the biology. And we'll see that something useful comes out of this. Let me pause. Any questions? Oh, I meant that mathematically. I meant we had four unknowns and three equations, so it was under determined. And so now I have four equations. Yeah, it means that this is a reached steady state. This is reached steady state. And this is reached steady state. And then under those conditions, we can then relate the unbound ribosome fraction to the growth rate. So we have four unknowns, four equations. Yeah. Yeah, exactly, exactly. And just like you said over there, the units here are in concentrations now, so we're going to need to do a little unit jumping. Any other questions? So now if we solve all four of these equations in steady state, so actually let me pause because the rest of this is just algebra. So it's important that the equations are sensible. If not in detail, at least in spirit. I mean that's really what I want you to take away from this. It's not that you could reproduce this argument on an exam, for example. What I'm interested in is that the spirit of the argument is sensible so that sometime in the future you're in the business of doing this. You also are in the business of doing this, looking after the physiological constraints that are at work in the system. Is that okay? Are there any questions? Okay, let me... Alright, so then let me talk about what happens at steady state. So as I said, the system at steady state non-dimensionalizes in a really nice way. So at steady state the four algebraic equations reduce to a single cubic which is non-dimensionalized by two parameters. So do I talk about that here? Yes. One that sets a concentration scale. So I'll call that IC50 star. And I'm going to write it out. You don't necessarily need to write it out. I'm going to write it out because I want to draw attention to what these things are telling us. And then a critical rate. So this is a concentration scale. And this is a rate scale or a time scale if you like. And this guy is 2 square root of p out, this kappa t. And then kd which is the ratio of these two binding rates. And so this one is not too important. It's what sets the concentration scale. So if that one takes millimolar and this one takes micrograms per mil, that's what's going to be decided by this. This one is the important one. And it's made up of the transport out, the rate of protein synthesis, and this constant which is called the binding affinity. And if this is high, then it means that you need a very high concentration of antibiotic to have half of your ribosomes bound. If it's low, you need a low concentration to have half of your ribosomes bound. So this is in units of concentration. And it's the concentration threshold that gives you half of your stuff bound. So lower it is, the better this antibiotic is at binding. The reason that I write this all up is because you end up with a cubic equation that's written in terms of these, and I'll show you, maybe I won't. I'll show you a limiting form of the cubic equation in a second. But I want you to see all the pieces here. So this stuff, this stuff, this stuff is part of the chemistry. It's what the pharmaceutical companies fiddle with. So this is antibiotic chemistry. This part and this part, oh, maybe I'll make them squares. This is hardwired into the pathogen into the bacterium. This is given by the physiology. And finally, this drug-free growth rate, which I don't have here, is given by the environment. It's given by how happy the bacterium is in your body or in a sore that's on your heel or whatever. And this is given by the environment. How happy the bacterium is to be where it is, how quickly it's growing when you don't add antibiotic. And so we have three things coming together in this model. We have drug chemistry, we have bacteriophysiology, and then we have the growth environment. Two of those come from, so bacteriophysiology and the growth environment is really the focus of what we've been talking about in this course. And we'll see that it colors the data that we get out of this. Okay, let me pause. Any questions? I haven't really said anything. These are the non-dimensionalizing parameters. But what I want to look at is at the IC50, at the IC50 concentration, so this is now A out equals IC50, lambda over lambda naught equals one-half by definition. And so what I can do is take the cubic equation, which I'm not going to write. It's in the lecture notes if you're interested. I'm going to make this substitution. And what you end up with is a very nice expression here. The scaled IC50 looks like this, lambda naught over lambda naught star plus lambda naught star over lambda naught. I mean, you really couldn't ask for a finer. And so this is now the cubic reduces to this. And now we're in a position to at least address the growth dependence in the IC50s between these antibiotics. And I would suggest, although I don't think it's in the lecture notes, that this also is going to answer the question about the shapes of these antibiotic curves. And so if I take that cubic equation that I've suppressed here and I fit it to those inhibition curves that I showed you earlier, I can get an estimate of each of these parameters over different growth rates. So I change this lambda naught. I insist that these two parameters are the same and then I try to fit the cubic to the inhibition curves. I do that because I want to get estimates for these parameters for these different antibiotics, comaprenical, tetracycline, streptomycin, canomycin, whatever it is. And then I can plot on a plot the IC50s that I read off of the inhibition curves and their drug-free growth rates. And what I find is the following. So this is the last thing that we'll talk about probably. So let's look at a log plot of this IC50. This is just going to be lambda over lambda naught star. In a log-log plot, that expression is going to look like two straight lines. It's going to go like this, an intercept chain. Okay, that's this thing. Does that make sense? Here the slope, the power is positive one. Here the power is negative one. So I'm going to get a negative slope when this guy's winning and a positive slope when this guy's winning. And what I find is that the antibiotics cluster very nicely so that you end up with so many, bless you, so many antibiotics that are here and some that are here. And so chloramphenicol, tetracycline is over here, streptomycin is over here. When you take the non-dimensionalizing out, here you find that the IC50 is directly proportional to the drug-free growth rate. Whereas over here the IC50 is inversely proportional to the drug-free growth rate. And just as we saw in these inhibition curves, which is, so then it gives us this rationalization for the way these antibiotics behave. And the way that this clustering is accomplished is by this rate scale. And this rate scale, if you want, is an inverse of a time scale. And the time scale is the time given that an antibiotic is bound to the ribosome, how long does it take for that antibiotic to unbind and go outside the cell? If that time is long, then this thing is small and we're in this regime. If that time is short, then we're in this regime. So these guys are reversible binding, reversible transport. These guys are at least one of transport or binding is irreversible. They come in and they stick and they never leave. Here they boom, boom, boom, boom. And so by looking at the difference, just qualitatively the differences between these antibiotics, we can say a lot about their clinical applicability. Okay? And as we run out of time, I'll say one more thing, which is we can take the limits as this parameter gets big or small and look at the two regimes of the cubic equation. And we recover very nicely the two inhibition curves. Bless you. So last point is the following. In the limits, this lambda star goes large. We're in this regime. Our inhibition curve looks like this. Or no, it goes large. We're in this regime. It looks like this. So this is lambda over lambda naught. And then in the other regime where this lambda naught star goes to zero, we end up with the sideways parabola, which is our other limit. And so not only do we have a rationalization for the growth rate dependence, but we have a rationalization for the inhibition curves that we see. It's all nice. But again, the point that I want to drive home is that we have the chemistry on the one hand, but we always need to remember that that's immersed in the physiology of the bacteria. And that's really the purpose of what I wanted to talk about today and the week's past. All right, let me pause. Any questions? So we scooted through that pretty quickly. And if you're interested in the details, I'll put them up on the website. They're on the website. You can take a look at them. But maybe just in terms of spirit of the discussion, does anybody have any questions? The thing that I was most concerned with was this closure. How do we determine a synthesis rate and how do we close the system based on the physiology of the organism? Hopefully that came across. If not, we should talk about it. Any questions about any of this? It's okay. All right, think about it. Take a look over it. And then if you have any questions, let's either talk about it at lunch or we can talk about it tomorrow. So if you missed it, I'll be here at 4.30 tomorrow and we can have an impromptu tutorial. All right. Yeah.