All right, so there is no time to do, we'll probably have two more assignments before our next exam. Alright, so last time we were, we quickly reviewed some elementary solutions to the Schrodinger equation for one-dimensional problems. So we're looking at the energy eigenvalues and eigenstates, that is, the solutions to the independent Schrodinger equation. And so we have motion of a particle along a line, 1D. And there, of course, are two kinds of motion that one has. One has bound motion where the particle is bound between two turning points because at some point it's kinetic energy goes to zero and then it turns around. And then I hit another wall where it's kinetic energy goes to zero and turns around again. Or the particle might be unbound. There's no two turning points that bind bound motion. And in the quantum world there are thus two kinds of energy eigenstates, bound states, which typically we call the energy levels of the system, and then the unbound states, which correspond to situations where the particle scatters off of whatever is inducing the force. And the spectrum thus, the set of energy eigenvalues of this equation come in two flavors, the bounced eigenvalues, which are discrete because of the boundary conditions. They're only allowed, circle-loud solutions. Whereas in the unbound case there is no restriction set by the boundary conditions on the eigenvalue. And thus for the unbound states you have the continuum, we call it the continuum of energy eigenvalues. So when one is solving for something related to the unbound states, when it's not solving for the energy eigenvalues, because all of them are allowed, right? What one is solving for is what is the nature of the eigenfunction, and in particular we want to extract certain aspects about the scattering problem associated that are encoded in the eigenfunction associated with the solution of that energy, okay? And there are many different solutions at that energy. They're degeneracies. So we look at a particular physical context we might be interested in. For example, we might be interested in the problem of I send in waves from the left asymptotically far away as a plane wave, and I want to know what fraction of that say we send it in with a definite energy and far enough away, it's a free particle, so the wave function looks like a plane wave. And then we want to know what fraction or what probability amplitude is there for the wave to be reflected and what is the probability amplitude to be transmitted, right? So how do you solve that? How do you find those positions? So I have a question. That is a question to you. You solve the short equation with a boundary condition. So in this case, you have to find what the wave is inside the valve. You imagine this valve, you imagine that valve. That those boundary conditions involve the continuity of the wave function, the continuity of its derivative, and you solve it and you find those coefficients, okay? And that's a problem that we have in homework, which you've all done because it's due today. All right? We're going to study this kind of problem in more detail in three dimensions next semester where there, again, the problem isn't what is the energy item that is all or allowed. The problem is about something related to the strength of the scattering of a particle off of another particle or off of a scattering center, as we call it. And typically what we're interested in solving for is the scattering system. So we're going to talk about that next semester in greater detail. We might touch on it a little bit later on this semester. For the bound states, what we're interested in is the bound state energies because there are only certain ones are allowed. We're interested in what the energy item values are because not all are allowed. And also typically we're interested in what the eigenfunctions are as well. So we have a few factoids. One is that in 1D there are no, it's impossible to have a degeneracy unless the wave function vanishes in some finite breakage between the two. That is possible. But otherwise the eigenfunctions are not in general. In 1D, that's not true in higher dimensions, higher spatial dimensions. Moreover, the lowest energy level, what we call the ground state, the wave function has no nodes. And each successive bound state has one more node.  If the potential is reflection symmetric, then what we can say in that case is that the Hamiltonian is invariant or commutes with the parity operator. And because the Hamiltonian commutes with the parity operator, and because unless under some very bizarre circumstances there are no degeneracies, the eigenfunctions of the Hamiltonian must also be eigenfunctions of parity. So the wave functions corresponding to bound states are parity eigenstates. As I say, they're symmetric or anti-symmetric functions under reflection of the energy. And because of this fact that they go from no nodes to one node to two nodes to three nodes, they alternate. Ground state is an even parity solution, even odd, et cetera, et cetera. Of course, the simplest problem is the particle in a box where we think about hard wall boundary conditions in that case. And there has to be that the wave function has to go to zero at the walls. And then we have simple alternating cosines and sines, right? And the energy eigenvalues are just h squared, h bar squared, two n times the wave vector for the n eigenvalues squared. And I just want to emphasize one point here. Of course, we should be able to use the uncertainty principle just qualitatively to estimate what we expect the ground state energy to be in the problem. We should be able to do that without going through matching boundary conditions just from our intuition about the problem. Just from the uncertainty principle. How would you think I have any suggestions about that? Yes, please. Would the uncertainty and the position be the width of the wall? Right. So the particle is localized. So just qualitatively, qualitatively, for the ground state, we expect the uncertainty and the position is on the order of the width of the box, right? So how would you then estimate what the ground state energy is? Well, the energy is, yeah, go on. Do you know something like using the weight number? Sure, exactly. It's in the weight numbers related to the momentum, right? I mean, just physically, the energy of the system is the kinetic energy and the potential energy, right? Now in here we can say let's call this energy potential zero, okay? So inside this well, the energy is just connected. What would you say then? What can we say with this value of p squared? Well, for the ground state, this is the expected value of p squared for the ground state. Yeah. Go back to the uncertainty principle. Yeah. If we assume that this system is in a minimal uncertainty position, because we're in a box, we already know what the momentum localized in the box must be, right? Uh-huh. It's just H4 to L, right? So we don't have to, we don't know for sure if it's a minimum uncertainty state, but we could say it's on that order and that's a good thing. So we could say just from the uncertainty principle that this is on the order of that. And so what is the energy of the ground state there? Well, this is just the spread in momentum, right? Because the mean momentum is zero. So this is HR squared over 2 mL squared, which is basically what this is. Wait, back to the pie. It's not exactly that because, you know, inside of the uncertainty state and all that, but that is a kind of back of the envelope calculations everyone should be able to do. Okay? Now, what would you say happens, suppose I have a finite square well? And so now this is some finite depth, not infinity. Same thing. What would you say happens to the ground state energy? Does it go up or down relative to the infinite square well with hard walls? Ground states below the V dot where it's cousin on one's graph, they not notice the change in the tetrahedral rule? It will because of tons, right? So what does the ground state look like in this case? Well, it's got a little bit of an eminence in tail, okay? So what happens to the uncertainty in X in this case? Does it go up or down relative to this case? It goes up. It's slightly bigger, right? So the uncertainty in X is now, you know, it's bigger than that. The details are there. So what happens to the uncertainty in P? It goes down. It goes down when it was for the infinite square well. Which means the ground state energy does what? It goes down. So the ground state energy for the finite square well has to be less than the ground state energy for the infinite square well. This brings up an interesting concept that if the wall gets shallower and shallower, the momentum will go to zero. It will get smaller and smaller and smaller. It doesn't go to zero in this case, in fact. There is always one bound state. No matter how shallow the well, there's always one bound state. That's true for this. There's always a solution. No matter how shallow the well is, there's always a solution. It's barely bound, but it's bound, okay? Always one bound state in this case. You can always find a solution to this kind of problem. At least one. In fact, you're looking at the opposite problem in your homework where you have infinitely small width, where you might think the momentum gets infinitely high that it doesn't because of the way the width goes. There's always, you know, this bound state for the delta potential. Okay, now let me ask you one last question about this kind of problem. Suppose I have this kind of well. So this goes up to infinity. It's got a hard well right at the origin, but then a finite well will be not at this distance. Can you tell me what you expect the solutions to that to be like? Well, this is not reflection of symmetric, right? This is not a, you only have sines and cosines inside the well if their eigenstates are parity. This is not a reflection of symmetric potential. But it does have to be, the wave function does have to go to zero right there, I agree. But maybe I could give you a hint. This potential is what I have in the finite well and then I put a wall right in the middle, right? So given the solutions to this, can you tell me what the solutions to this problem are? Does that have to be sined just because we need that node? It's the odd parity pieces. So there is a solution that looks like this, which would have been like that. So the odd parity solution of the finite well has exactly the right boundary conditions to put a hard wall right in the middle. So this problem has all the odd parity solutions, all the ones that were reflection symmetric guys. So I'd rather just do it a little bit higher. Up here there's another guy, right here. Yeah. I don't think you could start the well where you only have the even parity solution then. That's not symmetric. Impossible. Okay, so if you have even symmetric, but this of course is odd symmetric. I'd be a little bit careful. V solutions are not odd parity with respect to the center of this well. Right? They're not, they're V solutions. This well does not have reflection symmetric, but they are the odd parity parts of this on that half line. There are no ones on any part of the line which are just the even solutions that doesn't exist. I don't know if it's an artifact in the picture, but was there like an adult potential in the middle of that one that you drew or no? It's just like, I mean you were just drawing the other half of the symmetric well. It wasn't like a delta function with potential infinity right at zero. No, a delta function potential is not an infinite part of the wall. Okay, so it's just... This thing is a wall. There's a potential here, and then it goes up to infinity. And this goes on forever. Right, okay. The delta function is just a spike. The delta function would have been this goes back down, and then it's a spike. I think I was confused by the axis. Right, and then it could turn around. That's that. Okay, last question about this do hit. I said in this problem, no matter how shallow the potential is, there is always one bound state. You could always find a solution that would satisfy the boundary conditions. What about this guy, just going up to infinity? The ground state always even though? No, the ground