 OK, so let me try not to forget this. Can you hear me OK? Yeah? OK. Because yesterday I was told that I didn't speak loud enough, or the sound wasn't loud enough. All right, so let's start where we left. Remind you, last time I told you about a bound on the detection of concentrations, all right? And so one thing you could ask yourself is if you look at this bound and you do the math, can a bacterium actually sense a concentration gradient? So let's put the concentration we had. So we'll do a numerical application. So what we know also is that bacteria, they tend to make the decision on the order of a second. So that's the reaction time of the order of a second. So I can put in all these numbers. And if I do this, 23, let me see if I can do this right. This is really an order of magnitude, right? So again, this is really the best anybody can do. Not just the bacterium, this is the physical limit. You can't beat that. And so if you put in the numbers with the size of a bacterium, you get that the best accuracy you could get is 5% in the estimate of the concentration. But remember, the bacterium what it cares about is the gradient of concentration. So one thing you could ask yourself is how does it do that? How does it estimate the gradient? Do you have any idea of how it should do it? Yes? OK, OK, but it's a measurement device, right? So how does it know what this direction is? The sort of estimates I've been making now is just to estimate what the concentration is. How do you estimate what the gradient is? The gradient is a spatial gradient, right? So it wants to know where it's better and where it's worse. So how does it do that? OK, so that's the right answer. It does it as it moves. Now it's not obviously should be that way, because in fact, that's what bacteria do. But eukaryotic cells, such as amoebas or the neutrophil I was showing you in a movie yesterday, they don't do that. What they do is that they have a big cell and they estimate the concentration here and they estimate the concentration there, right? And then actually make some sort of a difference between the two. So they make a spatial difference between the fronts and back, for instance. And if it's larger here, then we start moving in this direction. If it's larger there, it's moving to me in that direction. That's for most eukaryotic cells. So eukaryotes, just to, I mean, eukaryotes is anything that's not a bacterium, essentially, right? There's also archaea, which is also close to bacteria. But we eukaryotic organisms, but yeast is also eukaryotic. So anything that's, it's a difference of scale, right? Because in fact, that's important because the bacterium, that's not what it does. And you see why from this estimate, right? Like you have, the bacterium is about one micron across, right? And it can only detect differences up to 5%, OK? So what it means is that it cannot, you know, if it were to measure what's in the front and what's in the back and make the difference, then this difference could be no larger than 5%, right? So if you get 5% decrease here at most, that means that your gradients, the scale of your gradients is over 20 times the size of the cell at most, right? So what this means is that if, so it's not what it's doing, right? But if the bacterium was trying, if the bacterium was trying to detect special gradient by making a difference between front and back, then it could not detect gradients that are larger than 20 microns, right? So it would be very, very sharp gradients on very, very short scales. Now, in the experiment I showed you, for instance, but there are many other examples, you can see that the scale here of the gradient is more of the order of the millimeter, right? So these are fairly shallow gradients. At least from the bacterium perspective, the bacterium is one micrometer across. A gradient of a millimeter is like 1,000 times its size. So it needs to be able to detect these very small relative differences of the order of one in a thousand. And the best it can do is 5%. So that means that it cannot use this strategy. So the strategy actually uses is the one that somebody said. And it's using a run and toggle strategy. So the way the bacteria will sense the gradient is by moving, right? So let's say you have the gradients, the bacterium will go forward. And instead of making special gradients, it will calculate temporal gradients. So the bacterium is asking itself as it's going forward. So the bacteria swim using some flat gel which I'll show you in a second. And when they swim, they swim more or less straight. And as they swim, they ask themselves, I mean, they measure the concentration. And they measure the concentration differences in time. And as they do that, they ask themselves, is life getting better? Do I see more of the ligands I'm interested in? Or do I see less? If they see more and more, means that things are getting better, well, if things are going good, what should you do? Should you change the reaction or should you continue what you're doing? Come on. This is not a hard question. You should continue what you're doing. If things are getting good, you continue. And if things are getting bad, so you see you're going in the wrong direction because basically the concentration is decreasing from your point of view, then what should you do? You should change the reaction. So the bacteria, E. coli in particular, but many others do as well, have a mechanism such that when they sense that things are getting worse, they will randomly change direction. So they stop running. So it will be done by changing the direction of a few of their models. And when they do that, they will tumble, which means that they randomly reorient. So you can see that on that movie, which I'm on that movie here on the right side. So this movie is not working, but if it were, you would see moving bacteria. Here what they did is that they fluorescently labeled the flagella of these bacteria. And so you can see them swim in. So you can see here, each of these are a bacterium, and you see the flagellum. And every once in a while, like here, you see like here, the flagellum, each bacterium has a few flagellum. When they all spin in the same direction, it's going straight. But when a few of them change direction, then the bundle that the flagella form unwinds. And then it starts tumbling, meaning that it's doing a random reorientation. And so people have been studying this. Biophysicists have been studying this with great accuracy, because you can actually measure many things about this behavior. And in particular, you can show that the lengths of each of the episodes of running versus tumbling are very precisely, exponentially distributed. It's experiments from the 70s. You can show, and these are more recent experiments with the track, many, many bacteria at the same time. You can also show that there's a bias towards the gradient. So this is exactly what I was saying. Like