 Thanks. Thanks a lot. I'm really glad to be here. I don't know if I have any... Do I need to worry about the microphone or anything? It's good. It's good. Okay. Thanks for coming. So, let me tell you a little bit about bacterial physiology this morning. The the the thing that I find most compelling about about physiology in general is is both experimentally and theoretically is we take a bird's-eye view of the phenomena. I mean biological systems are undoubtedly complex systems and sometimes overwhelmingly so. So taking this macroscopic view affords us some clarity of vision in terms of formulating the constraints that are operating the system. And so there's very much is this analogy between say statistical mechanics on the one hand where you have these mechanistic microscopic interactions and thermodynamics on the other where we have these macroscopic hydrodynamic constraints that are operating in the system. And the power that's afforded to this say thermodynamic view is that you can say what's possible and what's not. As I say, we know what the constraints are. And one of the great challenges in biology is that both of these viewpoints in physics, of course, we had this nice ordering with the thermodynamics for a few centuries before we started looking at the statistical mechanics. And so we got to climatize to the systems. In biology these arose almost simultaneously and then statistical mechanics overwhelmingly took over the research field. There's there's some effort to restore that balance in the last decade or so and I'll talk about that in the second half of this course, Larson. But for today, what I want to talk about, well first let me tell you what I mean by physiology and then for this morning we'll talk about what I mean by bacteria. Okay, so I think probably usefully you can think of physiology as constraints operating on living systems. And by constraints here I mean intrinsic constraints, something like the second law of thermodynamics or something like this, something you can't avoid. If the if the system is living then it must obey these types of laws. And now for the bacterial. So for the bacterial what I want to talk about this morning is some of the biology that we'll need as we proceed. But again, one of the aspects that I find so compelling about this subject is that we need a minimum of the biology. He'll tell you about the machinery that goes on inside of bacteria and maybe even more specific inside E. Coli bacteria, which will be our model system and it'll take maybe 20 minutes or so. But of course that means I go. So for for this morning, we'll look at the biology at the bacterial biology and specifically E. Coli. And so E. Coli is a typical model system for these types of studies because we know so much about it. For the last 60 years, 70 years, people have been studying every aspect of it. And so if we look at the thermodynamics, we have corresponding statistical mechanics information for virtually anything you want to know. Does it eat maltose? How does it eat maltose? Things like this. And that will become useful later on when we ask where do these constraints arise from? But of course let me tell you how they how they grow. So E. Coli is a is a bacterium that divides and grows exponentially. And so from the outside it looks like this. So how low can I go? Mateo, can you see is this as low as I should go? It's okay. Okay. I'll go up here. So from the outside the bacterium is somewhere between about, you know, one to three micrometers long. It's about a micrometer micrometer wide. But we'll see that that depends very much on the context within which it's growing. And it's a cylinder and it usually just grows lengthwise and it cuts down the middle. And what's important for us is that it's got a circular chromosome which contains the DNA of the organism. And the DNA is what serves essentially as a blueprint for the life cycle of this bacterium. Of course we'll go into more details as we go, but I want to just take a very bird's eye view of what's going on with these bacteria. So what happens is that the bacterium starts growing, elongating. This circular chromosome starts replicating. And so it will then go to a situation like this. So DNA replicates and then it divides and then keeps going. And it'll do that forever if you give it enough food. It keeps growing, elongating, replicating, dividing. So this is now cell division. And going on inside is for the large part protein synthesis. And I'll talk about that in a second. It's okay. Okay. And so that's from the outside. And from the inside we have is the DNA of the organism. And this is, as I say, where all the information is encoded for how the proteins are going to be made in the cell. And the proteins are what drive all of the reactions, catalyze all of the reactions that are going on inside this little sack of chemistry. And so this DNA is read by a molecular motor called the RNA polymerase. And this part doesn't matter. If you forget that it's called an RNA polymerase, it doesn't matter. The point here is that DNA is turned into an auxiliary molecule, a helper molecule called RNA. So this is a second type of molecule that's related to DNA. And this act is called transcription. So I'll put a little star here. That's transcription. And then that RNA is turned into protein. And as I say, that's what actively catalyzes most of the reactions in the cell. Let's say all of the reactions. So then there's a second molecular machine called a ribosome, ribosome that turns this RNA or reads this RNA as a template for, excuse me, polymerizing amino acids into protein. So this is protein. And this act, put another star here, is translation. And for now, that's all the biology that we'll need. We have these two events, transcription and translation. And for the most part, we'll be completely focused on, is the attendance sheet going around? I forgot to pass it on. But I think it's already in transit. Is that true? No? The attendance sheet? Oh, it's going. Okay. Thanks. So for the most part, this is all the biology that we'll need. And we'll be focused on this reaction here, this translation. Okay. And so let me say one more thing and then let's pause for some questions here. One last thing that I think is useful for thinking about the system is how these machines are made, specifically how this ribosome is made. So this is a remark and an important one. These ribosomes are made of RNA, which is called ribosomal RNA and protein, which is called R protein. So the machine that does this job is itself made of the product that it produces. And so we have an intrinsic positive feedback loop, if you like, that lies at the core of this living system. And we'll exploit that viewpoint. And so for us, at least for, well, no, probably for the duration of this course, it's useful to think of the bacterium as a sac of ribosomes that make more ribosomes. And then everything else that it does, it does in aid of that reaction. And of course, I'm going to be more specific as we go on just thinking this is in broad strokes. But let me pause. So, so again, this is really, I mean, go through the lectures, this is, yeah, this is more or less all the biology that we'll need. Okay, so it's worth pausing here. Are there any questions about, about anything that's going on on the outside? I mean, again, we'll talk about this stuff in more detail, I mean, quantitatively, but at least in cartoons, we have