state always has no nodes. It's only even if the potential itself is reflection symmetric. This is not a reflection symmetric potential. These eigenfunctions are not eigenfunctions of parity. Yeah, so there the ground state would be the lowest the u2. Yeah, and as I lower the depth of this potential, is it always possible to have a bound state? No, no, why not? Because this guy disappears. And this guy is not this guy. So because of that problem, as I try to squeeze this guy in, it just can't make the boundary conditions. There's just no way to do it. So it's not the case that every finite square well always has a bound state. This guy there is a minimum depth in order to minimum depth, such that this solution, once I lower it too much, boop, pops out of the well. There's just no bound states at all. This case with the hard wall of the origin is important because it relates to the motion in 3D where this axis is now the radius from the origin. If I think about the problem in 3D where I separate coordinates, it says variable coordinates, an R only exists on the half line from zero to infinity. And so when we talk about, as we will, about bound states in 3D and scattering in 3D, there's finite numbers of bound states, has important implications for, say, scattering in 3D. All right. Very good. So that was our lightning review of piecewise onset potentials in life. And what we're going to, now I'll turn our attention to for about a week or so, is one of the most important problems of a well in 1D, the simple harmonic constant. I didn't care so much about the simple harmonic constant. Well, there are a number of reasons why we care about the simple harmonic constant. One is that this is, you know, one of the most important problems. Well, firstly, if I had any binding potential, or I would say most any binding potential, not all, but most, has finite curvature. So I might have, you know, a binding potential, say, for a molecule that looks like this as a function of position. But near the minimum of the potential, this thing is approximately quadratic. So as long as the particle is not moving too far up the potential, near the equilibrium point, it undergoes simple harmonic. So when we talk about, for example, vibrational spectra of molecules, to good approximation, often we can approximate that by harmonically bound atoms bound together in the intercomic potential that binds them. So that's one reason. Another important reason is, of course, so in field theory, we talk about quantum fields where fields, the degrees of freedom are in the normal mode. So if I have waves on a strain, waves moving on, surface of water, I can always decompose them into normal modes. And those normal, the dynamics of those normal modes are the dynamics are those of a simple harmonic constant, which I will always call the SHL. So when we talk about field theory, which we maybe will do at the very end of the academic year, we talk about the modes of the field and they are described by simple harmonic constant. So that's a whole lot of important reasons. There's one other important reason that we'll see is that it's a natural forum for formulating quantum mechanics in the phase space, as I say, in the coordinates of x and p together. We already saw a taste of this, which I intended we would do with homework in the bigger function, but we'll get a little deeper into that discussion as we go along. All right. So let's do some more monotonic. We all know it, of course. So the Hamiltonian, let's talk about this problem first classically. The Hamiltonian, kinetic energy and potential energy. The potential energy of the system, well, for a simple harmonic oscillator, the force is, you know, the spring constant and x. That's a simple harmonic oscillator and it is a natural resonance frequency that's better with the capital over n and the potential energy is one half capital, right? And with that, we are always almost always by the return of the oscillation frequency, right? So we have a harmonic potential and we typically take the potential energy to be zero at the origin. The equations of motion, Hamilton's equations of motion, regions of motion. x dot is p over n and p dot is a minus the gradient of force this equation, okay? Right, okay, fine. Now, if we're going to get somewhere in this problem ultimately, and this is something generic that I encourage you to think about always when you're getting first presented with a new problem versus a new one, that might be new in some ways. The first thing you should do is to find out what are the characteristic characteristics scales of the problem? That's the first thing you should do in analyzing any physics problem. Any physics problem has associated some characteristic units, some characteristic scales, and one should always express all physical quantities in terms of those characteristic scales. So now I'm going to go on my diet or I'm against SI units, which has to happen once a year. If you want to have a standardized set of units, it's fine if you're an engineer and you want to make sure all the screws fit into the space level the right way. But this doesn't mean you should measure everything you're doing in SI units all the time, because there are other scales. There are natural units. We're talking about high-energy field theory at the Grand Unified Scale. We're talking about the plan scale. We don't talk about things in terms of kilometers. If we're talking about atomic physics, there's the more rakingness. There's characteristic scales in the problem, which is why Fahrenheit is so much better than Celsius. Why in the world do I care where water boils when I'm talking about the weather? Hot is 100, not whenever. And I should measure my size in feet because a foot is how big I am. What's a centimeter? What does a centimeter have to do with my height? So that's why it makes much more sense in my view. So, come on, Cameron. I've said it a few times.  All right, so... That's all his decision. All right. So, what are these characteristic scales?  I mean, one way to do it is sort of systematically is to say the following. Let's say there's a characteristic... So there's a... Let's call it Xc, the characteristic length, Pc, the characteristic momentum, and dc, the characteristic energy. There's some characteristic scales of the problem. So I'm going to define dimensionless. The position I'll call capital X is the position in units of the characteristic. And the momentum coordinate in phase space is dimensionless. And I'll call the dimensionless Hamiltonian. So let's then plug that in over here. And what it says is dimensionless Hamiltonian is equal to the characteristic momentum squared over the characteristic energy and the mass times 1 half e squared plus the mass omega squared, the characteristic length squared over the characteristic energy. Just factoring out all the units. Now, where you put that factor of 2 is the mean of my existence. I have many things, but that's one of them. Because it just drives you crazy when you see there's a square root of 2 to float around. But so I'm going to not put the factors of 2 in here. So one way to define the characteristic units is to say, well, this is of order 1. That just defines the relationship between what I'm calling the characteristic energy and the characteristic momentum and the characteristic energy. So I'm going to define the characteristic units, define the scales. The characteristic energy, whatever it is, is whatever the characteristic momentum over which it's got to be the same. So this is the relationship between these this sets the scale. It doesn't define them yet because I have two equations with three unknowns. I need one more thing. We'll have that soon. So with that set, we can say the following. Let's look at what the dimensionless position in units of the characteristic scale is what is its time derivative as a function of time. So that's equal to the characteristic momentum over mass over there divided by the characteristic scale times, according to that, that's the scheme of putting all that together. You get this. Similarly, p dot is equal to minus m omega squared the characteristic this over the characteristic momentum x, which is minus 4. So in dimensionless units, we have the dimensionless Hamiltonian is 1 half x squared plus p squared and x dot in a dot plus omega p and p dot. So what's the solution to this equation? Well, if I take the second derivative of this p double dot, so what we get is that x double dot plus omega squared x minus x is 0 and p double dot plus omega squared p. Of course, that is the differential equation of the simple harmonic oscillator as we know it. And the solution is there is two initial conditions. As usual, x dot is omega p dot, so this is p dot is 0 times omega, so we have the following solutions. x of t is x of 0 plus omega t plus p of 0 sine omega t and p of t is equal to p of 0 plus omega t and p dot is minus omega x so this is minus x dot. So the trajectory in phase space written in terms of these dimensionless canonical coordinates is what? It's a circle. Indeed, it's a circle and the circle has an amplitude here which is the radius of the circle which is what? Well, it's equal to the square root of x 0 squared and p 0 squared. Is this circle going clockwise or counterclockwise? So what's happening to p as a function of time? p is decreasing so it's going in this direction, right? It's going clockwise. So this phase this is phase space but I can also think about this as a complex space. This is a phasor and its phase is changing as e to the minus i. That's why in physics we always choose e to the minus i. Because that's the direction it's going in phase space. The phase is decreasing. Well, this gives us a hint that we can define this in terms of complex amplitudes. Whenever you have signs and code signs you can always define that in terms of the real or imaginary parts of some complex amplitude. And that complex amplitude so this is the trajectory in phase space it's a circle. The energy of the system by the way is what? 1.5 is constant. This is the energy at all times. So this says that we should define a complex amplitude and this is where that student factor 2 becomes confusing. What is in here? Let me define alpha. It's a complex number it's a function of time whose real part is related to x and the imaginary part is related to p. If it were just that it would just be that vector but we're going to put in the square root of 2 inside there so that x of t is the real part of alpha kind of term. And p of t is the imaginary part of alpha. So this is where p of t is square root of 2 there. I'm just saying that that's p. You're right. If you're actually right it does that must be the case because that's what's constant. It is over root 2 because that's where the energy is constant. Right? Because that is what you get right? Right. And so that's exactly why we're putting it this way. So the energy of the system is in fact equal to alpha star alpha squared A squared. That's equal to 0. So of course it's true at any time that this case alpha is constant. What is alpha of t? Well let's plug that in. Let's plug in these guys. We have x of t and p of t. When you plug that in you get the following assignment for you to try that. This is what we call so this is A into the i phi where A is the square root of the magnitude of alpha that's the square root of and phi is the argument. It's the art tangent of the imaginary part over the real part. So what we have is the solution to the motion of the particle in phase space alpha of t is equal to alpha of 0 e to the minus i omega t where we have alpha of 0 is x plus i t at time equal 0 over 2 and the position of the particle at time t is the real part of alpha and the momentum is the imaginary part. We always think about harmonic motion as the motion of a phaser in phase space that just rotates at the angular velocity omega crop wise. Projected on the x axis we have harmonic motion back and forth at frequency omega projected on the momentum axis we have 90 degrees out of phase harmonic motion. So if we have two harmonic motions that are 90 degrees out of phase with one another that's a circle. Of course we know that the momentum is 90 degrees out of phase with momentum because when