here, this is the mean run length. So how long it stays in the running mode as a function of direction. And here, the gradient is in this direction. So the bacteria actually want to go that way. And you can see here that when they are in this area of the angles, they do longer runs. So when things are good, they tend to stay longer in that direction. And when things are getting worse, they do much smaller runs. So it's a factor two on average. But they're all exponentially distributed. And so you can basically summarize this behavior by what you may want to call a response function of the bacterium. And the response function will be the probability that the bacterium will find itself in either of the two states of the model. So there are two states. One, it's called. So this is the run and tumble thing. It goes like this. This is a run. And here, you have the tumble when you are in. And the point is that when it gets good, it does longer runs, so it will go in general. So to simplify, these two modes correspond to two states of the model. One is called counterclockwise because it's the direction of rotation of the model. And this is more or less equivalent to the runs. And this one is clockwise. And these correspond more or less to the tumbles. And so you can characterize the response by the probability to be in the counterclockwise regime. That will be your response. So among the experiments that people have done, they've been trying to characterize this response here as a function of what the bacterium actually sees. So to do this, instead of doing these kind of experiments where these are freely running bacteria, what they did is that they stuck the bacterium to a glass slide. And then they attached a gold bead to all polystyrene beads to one of the flagellar. And then they observed the bead. If the flagellar is spinning one direction clockwise or counterclockwise, they will just be able to record that spin. And so that way, you can see in what fraction of the time the bacterium is in either of these two states. And this is the kind of data that they collect. And this is a somewhat old experiment from the 80s. Here what they did is that they really wanted to characterize the response in a, what they wanted is some sort of the Green's function of the response. So the idea is that you have some sort of linear response theory. And in linear response theory, what you think is that this response would depend on the past experience of the past concentration experience by the bacterium as it moves. So here it's actually not moving, but what you can do is that you can control its environment to change the concentration it's seeing. So you have some concentration C of lagons here. This guy has receptor, so they can sense. And you can change that. And the assumption is that in a linear response theory, this response will depend, by definition, you linearize the response linearly on, in fact, it's the log. It doesn't really matter. It's because it will be sensitive to the log of the concentration, not the concentration itself. But it's linear in the log. So you can view this as a Green's function. It's the propagator, if you like, or the gain function. There are many ways of coding it. And here you integrate from minus infinity to C. So what this means is that you integrate the past of the concentration with some kernel or Green's function. And how do you measure this guy here? Well, it's a simple way of doing it, which is to, and it's something that's often done in physics, which is to subject your bacterium to an impulse. So what you do is that you have your C, which you can control. And the way you will control it is that you add an impulse. And impulse, in mathematical language, is just a delta function. I shouldn't call that delta C. That big K doesn't matter. So if you plug that into this, you'll see that your response would be equal to r0 plus g of t minus t0. But let's say it's a constant here. I mean, it's not necessary. Yes, no? Sure. Yes, sorry. I'll try and stop writing in the bottom. I tend to forget. OK, so if you plug in an impulse, then what you get, essentially, that you probably know from linear response theory, you actually get the response kernel up to a constant. And this is exactly the experiment they did here. Had five seconds, they added this impulse, and you can see this response. So if you look at this, I'll tell you this is the response kernel. What does that tell you about what the response does, what the bacterium does? If you just inspect it, what do you think it does? Anybody? I mean, you know there's something about this function, right, which has it becomes positive and then negative. What do you think that is? What did I tell you the bacterium really cares about? It cares about the gradient. So it cares about making differences, OK? Do you see why this is a difference? Why is it a difference? But exactly, right? But the point is that when you look at this, this is like you make an average of the past, right, with some kernel. What this tells you is that first you have a positive lobe. So the positive lobe is taking an average of the very recent past of the concentration, OK? So if there was no negative lobe, that would be it. That the response would just be a smoothing, if you like, of the past concentration. But you do have this negative lobe. So what is it doing? It's taking an average of the recent past, but then it's subtracting from that an average of something that's a bit more into the past, OK? So what happens when you take the value of something a little bit in the past and you subtract from it the value of the same thing a little bit more in the past? What happens when you make a difference? Come on. This is not a difficult question. Maybe you don't see what I'm getting at. Come again? The term, the middle term, so does it get the difference? Yes, what happens if you make the difference between almost now and almost a little bit in the past? How do you call that? I write it down for you when you see why it's easy. So I'm simplifying, right? But the idea is that this response, it would be like c of t a little bit in the past. So let's call it delta t minus, and so there's some constant here, minus the value a little bit more in the past. What does that give you when delta t is small? It's a derivative, OK? OK, next time you see this, you should immediately say derivative. This is what this function does. It's taking a derivative. And it's taking a derivative because, in fact, if you do, you can maybe see it by eye. But if you take the area in this positive lobe and you look at the area in the negative lobe, they're exactly equal to each other. So one exactly compensates the other. So it responds almost exactly to the derivative. But remember, this is what we want, right? This is what we want the bacterium to do. Because the bacterium only cares about things, whether things are getting better or whether they're getting