NMR Spectroscopy. Structural Analysis. Rainer Wechselberger 2009

Transcription

1 NMR Spectroscopy in Structural Analysis Rainer Wechselberger 2009

2 ii 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 by me and adopted to my lecture 'NMR Spectroscopy in Structural Analysis' I thank Hugo van Ingen, Marloes Schurink and Hans Wienk for proofreading, discussions, suggestions, and continuing to improve the course material. Rainer Wechselberger, Utrecht in the summer of 2009 Please report errors in the text and/or explanations or any 'unclear' passages to: rwechsel@its.jnj.com or hans@nmr.chem.uu.nl

3 iii I Introduction Typical applications of modern NMR Some history of NMR Aim of this course General outline... 4 II Basic NMR Theory... 5 III An Ensemble of Nuclear Spins Ensemble of spins Effect of the radio frequency (RF) field B IV Spin relaxation Molecular basis of spin relaxation V Fourier Transform NMR From time domain to spectrum Aspects of FT-NMR VI Spectrometer Hardware The magnet The lock The shim system The probe The radio frequency system The receiver VII NMR Parameters Chemical shifts Effects influencing the chemical shift Protein chemical shifts J-Coupling... 40

4 iv Equivalent protons VIII Nuclear Overhauser Effect (NOE) Dipolar cross relaxation NOEs in biomolecules IX Relaxation Measurements T 1 relaxation measurements Calculated example for an inversion-recovery experiment Applications of T 1 relaxation T 2 relaxation measurements Applications of T 2 relaxation X Two-Dimensional NMR The SCOTCH experiment D NOE D COSY and 2D TOCSY XI The assignment problem Chemical shift Scalar coupling Signal intensities (integrals) NOE data XII Biomolecular NMR Peptides and proteins Assignment of peptides and proteins Secondary structural elements in peptides and proteins Nucleotides and nucleic acid Assignment of oligonucleotide and nucleic acid spectra XIII Structure determination Sources of structural information... 86

5 v NOEs J-couplings Hydrogen bond constraints Structure calculations Distance-Geometry Restrained molecular dynamics Appendices Appendix A a) Typical 1 H and 13 C chemical shift of common functional groups Appendix A b) Random coil 1 H chemical shifts for the 20 common amino acids Appendix B: 1 H chemical shift distribution of amino acids Appendix C: Nuclear Overhauser Effect Appendix D: 2D NOESY experiment Appendix E: Expected cross-peaks for COSY, TOCSY and NOESY for the individual amino acids Appendix F: Typical chemical shift values found in nucleic acid Appendix G: Typical short proton proton distances for B-DNA

6 1 I Introduction All spectroscopic techniques are based on the absorption of electromagnetic radiation by molecules or atoms. This absorption is connected to transitions between states of different energies. The nature of the electromagnetic radiation varies from hard γ-rays in Mössbauer spectroscopy to very low energy radio frequency 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), IR spectroscopy, where vibration states are excited, 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 lines are quite narrow and therefore the resolution is usually so high that hundreds of lines can be 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 quickly an indispensable tool for structural studies in chemistry and later also in biochemistry. NMR is the only available method to date to determine the structure of proteins and nucleic acids in solution on an atomic scale so that it became a well-established method in the field known as "Structural Biology".

7 2 1.1 Typical applications of modern NMR Structure elucidation Synthetic organic chemistry (often together with MS and IR) Natural product chemistry (identification of unknown compounds) Study of dynamic processes Reaction/binding kinetics Chemical/conformational exchange Structural studies of biomacromolecules Proteins, protein-ligand complexes, DNA, RNA, protein/dna complexes, Oligosaccharides Drug Design Structure Activity Relationship (SAR) Magnetic Resonance Imaging (MRI) MRI today is a standard diagnostic tool in medicine. 1.2 Some history of NMR NMR was discovered in 1945 by Bloch at Stanford and Purcell at Harvard University. For this Bloch and Purcell received the Nobel Prize for physics in Initially, it belonged to the realm of physics but after the discovery of the chemical shift (nuclei in different chemical surroundings have different resonance frequencies) the technique quickly became very important as an analytical tool in chemistry. The development of