the kinetic energy goes to 0 potential energy is maximum when potential energy goes to 0 the energy is maximum. It's called classical physics. However now we want to quantize the phi right there is that like the initial displacement? That's exactly what it is. So it's just the initial if we go back to this picture let's say at time equal 0 we have some initial we have our spring it has some initial displacement and then we have a ball p hammer and we give it a kick. So we have some initial momentum and some initial position. This is phi 0 and then the guy rotates. So this phi is the initial phase in the space plane and then at a time oscillates the question is excellent. Alright quantize what it means is that now X and P are not no longer commuting numbers they're non-commuting operators. So our dimension full X and P satisfy the canonical commutation relation. So we have a new constant in the column before we only had two constants and that completely defined the problem. Now we have a new constant H bar. So we can define all of our characteristic scales now. What is the natural union? What we should have is that the characteristic of the problem should be H bar. Put that together with what we have over there. So we had EC is equal to H bar over M is M omega squared times the characteristic scale of length. You put all that together and you get the following. The characteristic energy scale is exactly how you would expect it. H bar omega. The characteristic momentum scale is whatever the mass is and the characteristic energy scale is squared that's the square root of H bar and omega. And the characteristic length scale is the momentum scale over H bar. So just from this kind of dimensionless analysis analysis of scales we have a kind of sense of what we expected. From the same picture of a particle in a box and just thinking about the characteristic scales from the outside. Oh, there's a length so there's a momentum on certainty. We kind of can guess what order, what's the ground state energy of the harmonic oscillator approximately. And what is the width in position momentum and here they are. And we'll see if, of course, you know that's starting close. We'll get to those. So that's a kind of analysis of every good physicist. So, our Hamiltonian again, we're going to define a dimensionless position in momentum. It's equal to whatever the position operator in units of this characteristic length scale. And a dimensionless momentum operator give them my scaling. The dimension full momentum operator by that scale. And the commutator of dimensionless x with the dimensionless p relative to these scales is i of h bar divided by the product of those scales which is i. Hamiltonian operator kinetic operator and momentum operator plus sorry, kinetic operator and potential energy. In scaling out the units is h bar of omega times x squared plus p squared over. Notice the thing about the harmonic oscillator and this goes to the point I made about phase space plus the potential energy is quadratic and the kinetic energy is quadratic. It's symmetric in x and p which what makes it kind of a natural form for talking about phase space and quantum mechanics. Alright. Now, what about this, we have a non-permission operator an important role here and that is the quantized version of the complex amplitude. The complex amplitude is a complex number classically quantumly it's a non-permission operator. So I'm going to define the operator a as 1 over root 2 x plus i p. It's just the quantized version of the complex amplitude. I could do this in the Heisenberg picture or the Schrodinger picture I don't care. Okay, that operator remember that a does not equal a dagger. A dagger is x minus i p. A is not a normal operator meaning it does not commute with its adjoint. Let us calculate the commutator of a with a dagger. Plugging it in. So I get backering out the root 2s. So let's do it. x can be x but it doesn't commute with p. Right? So this is equal to minus i over 2 the commutator of x with p and then plus i over 2 the commutator of p with x. And the commutator of x with p in dimensionless units is i. Thank you very much.  Or a and a dagger no longer commute and the commutator is a famous one. If we look at our Hamiltonian and we plug in a and a dagger over here we get the following. This is equal to a 4 omega a dagger a plus a a dagger over 2. Classically we said somewhere here oh thank you that the Hamiltonian which is alpha star alpha quantumly because these guys don't commute you have to symmetrize it. So you have the equivalent of alpha star alpha plus alpha alpha star over 2 I mean I don't do that by fiat that's how it comes out when you plug in a and a you plug in x and p in terms of a and a dagger remember x is the real part which is alpha plus alpha or the Hermitian part I should say times root 2 and p is the anti-Hermitian part which is a minus a dagger over 2i times root 2 so you plug these guys into this guy and you get this which if I use the commutator a with a dagger put it the other way I get a dagger a plus a 1 so I get this hr omega a dagger a things you have all seen before but in a different light alright so as you know the equivalent of the complex amplitude squared which is the magnitude of alpha squared quantizes a dagger a this operator we call n so n is the quantum version of the magnitude of the amplitude squared related to energy of the oscillator so n is defined to be a dagger a this is Hermitian we take the adjoint so this is a Hermitian operator and thus it has eigenvalues and eigenvectors Hermitian we can always diagonalize so there exists a set of eigenvectors with eigenvalues I'll call lambda to start I guess Hermitian implies that ok moreover we know some important commutation properties of n n commuted with a well that's a dagger a commuted with a how do you do that yeah you pull out the one that the first guy with that commutation that's ok you put that guy on that side so that's a dagger a a and this is a negative one and the commutator between n and a dagger do the same trick or rune pull out the a dagger you're left with the commutator of a with a dagger that one's a plus one so with that said we can solve for the eigen values of n and thus the eigenvalues of h by the way go back to this picture of the harmonic gospel here what can you tell me about the spectrum the set of eigenvalues of the harmonic gospel here a continuum this potential as a harmonic potential goes off to infinity means of all the states all the in a harmonic potential everything is a bound state there are no unbound states it's true but they're not it's not a continuum it's disappeared so all that you have only bound states there are no unbound states but a harmonic gospel so the entire spectrum of the harmonic gospel of the potential is a square well but not like the finite square well the finite square well has both discrete and continuous eigen all right back to where we were all right so lemma lemma if I apply a to this eigenvector of n it is also an eigenvector of n with eigenvalue lambda n minus one similarly if I apply this this guy is an eigenvector of n with eigenvalue how do we do that well it's easy to check that's what we're deriving that's what we're just deriving indeed that's what it means so acting on this is equal to the commutator of n plus how do you use the commutator you write it up in the other order and you add the commutator and we just said that this was equal to minus a so this then says that this is even since this is an eigenvector with eigenvalue lambda n this is lambda n minus one by the same argument using the other commutator this is lambda n plus one so it increases the eigenvalue by one or lowers it now what are the eigenvalues well to show that that fact they are the integers non-negative integers use the following fact note and the positive operator what happened to the a there you're ending on a dot a you're going to get a lambda minus one times a u where the a comes from sorry there shouldn't be any dagger for me so I activated n on this I thought that you made the a by accident but you see the a is there for the following reason when I do the commutator nx on that it gives me lambda so then I get lambda minus one alright so what I'm saying here is that n is a positive operator how do we know that well to be a positive operator it means the expectation value of this is greater than or equal to zero for all how do we see that well if we plug in for this equals that but this is equal to the norm squared of that and that's a positive number or a non-negative number I should say it can be zero well what it says is that if this is zero it's annihilated it's the only possible way that's why a is known as the annihilation operator what state does it annihilate well how can we see that what we said is that if we apply a to one of the eigenstates of the number operator we get another eigenstate with eigenvalue decreased by one if I do it twice the same thing I get an eigenvector with eigenvalue that's decreased by two units if I do it three times three units if I do it n times it's decreased n units however however because n is a positive operator it has positive eigenvalues which means that at some point this screws up because this will start being negative which means there's got to be a minimum n below which I can't lower anymore there must exist a minimum value of lambda it can't go to minus infinity because this is a positive operator what this implies is that there exists a minimum case such that a does not lower it to any lower eigenvalues the only possible way to be consistent so if that's true then what is the eigenvalue of the number operator on this state I'm calling 0 well that's a dagger a acting like u0 but u0 is annihilated by a so this is 0 so that implies that lambda0 is 0 lambda1 is one more and lambdan is n QED thus n are the natural numbers and thus we call n the number a and a dagger are played a role here of annihilation and creation or the ladder operator that is to say so by the way for shorthand from now on we'll just label this head by it's label n so this has got to be proportional to n-1 and a dagger gives us something proportional it raises and lowers and a on load n equals 0 finally what is the proportionality constant so let's say you want to know what that proportionality constant is what we can do is multiply it by its adjuvant this is a normalized vector so this is 1 this is the number operator so what is this expectation value it's just n right so that helps me that this proportionality constant is some phase times the square root of n and by convention we choose that to be that by convention there's nothing that fixes that phase we just choose it by convention so what we have is that this is equal to square root of n n-1 and if you go through the same algebra this is the square root of n-1 finally if you do this over and over again what you could show is that n is equal to will you raise this n times on the ground state and then you know the most energy state of n is the ground state of our monocosal n equals 0 the energy eigenvalues so h acting on n is e to the n n according to that is h or omega n plus a half n equals 0 1, 2, dot dot dot ground state of course you know what that says is that the energy eigenvalues equally space in the energy separated by h or omega and the zero point energy the energy of the ground state is h or omega over


