 The first few days, we talked about historical papers, probably for about the first four lectures. We looked at this seminal work by MOLA, linking macromolecular composition and growth rate. We looked at this seminal work by Neidhardt and Makassanik that rationalized this linear correlation between the RNA to protein ratio and the growth rate by saying that ribosomes are catalytic in protein synthesis. Again, that wasn't obvious, but it took a little bit of unpacking when we got to the bottom of that. Then finally, we spent two days looking at Cooper and Helmsteader's rules or experiments plus rules on DNA replication and bacteria, particularly DNA replication and E. coli. This took a lot of jumps, a lot of baby steps to arrive at the idea that to replicate the chromosome, so E. coli, I remember, has DNA stored in a circle, to replicate the chromosome takes 40 minutes and then 20 more minutes to separate it and divide the cell, which means that if the cell wants to double faster than 60 minutes, it needs to initiate or somehow speed up that process. And the way it does that is by initiating parallel rounds of DNA replication. So it still has one chromosome, but it's got several replication forks running at the same time. And so that's what we talked about probably in Thursday's lecture. And on Friday, we moved to present day. And the main focus of what we were talking about in the present day was this proteome partitioning constraint. So we said, all right, in the 1960s, Knight-Hardermagasonic got this part, that when the growth rate goes up, well, the ribosome fraction needs to go up as well. But what wasn't recognized at the time and what we recognize now is that if you make more of some fraction of protein, in this case ribosomal proteins, so this is ribosome protein fraction, this is, you know, ribosomal proteins to total protein. If you make more of that fraction, you necessarily need to make less of other things. You can't do it any other way. It's a pie chart. You make a bigger piece of pie, you need to make smaller pieces of pie. And what we're going to talk about today and tomorrow is the implications of this pie chart. So one thing that we saw was that if you had an unregulated protein, that's something that you need to make smaller. And the ribosome fraction goes up either by increasing the growth rate or decreasing the translation rate or the protein synthesis rate per ribosome. This unregulated protein does the opposite. It's pulled along for the ride. So whatever the regulation is to set this little flipper, the P-protein does the opposite of what the R-protein does. And that's what these two experiments were showing us, well, two sets of experiments. All right, and then what I suggested is that this isn't the whole story. If we calculate how many proteins or what mass fraction does this and what mass fraction does this, we still have 50% left over. And that 50% that's left over is growth rate independent. And we can well ask, what is that? And if we do proteomic experiments and go in and check what is in this growth rate independent fraction, what we find is that any proteins that are in P and R have a small offset in the growth rate. And so they have some part of their expression, which is growth rate independent, and some part which is growth rate dependent. And so these are the sums of all the DC offsets, if you like, in these proteins. So they're not new proteins. They're the same as the ones that are here. It's just that half of it is growth rate independent and half of it is growth rate dependent. It's very dependent in that seesaw way. All right, so that was, let me see, probably the first of Friday's lectures. And then we went downward and we said, what could be, how can we interpret these experiments in a self-consistent way? And this is what's sometimes called an inverse problem. And it's not well-defined. I mean, it's not, you have to be careful with these things. So what I suggested is that one way to rationalize all of this data, not the proteome partitioning constraint so much, but at least these two pieces of data, is to imagine the ribosomes as being responsible for protein synthesis with some rate per ribosome of incorporation of amino acids. And then at these non-ribosomal proteins, these so-called metabolic proteins are responsible for nutrient assimilation, turning whatever's out in the environment into amino acids in order to be incorporated into proteins. And the reason that I say we need to be careful with this, so I opened the window because it was stuffy, but now it's too loud, so I'll just shut it, all right. So the reason that I say that we need to be careful about this is it's nice, it gives us a mental scaffolding about what's going on, but we can't fall in love with this interpretation because there's no guarantee that it's the unique solution that will give rise to these phenomenological or higher-order relationships, all right. It gives us a sense of how we might fiddle with these parameters, kappa t and kappa n, but it's by no means the only way that you could describe this data. So it's like in thermodynamics, you have some pressure, some volume, some temperature. The microstate could be, you know, one of uncountably many different scenarios that give you the same pressure, volume, and temperature, okay. So I put this up there as a way to think about this, but I don't want it to become, definitely don't take it as gospel, don't take it as truth. It's plausible, okay. These empirical constraints are much more important because they're much more reliable, they don't require any inference. And so what I want to talk about today is instead of this inverse problem, which we ended with last week, is a forward problem. And that's given that these are our empirical constraints, what can we use them for? For example, and I'll go through a couple here. What are the consequences for coupling between protein expression and growth rate? Expression and growth rate, and I'll be more, let's see, more specific in a moment. Another set of consequences that I want to talk about, and this will probably be our last example, which we'll get to either later today or definitely tomorrow, is antibiotic susceptibility. So consequences for antibiotic treatment. Okay, and as I say, this is a forward problem and it's much more reliable. But we use these empirical constraints and we ask what are the consequences? Okay, so in a whirlwind, that's what we talked about on, wow, the twilight of last class, or the last round of lectures. So a couple of people asked if we could clarify the meaning of this parameter. And so maybe I'll do that for a moment and maybe clarify that electrical circuit analogy that we used at the end of last lecture. But before I do that, let me pause. Are there any other points of clarification that people would like? Or any questions that lingered from last week? Okay, so I'm gonna come at this, let's go backwards here and look at some, or sometimes called loose ends. Okay, and the first was, what is this kappa n parameter? And so we had at the beginning of last week, this interpretation that this kappa t parameter had something to do with protein synthesis. So here we had. And then I suggested that this guy had something to do with the nutrient quality. But that's a very qualitative assessment. And so the suggestion or the argument at the end of Friday's lecture was that what we mean by this is that it's a proportionality constant between the growth rate and the non-ribosomal protein fraction. So if we now assume that, or we now have that proteomic constraint. I'm gonna write this up and then let's talk about it. This Phi max is equal to