 No, we need to press something. This is on, yes. Okay, so let's start where we left off. So, where did we leave off? Okay, so we started off with the master equation really. So, now let's consider what's called the self-activating gene. Okay, so we had the same picture we had before, we produced the proteins, but now we're going to assume that these very proteins add the transcription factors for the same gene. So, we have a system with feedback. A gene that produces proteins that activate its own expression. And so, we can write down the master equation for this, just like we were doing. But now, what changes is that the production rate is a function of the number of proteins that it produces. So, the degradation doesn't change, okay? But the production does. So, now, just in case, I mean, I really mean this is a function, I mean, both of them, that this is a function of the number of proteins. How much proteins you make depends on how much you have. So, remember before, we had the production rate with a function of this external concentration. Now, it's no longer an external concentration. Now, it's the number, this concentration actually depends on the number of proteins that are actually produced. And we're going to now look at the case of when the mean number of proteins is large, okay? Specifically, what I mean by that is larger than one. And so, that the change of one protein is not a big change. So, we're going to expand all functions that we have in the problem around one. So, we're going to tailor, expand, okay? And what I mean by f of g, I mean, are all of these functions. So, this is going to be an f of g minus one and this is going to be an f of g plus one. So, I can just do that. If I do that, you see that first of all, the first terms will go away because they'll cancel with that, right? Well, so, I mean, let's do one. So, if I expand, well, okay, let's not do it because it's easy. I mean, I can just, if I do it, then I get an equation where in the first derivatives, I put a minus sign out. So, with a minus sign, I'm going to get a term from here, which is going to be Rg. And in this term, from the first derivative, I get something with a plus sign, which is one over tau g. So, I'm going to have a Pg and then I get a one half Dg Rg. Okay, I'll put parentheses around it so there's no doubt about what this means. And I get the same sign now in both of this, Pg. Okay, if I expand to second order, where this thing is V, we call V, and this thing we call D of g. And V stands for drift and D stands for diffusion. Okay, so what you should be seeing here is a Fokker-Planck equation. So, drift is what determines the deterministic mean. If we set V of g equal to zero, then we get that the mean g is equal to R, which is now a function of the mean g times tau. And if you think back to what we saw before the break in terms of N0, or rather what I kind of motivated, this will be R0 k minus plus R1 k plus in this case g to the h, k minus plus g to the h. Okay, if it has that form, well, very, it has, right. So the drift gives us, again, recovers the deterministic equation. And diffusion gives us a temperature scale. Okay, so this is like an effective temperature, which effectively what it does is it sets the scale of fluctuations. So the thing that is different maybe from the kind of diffusion that you're used to in these processes is that now diffusion depends on the number of particles in the system, right? Diffusion is not a constant. It actually depends in our case on the number of proteins. And even if this was a constant like we had before, then it would still depend on the number of proteins through this production, right? So we have a protein in general, a particle-dependent diffusion. But in general, this should be, you shouldn't be terribly surprised by any of this, right? So since it's a Fokker-Planck equation, just as a reminder, it generally has the form of a continuity equation. We can write it in terms of a current where everything on this side has the form of the current. And specifically the current looks like V of G, P of G, okay? And so then if we want to set, since we are in 1D, if we do want to look at the, if we do want to look at the steady state, then that means we can ask the question of when is the current constant. And since we're talking about the probability distribution, so as G goes to infinity, the probability distribution has to go to zero, but the, and so does the derivative. And the current has to be constant. That means that in 1D the current has to actually vanish for all G, okay? This is because we're in 1D. In higher dimensions, of course, it wouldn't have to be like that. So, but this allows us just to solve for when the current vanishes. So I'm just going to solve this for zero. I have D of G, P of G. I put the two to D of a side. And on this side, I have V of G and P of G. I'm going to multiply and divide by D of G to make my solving easier, okay? So I'm multiplying by one. This is just one, which allows me to solve for D of G, P of G. So this is now an simple equation to solve. And so I can solve it for that. And then just write it as N over D of G. So, of course, it's just an exponential in D of G, P of G, of two integral of zero to G, DG prime, DG prime, okay? Plus normalization, which determines N. Okay. So what you're left with, so this is the formal solution. What you're left with is having to do this integral, which in general for any form of regulation. So remember that R of G, wherever I wrote it out, is generally, so if we get rid of this brackets now that we're not looking at means, it's a non-linear function of G, okay? So in general, this has a lot of non-linearities in the integral and it's a hard integral to solve and