So, ydych chi'n gweithio? Yn ystod, mae'n gwneud James Brisco. Mae'n... Lai cat, mae'n cyfnodd bywyd. Mae'n ffordd i'r cyfnodd o'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd. Ac, oherwydd, y cyfnodd bywyd. Mae'n ffordd i'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd, ac mae'n ddod i'r cyfnodd bywyd. Mae'n ddod i'n meddwl i'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd. Mae'n gweithio gyda'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd i'r cyfnodd bywyd. Mae'na gweithio, fel y dyma, yn gallu cyflwiğii yda i fi ffocwsio ar y cyfnodd bywyd. Mae'n ddod yn gwneud bod ddim yn y peideud o gwnaetio. Dwi'n mynd i gael, am ydw i'n ei wneud i ni fydd ystod am gyda fydd, a fyddwn i wneud o'r Ddau, mae'n ystod sphyf yn ymgwrthion. Felly, rwy'n sy'n cyfle i mi ddechydau gydag. Felly, os ddewch chi'n fynd wedi cael digwyddol, neu'r gellent yn llwy, oeddwn yn fawr yn bach? Ond yr ymarferwyr, oeddwn yn gweithio'n ddweud o gael. ac mae'n rhoi, dwi'n garfa'n mynd i wedi weithio beth, eu bod從i eit gnwysig ar y cyntaf eich gwahanol, hynny ymorth cynghiliadau penderfyniadol agor dyma, a ymwysig ar y mynd i gwahanol, ychydig o mhercyngiol yma,dweud fe wneudau effeithio deithlach, The biological sciences. And how many people are studying invertebrate models? And how many are invertebrates? And how many don't know? So you're about equally split between drosophor and then vertebrate organisms in terms of what your knowledge is. So thanks for that. Mae'r rhesydant yw rhaid yn ymdill o'r fath yw'r Llywodraeth yn ymddillaw'r Llywodraeth, ond yn fath yw rhaid o'r Llywodraeth yn ymdillaw'r Llywodraeth. Yn cymdeithasol, yw'r Llywodraeth yw'r Llywodraeth, yw'r Llywodraeth, mae'r gwahanol yn ddifo'r hyn ychydig yn ddechrau'r hyn ar gael y ddweud. Mae'n dweud o'r Llywodraeth o'r Llywodraeth mewn meddwlach i'r Ymddill Ymddill, A ydych chi'n yrhaf i'r hunain... ... Wrth i ni'n gallu ar y bôn iawn... ... ac mae efallai rannu gwahanol ni i fynd a'r hyn iawn. Felly piwn i'n ddisgadd i'r rai, fyddwn ni'n gallu. Felly dyna'r gwahanol ni i ddim yn fawr flynyddo i siarad arboed. Diolch yn ôl i ddiogel arnynno. Siarad iddyn nhw efalla ymddangos yn rwyf yn y ddechrau. Y dyma'r Abertaeth Llyw Skleiffo... ...i'r newydd yn yng Nghymru. space. I thought I'd take a picture of this because on Monday it's no longer going to look as clinical, clean and as uncluttered as it does right now. So, okay, so I'm going to, sort of for me, let me sort of step back and introduce what I think is sort of the principal problem, the major question in developmental biology and this is what I think it is. So really what most of us as developmental biologists want to understand is how you start off with a small number of seemingly equivalent cells, how you use that small population of cells to generate a functioning tissue and that functioning tissue is comprised of many more cells and those cells are different and they're organised in particular spatial arrangements. So we think about that problem, there's sort of three things, at least three things going on to accomplish this. So the first thing is growth. There must be some mechanism of proliferation that results in increasing cell number and that has to be controlled and regulated in some way to end up with the right number of cells. So we want to understand questions about growth control. The second thing is there's differentiation. You've got initially equivalent homogeneous progenitor population and from that homogeneous set of progenitors you want to generate different, functionally distinct cell types which do different things. So how do you generate, how do cells differentiate, how do you come different to one another and finally you want to do that differentiation in some kind of temple and spatial arrangement? So making the right number of cells, the right type in the right place. So how do you control and coordinate those processes of growth and differentiation to generate spatial patterns? So if we think about that problem then what are the questions we really want to understand? So the major sort of simplifying assumption underlying most of developmental biology is this idea that different cell types are equivalent to asking about gene expression. So a cell type can be identified by the profile, the combination of genes which are active in that cell. Therefore this problem becomes one of asking how do you set up differential spatial patterns of gene expression? And if we think about that problem then what we need to understand, what is required is some kind of symmetry breaking event, so you need that initial homogeneous population of cells, you need some kind of symmetry breaking event to provide spatial polarisation across the forming tissue. That has to be communicated across the tissue, so cells within that tissue have to acquire some knowledge, some idea of position information relate where they are within the forming tissue. And then a mechanism to convert that position information into discrete regions of gene expression and the discrete aspect is important as well, you want discrete switches in gene expression because it's those discrete switches in gene expression which give us distinct cell types. So it's these sort of questions that we've been interested in understanding, how do cells know where they are in a forming tissue and how is that information converted into cell type identity, converted into discrete spatial domains of gene expression. And as you all know, a underlying, a common mechanism by which this is achieved is the use of so-called morphogen gradients. So this is my sort of summary of how I think about sort of a classical definition of a morphogen gradient. So the idea is that a signal, usually a secreted molecule emanates from a localised source within or adjacent to a tissue forming a concentration gradient. So because it's emanating from a distinct location, that concentration gradient becomes a correlative position. If you know the concentration then you can estimate how far away you are from that localised source. And that information then is used by the cells so it's converted into discrete patterns of gene expression that then impose cell identity. So it must be some mechanism for how cells perceive and respond to that graded information to control differential gene expression. And I know Thomas over the last couple of days would have introduced this idea to you. And one of the most influential pieces of work over the last generation, the last almost 50 years now, is the theoretical work that Lewis Wolpert's building on work of others prior to him. His theoretical idea which has become known as the French flag model, with the idea that you can see this, you can sort of an analogy to the idea that you want to create a French flag of stripes of gene expression in response to a graded input. And this idea, this work by Lewis has dominated the field and dominated our thinking over the past 50 years or so. So what I want to talk about today, so structure for what I'm going to sort of try and cover today, we can break this problem of morphogen interpretation down into sort of four themes. The first is really sort of a purely sort of molecular genetic question about the identification of morphogens. So if we're thinking about the pathing of an individual tissue, is there a morphogen involved? If so, what is it, what is its molecular identity? Where and when is it being produced? Then a second set of questions arises around how is the gradient formed? So how do these are questions about intercellular transmission? What is the mechanism, the molecular and cellular mechanisms that result in the spread of that signal from its source through the tissue? Then we start to think about the cells themselves responding to that. So first sort of questions is then the perception. So in most cases, these morphogens are extracellular signals. So somehow that information has to be communicated across a membrane through a signal transduction pathway into the nucleus to control differential gene expression. So how are signals perceived? How is the signal transduced from extracellular to regulate differential gene expression? And then fourthly, but equally importantly, interpretation. So how do you convert that graded continuous information into the discrete switches in gene expression, which are necessary to define those spatial domains within the tissue and ultimately define cell type identity? So how do you convert continuous graded inputs into discrete responses? So what I want to do is sort of talk a bit about each of these ideas. So really in terms of, I don't want to talk too much about identification of morphogens, rather what I want to start off doing is just introduce you to some tissues which have been extensively used to study the action of morphogens and just make a couple of points as I introduce these to you. So I'm sure you're familiar with many of these. So Thomas will have talked to you, and I guess Defano as well, you must have introduced the idea of anterior posterior patterning, bicoid and gap genes in the early Drosophila embryo. Frank will have talked to you about the Drosophila wing disc. In addition, I want to talk about, just introduce you to two vertebrate tissues, which are commonly used. First is briefly sort of touched on by cat is mesoderm induction. In particular, I'm going to just introduce in frogs and zebrafish mesoderm induction. And then finally, the vertebrate neural tube, which is a system that we study. So let me just briefly introduce each of these to you again and just make a couple of points while I do that. So if we think about bicoid and early anterior posterior patterning, so as you're aware in the Drosophila embryo, along the anterior, so anterior is high bicoid to posterior low bicoid, there's a gradient of bicoid, and that gradient prefigures patterns of gene expression, notably initially the gap genes, which later give rise to the body segments of the Drosophila larvae. And so this anterior posterior bicoid gradient is responsible for establishing that AP body plan. So there's a couple of points I want to make about bicoid. So first of all, as Cat made the point, this has happened very rapidly. So in Drosophila this all happens within about the first hour of life after fertilisation, after egg laying. So this is very rapid, it's time scale of minutes. In addition, this is really unusual tissue in that it's a syncytium. So the stages where the bicoid gradient is being established, there are no membranes which divide adjacent nuclei. So this allows the direct spread of what would otherwise be cytoplasmic molecules. There's one big sack of cytoplasmic effectively, and bicoid is a transcription factor. So unusually compared to the other tissues I'm going to talk about, this syncytial nature of the early embryo allows a direct spread and formation of a transcription factor gradient within the tissue. So the time in syncytium and those two things could well be related to one another. So the speed of development of the early embryo may have necessitated this syncytial early development to allow to give you enough to make it rapid to form gradients. So as I said, this early AP gradient is responsible for controlling the stripes of gene expression along the AP axis that ultimately prefigure the major anterior posterior divisions of the animal. So we think about the Drosophila wing disc. So this again is a Drosophila tissue but it's in the larvae where it's from. So Drosophila, as you're aware, for us mammals are an unusual animal because they have this discrete larval stage of development and it's important from their life cycle perspective. So a larvae doesn't fly around, its main function is to eat and grow and these distinct life cycles of larvae in adult mean that the two don't compete with one another for the same viennish. But what it means is that in the larvae they have to set aside cells which are dedicated to form tissues which are only present in the adult. These are set aside in so-called imaginal discs. One of those discs is the wing disc which will in the adult give rise to the wings of the adult animal. So these wing discs are epithelial structures, epithelial bags of cells and the wing disc itself has a region within it called the wing pouch and these cells within the wing pouch will give rise to the adult wing and there are these wing discs are patterns along two axes. Are people familiar with how you convert a wing disc into a wing itself? If not let me try and explain. So there are two principal axes in the wing disc, so anterioposteria and dorsal ventral. So the anterioposteria axis will result in the if you like the equivalent of thumb to finger axis of the wing whereas the dv axis will result in the middle of the wing disc will eventually be the outer edge of the wing. So what you have to imagine is that the wing disc is like an open umbrella. So it's a flat sheet of epithelial cells, an open umbrella and to convert that wing pouch into the wing itself you have to imagine trying to close the umbrella by pushing up the umbrella handle while holding the spokes of the umbrella, two spokes of