8 3 stronger magnets (maximum proton frequency is now (2006) 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 prices for NMR followed in 1991 (Richard Ernst) and in 2002 (Kurt Wüthrich). 1.3 Aim of this 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 the three-dimensional structures of biomolecules by NMR. 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, but also the intensity of the Nuclear Overhauser Effect, which in turn 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 R.R. Ernst received the Nobel prize for chemistry in 1991) also allows the measurement of two-dimensional (2D) NMR spectra (and even 3D and 4D). 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 (3- dimensional) structure out of NMR data is given.

9 4 1.4 General outline I II III IV V VI VII VIII IX X XI XII XIII Introduction (what is NMR and what do you study with it) Theory (how does it work) Ensemble of spins (from single atom to real samples) Relaxation I (after the experiment: back to equilibrium) FT NMR (with a single RF-pulse to a complete spectrum) Hardware (what kind of device do you need for FT NMR) NMR parameters (what you can see in an NMR spectrum and why) NOE (How does the Nuclear Overhauser Effect work) Relaxation II (experiments to measure relaxation properties) 2D NMR (how to add an extra dimension and what's the good to it) Assignment (which signal comes from which atom) Biomolecular NMR (nucleic acids and proteins, spin systems and (structural) parameters, sequential assignment) Structure Determination (which parameters to use, how to calculate a structure)

10 5 II 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. Most nuclei have such a spin angular momentum, which is represented by a corresponding spin quantum number I, which can be integer or half-integer (I = 0, 1/2, 1, 3/2...) and we may simply speak of a nucleus with spin I. The magnitude of the spin angular momentum is given by (2.1) I = h I( I +1) (2.1) where h is h/2π ( 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 = h m I (2.2) 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 equations 2.1 and 2.2 leads to the spin-state diagrams for nuclei with different spins (e.g. ½, 1, ³/ 2 ):

11 6 A combination of their spin angular momentum and their 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 = γ I (2.3) γ is called the gyro magnetic ratio. Since I is quantized, accordingly also μ is quantized and we can express μ in terms of the spin quantum number, I or μ z in terms of the magnetic quantum number, m I : μ = γ h I( I + 1) (2.4) and μ = γ h (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 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. Expression (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 μ in an external magnetic field, B 0 as E = μz B 0 (2.7) and with Eq. 2.5 it becomes clear, that the energy of a nuclear spin depends on the magnetic quantum number m I : E = γ h m (2.8) For a spin ½ nucleus this results in two states, denoted α for m I = +½ and β for I B 0

12 7 m I = ½. As a consequence, the nuclear spins can not be found turning around in order to orient in an external magnetic field, but they rather can be found only in two different orientations, corresponding to the two possible energy levels described by the two magnetic quantum numbers: In all forms of spectroscopy, transitions between energy levels are induced by electromagnetic radiation of a particular frequency ν 0, provided that the frequency matches the energy difference between these energy levels: Δ E = h ν 0 (2.9) This is sometimes called Einstein's equation and it is a basic relation in spectroscopy. Based on Eq. 2.8 we find that ΔE = γ h B 0 = hν 0, or 0 γ B 2 π ν = (2.10a) 0 or even simpler: ω 0 = γ B 0 (2.10b)

13 8 expressed in terms of the angular frequency ω 0 = 2πν 0 (2.10c) In NMR the relation (2.10b) is often called the "resonance condition" i.e. the condition where the frequency of the radiation field matches the so-called Larmor frequency ω L = γb 0. For nuclei with spin I larger than ½ we have multilevel energy diagrams. However, the selection rule Δm 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 (equation 2.6), we can modify the figure from above and adapt it for different magnetic fields B 0 : E E = +½γ h B 0 (β-state, m= ½) ΔE = γ h B 0 E = ½γ h B 0 (α-state, m=+½) B 0 B 0 The separation ΔE of the energy states α and β and thus the resonance frequency, depends on the sort of nucleus (γ) and the strength of the external magnetic field B 0 or, in other words, on the strength of the NMR magnet (compare 2.10).

