NMR Spectroscopy. Structural Analysis. Hans Wienk & Rainer Wechselberger 2011

Transcription

1 NMR Spectroscopy in Structural Analysis Hans Wienk & Rainer Wechselberger 2011

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3 This reader is based on older versions which were written and maintained by a number of people. To the best of my knowledge the following persons were involved in the history of this document: Rob Kaptein, Rolf Boelens, Geerten Vuister and Michael Czisch. The current version was completely revised and adopted to the lecture 'NMR Spectroscopy in Structural Analysis' We thank Hugo van Ingen and Marloes Schurink for proofreading, discussions, suggestions, and continuously improving the course material. Rainer Wechselberger & Hans Wienk Utrecht, 2011 Please report errors in the text and/or explanations or any 'unclear' passages to: rwechsel@its.jnj.com or hans@nmr.chem.uu.nl The course material is also online: NMR Spectroscopy in Structural Biology iii

4 CONTENTS READER INTRODUCTION Typical applications of modern NMR Some history of NMR Spectroscopy Aim of the course General outline of the course BASIC NMR THEORY AN ENSEMBLE OF NUCLEAR SPINS Ensemble of spins Effect of the radio-frequency (RF) field B SPIN RELAXATION Molecular basis of spin relaxation FOURIER TRANSFORM NMR From time domain to spectrum Aspects of FT-NMR SPECTROMETER HARDWARE The magnet The lock system The shim system The probe The radio-frequency system The receiver NMR PARAMETERS Chemical shifts J-Coupling NUCLEAR OVERHAUSER EFFECT (NOE) Dipolar cross relaxation NOEs in biomolecules NMR Spectroscopy in Structural Biology iv

5 9 RELAXATION MEASUREMENTS T 1 relaxation measurements T 2 relaxation measurements TWO-DIMENSIONAL NMR The SCOTCH experiment D NOESY D COSY and 2D TOCSY THE ASSIGNMENT PROBLEM Chemical shift Scalar coupling Signal intensities (integrals) NOE data BIOMOLECULAR NMR Peptides and proteins Nucleotides and nucleic acid STRUCTURE DETERMINATION Sources of structural information Structure calculations EXERCISES OEFENTENTAMEN APENDICES Appendix A: Vector and matrix features and other mathematical functions Appendix B: 1 H and 13 C chemical shifts of common functional groups Appendix C: Random coil 1 H chemical shifts for the common amino acids Appendix D: Nuclear Overhauser Effect Appendix E: 2D NOESY experiment Appendix F: Amino acid COSY, TOCSY and NOESY cross-peaks Appendix G: Typical chemical shift values found in nucleic acid Appendix H: Typical short proton proton distances for B-DNA NMR Spectroscopy in Structural Biology v

6 1 INTRODUCTION All spectroscopic techniques are based on the absorption of electromagnetic radiation by molecules or atoms. This absorption is related to transitions between states of different energies. The nature of the electromagnetic radiation varies from hard -rays in Mössbauer spectroscopy to very low energy radio-frequency (RF) irradiation in NMR spectroscopy. Other spectroscopic methods, applying electromagnetic radiation of intermediate energy, are microwave spectroscopy (vibration and/or rotation of dipolar groups in molecules), infrared (IR) spectroscopy, where vibration states are excited, ultraviolet-visible (UV-Vis) spectroscopy, where the electronic orbitals of atoms are involved and X-ray or atom absorption spectroscopy involving the inner electron shells. In Nuclear Magnetic Resonance (NMR), transitions occur between the states that nuclear spins adopt in a magnetic field. Since the energy differences between these spin states are extremely small, long-wavelength radio-frequency matches these differences. Accordingly, NMR is a rather insensitive method and more sample material is usually needed than for most other spectroscopic methods. On the other hand, however, NMR signals are quite narrow and therefore the resolution can be so high that hundreds of signals are resolved in a single NMR spectrum. Also, the interaction between different nuclear spins is manifested in NMR spectra, for instance, in the form of J-coupling or the Nuclear Overhauser Effect (NOE). These properties have made NMR an indispensable tool for structural studies in chemistry and biochemistry. To date, NMR is the only available method to determine the structure of proteins and nucleic acids in solution on an atomic scale. Therefore it is a wellestablished method in the field known as "Structural Biology". READER 1

7 1.1 Typical applications of modern NMR Chemical structure analysis Synthetic organic chemistry (often together with mass spectrometry (MS) and infrared spectroscopy IR) Natural product chemistry (identification of unknown compounds) Study of dynamic processes Reaction/binding kinetics Chemical/conformational exchange Structural studies of soluble biomacromolecules Proteins, protein-ligand complexes DNA, RNA, protein/dna complexes Oligosaccharides Solid state NMR (ssnmr) Membrane protein structures Membrane lipid packing Solids like metals, silicates, soils and clays, polymers, etc. Drug design Structure Activity Relationship (SAR) Magnetic Resonance Imaging (MRI) MRI today is a standard diagnostic tool in medicine READER 2

8 1.2 Some history of NMR spectroscopy NMR spectroscopy was discovered in 1945 by Bloch at Stanford and Purcell at Harvard University. For this both received the Nobel Prize for physics in Initially, NMR belonged to the realm of physics but after the discovery of the chemical shift (i.e. nuclei in different chemical surroundings have different resonance frequencies) the technique quickly became very important as an analytical tool in chemistry. The development of stronger magnets (maximum proton frequency is now (2010) about 1000 MHz) and of multidimensional NMR methods allowed its entry in the field of biology. As a result of its continuously increasing importance in modern chemistry, biochemistry and medicine, two more Nobel Prizes for NMR followed in 1991 (Richard Ernst) and in 2002 (Kurt Wüthrich). 1.3 Aim of the course This course will bring the student up-to-date with the principles of modern NMR methods and provide a basic understanding of how these methods work and how they can be applied to derive three-dimensional structures of biomolecules. After a brief theoretical introduction of the basic physical principles of NMR, we will discuss the origin of the parameters that determine the appearance of an NMR spectrum such as chemical shift, J-coupling and line width. Spin-relaxation (i.e. how a spin system returns to equilibrium after excitation) is important as it determines the line widths of the NMR signals and how fast experiments can be performed, but also the intensity of the Nuclear Overhauser Effect, which is the major source of information for the structural analysis of biomolecules. The modern way of recording NMR spectra is by applying short radio frequency pulses and analyzing the response by Fourier transformation. This so-called pulse Fourier Transform NMR (for which Ernst received the Nobel Prize for chemistry in 1991) also allows the measurement of two-dimensional (2D) NMR spectra (and even 3D and higher- READER 3

9 D). An introduction is given to the FT-NMR technique and the principles of multidimensional NMR are reviewed. Exemplary, some basic 2D NMR experiments will be discussed in more detail. Finally, the important process of assignment of biomolecular NMR spectra is explained and an overview of the possibilities to extract a 3D structure out of NMR data is given. In the course all steps that eventually lead to an NMR protein structure are described in detail. Essentially, certain nuclei have spins that orient themselves in an external magnetic field. Manipulation of these spins by electronic pulses and delays gives rise to NMR spectra. From these, several parameters are influenced by properties that depend on the structure of the molecule under investigation. In the end, all these parameters are combined in a structure calculation. Figure 1.1: Protein structure determination using NMR equipment. READER 4

10 1.4 General outline of the course 1 Introduction (what is NMR and what do you study with it) 2 Basic NMR theory (how does it work) 3 An ensemble of spins (from single atom to real samples) 4 Spin relaxation (after excitation: back to equilibrium) 5 Fourier Transform NMR (with a single RF-pulse to a complete spectrum) 6 Spectrometer hardware (what kind of device do you need for FT-NMR) 7 NMR parameters (what you can see in an NMR spectrum and why) 8 Nuclear Overhauser Effect (How does the NOE work) 9 Relaxation measurements (experiments to measure relaxation properties) 10 Two-Dimensional NMR (how to add an extra dimension and why) 11 The assignment problem (which signal comes from which atom) 12 Biomolecular NMR (nucleic acids and proteins, spin systems and (structural) parameters, sequential assignment) 13 Structure Determination (which parameters to use, how to calculate a structure) READER 5

11 2 BASIC NMR THEORY The energy states, between which transitions are observed during an NMR experiment, are created only when a nucleus with magnetic properties is brought into an external magnetic field. These magnetic properties of nuclei can be derived from a quantum mechanical property, the spin angular momentum, I, a vector with magnitude and direction (for an introduction in vector and matrix calculus, see Appendix A). Most nuclei have such a spin angular momentum, which is determined by a corresponding spin quantum number I, which can be integer or half-integer (I = 0, 1/2, 1, 3/2,...). Usually we simply speak of a nucleus with spin I. The magnitude of the spin angular momentum is given by Eqn I I( I 1) (2.1), where is h/2 (h = J s is Planck's constant). Due to its quantum mechanic nature, any component of I along an arbitrary axis of observation, for instance the z-axis, which is by definition the direction of the external magnetic field, is quantized: I z m I (2.2). Here, m I is the spin quantum number which can adopt values from I to +I in steps of 1 (a total of 2I + 1 values). A combination of Eqns. 2.1 and 2.2 leads to the following spin-state diagrams for nuclei with different spins (e.g. ½, 1, ³/ 2, ): Figure 2.1: spin-state diagrams for I = ½ (left), 1 (mid) and 3/2 (right). READER 6

12 A combination of their spin angular momentum and positive charge causes nuclei to have a magnetic moment (compare the effect of an electric current in a circular wire). This magnetic moment is directly proportional to the angular momentum: μ I (2.3), where is called the gyro-magnetic ratio. Since I is quantized, accordingly also is quantized and we can express in terms of the spin quantum number, I or z in terms of the magnetic quantum number, m I : μ I( I 1) (2.4) and μ (2.5). z m I Classically, if we bring a bar magnet (compass needle!) in a magnetic field, denoted B, the magnet will tend to turn and orient itself in the field. This is a consequence of the fact that its energy, given by the inproduct E μ B (2.6) will then reach a minimum. For quantum mechanical objects such as nuclear spins the situation is similar except that now only a limited set of discrete orientations (quantum states) are available. Eqn. 2.6 still holds for nuclear spins. With the convention that B lays along the z-axis we get the energy of a nuclear spin with a magnetic moment of in an external magnetic field, B 0 as E z B 0 (2.7), and with Eqn. 2.5 it becomes clear, that the energy of a nuclear spin depends on the magnetic quantum number m I : E m (2.8). I B 0 READER 7

13 For a spin-½ nucleus this results in two possible energy states, denoted for m I = +½ and for m I = ½. As a consequence of their quantization, the nuclear spins cannot be found rotating in order to orient in an external magnetic field, but they rather can be found only in two discrete orientations, corresponding to the two possible energy levels described by the two magnetic quantum numbers. In an energy diagram: Figure 2.2: Energy diagram for a I = ½ spin system. In all forms of spectroscopy, transitions between energy levels are induced by electromagnetic radiation of a particular frequency 0, provided that the frequency matches the energy difference between these energy levels: E h 0 (2.9). This is sometimes called Einstein's equation and it is a basic relation in spectroscopy. Based on Eqn. 2.8 we find that E = B 0 = h 0, or 0 B 2 (2.10a). 0 Even simpler: 0 B 0 (2.10b). READER 8

14 expressed in terms of the angular frequency, using 0 = 2 0 (2.10c). In NMR the Eqns. 2.10a and 2.10b are often called the "resonance condition", i.e. the condition where the frequency of the radiation field matches the so-called Larmor frequency 0 = L = B 0; 0 = L = B 0 /2. For nuclei with spin I larger than ½ we have multilevel energy diagrams (see Figure 2.1). However, the selection rule m I = ±1 still holds so that we arrive at the same resonance condition. Since the energy of the spin states depends on the strength of the external magnetic field B (Eqn. 2.6), we can modify Figure 2.2 as follows: E E = +½ B 0 (-state, m = ½) E h 0 = B 0 E = ½ B 0 (-state, m = +½) B 0 B Figure 2.3: Field strength dependent energy difference for an I = ½ spin system. It is evident that the separation E of the energy states and and thus the resonance frequency, depends on the sort of nucleus () and the strength of the external magnetic field B 0 or, in other words, on the strength of the NMR magnet. Table 2.1 shows some properties of nuclei that are important for applications in organic chemistry and biochemistry. READER 9

15 Table 2.1: Properties of selected nuclei. Isotope Nuclear spin I Resonance frequency Gyro-magnetic ratio [T 1 s 1 ] Natural abundance [%] (MHz) * 1 H 1/ H C C 1/ N N 1/ O O 5/ F 1/ P 1/ *resonance frequency at a magnetic field of T (Tesla). In this course we will focus on spins with I = ½ because they have only two possible energy states and accordingly give only a single spectral line, which makes them the most popular spins in high-resolution NMR. For protein studies most frequently used nuclei are 1 H, 13 C and 15 N (note that 12 C has no magnetic moment, and 14 N has a spin 1). For nucleic acids 31 P is an additional important nucleus. So far we have given a quantum mechanical treatment of the nuclear spin in a magnetic field considering discrete energy levels. This led to the resonance condition Eqn Interestingly, the same expression can be obtained from a classical description of the effect of a magnetic field B on a spinning magnet with a magnetic moment that is tilted with respect to the field direction. While a classical non-spinning bar magnet would just READER 10