the umbrella. So no spokes are the the outside edges of that dv axis so you're pushing up holding that still so you're going from a flat sheet of epithelial and folding it over onto itself like that. So you end up with a the wing itself is a bilayered structure coming from one the one side is the former dorsal half the other side is the formal ventral half and then the dv axis then folds in on itself so you need to have matching between those two sides in order to have the precise positioning of those veins within within the wing. So if we look at those two axes there are two important signals involved in patterning. So along the anterior posterior axis there's DPP which is a BMP family molecule expressed at the anterior posterior boundary and it spreads both anteriorly and posteriorly controlling the expression of domains of gene expression and ultimately determining where the veins will be positioned along the anterior posterior axis orthogonal to that from the dv boundary then wingless is expressed along the dv boundary and wingless signalling is again important for the expression of domains of gene expression and in particular just adjacent to the dv boundary wingless signalling is important for specifying the wing margin and specialized sensory hairs which then will decorate the wing margin in the adult fly. So okay so that was a long explanation so let me give you sort of two important where I think are two important interesting facts about the wing. So it's a very structured epithelium so unlike the early Drosophila embryo now we're in a cellularized tissue and it has a very structured arrangement of apical basal polarity. Sister cells tend to stay next to each other so you don't see much cell movement in this tissue and this process of wingless patterning takes several days so I think it takes about three to four days so it's really at the other sort of time scale from the early Drosophila embryo so we're going from minutes to days here. Okay so let's move into vertebrates so mesoderm induction in vertebrates is another system that has been used extensively to study morphogen signalling. Mesoderm is a tissue that gives rise to your blood, your muscles and the bone so much of your sort of internal structure is dependent on mesoderm and it's induced early in development by signalling in here diagrammatically in Xenopus embryo so signalling coming from the bottom or vegetal half of the embryo so signals of the nodal family so another TGF-beta family member nodal is expressed in the vegetal half and signals to overlying cells to induce gene expression that is responsible for mesoderm specification and different levels of TGF-beta signalling will induce different types of mesoderm so from blood to muscle to bone. So here so in contrast to the Drosophila wing disc here the this is a cellarised tissue but there's much less structure to these cells at this stage and much more mesonchymol-like and there's a lot of cell movement and neighbor exchange going on so this is quite a fluid-like tissue at these stages of development. In addition in both Xenopus and Zebrafish the time scale here is a small number of hours so say one or two hours so this is somewhere in between those the two extremes we've just encountered it's relatively fast and it's in an unstructured tissue. Finally the neural tube so I'm going to talk a lot more about this but the key points are that the neural tube is patterned along both the anterior posterior so head to toe as well as the dorsal ventral so belly to back axis I'm going to focus on the DV dorsal ventral axis so if we cut a spinal cord an embryonic spinal cord along that transverse axis then you can see the DV ventral to dorsal axis laid out and along that dorsal ventral axis distinct neuronal subtypes are generated so these include things such as your motor neurons which occupy the ventral half of the neural tube but also a whole plethora of a molecular and functionally distinct interneurons which are responsible for forming the functional circuits which coordinate motor output and receive and process sensory information from our periphery. These are molecularly distinct and generated at stereotypic positions along the dorsal ventral axis and that pattern of neuronal subtype generation depends on signals emanating from the ventral and dorsal poles so dorsally BMP and wind signaling is important for specifying positional identity eventually hedgehog signaling so sonic hedgehog expressed in the notacall and at the ventral midline of the neural tube is essential for specifying pattern in the ventral neural tube and that pattern is within the proliferating progenitors so the progenitors which will then later generate post-mytosic neurons and that pattern is represented by the expression of a set of transcription factors which have discrete distinct domains of expression along the dorsal ventral boundary. So the embryonic neural tube is a similar to the Drosophila wing disc is a pseudostratified epithelium it's not quite as rigidly structured as the Drosophila epithelium but nevertheless has distinct apical and basal polarities that apical is what will be the central canal and basal is the lateral edges of the neural tube and that is maintained sister cells stay relatively close together but there is some dispersion of sister of sister cells. This process of pattern formation in amniote so in chicken and mice takes at the order of 24 to 48 hours so it takes a longer time than mesoderm induction but not as long as Drosophila wing disc patterning. So we think about those four tissues then so one way of distinguishing them is to think about those time scales so from very rapid Drosophila embryo takes a few minutes to the Drosophila wing disc which takes several days in addition there's those differences in tissue structure cell biology that I've emphasised as we've gone along. So if we think about these these sort of four examples then if we think first of all about the similarities then in each case the patterning happens along a particular axis and when there's more than one axis those axes tend to be orthogonal so that gives you some kind of sort of cartesian like idea of patterning axes within the tissue. In each case there's a graded signal usually a secreted molecule again the exception being bicoid in the early Drosophila embryo so there's some kind of graded signal that is produced from a localized source and spread through that tissue along the axis which is which is the patterning axis and there's gene expression discrete domains of gene expression that then divide that tissue into spatial pattern spatial blocks of tissue and again I haven't gone into this but in each case there's evidence that those graded signals are responsible for controlling that the spatial pattern of gene expression. But there are differences so there's these timescale and tissue structure and cell morphology differences and also there's no evolutionary connection these are very distinct evolutionary processes it's not that there's a there's one sort of common sort of origin for morphogen patterning in each case there are different signals involved the origin of these goes almost certainly into pre-medazoan evolution so it doesn't look like there's a common origin to this so given given the differences then I think it's interesting to think about if we look at in each of these cases compare the underlying mechanisms responsible for establishing pattern then I would argue that that should reveal some sort of fundamental principles about how you can do this so there's some the differences suggest that that any any similarities in the process must be um sort of design constraints okay how are we doing was that all right for an introduction to those tissues okay so now what I want to do is just sort of talk about these last three issues so a little bit about the formation intercellular transmission then talk more about the perception and interpretation but if we start off by thinking about the the formation of these gradients so again if we go back to sort of first principles and think about what is the simplest possible way in which you could establish a a gradient of a secretive molecule so there's sort of three processes involved um and it's simplest there's some kind of uh production of the signal some rate of secretion of flux of that signal into the tissue then there must be some mechanism of spread and the simplest possible mechanism one can imagine would be diffusion so a random walk-like process which just results in the um diffusion of that secretive molecule away from its source and finally some mechanism to clear that molecule from the tissue so that could be some kind of extracellular degradation maybe secretive proteases degrading the protein but it could also be the binding of that ligand to receptors on surface the internalization and then the degradation of that molecule internally so those are three sort of the simplest possible scenario containing sort of three processes some rate of secretion a diffusive like random walk process to distribute that away from the source and then some kind of degradation mechanism and again the simplest possible degradation to mechanism to think of is just some spatially uniform degradation so some spatially uniform degradation mechanism which results in the linear degradation of that so we put those together then you uh it's that those simple assumptions come up with a uh a a description of of um the distribution of the molecule away from its source within the tissue and you can solve this to steady state and you get an exponentially decaying um distribution which is dependent on two parameters right so well x the position within the tissue but the two key parameters here are what I've termed c naught which is the um concentration of the morphogen at the source boundary which is related to the rate of secretion diffusion and degradation rate and then lambda here which is some characteristic length scale the slope of the exponential which again is just dependent on the diffusion and degradation rates so the key assumption here is that the diffusion is a is a diffusive like process a spread is a diffusion like process a random walk like process and of course then you have to question that assumption so is it how valid would that assumption be and of course it's going to be more complicated in this so a tissue is not a uh a homogeneous uniform space in which the molecule is secreted and diffused we know it's composed of cells so that introduces tortuosity to the tissue itself and of course the molecule can be interacting with components of the extracellum matrix or with um self surface proteins interacting binding unbinding which will also stop the um the simple diffusion of of those molecules however as long as each of those processes is homogeneous is isotropic then it's not it's doesn't break the assumption of being a random walk process those it will be an effective diffusive like process even though it's complicated by um additional molecular mechanisms so in many cases where the distribution the spread of morphogen ligands has been analyzed they do appear to fit a exponentially decaying profile now of course the data is always noisy and you can always make the argument that things are much more complicated than this but nevertheless if we look at for example hedgehog protein within the forming neural tube here we've labeled um a neural tube with antibodies against hedgehog protein and you can see that there's a ventral to dorsal gradients of hedgehog protein within the neural tube so hedgehog is being produced by the notochord here and a few cells at the floor plate but as you move further away from the ventral midline you can see lower levels of hedgehog protein and indeed if you measure that then it's very easy to fit a exponential uh to that gradient of hedgehog and that would allow us to then estimate these parameters in particular lambda the characteristic length scale of the gradient and in the cases we've looked at in yes sorry I didn't hear you so again yeah so exactly so there's many complications here so remember this is a pseudostratified epithelium so the central here is is apical and then the lateral edges are the basal uh processes of the cells and what we've done here