the DNA is all stored as a loop. That loop initiates replication at one spot and then kind of unzips, replicates into two chromosomes, cell divides, and then the whole process continues. Here is a close-up of what's going on, on this chromosome, that the DNA is being turned into RNA, which is a template or an instruction set for how to make proteins. And it serves as a, as a template for polymerizing amino acids into protein. And maybe it's worth drawing that out. And then I'll pause. So this is translation in more detail. You have the RNA. You have this ribosome. And then you have protein, which is made out of these amino acids. They have different identities, but they're strung together like a necklace. And the, the sequence of amino acids tells you the protein structure or dictates a protein structure. It's very hard to tell what the structure is going to be, but dictates how these proteins fold. And then consequently, what their catalytic function is. Okay, so this is the ribosome catalyzes polymerization of amino acids. And so by catalyze, I mean facilitates, makes it very fast and very easy. And by polymerized, I mean sticks together, irreversibly, covalently bonded. So it takes these free amino acids, strings them together in a necklace that forms a protein. And then this protein itself, what we call an enzyme, catalyzes further reactions in the cell. For now, this is the, this is a reaction that I'm mostly interested in. Okay, so again, let me pause. Any questions about? Yes, sure. So this, this RNA is, is a string of mucleic things. Okay. So the, so the question is, what's the, what's this picture saying, which is totally legitimate. It doesn't make any sense like this. So the RNA, the RNA, and so now this is the difficulty. I can't remember what the code is. But anyway, there is a code. So the RNA is made up of bases like U, A, G, T, no, not T, C, G, G, A, something like this. Okay, these are, these are themselves chemicals that are called nucleotides. And it's this long string. And one of the miracles of biology is that, that this string encodes for 20 different amino acids. Again, I'm not sure exactly what, how it codes, but these are in triplets. And so three bases, which are called codons, encode for one unique amino acid, encode one amino acid. And some of these bases, these triplets also encode syntax like start here, finish here. And so as the, as the ribosome reads this molecule, it looks out into the wide world and it assembles amino acids that correspond to this codon sequence. So this one might encode for say, that amino acid. And then this one would encode for that. And so it gets to one, reads the codon, looks for its, its cognate amino acid, sticks it onto the chain. And so it's very much an assembly instruction. Okay. And so this is the instruction set. This is the product. And then this is a machinery that assembles that product, if you like. Are there any other questions? So we'll talk about that reaction in more detail, probably toward the end of this week. Okay. Terrific. So this picture that I, that I, or this say schematic of bacterial growth is not very old. So it's probably at most about 50 years old. It's the picture emerged and evolved from about 1950 up to about say the mid-70s. All right. And so what I want to talk about for the first, say, three or four lectures is how we got this picture. I think it's a lot easier to see where it comes from in hindsight for the, for the poor scientists who were there in the middle, who had no idea what was going on and had to formulate these, these ideas out of nothingness. But with this, as our background, we'll go back and look at some of the historic experiments and see where this, this picture comes from. Because in the 1950s, I mean, the technology just wasn't there to observe, I mean, single molecules in real time, for example, even single cells in real time. So there's a lot of really deep quantitative mathematical thinking that goes on behind the, this picture. Okay. So what I'll do today this morning is look at, at a historical paper by Manode, Jacques Manode is a famous biologist. He won the Nobel Prize for figuring out how enzymes, which are these proteins, are regulated in the cell, how the cell knows how to turn them on or turn them off. But he also did a lot of work on bacterial physiology. So we'll look at his review from 1949. Let me write some of this stuff up. So how did this picture emerge? And I think without hyperbole, it's some amazing experiments from about, say, 1940 to 1970, in particular, a golden decade from 1958 to 1968. And so we will focus this afternoon and, and Tuesday's lecture on this golden period, 1950 to 1968. But today at least we'll start with the preamble, what was known at the time through the state of the art, 1949. Okay, we'll start with Manode, 1949, a review called growth of bacterial cultures. It's, of course, 1950, so almost 70 years old, but it has a lot of really great science in it, and it really sets the stage for what we want to talk about. So this review came out in 1949. It's largely a review of what Manode did for his PhD thesis work during the war. And it's well worth reading even now. So it's posted on the webpage. It has the lecture notes and things like that. And so that brings me to another point. If you have any trouble accessing that page, let me know. So if you've tried to get the lecture materials for this course and you failed, let me know after the lecture and we'll figure it out. All right, so what Manode did was he went through the, what was known at that time. And so people had been growing, scientifically growing bacteria for about 100 years by this point, but they really didn't have a good sense of what was going on. And that didn't change with this review. I mean, you'll see in a second that there's glimmers, that there might be some deeper order or some thermodynamic description at hand, but it's not quite there yet. Okay. And so before we talk about this, let me talk about how people grow bacteria. Okay. So first, two ways to grow bacteria. Yeah, it's really not anything else you can do. So one way is called a batch culture. And that's what we'll focus on in this course. So the idea here is you, I can't remember how to spell inoculate, but you put bacteria into a, into a growth medium. So something that they like to eat. And we'll talk about that in more detail in a second into a growth medium in a test tube or a flask. And then you keep it at a constant temperature. Right. So, and you give it whatever it needs to survive. So if it's, if it's what's called an anaerobic bacteria, so a bacterium that cannot deal with oxygen, of course you, you keep it away from oxygen. If it's an aerobic bacterium, you feed it oxygen. And so for the rest of what I'm going to write here, I'm thinking specifically of E. coli. And E. coli lives in your gut. And so it, it likes 37 degrees and it likes air. I mean, it can handle not having air, but it prefers having air. So from here down, this is general, but from here down, or from here down, I'll be talking specifically about E. coli. So for E. coli, we set it at, this is 37 degrees centigrade and you shake it like crazy. So you put it in a water bath, it's at 37 degrees and you shake it back and forth like crazy. And the idea is that you have turbulent mixing in your test tube that brings in oxygen. If you're very fancy, you would bubble oxygen through. And so it depends on