worse. They don't care about the absolute value of the concentration, but this really shows it. There are more recent experiments based on tracking. So this was based on the behavior of the model. Then you can also do it directly at the level of runs and tumbles. And this is a recent experiment by the group of Massimo Vergassola. And you can find almost exactly the same result. Positive lobe, negative lobe. Another way of checking that things are so OK. So the fact that the positive lobe and the negative lobe balance each other exactly, this is what people call adaptation. They call it perfect adaptation. So why is it perfect adaptation? It's because you can see from this response that if you put a constant C, let's assume that C is constant now. If C is constant, these two terms would cancel out. So the response, if C is constant, the response is always the same, no matter what C is. So what that means is that let's say instead of putting an impulse, let's say you put this for concentration. You put some sort of a step function. What do you think will happen? Can you predict what the response should be? But delta, OK. It starts at r0, right? And then what happens? We think it's a dead, OK. But everything is a bit smoothed out, right? I mean, it's not as clear as a, so it first goes up and then what happens? And it goes down and it goes down to where? To the same level, OK, exactly. So you can do this experiment and you see what you get, OK. So it's working. It's adapting, right? So here what they did is that actually changed the concentration. But first it reacts and then it goes back to the basal level, always the same. So how does the bacteria do that, right? Because taking derivatives, how do you take derivatives? If I ask you as an engineer, how would you take a derivative? How would you do it? I don't know the answer to that, but I know how E. coli does it, at least. I'm going to explain very briefly to you and just do a very simple calculation to show you how it does it. And in the problem set, you'll see there's a problem, which is assuming that it's sensitive to concentration and the way I explained, the problem is about showing how this actually allows the bacterium to move in the direction of the gradients, by calculating the mean drift of the bacterium doing this run and tumble strategy. But how does it do it? Well, yes, sorry. How does it do it? It's quite complex. There's a network, but basically it starts. Let me just give you the broad picture. What happens is that there are receptors at the surface of the protein, which are sensitive to the ligands, which are the chemo-attractors. So this is what we've been talking about from the beginning. The bacteria have some receptors on the surfaces. These receptors will bind the chemo-attractors. And based on this binding, they will know something about the external concentration. Now, these receptors bind the ligands. And when they bounce, in fact, it's the opposite of what you would think. They become inactive. They tend to become inactive. So it's like a negative influence. And when it's active, it's actually then activating some signaling pathway that then controls the molar. So you can see this directly. The more ligands there are, it would tune up the concentration of some protein species inside the cell, which itself will control the activity of the molar, and namely, the activity of the molar, meaning whether it's going clockwise or counterclockwise. So in the simplest thing, you could say that the response of the molar is proportional to the probability, sorry. So I said when the receptors are bound, they become inactive. So the response is about the activity of the molar. So it's like a repression, and that you saw with Alexandra. It would be something like this. In fact, you saw also maybe with Alexandra, there's a possibility of having a cooperation between receptors. It's called a hail function. So you put this to the power L. But it doesn't really matter for our purpose. Really what's important is that the response is a decreasing function of the activity. Sorry, the response is a decreasing function of the concentration. But if it did just that, if that was the response and there was nothing else, would the bacterium take the derivative? And the answer is no, right? Obviously, then the response you see here does really depend on the absolute concentration. So the way it works is that, in fact, there's something that's called integral feedback, which is that the KD of the receptors can itself be modified as a function of the activity. And the way it works is the following. There's some you don't need to remember. The receptors can be modified by adding some methyl groups to the receptors, methyl groups like CH3 here. And these methyl groups can be added by two enzymes. And they can only be added to the active sites and removed from the inactive sites. So because of that, to make a long story short, this methylation level you can show follows this sort of equation. So basically, the evolution of the methylation level will depend on the activity itself. I'm not going into the mechanism, but this is really the one thing you need to remember. And then you see that this methylation level will change the KD, right? So there's some sort of a feedback that if you like, you have the concentration of the ligands here. Then this goes, you know, you have activity, which is basically the level of this messenger proteins inside the cell. And here you have the model that controls the flagellant. So what I wrote here is just this part. That's the actuation part. And now you have this part, which is the feedback. This activity will feedback negatively on the receptors. And this is called, so this is a very simple network. You just have two nodes and just a repression. And it's called integral feedback because you can see that M, which is what mediates this repression here, essentially is proportional to the integral of the activity, because of this equation. So how does that help me achieve adaptation? So remember, adaptation, if you think about it, like really what we mean by it adapts, meaning that if you change the concentration level, there will be some response. And then we'll adapt back to the basal response. This is what we said. And this was the result of that experiment. No matter what you do, then if you keep a constant concentration, it should always come back to the same level. You see from this equation, if you want to solve this at steady state, so c equals constant. So you put your bacterium at constant concentration and then look at these equations. Steady state means that this is 0. So steady state, constant response, just by virtue of this. It's the motor response. So it's what I call it's more or less the probability of being clockwise. No, no. I mean, in this one? Yes, right, OK, sorry. That different convention. Yeah, OK. Right, OK. So it's a good point. I always get this