16 orient itself in the direction of the field (compass), a spinning magnet cannot do this but instead performs a precession about the direction of the field. In essence this is a consequence of the conservation of angular momentum. There is a perfect analogy with the motion of a spinning top (Dutch: "tol") in the gravity field of the earth. The spinning top will also undergo a precessional motion. Note that the direction of the precession should be perpendicular to both B and. Mathematically, the effect of the torque acting on a spinning magnetic moment is given by the cross-product, which can be written as the determinant of a matrix (appendix A). This is the equation of motion : dμ( t) dt B μ e B x x x e B y y y e B z z z (2.11), where e x, e y and e z are unit vectors (i.e. vectors with length 1) forming an orthogonal coordinate system along the x-, y- and z-axis, respectively. With the usual convention that B is along the z-axis (B z = B 0 and B x = B y = 0) the equations of motion for the magnetic moment become: d x( t) B0 y ( t) dt d y( t) B0 x ( t) dt d z( t) 0 dt (2.12). It can be derived that a correct solution is given by: READER 11

17 ( t) (0) cos( B t) (0) sin( B t) x ( t) (0) sin( B t) (0) cos( B t) y ( t) (0) z x z x 0 0 y y 0 0 (2.13). And indeed these equations describe a precession of about the z-axis with an angular frequency 0 given by: 0 B 0 (2.14), an expression identical with Eqn. 2.10b. Thus, we have found that the quantum-based resonance frequency corresponds exactly with the equation for the classical precession of a spinning magnet in a magnetic field. In NMR we will often use the classical mechanics analogy for a description of nuclear spins. Sometimes such an analogy must break down, however, since the spins really are quantum mechanical in nature. READER 12

18 3 AN ENSEMBLE OF NUCLEAR SPINS 3.1 Ensemble of spins In a sample of identical molecules we are dealing with a large number, an ensemble, of equal nuclear spins. In the quantum-mechanical picture these are distributed over the spin states according to Boltzmann's law: n β n E e kt (3.1), where n and n are the populations of the and state, respectively, and k is Boltzmann's constant (k = J/K). For I = ½ such a distribution is shown on the right. Due to the very small energy difference E = B 0 transitions are easy and the populations of the and states are almost equal. For a field B 0 = 14 T (Tesla) (proton frequency 600 MHz) the relative excess of spins is only one in This is one of the main reasons for the low sensitivity of NMR spectroscopy. In the (classical) vector model an ensemble of spins ½ can be illustrated as shown on the right. The individual spin vectors all make an angle with the field B 0 and slightly more are READER 13

19 aligned parallel to the field than antiparallel. In equilibrium the "phases" of the individual spins (their xy-components or positions) are randomly distributed. Therefore the resultant magnetization vector M (the sum of all spin vectors) is aligned along B 0. Like all the individual spins that add up to M, also M has to be imagined as spinning. So, if M is somehow tipped away from B 0 (see Figure on the left) it will perform a precessional motion about B 0 with frequency = B 0, just the way the individual spins do. This tipping can be done by way of a radio-frequency pulse and involves transitions between the and spin states of the nucleus! 3.2 Effect of the radio-frequency (RF) field B 1 In all FT-NMR spectrometers an additional magnetic field B 1 can be generated perpendicular to the static magnetic field B 0. This B 1 -field is created by means of radiofrequency (RF) pulses (it s actually the magnetic component of the electromagnetic radio waves). What happens during the pulse can best be compared to what happens to the spins, when they are brought into the magnetic field of the spectrometer. We are just dealing with an additional magnetic field, which tries to orient the spins in a new direction. The result of this additional field is a rotation of the magnetization vector M around the axis of the additional field. This rotation lasts only as long, of course, as the additional field is present. In other words: only during the duration of the radio-frequency pulse. In analogy to the precession around the B 0 -field, the speed of the rotation of M due to an RF pulse (its angular frequency) can be described as 1 B 1 (3.2). So, acting on the equilibrium magnetization M, the effect of B 1 is to tilt the vector away from the z-axis. How far the magnetization is tilted depends on the sort of nucleus READER 14

20 (), the strength of the B 1 -field and the duration of the pulse. Like the precession in a static magnetic field also this rotation can be described by the equation of motion (Eqn. 2.11). Assuming a pulse given from the xy-plane, the net magnetization vector is described by: dm dt B M 1 (3.3a). When the pulse is given from the x-axis, the solution of the equation of motion simplifies to: M ( t) M M z y ( t) M 0 0 cos( B t) 1 sin( B t) 1 (3.3b). Illustrating the effect of a radio-frequency pulse is a bit tricky (Fig. 3.1). Actually the magnetization vector remains precessing around the B 0 -field (with ), but at the same time, during the RF pulse, the magnetization vector is tilted away from the z-axis (and is rotating around the x-axis with 1 ; right figure)! Fortunately there is a simple way to describe this: the rotating frame. Here, the x- and y-axes rotate with about the z-axis, so the B 1 -field appears stationary (left). In the rotating frame the axes are denoted x', y' and z'. Now, at resonance, the motion of M becomes very simple (left figure): Figure 3.1: Rotating frame (left) versus laboratory frame (right). READER 15

21 While in the laboratory frame the M vector performs a complex spiralling motion, in the rotating frame it simply precesses about the direction of the stationary B 1 (in the z'y'-plane) with an angular frequency 1 = B 1. Thus, by going over to the rotating frame of reference we do not need to bother about the precession about B 0 anymore. The effect of an RF pulse can easily be described now: during the duration of a pulse (and only that long!) the magnetization M is rotating around the axis from which the radio-frequency pulse is applied! The tilting of the magnetization vectors can also be described using the equation of motion given in Eqn If we keep the B 1 -field on for a short time such that t / 2 (3.4) 1 and then switch it off we have tipped the magnetization along the y'-axis. This is called a 90 pulse. If we keep it on twice as long M ends up along the negative z-axis (180 pulse): 90 pulse 180 pulse Figure 3.2: The effect of a 90 x-pulse (left) and a 180 x-pulse (right) on +z-magnetization. Remark: The concept of the rotating frame makes the description of NMR experiments much easier. It is so convenient that we will use the rotating frame throughout the complete course if not explicitly stated otherwise. For simplicity, we will use the notation x, y, z instead of x', y', z' for the rotating frame in the following. READER 16

22 4 SPIN RELAXATION It was seen in the previous Chapter that after a 90 pulse is given towards the +x-axis, M ends up along the +y-axis. If we leave the system now, it will reach the equilibrium state again after a while by a process called spin relaxation. Actually this relaxation involves two independent phenomena: i) magnetization will grow along the z-axis until the equilibrium value M eq has been restored, ii) the M x and M y components decrease to zero (usually faster than the equilibrium magnetization is restored!). The characteristic times for these processes are called T 1 and T 2, respectively: T 1 : longitudinal or spin-lattice relaxation time (increase of z-magnetization), T 2 : transverse or spin-spin relaxation time (decrease of x,y-magnetization). In mathematical terms this can be described as follows: dm ( t) ( M z dt dm x ( t) M x ( t) dt T dm y ( t) M y ( t) dt T z 2 2 ( t) M T 1 eq ) (4.1a) (4.1b) (4.1c). The solution of Eqn. 4.1a is: t T M (0) M e 1 M ( t) M (4.2), z eq z eq READER 17

23 t T and of Eqn. 4.1b (and similarly for M y (t)): M ( t) M (0) e 2 (4.3). x x Thus, with time all components of M return exponentially to their equilibrium values. T 1 relaxation involves changes in z-magnetization and hence, transitions between and spin states that are accompanied by an exchange of energy with the "lattice" (environment); T 2 processes involve the loss of phase coherence in the xy-plane, and energy is not exchanged with the environment. Note that spin relaxation is a random process and should not be confused with the coherent rotations around B 0 - or B 1 -fields. For example, T 1 relaxation only affects the z- component and does not create x- or y-magnetization (see Eqn. 4.1)! 4.1 Molecular basis of spin relaxation What causes nuclear spins to relax? A short answer is: the exchange of energy with their environment. However, the energies related to the corresponding processes must match E between the involved energy states. In most other spectroscopic techniques, this is achieved by collisions (with other atoms or molecules), but for the small energy differences in NMR this is not an option. Therefore, we have to look for processes with energies (i.e. frequencies) comparably small as the resonance frequencies in NMR. It seems that we can find these in the small inhomogeneities of the static magnetic field and in the interaction between fluctuating magnetic dipoles. In the latter, the magnetic dipoles are the NMR-active nuclei themselves and the fluctuations come from the translational, rotational and vibrational motions of molecules. There are two fundamental differences between T 1 and T 2 relaxation. Firstly, as mentioned above, in T 1 relaxation energy is exchanged with other molecules (the environment or the lattice ) causing transitions from the - to the -state and this influences the length of the net-magnetization vector M along the z-axis of the system (see Eqn. 3.1). The time-dependence of T 1 relaxation comes from local magnetic fields, which, READER 18

24 if they contain a component at the NMR frequency of nuclei, can induce transitions to return the spins to equilibrium. With T 2 relaxation, however, energy is exchanged with different spins of the same molecule, which introduces a small variety in the precession frequency of an ensemble of identical spins. Therefore, during T 2 relaxation the netmagnetization vector M is split up in many small components rotating with different frequencies compared to the rotating frame. If we wait long enough, these components shall be evenly distributed in the xy-plane, and no measurable transversal magnetization is left. One speaks about dephasing of magnetization. The second difference between T 1 and T 2 relaxation is that a static distribution of B z -fields causes small different resonance frequencies (in different locations of the sample), which also contributes to the dephasing that underlies T 2 relaxation. In a sample, the magnetic dipoles (i.e. the atoms of the molecules) move with respect to each other in a multitude of different frequencies and in different directions (B dip x (t), B dip y (t) and B dip z (t)). Depending on their direction, these motions contribute to relaxation. For instance, like when a pulse along the x- or y-axis disturbs magnetization that exists along the z-axis by converting spins between their - and -states, the fluctuating dipolar fields B dip x (t) and B dip y (t) contribute to T 1 -relaxation. These B dip x (t) and B dip y (t) fluctuations also influence T 2 -relaxation; the small local differences in the magnetic field experienced by identical spins causes the small differences in the precession frequency leading to dephasing. Mathematically, the total additional field (B dip ), that is felt by 2 and slightly changes its precession frequency, depends on the orientation of 2 relative to 1 (the angle and the distance r between the nuclei: B dip = 1 / r 3 (3 cos 2 1) (4.4). READER 19

25 One can imagine that with small molecules, due to fast rotation averages out to zero in time, while in bigger particles this need not be the case. In an NMR sample, different motions with different frequencies () are present that may or may not contribute to the relaxation events. For instance, for T 1 relaxation only motions with a frequency near the resonance frequency are effective. The distribution of the frequencies of all these motions is represented by the spectral density function, J(): J() = 2 c / (1+ 2 c 2 ) (4.5). One major factor contributing to the time-dependency (frequence-dependency) of the dipolar interactions underlying the relaxation efficiency is the speed of the rotational diffusion (the thermal motion ) of the molecules, which can be described by the rotational correlation time c as explained in the following Figures. The sphere represents all possible positions of a certain atom that is connected to an other atom in the center of the sphere. The black line represents the pathway of the atom on the surface as the result of rotational diffusion of the complete molecule. The dots indicate the position of this atom at certain time intervals. For waiting periods smaller than the rotational correlation time c, the orientation of a molecule has not changed much compared to the starting point (Figure below, left); for t >> c the correlation between different orientations is completely gone and all memory of the original orientation is lost (most right Figure). Thus, c is a way to describe at what moment the correlation of the rotational diffusion disappears. READER 20

26 For small (i.e. fast tumbling) molecules the correlation between different atom positions is quickly lost, so c is quite short ( s). For large biomolecules that tumble slower, c is much longer ( s). An approximate relation of c with the molecular volume V and the viscosity is given by V c (4.6a). k T For macromolecules of molecular mass M r in aqueous solution at room temperature a useful approximation is M 2.4 r 12 c 10 (4.6b). The unit of M r is Dalton (1 Da = one twelfth of the rest mass of an unbound atom of 12 C in its nuclear and electronic ground state and has a value of ). Now, let s see if we can understand the spectral density function in more detail. In a figure such functions look as follows: Figure 4.1: Spectral density functions as a function of motional frequency for a small molecule (dashed line) and a big molecule (solid line). In an aqueous system, smaller frequencies occur usually more often than large frequencies. The frequency = c 1, where the bending point of the curve occurs, acts much like a cut- READER 21