you can't do it analytically in most cases. You can do it analytically in the case of when H equals zero. If you want to do that, you can do that later. But in general, this is there and it's a complicated thing. However, there is an approach. So this is the solution of the Fokker-Planck equation, just to be clear. This is the general solution of the Fokker-Planck equation. There is an approximation we can do to solve this more generally and this is called the small noise approximation. An important thing to remember is that this is an approximation. So we're going to assume now that R number of proteins is given by some mean, like its steady-state value, plus fluctuations around that mean and we're going to assume that these fluctuations are small and the mean, as I said, is the mean of this distribution. So the thing before we've been calling G. So this, we can calculate the Jacobian and since the Jacobian of this transformation is one, this means that the probability distribution in G can be replaced by the probability distribution in eta, in the fluctuations. So because this is the deterministic part, so if we're interested in the fluctuations in this, this is fixed, so we only need to worry about fluctuations in eta. So the Fokker-Planck equation now becomes an equation in eta. Yeah? When we say fluctuations, what do we mean in the self-activating jet system? Okay, so in general, what do we mean by fluctuations? Yeah, so that, I mean, okay, maybe we should start without having a self-activating gene, right? If you just have a constitutively expressed gene, what will happen? I don't have blackboard room. I need to erase something. So if you look at the time trace, okay, if you look at the number of proteins produced as a function of time, since it is stochastic process, it will look something like that, even once you've, okay, because I cannot draw a stochastic process even, okay, around some mean G bar, okay? So at a given time, the number of process will deviate from that, and so these are these fluctuations. It's really just the, at a given time, minus the mean. Now, so you're asking where does this come from? It comes from because there's a rate, and at any time, you will either have a production or you'll have, you'll either make a protein or you'll have a degraded protein, but you can also have, I mean, even if it was one after the other with regularity, the number would still change constantly, but it doesn't necessarily happen that it's always one birth, one death, one birth, one death. You can have a few births go up or a few deaths go down. You can even have, you know, fairly large deviations. So it just means in terms of temperature that this isn't an externally set variable, but it's, okay. So in general, even if we have a gene like this, it's not an externally, so in that case, diffusion is given by a constant, so this is constant because it's given from outside, plus one over TG, right? So the degradation term already depends on the number of proteins, so it depends on the internal variable. So temperature is set by the same, by its system. Does that answer your question? Okay, and then if itself activating on top of the production also depends on the parameter of the system. So I think it's just, it's a property of the system that it internally, you know, that the noise is internally generated, and that's also why you can't get rid of it. So when we talked about the limits, that, you know, like above the process, has the variance equal to the mean, as described by Poisson distribution, you can never get the noise, well unless you have very large number of proteins and you're in the deterministic limit, you will always have the noise because it's just the property of the system itself. Okay, any other questions? Okay, so if we go to the small noise approximation, I actually have something else to say about that. Let me just finish the equation. Okay, so now we're essentially, well, for now I'm just rewriting, and okay, so for now I just rewrote and let me get back to this question. So another thing that you could do is you could do a simulation of this process, right? It's a stochastic process. You can do random Monte Carlo simulations. These simulations are called, it's actually a time-varying Monte Carlo simulation because you have two things to choose. One is the time scale of your next reaction, which depends on these rates, and then the other one is you do normal Monte Carlo of what reaction will happen next, which one of the two. But the time is also a random variable because it varies with the number of proteins you have in the system, right? Because the time is given by the inverse of these two rates. If you have more proteins, those rates are higher. If you have less, those rates are... So these things are called time-varying Monte Carlo or Gillespie simulations. And there I can show you, I can look up a trace in a second if you want to see. Well, maybe I'll do that right now about what that looks like. So this is just to show you that you can model this as a random walk. We already said that in equations, and in each time step you make the decision, so you see that it is, in fact, random. This is the steady-state version of the trace, and so the way you do these simulations is you make a list of all the reactions that can happen, and then you find the probability that the reaction... For each reaction, what's the probability of that reaction happening in a given time interval? And that probability depends on the actual number of proteins you have at a given time. And then you do a trick that you can... So the probability of a reaction happening is the probability of the reaction and the probability of all the other reactions not happening, and then you