wrong. But I don't think you should worry about it too much. The point is that at the end of the day, we have many minus signs. And in the end, it should always be that if things get better, you should not turn. So I think this is what this is saying. So maybe I got it wrong last time. There was a probability of being counterclockwise. But there are different conventions. Because I think here, I think here they actually looked at the probability of being clockwise. So you have to be careful. But don't worry about the sign, OK? Let's agree not to worry about the sign. Well, OK, why the logarithm? You have to, why not? I want to say it's linear. I can say it's linear, whatever I want. If you want a motivation and you look at this, I'm going to expand on this a little bit, right? You can write it as this way. So it's in my logarithm. If I want to linearize in the logarithm, why not? What you see is that here, this Kd, because of the way I wrote it, maybe this is the reason why. Here, I have log of c and of n. So what I'm going to do is I'm going to linearize this. I think the short answer is that you should take my word because there's a reason why this is always the right regime. But I don't want to go too much into detail of the biochemistry. So let's assume that I can do this linearization. And here, my linearization would be the following. So here, alpha and beta can be expressed as a function of my parameters here. And I still have my equation from before. So now I want to solve for this for any. I want to solve for this for any function c of t. I want to know what's r of t. So let me define delta r is equals r minus r naught. I'm going to do is I'm going to take its derivative. So it's the derivative of this function. I'll call that x of t. So that's my input. And then what I have is just replace dm dt here. So you all know how to solve this. I want to solve this for delta r. So it's a linear equation with a non-homogeneous term. I'll just write quickly how you do this and this is important. So again, you can see that the response only depends on the derivative. And if you interpret this, you can see that this is basically taking some sort of an average with an exponentially decaying kernel over the recent past of that derivative. Now if you want to calculate the g we had before, you can do it. You just need to do an integration by parts. Sorry I wrote very low again. I hope you can see. So my g, remember my g, that's my kernel function, is simply minus alpha of Dirac. So this is what I want. What g is, is it takes the value now and subtract from it the value a little bit before. So this is like an average. And so if you plot this out, then we run into the problem you mentioned of the sign. So if you take the inverse of the response, so it then changes the sign, you get that the response to a pulse is a delta function. And then you have an exponentially decaying negative load. So that's the delta function. That's the negative load. The weight of the negative load here is about 1 of the beta gamma. It's consistent with what we see experimentally. Remind you, first a big peak and then a negative load. So this is how bacteria take derivatives using this integral feedback. In the homework you'll see in the first sheet, nothing of the homework, there's an example of another way of taking derivatives. OK, so just a few words again about experiments. So this was the experiment from the 80s where they looked at the response to an impulse. So more recently, people have been revisiting this idea saying, OK, they're really taking derivatives. And here, really, since I'm working in x-pace, which is log c, people wanted to know whether it's really true. They're taking, if you like, logarithmic derivatives, meaning derivatives of log c. So what they did for that is that they assumed that they had a system in which they could measure the activity. So they didn't measure the activity by looking at the motor or the behavior. They looked directly at the molecular level what happened inside these cells. And they measured it. Now, non-response to an impulse or to a gate or to step function or whatever, they looked at the response to an exponential ramp. So they put some exponential ramp of this form, which is like saying that x is linear in time. And the idea is that if that's the case, then we put this kind of stimulus, then the response of the bacterium by virtue of this should only depend on the rates R of the ramp after some adaptation time, of course. So they looked for some sort of fake steady state. They saw a real steady state. But they want to see where the response stabilizes when they expose the bacteria to this ramp. And this is the result here. In blue, you see the exponential ramp of concentration. And see, you see the response. In the beginning, there's some adaptation, short adaptation. And then the response stays at some slightly negative level. So this response here is exactly my delta R. And if they put a bigger ramp, meaning with a larger R, then you see the response is larger. And you can calculate how the response depends on the rates of that ramp. And what this really shows is that the bacterium is really sensitive to the exponential, to the logarithmic derivative of the concentration. You can go even further. You can do a Fourier analysis experimentally. So what you do is that now instead of putting exponential ramps or whatnot, you put oscillatory stimulus. So you make the concentration oscillate. And then you look at the response. And once you have the input and the response, you can do a bold diagram for the transfer function. So what is the transfer function in Fourier space of taking the derivative? If you take the derivative, put that into Fourier space. What does it mean? Come on. You take the derivative. In Fourier space, what does it mean to take the derivative? In physics, what do you do? Is it that people who know don't want to say it, and people who don't know don't want to say it? If you know, just give me some hints. You take the derivative in Fourier space. What does it mean? Yes. More precisely, you multiply by i omega. Taking the derivative. You know that. I mean, I'm not. Come on. So this should be my g. This is the Fourier transform of my g. It should be proportional to i omega. This is just a bold diagram of response function. So here you can calculate this response. And you can see that unless you go to very high frequencies, fairly high frequencies, you do see a linear dependency of the function of frequency. And also, quite importantly, you see that the phase of this, so the angle of this, is actually pi over 2. So the 1 half means pi over 2. So it's another evidence that's really taking the derivative. OK, I think we'll take a break now. It's a good time. And five minutes after the break, we start talking about other things and chemotaxes. OK, so now we've seen chemotaxes. One motivation was to look at this physical bound