how fancy you want to be. It typically doesn't make very much difference from where the other, but the point is you need to keep them constantly aerated and shake like crazy, shake to aerate. That means supply with oxygen. Okay. And, and if, if you're doing this, depending on what you feed the bacterium, so depending on what kind of, for example, carbon source you give it or nitrogen source, E. coli will grow very, very quickly. So it will double every 20 minutes. So you get about an order of magnitude increase every hour or so. It's incredible. Or you can make it grow every, you know, it'll double every 10 hours or maybe every 20 hours. If it's in your gut, it doubles say about every day, 24 hour doublings. Okay. And so you feed it a carbon source and a nitrogen source. And broadly speaking, the carbon source supplies it with energy, but it also supplies it with carbon. And the nitrogen source is important for making proteins. So when I call those, those pieces, amino acids, the amino means nitrogen containing. Okay. And so this is largely for protein. This is largely for energy. And then the third piece that E. coli needs is a buffer. And so buffer just keeps the, the acidity of the medium constant. And the problem with, with not doing that is that E. coli generates a lot of waste and it just excretes it into the medium. And if you don't have a buffer, it's eventually going to make the, the test tube so acidic it kills the bacteria. Right. And this is something you know this from yeast. The yeast will, will start to ferment sugars into alcohol to such an extent that it kills all the yeast. And so that's what gives you beer and dead yeast. So we put carbon source, nitrogen source, and a buffer at minimum. You can add extra stuff and it'll grow faster and faster. But my point here is that E. coli is very cheap and very easy to grow. And that's part of its appeal as a model organism. So we have nitrogen and buffer and it will double E. coli will double say once per 20 minutes down to, you know, however slowly you want to, you can basically make it double however long you want down to say double every several hours. You have a huge dynamic range. Yeah, exactly if it builds up. Oh, they definitely, it becomes more and more. So the question is, as the waste builds up in the medium, will they get, will they slow down their growth? They will start to feel it starts to inhibit their growth. And so we avoid that experimentally. And I'll show you well, let me give you a preview. So what you do then is you dilute the samples. So you have a buffer, which is a chemical system inside the test tube that absorbs that waste and converts it into something that's not acidic. And so that'll help you for a while. But eventually, the waste is going to build up, the cell density is going to become too high, and the cells are going to slow down. And so you'll get, I'll show you in a moment what you get, but it's something like this logistic growth that you might have seen in differential equations. So exponential for a while and then it sort of peters off. The way that we avoid that is we just dilute it into fresh media. And you can keep it exponentially growing forever. I mean, as long as you feel like diluting it. Does that make sense? Yep. Any other questions? That's good. Okay. Ah, okay. And so I was, I started talking about this because I want to contrast two ways of growing. This way of growing seems perhaps the simplest. And in this way, the bacterium sets its own growth rate. And that statement may not make sense until I tell you about the second way of growing these bacteria. So in this mode, the bacterium, its own, and I want to contrast that with something that's called continuous culture or chemostat growth. But this at least chemically or sort of scientifically is easiest. You mix up some recipe and put it into a test tube. It's just chemicals off the shelf, stir it up, add some bacteria, and then let it go. And then so it's just growing, growing, growing, growing. And they, and they determine how quickly they're going to grow based on how they process and nutrients that you feed them. And so for E. Coli, if you feed it glucose, it will grow very quickly. If you feed it glycerol or some other carbon source, it will grow more slowly. And so that's how you toggle its growth rate. So I want to contrast that with continuous culture in a second, but is that sensible? If not in detail, at least in spirit, we'll talk in more detail as we go. All right. So contrast that with continuous growth, continuous culture. So this, this is the way that we'll be talking about for all the experiments that we'll consider in this course. They'll be grown in this way. But Menod talks about the second method, which was just coming onto the scene in 1949, which is continuous culture, sometimes called chemostat or turbidostat, turbido stat. And so this is something you might also have seen in the differential equations course. We have this big mixing tank or a small mixing tank where you have media coming in here. So and then you have media and cells coming out here. And then you have a stir bar here that's whizzing around this. Sorry, this is inflow. There's stuff dripping in and dripping out. Okay. So you drip, you have some inflow outflow rate to keep the volume constant and you keep the, the bacterium in this tank in a constant state of starvation and you feed them just a little bit to keep them growing. And so their growth rate in this tank is determined by how much food you have dripping in per unit time. So the flow rate essentially of your, of your chemostat. And so they grow more slowly than they would if you had saturating levels of nutrient in a test tube and you control that. So in this, this batch culture mode, it's more like discreet. You choose this chemical, this chemical, this chemical, put them in a test tube and set the system going. Here it's like a dimmer switch. You can control the flow in and out of this chemostat and control the growth of the bacterium. So here keep the cells in a continuous state of starvation and control their growth rate by the flow rate of some growth limiting nutrient. It's usually glucose or some other essential nutrient by the flow rate of some growth limiting nutrient. So when this first came out, people were very excited because again, it gives you this literal dial for changing the growth rate of the bacterium. It's not so obvious that this is a natural way to grow bacteria. As I say, they're starving the whole time and it's not clear that it's not just some small pocket of bacteria that gets hit by this drop of nutrient, grows very fast and then everybody else just doesn't grow. It's not obvious what's going on inside these chemostats. So we won't talk about this. We'll talk about the other mode of growth. Let me pause though. So we have these two very different ways of growing. One, we again add all the chemistry to the test tube. We let the bacteria do their own thing. In the other, we set it up so that the bacterium can't grow unless we drip in something that they need to eat and we drip that in so slowly that we control how quickly they grow. These are the two modes of growth and you very rarely see any sort of combinations or deviations. And as I say, we'll talk about the batch culture mode for this course. Let me pause though. Any questions? Okay, all right. So now let's come back to the question that she asked about how these