do a trick that you just multiply and divide by one, which is a naught, which are all the reactions, and this allows you to rewrite the whole thing as when the time at which the next reaction occurs and then which reaction occurs, okay? Because mu is the specific reaction. So that just inverting the first part gives you the time, so you pick a random number, you compare it to a time, you pick a random time, and then you again make a list of all the reactions that happen and pick one of these reactions using... Just doing normal Monte Carlo, you update the number of molecules and you start again. And this is how you generate these kinds of trajectories. Okay, but I think that maybe gives us also a better idea. Okay, and then you collect statistics and you get these probability distributions. But that maybe gives you a better idea about where the... You know, what is temperature and where it comes from. Okay, so getting back to the small noise approximation. So then you make... We said eta is small, so we can say that. And if it's small, then again we can expand to first order around the non-zero tater, well, around the zero tater. So that gives us the mean plus the first derivative plus additional terms. And this gives us... Okay, so the thing is that this is by definition zero, right? This is how you calculate G bar. G bar is the solution of the deterministic equation. So you're only left with eta v bar G bar here. And here this is non-zero. There's no reason for the noise to be zero. And so the leading order in this equation, you're just going to be keeping the leading order is diffusion. Okay, just to be sure everybody's on the same page. The prime means dv with respect to gg. Okay, so then we have dp d eta equals minus v prime g because this no longer depends on eta. eta p eta. Is that right? Yeah, that's right. Plus one-half d of G bar d eta. Okay, and then in steady state, we can solve this. And we can again solve this now by direct integration. So we have v prime over dg, eta p eta equals d eta p eta. So I move this to this side and eta to that side. And I get that p of eta. It's again some normalization constant, exponential of 2. These are constants with respect to eta. Zero eta d eta prime eta prime. Okay, so this is now an integral that I can easily do. It's eta prime over 2, which gives me n exponential of 2 v prime of g d of g prime eta squared over 2. Let's keep it that way and I can rewrite it. So I can rewrite this whole thing as e to the minus eta squared over 2 sigma z, where I've defined this to be 1 over sigma squared. So I'll write it down in a second. But since this is a Gaussian, I know how to normalize Gaussians. So I can write down the normalization precisely. And so specifically sigma squared is minus, because I need a minus, dg 2 v g. And v prime of g is negative for stable fixed point. So we get a Gaussian distribution. Now a point to make, and I really want to make this point. So this is the small noise approximation. This is an approximation. This gives you a Gaussian. This is easy. This is the full solution. This does not have to be a Gaussian. This does not have to be a peak distribution. These are two different results, and people sometimes often forget that they're two different results. But in fact, it's a very strong approximation. So one thing that I'm going to tell you, and again you can convince yourself by playing with it, or maybe you'll hear about it later, is that one possible solution of a self-activating gene is that it's a bi-stable system. So if you look at the probability distribution, then it's a bimodal distribution. It has two fixed points. I mean, the deterministic equation has two fixed points. The stochastic equation has two bumps. So this will not describe that situation. It will approximate it by something that looks like this. Okay? That gets the mean right. And so this is wrong. But the solution of this equation will get you that. What did I want to show you? Okay. So in fact, the Fokker-Planck equation, so the solution of this and the solution of the master equation are essentially completely equivalent as long as the mean number of proteins is relatively large. And it's not very hard to get a large number of proteins. So let me maybe get out the right. I have it here. Okay, I can just write it from my head. So what's being plotted here is a comparison of the Fokker-Planck equation and the master equation. Sorry, this board is slightly too small for me. And let's delete that. Okay? So it's a comparison of the solution of the Fokker-Planck equation and the master equation as a function of the mean number of proteins as a simple birth-death process. So when this is constant. And what's being plotted is something called the Kullbach-Liebler divergence. Okay, so this is an information theoretic measure and we'll get to information theory later on in the course. But what it is, is a way of comparing two probability distributions. So let's say I have a probability distribution P calculated from a master equation. Okay? And I want to ask how different is it from the probability distribution Q calculated from a Fokker-Planck. I can call it, well let me call it Q just to give it a different name. But it's exactly the same probability distribution I have here. Okay? So I want to calculate the difference between these two. So what this is is the Kullbach-Liebler divergence became the P from the master equation and the Q from the Fokker-Planck equation. Where the two equations have the same support because they describe, well, okay, no, they actually, they have to have the same support so then I have to do something so that they do, so I discretize the Fokker-Planck equation in practice. But if the Kullbach-Liebler divergence is