on concentration detection, which was derived by Berg and Purcell. So let me go back to that problem, because there will be a motivation to look at another example, and to see the link and the usefulness of maximum likelihood and Bayesian thinking. So as I said, I've been talking, when I talked about Berg and Purcell, I told you about a perfect idealized monitoring sphere that was measuring the concentrations. Now, in fact, the ligands are really sensed by receptors. So the receptors, they bind the chemo-attractors, and then the chemo-attractors get unbound. And in the simplest approximation, you will have to admit that the rate of arrival of new ligands, assuming that the receptor is perfect, will be equal to 4Da. A is the dimension of my receptors, of my receptors. You can actually show this more rigorously by assuming that my receptor is on my surface of my cell is a perfect absorbing disk of radius A, diameter A. I'm not going to show it now, but this is something you can show. So the rate of binding depends on diffusion, of course. The faster you choose, the more it's likely to hit the receptor. It also depends on the size A. The bigger the receptor, the more likely it is to actually bind the ligands. So this is the rate of binding, and then there's also sometimes, spontaneously, with some rate k of, it will unbind. So if you want to know what's the probability for the receptors to be bound, for instance. So if I look at any given time, what is the probability that my receptor has a ligands bound to it? Do you know the answer? Can you guess? You can sort some sort of a simple master equation, which is the following. The probability of being bound as a function of time will evolve according to this equation. When it's unbound with probability 1 minus bound with the rate for dA times C, because the rate of binding that's for a given molecule. But then you multiply by how many molecules there are. So here I should have the C, really. And then once it's bound with probability p bounds, it can get unbound with rate k of. So if you solve this as steady state, you found that the following results. So now one can ask the same question as we asked for the perfect monitoring sphere, but not the level of a single receptor. If you're asking about a single receptor, let's say my measurement device now is just the receptor. Of course, the cells are many receptors, so you can integrate for many receptors. But let's focus just for a moment on one receptor. And ask yourself, knowing the state of the receptor, how could I estimate the concentration? So it's again, we want to know it's some sort of, you can view it in a Bayesian way, or in maximum likelihood way. The receptor, what it sees, it sees whether it's bound or not. That's all it sees. But what the cell wants to know about is the concentration. So really, if you think about what the receptor sees, or what the cell sees eventually, it's something like this. It's just the state of the receptor. So sometimes it's bound, right? So then it takes value one, let's say, and it's a so-called telegraph process. It takes very zero and ones. And this is what you see. What you want to know, really, is the concentration. So if you see this kind of thing, and you want to estimate the concentration, what would you do? What would be your first guess? Yeah, OK, very good. You say it's the fraction of time. It's because you have this formula, right? So if you say, OK, I look at the fraction of time I was bound. So I look at this, plus this, plus this, plus this. I'll call it this t-bound. And I take t-bound over total time of my observation. And I say, OK, this is about equal to probability of being bound. And you get this, OK? So that would be a good guess. You estimate how often you were bound. And then you invert this relationship to get c. And in fact, this is what Bergen per se considered. So it's the same paper that I talked about talking about the perfect monitoring sphere. But they also applied it to Sanger receptor. And what they found is if you do this, so sorry, now I'll write what they found is the following results. I'm not going to do the calculation, but you can do it by calculating the fluctuations on the total time that was bound. And see how it translates into fluctuations of c. But this is what they find. But in fact, now you can ask yourself, what if now to do this I use some heuristic guess. And so far, you've seen that the heuristic guess, like taking the empirical frequency or this sort of thing, was usually the right thing to do in terms of maximum likelihood. It would give the same result as the maximum likelihood estimate. Here, however, it turns out not to be the case. If you do maximum likelihood, this is not the estimate you would end up with. It's not completely obvious, right? But let's write it down explicitly. If you write down the probability of a given trace, so a given trace I'll write is like the entire trace, right? I'll write it this way, given c. So if I want to write this down, I'll just write that at each event of binding, I'll have a factor, I'll use maybe a different color, d4 DAC. At each unbinding, I have a k of, right? So n is the number of binding events, right? But that's not it. I also need to take into account the probability that nothing happens in this period. So here in this period, it didn't get unbound. So that goes with probability and exponential minus k of this amount of time. So sorry about the heavy notation, but this will be t1 plus. This will be t1 minus t2 plus, t2 minus, et cetera. The plus are just the binding events. The minus are the unbinding events. And here, I just have the probability that nothing happens during this time. I mean, there are no binding events during that time. What's geometric? Exponential? It's because when I look at something that happens in a Poisson manner, like a Poisson point process, the probability that happens with rate k, the probability distribution for its time of arrival is equal to this. And you can interpret this that this is the rate that actually happens during the delta t. So there's always a delta t here, right? But here, I put proportion to forget about the delta t otherwise. They won't matter anyway. This is the probability of binding. And this is the probability of not having bound so far, right? So here, I'm just putting all these factors here. And then it somewhat simplifies. I have my 4dac k of to the power n. And here is basically, I just add up all the time it was. So this is the time it was unbound. This is the time it was bound. And this is where I use maximum likelihood. Now, I'll take the derivative of the logarithm of that monster with respect to c. And I set it to 0, right? And this will be. So all I need to care about is the terms that depend on c. So if I look, I just have other c here. So I just get the n logarithm of c. And here, I just have 4dac t unbound. What does it give me? Just give me c star