cells actually grow. All right. And so Manoud looked at what happens when you first inoculate a test tube with bacteria and then you let the bacteria grow. There was precedence for this. I mean, people have been looking at this for, by this point, about 50 years, considered what he called the bacterial growth cycle, bacterial growth curve. And so what I'm going to do is plot the natural logarithm of the number of bacteria that are inside your test tube. So here I have log, it doesn't really matter. I'll use a log-based tube, number of cells. I will plot this as a function of time. And what you see initially, so I have a clean test tube. It has a bunch of chemicals that the cells need to grow. I add a little bit of bacteria. And then what happens? For a while, nothing. So the cell numbers just stay the same. And then you have a very brief period where the growth rate changes. And I'll talk about what the growth rate means in a second. And then you have what's called exponential growth. And again, I'll show you that in a second. And then you have a deceleration. You have nothing happening. You have another change where often get exponential death. So this is looking at the number of cells. And what you can do is look at the derivative of this curve. So take the d by dt. And just so that this guy lines up, here I'm taking the d by dt of the log 2, the number of cells. Here's 0. Here's plus. Here's minus. Good point. So let me write it. And then I'll tell you the two ways. And so here you'll have no growth. Then it will go here. It goes up and positive, up and zero, up and negative. All right. And so let me, let me first, no, you know what? So first let me identify or answer Matan's question. So, okay. So actually first let me come back to your question. So this is, this is that logistic growth. So you have nothing going on while this is not logistic growth yet. This is a logistic growth. So you have exponential growth here. And then you have saturation, stationarity. And so let me quickly annotate this. This thing is called the lag phase. This is called the exponential chill phase. And this is sometimes called the stationary phase. And this is called the lag time. This is called the growth rate. So I'll, I'll circle the concepts. And this distance here between the top and the bottom of the curve is called the total growth. Exactly. So the question is, what dictates this lag time? And, and that's what, to, to an extent, what my node one is Nobel Prize for. It depends what the bacterium had been growing in before you put them into this new food. And so if you're growing them, say, exponentially in one nutrient source, say some, some carbon source, and then you move it to another, it's the, the time it takes for it to reprogram its, its proteins to be able to eat that other carbon source. Okay. So that, that one's easiest one to understand going from exponential growth to exponential growth. You have a waiting time. This one, it's less clear because for one thing, we don't know what happened the night before, you know, before this experiment happened. And, but at this point, people didn't know that. But you're exactly right. You just look at this, you say, what, what's going on with this? And we'll talk about why this is a problematic picture in a second. But this is how people were thinking of it in 1949. But you're asking the perfect quantitative questions. Mateo, I will come back. I promise I'll come back to the same question. Any other questions about this annotated? And so what we'll do is look. So I'll say a couple more words about the lag time, but we will focus primarily on the growth rate. And I'll say a couple words about the total growth. These are two things that Minode was, was most noted for at this period. So in 1949, coming back to Mateo's question, how do you measure, you determine the number of cells? And so there are two ways of doing this. The way that, that is perhaps the most proper is counting viable colonies. So this is the most laborious, but it, it tells you really what you're, what you're looking at. And so the idea is the following. You, you take a Petri dish, so it doesn't have to be Petri dish. You take a container. So this is some dish and you add to it some jelly, some agar. So in the very, very olden days, when, when Robert Koch came up with this plan, he noticed that bacteria would grow in colonies on a sliced potato. And so for a long time, people were using slices of potato, which weren't just fine. But now if you, I mean, nobody wants to go into a lab and see a bunch of sliced potatoes. It looks very fishy. But here, what we do now is we make an agar, a jello broth. So this is, you fill it with, with agar jelly, which is like a gel and some nutrients. And so, and usually what's done is you use, what's called yeast extract. So and the reason is yeast extract is because if you grind up yeast, it has all the nutrients that, that bacteria enjoy eating. And it's incredibly cheap. So to make these dishes is not very expensive. And so now what you have is a solid dish. So if you like, you can think of it as a sliced potato or a piece of bread. And then you pour some of your, your bacteria onto here. So you've got your test tube. You sample some of it. So you've got a, what's called a pipette. So here's your bacteria. Here's your sample. You put it here. And you dilute it about say a million times. So dilute a sample about 10 to the six times. And you spread it out on the dish. And then you incubate the dish at 37 degrees. Then incubate, you will get little colonies. So incubate at 37 degrees for say, you know, 12 hours. What happens is that every individual in that initial sample that you put down forms its own civilization, basically, on this solid agar. And they're spatially isolated from one another because there's no mixing. It's just a jelly slab. And so it's very much like, like I said, a potato, but we usually don't let those rot. But cheese, when you look in the fridge, you'll see mold and isolated colonies on your cheese or bread. If, you know, you've had the bad luck of leaving it out on the counter or something like this. Again, you get these isolated pockets of mold. Here you get isolated pockets of bacteria, all arising from individuals. And so in this case, so what do I have here? One, two, three, four, five, six, seven. I can conclude that if say I diluted 10 to the six times that I have something like seven times 10 to the six bacteria per, I don't know, milliliter or something like that, depending on what my dilution was. So here we have seven times 10 to the six bacteria one milliliter, for example, depending on how much I took in my original sample. Is that okay with that? So that's that's way number one. It's quite laborious. So if you wanted to do this in time, depending on how quickly these bacteria are growing, you're taking samples and plating them out at a ferocious pace. It can be done and people do do that. And this afternoon we'll look at experiments that are exactly that. But there is another way that we can do this, but I'll come to that in a second. Yeah. No, no, it's okay. So here, so the question is here, are the colonies going to be close to one another? The colonies are, so the number here doesn't change. That's what's happening here. And so provided you dilute them enough, you'll get, say, still 10 colonies. But then you do it again, you'll still get 10 and 10 and 10. So they're still spatially