a measure of comparing two distributions and if they're the same then this log ratio is, so the ratio is 1 so the log ratio is 0. So if the distributions are the same this is 0 and if they're not the same then the number grows. Okay? So here it goes to 0 because for very large numbers the Fokker-Planck equation and the master equation gives you the same thing. And for very small numbers the Kullbach-Liebler divergence goes up because the distributions are different and what you see with the stars is different points along here. So if you look at the two distributions here you can't tell them apart but if you start going to small numbers means of under one they start looking different and they look more and more different as the number gets small. The point is that you really have to go below one to see a difference. So for most practical purposes it's really, there's no difference of solving the Fokker-Planck equation and the master equation and since the Fokker-Planck equation is a continuous equation it's easier to solve in practice. Okay? So that's it for the measure. So you understand why we're taking the ratio, we're taking the log to amplify differences and then what we're doing here is we're weighting it by the correct probability distribution. So here we know the master equation is in a way more principled so we trusted more so we're going to reweight it and in regions of space where this is not 0 we want these differences to be more important so that's why we remake it. There's a thing you can do if you don't actually trust either one of your distributions more than the others that's called the Jensen-Schanen divergence which is just, what's the word? Symmetrized form of this distribution. And so, but that one I'm going to screw up if I write it from memory because let me see what I did with this piece of paper where I had it defined. Okay, anyway, any questions for this part? Okay, yeah. So for concreteness the symmetrized form of this is called the Jensen-Schanen, well one symmetrized form. So the Jensen-Schanen divergence between two distributions p and q is one half of the dkl between p and m. I'll define m in a second and the dkl between q and m where m is one half p plus p plus q. Okay, and we can show that they are the same so maybe I'll do that right now. Okay, so just take one half sum over p log p, p plus q divided by 2 plus one half q. Sorry, okay, I mean I'm doing it on this great distribution so let me stick with that. log q p plus q divided by 2. And then you can just manipulate the measures and see that this is p log p, yeah. Yeah, it makes you happier. I mean if anything I don't know. Yes, it's Jensen-Schanen. Yeah, okay, I don't know. If it makes you happy I can call it this. Okay, doesn't matter. I'm actually, I'm going to stop here because what I can show later that this isn't related to the entropy but we haven't really gone into defining information theory to get entropy so I'll just stop here. So for the time being all I guess I want to say is that there's ways of comparing probability distributions using these measures and they're kind of useful and we'll probably get back to that. So now I guess I should take a break, right? I'm a bit confused about how, oh no because yeah, no it's fine. I can take a break of 15 minutes, right? This will be the way it goes and then I teach for two hours, right? Well then we end at no. Okay, so we can go on. But wait, let's just get the time straight. So we started at 230. We're supposed to end at 430 altogether, right? Okay, with the break. No, but they, I mean, okay. So let's end at 415 and let's take a break. So let's do one more thing before we take the break. That's what you want? Yeah, okay. So 415, what else do I have? So that's good. We'll finish noise and then we'll take a break before evolution. And I'll figure out my confusion about time scales. Okay, so this is really the last thing. It's going to be somewhat heavy computationally, but hopefully you'll see something that really most of you have never seen before. So we can write this, as you know, there's the third level of description of all of these equations. Which is the Langebaum equation, right? And you probably know that you can go from the Fokker Planck to the Langebaum using something like maths and seizure rules. And we, you know, we won't get into the field theoretical aspects of it. But we're going to write down the Langebaum equation with the zero noise correlations and the noise, right? So since this is temperature, this now comes out here just the way this temperature does. Only this is the same term we wrote down on the board before. That now depends on the G's and all that. So let's do it. Okay, so this is the general formulation. And now what we're going to do is we're going to do it for a two-state system that we had before. We're going to do it for the gene. Remember that there was the gene that has the binding side and the binding side can either be occupied by the transcription factor or can be empty. So we're going to write down Langebaum equations for this. So we're going to scroll like before the state of this gene N and G is going to be our protein which can also die and it has a rate of being produced. Okay, so the way to do it is that the state of the system can change and it's going to be external. It's not going to be self-activating now. There's noise and then proteins are produced but now they depend on the state. There's a general rate but then we're going to say that this one is really zero. It's just going to be a mean between the two states. Yeah, so the D is the thing we had before. It's in general the R and G1 over tau and G. That's the same diffusion coefficient we had in the Fokker-Planck equation because it's