this Phi p plus Phi r. And here I'm subtracting off the min, if you like. Let me show you what I mean in a second. Again, this constraint looks like this. Phi max, which is about 0.5, let me put in about equal to, is the sum of these two fractions. Phi r minus Phi r min, which I call, well I'll just leave it like that. Plus Phi p. What I mean here is that there's some constraint in the growth dependent fractions here. So I've subtracted off from R the growth independent fraction. And the sum of these two sectors has to be equal to 0.5 or some constant. Okay, let me pause. So that's everything on the table and then let me talk about this cap at n in a moment though. Is that okay in terms of the meaning of all the pieces? Does anybody have any questions about the notation? So this I variously called Delta Phi r, Delta Phi p. Just to denote them as growth dependent. Okay, so if we have this constraint then, I can take this second equation. A great substitute into the second growth law. This growth law I'll end up with is this, now what? Minus Phi p is equal to, so this is now what? So I'm going to have to do the algebra in my head and I'll take a second. So I'll have Phi r minus Phi r max is equal to negative lambda over cap at n. Sorry, no, Phi r minus Phi r. Yeah, so, yeah, let's see. Okay, so that's this guy. And now Phi max is equal to Phi r max minus Phi r min, which is what I'm sort of assumed. There's a lot of maxes and min, so let's talk about them in a second. Then finally, what do I end up with? This is then going to be equal to, so Phi r is Phi max plus Phi r min. So here I'm solving for Phi p. A Phi plus minus Phi r minus Phi p. Okay, so this guy is Phi r. Okay, I promise you I'm getting some over this. I mean, it doesn't look like it yet. This is Phi r max, and so these two sum together give me, or sorry, the difference of this guy and this guy give me that. And so they cancel. So this is now going to be equal to Phi max minus Phi r max minus Phi r min minus Phi p is equal to negative lambda cap at n. All right, this is the last step. So these cancel, and I'm left with lambda is kappa n Phi p. And so what I'm suggesting is that there are two things that are driving growth rate in this cell. One is the synthesis of protein, and one is the supply of amino acids to feed that protein. And in balance growth, we need a balance of these fluxes. Consumption of amino acid into protein synthesis? Yes, but also supply of amino acids to feed that protein synthesis. And so one interpretation of this constraint is that kappa n is a proportionality constant that tells you how good these proteins are at supplying amino acids. If kappa n is very high, then you need very few proteins in order to maintain a high rate of protein supply, or amino acid supply. If kappa n is low, then that tells you that you need many proteins to maintain this amino acid supply. And that may be because your proteins are bad at what they do, or it may be because your environment is very, very difficult to metabolize. So you need many proteins to break them down and then build up amino acids, whatever the case may be. The point here is that one rationalization of this kappa n then, given this proteome constraint, is that it's a proportionality between non-rythosomal proteins and growth rate. Now let me pause. So that was an orthogonal way, or say a complementary way of talking about what we talked about on Friday. So on Friday what we did was wrote out explicitly the amino acid supply and demand, or supply and consumption if you will. And then we made some ad hoc, or I made some ad hoc assumptions about that supply rate being related to the external amino acid concentration and so on. And then argued that if the transporter was constitutively expressed, then you would get something like this. So that may have been unsatisfactory to you. And if it was, that's fine. Just throw it away. The first important part here is that there are two things that are making the growth rate. So we have a balanced situation, and that balance also gets balanced growth, I mean. That balance implies not only that everything's growing exponentially at the same rate, but that protein synthesis consumption of amino acids is balanced by protein synthesis supply of amino acids. Let me pause. That was a whirlwind sort of recap of what we did last Friday. Is that better? All right. Okay, so this is going to feed into a larger point that I want to make, which is the exams on Friday, if you're worried about that, in the lecture notes, in the course notes, the typed stuff, there are practice problems at the end. Give them a try and see how you feel about them. Those would be good practice. And then on Thursday, I'll check the schedule. I don't know what's a good time for you guys, but I can have office hours. I can just be in the TV room and we can have a tutorial session if you like, as long as we need it. So if this doesn't make sense, think about it and then come and talk to me maybe at the break. And then definitely let's talk about it on Thursday sometime. I think that fits your schedule. We'll see. Okay, if not, we'll pick a better time. Yeah. Yeah. Oh, yeah. So let me tell you about it. So I thought maybe 10 multiple choice and then five short answer where like, by short answer, I just mean right out the mathematics. Don't give me the whole derivation. Just do that on rough paper. Just give me the answer. So we've got a little rectangle. You write your answer in it. It's like that so I can mark it quickly. All right. So but it's gonna be mostly multiple choice. No. And I don't want it to be tricky. So yeah, I don't want it to be tricky. I'm not going to give you any weird. I don't know how to tell you. I mean, I don't want, I don't want, I want you to do well on the exam. I also would like it if you retain some of the material. But so those are the two things that are coming together for that. Yeah. This guy, this guy. At the beginning it was a theoretical constraint that linked this plot to this plot. But it turns out subsequently that it's an empirical constraint. I don't think I mentioned that on Friday. So since this work was done, we've done a lot of proteomic experiments where you actually go in and count the abundance of different proteins. And this is, this is not really a hypothesis. It's an empirical constraint now. Yeah. So this is the dynamic range of the R proteins here. And so this is telling you how much the R, the growth rate dependent R part can swing. And it's about 50%, 45%. And that's, that's empirical now. And mostly growth rate independent. So for the growth rates that we're talking about, it's growth rate independent-ish. But if you go to very slow growth rates, then what you find is that this part shrinks up. But that's, that's sort of a second order constraint compared to what we're talking about here. So for the, for the sort of first order to first order, this thing is constant growth rate independent. And it's given by this dynamic range, which is fireman, the difference between firemax and fireman. Any other questions about this? Fire can be zero. Yeah. Well, if fire can go down to fireman, in which case this whole piece of pi is zero. But that's a theoretical limit, because that means there's no protein synthesis and no growth. And so this comes in back to the electrical circuits, which is the, I mean with electrical circuits, you can short it out, or you can make an open circuit. For these, you can't. So a short circuit would mean infinitely fast reaction rates, which we can't have. And then an open circuit is dead, in which we also is not experimentally accessible. So we do, or we have to do interpolations