separated. And the way you ensure that is by this dilution. But the number doesn't change with time. That's the important thing. You've got complete control basically on the density. But technically, I mean, technically it's not difficult, but it's a real labor, labor of love depending on who you are. Any, any other questions about that method? And there you really know what you're counting. Yeah. All the continuous culture, yeah, you can fix the temperature. So you take that whole thing and put it at 37 degrees. So here, here this stuff is all up to you. Because once they're on the plate, nothing, nothing changes. They are just individuals. And now they are not eating what you fed them in a test tube. They're eating what you put on the dish. And they're growing at whatever temperature makes your life the easiest. So you could put this test tube at, say, 30 degrees and then still grow your plates at 37 degrees. And there would be no contradiction. Because you've got them on the plate or even in continuous culture, you've got them on the plate. Your experiment is over. Now what you do is put it in the oven. They grow up and you count them. But as soon as they hit the plate, your, your experiment is basically over. Does that make sense? It's like a bubble chamber or something. Yeah. Exactly. Exactly. So, so now we come to to a physicist thinking about this. You want them well separated. But then the, the smaller the number, if you're assuming this is say Poisson, say Poisson statistics, then your variability is going to go like one over the square root of the number. And so this, I mean this for illustration is fine because you can count how many colonies there are. But the deviation, the variance, if I did this two or three times would be astronomical because there are only seven colonies. And so one plate would have 15 or would have seven. One might not even have any colonies. So what you do is you aim for, I mean, and this very much depends on the patience of the, of the person counting, but typically you, you aim for something higher than 150. Okay. So that the, so you've got about 3% error. Yeah. So here they're liquid. So it's sloshing around. Here it's jelly. And so when you, when you take a drop and you put it on, you spread them out. And so there, there's that, that liquid gets spread out and then absorbed into the jelly. And so it's, it's not quite, I mean, we call it a solid medium, but it's not quite. It's sort of a, yeah. Exactly. Exactly. Exactly. And, and so, oh, no, no, no, no, it's, I mean, now that we make it, it makes it sound difficult. It's not bad. So what you do is you make these jelly plates, you buy them out, I mean, not so that they're desiccated, but just so that they're dry. So there's no film of liquid on top. And then when you put the liquid on, you, it depends on how you want to do this. Some people spread it out with a manual spreader, but you spread it out until there's no observable liquid. It gets absorbed into the gel and it dries very quickly and then you let it sit dry for a few more minutes and then you throw it in the oven. And so the gel itself does contain some liquid that allows the nutrients to diffuse, but the gel itself is dense enough that it doesn't allow the bacteria to travel around. I mean, it's amazing that it, that it works, but, but it's, it's not as bad as it sounds. It's really technically not too bad. Are there any other questions? Is that okay? All right. So let's go to the second one. So the other one is what's called turbidity. And, and the, the idea here, so I'm trying to think if there's an obvious, I mean, hopefully you haven't encountered this, but it's not too difficult to imagine. If you have, say, apple juice or something and you leave it on the counter, it's going to start to get cloudy and, and probably stink. But as the days go by, it's going to get cloudier and cloudier and cloudier. And what's going on is that there's some infection in your apple juice that's just growing. And so the exact same thing happens in these test tubes. You inoculate them with a little bit of bacteria and it's, it's perfectly transparent. But as the hours go by, it becomes more and more cloudy. And that's because the number density of these bacteria is increasing in the test tube and they're scattering more light. Okay. And so we can use that as a proxy for number density, that light scattering. So the idea is that high number density of bacteria scatter more light and said sort of in less scientific words, the, the test tube gets cloudy. And so as I say, we can use that as a quantum, quantitative measure for number density in the following way. So we use as a, as an instrument called the spectrophotometer. So you send light with some initial intensity through your sample of bacteria. So let me, let me draw the scenario and then I'll explain it. So again, you've got your cells growing here. You take a sample, something like say 100 microliters or, you know, one milliliter. Those would be typical sample sizes. And then you would put them into something that's called a cuvette. This is like a little, fixed volume cell. So this guy goes to here, that goes to here. And then you measure the, the amount of light that gets through. Maybe I'll change this to in and out just so that it's easier to imagine. Now chemists use this as a way to measure how much light of a particular wavelength is absorbed by the chemicals. So you get an absorbance spectrum. Biologists use it as a way of measuring the scattered light. And so we typically use a wavelength that doesn't have a significant amount of absorbance. And so that we're really measuring as a scattered light. And so the standard is about 600 nanometers. So you shine a beam of 600 nanometer light here. A bunch of it gets scattered. And then you measure what's called the absorbance at 600 nanometers. And that's going to be the logarithm of the intensity. I was, mix up the order here. Intensity in over the intensity out. And so the, the higher that number is, the more the light is scattered. Okay? Does that, does that make sense? Now the key here is that you, you fix the path length. So the, the volume that the light is passing through is always the same. So you empty this out and then you take another sample 10 minutes later, you empty it out and then 10 minutes later you get another sample. But the point is that the volume here is always fixed. And so the absorbance here is proportional to the number density and some shape factor which depends on the wavelength. And this is some shape factor. Okay? So I put this up here for two reasons. One is that the, the shape of these bacteria is rod-like. And so they don't scatter like spheres unless they're growing very, very slowly. Which brings me to another point, depending on how fast they're growing. Sorry, this is not the wavelength. This is, this is the growth rate. So depending on how fast they're growing, their shape changes. Alright? And we'll come to that in this afternoon. Yeah. Yeah, no, no, no. So it, those are just chemicals. So they're, they're not, they might have absorbed light, but we, we, but they won't scatter. It's not, they're not particulate excretions. Does that make sense? So they're not, they're, they're small enough that they don't scatter light. Okay? We also dilute these. This brings me to my second point. This needs to be diluted enough that you don't get secondary scattering events. Okay? So