14 9 Table 1 shows some properties of nuclei important for applications in organic chemistry and biochemistry. Table 1: Properties of selected nuclei Isotope Nuclear spin I Resonance frequency gyro magnetic ratio γ [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) We will focus here on spins with I = ½ because they have only two possible energy states and accordingly give only a single spectral line and thus are the most popular spins in high-resolution NMR. For protein studies these 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 in addition 31 P is an important nucleus.

15 10 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 (2.10). Interestingly, a similar expression can be obtained from a classical description of the effect of a magnetic field B on a spinning magnet with a magnetic moment μ that is tilted with respect to the field direction. While a classical non-spinning bar magnet would just 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. 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. This is the equation of motion : dμ dt e x x x y y y z = γ B μ = γ B B B (2.11) μ e μ e μ z z where e x, e y and e z are unit vectors forming an orthogonal coordinate system along the x-, y- and z-axis, respectively. Note that the direction of the precession is perpendicular to both B and μ. 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

16 11 dμ x dt dμ y dt dμ z dt = γ B μ = y = γ B μ x (2.12) It can be shown that a correct solution is given by μ ( t) = μ (0) cos( γ B t) + μ (0) sin( γ B t) x μ ( t) = μ (0) sin( γ B t) + μ (0) cos( γ B t) y μ ( t) = μ (0) z x z x 0 0 y y 0 0 (2.13) This indeed describes a precession of μ about the z-axis with an angular frequency ω 0 given by ω 0 = γ B 0 (2.14) an expression identical with (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 a 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.

17 12 III An Ensemble of Nuclear Spins 3.1 Ensemble of spins In a sample of identical molecules we are dealing with a large number of nuclear spins. In the quantum-mechanical picture these are distributed over the spin states according to Boltzmann's law: n n β α = e ΔE kt (3.1) where n α and n β are the populations of the α and β state, respectively, and k is Boltzmann's constant (k= (24) J/K). For I = ½ such a distribution is shown on the right. Due to the very small energy difference ΔE = γ h B 0 the populations of the α and β states are almost equal. For a field B 0 = 14 T (Tesla) (proton frequency 600 MHz) the relative excess of α 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 described as shown on the right. The individual spin vectors all make an angle with the field B 0 and are slightly more aligned parallel to the field than antiparallel. In equilibrium the "phases" of the individual spins

18 13 (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 tipped away from B 0 (see figure on the left) it will perform a precessional motion about B 0 with frequency ω 0 = γb 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 spin states of the nucleus! 3.2 Effect of the radio frequency (RF) field B 1 In all FT-NMR spectrometers an additional field B 1 can be generated perpendicular to the static field B 0. This B 1 -field is created by means of radio frequency pulses (it s basically 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 perfect analogy to the precession around the B 0 -field, the speed of this rotation (its angular frequency) can be described as ω 1 = γ B 1 (3.2) 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 (γ), the strength of the B 1 -field and the duration of the pulse. Illustrating the effect of a radio frequency pulse is a bit tricky. Actually the magnetization vector is still precessing

19 14 around the B 0 -field (with ω 0 ) and at the same time, during the RF pulse, is tilted away from the z-axis (is rotating around the x-axis with ω 1 )! Fortunately there is a simple way to describe this: the rotating frame. In a frame of reference in which the x- and y-axes rotate with ω 0 about the z-axis, 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: Rotating frame Laboratory frame 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 ω 1 = γb 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 just follows the equation of motion given in If we keep the B 1-field on for a short time such that ω t = / 2 (3.3) 1 π

20 15 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 is along the negative z-axis (180 pulse): 90 pulse 180 pulse 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.

