 Welcome to this course on NMR in Structural Biology. NMR has emerged as a technique with applications in all branches of science. It was discovered way back in 1945 in the solution phase by Professor Felix Bloch at Stanford University and his group and Professor E.M. Purcell at MIT Boston and that was the beginning of the NMR in the solution phase. Previously of course some observations were made by Rabi in the gas phase and it was such an important discovery is that all of these people got Nobel Prizes. Felix Bloch and E.M. Purcell they shared the Nobel Prize in 1952 for physics for the basic discovery. Subsequently a whole lot of discoveries have happened and that have led to more Nobel Prizes and structural biology has emerged as one of the most important contributions of NMR. In the initial phases it was mostly applications in chemistry, initial days and with developments such as Fourier transform NMR in 1966 by Richard Ernst and Anderson. Two dimensional NMR in 1974 by again Richard Ernst and J. J. Nair a Belgian scientist. This revolutionized the applications of NMR in chemistry and structural biology opened up the floodgates in biology. While these were technological advancements there were basic discoveries of various types like chemical shift and this was discovered by an Indian scientist Dharmathi, Dharmathi SS Dharmathi who was initially a postdoc at Stanford with Felix Bloch and he joined Tata Institute of Fundamental Research in the 1950s. That was the time when the chemical shift was discovered. Then we had discoveries like spin-spin coupling. These were important parameters, NMR parameters. Then we had things like spin echo. This was discovered by Hahn again in the 1950s. The 1950s saw a whole lot of developments, discoveries in NMR. All of them have led to great contributions in NMR and its applications. We will start this course with the basics of NMR and eventually progress towards the applications at the highest level namely in structural biology. Because this is very important to understand the fundamental principles of the technique so that we can appreciate the applications. You can design new techniques as per the needs, as per the problems and that is what has led to unimaginable developments and applications of NMR. So let us start with the basic concepts. So we will start here. And what does NMR stands for? NMR stands for nuclear magnetic resonance. All nuclei have two fundamental properties, intrinsic properties. One, spin angular momentum. This is a quantum mechanical concept. It did not start from the nuclear. Of course it also was there for the electrons and slowly, slowly one discovered that the spin angular momentum is a very general concept. This is a quantum mechanical concept. There is theory of angular momentum. We will only pick up the results from this theory. Spin angular momentum, the nuclei have different energy states depending upon the angular momentum it processes. This is quantized. What does that mean? It has very discrete values. Though spin angular momentum, we said this is quantized which means it can only take discrete values. And it can have various values. It is indicated by a particular number known as the quantum number. There is a spin angular momentum quantum number. It can take various values 0, half, 1, 3, y, 2 etc. various values it can take. Now every nucleus can exist in many different states. The nuclear states, the so called nuclear states corresponding to different quantum numbers just as it happens in atomic theory. You have the electrons in the first orbital, second orbital, third orbital, etc. And they have different energies. So all of these have different energies. Most often we deal with the ground state of the nucleus, ground state which is the state of lowest energy. And the spin quantum number, here the spin quantum number corresponding to the ground state of the ground state is referred to as nuclear spin. In the common language, it is referred to as nuclear spin and is represented by the symbol I. Angular momentum is obviously a vector. A nuclear spin is a value which is a number. This is the quantum number which is a number. And the angular momentum corresponding to this will be a vector. So angular momentum itself is a vector. So we represent it like that. Now what is the magnitude of this vector? The magnitude of this vector is h cross under root i into i plus 1. i is a number here. And what is h cross? h cross is h by 2 pi. h cross is equal to h by 2 pi. And h is the Planck's constant. Now once we said it is a vector, it has a orientation. So the angular momentum has a orientation. Orientation in 3D space, it turns out from the theory of angular momentum that the z component of i is also quantized. It is also quantized. What is the meaning of this? So if I were to draw the angular momentum vector like that, suppose this is angular momentum vector, it has a magnitude which is h cross under root i into i plus 1. And it makes an angle, let us say call it as theta. If this is the z axis, x, y, z axis, if they are, then we have a component here. This is the i z component. This is quantized. So i z is quantized or i z is discrete. So the z component is often represented. The magnitude of the z component, i represent like this. Magnitude of i is represented by modulus of i. Magnitude of z component is represented by like this. And this is given by m h cross. And what is m? m is a number, is a quantum number, another quantum number. And this is called as magnetic quantum number. And what values it can take? It can take a certain number of values. It can take values m is equal to i, i minus 1, i minus 2, so on up to minus i. That means it can take 2i plus 1 values. So in other words, if I have a spin of i, if we have a nucleus which has a spin of i in the ground state, nucleus spin i, it can have 2i plus 1 orientations in space. For example, if i is equal to half, how many will be there? 