the temperature of the problem. It's the thing that sets the scale of the fluctuations of the problem. Okay, think about Einstein relations. Think about calculating the correlation from a Fokker-Planck equation and then you'll see that you get this out of the correlations and that's the scale that sets the temperature. I mean, okay, whenever you have a... What I did derive is this form for the diffusion equation. So you don't have a doubt that that is actually the form that goes into the diffusion equation. Then what I made now is I made a jump from Fokker-Planck to Launcheville. When we go to Launcheville, is there any more serious way I can motivate this that we are assuming that there's just noise added and the correlation of this noise is set by the correlation of the noise from the Fokker-Planck equation and that's what's given by, well, generally what we call temperature but temperature is in fact related to the diffusion coefficient. So that's why I'm saying if you calculate the correlation from the Fokker-Planck equations through what's the... I'm blanking out on the name of this. Somebody help me. The response, the thing that links correlation to response fluctuation dissipation theorem gives you this. It's a general part. This is the general formulation. This is the specific formulation for our system except that it makes no sense to give a Launcheville equation if you don't say what the noise correlations are. So in G, so the means are going to be zero in both cases. We can set the means to zero and then we have to say what the correlations are. Now, again, we're going to use the fact that we do know what the correlations are because of the, you know, what sets the scale of fluctuations is the noise in the birth-death process for the genes and what sets the scale, so this we did in the rife, but we can easily intuit it. So what sets the scales of the fluctuations for the binding and unbinding? It's a two-state process. So what sets the scale is the sum of the two rights. This is just, I now no longer have n zero and n one, so I'm writing it as one minus n. Yes, exactly. And since n is, well, okay, it's a binary variable, I mean it's a continuous version of a, right? It describes, it's the probability of being in one of these two states. It's not a binary variable, but it's the probability of being in the two states, but because of that, that gives the form of this. And n is, so you bind, if you're not n, yeah, you bind if you're not n. And you go out, yeah, and n describes the probability of being in this bound state. Okay, so now what we're going to do is we're going to linearize these two equations. So we're going to say that n is described by its mean plus fluctuations around it and I'm going to write down equations for the fluctuations around the mean. Because they're equations, as I said before, are nonlinear, so in general it's hard to solve them. And the reason we're doing this is because I want to show you the usefulness of the Langevin approach as opposed to the other one. Well, there's some things that are easier in each of these approaches. So in general, the Fokker-Planck equation gives you the full distribution, but if you're not interested in the full distribution, then the Langevin equation is completely equivalent and often easier to solve. Okay, so all I did is I linearized these equations. Remember that the means are defined by setting the steady state, so some things go to zero. And now we're going to go to Fourier space, so I'm going to define the Fourier transform of these variables. Okay? And now we Fourier transform. So I hit this with the Fourier transform. I'm going to get minus i omega from the t. And then, okay. And then I'm going to write, so here these are just writes. So I'm going to define a timescale 1 over tau c, which is just what you have there. And this also gets Fourier transform, of course. So I'm going to say that this is equal to 1 over tau c. And here I have minus i omega g omega dn minus 1 over tau delta g plus, okay. And so now we want, okay. What we're going to try and calculate here is overall the variance, the fluctuations in these systems. So we're going to try and calculate the variance, which is related to the power spectrum of the system. Right? The noise power spectrum, if I take the correlation in Fourier space, this correlation is called the noise power spectrum density, right? I mean, you may not remember this, but I'm sure you've seen this, okay. Either in some optics lab or in actually doing statistical mechanics when you solve a large equation, right? We usually go to Fourier space when we calculate, when we look at response theory. What I'm doing here is linear response theory, right? And you calculate when you do fluctuation dissipation, when you derive the fluctuation dissipation theorem, you do similar things. So we want to get this. This is our goal for this system, okay. So in order to get that, I need to calculate this object. I need to calculate the Fourier space of the correlator. So I'm going to take my two equations and I'm going to start on solving them. So first I'm going to write them in matrix notation. Hopefully not get it too wrong, okay. And now all I have to do is I have to solve it, so I have to solve for that. I have to invert this matrix, okay. And so that's the general solution. What I'm really interested in is G, because that's what I need for this. So from the first one, well, the first term I have R xi n times the 2. And for the second one I have xi g. And this one cancels with that. So I have this. And if I'm not going to correlate it with its complex conjugate, I need to remind myself of what xi n, xi n star is. And, okay, so what is it? Where was it? It was here. But I'm going to be