21 16 IV Spin relaxation After a 90 pulse from the x-axis M lies along the y-axis. If we leave the system now undisturbed we will reach the equilibrium state again after a while by a process called spin relaxation. Actually two things will happen: 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. Thus we have T 1 : longitudinal or spin-lattice relaxation time (z-magnetization), T 2 : transverse or spin-spin relaxation time (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) M ( t) M = M (0) M e (4.2) The solution of Eq. 4.1a is: [ ] 1 z eq z eq t T and of Eq. 4.1b: M ( t) = M (0) e 2 (4.3) x x t T

22 17 and similarly for M y (t). Thus, all components of M return exponentially to their equilibrium values. An important difference between T 1 and T 2 processes is that the former involves changes in z-magnetization and hence, transitions between α and β spin states that are accompanied by an exchange of energy with the "lattice" (environment). In contrast, T 2 processes involve loss of phase coherence in the xy-plane, no energy is 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 which affects the z-component does not create x- or y-magnetization (cf. Eq. 4.1). 4.1 Molecular basis of spin relaxation What causes nuclear spins to relax? The simple answer is: exchange of energy with the environment. But the energies of the corresponding processes must match the ΔE between the involved energy states. In most other spectroscopic techniques, this is achieved by collisions (with other atoms or molecules). For the small energy differences in NMR this is not an option. We have to look for processes with comparably small energies (i.e. comparable frequencies) as the corresponding resonance frequencies in NMR. We can find these in the interaction of moving magnetic dipoles. The magnetic dipoles are the NMR nuclei themselves. The movement comes from the diffusional motion of molecules. This process contributes both to T 1 and T 2 relaxation. But while with T 1 relaxation energy is exchanged with other molecules (the environment, the lattice ) causing transitions between α and β states, in T 2 relaxation the energy is exchanged with spins of the same molecule, leading to a small variety in the precession frequency of otherwise identical spins. The net-magnetization vector M is split up in many small components rotating with different frequencies. Eventually these components are equally distributed in the xy-plane and no measurable transversal magnetization is left. One speaks about dephasing of magnetization. It is important to note that also a

23 18 static distribution of B z -fields causes different frequencies (in different locations of the sample) and also leads to dephasing i.e. T 2 relaxation. To understand the effect of motion, it is important to consider the time-scale of it. For T 1 relaxation only motions with a frequency near the Larmor frequency ω 0 are effective. After all, they have to induce transitions between spin states and therefore must have a frequency, which coincides with the ΔE between the states (thus, the Larmor frequency). The time-dependence of the dipolar field comes from the rotational diffusion (the thermal motion, see figure to the left) of the molecules which is characterized by a rotational correlation time τ c. For times smaller than τ c the orientation of a molecule has not changed much (leftmost figure below), while for t >> τ c the correlation between different orientations is lost (rightmost figure): For small fast tumbling molecules τ c is quite short ( s) while for large biomolecules it is much longer ( s). An approximate relation of τ c with the molecular volume V and the viscosity η is given by η V τ c = (4.4) k T For macromolecules of molecular mass M r in H 2 O solution at room temperature a useful approximation is

24 19 M 2.4 r 12 τ c 10 (4.5) In a randomly tumbling motion many frequencies are present. The distribution of the frequencies of the motions is represented by the spectral density function, J(ω). J(ω) 1/τ c ω -1 In this distribution the frequency ω = τ c acts much like a cut-off of the spectral density function, in other words: motions with frequencies ω > τ -1 c are quite rare. The area under the curves is constant. Only the shape of J(ω) differs for molecules of different size. For a smaller molecule for instance, J(ω) would look like: J(ω) 1/τ c ω τ c for smaller molecules is shorter and their frequency distribution extends to higher values of ω. On the other hand, J(ω) for low values of ω is relatively small. With increasing size of the molecules J(ω) is getting larger for slow motions but at the same time the cut-off moves to lower frequencies. In other words, slow motions (small values

25 20 of ω) are more likely to occur for (big) biomolecules than for small organic molecules. Not very surprising indeed! Now what does this mean for T 1 relaxation? Here, the fluctuating fields have to induce transitions between α and β states separated by the Larmor frequency ω 0 = γb 0. Therefore, the efficiency of relaxation will depend on how much this frequency is present in the distribution of frequencies of the molecule, thus on J(ω 0 ). This is maximal for molecules that have τ -1 c = ω 0, and the efficiency will drop for both larger and smaller molecules (longer and shorter τ c values). This explains the behavior of T 1 versus correlation time τ c as shown in the next figure. The minimum in T 1 (most efficient relaxation) is at ω 0 τ c = 1, which for common NMR fields occurs for intermediate size molecules of molecular mass M r 1000 D. In terms of the fluctuating fields B x (t) and B y (t) a general expression for the efficiency (or rate) of T 1 relaxation is