is equal to n over 4. Just did algebra to the derivative, OK? But what you notice is that this was my estimate from Berg and Purcell. You see, it's quite different actually. Here, I use t bound and total t. Here, I actually use the number of binding events and t unbound. So it's actually used different pieces of information. Of course, t bound plus t unbound is equal to t. But clearly, here, I don't use the total number of binding events. And here, you do. But now, more interesting, let's look at the error. So the error, as I said, what you need to do is that you need to calculate. So this is my, of course, this is my likelihood. So I call it L. I take the second. I can calculate the error I will make from the second derivative of the log likelihood. And this is also easy to do. The second derivative of this, I just get n over c squared. OK? I remember the error I will make, which is the difference between my estimate and the truth, on average, is equal to the inverse of that with the absolute value, so c2 over n. Now I get something familiar again. My relative error is proportional to 1 over n, number of binding events. You see, it keeps coming back, right? You have n measurement. Your relative error is going to go like 1 over n. But what's interesting is that what's the error in terms of the physical parameters and the number of binding events? So to calculate the number of binding events, all you have to do is to look at the rate of binding and multiply it by the probability of being unbound, because you can only bind, right? So that would be the effective rate of binding and you multiply it by t. Sorry, actually, here I made a mistake. There's a 4. OK? So let me rewrite this 1 minus p bound, t. And now my error goes like 1 over 4 dac 1 minus p bound, t. So that was Bergen-Passeur. And now my maximum likelihood estimate is this. What's the difference between the two? There's a factor 2. Yes. It's not a mistake. Usually when you get a factor 2 wrong, you have to do your calculation again a few times. But here's actually not wrong. And there is a big difference between these two estimates. And let me just explain intuitively what the difference is. And it would be useful to understanding something really fundamental about vision, which I'll come to in a second. Basically, when you're looking at the fraction of the time you're bound, you're adding two sources of uncertainty. One is you have to wait between two binding events, right? This is a random variable, how much time you wait. And as we said, it's distributed exponentially in this way. So you have one source of uncertainty here. And this is really what depends on concentration, these binding events. And then when you get the fraction of bound events, you also add another source of stochasticity, which is the time it remains bound. But the time it remains bound is also stochastic. It's also distributed exponentially at this rate. But it has nothing to do with the concentration. So you add this new source of stochasticity to your estimate that you didn't have to, right? And you can see it when you write a likelihood. This doesn't depend on concentration. When you take the derivative, it will just drop out, right? Same thing here. But when you do the estimate when you ask what's the fraction of the time it was bound, you do care about how long it stayed, right? And in fact, you shouldn't care. What you should really care about is how many times it bound in the time that it was actually allowed to bound, right? This is exactly what this is saying. How many times does it bound, did it bound, divided by the entire time where it could be bound? And so this is where this factor 2 comes from. So if biology were to solve this problem, because what E. coli actually does, it does what Burg and Purcell suggested. E. coli, it does this sort of integration over the recent past of the activity of the receptor. So it has no way of actually removing the noise from the binding events. So there's the binding events. Whenever it's bound, it's going to suppress activity, and then it gets unbound. And the time it will remain bound is stochastic, and there's nothing can do about it, right? If it were to solve the problem, instead it would count the number of binding events. This is how really it should do it if it were doing maximum likelihood. So the way E. coli signals and the way most receptor signal is that the signal, whenever it's bound, it will signal. So this would mean a signal is being generated in the cell whenever it's bound. But really what it should be doing is counting how many times it got bound. So what you should really be doing is, for instance, instead, signaling a completely constant amount each time. And what's important is that it's constant, right? So this would be the signaling the way it does it. But if it were to do maximum likelihood, then it would signal a fixed amount for each binding event instead of signaling an amount that depends on the bound time. So E. coli doesn't do that. But there's another important example in which it does that and it's in vision. This is what I'm going to talk to you about next. And there are other things to say about this. But I think for today maybe I just stick to that. So let me just give you a bit of biological facts about the human eye, about vision in general. So the way vision works essentially is that it's through the retina. The retina is just a piece of the brain. We'll come back to that later. And you can view it as some sort of an array of photoreceptors. So the light comes through the pupil, then it's focalized by the lens onto a given point on the retina. And then it will activate some photoreceptors, which will generate an electrical current. These are really neural cells. And this electrical current will be then transferred to other types, sorry, the photoreceptors are actually on the back of the eye. There are three types of photoreceptors, four types, the roads, and then three types of cones. And then it's photoreceptors that transmit electrical currents to an intermediate layer of cells, and then to gangon cells, which the gangon cells are neural cells whose axons make up the optic nerve. So basically whatever comes out of the gangon cells, we go into the optic nerve. And then the optic nerve is like a big bundle of cable. And it goes, it's sent to the back of your brain where you have a visual cortex where it's later processing. It's important to know that actually a lot of, this is really a neural network, and a lot of processing already happens at that level. But it's already interesting to see what happens in even at the level of photoreceptors. So this is a blow up. You see the roads and the cones. Then you see this intermediate layer of bipolar cells, and then the gangon cells. This is just for the fun of it. And it's kind of weird if you think about it. The light comes here, and then it hits the photoreceptors which sit at the back of