taking, I've linearized, so now I need to take it in steady state. And in fact, in steady state implies that the two of these are equal. If I look at this equation, the deterministic steady state means the two are equal. So if I sum the two of them, I just have 2k minus n. Just to remind you where it came from. And the same thing for G. Right, now the two of these are equal. So if I add them, I can take either one and I choose to take n. Okay, we're nearly there. I just have to erase something that I will not need. So now we have to calculate. Yeah, I didn't erase this. Okay, so we want to calculate the power spectrum. So we want to calculate dG of omega dG star of omega. So I just have to multiply this with a complex conjugate. Remember, there's no correlation between the noise in n and the noise in G. So that means I only have to square the terms. This is mean, omega squared tau c minus plus 2rn omega squared plus tau minus 2. Okay. And then we want to integrate this over omega. And that's what's called sigma G. Okay, so sigma G, I am now going to rewrite as, I'm going to do a little bit of rewriting and collecting terms. So it's just going to be algebra here. So I have 2rk minus n, which is a pre-factor. And here I'm going to take Tc and omega together, which means that I'm going to get both a Tc and a T squared up here. And same thing here. 2rn T omega T squared plus 2. Okay. I'm going to do a change of variables, introduce x, which is going to be omega T, meaning dx is d tau d omega. So I have dx to pi, change of variables and the two taken out. R squared k minus n tau c squared tau c squared x tau c over tau squared plus 1, x squared plus 1 plus squared. Okay. And now we're going to do something vaguely exciting, which means we're going to make an approximation. The binding, unbinding is much faster, time is much faster than degradation. Okay, so this means that tau c over tau is much less than 1. And this is a reasonable assumption in many cases. Degradation usually happens by dilution in cells, especially in bacteria, there's very little passive degradation. So, but binding and unbinding can happen from time to time. So we're making a separation of time scales, but this means that this goes to zero. So get rid of that. So with that, okay. And there's another thing that I'll remind you of, that dx over x plus 1 from minus to plus infinity is pi. So we can now do these integrals. We get a pi from the two remaining ones, since this one went away. So we have pi over 2 pi. From the first term, we have r minus r squared k minus n bar tc squared tau. And from the second one, we have r n tau. Okay. And so we're nearly, nearly done now. r n tau, and this, I fortunately, I erased. This is g bar from the deterministic equation, if you look back. And there's another useful identity that k minus times tau c, which is what we have in a way here, is k minus over k minus plus k plus c. And this is by definition of n 1 minus n bar. So that means, and I would really love to have some colored chalk, is that this whole thing becomes n bar 1 minus n bar. I have r squared left. I have a tau squared, because I'm going to do this twice. So, at the end of the day, when I put all of this together, I get sigma squared is g plus, that's the second term, r tau squared k minus tau, and 1 minus n squared is our bars. Okay. So what this is to remind you is the variance. I guess if it's the variance, I should have been calling it squared all along. Of above death process plus promoter state fluctuations. Okay. So now comes the reward for the hard work. You can, you know. So how can this be used? So this is, what this is, is the developing fruit fly, embryo. Okay. What happens in the fruit fly is that the fly mother lays some mRNA at the tip of the fruit leg, egg, and then this mRNA turns into proteins and diffuses and forms a protein gradient. That's what you're seeing stained here. You're seeing actually a full fly egg with the different dots at different nuclei. They haven't turned into cells yet. And you're seeing where it's really bright green, there's a lot of this protein, and as you can see that the protein decays and forms a gradient. And so this is development. So this is what gets you from having a fruit fly embryo to actually having an adult fly. Okay. And so the information from the very beginning is contained in the embryo. The embryo knows what to do. It has a blueprint. But this blueprint relies on all the things we've been talking about this morning, and it's a noisy process. But at the end of the day, the flies, as we do, develop in a reproducible way, right? We all look the same. There's no difference between any of us except for like cosmetic differences, right? Depending on which hairdresser you went to. But, you know, it's all reproducible. It works. But everything is noisy, right? So, okay. So that's one of the questions that, in general, biophysics is trying to look at. And this is a real, real research question. So, okay. So the gradient, what the gradient does is it actually controls a set of genes directly. It regulates them exactly in the way we've been writing down. This is one of the first genes to be turned on. It's called hunchback, because if you knock it out, the fly has a hunchback like Quasimodo. And then hunchback and croupole and other genes control these, what are called perul genes. These are the striped genes that form these very precise stripes, okay? And it's, they're very precise. That means that if you normalize by the length of the embryo and take different embryos, they'll always be in the same place. And it's very important that they're in the same place, because then these stripes determine how the fly body will be partitioned, okay? They'll determine