and extrapolations basically. But you're right, the extrapolation here would be to fireman, in which case metabolic proteins would be the whole of the pi, and we'd have zero growth. Does that make sense? So we'd be putting all our resources into scavenging nutrients, and we wouldn't have any resources for turning those into proteins. And that's what we mean by a terrible growth environment. Is that okay? Any other questions? Is that okay? It's okay? Okay, so then the other question that I had before the lecture was, what does any of this have to do with the electrical circuits? And my motivation with the electrical circuit analogy was to show you that these two linear empirical relationships and this proteomic constraint, so basically this whole line is mathematically identical to Ohm's law and Kirchhoff's loop law. So two resistors in series. And the reason I did that was two-fold. One, it gives you a little bit of a computational edge and you want to do a sort of a complicated computation-like co-utilization of carbon. Why you would want to do that was sort of as a, I mean I don't know individually if anyone would ever want to do that. But that was an example of how the circuit analogy is helpful. The other side of my motivation there was to show you at what level we're dealing with here. So here we're dealing with exactly the level that electronics were at in about 1870. So we've got empirical constraints, Ohm's law. We've got hypothetical constraints or just rules of operation, Kirchhoff's loop laws. And we don't really care or know about the underlying mechanisms. We have no quantum mechanics. We don't really know about, we sort of know about conservation of energy. But we're using these things and we still use these Ohm's law and Kirchhoff's loop laws constructively for rapid computation of complex problems. For some kinds of questions you don't need quantum mechanics. Electrical engineers don't really care. Oh wow, don't really use it that often. Depending upon the types of questions they're asking. And so what I'm suggesting or what I meant to suggest at the end of last week's lecture is that for certain types of questions you don't need to know all the mechanisms and all the molecular details which to my mind is I'm grateful for because these huge wiring diagrams of this protein interacting with that protein can just become overwhelming. And so what I wanted to do is draw your attention to this coarse-graining approach that allows us to answer questions at a different level. But let me come back to that. Who asked me about the electrical circuit? Okay, we'll come back to that. Yeah. Can you remind me who is it? Is it Minode? Was that the first lecture? Is that what you're referring to? Yeah. Minode. It's okay? So this is not the same situation because you see that the growth rate here depends on the loop in that quantity, right? Yeah. Let me address that. Can I erase this? Does anyone have any questions about what I have written on the board? Okay. No? Okay. Let me talk about it. So that's loose end, let's talk to loose end number two then will be Minode and loose end number three will be the electrical circuits. All right, loose end number two. So Minode happened very early in the lecture series where we didn't, I mean it's useful to go back and see what we were talking about. I don't think that that expression that we had will make sense or made sense then. Now I think in retrospect it will make more sense. And this is what Minode found. I think this is what you're talking about, yeah? Yeah, yeah. Oh, okay, okay, so that's different. So this is Minode's relation. Does anyone have any questions about that? So this lambda max, so this is what I thought your question was. This lambda max will depend upon what the substrate is. This is clear. Apologies. All right, so this one is the molar experiment. So here we had square, here we had circle. They're very different, so chemically very different. All right, so one of them has, I don't know, like she said, and like I said at the time, maybe has some given carbon source like glucose and then some nitrogen source like alanine. The other one has some other carbon source, glycerol, and some other nitrogen source, ammonium. So they're very different chemically. So chemists would run these through spectrophotometer or atomic absorption spectrophotometer and say these are very different. But what I'm saying is arrange these so that biologically the E. coli in either one of these flasks will double once an hour, once per hour. So I'm saying, you know, chemistry aside, the biological fact is that if you put E. coli in either one of these flasks and grow them up, they'll be doubling once per hour. And I think a naive, I mean not naive, but totally reasonable thing to think is that because the chemistry is so different in each of these flasks, the biology, I mean irrespective of the fact that they double once every hour, has to also be very different. You need to turn on this protein or that protein and there's very much a difference in what's going on mechanistically to process the nutrients in these two flasks. I mean, we'll be an analogy. That's clear? That's okay, yeah? Okay, but now what I'm saying is contrary to that expectation, they're indistinguishable, biologically indistinguishable cells, yeah? So what you then look at, for example, is mass per cell. The growth rate is once per hour, say? So that's a steady state growth rate. But by indistinguishable, I mean that if you took these cells out and you measured their size or you looked at them under the microscope, they would look the same, which maybe is not too surprising, but now you break them open and you count the RNA per cell, it's the same. DNA per cell, it's the same. Protein per cell, same. So by any large-scale measure, they're indistinguishable. So if I took some of these cells and I put them on the microscope and I gave you all the chemical breakdown at the level of DNA, RNA, protein, you wouldn't be able to tell me which flasks they came out of. Is that okay? Does that make better sense? Exactly. I mean, I think that's exactly the way to think of it, that the chemical composition of these cells is predominantly determined by the growth rate. So however you get there, it doesn't seem to matter all that much. And so here, what are we doing? Here's, say, square and circle would be right on top of each other. And so as I say, if I took one, you wouldn't be able to tell which flask it was growing. Does that make sense? Does that make better sense? Do you know what I'm saying? If this is still balanced growth, what I'm saying is that more or less what I'm saying is that the biology is very different from the chemistry. So if I gave you these two flasks, you could, or if I give you a little bit of this, the flask water as a chemist, you could tell me which flask it came out of. If I gave you a little bit of the bacteria as a biologist, you couldn't tell me which flask it came out of, unless you measured something more detailed than mass per cell, RNA per cell, DNA per cell. That's an important point. That underlies basically most of what we've done. So the idea is that the growth rate is a macro variable and everything is, or seems to be, a slave to that. But in very simple ways, like these empirical constraints. Any other, is that better? This? Yeah. Yes, that's right. So in that formalism, these two guys would have the same kappa n. Even though they have different chemistry, these would have what we call, we would say, same kappa n. That's a connection. Okay, so kappa n isn't a unique feature of the chemistry. No, I feel like you're unhappy with that. Is that okay? Let