27 off of the spectral density function. In other words: motions with frequencies > 1 c are increasingly rare. J(0) describes the non-fluctuating, static effects ( = 0) that influence spin relaxation, like magnetic field inhomogeneity or being in a solid state. For molecules of different size the shape of J() differs, while the area under the curves remains constant. For small molecules c is short, so 1/ c is long and higher values of occur more often than for bigger molecules On the other hand, the presence of motions with low values of is relatively small for small molecules With increasing size of the molecules J() is getting bigger for slow motions (small ) but at the same time the cut-off moves to lower frequencies. In other words, slow motions are more likely to occur for large biomolecules than for small organic molecules. Not very surprising indeed! To understand the effect of molecular motions on the relaxation properties, it is important to consider the time-scale of events. For T 1 relaxation the motions with a frequency near the Larmor frequency 0 are effective, since they have to induce transitions between spin states and therefore must have a frequency, which coincides with the E between the states (thus, the Larmor frequency). So, the fluctuating magnetic fields have to induce transitions between - and -states which are separated by the Larmor frequency 0 = B 0. The efficiency of relaxation will depend on how often this frequency is present in the distribution of frequencies of the molecule, thus on J( 0 ). This efficiency is maximal for molecules that have 1 c = 0 and will drop for both larger and smaller molecules (longer and shorter c values). This explains the behavior of T 1 versus correlation time c as shown in the Figure. The minimum in T 1 (most efficient relaxation) is at 0 c = 1, which for common NMR fields occurs for molecules of molecular mass M r 1000 D. READER 22

28 In terms of fluctuating fields, a general expression for the efficiency (or rate) of T 1 relaxation is 1 T 2 2 B B J( ) 2 x y 0 (4.7), 1 where the average of the square of B x dip (t), <B x 2 >, reflects the overall strength of the fluctuating magnetic field. Assuming that the fluctuations are similar in all directions (<B x 2 > = <B y 2 > = <B z 2 > = <B 2 >) Eqn. 4.7 becomes: B J ( 0) T (4.8). 1 An expression similar to Eqn. 4.8 for T 2 relaxation is: 1 T B J(0) B x, y J( ) B J(0) J( ) 2 z 0 0 (4.9). This formula contains terms for the two mechanisms that contribute to T 2 relaxation: the static distribution of B z -fields (no or zero -frequency motions), which is proportional to J(0), and the effect induced by fluctuating dipoles B dip x (t) and B dip y (t), proportional to J( 0 ) due to (intra)molecular motions. For larger biomolecules the J(0) term will dominate and becomes approximately equal to c. Thus, for slowly tumbling (bio)molecules we have the simple expression B c (4.10). T 2 The linearity of 1/T 2 as function of c implies that T 2 becomes progressively shorter for larger molecules and explains why, in the Figure on the previous page, T 2 unlike T 1 does not go through a minimum. In Chapter 9 we will describe how to measure the T 1 and T 2 relaxation times experimentally. READER 23

29 5 FOURIER TRANSFORM NMR 5.1 From time domain to spectrum Nowadays, all NMR machines are so-called Fourier Transform NMR (FT-NMR) spectrometers. Earlier we have seen (Chapter 3.2) that the magnetic component of an electromagnetic field (by a Radio Frequency or RF pulse), applied along the x-axis of the rotating frame works on equilibrium z-magnetization and results in a rotation around the x- axis when = RF. For the rotation frequency 1, we found 1 = B 1. If we apply this RF field for a period t = /2 1 we create 'pure' y-magnetization. An RF pulse of this duration was called a 90 pulse. The RF transmitter of an NMR spectrometer is operated by a pulsecomputer, which can generate a single RF pulse, but also a series of RF pulses of arbitrary length, frequency, phase, and amplitude separated by delays of adjustable length. In general, an RF pulse of length p excites the whole frequency range from RF 1/(2 p ) to RF + 1/(2 p ), since 1/ p = RF (5.1). To excite or select a certain range of frequencies, p must be adjusted to be sufficiently short or long. For example, at 600 MHz proton resonance frequency a complete excitation of an NMR spectrum of 10 khz requires p << 100 s. In practice, pulses of p = 2 20 s are used. rf rf p rf ½ p rf rf +½ p Figure 5.1: RF pulse of duration p (left) and the corresponding excitation profile (right). READER 24

30 What will happen with M eq after a single RF pulse of 90 along the x-axis? Earlier we have seen that the magnetization vector ends up in the xy-plane and now is oriented along the y-axis (phase coherence of the spins). In the xy-plane, two detectors are tuned to the radio-frequency RF to record the intensity of transverse magnetization M x and M y. If the resonance frequency of spin j, denoted by j, is equal to RF, the magnetization will remain along the y-axis (of the frame rotating at speed RF ). The magnitude of the magnetization will decrease in time due to T 2 relaxation. In case spin j has a resonance frequency j, which is different from RF, then the magnetization of spin j will precess in the frame rotating at RF with the difference frequency Figure 5.2: Detection of transfer magnetization. = j RF (5.2). The in the xy-plane rotating magnetization vector induces a current when it passes the receiver coil. This is the actual signal recorded. In order to be able to distinguish between positive and negative frequencies (vectors which are with the same speed compared to the rotating frame, rotating either clockwise or counterclockwise), both the x- and the y-component of the rotating magnetization are recorded simultaneously. The corresponding signal induced in the receiver coil has the shape of a decaying harmonics (sine and cosine waves) and thus is called the free-induction decay (FID). For a number of different spins j (for example for a protein: H, H, and H N protons), each with their own equilibrium magnetization M eq j, frequency j, and relaxation time T 2j, the FID consists of the sum of all magnetizations: READER 25

31 M y ( t) M ( t) x j j M M j j (0) cos (0) sin t t T e 2 t T t e 2 j j (5.3), where M j (0) = M j eq in the case of a 90 excitation pulse. Figure 5.3 illustrates the difference in appearance of an FID containing only a single frequency and an FID containing multiple frequencies: Figure 5.3: Recording a single versus multiple frequencies. The FT-NMR signal (the FID) is recorded in the time domain. The signal is digitized by an analog-to-digital converter (ADC) and stored in the memory of a computer. The resonance frequencies, j ', are then extracted by Fourier Transformation (Eqn. 5.4). This transforms the signals from the time domain, f (t), to the frequency domain, g(): i t g( ) f ( t) e dt (5.4), where the recorded signal f (t) is defined as f ( t) M y ( t) i M x ( t) M j eq j t it T j e e 2 (5.5). READER 26

32 So, in practice Fourier Transformation results in a sum of complex frequency signals with a real and imaginary part. The real part describes an absorption signal, the imaginary part describes a dispersion signal: Figure 5.4: NMR absorption and dispersion signal. In NMR we record both, but since it is narrower (the flanks of the absorptive signal drops with, whereas the dispersive signal goes with ) and does not change sign, we are only interested in the absorption signal. The shape of this signal is a so-called Lorentz line shape. One can calculate the relationship between T 2 and the line width at half-height, : 1 1 (5.6). Figure 5.5: NMR line shape. 2 T 2 Some important Fourier pairs are shown below. Basically these are the ones responsible for the shapes (or their distortion) of most signals observed in an NMR spectrum. READER 27

33 Figure 5.6: Several Fourier pairs. When the FID would not decay, infinitely sharp lines appear (top). Decay of the FID due to T 2 relaxation causes line broadening (middle); clipping the FID from top and/or side causes artefacts (bottom). 5.2 Aspects of FT-NMR It might be that the signal-to-noise ratio sino is not good enough after a single scan. By co-adding n successive NMR measurements the signal increases by a factor n. The magnitude of the noise however, fortunately increases with n only. This is due to the random nature of noise. Hence the signal-to-noise ratio improves as sino ~ n (5.7). FT-NMR is a very flexible technique. A large number of different experiments can be done, each aiming at different parameters of the molecules in study to be extracted, such as relaxation measurements (discussed in Chapter 9), multi-dimensional NMR (discussed in Chapter 10), heteronuclear NMR, etc. READER 28

34 6 SPECTROMETER HARDWARE 6.1 The magnet As one can expect, the high field magnet is crucial for an NMR system (see also Fig. 1.1). Nowadays mainly superconducting coils are used to generate the high magnetic fields necessary for high resolution NMR (permanent magnets are only used for instance in food sciences or on older, lower field NMR imaging systems). The superconducting coils consist of NbTi, NbTi-Nb 3 Sn, or NbTi-(NbTa) 3 Sn wires (alloys of niobium, titanium, tin, tantalum) which are superconducting at 4 K ( 269 C). Several layers of coils generate an increasingly higher field towards the innermost section where the final magnetic field strength is reached. In this section the field must be extremely homogeneous over a volume of some cubic centimeters, otherwise the resonance frequencies would vary at different locations in the sample leading to broad and asymmetric signals. The coil wires are also the READER 29

35 most expensive part of the spectrometer. To reach higher field strength, larger (and more complicated designed) coils have to be used. This makes most of the price difference between for instance a 700 MHz and a 900 MHz machine, which is approx. a factor of 4 (in 2006). The necessary low temperature for superconductivity is reached by submerging the coils into a dewar containing liquid helium (at 269 C). This inner dewar is surrounded by a second, outer dewar containing liquid nitrogen ( 196 C) to reduce helium evaporation. In the course of time, helium and nitrogen do evaporate, therefore the magnet (actually the dewars) has to be refilled periodically (typically weekly for N 2 and monthly for He). The strength of a magnetic field is normally given in Tesla or Gauss (1 G = 10 4 T). However, the strength of an NMR magnet is usually described by the corresponding resonance frequency of hydrogen atoms (proton-frequency). A field of 14 T corresponds to a field strength of 600 MHz. Today (2010) the typical field strength used in biological applications are 600 MHz 950 MHz. The highest currently available field is 1000 MHz (CRMN Lyon, France). The need for higher fields is explained by the gain in resolution and in sensitivity. The sensitivity of an NMR experiment is usually described by the signal-to-noise ratio: sino ~ N 5/2 B 0 3/2 n 1/2 T 2 /T (6.1), where N is the number of spins (concentration of the sample), the gyro-magnetic ratio of the nucleus, B 0 the field strength, n the number of scans per experiment, T 2 the relaxation time and T the temperature of the detection circuit. How a bigger field affects the signal-tonoise ratio and the resolution (which follows a linear dependence on B 0 ) is shown below. Table 6.1: Magnetic field dependence of signal-to-noise ratio and spectral resolution. B 0 (T) (MHz) sino resolution READER 30

36 For solution state NMR, the Biomolecular NMR laboratory at Utrecht University is housing two 500 MHz, two 600 MHz, a 750 MHz and one 900 MHz spectrometers. One of the 600 MHz spectrometers and the 900 MHz spectrometer are equipped with cryogenic probe systems for additional sensitivity. For solid-state NMR experiments 500 MHz and 700 MHz NMR machines are present. 6.2 The lock system Figure 6.1: The coil surrounding the sample delivers the pulses to the sample and also allows the FID to be recorded. Even in a very well designed magnet the B 0 field is not perfectly stable over the period a single experiment can take (up to one week). After initial charging of the magnet the magnetic field slowly decreases. This drifting of the main magnetic field can be compensated by the lock system by applying correction currents in a coil which is part of the room temperature shim system (see below). The lock system exploits the NMR phenomenon itself: a reference NMR experiment is continuously performed on a nucleus different from the one being studied. In most biological experiments deuterium ( 2 H or D) is used for this purpose. The deuterium spectrum is continuously acquired and the frequency of the single deuterium line is observed. When this frequency shifts, small correction currents are applied to the lock coil to compensate for this change. As a consequence of this procedure, in most biological applications deuterium is introduced by dissolving the sample in a mixture of 5 10% D 2 O in H 2 O. READER 31

37 6.3 The shim system As stated above, the magnetic field experienced by the sample must be constant, but also very homogeneous to keep the NMR signals as narrow as possible. Since the homogeneity of B 0 is not only a matter of the coil design, but is also influenced by the sample itself (filling height, quality of tube, etc.), for each individual sample additional field corrections have to be applied. This is achieved by a number of correction coils (the shim system) in which adjustable currents produce field gradients to compensate static field inhomogeneities. There are two sorts of shim coils: superconducting ('cryo-shims') and room temperature coils. The currents through the superconducting shim coils are usually only adjusted, i.e. during the installation/charging procedure of the magnet. The room temperature coils are the ones used by the user for the 'shimming' of each individual sample. In practice the optimization of the field homogeneity exploits the lock experiment. The D 2 O in our sample gives a single signal in the NMR spectrum. The integral of this signal is constant (as it only depends on the number of nuclei in our sample, which is constant) but the height of the signal is not: the narrower the signal the higher its maximum. The NMR operator can now manually adjust the different currents in the different shim coils to optimize this value. In the past, this was a very time consuming procedure which requires some experience. Now, automatic shimming methods are available which employ pulsed field gradients (PFGs) to reach good results within minutes. 6.4 The probe The probe (or probehead) is in many ways the most critical component of the NMR system. It is connected to the console and has two main functions (see Fig. 1.1 and also Fig. 6.1): a) To convert radio-frequency power from the amplifiers into oscillating magnetic fields (B 1 -fields) and to apply these fields to the sample. READER 32