where the fly will have its abdomen, will have its head and so on. So if you get this wrong, the fly will not be okay, okay? So somehow you need to transmit this information very reliably in the fly. And so people are studying this a lot. And one of the things they did is they actually went in and they measured the noise in the expression of this gene, of this hunchback gene as a function of this gradient, okay? There's a lot of, again, there's a lot of interesting things going here. You can calculate the diffusion process. You can calculate the time scale. We happened to, somebody has happened to done all of this. But I just want to tell you why you've just suffered for what you've suffered. And so this is because these people did this calculation before you and they did exactly that. So what you're looking here is the variance in this hunchback gene plotted as a function of the mean number of hunchback molecules produced, okay? So the mean number of hunchback varies as a function of bicoid. So G is hunchback and C is bicoid. C is the green gradient you saw. And so these are the experimentally measured points. And the different lines are different kinds of noise models. Where is the one we just derived? This is the one we just derived. It's the blue line where you have the birth-death noise plus the noise from binding and unbinding. And it actually doesn't agree exactly. This is another noise. This is the binding and unbinding but with diffusion noise, which I think Thierry will show you how to calculate. And that one also doesn't agree. And it turns out that if you take the birth-death noise with the diffusion noise, where the diffusion noise is noise coming from the fact that you need to actually diffuse and hit the transcription factor site, right? So the diffusion is also a noisy process. It has some noise. Then that's the black line and you see that it agrees. So it actually, what I put you through isn't necessary for understanding the fly, but it was a hypothesis that didn't work out and it's a useful calculation to know how to do. But here you see that this is actually a very good fit. And we can actually now say that we understand the important sources of noise in this fly embryo. So with that, we'll end with noise. We can take a break and 10 minutes. And then it's very short after the break. Okay. Okay. So we're going to do, we're going to talk in the remaining 10 minutes then, right? Is that it? About evolution. So first of all, so what is evolution? Evolution is generally the study of diversity and how we got there and how organisms change with time and then those changes stabilize. So much longer time scales, much sort of longer scales in general than we looked at before. But it's also, it's interested in diversity and so, and it started with Darwin, of course. So first of all, I want to show you this. These are Darwin's finches. So you probably know that Darwin went to the Galerapagos and then he looked at different finches, which are these birds, and he figured out that something must be going on, that this isn't, you know, divine intervention giving us all these weird birds. And so I just want to show you this because I'm guessing most of you have never seen a finch dead or alive before. And so, you know, since we talk about evolution and so on, this is what, so this is what Darwin drew and this is what they actually look like in real life, okay? So they are very, very different, there's very different kinds. Now you can say you've seen a finch. So anyway, so Darwin and many others including Wallace went around the world collecting these different species of birds and, you know, just figuring out that there are, you know, you can sort of see differences and on, you know, on different islands depending on environments that they look differently. So Wallace is the loser in the game. He actually went and collected an amazing collection before Darwin came back and his ship sank, so he was in the Caribbean collecting stuff and then his ship sank off the coast, you know, of what is now the United States and all his specimens went to the ground and he made it back to England but without much to say for himself. And so that's a very sad story but just also to say that, you know, we remember Darwin but there were many others and they were all writing letters to each other. They were traveling around the world on these ships and writing letters to each other. So they were all in touch, right? It wasn't just one guy sitting in his armchair and then going to Argentina. It was really much more of a collective effort. So anyway, so we end up with diversity and we realize there's diversity in nature and then we're trying to explain how, first of all, what is the diversity in a population and then how it got there. So let me give you some numbers about diversity and this is about humans. So our genome length is 3 times 10 to the 9 base pairs, okay? So, right? Our genome is all the DNA that we have in a given cell. A base pair is the letters in the DNA. So we have two chromosomes, one from mom, one from dad. And that means we have 10 to the 9 base pairs together. And then I'm going to give you a fact that there is one difference, the one different or one different base pair per 1,000 base pairs, okay? So if we compare two haploid chromosomes, haploid just means half of ours. So either if you compare one of yours with the other one of yours or if you compare yours with your friends or your non-friends or with whoever, they will differ by 10 to the 9 divided by 3. They will differ by so many differences, okay? So the mutation rate in humans, right? That's how changes happen. These base pairs mutate is 2 times minus 8 per base pair per generation. Now there are roughly 6 times 10 to