me know if it's not satisfactory in a second. Any other questions? I'll let them talk it out. It's okay? That means, or what we just saw, that they have the same kappa n of the square times 5p of the square is equal to kappa n of the circle times 5p of the circle, the growth. Yeah, exactly. Since they also are biologically the same, that also means that 5p of the square is the same as 5p of the circle, right? That's right. Because they're producing protein at the same rate also. Yes, exactly. So now here's what's important. That 5p is a mass fraction. So micrograms of p protein for total protein. But the makeup of those proteins are different. So the mass is the same, sure. But they're very different proteins that are, yeah? Does that help mix it together? Yeah. Okay, so let me keep going with his thought because it's very good. So i.e., this 5p square is equal to 5p circle. That is to say the micrograms of non-liposomal proteins are milligrams of metabolic proteins or milligrams of total protein. Those are the same. So the weight, the weight of the proteins. The identity of the proteins is very different. Well, not very, very different, but is different. Okay, so let me pause. This implies this, which implies this. Does that, is that chain of inference reasonable? Now let's go into interpretation. Interpretation is, but the identity of these metabolic proteins is not the same. It's not necessarily the same. The identity of individual metabolic proteins. This guy might have some certain enzymes that are on to chew up that carbon source. This guy might have different enzymes that are turned on to chew up this other carbon source. But then when it all shakes out, the mass of those proteins is the same. The mass fraction of those proteins is the same. Which is odd, I think. But it's a consequence of what we've seen. The fact that the doubling rate almost 100% determines the macromolecular composition is what leads to this. Is that better? Okay, so are you guys okay? Is that okay? Did you guys rectify it? It did. Oh, great. Okay, all right. Any other questions? Think about it. Let me know if anything feels uncomfortable. I mean, not, you know, it just sort of isn't sitting well with you. I'll do, I'll do one more example. So what time do we start? So I'll do one example and then, you know, let it stew and then we'll have a break and then we'll come back and do another set of examples. But I think the set of example or the first example that I want to do is a nice warm up to what, well, what we're going to do next, but also what we did previously. So warm down, I guess. All right. Any questions of this, about any of this before I erase it? All right, so that was loose end number two. Let's go to the first consequence that I want to talk about, which then ties into loose end number three. Let me draw it and then let's talk about it. So here was the, this electrical circuit analogy was that there's some fixed proteome fraction that's available to us to drive protein synthesis into supply amino acids. And that guy is Phimax. And then we can think of those two empirical relationships, well, that are buried inside those straight lines over there, as two resistors in series. One with a resistance one over Kappa t. So Kappa t is a conductivity of the resistor and one over Kappa n. And then we had this potential drop, which is Phi P, which I, I call Delta Phi P because I wanted to, to, to dry your, or maybe make the suggestion that it's like a drop across the resistor. Up here we had Delta Phi r, which is the, the protein fraction, but also in this analogy, the voltage drop, which was equal to Phi r minus Phi r min. So here I'm just thinking of the growth rate dependent part. And then this Phi max was equal to Phi r max minus Phi r min. Okay, so that was the scenario and that was the connection between this, this voltage analogy or this, this electrical circuit analogy and those constraints that we had previously. Before we talk about the first consequence, so we looked already at one set of consequences, which was, what if we had co-utilization of carbon sources? And we assume that they use totally different proteins, some set of proteins to chew up that carbon source and some set of proteins to chew up that carbon source. Well, what we could do is imagine that as parallel resistors. And then we can characterize the rest of the circuit having shorted out those two resistors and then individually with one resistor and then the other resistor. And then that would give us an expression for this, the circuit with these two resistors in parallel. So that's to say that we could look at the individual carbon sources and then bring that information together to get an expression for the co-utilized carbon sources. And what we saw, well, what I suggested with a cartoon drawing was that the experiments agree very well with that idea, with that conception, giving us at least some credibility to this picture. But what I want to talk about now is an even simpler example, which is what if we make a protein that the cell can't use for growth? Like insulin, for example, but we can just force a cell to make a protein by engineering it into its DNA that it can't possibly use for growth. It doesn't serve any metabolic function, it doesn't aid protein synthesis at all. It simply uses up available resources and ribosomes for its own ends and causes a growth defect. What would that look like in this picture? So that's what I want to talk about now. But before I do that, there were questions, did you want to ask, do you have any questions about this guy specifically? Okay, maybe it's better, we should talk one on one now, we'll go through it one on one. Are there any questions though about this electrical circuit analogy? So my hope with this is that it would act as a secondary analogy that would make some of the ideas a little bit more memorable or clear. If it didn't serve that purpose, ignore it. It's just another way to write the same set of constraints. So if it's not, what do you call it? Well, if it's not useful, just ignore it. I don't know if that made people happier or less happy. Okay, so the first example that I want to look at, so what I want to look at now is I want to use these constraints to explore the coupling between protein expression and growth rate. So protein, when I say protein expression, I mean synthesis of a particular protein and a growth rate. So we've looked at the, not the commerce, but maybe the inverse of that. We've looked at how growth rate affects some unregulated protein. Now I want to ask a different question. If I make a particular protein, what can I expect as an outcome on the growth rate? And I think the easiest thing to imagine is a protein that doesn't help growth, it's just a burden. So the first example is going to be useless protein synthesis. So here I'm going to denote by phi u, the mass fraction, so this is milligrams of useless protein, milligrams of total protein. And by useless, I mean not necessarily useless to us as humans, but definitely useless for the bacterium in the sense it does not contribute to the growth of the bacterium. It serves no metabolic function, it serves no protein synthetic function. It's just made because we want it made and not because the bacterium would ever want it made, growth of the bacterium. And the first, the first example of this, I mean historically, was people using E. coli to produce insulin. And you can imagine insulin, which is what we use to treat diabetes in humans, has absolutely no purpose in the bacterium. It's just made, it's a protein. Sure, it's made out of amino acid, sure, but it doesn't help the bacterium in any