must be dilute enough that we only have single scattering events. That is to say that there are so few cells in here that the light beam bounces off and goes away. You never have a light beam bouncing off, bouncing off another cell and then bouncing out. Okay? And so I say this because if you read modern papers, very often these types of measurements are not done in a spectrophotometer. They're done in what's called a plate reader where you have a plastic dish with little wells. You put your cells in and you put them into this plate reader. In that case, the, first of all, that the path length is not constant. So this doesn't make any sense. Second of all, the density is not necessarily controlled in a way that you get single scattering events. Okay? So this is faster and much easier than the solid plate counting. But you need to be mindful that this proportionality between the scattered light and the number density is only over a very narrow range of densities to avoid these secondary scattering events and only under very well controlled conditions that the light path length is, is fixed and falls out of the equation. Okay? So those are the two ways to do it. You count on the plates or you use the turbidity of the culture. Exactly. Exactly. So this shape factor for a given growth scenario, you calibrate by plating and that's essential. Okay? If all you're interested in though is the growth rate, then you're, you're only interested in the changes of this absorbance and so that factor falls out. So it depends what your motivation is. In the experiments that we talk about this afternoon, they'll be calibrated to the, to the plate method. So they're in absolute numbers. How does that sound? Is that okay? Okay. So what I'll do, maybe let's take a five minute break, get a glass of water or something and then I'll, I'll tell you about what Minot did with these growth curves and why scientifically these curves are a sort of a turning point in bacterial physiology. All right. So I'll see you in five minutes. Let's say features of the bacterial growth cycle that Minot highlighted in his review. So the growth rate was something that people had been looking at for quite some time. I'll come back to that and in fact, that'll be our focus for most of this course. But the lag time and the total growth were largely Minot's invention. And so let me show you a little bit of that data in a second. I'll say it like this. Minot's, Minot's most lasting or probably largest contribution from this period, bless you, bless you was to identify features of this bacterial growth cycle that we're worth studying. And we're studying quantitatively, identifying features of this cycle worth studying. All right. And so the lag time, I'll only say a few words about it. Total growth, I'll say it's talking considerably more detail. So the lag time, he determined, was largely dictated by how long it takes a bacterium to sense a change in nutrient conditions and to adapt to them. And by adapt, I mean, make all of the proteins that would be necessary to eat that particular nutrient. Okay. And this, although he didn't know at the time, has to do with how the cell regulates a production of protein. So it turns off, turns off and on proteins depending upon what it's growing in. This is something that's called enzyme induction. And that's what he won his Nobel Prize for about 15 years later. So this has to do with turning on or off protein synthesis, particular enzymes. And these enzymes, again, are particular proteins that are used to eat particular nutrients. Okay. I don't want to say anything about that. That's, that's, you know, not, not really the, what I want to focus on. I want to focus on more of this, this growth rate. Let me say one more thing about, now, total growth. Because here for the first time, we, we'll see an example of truly quantitative data where you look at it and you say, there's something I could probably do with that. I mean, as a physicist, you, you'll see it and I'll show it to you on the screen. You say, that looks, that looks physicsy. All right. So let me pause though, actually. Let me talk about lag time. Are there any questions about lag time? Again, I haven't really gone into any detail, but depending on what it was growing in the night before your experiment and what you want to grow it in the experiment that you're doing today, that'll dictate how long this lag time is. And so for example, if you take something that's growing exponentially in glucose and then you move it into a test tube with glucose, there is no lag time. It just goes exponentially, exponentially, exponentially and so on. Okay? But if you move it from, say, glucose to something it doesn't like to eat, it takes a lot of proteins to chew up that particular nutrient, you'll see a long lag time before it starts growing again. Yeah. This, this thing? Yeah, it can be anything from, it can be instantaneous if it's the same thing, up to, say, hours, probably less than that. It's usually order of minutes, just as a rough, rough idea. It seldom anything longer than an hour, unless there's something really, really hard going on. I mean, it has to make a lot of stuff if it's an hour or something like that. Exactly. So you can change both the slope of that exponential and how long it stays in that just by diluting and feeding it different things. And we'll talk about that in a moment. You're exactly right. Is that any other questions about lag time? All right, let's talk about total growth. So total growth is you add a little bit of bacteria and you set one particular nutrient to be the thing that limits its, its maximum density before it hits stationary phase. So you limit a single nutrient. It's usually the concentration of the carbon source. And then what you end up with is something that looks like this. So I'm going to redraw this, this plot. You start with more or less the same amount. You go up, you go here. This would be, say, triangle. So this is time. And this is some low initial concentration of carbon. And then you repeat the experiment. You add the same amount of bacteria at the beginning of the experiment. And then you let this thing go until it stops growing with a little bit more carbon. So now this is, you know, a bit more. And then this would be a bit more. So does everybody see the experiment? It doesn't necessarily, you don't need to actually leave it for longer times. I just want you to be able to differentiate it. The bottom line is from some low concentration, middle concentration, high concentration. But the point here is that by toggling how much of the carbon source you start within your test tube, you dictate how many bacteria you end up with at the end of the experiment in the late afternoon. Okay. And what's amazing here, so this is a total growth. What's amazing is that if you plot now this total growth, and it's useful to measure this number density, like a number of cells, let's say a milliliter. And you plot that versus the concentration of the carbon source. So let's say milligrams of, let's say glucose per milliliter, you get a straight line typically through the origin. So you have, so here's square. Here's, oops, sorry, here's triangle. Here's square. And here's circle. And the slope here is called the growth yield. All right. And now let me show you what that data looks like. And so I'm not going to use the projectors by and large, but I will show you the original