is we've measured a region of interest along the apical surface of the cells and recovered this so of course you can measure um in different dimensions and in each case you recover data that can be fit by an exponential and in fact it gives pretty close similar values of lambda so we're again I think that's you know that was one of the assumptions that I sort of introduced to you this idea that these systems look like sort of one-dimensional um uh patterning mechanisms and here we're simplifying this not thinking about the apical basal dimension here just the dorsal ventral so the the the characteristic length scale we got when we did this in the vertebrate neural tube was approximately 20 microns um so a cell diameter along this dorsal ventral axis in the neural tube is about four or five microns so a decay length of 20 microns means your the concentration is decreasing to one over e so about a third of the concentration in about 20 microns about four or five cell diameters so over a over about 10 cell diameters you would decrease to about 10 percent of your starting concentration and in other tissues where similar measurements more accurate and more sophisticated measurements than we've done so in the drosophila wing disc for example where this has been analyzed for BMP in wingless then uh lambda so characteristic length scales of a similar order have been found so a few microns to a few tens of microns so this again seems to be a common feature in each of these cases yeah right so I think that's an important question so I I think I will touch on this on saturday in the neural tube where we would argue that in the neural tube the strategy is that you pattern when you're small and then you grow to elaborate that pattern that yeah that has some trade-offs that I'll get to but that's one way of doing this um it would have been interested to get thomas's maybe you have some insight into this in bicoid so where the drosophila embryo is is longer and I think the decay length of bicoid is slightly higher yeah yeah within a certain range though right because you you never have twice the size right you're always uh-huh and the decay length increases so what what kind of numbers you're talking about yeah so the yeah so if you go back to yeah oops right so if we look at so lambda is given by diffusion over degradation rate so manipulating either and I guess degradation rate is um perhaps one can imagine molecular mechanisms easier to manipulate the degradation rate than one can imagine how you would manipulate change of diffusion rate yeah okay but we're still a question so we can see the spreads of these signals away from the source but how do we really know it's acting at a distance so um I want to ask that question and show you some evidence from um from again using the neural tube as an example so you can so is there evidence of direct long range action of morphogens so this question arises because you can easily imagine establishing a pattern a spatial pattern with a direct long range morphogen action but equally with some kind of signal relay mechanism where signals are acting at short range to induce their neighbours to secrete a signal similar or different signal which then again propagates through the tissue in that way and so distinguishing between these possibilities is difficult to do unless one experimentally perturbs the system so really what you want to do the strategy is to specifically block signaling at a distance from the source and ask whether that perturbs pattern or not so we did this by taking advantage of a feature of the hedgehog signal transduction pathway and um to introduce this to you I just need to um introduce one feature one aspect of hedgehog signaling and so I just want to concentrate at the moment your attention on the transmembrane component of hedgehog signaling so there are two important proteins transmembrane proteins involved in hedgehog signaling patched which is the ligand receptor for hedgehog so hedgehog protein binds to patched and then the other transmembrane protein is called smoothened and smoothened is responsible for the intracellur signal transduction so it's responsible for communicating the signal intracellur and the unusual thing about hedgehog signaling is it's a derepression mechanism for initiating signal transduction so in the absence of ligand patched is active and it's repressing smoothened therefore smoothened doesn't signal intracellur in the presence of hedgehog hedgehog binds patched that blocks the activity of patched relieving smoothened of inhibition allowing smoothened to initiate intracellur signal transduction so it's a backwards mechanism you just have to flip what you're used to thinking of when you think about signal transduction however this derepression mechanism for initiating signaling we were able to exploit to generate a useful reagent so again just to emphasise so both patched and smoothened are multiple transmembrane proteins in the absence of ligand patched inhibit smoothened hedgehog binds to patched it binds through two specific extracellur loops yeah that's a whole different question so the answer is it's not clear and in fact aspects of sort of subcellular signal transduction of hedgehog is very unclear and there are lots of complications so can we park that question I'm happy to talk to you about it afterwards so what I wanted to emphasise here is the idea that hedgehog is binding to patched through two extracellur loops and when when it's liganded patch no longer represses smoothened so what we were able to do is design a version of patch we've called patched delta loop two which lacks the second large extracellur domain so it's no longer able to bind hedgehog ligand but this mutant patched delta loop two maintains its ability to inhibit smoothened so it's not able to bind hedgehog but it is able to inhibit smoothened so it becomes it's a dominant active protein which means it's a dominant inhibitor of hedgehog signal transduction so now this provides a reagent in which we can if we were to express this in individual cells this blocks hedgehog signaling even in the presence of hedgehog ligand so we can test this question of whether blocking hedgehog signaling at a distance from the source of hedgehog whether that affects patterning in the neural tube so we've done that I'm not going to go through experiments in detail but let me just give you sort of one example so we've taken advantage of chick embryos to perform these experiments and a technique called in ovo electro operation which allows us to transfect in in vivo individual neural tube cells in a mosaic manner and we've uh the patched delta is hooked up to a gfp so we can identify transfected cells based on the expression of gfp and this technique results in the unilateral transfection so one side of the neural tube is mosaically transfected the adjacent side acts as a control an untransfected control so here we're looking at the effects of inhibiting hedgehog signaling in the green transfected cells by looking at the expression of a a gene called pack seven a transcription factor called pack seven which in the control neural tube in control neural progenitors is not expressed as repressed throughout the ventral neural tube if we inhibit hedgehog signaling on the experimental side here we see ectopic expression of pack seven so these cells are now expressing pack seven and you'll notice it's in mosaic fashion and it's only those cells which have received patched delta only those cells in which hedgehog signaling is inhibited do you see ectopic expression of pack seven so neighboring cells ventral and dorsal to transfected clusters of cells continue to express their normal complements of of genes so this argues that all of the cells within the ventral neural tube have to receive hedgehog signaling to adopt the appropriate gene expression pattern so that argues in favour of this direct long range action of hedgehog and it argues against signal relay mechanisms where cells are handing or adjacent cells are inducing neighbors to propagate a signal in a relay mechanism across the tissue so in the case of hedgehog signaling and in some other cases notably in the drosophila wing disc there's evidence to support the direct long range of zone the drosophila wing disc in particular DPP there's evidence to support along the anterior posterior axis direct long range action of these morphogens but this isn't the case in all tissues so for example if we look at nodal signaling the nodal induction of mesoderm in xenopus embryo so this is work by my colleague Caroline Hill at the Francis Crick Institute she's recently provided evidence of a relay like establishment of nodal signaling within a drosophila embryo so just to remind you nodal ligands are expressed in the vegetable half of the embryo and induce overlying cells to become mesoderm what Caroline's lab found is that this induction of nodal appears to be the result of a relay so initially cells just adjacent to the source of nodal response to nodal and one of the responses is to induce expression of nodal which then is signals to adjacent cells and so on to result in this relay induction of nodal signaling so if you think about that that relay would eventually result in the whole tissue expressing nodal signaling if there was nothing to break it if there's nothing to inhibit the further propagation of that relay and in this this nice study from Caroline what they found is that in addition to inducing nodal at the same time a secreted inhibitor of nodal called lefty one and its paralog lefty two are simultaneously induced and those lefty inhibitors are secreted and actually secreted further than the adjacent action of nodal resulting in the eventual break in that propagation of nodal so you can see that as some kind of Turing-like mechanism reaction diffusion a the propagation of nodal is then restrained by the secretion of an inhibitor but in this case in contrast to the neural tube it looks like there is cell-to-cell propagation resulting in the expansion of of the morphogen expression yeah yeah so these cells up here have I'm not quite sure so what instead of what's the what would be the the alternative interpretation yeah so so normally this gene pack seven is repressed so out the ventral neural tube if we inhibit hedgehog signaling you see the cell autonomous expression of that responding gene yeah we can talk more after and yeah so coming back to this question so there are there's evidence in some cases of relay mechanisms also within literature there's evidence of more complicated cell biological processes involved in the propagation and spread of ligands and this is a very active and sometimes contentious area of research and one of the reasons it's active and contentious is that the the tools to look at the spread of signals are really now only just giving us the the resolution we need and one of the challenges here is that it's a multi-scale aspect of that you really want to be able to to really address these questions you want to be able to look at individual molecules but view those individual molecules at the length scale of a tissue right so you need to go from sort of molecular scales to whole tissue scales to be able to follow what individual ligands are doing how they're spreading across the tissue and that that technology the tools to do that are only really just becoming feasible but in addition to this sort of the simple idea of spread through the the extracellular space additional mechanisms may play a role certainly in in some cases so for example transcytosis where cells take up in vesicles the morphogen and then transport that vesicle across the cell to resecrete it may play a role in influencing the spread of ligands through a tissue and then you know more exotic cellular processes cytonyms these phylopodio like long extensions either coming from the morphogen source into the tissue or from the tissue back to the morphogen source have also been implicated in influencing the spread of ligands and as I said I think this is a very active area of research continues to be a very active area of research yeah yeah right I completely agree so that that's that's certainly possible so one one possibility of course is that the the the length the distance