way. Now the question that I want to ask is how does the growth rate depend on this fraction of useless protein? So how does growth rate depend upon this useless protein fraction? I.e., what's this expression? If I tell you that I'm making 30% of my proteome is useless protein, can you then tell me what the growth rate of my bacterium is? Okay, that's the type of question I want to ask. Let me pause though. Before we get to the analytic, the expression and things like that, we'll come back to this, but is the scenario, does everybody see the scenario that I'm talking about here? Does the term useless protein, do you see what I mean by that? So it does not increase growth rate, it can only decrease growth rate. Or keep it the same, we don't know yet. I mean, could it be just like an E. coli would just like it out of DNA? Yes, you'd think, right? And so if all that was growth limiting was the replication rate of the DNA, then yes, then the only burden would be the small, say, seconds it would take to reproduce that piece of DNA. So you don't really know how to use the DNA in a pool, and all the other things. But assuming that they're already there, and it's just a variation in what protein you're making, but not really like, generally very different from what you've already made, you would expect that it's just an equal and slightly larger DNA. Yeah, that's a good, I mean, what would you think about this lambda? Then it would be this? I guess. No, it's a slightly larger DNA, so I would say it will be very much the same. Yeah, okay. So his suggestion I think is a perfectly reasonable suggestion, is that the only growth defect is going to be associated with the time it takes to reproduce the chromosome. And here's something weird about that. Is it, if you make the chromosome bigger, you just put junk in, or you, let's talk about it like that. So say you just put in junk that is not made into protein, it's just junk DNA bases that are replicated but never transcribed. The growth rate doesn't change. And the reason it doesn't change is because the cell to compensate for that just initiates its fork of replication earlier than it normally would. So the age of initiation shifts, but the growth rate doesn't. So for some weird, well, I'm making it sound weirder than it is, the DNA replication dynamics and the growth are intrinsically uncoupled, which is weird, okay? So your suggestion is valid and would even be true no matter how long a piece of DNA you stuck in. I wouldn't say that. Okay, yeah, but then it would just make itself bigger. Yeah, well, yeah, okay, I mean, there's probably, there's going to be a limit. In terms of the, say the density inside, some folks would argue that the volume fraction is growth limiting. But then there's, there may be other growth limiting factors that we can get our hands on, which I'll show you in a second. Any other questions or suggestions? Yeah. No. Oh, you're equalized. So what happens, I should tell you, this is, this is, this is outside of, of the constraints, but it's an empirical observation that I'll tell you about. If you do make this useless protein, the cells swell. So they just get bigger. Does that make sense? I mean, it doesn't need to make sense. What I mean is what I said is what I said sensible. So it's not, so they're not constrained by volume. They keep their density the same. And so to compensate for this useless protein, they're, they're bigger for whatever reason. The mechanisms on them. Now my question to you is, how does this pie chart change if you start to ram in a wedge of useless protein? And then what effect does that have on the growth rate? So how can we model this? And there are two complementary ways. So how, how can we, how can we, I don't want to say model, but yeah, let's say model, model this. And so there are two ways. One is, like I suggested just now, we can ram a wedge into this pie chart. And I'll tell you what I mean by that in a second. Variable resistor. Or we can make this electrical circuit analogy. We can think of a voltage drop external to the, to the original circuit. Where we force some potential drop across this resistor by fiddling with it, which is equal to this phi u. And so as I make more phi u, what do I get? I'm not so artistic. All right. As I make more and more and more phi u, and I can do that by changing the DNA, making it more and more transcribed or translated. The point is that what I'm allowed to read out is this phi u and the growth rate. So where the growth rate is a current through here. And now I want to plot these things. What should I see? So here's growth rate. And here is a useless protein. And suppose I start here. So this is my growth rate with no useless protein. And then either I have a system where I can add a chemical that increases the amount of this useless protein. Or I have just a bunch of different E. coli with different pieces of DNA jammed in that make different amounts of useless protein. And I now measure the growth rate along the vertical in this useless protein along the horizontal. What should I see? Monotonically decreasing. Monotonically decreasing. All right, more and more. So you drew it with your hand. Why did you draw it like that? Can anybody say anything more than monotonicity? The size of the DNA won't be the only difference. The size of the DNA is no constraint. The other thing is to produce the growth rate. Yeah, yeah, yeah, exactly, exactly. The other thing is RNA. Yeah? Yeah. So I would say it's a higher quality of DNA. All right. Okay. So remember that we had for this circuit, but also with the combination of these phenomenological laws. When we brought them together we had this expression. Where this thing was phi r max minus phi r min, which was this battery here. Now, suppose you start making this phi u, what can you say about that voltage drop? Exactly. So with useless protein, this is the sort of the first order suggestion is that you would have this. You don't change the translation rate for ribosome. Possibly. Possibly you do. But to the first approximation you say, what does this useless protein have to do with translation? What does it have to do with nutrient assimilation? Nothing. What it does do is take away resources that would otherwise be occupied making proteins, i.e. ribosomes, or supplying amino acids. Okay. And so then, if I pulled this apart, so let me write this as, so now I have phi max minus phi max over phi max by u over all of this stuff. I can rewrite that. I'll just pull everything out and I will have phi max over 1 over kappa t plus 1 over kappa n, 1 minus phi u over phi max. Did I do that right? Does everybody believe that? Yeah? Then this is my growth rate when phi u is zero. This is my say, nominal growth rate. So what can you tell me then about what you would expect for this growth rate? Linear. So the idea is, or the sort of the kind of parameter free estimate would be that you get a linear decrease and then what's this intercept, phi max. And now if I take a different growth rate, so say I'm now growing in a different nutrient environment, what's your expectation then? Oh, yes. You're not replacing those. Oh, to here. Experimentally? Is that what you mean? Yeah, no, your experiments. No, this is not DNA. This is just a protein it makes. So the DNA that encodes for this protein may only be, say, yeah, that's true. It's a genetically engineered E. coli. But so was this one, the one that I used for this experiment. So there I put in a piece of DNA that had a synthetic promoter that was making an enzyme that I could read out. So this one is as non-E. coli species as this one is. It's true. But I mean, the amount of