figures whenever I think, you know, whenever it makes sense. Because I want you to see that the cartoons that I draw here, for the most part, are high fidelity reproductions of the actual data you get. It's incredibly linear. So these dots are different concentrations of some carbon source. In this case it's mannitol, but that's completely irrelevant really. And then this total growth, although he doesn't put the units there, is the number density per, well, the number of cells per unit volume. He's using cubic centimeters though. It doesn't matter. Does this make sense? Yeah? Okay, then what does the growth yield tell you? That's my question. And so you look at this and it looks very much like something like Boyle's law or something. Perfect linear relationship between two variables. If you look at the units of the axes, what does the growth yield tell you then? You just shout it out. Exactly. And so then what would the slope, what units would the slope have? Exactly. So this would give you number of cells per, in this case, milligram of glucose. So this is telling you the efficiency of the organism to turn that nutrient into cells. It's a very simple experiment to do. The data is unambiguous and the interpretation is right in front. It's tangible. There's no layers of inference. I mean, this is what it is. This is telling you the efficiency of the organism to convert, say, in this case, glucose to biomass or cells. And this is an operating parameter of this cell. You change the species, maybe that'll change. You change the carbon source, that's local change, obviously. But the interpretation is the same. It's like measuring the efficiency of your engine in a car. All right. Let me pause. Is that sensible? It's okay. All right. And as I say, the linearity is compelling. Let me just close this up. All right. Okay. And so finally, the last thing that I want to talk about from this review is a growth rate. Okay. So if we now look at the growth rate. So this is the slope of the number density versus time on a log linear plot. Okay. And so, yeah, it depends how much you would have had, not from first principles. You would have had to have a catalog of, say, seven different bacteria and all the same carbon source or something. Then it's very unlikely that any of them will have the same slope. And then you could read backward. But if, for example, I showed you that picture, there would be no way to know that that was E. coli, for example. It could be any number of bacteria that would exhibit that type of linearity. Not unless you had some sort of experimental catalog that gave you that mapping. So it would be very difficult, as I say, from first principles. Just say you knew all the chemical reactions that were going on in the cell for this species and that species and that species. To reverse engineer what the efficiency is would be, or for now, is impossible. There would be, so for a given species and a given carbon source, that's unique. But there's no way of knowing that without doing the experiment first, for the most part. Is that sensible? Yeah. It's like a fingerprint. I mean, once I know it's fine, but I can't guess. Yeah. Oh, I mean, once you've run the experiment, or no, no, no, no. For E. coli, it's definitely its most preferred, but that's not a universal. So there would be, say, some soil bacteria that never see glucose whose slope here might even be flat. So they wouldn't even be able to eat glucose. There would be some that are very, very, they metabolize glucose very poorly. Oh, I see, I see. Yeah. Yeah. But then there's also a lot of transport issues and things like that that also go on. So, I mean, so there's that also before glycolysis. But you're right, the glycolysis path is fairly, I mean, that's largely universally used. But then there are other guys that prefer to ferment rather than respirate, for example. So that it could also depend on whether you have or will definitely depend on whether the cell is aerobic or anaerobic. So if it doesn't like oxygen, it's not going to respirate. Does that make sense? I don't think I answered your question. I think it just dodged around it. Yeah. Yeah. And we'll talk about, as the course goes on, is that there's intrinsic constraints. So if you want to up-regulate some set of proteins, you have to down-regulate some other ones. And so this is going to incur costs as well. There's a question up here, though. That's a good question. And I don't know. So then the question is, does this have to just be a carbon source? In Minnode's day, that was the only thing that they considered. He suggests that you could maybe do it with inorganic sources like oxygen. And I suppose that you probably can. I just don't know. So to be say, partial pressure of oxygen. This is very much a measure that bioengineers favor. Because it tells you, for example, if you're feeding it, I don't know, some weird feedstock, and you're trying to get out, I don't know, insulin or something for themselves, this slope is paramount. And so you just, you try to make that slope as steep as possible. So it's very cheap to feed these things. And so I suppose you could try any kind of growth limiting horizontal. Any other question? Exactly, exactly. And you make it so that's not an assumption. Ensure that that's true. Exactly, it's exactly a partial derivative. And you would know, wow, this is a circular argument, but you would know that you've messed it up if this thing starts to flatten out. And then you say, oh, wait a second, that's not the growth limiting. You know, my derivatives are mixed. Does everybody take his point? So you have to manufacture this so that when you change the concentration of one single thing, this plateau gets higher and higher, or lower and lower. If you notice it, it stops changing. Well, then that means there's something else limiting the growth. And you can try and hunt for it. But that's a biochemistry question. Okay, let me pause it. Any questions? All right. So let me talk then about the growth rate in a little bit of detail. We're perfect for timing. Okay. And so here, depending upon what you choose as your base, I think as humans, it's natural to think of log 2 because this is a doubling. It's a natural base to choose for an organism that grows, divides, grows, divides, grows, divides. And so the numbers go from one to two to four and so on. But mathematically, it's a bit fussy because you're going to have these natural, you want the natural base. And so the relationship is straightforward. So if you have, say, exponential growth like this, it's the same thing as something like this. And this thing we will call the exponential. Sometimes you see this called the specific, although that is bizarre term growth rate. And this guy is your doubling rate. And so on this graph, the slope of this line is that mu. And of course, a conversion between the two comes from taking the natural log of both sides. So take the natural log, and you'll carry this guy down. You'll get lambda t is equal to logarithm of 2, natural logarithm of 2. And you'll bring down the mu t. Is that okay? And so then you can just solve. So that means that lambda is equal to mu times long 2. And so there's a one-to-one mapping, which it should surprise nobody. But the other point here is that this guy is about 0.7. And so if you see numbers that are small, it's probably this. It's bigger if you're talking about doublings per hour, which makes