26 21 1 T ( B B ) J ( ω ) 2 = γ x + y 0 (4.6) where the average of the square of B x (t), < B 2 x >, is a measure of the strength of the fluctuating fields. Assuming that the components in all directions are the same ( < B 2 x > = < B 2 y > = < B 2 z > = < B 2 > ) Eq. 4.6 becomes γ B J ( ω0) T = (4.7) 1 A similar expression for T 2 relaxation is 1 T 2 γ ( J (0) + J ( ω )) 2 2 = B 0 (4.8) This describes the two mechanisms that contribute to T 2 : the static distribution of B z - fields (no or zero frequency), J(0), and the effect induced by B x (t) and B y (t), which is proportional to J(ω 0 ). For larger biomolecules the J(0) term will dominate and becomes approximately equal to τ c. Thus, for slowly tumbling molecules we have the simple expression γ B τ c (4.9) T 2 This means that T 2 becomes progressively shorter for larger molecules and explains why T 2 unlike T 1 does not go through a minimum.

27 22 V Fourier Transform NMR 5.1 From time domain to spectrum Nowadays, all modern NMR spectrometers work as so-called Fourier-Transform NMR spectrometer (FT-NMR). Earlier we have seen (chapter 3.2) that the magnetic component of an electromagnetic field (RF), applied along the x-axis of the rotating frame on equilibrium z-magnetization, results in a precession around the x-axis when ω = ω RF 0. For the precession frequency, ω 1, we found ω 1 = γ B 1. If we apply this RF field for a period t = π/2ω 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 pulse-computer, which can generate a single RF pulse or a series of RF pulses of arbitrary length, frequency, phase, and amplitude separated by delays of adjustable length. An RF pulse of length τ p excites the frequency-range ν RF - 1/(2τ p ) to ν RF + 1/(2τ p ). To excite a certain range of frequencies, τ p must be adjusted to be sufficiently short. For example, at 600 MHz proton resonance frequency a good excitation of an NMR spectrum of 10 khz requires τ p << 100 μs. In practice, pulses of τ p = 2-20 μs are used. ν rf Δν rf τ p ν rf -½τ p ν rf ν rf +½τ p A radiofrequency pulse of duration τ p (left) and the corresponding excitation profile (right). Details in the text above. What will happen after a single RF pulse of 90 along the x-axis? Earlier we have seen that the magnetization vector has been rotated and now is oriented along the y-axis (phase

28 23 coherence of the spins). The receiver is tuned to the frequency ω RF. If the resonance frequency of spin j, denoted by ω j, is equal to ω RF the magnetization will remain along the y-axis (of the frame rotating at speed ω RF ). The magnitude of the magnetization will be decreased by T 2 relaxation. In case spin j has a resonance frequency ω j, which is x component different from ω RF, then the magnetization of spin j will precess in the frame rotating at ω RF with the difference frequency Ω = ω j - ω RF (5.1) Needless to say, also in this case the y component magnetization decays due to relaxation processes. The in the xy- plane rotating magnetization vector induces a current when it passes the receiver coil. This is the y y y actual signal recorded by the receiver. In x x x order to be able to distinguish between positive and negative frequencies (vectors which are rotating clockwise or counterclockwise with the same speed), 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 of the H α, the H β, and the H N ), each with their own equilibrium magnetization M j eq, frequency ω j, and relaxation time T 2j, the FID consists of the sum of all magnetizations: M y ( t) = M ( t) = x j j M M j j (0) cos (0) sin [ ω t ] t T e 2 t T [ ω t ] e 2 j j (5.2)