of which the cytonym influences extends is related you could imagine if if the extensions are isotropic and the probability of extensions follows some kind of decreasing probability then in fact you're just back to a random walk like mechanism so it extends sort of it's an important cellular mechanism in that view of long range spread but it doesn't provide any polarity right it could be isotropic in that sense um I think some of the proponents of cytonyms wouldn't agree with that interpretation and would like to emphasize the idea that they provide directionality in which case I completely agree with you that you have to answer the question of what provides that long range direction then okay so yeah the last point I wanted to make about gradient formation is the idea don't forget dynamics and I think a good example of this is again another colleague of mine Francis Crick Institute recent work from John Paul Vincent looking at wingless in the Drosophila wing disc so remember wingless is responsible for the dv patterning in the embryo and most of the work and images you will see a wingless expression is at this late stage of of wing this development where wingless expression is restricted to the future dorsal ventral boundary but if you look back at earlier stages of development in fact wingless expression is initially expressed in almost all if not all of the cells of the forming wing pouch and then over time gradually refines down to the the margin the dv boundary so this is and serial Alexander so John Paul Vincent's work has provided evidence that in fact you don't need the long range spread of wingless in the disc in order to get this pattern instead what may be going on is you've got a a temporal gradient so cells of the margins cells of the lateral edges of the wing disc see wingless early for a short period of time and it's the cells closer to the future dorsal ventral boundary see wingless for increasing periods of time so instead of a spatial gradient you're generating a temporal gradient I want you to remember that because I want to come back and just mention that briefly again tomorrow okay so that was a bit about the intercellular transmission the formation of morphogen gradients so now I want to switch to thinking about how the cells within the tissue are responding to that and first of all talk about signal transduction so how does the information get from outside the cell into the nucleus and of course as we discussed we don't have to think about this at all for bicoid because of this unusual sensitial nature of the early jesofler embryo but for most other cases we do have to worry about this information transmission and of course if we think about this in again simplistic terms the ligand arrives at the cell binds to a a membrane bound receptor which then communicates that signal through a signal transduction pathway eventually resulting in the regulation of some kind of transcriptional effector which then controls gene expression so in this view it's a nice simple direct linear pathway from ligand through signal transduction pathway to regulation of gene expression but of course the reality is is much more complicated in this at every level of at every step in this process there's potential for feedback and interactions so for example within the signal transduction pathway itself there may be feedback where some components of the pathway alter other components so changing sensitivity and more noticeably the some of the target genes themselves may regulate components of the signal transduction pathway so negative feedback for example and in addition some targets of the pathway may affect ligand spread itself so the example I gave from nodal would be an example of that so nodal induces lefty which is a secreted inhibitor of the pathway so you have feedback from the signal transduction pathway which will affect the spread of that ligand through the tissue so all of these processes will introduce dynamics and non-linearity system and potentially timescales between you know we have to think about the this sort of the biophysical processes of kinetics of ligand interaction signal transduction and the timescales of gene expression as well so how do these things what what are the what are the key things here to to think about in terms of the perception how cells convert the extracellial gradient into differential gene expression and again I'm going to take an example using hedgehog signaling and neural tube patterning and now I just want to focus your attention right at the bottom of the pathway downstream of smoothen there's a still fairly poorly defined signal transduction mechanism but what is clear is that signal transduction mechanism culminates in the regulation of a set of transcription effectors of the glee family so the glee family zinc finger containing transcription factors there's three of them in vertebrates in the absence of hedgehog the absence of signaling these glee proteins are proteolytically processed so they're either completely degraded or processed into a truncated form which acts as a transcriptional repressor in the presence of hedgehog that processing is inhibited smooth and is active and the proteolytic processing is inhibited resulting in the accumulation of full length glee proteins and these full length glee proteins are able to activate transcription so important thing here to remember is that the glee's are the effectors of hedgehog signaling in presence of hedgehog they act as transcriptional activators in the absence of hedgehog they function as repressors so we've been interested in trying to look at glee activity in the neural tube and to do this we've taken a fairly simple minded approach of just trying to make transcriptional reporters so briefly what we've done is we've concataromised the canonical binding sites of the DNA motif to which the glee proteins bind concataromising that and hooking that up to a fluorescent reporter GFP protein and then we've made transgenic animals so here mouse embryos containing this reporter so just looking at an overview of this reporter within mouse embryos we can see this reporter is active in tissues where we know hedgehog signaling is active and controls i'm not going to get into demonstrate that it is indeed responsive to hedgehog so we just looking sort of a cross section at two different embryonic stages we can see activity in the ventral neural tube but also in the adjacent somites which you'll hear more about this afternoon okay so now this gives us a reagent where so the fluorescent protein is now a reporter of glee activity and we can look at the level of GFP in the neural tube over the developmental time window in which the pathing of the neural tube is being established so we just look at individual embryos we can see the highest levels of fluorescent protein ventrally where there's high levels of hedgehog ligand and decreasing amounts of fluorescent protein as we move more dorsally in the neural tube so similar to the sonic protein quantification we're just taking a a region of interest a line of interest along that dorsal ventral axis and measuring fluorescent intensity and we've done this over the sort of developmental time window in which pathing is established so this x-axis here is developmental time along the y-axis here is ventral to dorsal so ventral is 0% and dorsal is 100% in the neural tube and then the colour map here is the GFP intensity so if we look at every individual embryonic stage we see high levels of signaling ventrally decreasing as we move dorsally in the neural tube but what's immediately apparent is the amplitude of that gradient changes really markedly over time so quite rapidly it reaches a peak amplitude and then over the next few hours decays back to to basal levels so we can take these data just plot it in a slightly different way it's the same data still developmental time along the bottom here but now we're plotting average GFP intensity so average level of glee activities and each of these lines represents a different position in the neural tube from ventral down at 5% to dorsal 50% in the middle of the neural tube so again you see the same thing at every individual time point you see a ventral high to dorsal low gradient but the amplitude of that gradient changes in time so increases to a peak quite early on and then sort of relaxes back down to to low levels over the the next day and a half so okay so what does that mean so that means that cells within the neural tube are not exposed they don't contain a constant level of glee activity that glee to that level of glee activity is changing over time i'm going to wait one minute wait one slide in fact so so one possibility so just before i get to that question so one possibility is that this could be just about changing levels of heteroprotein so one piece of evidence that argues against that is we can do an ex vivo and in vitro experiment so we can transfect that same reporter into chick embryos and now cut out a little region of the forming neural tube and culture that in vitro and we can expose that to define concentrations of recombinant heteroprotein and we can look at the response of the the cells to a defined fixed concentration constant concentration of heteroprotein so if we do that again this is over about 24 hours you see the same kind of thing happening so initially you get a rapid rise to peak and then decreasing level of gfp of glee activity over time so it's not simply there's not a simple relationship between glee activity and and the external concentration of hedgehog so what is this relationship then so how do cells within the neural tube transform um hedgehog concentration into glee activity so now we have yeah yeah yeah so this is a technique we've developed over you know my career 20 years so it's it's using i mean it's not technically it's not difficult because now we know how to do it okay is that the question or do you yeah i guess i don't think there is a generic answer to that right so i mean this this idea of x planting it's sort of related to some of the work that cat is doing as well is it's this strategy is very established right we we try and recapitulate in vitro um conditions that we see in vivo but in a more defined way yeah and i i've talked more about i'm sure cat will as well this i mean one of the sort of recent advances the last few years is um going one step further and doing this with es cells so taking es cells directing those down particular differentiation pathways in vitro and then manipulating and perturbing the system there so again trying to reconstruct aspects of development in vitro and being able to then manipulate the parameters we're interested in so what's the relationship here so we've got the two data sets so we we've got measurements of the sonic gradient over time i've just shown you the measurements of glee activity over time we've got these in in stage matched embryo so we can ask what is this relationship so okay so this this is a little graph is just a little complicated to get so we're looking here on the x axis at the concentration of hedgehog is measured from those embryos and along the y axis gfp intensity intensity or glee activity levels and each of these lines represents a different staged embryo from young embryos in blue to old embryos in red so if we look at the young embryos first then um the amplitude of the hedgehog gradient is relatively low at these stages yet you've already got high levels of glee activity and there's a relatively straightforward relationship between glee activity and uh hedgehog concentration the more hedgehog you have the higher level of glee activity as we look at older embryos you can see that the the amplitude of the hedgehog gradient increases but at the same time the amplitude of glee activity decreases and there's no longer such a straightforward relationship between hedgehog concentration and glee activity so yeah so this is just trying to summarize that so if we look at glee activity in response to different concentrations of hedgehog over time we see this adapting response where cells are initially sensitive to hedgehog and the level of glee activity rapidly ramps up in those cells and then they become desensitized so they adapt over time to hedgehog signaling