sense. I mean, exponential would be basically going up by a factor of 3. Anyway, let me pause. Does that make sense? So it doesn't matter what base you use here. You just need to be sensitive that somebody might be plotting one piece, and you might be plotting another one. But anyway, it should be cleared by the caption. So that's just a mathematical technicality. What I want to talk about is how you change that growth rate. It's okay. All right. Okay. And so the first point that I want to make is something that Menod was famous for from his thesis work. So this total growth is something that's used all the time in bioengineering. The next thing that he was probably most known for is something called Menod kinetics. And the idea here is that if you plot the growth rate, so this is now growth rate against the concentration of a, again, growth limiting being nutrient, for example, glucose. Then if the concentration is saturating, it's very, very high. Well, then you'll have some limiting growth rate, say, lambda max. But for low concentrations of that growth limiting nutrient, your growth rate is going to be depressed. And it's going to go something like this. And empirically, so you have some scatter, of course, but it looks like this. And so you have that the growth rate is something like this is some concentration of the nutrient. This is some affinity, affinity constant. And then this is some maximal growth rate. And this is all in batch culture. So you can have a sustained exponential growth at low concentrations of glucose, but it will be lower, it will be a smaller growth rate than you would have if you saturated the system. And so if you've seen any enzyme kinetics, this looks very much like what's called a Michaelis-Menten kinetics scenario. And we'll talk about that probably at the end of this week in more detail. But for now, this is just an empirical fit that Menod noted. So this lambda max will depend on what species of bacteria you have and what the nature of this nutrient is, like its glucose or whatever it might be. And this affinity is telling you something about what the critical concentration would be in order to get growth from this substrate. It's setting an intrinsic concentration scale. And so again, let me show you its original data. You can see it's not quite as compelling as a growth yield, but it's not terrible. And so this came to be an empirical relationship that people use and continue to use, especially in bioengineering, relating concentrations of some growth-limiting substrate and the growth rate of the bacterium. All right, let me pause it. Any questions? Okay, this form, as I say, will come back again. All right. All right, so then let me say one last thing about the growth rate. So this is under conditions of saturating or sub-saturating nutrient. In exponential growth, we can adjust the growth rate by adjusting or changing what we feed the bacteria by changing the nutrient source. And we keep it saturating. I mean, we add a lot of it so that the concentration effectively doesn't change throughout the course of the experiment. Otherwise, you would have some weird behavior like these Menod kinetics. So we keep the concentration high. The cell uses this stuff, but it doesn't change the effect of concentration over, say, two or three hours. But the quality of what we feed it is what's important. So, i.e., we adjust the nutrient quality, not the quantity. And what you end up with is base two. Let's say this is the number density. This is time. You can toggle this slope through, as I said, the kinetic range or dynamic range is very large. So this would be poor nutrient conditions. And this would be rich conditions. But one point that I want to make here, does everybody follow what I'm writing? And so the slope here is a very high growth rate. This guy is a very low growth rate. But I can control that. And the point that I want to make is that these cells really do grow exponentially. Once you've let them adapt to things like that, I mean, they're ramrod straight. So this is a, I mean, the reference doesn't really matter all that much. It was just one that I had on hand. But what I'm plotting here is the log two of the number density normalized to the number density at time zero. So everybody starts out at zero here. And then this is time. And what's difficult to see perhaps is that each of these lines has two different symbols, squares and circles. And those are experiments done on different days. There's tremendous, I mean, the reproducibility here is exactly what you would want for a quantitative study. So typically from lab to lab, day to day, decade to decade, there's less than 5% variability in what you measure in the growth rate. Okay, so this is tremendously reproducible. So less than, less than 5% variability typically. And so that makes it an extremely compelling or useful parameter for people to use in quantitative modeling. So here I'm not changing it by nutrient source. I'm adding an antibiotic. But we'll talk about that a little bit later in the course. The idea is still the same. We have exponential growth that we can toggle. But day to day, there's almost no variability at all. Okay. And so, so, yeah, let me let me then finish off by saying two more things. All right, so the picture that Manot had of this growth cycle was what people were thinking of from say the 1850s on. And so the focus was on this life cycle and very much thinking about human life cycles. It's born, it has a lag time, it has vigorous sort of adulthood, and then it declines into senility and old age. And that's not, that wasn't thought of as an analogy. That was really sort of what was underlying this bacterial cycle. We'll see that that's crazy. But that's always an essential step in scientific development. This anthropomorphization, this kind of putting yourself into the place of what you're studying and then slowly taking yourself out again. And so we'll talk about that this afternoon. Okay, will we pause any, any other questions? Yeah. Okay, yeah, let's talk about that. Maybe offline. Is that okay? Any other, unless, yeah, if you're interested, so her question is, is the exponential unique? Could you have another choice if you were trying to maximize the number density at the end of the day? At the end of the day, yeah. Any other questions? Yeah. Yeah. The growth rate. So, ah, yeah, so let's go back to the analogy. The, the growth yield was the efficiency, like kilometers per, per liter of gasoline or something. This is your top speed. This is like, you can go, when you've got your foot flat on the gas, you're going 180 kilometers per hour. You're going 10 kilometers per hour. So there are two different things. It's important that very often these ideas are mixed together in, in research articles. They're very separate. One is the efficiency of your engine. One is the top speed. And they're not, they're not this, you can obviously have very efficient, very slow engines like in a tank. And you can have very, very fast, very inefficient engines like a Formula One race car. And this is very, I mean it depends on the ecology of the vehicle that you're driving. Same way that it depends upon the ecology of the bacteria in which you're grown. Does that make sense? Any other questions? All right, if you have any other, we'll, we'll, we'll pause at the beginning at this afternoon and we can talk about again. Okay, so I'll see you after lunch.