the retina. And it's only made possible by the fact that all these cells here are transparent. If you think about it, it's really a dumb way of doing things, because first of all, you have to make everything transparent. But then on top of that, then all these cells have these axons that make the optic nerve. And where there's the optic nerve, then you cannot have photoreceptors. So maybe you notice there's a point in your vision where you don't see anything. This is exactly this point. And you think you see something, because your brain reconstitutes in a Bayesian way what should have been there. Anyway, so what's funny is that not all eyes, this is basically a flow of design. Like the octopus eye is designed in the right way with the photoreceptors in the front. And then your photoreceptors along the entire surface. And then the optic nerve is here. You see it doesn't get in the way of photoreceptors. There's another design of eye, which is the fly compound eye. I'm mentioning it because I'm going to also show some data from the fly retina. This is very different design, because here you basically have one lens. So in our retina, we have one lens that focuses light in different directions to different photoreceptors. But you just have one lens, right? The fly has one lens per photoreceptor. You could think it's not very smart, but this is the way it works. So the first thing you can ask yourself, and it's a bit in the line of what I talked about about Burke and Poster, which is what's the best possible performance you can achieve. And here we'll have to do with how many, if I try to see in a very dark room and ask myself, what's the smallest amount of light I can detect. So people have been asking themselves this question. Psychophysicists try to answer this question by designing experiments in which they put some people in the dark. And they put very dim flashes and ask them, did you see something or did you not see something? So this is an experiment from the 40s. And there were several people doing similar kinds of experiments. And I don't want to go into the details too much, but they put these people in dark rooms. And they asked them when they saw a flash. And the conclusion they came to in a quite indirect way is that all subjects could detect dim flashes of light that had as few as six photons. So of course, the photon goes through your lens. Then some of them might be lost. Then they go through the photoreceptor. There are some of them might be lost as well. But once they hit the retina, then almost all of them would be detected. And people can actually detect as few as a few photons. So we're very close to the single photon limits. That's quite impressive, if you think about it. And to see why it's impressive, you can look also into more precisely what happens at the level of photoreceptor. So here I'm showing some experiments that were done I think in Toads. While they isolated photoreceptors, this is the small, actually rods. So rods are the photoreceptors that are responsible for night vision. So they're really the ones that you want to care about when you talk about ability to see very, very few photons. So they took these rods. And they put them in a pipette. And that way they could actually measure the current that comes out of these rods. And then what they do is that they subject, in the experiments, these rods, which are still alive and still still functioning cells, to very dim flashes of light. So here in this experiment, you can see each tick here corresponds to a very dim flash of light. And what you can see here is that sometimes you see a response, meaning a small current. And sometimes you don't. You can forget about this one. What's really remarkable is that if you plot out the distribution of intensities you see come out of these cells. So these are, forget about this one, this is like a response to one of these dim flashes. There's sometimes nothing, a lot of the time nothing happens. And then every once in a while, you get maybe a response. And sometimes you also get a larger response. And so if you take the peak of these responses and you draw a histogram of it, you get a curve like this. These are the experimental points here, and the circles. And you can clearly see the single photons because you can see a peak here at zero, which corresponds to probably no photons has been absorbed. And here you can see the response to a single photon. And here the response to two photons, and so on, and so forth. So you can really separate these peaks, which means they can really delineate single photons. That's the important point. But it goes even a bit further than that. It's like when you put these very dim flashes that are generated by laser, what you actually know is that the distribution of the number of photons that should impinge the photoreceptor should be a Poisson distributed. So if n is the number of photons, p of n should be something like this. And what this means is that the height of these peaks, let's say rather the area under each of these peaks, if I look at them, they should actually correspond to this Poisson distribution. And it turns out to be the case. So you can fit this curve with a Poisson distribution, which you have to smooth because there's some noise in the response. So that's one first line of evidence that if you look at these single photoreceptors, they can really count photons one by one. There's another series of experiments in fly photoreceptors this time. I'm not going to go into the details, but the experiments showed that it's not just single photons. It's that these cells, you can show that the accuracy with which they respond is such that they can really count up to 1,000. So meaning that up to 1,000, the estimates you can make from looking at the response in the number of photons that impinge the cell, the error you will make is of the same order of magnitude as square root of the total number. So not to go into detail what this means, but essentially these cells can count up to 1,000. So it's quite remarkable that it can really count single photons on such a large dynamic range. So why is that a bit surprising? To know why it's a bit surprising, you need to know a bit more about how the, because here I told you there are single photons that are being absorbed by the photoreceptor. And then I showed you the response, which is the current that goes out. And maybe I need to tell you a little bit about how you go from the photons to the current. So this is your retina again, sorry. So the way photons are absorbed is they're absorbed by a big molecule, which is called rhodopsin, which sits here in the photoreceptors. And they're arrayed in some bilayer liquid membranes. So you have many, many, many copies of these different cells, of these different molecules sitting in the photoreceptors. And what happens to them is that when a photon hits them, they will change conformation. There's just