38 b) To convert the oscillating magnetic fields generated by the precessing nuclear spins of the sample into a detectable electric signal that can be recorded in the receiver. Both functions are achieved by a parallel tuned circuit having a coil surrounding the sample. The tuning of this circuit, i.e. finding the exact frequency of the nucleus, depends on the sample position, volume, solvent, and ionic strength (remember: the frequency of the B 1 -pulse should match the Larmor frequency). This adjustment of the circuit is important in two ways: first, we want to transmit the maximum possible B 1 -field strength to the sample. This ensures that our pulses are as short as possible to ensure a good excitation bandwidth (Eqn. 5.1). Second, since NMR is a very weak phenomenon, we do not want to loose any signal coming from the sample by picking up only a fraction of the oscillating magnetization. There is a variety of NMR probes. For 1 H spectroscopy typically probes are used which can hold sample tubes of 5 mm diameter (with a sample volume of ~500 l). Beside the proton channel there is another coil for the lock system tuned on 2 H (sometimes a single double tuned coil is used for both frequencies). The general sensitivity of NMR probes is still improvable as reflected by the increased sensitivity over the past ten years (nearly a factor of 2). This shows how critical coil design is for NMR purposes. In a quite recent development, cryogenic probe systems were introduced, which consist of a probe which can be cooled with cold helium in order to reduce the amount of electronic noise in the receiver coils (and preamplifiers) to a minimum, thereby increasing the signal-to-noise ratio (Eqn. 6.1). The technological challenge of such a system, among others, is the fact that the temperature of the sample must still be adjustable to as high as about 80 ºC without heating the cold part of the probe. Since, on the other hand, the coils of the probe are supposed to be as near as possible to the sample, the difficulties of designing such a system are obvious. The gains in sensitivity with the installation of such a system to an existing spectrometer are remarkable and can be more than a factor of 2.2. READER 33

39 6.5 The radio-frequency system The RF or transmitter system mainly consists of pulse generation units and subsequent power amplifiers. It is generating the excitation pulses at the frequency of the nucleus of interest. On modern spectrometers the frequency can be set with a precision of 0.1 Hz across a band many megahertz in width. Since we may want to apply RF pulses out of several directions in the rotating frame, the phases of the RF waves are also adjustable (typically to 0.5 degree accuracy). These settings are under extremely fast computer control with setting times of only some microseconds. The power amplifiers boost the transmitter output to high levels (from several tens up to hundreds of Watts). This assures that short, non selective pulses can be applied to the sample. Schematic overview of NMR spectrometer components. Most transmitter and receiver electronics are built into the console (Fig. 1.1). Figure 6.3: The electronics of NMR pulse generation and signal recording. READER 34

40 RF generator (radio sender): creates RF signals with a frequency of less than 100 MHz up to about 1000 MHz. Pulse generator: creates RF pulses of a duration of several µs up to several seconds. RF amplifier: amplifies the pulse signal up to several 100 Watts. Magnet: the B 0 -field (from about 1 T up to about 23.5 T). Probe: holds the sample and houses the send and receive coils. RF amplifier: amplifies the received signal from the probe. Detector: subtracts the base frequency from the signal, resulting in an audio frequency (up to several khz), containing only the differences of the resonance frequencies from the base frequency RF. AF amplifier: amplifies the audio signal. ADC: analog-to-digital converter. Computer: controls all the electronic parts, receives, stores and processes the NMR signal. 6.6 The receiver The final stage in an NMR experiment is the detection of the precessing magnetization (xand y-components) in the sample. As stated earlier the same coil is used for this purpose as for excitation. This means that directly before data acquisition the transmitter system has to be closed and the receiver has to be opened (this ensures that no strong RF pulses are applied while the sensitive receiver system is on). The detection of the high frequency signal (MHz) is actually quite involved. First, analog filters are applied to the signal to reduce it to the relevant frequency range. Then the weak signal is amplified. The incoming signal is now mixed during several stages with reference frequencies. This mixing reduces the frequency of the signal from several MHz to the audio range (khz). Finally, the signal is digitized in real-time and stored. For digitization the continuous NMR signal is sampled at evenly spaced time-points. The Nyquist theorem requires that if we want to detect all resonance frequencies up to a certain value, we have to digitize the highest frequency twice READER 35

41 per sinus period. This means that time points must be stored every 'dwell time' dw or faster. In a formula: 1 dw (6.2), SW where SW is the spectral width, i.e. twice the distance from RF to the maximum resonance frequency. An example (see Fig. 6.2): assume we digitize our FID every 10 ms. This means that frequencies we can measure cover a spectral width of 100 Hz, the distance from 50 to 0 Hz plus the distance from 0 to 50 Hz. A frequency of 50 Hz will be sampled exactly twice per period (solid line), which is enough to characterize this frequency. A resonance of 60 Hz (dashed line) is sampled less than twice per period, thus the frequency cannot be distinguished from a slower frequency (40 Hz in this case, dotted line). It can be seen that at each digitization moment the magnitude of the FID is the same for the 60 Hz and the 40 Hz frequency. This means that frequencies of 60 Hz or 40 Hz would both give a signal at the same position within the region from 50 to 0 Hz! So in the spectrum, a signal that would resonate at 60 Hz ends up at a position of 40 Hz. This phenomenon is called folding. Figure 6.2: Dwell time: the FID should be digitized at least twice per wave period to be able to determine its frequency unambiguously. All further operations after the digitization and storing of the signal are performed in the data processing system of the computer. This includes application of window functions, zero-filling, Fourier Transformation, phase corrections, baseline corrections, and other data manipulations like signal integration, displaying and plotting the spectrum. READER 36

42 7 NMR PARAMETERS 7.1 Chemical shifts We have seen that the resonance frequency of a nucleus depends on its gyro-magnetic ratio and the magnetic field B z. If all nuclei of the same kind (e.g. protons) would have an identical Larmor frequency, NMR would not be useful for studying biomolecules: we would observe a single signal per type of nucleus. Fortunately, this is not the case since in practice different spins, even from the same sort, have a slightly different Larmor frequency. This is because not all nuclear spins experience the same effective static magnetic field B eff. Instead they experience the superposition of the external field B z and a local field B loc. The static field B z induces currents in the electron clouds surrounding each nuclear spin. These induced electronic currents result in local magnetic fields. The induced current will counteract its cause ( Lenz law, electromagnetism), so the induced field will be opposed to B z. The nuclear spins will be shielded from the external field. The strength of this shielding depends on the electron density around each individual nucleus and the strength of the static field: B (7.1), loc B z where is a quantity expressing the amount of shielding. The net, or effective field experienced by the spin becomes B B B B ( 1 ) (7.2) eff z loc z and the new resonance frequency is given by (compare to Eqn. 2.10a): READER 37

43 B z (1 ) (7.3). 2 The shielding factor is different for different types of nuclei in a molecule because the electron density around a typical nucleus is very sensitive to the chemical environment of this nucleus (e.g. chemical bonds and neighbours). The amount of shielding is usually given as a dimensionless parameter, the chemical shift, which expresses the difference between the actual NMR resonance frequency of a spin and a reference signal, which is usually the operating frequency of the magnet or the frequency with which the rotating frame rotates (see also Eqn. 5.2). Mathematically: ( 6 10 ref ref ) 10 6 ref 6 ref ( ref 1 ref ) (7.4) since 1. With this, when spectra from two different magnetic fields are compared, ref individual resonances have the same position on the -scale, although they actually do have a different resonance frequency in Hz. For the reference of the -scale the single line of the methyl-protons of Si(CH 3 ) 4 (TMS, Tetramethylsilane) can be used ( = 0 ppm. For biomolecules, slightly different compounds (e.g. TSP, (CH 3 ) 3 SiCD 2 CD 2 CO 2 Na, the sodium-salt of trimethylsilyl-propionic acid) are used since TMS is not soluble in water. Sometimes also the water resonance itself is used as a reference, and it s exact position is temperature dependent: T [ Kelvin] ( H 2O) 7.83 (7.5) A small value of corresponds to a shielded proton, or a low resonance frequency. The dimensionless -scale (ppm = parts per million, represents a fraction and it has the important advantage over a frequency scale (Hz) that the chemical shift values become independent of the external magnetic field B z. READER 38

44 7.1.1 Effects influencing the chemical shift While s-orbital electrons generate a field opposing the static field, i.e. a shielding effect, p- orbital electrons and other orbital electrons with zero electron-density at the nucleus result in a weak field reinforcing the static field. The contributions of these two effects are of well-known magnitude for different functional groups of chemical compounds. The Table in Appendix B gives common chemical shift values for a number of such different functional groups. In addition to the constant effects of the s- and p-orbitals of an atom to the chemical shift, there are also variable contributions resulting from the local surrounding of the nucleus (e.g. solvent effects or interaction with other parts of the same molecule) and the local conformation. Some are mentioned in the following. 1. Aromatic and carbonyl groups have an extensive conjugated -electron system comprising delocalized molecular orbitals. Also in these systems the B z -field induces currents, the so-called ringcurrents, resulting in quite large magnetic moments. Their effect on the chemical shift of a particular nucleus depends strongly on the distance and orientation with respect to the aromatic system: above and below the aromatic ring-system an opposing field is generated and is negative ( upfield shift ). In the plane of the ring-system a reinforcing field is generated and is positive ( downfield shift ). These effects can be as large as 2 to +2 ppm and all nearby nuclei are affected. The effect decreases with 1/r 6, where r is the distance from the -electron system. 2. Paramagnetic groups, like Fe 3+ in the heme-system of hemoglobin, can have a pronounced effect on chemical shifts. An unpaired electron influences the proton chemical shift through spatial interactions (through the electron magnetic moment) as well as by direct electron-proton hyperfine interaction. 3. Charged groups or electric dipoles can cause electric fields that can polarize electron clouds and thus influence the chemical shifts. 4. H-bond formation strongly influences the chemical shift. The proton in an X H Y READER 39

45 hydrogen bond is little shielded and has a large value. For example, the imino protons in the Watson-Crick hydrogen-bonded bases of a B-DNA fragment resonate at ppm, whereas the non-hydrogen-bonded imino protons resonate at ppm Protein chemical shifts From section it will be clear that 1 H spectra of certain classes of molecules, for instance proteins have a 'general' appearance, since they are built up from similar building blocks (amino acids) with similar functional groups (see Chapter 12 and Appendix F). An example of a protein 1 H spectrum with typical spectral regions is given below. Figure 7.1: Protein 1 H-NMR spectrum (the water signal around 4.7 ppm is suppressed). Indicated are the typical regions where the different protein proton resonances are found. For several amino acid residues it is even possible to split out the different chemical shift preferences. This is shown in Appendix C. READER 40

46 Although the theory of chemical shifts is well-known, in practice it is still quite complicated to accurately predict the chemical shifts for protein protons. Partially this is the result of the inaccuracy of protein structures and their internal mobility. On the other hand the range of proton chemical shifts is fairly limited (ca. 12 ppm) and the exact geometry is relatively important. The range of 13 C chemical shifts is much larger (ca. 200 ppm) and the effects of the exact protein geometry are less important. 13 C chemical shifts are therefore easier to predict and can be used in a more straightforward fashion for the interpretation of spectra. 7.2 J-coupling J-coupling occurs when magnetization is transferred between two spins via the electrons of the chemical bond(s) connecting them. The resonance frequency of a certain spin A depends on the spin-state of a second spin B and vice versa. If spin A is in the -state spin B will resonate at slightly lower frequency whereas it will resonate at slightly higher frequency when spin A is in the -state. For spin B, in the NMR spectrum we observe a doublet (two lines with equal intensity) centered on b, the resonance frequency of this spin in absence of J-interaction. Since the effect is mutual, also for spin A a doublet is observed, centered around a. The size of the coupling, J AB is the distance between the components of the doublets and is called the coupling constant (unit Hz). READER 41

47 If besides spin B, there is a third J-coupled spin C connected to spin A (see Figure previous page), spin A also splits up due to its coupling with spin C. The result is a doublet of doublets. This means that the coupling of the third spin is independent of the coupling between the first two spins. It just introduces another splitting on the first splitting. However, if the spins B and C are magnetically equivalent the coupling J AB is equal to J AC and the doublet of doublet collapses into a triplet (three-lines with intensity ratio 1 : 2 : 1). READER 42

48 Three equivalent neighbours (e.g. the three protons of a methyl group) give rise to a quartet (four signals with intensities 1 : 3 : 3 : 1) for the signal of a connected proton spin. The intensity ratios, for a spin coupling with increasing numbers of magnetically equivalent other spins, follow the well-known Pascal triangle: Figure 7.2: Pascal triangle explaining coupling pattern intensities Equivalent protons In general, equivalent protons are protons which are chemically and magnetically equivalent. Chemical equivalence means that there is a symmetry axis in the molecule for the protons under consideration. Protons having this type of equivalence resonate at the same frequency. For magnetic equivalence the nuclei must a) be chemical equivalent and b) experience exactly the same J-coupling with all other nuclei in the molecule. Magnetically equivalent nuclei are a very special case in the coupling network: they do not couple with each other. This explains why for instance the benzene spectrum shows only one signal. Due to the high symmetry of the molecule all six benzene protons are magnetically equivalent, thus showing only one frequency and no coupling with each other. Figure 7.3: Chemical vs. magnetical equivalence. READER 43