29 24 where M j (0) = M eq j in the case of a 90 excitation pulse. The following figure illustrates the difference in appearance of an FID containing only a single frequency and an FID containing multiple frequencies: The FT-NMR signal (the FID) is recorded in the time domain. The signal is digitized by an analog-to-digital (ADC) converter and stored in the memory of a computer. The resonance frequencies, ω j ', are extracted by Fourier analysis. The Fourier Transformation (Eq. 5.3) transforms the signals f(t) from the time domain to the frequency domain, g(ω): i ω t g( ω ) = f ( t) e dt (5.3) where the complex signal f(t) is defined as f ( t) = M ( t) + i M y x ( t) = M j eq j e iω t j e t T 2. (5.4) Fourier Transformation results in a sum of complex frequency signals. The real part of

30 25 which describes an absorption signal: T eq 2 j Re[ g( ω)] = M j (5.5) 2 1+ T ( ω ω ) j 2 2 j j The imaginary part describes a dispersion signal: eq T ω ω j ) Im[ g( ω) ] = M j (5.6) T ( ω ω ) j 2 2 j ( 2 2 j j The resonance line shape is a so-called Lorentz line shape. Usually we are only interested in the real part of g(ω), the absorption response, because the flanks of absorptive line drops with ω 2, whereas the dispersive line goes with ω 1. This means that the absorptive line is much narrower. From Eqn. 5.5 we can calculate the relationship between T 2 and the line width at half-height, Δν 1/2 : 0.5 = 1+ T 1 π Δν (5.7)

31 26 which can be rewritten as: 1 Δ ν 1 = (5.8) 2 π T 2 Some important Fourier pairs are shown here. Basically these are the ones responsible for the shapes (or their distortion) of most lines observed on a FT-NMR spectrometer I f(t) t FT g(ω) M t FT I FT t 5.2 Aspects of FT-NMR It might be that the signal-to-noise ratio is not good enough after a single scan. By coadding n successive NMR measurements the signal, S, increases by a factor n. The noise,

32 27 N, however, fortunately increases with n only. This is due to the random nature of noise. Hence the signal-to-noise ratio, S/N, improves as S ~ n (5.9) N FT-NMR is a very flexible technique. A large number of different experiments can be done with the FT technique, each aiming at different parameters of the molecules in study to be extracted, such as relaxation measurements (discussed in chapter 9), multidimensional NMR (discussed in chapter 10), heteronuclear NMR, etc.

33 28 VI Spectrometer Hardware 6.1 The magnet The magnet is the core of the NMR spectrometer. Nowadays mainly persistent superconducting coils are used to generate the high magnetic fields necessary for high resolution NMR (permanent magnets are only used e.g. in food sciences or on older, lower field NMR imaging systems). The coils consist of NbTi (or NbTi-Nb 3 Sn, NbTi- (NbTa) 3 Sn) wires which are superconducting at 4 K ( 269 C). Several layers of coils generate a higher and 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 unsymmetric lines. The coil wires are also the 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 e.g. 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 ( 200 C). In the course of time, both

34 29 helium and nitrogen evaporate therefore the magnet (the dewars actually!) 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). The strength of an NMR magnet is often 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 (2006) the typical field strength used in biological applications are 700 MHz and 800 MHz. The highest currently available field is about 1000 MHz. The need for higher and higher fields is explained by the gain in resolution and in sensitivity. The sensitivity of an NMR experiment is usually described by the signal-tonoise ratio S/N: S/N ~ N γ 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), γ the gyro magnetic ratio of the nucleus, B 0 the field strength, n the number of individual scans per experiment, T 2 is the relaxation time and T the temperature of the detection circuit. How a bigger field affects the S/N and the resolution (which follows a linear dependence on B 0 ) is shown in the following table. B 0 (T) ν (MHz) S/N resolution The Biomolecular NMR laboratory at Utrecht University is housing a 360 MHz, two 500 MHz, two 600 MHz, a 700 MHz, a 750 MHz and one 900 MHz high-resolution NMR

35 30 spectrometer, one of the 600 MHz spectrometer and the 900 MHz spectrometer are equipped with cryogenic probe systems for additional sensitivity. 6.2 The lock Even in a very well designed magnet the magnetic field is not perfectly stable over the long time a measurement can take (up to one week). Small deviations 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