a chemical reaction that provokes isomerization of a small group of this big molecule. So it would just change conformation. It would just change state. It's called isomerization because it's just a cyst that goes into a trance chemically. And then what happens is the following is that the rhodopsin in the beginning was sitting there doing nothing. And then there's a photon absorption. The photon absorption will make this rhodopsin change states, which will denote by rH star. And once the rhodopsin in this star state, it changes its catalytic activity. So it can become an enzyme that can catalyze other reactions. So it's become active if you like. And once it's active, it will catalyze this reaction which makes another molecule active, which itself catalyzed the reaction that makes another molecule active, which itself will degrade the product of some molecule and we control the opening of the channels that we create the currents. So you have this complex signaling cascade. Why do you need a signaling cascade like this? It's because what's really amazing about single photon absorption is that in the beginning, you just have one single event, like one single photon, one single molecule. But in cells, you have typically thousands of molecules. So how do you get from one photon to then a macroscopic current flowing through the membrane of the photoreceptor? Well, you need to multiply the signal. And this is a signal multiplication scheme. So here, it activates. Then this guy will activate a few of these molecules, which itself will activate a few of these molecules. So each time, you get a multiplicative effect. And each of these guys will degrade a few of these molecules. Each of these steps here can give you a 10 to a thousands fold a multiplication factor. So that at the end, you get thousands and thousands of ion channels that will open to let the ion currents flow. So you need this photomultiplicative effect. In fact, in a CCD camera, you also have a photomultiplicative effect. It works with electrons instead of working with molecules, but it's the same idea. So you have this photomultiplicative effect. However, there's still a problem with this if you want to be able to detect single photons. And the reason is the following. It's that you see the link with this. Once it starts getting activated, it gets activated. And then, just because of relaxation, it gets inactivated in a state of higher energy. And at some point, it will relax back to the ground state, which is this one. So the photon allows it to be kicked up into a configuration of high energy, where it's active. But then at some random time, it will go back to this state. Actually, it will not go back to this state. It will go back to a state where it has to be recycled. But the bottom line is that the time it will remain active is stochastic, because the reaction that makes it inactive is a random reaction. It's a Poisson process. So the time it will remain active is stochastic. So it's exactly the same as what happens here, where the time it would remain bound is stochastic. But you don't want that, because if it's, so in fact, it would be in the simplest approximation. The time rh is active is distributed exponentially. So there's some rate, so a rate of, let me simplify and say, a rate k of becoming inactive. The probability distribution of the time it stays active is equal to this. That's just an exponentially distributed time. So of course, the average time it will remain active. So it's just, well, you know the answer. It's 1 over k. This will be the average effect of each photon absorption. It will be multiplied, but at the end of the day, you expect something like this. It will be proportional to the time it remains active. But now, if you want really to be able to be reproducible, you also need to control the fluctuations of this. If you want to be able to detect single photons or to be able to count photons, you also need to have some control over this time. Because the idea here is that the response is proportional to this time t that will remain active. The longer it stays active, the more it can categorize this guy and the more it will create output products. So the variation in the response, sorry, the variation in delta t will also give you an indication of what fluctuations you expect in the response. So in this simple model, where the rhodopsin simply deactivates, you know what actually what the variance of something like this is. It's 1 over k squared. The point is that it's equal to t squared. So if you look at delta t over t, it's on average, meaning I take it to square root and stuff. This is the same as what we had before. Like when you have something that's exponentially distributed, the variations you have over it are of the order of 1. And the point is that this, since the response is proportional to that, this will also be of order 1. Actually, it has to be larger than 1, because then there could be more variations as you go down this signaling cascade. There could be more noise added to this. But if you have fluctuations in the response of the order 1, it's like a 100% error. So you're not going to be able to detect single photons. In fact, when you do the experiment, and this is another way of redrawing what I already showed you, which was this, you look the peak corresponding to single photons, it's very well resolved. The variations are small. You can quantify this and show that the variations, the relative variations in the response, so as measured by the output current, are of the order of 25%. So there's a bit of a contradiction I just said. If I assume a model like this, where the things deactivates spontaneously like this, then I must have variations in the time of activation of 100%, of the order of 100%. And because this really is what determines and the entire signaling cascade, I must have variations in the response, which are also of the order of 100%. And yet, when I do the experiments, I find 25%. So what's going on? And we see the answer next week. But essentially, you can think of it in the light of the argument I made about maximum likelihood. It's the same thing. Here, you really want to have a fixed response for each photon event. But when you have a system like this, where things degrade spontaneously with a Poisson process, you're not going to get that. You're going to get exponentially distributed responses. And what the retina has developed is a way to implement this, essentially. So we see next week how it does that. And then next week, also after presenting this and then something else about the early stages of retina, I'll be moving on to something else. I'll be moving on to learning in more general settings. And in particular, I'll be talking about maximum entropy modeling and how to use that to model biological data. And one example that I'll give is actually in the retina so that you'll be in known territory. OK. Enjoy your weekend.