49 In proteins magnetic equivalence due to symmetry is rare because of the high complexity of the biomolecules. But also here protons can be equivalent. If a group of atoms rotates fast enough they become magnetically equivalent as a result of dynamic averaging. This is the case for instance for protons in methyl groups or fast rotating aromatic rings. The magnitude of the J-coupling depends on the number of intervening chemical bonds, the type of chemical bonds, the local geometry of the molecule, and on the -values of the nuclei involved. Proton-proton couplings in biomolecules are usually observed for protons separated by two or three chemical bonds. The magnitude of these proton-proton J- couplings is relatively small, typically 2 14 Hz. Often the patterns resulting from these 2 J HH (two bonds) and 3 J HH (three bonds) couplings are not resolved because of the large line width in biomolecules. The magnitude of heteronuclear one-bond couplings is much larger than for 2 J HH and 3 J HH. For instance, the J-coupling between the amide proton and the directly bond 15 N nucleus, 1 J NH is ca. 92 Hz; the one-bond coupling between a proton and its directly attached 13 C nucleus, 1 J CH is ca. 140 Hz. J-couplings involving 15 N or 13 C nuclei also exist between nuclei two or three bonds apart. For example, there is an interaction between H and 13 C over three bonds in a 13 C (C=O)N H H N fragment ( 3 J CHN ). Because the splitting causes the presence of extra signals in the spectrum that may overlap with each other, often the occurrence of scalar coupling is not desired. In some cases, however, especially using 2D and 3D NMR techniques, it is possible to record the frequency of one nucleus, while the frequency of the coupled nuclei are irradiated. Hereby these nuclei get saturated, i.e. the - and -states are made equal, hence the splitting with the other nucleus does not occur. READER 44

50 8 NUCLEAR OVERHAUSER EFFECT (NOE) By applying realatively long, low-power RF-pulses it is possible to saturate magnetization, since during such a pulse, dephasing of magnetization, i.e. equalization of - and -states, occurs in time due to magnetic field inhomogeneities (T 2 relaxation). It appears that the observable intensity of a signal of a nucleus depends on the intensities of other nuclei when they are in close spatial proximity. If, for instance, two protons are situated at a distance of less than 5 Å (i.e. 5 Ångström or m) and the signal of one of them is saturated by selective irradiation, the other signal will change in intensity. This effect is called the nuclear Overhauser effect (NOE) and is the result of a relaxation process, caused by a dipole-dipole interaction (dipolar coupling) between the two nuclei. We are talking here about cross-relaxation, because the population of the spin-states of one nucleus depends on the population of the spin-states of another one. 8.1 Dipolar cross relaxation Let us consider a two-spin system with spins A and B which are dipolar coupled because they are close in space. In the steady-state NOE experiment one resonance, let's say spin B, is selectively saturated by RF irradiation. This disturbs its equilibrium magnetization, and therefore spin B tries to reestablish an equilibrium by exchanging magnetization with its environment. This can either be the lattice (environment, H 2 O) or another nucleus that is closeby. The NOE between the saturated spin B and another spin A is defined by the relative change in the intensity of spin A: Figure 8.1: The NOE. READER 45

51 NOE = 1 + with ( M M ) eq a a (8.1), eq M a where can either be positive or negative. The energy diagram for this two-spin system is shown on the right. Both spins A and B can undergo transitions between its - and -state resulting in the resonance lines which are observed. The rates of these transitions are W 1A and W 1B for the transitions of the spin A and B, resp. The dipolar interaction between spin A and B introduces two more possible transitions: W 0 and W 2. These transitions involve simultaneous changes in the spin states of both the A and the B spin (i.e. and ). How the NOE can be understood is schematically shown in Figure 8.2. A A = 1.5 W 1A W 1B A W 2 W 2 > W 0 small molecules A W 1B W 1A W 0 A W 0 > W 2 large molecules A A 0 = B 0 = A = A 0 = B = 0 A A = 0.5 Figure 8.2: Schematic explanation of the NOE using energy diagrams. READER 46

52 The most left diagram represents the situation at equilibrium, the number of lines represents the number of spins in each spin-state. A 0 and B 0 are defined as the relative population differences in equilibrium (corresponding to W A and W B ); A and B are the relative differences in population of the spin-states of the A and B nucleus after saturation of spin B (diagram in the middle) and finally after cross-relaxation (diagrams on the right). By applying a long, selective pulse with the frequency of nucleus B, the population difference for the transition W 1B is removed. This leads to the situation in the middle, where it is seen that the population differences for the W 1B transition are now indeed equal. To re-establish equilibrium magnetization the cross-relaxation mechanisms W 0 and W 2 come into effect. Two different cases are distinguished here: for small molecules W 2 dominates over W 0 and the result is shown in the upper right diagram. It is seen that compared to the original distribution for W 1A, A 0, the relative population difference for the states of nucleus A has increased. Consequently, also the observed signal for A will be increased. On the other hand, for large molecules W 0 dominates over W 2 and the result is depicted in the lower right diagram. The relative population difference for the states of nucleus A has now decreased and consequently also the observed signal for A will be decreased. Of course, both relaxation mechanisms are always simultaneously active, and one can imagine a situation where the two cross-relaxation mechanisms cancel each other. Indeed a zero-crossing of the NOE is observed when (in water at room-temperature): c Suppose for a particular molecule with a particular size on a particular NMR spectrometer the 'observed' NOE turns out to be zero. Are there any options to still observe NOEs within this molecule? Well, obviously we can do nothing about the size of the READER 47

53 molecule, but the tumbling speed of the molecule, of course, depends on the viscosity of the solvent. The viscosity is usually very sensitive to changes in the temperature. So, when we change the temperature of the sample, the tumbling speed will change and we have a chance to observe NOEs. The other parameter which we might change is 0, which depends on the spectrometer frequency. So we could just repeat the experiment on a spectrometer with a different field and chances are high that we moved away from the zero-crossing situation. A more formal derivation of the NOE is given in Appendix D. 8.2 NOEs in biomolecules T 1 relaxation times are rather uniform for biomolecules. In contrast, cross-relaxation rates vary a lot since the strength of the dipole field is strongly dependent on the distance between the protons. A good approximation for the cross-relaxation rate (this is of course different from the shielding factor in Chapter 7.1) in a biomolecule is: = W 2 W 0 ~ c 6 r (8.2), where c is the rotational correlation time of the proton-proton vector (see also section 4.1), and r the distance between the protons. Since T 1 values are uniform we can also write for the NOE (Appendix Eqn. D.11): c ~ (8.3). r 6 Therefore, the measured NOE can be converted into a distance. We can calibrate the NOE by comparing it with a NOE of a fixed distance in the same molecule: ref r 6 ref 6 r c ref c (8.4). READER 48

54 If we assume that there is no internal mobility (i.e. a rigid molecule), c is uniform in the entire molecule. We can now directly calculate the distance with: ref r 6 (8.5). r ref Eqn. 8.5 provides the basis of the most important factor for structure determination by high-resolution NMR spectroscopy, namely the extraction of NOE-based distances between proton pairs. An example: In a protein we observe NOEs between tyrosine aromatic ring protons and other protons. In an aromatic ring, the distance between C H and C H is fixed at 2.45 Å. Table 8.1: Calculating atom distances from NOE intensity and reference distance. NOE Intensity Distance (Å) Calculation Tyr C H Tyr C H Tyr C H Val C H r = 2.45 (0.15 / 0.02) 1/6 Tyr C H Asp C H r = 2.45 (0.15 / 0.01) 1/6 Internal molecular motions are often fast ( c,intern < 1) and contribute predominantly to W 2. Because of Eqn. 8.2, the NOE in mobile sub-domains will be reduced in intensity. Therefore, in the case of internal mobility we can not give the exact value of the distance but only an upper limit. c ref c Since c,real c,rigid : 1, and r r 6 ref ref ref (8.6). c c READER 49

55 9 RELAXATION MEASUREMENTS 9.1 T 1 relaxation measurements Applications of T 1 relaxation As seen in Chapter 4, T 1 relaxation works on the z-magnetization until M eq has been restored. The T 1 relaxation time is a measure for how long we have to wait until this equilibrium magnetization is restored. This is important information for the setup of Fourier Transform NMR experiments, where repeated experiments are added up in the computer to improve the signal-to-noise ratio. Therefore T 1 gives us information on how fast we can repeat our experiments. Additionally, we can use T 1 to derive motional parameters for the molecule under study, since internal motions may be of the correct frequency to flip nuclei from the - to the -state, thereby giving a slightly different T 1 than expected for a completely rigid molecule The inversion recovery experiment The so-called inversion-recovery pulse sequence, detection, can be used for measuring the longitudinal relaxation time T 1. At the start of the sequence, the equilibrium magnetization M eq is inverted by a 180 pulse, after which the magnetization M z (=0) = M eq. The magnetization will return to its equilibrium value ('recover') with the relaxation time T 1, and after time we have: READER 50

56 M z T ( ) M eq e (9.1). This time dependency of the magnetization along the z-axis is shown in the Figure on the right. We can measure the magnitude of M z () (i.e. the signal intensity) by applying a 90 detection pulse, which will rotate the z- component into the xy-plane where detection takes place. The FID is recorded and the spectrum obtained by Fourier Transformation. Using this pulse sequence, the signal volume is directly related to M z (). For < ln(2) T 1 the spectrum intensity is negative, for longer values of M z () has recovered to positive values, so a positive signal is observed. By repeating the experiment with increasing values of, the relaxation behavior can be determined and T 1 extracted. The T 1 analysis is not limited to a molecule with a single resonance. In a molecule with more spins, each of the individual spins, j, with frequency j and relaxation time T 1j has the starting magnetization: M ( 0) (9.2). j zj M eq So for each isolated resonance j we can determine the relaxation time T 1j by monitoring the signal intensities j in the spectrum as a function of the relaxation delay The different time points of the inversion recovery sequence are measured one after another. In case more than one FID is recorded per value of (e.g. READER 51

57 for signal-to-noise improvement), or in multi-dimensional experiments (discussed in Chapter 8), care has to be taken that saturation is avoided and enough time is allowed between the individual experiments for the magnetization to relax completely. Typically, the repetition rate of the experiment should be adjusted so that at least a time of 5 T 1 is waited before the experiment is repeated Example for the inversion-recovery experiment We follow the intensity of a single resonance and detect signal intensities after varying between s and = 3.0 s. Let s assume that the inversion was incomplete as a result of an imperfect 180 pulse ( M ( 0) ). Therefore we have to use the more general Eqn. j zj M eq 4.2 instead of Eqn. 9.1: M z ( ) M eq τ T M (0) M e 1 z eq Table 9.1: z-magnetization as function of relaxation delay in an inversion-recovery experiment. (s) M z () M() = M z () M eq ln{m()/m()} M z (0) M eq READER 52

58 Since M ( ) ln M (0) T1 (9.3), T 1 can be determined by linear regression analysis: T 1 = / ln(...) = 1.64 / = s. 9.2 T 2 relaxation measurements Applications of T 2 relaxation In Chapter 4 we have seen that the T 1 and T 2 values depend on the motional behavior of the dipole-dipole vector and thus on the rotational correlation time c of the molecule. Thus, analogously to T 1, we can also use T 2 to determine motional parameters The spin-echo experiment The value of T 2 could in principle be extracted from the envelope of the FID or from the line width at half-height (Eqn. 5.6). However, T 2 values obtained this way do not only reflect (intra)molecular motions, but are also influenced strongly by static field inhomogeneities (the shimming, Eqn. 4.9). Remember, if the static magnetic field B 0 is not homogeneous over the whole sample, the same spin at different locations in the sample experiences a slightly different field. This of course results in slightly different resonance frequencies and the total signal for this spin, which is the sum over all individual spin contributions in the sample, will be broadened due to this effect. Due to such nonmolecular T 2 -relaxation contributions, the apparent relaxation time, T * 2, is always faster than the T 2 due to (intra)molecular spin-spin interactions. Usually we are interested in the physical properties of the molecule only. Obviously, the contribution of a 'bad' shimming READER 53

59 to the relaxation of our molecule is less interesting for us! Therefore, we need to find a way to extract T 2 from T * 2. 'Pure' T 2 times can be determined by the so- called 'spin-echo' pulse sequence, shown on the right. The equilibrium magnetization, M eq, is transferred from +z- into y- magnetization, M y, by a 90 x-pulse. The macroscopic vector M y can be considered to consist of a sum of macroscopic magnetizations M j at j different positions in the sample. Now we look at the rotating frame (which rotates with the average Larmor frequency 0 of all the spins j). As a result of an inhomogeneous field, we will find that some of the M j components will rotate faster than 0 and some will rotate slower, depending on the exact position in the sample. In the rotating frame ( 0 ), the fast and slow components will start to precess in the xy-plane with frequency Bz M j ) 0 ( B (9.4). j Since B varies with j, this frequency is different at different locations. Hence, the transverse magnetization M y will dephase due to the inhomogeneity of the field. It can be shown that the spin-echo sequence eliminates the dephasing that results from these constant static field inhomogeneities, but not the random molecular events that result from fluctuating dipole interactions. In order to explain how this works, we actually should consider the precession of READER 54