differences is insignificant. No, you're unhappy. We should talk about it. Physiologically they're the same bacterium. So it's true genetically there is a difference of about 1,000 bases. What I mean is, I guess that's what I know. Yes, that's right. It belongs to the white E. coli. It's decreasing every time it produces more and more protein that E. coli doesn't produce. No, no, no. You only insert one set of instructions. It's about 1,000 bases long. And then that will take you all the way along this curve. By just adding different, what they call inducers, which turn on that make the transcription rate higher. No, well, you are able to chemically amplify the amount of transcripts off that piece of DNA. And that piece of DNA is jammed into a place where it's like you've got the regular DNA. Then you cut it open. But this break point isn't in the middle of a gene, for example. It's usually at a place where, oh, it doesn't matter. It's a phage insertion site. So it's non-damaging that part. You could insert junk DNA and it would have no effect on the growth rate. So this exactly makes a lot of protein. So getting down here is quite tricky because the cell, of course, really resents having to make half of this proteome garbage. Enable fight. I mean, it's under a huge selective pressure to lose that piece of DNA. But if we're able to, all of that sort of evolutionary stuff aside, where am I here? So now if I start growing it in a medium that supports slower growth rate, what do you think about the intercept? Exactly. By that it shouldn't matter. So no matter what your nominal growth rate is, they should all converge to the same intercept. And this intercept is, like I say, the difference between this intercept and that intercept in the FIR plot. That's the self-consistent picture. Now whether or not that's true, we have to look at experiments. Okay, but conceptually, is everyone on board with me conceptually? See what I'm saying? So I'm not tall enough to take advantage of this, but this was our first experiment, Night Heart of MagaSanic. And then we started doing all these acrobatics in terms of proteome partitioning. And we end up with this situation when we say, okay, if you believe this proteome partitioning picture, then production of a useless protein should result in a linear decrease with an intercept that agrees with this Night Heart of MagaSanic picture. Whether or not that's true, we need to investigate. So far so good? Okay, I'll show you the data and then let's take a break. And so the right-hand side is the data, and different colors or different growth rates corresponding to the dots over here. And the straight lines are not best line fits, as you can see. They are using that phi max that you get from the left-hand panel and making a completely parameter-free guess like we did here. And I hope you can see it's not bad. So I'm looking now at the top panel here. This top panel is meant to be this guess. And again, the straight lines are not best fits. They're just, all right, I know phi max more or less. And you can see it doesn't do a bad job. We do see a more or less monotonic decrease with growth rate, and they are more or less linear. Okay, it's hard to tell once you start to get to very high fractions of the proteome. So this is 30% of the proteome, or 30% of the protein in the cell is garbage. And the cells really resent that, like I say, and under tremendous selective pressure to not do that. And so this is not a really sustainable situation. They quickly evolved to not do this. But anyway, we can then go back into the literature and look at other people who have done a similar experiment and plot their results. And again, so where's the intercept? The just pure guess would be about here. And you can see it's not terrible. So these are very different proteins being made by very different systems, all in E. coli, but they all have this feature that decreases, yes, monotonic with growth rate, but more or less linear. And so at best when you make a garbage protein, you can expect a linear decrease. But if there's any additional toxicity of that protein, it's going to be more than linear. So you'll get some kind of higher order terms. But at best, you can't escape this. Does that make sense? This is like the Carnot cycle. Maximum efficiency, if you like. You can't ever expect less of a growth burden than this. Based on this whole proteome partitioning picture. So let me say that really quickly and then let's break. So I'll say it in words. I've run out of chalk. So the idea is this is the best you can do. And probably reality is going to be somewhere down here. Especially at high production rates. But let me come back to any questions about this. What would you call it? This set of ideas. This consequence of this proteome picture. Anyone have any questions about the relationship between this pie chart and this electrical circuit? Another way to think of it is that this over expressed or useless protein is just like taking a battery and flipping it around and jamming it into your circuit. So it's a negative voltage drop if you like. Sucking up potential from this battery which would be otherwise used for good into just garbage. Fittering it away. I think this takes some thinking. This one is sort of inevitable. You just jam in another wedge. These wedges have to get smaller. But we already said that the growth rate is proportional to the area of these wedges. It's proportional to kappa t times delta phi r which is equal to kappa n delta phi p. And so if you make these guys smaller you necessarily make the growth rate smaller. Let me pause though. Any questions about any of that? Think about it. Let it stew. Maybe let's break till 12.30. It's okay. I'll see you guys shortly. Hi. So let me come back then so the suggestion is that if this protein is useless but non-toxic the best you can hope for is this linear decrease in growth rate because making a useless protein necessarily squeezes out useful protein. And we have this constraint that the sum of the fractions of useless and useful is equal to some fixed numbers. 0.5 is the case maybe. So the more of this you make or force yourself to make the less it can make of these and consequently the slower it will grow. Does that make sense to everybody? And again this is an ideal situation. At best you can hope for a linear decrease. In reality it's likely that you'll see a much more acute growth rate change something like maybe like that especially at very low growth rates where the cell is more or less breaking the pieces. Okay. So but again this is very much like in thermodynamics where we have these estimates of perfect efficiency of heat engines or something like that. This is the best we can hope for from an E. coli. Let me pause. Any questions about that? And we get it for free, right? I mean we got it from the night-heart-magazinic plots basically. Okay let me pause. Is the conception okay? All right. So now I want to go even more I think sort of to a higher level of self-consistency. So what we initially talked about was this connection that as we change the growth rate we change the expression of different kinds of proteins ribosome associated proteins, metabolic proteins and so on. We just now saw the case where forget about this what if we force it to make a particular protein? How does that feedback on the growth rate? Okay how does that change the growth rate? Now I want to bring both of them together. And so now I'm not thinking of a protein that is neutral i.e. doesn't affect the growth rate. I mean that isn't in and of itself toxic or beneficial. I'm thinking of a protein that specifically changes the growth rate. So