2i plus 1 is equal to 2. So there will be 2 possible orientations. So I can represent them like this. So I can have one orientation like this, another orientation like this. Because what are the values it can take? m can take values half and minus half. So the z component will be half h cross and minus half h cross. So if this is the z axis, therefore I will have to write this as on one on the positive z axis, other one on the negative z axis. So therefore the z components, the magnitudes of the z components are equal but they are on the z axis in the opposite direction. If I take the absolute value, then both these orientations will have the same absolute value. But the orientation are different. This angle will be different and this angle of course will be the same. And what will be this angle? Because cosine theta is equal to m h cross divided by h cross under root i into i plus 1. Because the length of this vector is h cross i into i plus 1 under root and the z component is m h cross. Therefore cosine theta is equal to m h cross. This h cross h cross will cancel. Therefore cosine theta is equal to m divided by under root i into i plus 1. So therefore you can see how many angles are possible. For the case of i is equal to half, I have two possibilities. So if I have i is equal to 1, then there will be three possibilities, m will be 1, 0, minus 1. And then I can draw this thing like that one here, one there and one here. Let me use a different color here to indicate this. So I have one here, one there and one here. You can actually again calculate what is the magnitude of this angle and your three possible orientations. If I have i is equal to 3 by 2, then there will be four possible orientations. So i is equal to 3 by 2. I will have four possible orientations here, here, here and here. So this will be 3 by 2, m is 3 by 2, half minus half minus 3 by 2. So these are the m values and correspondingly the z component will be m h cross. So this is the first important property of the nuclear which is very fundamental to NMR. What is the next property? The next property is, next property, fundamental property is magnetic moment. Nuclear have a magnetic moment. It turns out that the magnetic moment which is represented by the symbol mu is written as mu. This is proportional to the angular momentum. Mu is proportional to gamma times i. This is a vector. This is also a vector. And gamma is a constant which is called as the magneto-guirich ratio or sometimes it is also written as gyro-magnetic ratio. Both are the same. And this is the property of the nucleus. Gamma is a property of the nucleus. What does that mean? Different nuclei which may have the same i value may have different magnetic moments which means for example i is equal to half. How many nuclei are possible here? You have large number of possible. I will give you the rules for that. For instance you have proton, nitrogen 15, carbon 13, phosphorus 31. All of these are i is equal to half but they are different magnetic moments. And that is because the magneto-guirich ratio gamma is different. So therefore the magnetic moment is an important property. It is an intrinsic property of the nucleus which it determines the application of NMR to various systems. So that is a very important property. And also you notice that the magnetic moment is parallel, mu is parallel or anti-parallel to the parallel to i depending upon whether gamma is greater than 0 or gamma is less than 0. This can happen. Gamma can be positive or negative. Most nuclei they have it positive but some nuclei have it negative also. For example nitrogen 15 has negative gamma. Proton is positive, proton, carbon 13, P31, these are I am labeling only taking only those nuclei which are of common interest in biology, structural biology. So these ones have positive gamma. So this is the fundamental property of the nuclei. We have two entities. One is the spin angular momentum and the ground state of the nucleus is characterized by a number which is called as the nuclear spin and that determines its angular momentum. And nuclear has also an intrinsic property known as the magnetic moment. The magnetic moment is proportional to the I value. Now is it possible to know what kind of, are there any rules for the angular momentum nucleus spin? Can we calculate the angular momentum? There are certain rules, empirical rules, rules for I. How do we know what sort of a I a particular nucleus will have? So if one, if atomic number is E1 and mass number, mass number is the same as the isotope number. If the mass number is E1 then I will have values 0. For example carbon 12. If the I is 0 then of course it does not have any magnetic moment and therefore this nucleus will not be NMR active. We will say then it is not NMR active. Number two, if nucleus has odd mass number then I will have values half, three, four, five, six, three by two, five by two, etc. These are empirical rules which have been established. There are theories to calculate this of course but we cannot go into those ones in this course. Then number three, for odd atomic number E1 mass number I will take integral values 1, 2, 3, etc. For most of the time we will be dealing with nuclei which are I is equal to half. Occasionally we may come across I is equal to 1, deuterium for example has I is equal to 1. Sometimes we may deal with deuterium as well in biological NMR. So, this is the nuclear properties. Now what happens if you put this nuclei, an ensemble of nuclei in a magnetic field. I told you earlier that all possible orientations of the nuclei have the same energy. In the absence of any external perturbation all orientations have the same energy. Let me write that here. All orientations same energy. However, if we apply