60 each of the individual components M j, but fortunately the principle can also be shown by picking only two identical spins, rotating with different speed with respect to the rotating frame: a slow black and a fast white one. At point (a) in the sequence we have pure y-magnetization for all individual spins (see Figure previous page). At point (b) some phase coherence is lost because each spin has precessed with its own frequency. Due to field inhomogeneities the white one precesses a bit faster than the black one. After the delay a 180 pulse from the y- direction is applied (point (c)). This pulse will invert the x-component of the vectors M j but will not affect the y-components. During the second delay the vectorsm j precess again with their own frequency B j, so the white one still a bit faster than the black one, and, because the static inhomogeneities are unchanged, still in the same direction as before the 180 pulse. Consequently, in point (d) both magnetization vectors have returned to the y-axis creating a so called 'spin-echo'. In this way, we have eliminated the effect of field inhomogeneity. By repeating the experiment for several values of in the range 0 4 T 2 we can determine T 2 from the decay of the intensities of the resonances in the spectrum, which now results purely from the relaxation by random fluctuating fields and is independent from any static field inhomogeneity. The molecular contribution to T * 2, T 2, can be derived from M xy ( ) T M (0) e 2 (9.5). xy READER 55

61 10 TWO-DIMENSIONAL NMR In the seventies the development of two-dimensional (2D) NMR techniques has revolutionized NMR spectroscopy and has made structural studies of biomolecules possible. The basic idea is to spread the spectral information in a plane defined by two frequency axes rather than linearly in a conventional one-dimensional spectrum, which clearly provides a large increase in spectral resolution. Also, in a 2D NMR experiment interactions between many spins in a molecule (whether it be J-coupling or NOE-type interactions) can be measured simultaneously. This represents an enormous time-saving for large biomolecules. To illustrate the 2D NMR method we will first focus on a very simple experiment. READER 56

62 10.1 The SCOTCH experiment SCOTCH stands for spin coherence transfer in (photo) chemical reactions. It works as follows. Consider a photochemical reaction A h B (10.1). Under the influence of light molecule A is transformed into molecule B. A proton which first resonates at frequency A will after a light flash resonate at B. The SCOTCH experiment correlates the resonance frequencies in A and B for each particular proton. In other words, it enables us to find B in B that corresponds to A of the same proton in A. The pulse sequence is shown on the right. Before applying the light-flash, a 90 pulse creates xy-magnetization for any given proton, which precesses with A during the so-called evolution period t 1. After a period t 1, the light pulse changes the precession frequency for this proton from A to B with which it is detected as an FID during the detection period t 2. The trick to get information about A is to increase t 1 systematicaly in a series of experiments (so-called increments ) and to collect a large number (typically ~500) FIDs belonging to the different t 1 values. The result will be a data set S(t 1, t 2 ) depending on both t 1 and t 2 (the time-domain FIDs). A first Fourier transformation t2 F2 leads to a so-called interferogram S (t 1, F2) and after a second Fourier transformation t1 F1 we arrive at the 2D spectrum: FT FT S( t1, t2 ) S( t1, F2 ) S( F1, F2 ) (10.2). FIDs Interferogram 2D spectrum READER 57

63 Figure 10.1: 2D NMR spectrum showing time domains in both dimensions (left) and a spectrum with time domain in F 1 and frequency domain in F 2 (right). Projections are drawn along both axes. The effect of incrementing t 1 is to "sample" the frequencies present in the molecule during the evolution period. How this works in practice is illustrated in Figure By incrementing t 1, in t 2 the FIDs are acquired with different phases. The recorded FIDs all have the same frequency ( B ), but their phase (i.e. how far the magnetization vector rotated in the xy-plane) depends on the evolution time t 1. After Fourier Transformation of t 2 this leads to a peak at the position B in the F 2 -dimension with an intensity that oscillates in the t 1 direction with the frequency A. Looking along the t 1 -axis of the interferogram the signal actually is an FID oscillating with the frequency A. Hence, after additonal Fourier Transformation of t 1 this gives a peak at ( A, B ) (i.e. A in the F 1 - and B in the F 2 -dimension). This is called a cross-peak, in contrast to peaks with A B, which are called diagonal peaks. Thus, the experiment gives the connection (correlation) between resonance frequencies for protons in molecule A and molecule B. For a single proton this case is rather trivial, but if there are many, this method proves very useful. In general, all 2D NMR experiments use the following scheme: READER 58

64 In the example above the preparation period would include a relaxation delay and the 90 pulse, while the mixing period is formed by the light pulse. Figure 10.2: The SCOTCH 2D NMR experiment. On the left side the sampling of the A frequency is shown at different values of t 1. After Fourier Transformation of the FIDs (t 2 F 2 ) the interferogram S(t 1, F 2 ) consists of lines at B in F 2 with intensities dependent on t 1 oscillating with A. The second Fourier Transformation (t 1 F 1 ) leads to a 2D spectrum with a single cross-peak at ( A, B ). The representation is as a contour plot, where the signal intensity comes out of the plane of the paper. READER 59

65 10.2 2D NOESY 2D NOE or NOESY is one of the most important 2D NMR experiments, because it measures all short inter-proton distances in a single experiment, for instance for a protein. The first 90 pulse belongs to the preparation period and creates transverse magnetization. The evolution time t 1 is regularly incremented and mixing consists of two 90 pulses separated by a constant mixing time m. During m magnetization between neighbouring spins is exchanged via cross-relaxation (see Chapter 8). For biomolecules this crossrelaxation AB W 0 and therefore the transitions dominate the NOE. Figure 10.3: Pulse seqeuence and vector diagrams of the 2D NOESY experiment. READER 60

66 We will now look at the 2D NOE experiment in more detail. Let us assume that there are two spins, A and B, within NOE distance (i.e. r < 5 Å), and that the carrier frequency is chosen at the Larmor frequency of spin A, RF = A. The vector diagrams at various times in the 2D NOE pulse sequence then look as shown in Figure After the first 90 pulse the magnetization vectors of both spin A and spin B lie along the y-axis in the rotating frame. During the evolution time t 1 the A-vector precessing at RF will stay along the y- axis (at least for short times when relaxation can be neglected), while the B vector precesses with a frequency B RF. The second 90 pulse tips the A-vector to the negative z-axis and the B-vector into the xz-plane. The z-component of B is shorter than that of A and depends on the length of t 1. We see that the variable t 1 time acts to create z- components with different magnitudes depending on the different Larmor frequencies (i.e. how far a particular vector did rotate in the xy-plane during t 1 ). This is called frequency labeling and is a common feature of the evolution period (t 1 period) of 2D NMR experiments. Let s focus now on these z-components that correspond to populations of energy levels. We know that W 0 transitions during the mixing time m will tend to equalize the populations of the - and -states (because at equilibrium they are equal). Therefore, the z-components of the A and B magnetizations will become more equal (d). Finally, the third 90 pulse flips the vectors in the xy-plane where the signal can be observed (e). Now let us see how this leads to cross-peaks in the 2D NOE spectrum. In Figure 10.4 we look at the magnetization vectors at point d in Figure 10.3 (i.e. after the mixing time m ). Let us first consider a trivial case where no magnetization transfer between spins A and B occurs, for instance because the spins are too far apart. After that, we ll look at the more interesting case where cross-relaxation does occur between A and B. The vectors are depicted for various evolution times t 1 chosen such that the B- vector has rotated through 0, 90, 180, 270, and 360. If no mixing occurs the vectors precess at their own frequencies A and B during t 1 and continue to do this during the detection period t 2. Thus, this leads to a 2D spectrum after double Fourier transformation with diagonal peaks at ( A, A ) and ( B, B ). READER 61

67 Figure 10.4: Vector representation of spin A (solid vector) and spin B (dashed vector) at time point (d) in Figure 10.3 for various values of t 1.When no mixing occurs during m the 2D NOE spectrum contains only diagonal peaks. In the case of magnetization transfer during m cross-peaks arise at ( A, B ) and ( B, A ). In contrast, when mixing occurs in m the equalizing effect of the W 0 transitions causes the A-vector to borrow intensity from B and vice versa. Thus, the intensity of the A-vector is now frequency modulated with B for the different values of t 1! Since the vector will continue to precess at A in t 2 this will lead to peaks both at ( A, A ) and ( B, A ) in the 2D spectrum (F 1, F 2 ). This means we see the diagonal peak that we have seen before at ( A, A ), but also an off-diagonal cross-peak at ( B, A ) at the upper left half of the spectrum. In the same way, the modulation of the B-vector with A leads to a symmetry related cross-peak at ( A, B ) below the diagonal. Because mixing between spins A and B is reversible 2D NOE spectra are always symmetrical. And as all proton pairs within 5 Å will give rise to cross-peaks with intensities inversely proportional to r 6 (Eqn. 8.3), the 2D READER 62

68 NOE spectrum provides a map of all short proton-proton distances in a molecule. A more mathematical description of the NOESY experiment is found in Appendix E D COSY and 2D TOCSY In another important class of 2D NMR experiments magnetization transfer in the mixing period takes place via J-coupling. Here, the two mostly used of these experiments are shortly described. The simplest one making use of J-coupling for magnetization transfer is the COSY (correlated spectroscopy) with the pulse sequence shown in the top right Figure. The mixing period is formed by the second 90 pulse, which transfers magnetization between two spins whenever they are J-coupled (i.e. the spins are less than four bonds apart). For a two-spin system, this experiment eventually results in a 2D spectrum as shown in Figure Figure 10.5: COSY spectrum of two J-coupled spins A and B. The 1D spectrum is drawn above. The crosspeaks show the fine structure of the 1D spectrum (doublets in this case) which can be used to measure J- couplings. Note the typical negative-positive sign pattern in the cross-peaks (but not in the diagonal peaks). READER 63

69 COSY spectra are often recorded at low resolution so that the fine structure is not visible. In this way they are used to trace networks of J-coupled nuclei in a molecule. Schematic examples of such low-resolution COSY spectra are shown below for the two amino acids alanine and valine: Figure 10.6: COSY patterns for the amino acids alanine (left) and valine (right). As measurable J-couplings only exist between nuclei separated by less than four chemical bonds, the connectivity patterns can be easily predicted. For instance, in a protein a network of J-coupled protons does not extend beyond an amino acid residue because protons of sequential amino acids are at least four chemical bonds apart. Therefore, each of the amino acids forms a separate spin system and a COSY spectrum is a valuable tool to identify individual amino acids. Another important J-coupling based 2D experiment is the TOCSY which stands for total correlation spectroscopy. The pulse sequence is shown on the right. Here the mixing period consists of a complicated READER 64

70 pulse train, which allows magnetization to be transferred through a network of J-coupled spins. For instance, in the molecular fragment on the right we have non-zero J-couplings 3 J AB and 3 J BC, but the four-bond coupling 4 J AC is negligibly small. During the TOCSY mixing period magnetization of spin A is transferred to spin B via 3 J AB, but also transferred further to spin C via 3 J BC. As a result, a cross-peak will not only arise between H A and H B, but also between between H A and H C, even though there is no direct J-coupling between these spins! To illustrate this, a comparison between COSY and TOCSY spectra for the ABC -fragment is shown in Figure 10.7: Figure 10.7: COSY and TOCSY patterns of a three-proton spin system where 3 J AB and 3 J BC are non-zero and 4 J AC = 0. Because of symmetry the part below the diagonal is not drawn. It should be clear from this that a TOCSY spectrum must always contain the COSY as a sub-spectrum. Although the information content of COSY and TOCSY spectra is in principle the same, in complicated spectra with a lot of overlap the TOCSY spectrum is very useful. For instance, if B and C in the example of Figure 10.7 are in crowded spectral regions but A is not, then the whole spin system can be observed on a vertical line at the A-position, while this would be more difficult for B and C. Finally, it should be mentioned that the TOCSY is a sub-spectrum of the NOESY for almost all cross-peaks. We will come back to this point in the following Chapter. READER 65

71 11 THE ASSIGNMENT PROBLEM The interpretation of an NMR spectrum always starts with the identification of which resonance frequency corresponds to which atom in the molecule. This so-called assignment of resonances constitutes an essential step in the structure determination process by high resolution NMR spectroscopy that has to precede the actual calculation of structures. While the assignment of smaller organic compounds with only a few 1 H nuclei can often be solved easily by means of a single experiment (e.g. COSY), the assignment problem is much more complicated for bigger and more complex molecules like peptides, proteins and nucleic acids. Not only does the number of resonances increase with increasing size, also the line width increases as a consequence of the shorter T 2 relaxation times. As a result the signals become broader and overlap of signals becomes an increasing problem. In this Chapter, we shortly summarize which parameters of the NMR spectrum can be used to overcome the assignment problem. We will come back to the assignment of spectra of biomacromolecules in the next Chapter Chemical shift The chemical shift of signals gives a first indication of the surroundings of the corresponding nucleus in a molecule. We saw before that proton chemical shifts usually are grouped according to the chemical environment in which they are located in the molecule (Chapter 7.1.2). In our protein spectrum the amide and aromatic protons occur in the left half of the spectrum, the H protons just right of the water signal (in the center of the spectrum), the protons of aliphatic side chains more to the right and finally the methyl groups on the right edge of the spectrum. If you look at the vast variety of organic molecules you can find many more functional groups and chemical environments a nucleus can be situated in. Most of the protons belonging to such a group share their individual ranges of chemical shifts. Tables are available for 1 H and 13 C chemical shifts, which can READER 66