what happens if the protein I am making is not neutral, useless, but rather itself affects growth rate. And I think perhaps one of the simplest examples would be a protein that confers antibiotic resistance. Again I'll show you what I mean by that. So now what I'm thinking in contrast to the picture that I showed you earlier was not some passive wedge that just takes up space in our pie chart but something now that we need to be more we need to think more about where this affects this and this affects that and then we ask what's going to be the end result. And I'll be more specific as we go obviously. So let me tell you what I mean by this antibiotic resistance and then let's talk about, oh good, I see the attendance sheet being passed around because I forget it every single time. Antibiotic resistance. And this will be useful again on tomorrow's lecture when we talk about this particular set of antibiotics. But imagine an antibiotic that targets the ribosome, let's say targets protein synthesis. So here's my antibiotic and then here's my ribosome. And what I'm imagining is that this antibiotic either reversibly or irreversibly, I don't really care at this stage, it binds to my ribosome and renders it non-functioning. So that's the scenario that I'm going to talk about more tomorrow. But now I'm imagining that in addition to this antibiotic mode of action we have a protein being made by the cell that destroys this antibiotic. So this is the antibiotic mode of action and then on top of that I'm imagining a cell that has some resistance to this antibiotic in the form of some protein. So here I've got some protein X, which is a resistance protein. Here I've got the antibiotic. And now here I'm not, I don't really care what the mode of resistance is and there are typically two big groups, one where the protein in question pumps the antibiotic out of the cell or another group that just chemically breaks the antibiotic. So here what I'm imagining is something, this is now an ineffective antibiotic and then usually this X is just recycled. So usually X is an enzyme that chemically modifies the antibiotic so that it's useless. Does everybody see this scenario if I didn't make any spelling mistakes? So this is the mode of resistance. So far so good. So I have an antibiotic that I'm adding as a human. I have a protein that the bacterium is making to try and fight me to thwart my efforts. And now I want to ask how is this protein related to the growth rate of the bacterium? Okay. And on the face of it this is not a difficult question but I want to show you that there's something subtle going on here. All right, but does everybody feel comfortable with the scenario that I've just drawn? It's okay. So we can characterize these reactions what's called in vitro. That's to say we can do it on a chemistry bench. And if I look at all the players here, so I've got resistance protein, I've got the antibiotic, translation rate, cap of tea, and I have the growth rate. So these are the major players that I want to call your attention to. And the interaction among all these players we know. So for example, this translation rate has a positive correlation with growth rate because the growth rate is this by max 1 over cap of tea plus 1 over cap of n. If you make this translation rate higher, the cells grow faster. If you make it lower, they grow slower. Of course, there's a maximum which is when there's no antibiotic they're growing as fast as possible. The KT is as high as possible. But if you make this KT, this translation rate smaller, they will grow slower. Does that make sense? And so I'm going to use this pointy arrow to denote a positive correlation which means if this goes up, this goes up. Or if this goes down, this goes down. I'm never going to this point because I'm going to start adding connections and I don't want the logic to get lost. Is that sensible? In control engineering, these types of interaction diagrams are fairly ubiquitous, but you may not have come across them. So we'll take them slowly. Here, what do you expect? A positive or a negative correlation? Negative. So what I'm going to do is put a blunt arrow. This means negative. And if you want some analytic scenario here, you could think of KT being equal to some KT max and then one plus the antibiotic concentration. So the more of this antibiotic you have, the smaller the translation rate. And this KD is just some non-dimensionalizing parameter that tells you how sensitive your bacterium is to that antibiotic. And you could run this experiment in a test tube. You've got a bunch of ribosomes, you've got a bunch of antibiotic, and you just measure chemically how strongly they interact with one another. That's what you do in a chemistry lab. This is just by definition the antibiotic, the higher you make it, the smaller the translation rate. That's how this antibiotic works. And then here, of course, we have a positive interaction, or sorry, a negative interaction. So the more of this resistance protein you have, the less of this antibiotic you have. So if you want a sort of analytic description of that, maybe something like this A is equal to some A external divided by one plus some, let's call it, alpha x to the square, I'd say. And so A external is the amount of antibiotic outside of my cell. And so this protein either chews up the antibiotic or it pumps it out. I don't really care. Either way, the more of this I have, the less antibiotic I have inside of my cell. So far so good. And now if we look at this interaction map, it's not surprising. You have two negatives make a positive, and so you end up with an overall positive interaction. The more of this x you have, the faster you grow. Not at all surprising, right? Let me draw it. So I have this growth rate. Leave some space here. Growth rate. And here I'm going to normalize it by the antibiotic-free growth rate. There's one. Here's zero. And then this is going to be the efficacy or the efficiency of the resistance in units. Again, this alpha is just a non-dimensionalizing parameter that tells you how good your protein is. If it's very good at its job, then you don't need much of it to get rid of a lot of antibiotic. If it's very bad at its job, then you need lots of it to get the same effect. But in either case, alpha times x has no dimensions, and it's a measure of how good my protein is or how much of the protein I need. And based on that wiring diagram that I have there, we'll see something like this. Does that make sense? I promise you there's something strange that's going to happen in a minute. This should be sort of self-evident, and you should maybe be saying to yourself, why are you even bothering with this? Now, we'll draw your attention to two more things. One is this is what we call the phenotype or the outcome, the observable characteristics. So this is what's sometimes called a phenotype. That is to say what we observe in a bacterium, how fast it grows. And this is what we call a genotype. This we can change by fiddling with the DNA. So this is now what's called a genotype. And if you're thinking about evolution, then what you're thinking about in this scenario is the slope of this line. So the slope here tells you how much of a difference a small genetic change is going to make. So this is the, if you like, the fitness advantage of a mutation. It's saying a small change in the DNA, if that slope is steep, a small change in the DNA gives you a big change in the growth rate. And so if you're thinking about modeling antibiotic resistance and evolution and things like that, you would try to make it so that this slope was as shallow as possible. So that you can get, you