36 31 experiments deuterium ( 2 H) is used for this purpose. The deuterium spectrum is continuously acquired and the frequency of the single deuterium line is observed. Whenever this frequency shifts, small correction currents are applied to the lock coil to compensate for this change therefore slightly increasing or decreasing the total magnetic field. In most biological applications deuterium is introduced by dissolving the sample in a mixture of 5 10% D 2 O in H 2 O. 6.3 The shim system As stated above, the magnetic field experienced by the sample must be very stable and also very homogeneous to keep the lines as narrow as possible. Since the homogeneity 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 which can compensate 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 once adjusted during the installation procedure of the magnet. The room temperature coils are the ones used by the user for 'shimming' 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 line in the NMR spectrum. The integral of this line is constant (as it only depends on the number of nuclei in our sample, which is constant) but the height of the line is not: The narrower the line the higher its maximum. The NMR operator can now manually adjust the different currents in the different shim coils to optimize this value. This was and still is a very time consuming procedure which requires some experience. Fortunately a very fast automatic shimming method is available nowadays which employs pulsed field gradients (PFGs) which reaches very good results within minutes. This so-called 'gradient shimming' is available on all of our machines (except the 360).

37 The probe The probe (or probehead) is in many ways the most critical component in the spectrometer. It has two main functions: a) to convert the radio frequency power from the amplifiers into oscillating magnetic fields (B 1 -fields) and to apply these fields to the sample. 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 points can be achieved by a parallel tuned circuit having a coil surrounding the sample. The tuning is dependent from the sample (on position, volume, solvent, ionic strength). The coil has to be carefully tuned to the frequency of the nucleus of interest (remember: the frequency of the B 1 -field should match the Larmor frequency). This adjustment of the circuit is important in several 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, and therefore ensures a good excitation bandwidth. 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 5mm diameter (with a sample volume of ~500 μl). 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 same setup can be found in probes for 3mm and 10mm tubes. The sensitivity of the probes is still improvable as reflected by the increased sensitivity over the past 10 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. The technological challenge of such a system, among others, is the fact that the temperature of the sample, of course, 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

38 33 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 (compare the relative sensitivities at different fields in chapter 6.1). 6.5 The radio frequency system The RF system mainly consists of pulse generation units, the actual transmitters 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 also must be adjustable (typically to 0.5 degree). 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. 6.6 The receiver The final stage in an NMR experiments is the detection of the precessing magnetization (x- and y-components) in the sample. As stated earlier the same coil is used for this purpose as for excitation. This means that directly before the data acquisition the transmitter system has to be blanked 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 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

39 34 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 following relation is important: If we want to detect a certain spectral width SW we have to digitize the signal with a time dw ('dwell time') or faster ('Nyquist theorem'): 1 dw = (6.2) 2 SW An example (see the following figure): Assume we digitize our FID every 5ms, corresponding to a spectral width of 100 Hz. A frequency of 100 Hz will be sampled twice per period (solid line), which is enough to characterize this frequency. On the other hand, a resonance precessing with 120 Hz (dashed line) is sampled less than twice per period, thus the frequency can not be distinguished from a slower frequency (80 Hz in this case, dotted line). This means that both frequencies would give a signal at the same position! All further operations after the digitization and storing of the signal are performed in the data processing system (the computer workstation). This includes application of window functions, zero-filling, Fourier transformation, phase corrections, baseline corrections, integrations in several dimensions as well as displaying and plotting the final spectrum.

40 35 The components of a spectrometer at a glance: RF generator (radio sender): creates the RF signal with a frequency of less than 100 MHz up to about 900 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: generates the B 0 -field (from about 1T up to about 21T). 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. AF amplifier: amplifies the audio signal. ADC: Analog-to-digital converter. Computer: controls all the other electronic parts, receives, stores and processes the NMR signal.