72 help to identify the origin of particular resonances in NMR spectra (Appendices B and C) Scalar coupling From Chapter 7.2 we know that we can identify coupled nuclei (i.e. maximal three chemical bonds apart) with help of their coupling constant. If we look, for instance, at the signals of the methyl group of ethanol, we find it split due to scalar coupling. The number of multiplet components gives us an indication of what the neighbouring group looks like. The signal of the methyl group is a triplet due to the coupling with the two equivalent protons of the neighbouring CH 2 group. The distance of the components of the multiplets (the coupling constant) is exactly the same for coupled protons. This means that for the ethanol CH 2 -group a similar coupling must be found. This helps to identify partners with a scalar coupling between them Signal intensities (integrals) The intensity, or better the integral of a signal tells us to how many equivalent nuclei a particular signal corresponds to. For example, the integral of the proton signal of a methyl group is three times as large as the integral of a single proton in the same molecule NOE data NOE data can be very helpful for the assignment, especially when we already have an idea of the basic building blocks of the molecule we are looking at. It is quite straight-forward for example to identify neighbouring protons in an aromatic ring system, because the distance from each other is quite short and well known (~2.45 Å). Once we know one of the aromatic protons, we can relatively easily identify others with an NOESY spectrum. Especially in biomacromolecules, where the spin systems of sequential amino acid residues cannot be connected by proton scalar coupling experiments, NOE data is often the only outcome to get a complete sequential assignment. READER 67

73 12 BIOMOLECULAR NMR Although also lipids and carbohydrates play important roles in biological processes, in this course we will focus on biomolecular NMR of amino acids and nucleic acid. Nucleic acid carries the genetic information and is essential for protein synthesis, where it acts as a template containing sequential information for all proteins occurring in organisms. Each three consecutive nucleotides in a gene encode a particular amino acid in a protein. Besides this, genes also contain control regions, like stop codons and sequences where factors involved in transcription and transcription regulation bind. The role of proteins in nature is very diverse. We know them for example as enzymes and regulators, as building material of cells and their compounds, and as transporter of molecules within and between cells. Obviously both nucleic acids and proteins play a major role in the function of all living organisms and accordingly also many of their disorders can be traced back to the malfunction of proteins or to defects in nucleic acid. This makes these molecules very popular subjects of study in a multitude of research disciplines, since their structural and functional understanding gives insight into how they work and what goes wrong in the case of diseases. Knowing the exact composition and function of a particular virus, for example, can lead to the development of anti-viral drugs. Also, the exact knowledge of the structure and function of a particular enzyme can lead to the development of inhibitors which can deactivate the enzyme when needed. In this Chapter, we will have a closer look at the structural properties of peptides and nucleotides and how they show up differently in NMR spectra Peptides and proteins Peptides and proteins are generally built from twenty different naturally occurring amino acids. These all share the same basic structure and only differ in their side chain R: READER 68

74 H Amino group H 2 N C C R Side chain O OH Carboxyl group The different amino acids are usually referred to with either a 1-letter or a 3-letter code: Table 12.1: Amino acid nomenclature in three- and one-letter code. Glycine Gly G Histidine His H Alanine Ala A Proline Pro P Valine Val V Aspartate Asp D Leucine Leu L Glutamate Glu E Isoleucine Ile I Asparagine Asn N Serine Ser S Glutamine Gln Q Threonine Thr T Lysine Lys K Phenylalanine Phe F Arginine Arg R Tyrosine Tyr Y Cysteine Cys C Tryptophane Trp W Methionine Met M The chemical formulas of the natural occuring amino acids together with their COSY, TOCSY and NOESY spectra are shown in appendix F. Amino acids are often classified by the character of their side chain. One way to do this is: aliphatic (A, V, L, I, (G), aromatic (F, Y, W), other ring (H, P), negatively charged (D, E), amidic (N, Q), sulfur-containing (C, M), hydroxy-containing (S, T) and positively charged (K, R). Some residues can be READER 69

75 found in more than one category. For instance, the tyrosine side chain is aromatic as well as hydroxylated. In peptides and proteins, amino acids are linked via a so-called peptide bond, where the amino group of one residue is connected with the carboxyl group of another, while water is released: H O H H 2 N C C N C C R H R O OH The left amino-end end of a peptide chain is called N-terminus, while the right end with the free carboxyl group is called C-terminus. The positions of atoms of the backbone and the sidechain with respect to each other can be described by so-called dihedral or torsion angles. Each of these is defined by four atoms. For instance, a -angle is determined by a CO-N H C -CO fragment, the ψ angle is the one around the C CO bond, and around CON H. All side chain dihedral angles are usually called χ-angles. Note that the peptide bond is planar due to its partial double bond character, which means that the -angle is usually about 180! Polypeptide chains can fold themselves into higher order structures, and here we distinguish different levels. The amino acid sequence itself is also called primary structure. Groups of amino acids may cluster together to form defined local periodic structures, like -helix, -strand (contacting each other to form -sheets) or turn. These are called secondary structure elements. The absence of these secondary structure elements is often called random coil. The overall folding of a polypeptide chain, or how the secondary structure elements and loops are arranged with respect to each other, is called the tertiary structure of a protein. One speaks of quaternary structure when multiple protein subunits cluster together to form a single functional unit. READER 70

76 Assignment of peptides and proteins The assignment strategy based upon homonuclear 2D experiments (COSY, TOCSY, and NOESY) was developed in the 1980 s (K. Wüthrich, 1986). This approach is discussed in this Chapter. A more recent approach employs uniformly 15 N and/or 13 C labeled proteins. The strategy uses so-called double- and triple-resonance experiments (involving 1 H, 15 N and/or 13 C resonance frequencies) to transfer magnetization through the polypeptide chain making use of the large one-bond homo- and heteronuclear J-couplings. For bigger proteins several residues of a certain type are present, and in a COSY for instance, several alanines give rise to similar patterns. So, how can we decide which of the alanines in the protein sequence corresponds to a particular NMR signal pattern? The solution to this question involves three steps: 1. Find patterns of coupled interconnected spins (spin systems) belonging to amino acid residues using COSY and TOCSY (amino acid identification). 2. Connect neighbouring spin systems (in the amino acid sequence) using sequential NOEs. 3. Match stretches of connected spin systems with the (known) amino acid sequence for unique fits. Let us focus now on the individual steps. Step 1: Spin system identification The identification of spins belonging to the same spin system can be performed on the basis of the COSY and the TOCSY experiment (see the READER 71

77 example of an Ala-Ala peptide fragment on the right). In both spectra, most protons belonging to a certain amino acid can be identified. A summary of the expected patterns for the different amino acids is given in Appendix F. Some of the amino acids have a very typical pattern, for instance Gly where there is no side chain, but two H protons, or prolines where no H N is present. Some other amino acids share a common pattern, like the so-called AMX spin systems where AMX represents the H and the two H protons of an amino acid. To this group the following residues belong: Phe, Tyr, Trp, His, Ser, Cys, Asp and Asn. In the COSY and TOCSY spectra of Phe, Tyr and Trp no J-couplings between H and protons in the aromatic ring are observable (more than three bonds apart). This gap can be closed if in addition the NOESY spectrum is used since the distance between these protons is typically smaller than 5 Å. Other amino acid residues that usually can be recognized easily from 1 H-NMR spectra are alanine (its H -atoms form a methyl group, which gives an intense signal at about 1.3 ppm) and serine or threonine (both H and H are found around 4 5 ppm). For bigger proteins spectral overlap makes it harder to identify spin systems. Moreover, the NMR signals are broader in general due to the shorter T 2 relaxation times, which in addition also results in a reduced efficiency of the magnetization transfer during the mixing period in TOCSY since the magnetization is transversal during this time. Step 2: Identification of neighbouring residues When more than, for instance, one alanine is present in the protein, it is not a priori clear which alanine in the primary sequence corresponds to a certain alanine pattern in the 2D COSY and TOCSY spectra. In order to make a so-called sequential assignment, i.e. correlating the COSY/TOCSY patterns to individual amino acids in the primary sequence, we have to connect the pattern of the alanine to the pattern of its sequential neighbour. When using only proton NMR, no 1 H 1 H J-couplings of READER 72

78 appreciable size exist over the peptide bond since the shortest connection of two protons in neighbouring residues involves four bonds. On one side this is very convenient to identify the individual spin systems, on the other side, no information about amino acid neighbours is obtained from COSY and/or TOCSY spectra alone. Fortunately we can employ another mechanism of magnetization transfer, since the short sequential distances between consecutive residues result in cross-peaks in the NOESY spectrum. The Figure above summarizes the sequential assignment approach: the type of the spin system is identified using COSY and/or TOCSY experiments (bold arrows), whereas the sequential connectivity is established by sequential NOESY cross-peaks (dotted arrows). A crosspeak between the H proton of a spin i and the H N proton of the neighbouring spin i+1 results from a short distance between these two residues, often referred to as d N. Similar, the distances between H or H N of spin i to the neighbour H N of spin i+1 are called d N and d NN. Note that d N is not the same as d N ; the first one is usually observable in NOESY, the second one not! Of course also the tertiary structure of the protein causes intense signals in the NOESYspectrum. So how can we be sure to observe a sequential peak? The statistics of these short distances have been investigated on the basis of thousands of known protein structures. Table 12.2 shows for example that 98% of all H H N signals with distances shorter than 2.4 Å come from sequential proton contacts. Naturally, the score drops with increasing distance limit. Similar values are obtained for the H N H N and H H N contacts. This means that many intense H H N, H H N and H N H N cross-peaks most likely result from sequential NOEs. It can be shown that the probability for identifying a sequential connection increases dramatically when simultaneously two (or more) short distances between two spin systems can be found. Table 12.2 shows that if simultaneously two NOEs are found between two spin systems, they most likely result from sequential residues. READER 73

79 Table 12.2: Chances that intense NOEs between protons from different spin systems are sequential. Distance (Å) j i = 1 (%) d N (i,j) d NN (i,j) d N (i,j) d N (i,j) 3.6 & d NN (i,j) d N (i,j) 3.6 & d N (i,j) d NN (i,j) 3.0 & d N (i,j) Step 3: Matching to the sequence After detemination of amino acid neighbors, the next step in the assignment of proteins and peptides is to locate the fragment of connected residues in the amino acid sequence by matching connected spin system patterns with amino acids regions in the sequence. For bigger proteins there might still exist several possibilities. Naturally, we can try to link more patterns together and try to make tri-peptide, tetra-peptide, and even bigger fragments. The uniqueness of such di-, tri-, and tetra-peptide fragments in proteins with less than 200 residues has also been investigated. If all residue types could be identified unambiguously in the fragment from the COSY and/or TOCSY spectra, a given di-peptide fragment has a probability of uniqueness of 56%, but the tri-peptide and tetra-peptide fragments of 95% and 99%, respectively. As expected, increasing the length of the READER 74

80 fragment increases its uniqueness in the amino acid sequence. A tetra-peptide fragment is usually sufficiently unique to allow its identification in the polypeptide chain Secondary structure elements in peptides and proteins The intensities of sequential NOEs contain information on the secondary structure present in a protein because they depend on the local conformation of the polypeptide backbone. Take, for instance, the distance between an H proton of a residue i to the H N proton of the following residue (i+1). In an extended region (-strand) this distance is short (2.2 Å) whereas it is 3.5 Å in a helical conformation. Therefore, in a NOESY experiment, d N for an amino acid in an extended region is much more intense than for an amino acid in an - helix. This information, together with other characteristic medium range NOEs (between residues which are less than 4 positions apart in the sequence) is usually sufficient to specify the secondary structure element in which an amino acid exists. An overview of important sequential- and medium-range proton-proton distances is given in Figure Figure 12.1: Characteristic sequential and medium-range NOE connectivities. READER 75

81 The recognition of secondary structure elements (i.e. -helices, -strands and -sheets, turns) constitutes an important element in the structure determination process. Luckily, many of the observable NOE cross-peaks between different residues are due to this secondary structure. Besides these NOEs, also chemical shifts, the 3 J HNH coupling and the detection of amide proton that undergo slow proton-to-deuterium exchange when dissolved in D 2 O (see Chapter 13) have traditionally been used as markers for secondary structure. Figure 12.2 shows the short distances in an anti-parallel -sheet (upper part) and in a parallel -sheet (lower part). Figure 12.2: Anti-parallel (top) and parallel (bottom) -sheet. Sequential NOEs are indicated by open arrows, interstrand NOEs by solid arrows. Hydrogen bonds connecting strands are shown by wavy lines. READER 76