and if we look at the effect of this in the neural tube is that what we're seeing is this um very um yeah dynamic relationship between hedgehog concentration and glee activity yeah and some exactly yeah so some of those experiments you can do with that in vitro system so by x planting out a region of the neural plate and then you're in control you can add or remove recombinant hedgehog protein and I will mention hopefully tomorrow an experiment along those lines what was there a specific question would they read different yes so that's exactly so hold that question that's exactly what I'm going to answer tomorrow yeah yeah yeah okay so I think I can I can give you two possible reasons why you do it this way so one if we think of timescale here um so this is a timescale of tissue patterning but of course hedgehog has later roles within the neural tube which I'm not talking about at all so following this period of neurogenesis there's a period of gliogenesis which again different types of gliogenesis so you could reuse that signal later on in addition hedgehog is also used for axon guidance as well so you want hedgehog to stay around because it's involved in other processes but you may not want the neural progenitors to be responding at at the same time so I think that's one possibility I think there's a more um yeah there's another possibility as well which I think is interesting um but remind me about this question when we talk about the dynamics of the transcriptional network because it involves avoiding the dynamical slowing but we can talk about that tomorrow but one so okay so one way of thinking about this then is what cells are doing is converting a concentration of hedgehog into a duration of signaling so the higher the concentration the higher the peak amplitude but also the longer the duration that gli activity is maintained within cells and this will again become relevant tomorrow okay so so what is the possible mechanism of this so what we're seeing is this sort of desensitisation this adaptation so one possible mechanism is negative feedback and indeed there's a candidate for this that is well known within the pathway so one of the targets of gli activity within hedgehog signaling is patched remember patched is a receptor but a negative regulator of the pathway so initially uh head cells are responsive to hedgehog they induce gli activity which results in the induction of patched patched accumulates and then um desensitises cells to continue an exposure to hedgehog so i'm going to skip the next little section okay so the conclusions the ideas I wanted to introduce here about how cells perceive extracellur concentration so the idea that a signal transduction pathway intervenes between morphogen and gene regulation and that signal transduction pathway can have interesting consequences in introducing dynamics and non-linearities to the system but ultimately it's a transcription effectors within the nucleus that control gene expression so it's that's that's where we really need to focus if we want to understand interpretation so that's where I want to turn now and talk in the last few minutes about some ideas about how cells interpret interpret the morphogen gradient and by interpretation really what I want to focus on is this idea that you need spatial domains of gene expression so there's two issues there is how do you get differential gene expression so how do you get different genes expressed at different spatial extents and how do you do how do you generate those discrete switches so how do you convert the extracellur gradient into all or none switches in in gene expression so if we sort of again sort of think go back to sort of some sort of naive ideas so we think about um sort of a simple view of gene regulation imagine a gene controlled by binding sites for the morphogen effector if those binding sites for the morphogen effector are relatively low affinity you need a lot of that morphogen effector around to activate that gene conversely if the binding affinity for the morphogen effector is high then you only need a little bit of the morphogen effector to activate gene expression which will result in a larger spatial domain is everyone sort of happy with the basic ideas of gene regulation I know Thomas sort of introduced you to the idea of enhancers is that everyone kind of happy with that idea is anyone not happy with that idea okay so if we yeah so just to restate that idea so if we think about a very simple uh view of a morphogen that it activates a transcription effector that then activates target genes if we look at the binding affinity for that transcription effector then you can imagine a gene gene b that has high affinities for that effector will be activated to a greater extent within the gradient it only needs a little amount of that effector to be activated whereas gene a which has low affinity binding sites for that effector requires a lot of the the morphogen effector to be activated so is restricted to a much smaller spatial domain and of course in the corollary of that is if you increase the affinity of the binding sites you would expand the domain of expression if you decrease the affinity you would decrease the the spatial extent of expression so this is a a simple and elegant idea and indeed it was very prominent idea within the field and some of the early molecular evidence seemed to support this idea so really sort of groundbreaking work from Janine Nishloin-Volhard it's Wolfgang Driever and Janine Nishloin-Volhard in sort of the mid to late 80s after the molecular identification of bicoid and the identification of some of the target genes lent support to this affinity threshold idea so looking at a reporter for this bicoid target gene called hunchback what Wolfgang was able to show is that if you increase the affinity or number of bicoid binding sites within that reporter you see a a broadening a larger domain of hunchback expression so expressed further down the bicoid gradient towards the post area conversely if you decrease the number of affinity of bicoid binding sites then you see a restriction of the hunchback reporter to uh more anterior regions so that's that's in line with the model um so we haven't measured the degradation rate of hedgehog so in fact what you might expect to see so as I mentioned patched is abregulated by hedgehog patched binds to hedgehog and that binding is then internalised and degraded so in fact you would have you wouldn't expect to see a spatially uniform um degradation rate of hedgehog in the neural tube but being able to assay that directly is is is difficult okay back to the affinity threshold model so okay so this was the uh a simple idea but what I now want to do is introduce uh data which challenges that idea and argue that this is an oversimplification in the fact we need to think about much more complicated uh not much more we need to think about more complex ideas more complex mechanisms so so the hunchback affinity uh seemed to the hunchback affinity for bicoid seemed to support the idea of affinity threshold but when Steve Small looked more broadly at a larger range of uh bicoid targets and note this is many years later now what they found is that there was no clear relationship between the number or affinity of bicoid binding sites and the spatial extent of the associated gene so just sort of more cartoon style looking at the binding affinity of bicoid binding sites associated with particular genes and relating that to the extent of the target gene expression along the ap axis there was no um strong uh positive correlation between those two things so this argues that affinity isn't a simple predictor of spatial extent of the neural tube of course yeah so okay so the binding affinity is implied from the sequence of the bicoid motif so um bicoid binds to a short DNA motif which Stefano may well know in fact no it's a short motif of five or six uh nucleotides the um so the depending on that sequence uh that sequence will affect the affinity of bicoid for that sequence does that answer the question so this is so in this work here it's implied purely looking at by mathematically at the sequence so previous work has identified the canonical bicoid binding site and in vitro defined um the necessary residues and the effect of affinity of changing those nucleotides on bicoid binding affinity here uh target genes of bicoid were identified and the bicoid binding sites within close to those target genes were examined and those sequences compared to um and that's what PWN stands for actually position weight matrix so how close those bicoid binding sites were to canonical bicoid binding sites was assaid was documented here and I think you're picking up on an interesting point because there is clearly a question about whether one can directly extrapolate in vitro binding affinity on oligonucleotides so non-chromatonized oligonucleotides with what is the effective binding affinity in vivo I think that's a um a very live and important issue within within sort of transcriptional regulation in general but it certainly says as these data suddenly says there's no simple relationship between predicted binding affinity and the extent of gene expression so another challenge to the sort of the simple idea as well is just from looking at the gene expression patterns themselves so a simple affinity mechanism will result in graded distributions related to the graded distribution of the morphogen but as we all know we see nice stripes of gene expression whether we're looking at gap genes in Drosophila embryos or the neural progenitor transcription factors along the dorsal ventral axis of the neural tube so there's still there has to be some mechanism which results which converts the continuous gradient of the morphogen into the three spatial domains of gene expression moreover if you look at those patterns of gene expression over time they're also dynamic so it's not that stripes emerge in a fixed they actually change over time this is true even in um the gap gene system remember this is happening over a small number of minutes but still uh Yogi Yeager, John Reinerts and colleagues and many others have documented that the uh spatial pattern actually changes noticeably over time and this is much more noticeable in tissues which develop over a longer period of time so for example in the neural tube over that sort of 24 hour time window of development what one sees is sequential induction and sort of waves of gene expression across the tissue so looking in these cartoons here you can see over this sort of developmental time window here this would be equivalent to about 24 hours in mouse uh mouse neurogenesis we see sequential induction of more ventral genes so the more ventral you are the later you're induced in developmental time but also the spread of those genes dorsally so they're initially expressed more ventrally and then over time move into more dorsal positions so if we look over developmental time along the dorsal ventral axis more ventral genes are induced later and you see this gradual dorsal shift in gene expression uh again it's it's not simply proliferation proliferation does contribute to this and again I'll touch on this in a couple of days time but it's not simply proliferation and tomorrow I will talk more about where those dynamics are coming from okay so if we put this together so the affinity threshold model isn't sufficient to explain gene expression we need to explain so if it's not affinity threshold how is what is controlling gene regulation we need to explain the stripes of gene regulation what accounts for the formation of stripes and what gives us these dynamics and the solution I'm going to argue to this problem is that underlying each of these morphogen systems is a transcriptional network in which the morphogen transcriptional effector controls the expression of a set of transcription factors which are connected together in a network of cross-regulation and it's that combination of graded input filtered through a transcriptional network that is responsible for generating um the features the properties we see so I think I'm going to stop here because yeah I think it's a good place to stop and I think it's a good time to stop but I'm happy to answer questions