a magnetic field H naught, what happens? The magnetic moments will interact with them, magnetic moments will interact and this energy is E is equal to 1. Equal to minus mu dot H naught. These are vector quantities and H naught is conventionally considered to be applied along the Z axis. H naught is the magnetic field and H naught is conventionally taken along Z axis. So, as a result what happens? The different orientations will now have different energies. So, therefore different orientations will have different energies. So, let me draw that here. So, I have the nuclear spin oriented like this, oriented like this. The mu dot means E is equal to minus mu dot H naught means this interaction, this component and this component here they have different energies. So, this if I want to write this is equal to minus mu Z H naught. Mu Z is the Z component of the magnetic moment and this is then given by minus gamma M H cross H naught. The energy magnitude is given by this and therefore now we see depending upon the value of M the energy value is different. Therefore, these two have different energies. So, here I have M is equal to half and this is M is equal to minus half and what will be the energies? For M is equal to plus half the energy value is negative and for M is equal to minus half energy value is positive provided H 0 is along the Z axis and this is M is equal to half and this is M is equal to minus half and conventionally we label this as alpha and beta. These states discrete states are now labeled as alpha and beta states. Now, this is so far as the Z component is concerned. However, what about the orientation in space? The orientation in space when it is not only two dimensional it is three dimensional. What happened to the X and the Y components of the magnetic moments? Do we know them? No, we do not know them. Only the Z component is properly defined the X and the Y components what does it mean? Suppose I wait to draw a cone here like that. I draw a cone here. Similarly, I draw a cone here and now I can draw these vectors angular momentum vectors here, here, here, here, here, anywhere on the surface of the cone. So, if I have an ensemble of spins which is there in the solution we are larger number of spins. The molecules have millions and trillions. You have 10 to the power 19, 10 to the power 20. If you take molar sample you have 10 to the power 23 kind of things. That many spins are there and where are they located? You cannot change the orientation with respect to the Z axis. That orientation remains the same. So, which means they will all be distributed on the surface of the cone in a random way. X and the Y components are not well defined. I X, I Y are not well defined. Therefore, the I vector or the mu vector can be oriented on the surface of the cone in a random manner. If this is the case then what happens? Suppose I take an orientation like this. I take a projection of this. This is my I or mu. Both are same. Now, I take a projection of this on to the XY plane. This is X, Y let us say and the Z is on the top. So, then I have this projection. If I take a component here, if this angle is particular value let us say phi, I am taking the cone like this. Now, this angle will be different depending upon where I have my vector on the on the cone surface. Whether it is here or it is here or it is here, it will have a different angle on the in the XY plane. So, therefore, this phi is called the phase, phase of the spin. Since they are randomly distributed, since they are randomly distributed, this is called hypothesis of random phases. Let us consider a situation of I is equal to half. I is equal to half. I have this here, one orientation like this, other orientation like this and I have this cones here. So, I draw the cone here and all the spins are oriented on the surface of the cone here, here, here, here. Now, this energy is alpha is lower in energy and this energy is beta, this is energy here, this is the alpha state and this is the beta state and obviously, they have a different populations. So, if I want to write the populations of this as N alpha and I call it as N beta, what determines this populations? The populations are governed by Boltzmann statistics. So, there will be more number of spins along the on in the alpha state compared to the beta state. Now, let us look at one of those, let us look at the alpha state for example. Now, if I were to take the projections of all of these components on the XYZ planes, what happens? All the Z components will add, all the Z components will add because they are in the same direction. What about the XY components? XY components are along distributed in the 360 degrees phase, anywhere on the phase. So, if I draw this, if I draw this phase here, they have XY plane, they are anywhere present. So, for every one orientation, there is a correspondingly opposite orientation, therefore they will all cancel. So, there will be no XY component of the magnetization from either from the alpha state or from the beta state. Now, what about the Z component? The Z component, there will be positive Z component as well as negative Z component. So, there will be a positive Z component and a negative Z component, these are not equal. Therefore, they will not cancel entirely. As a result, there will be a net magnetization, net component along the Z axis. So, we will represent this as this thick arrow here, this we call as the equilibrium magnetization. No transverse component, equilibrium magnetization is along the Z axis, 0 XY components. This is often represented as M0 and XY component is 0. So, therefore, at equilibrium, the magnetization of the spin system is along the Z axis and there will be no components along the X Y plane. This is an important concept and we will take it forward from here in the next class. We will stop here.