would take many, many, many, many mutations before you had any appreciable change in the growth rate. Does that make sense to everyone? What's missing? So this is what I would call open loop. You say, this guy goes to this guy, goes to this guy, goes to this guy. But there's something missing. We know that this guy changes this guy. So imagine now that this resistance protein just arose in the cell. There's no particular regulation that's involved here. It's just a protein that maybe was doing something else, but now is being used as an antibiotic resistance. He's got no regulation. So suppose this guy is unregulated, as we called it before, constitutive. This is a simple as possible scenario. And if we're thinking evolutionarily, that would be where everything started, with an unregulated protein. Having said that then, this missing link along the left vertical side, what should its logic be? Positive or negative? So what do you think about it? Let me go previous. There we go. So this link here, positive, positive or negative? Positive. Positive. Who says positive? Who says positive and why? Why do you say positive? Anybody? Okay. But you guys would say positive. Why do you say positive? It's okay, it's okay. Where on this diagram are you looking? If you're going to look at this diagram, this one is the constitutive proteins. Which lines would you be looking at, the black or the, sorry, the kind of the pink line? No. Okay. The invisible line, it goes like this, or the colored lines? And why? So think of it that way, and then I'll come back to you then. Why do you say negative? Yes. Okay. Okay. So you're thinking two steps ahead. So then empirically, though, what does this picture suggest, positive or negative correlation and why? Anybody? Is my question answerable? I mean, does it make sense, is what I mean? Positive. Positive. You say positive, why? Exactly. So he's looking here along a colored line, which says that, and why he's doing that is because we're in a situation where we have translational inhibition. We have an antibiotic acting, which is these numbers. So we are at, for example, eight. And now we say, what happens if the growth rate goes up? If the growth rate goes up, this protein abundance also goes up. This line has positive slope under those conditions. So under conditions of, let me write that out and let's talk about it, because it's important. Both points are important. So his point and that point over there. So here, under conditions of translational inhibition, this growth rate is positively correlated with X. That's not to say that X, necessarily, well, it does in this case. X is going to increase the growth rate. But it also means that when the growth rate increases, X increases necessarily. So everybody see that? So that if we relieve some of that translational inhibition, well, then we start to make more constitutive proteins, which means we make more X, which means we relieve more of the antibiotic, which means we grow faster, which means we make more X and so on. Before we get to that loop, that and so on, does everybody believe this line? This comes from the plot that we had where we had this 5P. We had growth rate. We had this. And then we had these dashed lines under translational inhibition. Now, the point here is that these dashed lines have positive slope. These dashed lines have positive slope. So far, so good. Okay, so this guy was the open loop. This guy that I'm about to draw is the closed loop. So as was suggested, all right, pause here actually. Does everybody see why we need this vertical line? So two weeks ago, I would have said there's no need for it. But now that we've had this course, you see that it's inevitable. There has to be this other link. So this link is never included in these mathematical models. But anyway, we know to include it. But then if you do, what happens? Well, then we have two negatives, make a positive, makes a positive, makes a positive. And we have what's called a positive feedback loop. And you've seen this. I mean, you've seen this when somebody's trying to speak. Usually it's at a high school like gymnasium. And the speaker is picking up with this, or sorry, the microphone is picking up with the speakers putting out. And so they say, all right, everybody. And then they say, everybody gets confused and they unplug everything. So if you take a speaker and you point the microphone at the speaker, you get a positive feedback loop. You get production with amplification. And so just like that system, we have a danger of by-stability, which is that this thing is quiet until someone coughs, and then suddenly it flips to the high feedback line. So we have two possible states, silence and maximal moot out of the speaker. Does everybody know what I'm talking about? If you've never experienced it, at least hopefully you can imagine it. It's horrible and it happens all the time. And it inevitably happens in high school. All right. And so with a closed loop, we have a feedback system. And so now instead of this alpha x, we would just have this efficacy alpha beta. So again, it's efficacy. Again, we have the growth rate. But now instead of this smooth monotonically increasing curve, we can have situations like this, where this is a regime of what's called by-stability. Let me put it out like this, which means that there are multiple fixed points in our system. A low growth rate, low resistance x-point, and a high x-high growth rate point. So this in and of itself isn't terrible. It just means that we can expect in a certain range of antibiotics and efficacy of the resistance, subpopulations, some that are growing fast, some that are growing slow. So we have a heterogeneous distribution of growth rates in our population. What's horrible is right here, where we lose the heterogeneity. Now instead of some slope, however steep it might be, we have an infinite slope. A small, tiny change in the DNA gives you an infinite advantage, an infinite selective advantage. You jump from some low growth rate to almost maximal growth rate. And so it turns out that a map like this, versus a map like this, this type of map between your DNA and your growth rate facilitates evolution of resistance, many, many more, you know, thousands of fold more than this type of view. And so if we bring in the physiology, we end up with this extra cooperativity which facilitates evolution. Okay, so that's one hand, something I wanted to talk about, but on the other hand, whenever we talk about proteins changing growth rate, we need to remember that growth rate changes proteins. All right, and so what I want to talk about tomorrow then is some more examples of this type of feedback, and then the case of antibiotics targeting ribosomes but without a resistance mechanism. Okay, but before I do that, are there any questions about this scenario? So I think the top curve is almost self-evident to the extent that someone might be resentful that I even put it up there. But the bottom curve is surprising, and it's surprising for the reasons that he suggested that when you have a loop like this that reinforces itself, you can at the beginning have nothing, and then all of a sudden flip up to lots, saturation. That's a feature of these positive feedback systems. Okay, and again, the feedback you would never see in the DNA of the organism, you could do all the chemistry in the world and you would never find this loop. This loop comes from the biology, from the physiology, comes from remembering that these cells are growing. All right, let me pause. Any questions? It's okay? All right, think about it, and then let's talk about it tomorrow. All right, have a good lunch.