82 It appears possible to distinguish parallel from anti-parallel -strands by looking at the NOE pattern. The anti-parallel -sheet is characterized by short d N (i,i+1) and interstrand d (i,j) distances whereas the d (i,j) distance is much longer in parallel -sheet (4.8 Å). Also the d NN (i,j) distance in parallel -sheet is much longer compared to the d NN (i,j) distance in anti-parallel -sheet. In Figure 12.3 short distances are shown for an -helix. It is known that an -helix is characterized by the close proximity of residues i and i+3 and residues i and i+4 due to hydrogen-bond formation between CO(i) and H N (i+4). As a consequence, the presence of d N (i,i+3) and d N (i,i+4) NOEs is a clear marker for this element of secondary structure. In addition to the aforementioned short distances in an -helix, the sequential d NN (i,i+1) distance is also short, and strong sequential d NN NOEs can be found in the spectrum. Figure 12.3: Proton contacts in an -helix. The sequential d NN is shown (2.8 Å) together with d N (i,i+1), d N (i,i+2) and d N (i,i+3) (side-chains are not shown). The hydrogen bond CO(i) NH(i+4) is indicated. Protein backbone turns are characterized by short distances between residues i and i+2; in particular the d NN (i,i+2) distance. In practice, however, since there exists a large number of turns, which all have mirror images as well, e.g. I, I, II, II, III, III, the precise nature of the turn is hard to establish from NOE and J-coupling data alone. An overview of the READER 77

83 characteristic patterns and short distances found in -strands, -helices and different turns is given in Figure Figure 12.4: Characteristic NOEs for several secondary structure elements. The thickness of the bars reflects the strength of the NOE. The thicker the bar the stronger the NOE (and the shorter the distance between the protons involved). 3 J-coupling constants are given at the bottom. The short distances in secondary structures leading to observable NOEs in NMR spectra are listed in the following table: Table 12.3: Secondary structure specific atomic distances (in Å). Distance -helix helix p turn I a turn II a d N d N (i,i+2) d N (i,i+3) d N (i,i+4) 4.2 d NN d NN (i,i+2) d N d (i,i+3) a for turns, the two numbers apply for the distances between residues 2, 3 and 3, 4 respectively. READER 78

84 12.2 Nucleotides and nucleic acid Nucleic acid is mainly build from five different nucleotides. All of them share a common general structure: a nucleobase, a pentose-sugar ring (ribose in the case of RNA ; 2'-deoxy ribose in DNA) and a phosphate group which links the nucleotide units to each other. (Nucleo) Base Phosphate Sugar Figure 12.5: Common structure of nucleotides. There are two sorts of nucleobases: the purine bases adenine (A) and guanine (G) and the pyrimidine bases uracil (U, only found in RNA), thymine (T, only found in DNA) and cytosine (C). The base is connected to the sugar moiety via a glycosidic bond at the 1' carbon of the pentose ring (see Figure 12.7 for the atom nomenclature). In the common nucleotides the phosphate group is attached to the 5' carbon of its sugar. A nucleotide which has no phosphate group is called a nucleoside. In oligonucleotides and in nucleic acid, the phosphate group is linked between the 5' carbon of one pentose and the 3' carbon of the next. Evidently, NMR spectra of large nucleic acids are much more complex than NMR spectra of peptides of comparable size. While peptides are built from as many as 20 different building blocks (the different 'natural' amino acids) nucleic acid is built from only four different nucleotides. Consequently the number of occurrences for a particular kind of nucleotide is in average five times as high as for any particular amino acid, and pairs of sequentially assigned spin systems occur more often than in protein. Also, the presence of only four different building blocks usually leads to very crowded regions in the NMR spectra. All these factors complicate the spectral assignment for nucleic acids considerably. READER 79

85 Figures 12.6 to 12.9 show the different nucleobases and sugars with their numbering schemes and the eight different RNA and DNA nucleotides Adenine Uracil Thymine Guanine Cytosine Figure 12.6: The five different nucleobases found in nucleic acids. 5' 5' 4' 1' 4' 1' 3' 2' 3' 2' Ribose (RNA) 2'-deoxy ribose (DNA) Figure 12.7: The pentose sugar rings of RNA and DNA. READER 80

86 AMP Adenosine- 5'-phosphate GMP Guanosine- 5'-phosphate UMP Uridine- 5'-phosphate CMP Cytidine- 5'-phosphate Figure 12.8: Ribonucleotides (RNA). damp Adenosine- 5'-phosphate dgmp Guanosine- 5'-phosphate dtmp Thymidine- 5'-phosphate dcmp Cytidine- 5'-phosphate Figure 12.9: 2'-Deoxy-ribonucleotides (DNA). READER 81

87 The Figure below illustrates how nucleotide units are linked by the phosphate groups in oligonucleotides and in nucleic acids. If we want to analyze the spin systems of this molecule by NMR, we find that both the link between sugar and nucleobase and between the individual nucleotide units (via the phosphate groups) reach further than three bonds before the next proton can be found. In other words: the bases and the sugars form isolated spin systems. This is important when it comes to think about an assignment strategy for these molecules! Figure 12.10: The phosphate-sugar backbone of DNA. When we look at the structural properties of nucleic acids, the most prominent feature is the double-helical conformation it often adopts. The two strands of the helix adhere to each other by means of hydrogen bonding between bases of two complementary stands. These so-called base-pairs were found by James Watson and Francis Crick and accordingly named 'Watson-Crick base-pairs'. Base-pairs are always built from one purine and one pyrimidine base. Adenine either pairs with Uracil (in RNA) or with Thymine (in DNA), Guanine always pairs with Cytosine. The hydrogen bond patterns are shown in READER 82

88 Figure Note that the GC pair is more stable than the AT pair, because the GC base-pair has three hydrogen bonds, while AT has two. T A C G Figure 12.11: Watson-Crick base-pairs. AT on the left, GC on the right. There are two major conformations in which the double helices of DNA and RNA usually are found, the so-called A-form and the B-form (Figure 12.12). Both of them are righthanded (like ordinary corkscrews) and the difference between the two forms lies mainly in the width and height of their turns. A third conformation, the Z-form is left-handed and of minor importance. The important parameters of the A-, B- and Z-form helix are listed in the following table. Table 12.4: Structure parameters for different DNA double-helices. Geometry attribute A-form B-form Z-form Helix sense right-handed right-handed left-handed Repeating unit 1 bp 1 bp 2 bp Rotation/bp /2 Mean bp/turn Inclination of bp to axis Rise/bp along axis 0.23 nm nm 0.38 nm Pitch/turn of helix 2.46 nm 3.32 nm 4.56 nm Mean propeller twist Glycosyl angle anti anti C: anti G: syn Sugar pucker C3'-endo C2'-endo C: C2'-endo G: C2'-exo Diameter 26 nm 20 nm 18 nm READER 83

89 A-form double helix (RNA) B-form double helix (DNA) Figure 12.12: The two major conformations of double-helical DNA. Information about typical chemical shift values of A-DNA and B-DNA, and short distances that lead to observable signals in NOESY spectra can be found in Appendices G and H Assignment of oligonucleotide and nucleic acid spectra The assignment strategy for nucleic acid spectra is very similar to the one discussed for peptides and proteins. The major difference is that for the identification of a particular nucleic acid residue always the combination of COSY/TOCSY and NOESY is needed, because the bases form isolated spin systems that cannot be connected to their sugars without making use of NOE data. We will not go into the details of this. Tables to be used for this purpose with common shift values and short distances in nucleic acids can be found in Appendices G and H. READER 84

90 13 STRUCTURE DETERMINATION In this Chapter we will discuss the method to determine the 3D structure of a protein. Apart from internal amino acid bond lengths and several bond angles, which are known on the basis of the amino acid sequence, the most important source of structural information is the NOE. In particular, the so-called "long-range" NOEs (those between protons more than four residues apart in the amino acid sequence) provide important structural information. We will first see which experimental NMR parameters can be translated into structural constraints and then describe the computational structure calculation procedure Sources of structural information NOEs For a ~10 kd protein typically between 1000 and 2000 cross-peaks can be observed in a 2D NOESY. We have seen in Chapter 8 how NOE intensities can be converted into proton proton distances with the aid of a reference distance (for instance the distance of 2.45 Å between neighbouring protons on an aromatic ring). In principle the r 6 relation between NOE and distance should give very precise distances. For example, if there is a 10% error on the NOE intensity this translates in only a 1.5% error in the distance! However, there are two reasons why such a high precision cannot be obtained in practice. The first reason is the presence of local mobility. Remember that the simple relation of Eqn. 8.5 was derived on the assumption of equal c for the protons of reference and unknown distance. However, only for rigid proteins this assumption is really valid. Also, if there is a conformational equilibrium, e.g a rotating methyl group, the r 6 dependence leads to a distance value which seems shorter than the individual distances really are (see exercise 32). The second reason is the so-called "spin-diffusion" effect. This is the result of multiple transfer steps of magnetization A B C during the mixing time m that may READER 85

91 disturb the intensity of the direct NOE between A and C. Only for molecules with very short c, or when m is short this indirect transfer path can be neglected. For these two reasons NOE-based distance constraints are often used in the form of upper limits or distance ranges rather then precise distances: strong NOE medium NOE weak NOE Å Å Å This procedure works well in practice because it turns out that for a high precision of the structure a large number of NOE constraints is more important than precise distances J-couplings The magnitude of observed three-bond coupling constants, 3 J, is related to the dihedral angle (between the two outer bonds) and can provide a constraint on this angle (). This is expressed in the empirical Karplus relations J Acos 2 ( ) B cos( ) C (13.1), where A, B and C are parameters that depend on the particular situation. For instance, the J-coupling between the amide proton and the -proton, 3 J HNH depends on the backbone angle as follows: J 6.51cos 2 ( 60) 1.76cos( 60) 1.60 (13.1a). 3 READER 86

92 The form of the Karplus relation for this case is shown in Figure Figure 13.1: The Karplus curve for the protein backbone -angle. A complication is that several values of give rise to the same J. Also, conformational averaging within a molecule may lead to average values of J in the range 6 7 Hz. Therefore, the most reliable values for J HNH are 910 Hz for extended (-sheet) structure and 34 Hz for -helical structure (Figure 12.4), while values around 67 Hz are often difficult to interpret and ignored. Similar Karplus curves exist for CH-CH J- couplings from which side chain -angles can be derived Hydrogen bond constraints When a dried protein is dissolved in D 2 O many of the attached amide protons exchange rapidly with deuterium and therefore their signals disappear from the spectrum. However, often some H N signals remain in the spectrum for some time and exchange only slowly with deuterium. These slowly exchanging H N -atoms are invariably present in hydrogenbonds for instance to stabilize -helices and -sheets. If the H-bond acceptor is known READER 87

93 with certainty, for instance because we have several short- and medium-range NOEs defining the secondary structure, we can use distance constraints corresponding to the H- bonds. Usually, a range of Å is used for the distance between H N and O, and a range of Å for the distance between N H and O Structure calculations Several computer programs exist that are able to calculate the 3D structure of a protein based on distance and dihedral angle constraints. Often one starts using only geometric constraints (distances and angles) in a so-called Distance-Geometry (DG) procedure. The resulting structures are subsequently refined with Molecular Dynamics (MD) simulations which include also energy terms for electrostatic interactions etc. The calculation procedure will now be briefly discussed Distance-Geometry In theory, the structure of a macromolecule containing N atoms can only be perfectly described by specifying the N(N1)/2 distances between all atoms (except for the chirality). Since this is a very large number we will never have so many distance constraints in practice. There is, however, a method called Distance-Geometry (DG) that converts distances into Cartesian coordinates for a much smaller number of distances even if they are not precisely known. The algorithm for Distance Geometry executes the following procedures: 1. Set up matrices for upper bound (u) and lower bound (l) distances (and angles) for all atoms in the structure. These include the so-called holonomic values (standard amino acid bond lengths and angles) and the NOE distance constraints. For those elements for which no information is available the upper bound is set to a large value and the lower bound to the sum of the Van der Waals radii (1.8 Å). READER 88

94 2. Smoothing The upper and lower bounds are adjusted using triangular inequalities. Consider three atoms i, j and k with upper bounds u ij, u ik, and u jk, and lower bounds l ij, l ik and l jk. The maximum distance between i and j is when i,j and k are colinear. Thus we must always have uij (13.2). uik u jk A similar relation exists for the lower bound l ij : lij (13.3). lik u jk These relations are applied to all distances in the set until no further changes are obtained. 3. Select a distance matrix for all atoms with random trial distances between upper and lower bounds. 4. Embedding This is the procedure of finding the 3D structure that fits the distance matrix best. We will not explain this further here. 5. Optimization Because the distance matrix is not perfect, it is usually necessary to optimize the structure coming from the embedding stage. This optimization involves the minimization of the coordinates against a distance error function. Because of the random nature of step 3, repetiting steps 35 will produce slightly different structures. Ideally, if this is repeated many times, an ensemble of structures is READER 89

95 obtained, which samples the conformational space consistent with the experimental data. An example of such an ensemble of structures is shown in Figure Figure 13.2: Structures of the protein crambin calculated by the DG procedure Restrained Molecular Dynamics The Distance Geometry protocol makes use of NMR restraints, but not of the biophysical properties of the molecule, like hydrophobicity, vanderwaalscontacts or electrostatics. Refinement of structures calculated by DG is done by Molecular Dynamics (MD). MD programs were originally designed to simulate the dynamic behaviour of atomic or molecular systems. It turns out that a purely classical approach works well for this purpose. For the initial coordinates and velocities of all atoms i, Newtons equation of motion is solved: m i d dt 2 ri 2 F i (13.4), where m i is the mass, r i the position vector and F i the force acting on atom i. These forces are given by READER 90