the 9 people in the world. So that means there's 12 times 10 to the 9 gametes or individual chromosomes, okay? So that means that each base pair is hid this times the mutation rate, which gives you 240 times per human generation, okay? Each base pair is hid 240 times. So these are the two numbers to compare. So if each base pair in a genome is hid that many times per human generation, then how come if we compare any, if we look at the genomes, there's only a thousand differences, there's only one difference per thousand base pairs, okay? So really every position in the DNA is hid 240 times per generation. We've been around for quite some time, okay? So why are we so similar? Why are the not more differences, okay? So the question is why is this so small? That's basically if you want to know what is the question that evolution tries to answer, this is roughly the question, okay? Part of the answer is because of selection. So although the changes are random, then not all changes remain and survive in the generation. But again, we'd like to, and I'm no way am I going to actually make this work for you in the next couple of lectures, but as humanity, as scientists, we'd like to explain this number, right? If you want like a goal in life that you will not reach for sure, okay? It's like with this lag operon, right? We'd like to have the numbers agree. If we understand everything about evolution, it should agree, okay? So just briefly, well just briefly before do I have a slide on that? Yeah, so how do genomes evolve? So these lines are supposed to be genomes and so they evolve by mutations. So base pairs actually change. So the letters change and A goes to a T or a C or a G. And then mutations can be beneficial, deleterious or neutral. So they can be good for you, bad for you, or you don't care. And then comes natural selection and it figures out whether it's good, then it's probably more likely to stay bad or you don't care. Some of these will stay too, as we'll see. Then there's genetic drift, which is just a fact. It's again, it's the evolution word for small numbers. Every mutation when it appears, it appears as one copy, right? It's a random event, happens as one copy. So you have to go from one for it to spread in the population and that's what's called genetic drift. Other forces are demography, which is such as the fact that humans came out of Africa. That's a demographic, sorry, that's rather geography. There was a bottleneck, there was a geographic effect. Then there's demography, which are also bottlenecks that sometimes the population gets decreased and only a few individuals survive and only they get to spread the genes late. So there's forces like that and then there's recombination, which is the fact that you have two individuals who exchange DNA. And so you know that this happens for higher organisms, but in fact all organisms do that. Bacteria also exchange DNA, it's called horizontal gene transfer and so people get very excited because they say, oh bacteria have sex, but if we have time, I'll show you why it's actually very important from a computational point of view. Yes, say that again? Also yes, I mean not only you can think of other effects of bottlenecks, for example on selection, but you're right that a bottleneck is an important thing. If the bottleneck is very slow, then the population size gets reduced so much that genetic drift starts playing a role in a system that didn't have to play a role. Okay, so as I said there was Darwin, we've moved on from Darwin and the way we study evolution now is on one hand we go out, we essentially do what Darwin did, but in a slightly different way. We collect organisms, so these are flies for example. We no longer, I mean there's still some people collecting flowers and birds and so on, but most people collect a few chosen number of species and then they sequence them. We have sequences which are machines that take the DNA out of anything and then give you a set of letters. And we have high-fruput sequencing and this gives you 23andMe and whatever else you want. But this is a big source of information about evolution. Then we have fitness experiments which are done on yeast, E. coli. It also flies recently, so you take two things to two species like this in the lab and you compete them and so you see who grows faster. This is probably E. coli. We'll talk a bit about that. Then we have theoretical population genetics which is probably one of the oldest forms of studying biology by mathematicians. It was initially done by mathematicians and physicists joined in the game quite recently because what physicists like about it is that evolution is the ultimate random process. It's really one random event that then propagates and takes over. So there's cool stuff to do with that and you see from the point that we'll derive this equation but you'll see that it had best some analogies with what we've already seen. And then there's a lot of work on doing experimental evolution of networks and functions because the important thing to understand about evolution is that yes, what you do is you're breaking things, you're basically changing things, but you need to change them in a way that you don't break the functioning of the whole organism. The reason that I wanted to tell you about all this gene regulation stuff is that now you understand that everything is connected with everything, right? So if I break the gene, this gene will no longer produce proteins, then if these proteins are really essential for the functioning of the organism, the organism will die and it won't evolve. So how do you evolve new functions while constraining old ones? Okay, so that's a means of motivation and time's up. So we'll start from here next tomorrow.