41 36 VII NMR Parameters 7.1 Chemical shifts We have seen that the resonance frequency of a nucleus depends on its gyro magnetic ratio γ and the magnetic field B z. If all nuclei of the same kind (e.g. protons) would have an identical Larmor frequency then NMR would not be a very useful technique for studying biomolecules we would observe just one line per sort 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 currents result in local magnetic fields. The induced current will counteract its cause ( Lenz law, electromagnetism), thus 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 = σ (7.1) loc B z where σ is a quantity expressing the amount of shielding. The net field experienced by the spin becomes B B + B = B ( 1 σ ) (7.2) eff = z loc z

42 37 and the new Larmor frequency is given by (compare Eq. 2.10a) B z ν = γ (1 σ ) (7.3) 2π The shielding σ is different for different types of nuclei in a molecule because the electron density around a nucleus is very sensitive to the chemical environment of the nucleus (e.g. chemical bonds and neighbours). The amount of shielding is usually given as a dimensionless parameter δ, the chemical shift, which expresses the difference in NMR resonance frequency with respect to a reference signal since σ << 1. ref 6 ( ν ν ref ) 6 σ ref σ 6 δ = 10 = ( σ ref ν 1 σ ref ref σ ) (7.4) For the reference of the δ-scale the single line of the methyl-protons of Si(CH 3 ) 4 (TMS, Tetramethylsilane) can be used (δ = 0). For biomolecules, slightly different compounds (e.g. TSP, (CH 3 ) 3 SiCD 2 CD 2 CO 2 Na, Sodium-salt of trimethylsilyl-propionic acid) are used since TMS is not soluble in water; sometimes the water resonance itself is taken. The chemical shift of water is temperature dependent: T [ Kelvin] δ ( H 2O) = 7.83 (7.5) 96.9 The dimensionless δ-scale (ppm) has the important advantage over a frequency scale (Hz) that the chemical shift values become independent of the magnetic field B z.

43 Effects influencing the chemical shift As was mentioned above, the s-orbital electrons generate a field opposing the static field, a shielding effect. The p-orbital 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 the different functional groups. The table in appendix Aa gives the common chemical shift values for a number of different functional groups. A small value of δ corresponds to a shielded proton, or a low resonance frequency. In addition to the constant effects of s- and p-orbitals 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. Aromatic and carbonyl groups have an extensive conjugated π-electron system comprising delocalized molecular orbitals. Also in these systems the B z -field induces currents, the so-called ring-currents, resulting in quite large magnetic moments. Their effect on δ 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 Δδ is negative ( upfield shift ). In the plane of the ring-system a reinforcing field is generated and Δδ is positive (downfield shift). These effects can be as large as -2 to 2 ppm and all nearby protons are affected. The effect decreases with 1/r 6. 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, the electron magnetic moment, and by direct electron-proton hyperfine interaction. Electric fields resulting from charged groups or electric dipoles polarize the electron clouds and thus influence the chemical shifts. H-bond formation strongly influences the chemical shift. The proton in an X-H Y hydrogen bond is very little shielded and thus has a large δ value. For example, the imino

44 39 protons in the Watson-Crick hydrogen-bonded bases of a B-DNA fragment resonate at ppm, whereas the non hydrogen-bonded protons resonate at ppm Protein chemical shifts From section it will be clear that also the 1 H spectra of proteins will have a 'general' appearance. An example 1 H spectrum of a protein is given below: Indicated are the typical regions where the different proton resonances are found. It is even possible to split out the different chemical shift ranges on a per-residue basis. This is shown in Appendices Ab and B. Although the theory of chemical shifts is well known, it is quite complicated in practice to accurately predict the chemical shifts in proteins. Partially this is the result of the inaccuracy of protein structures and their internal mobility. On the other hand the range

45 40 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 geometry are less important. 13 C chemical shifts are therefore easier to predict and can be used in a more straightforward fashion for interpretation of spectra. 7.2 J-Coupling J-coupling is the interaction between two spins transferred through the electrons of the chemical bonds between them. The resonance frequency of a spin A depends on the spinstate of a second spin B and vice versa. If spin A is in the α state spin B will resonate at slightly lower frequency whereas it will resonate at slightly higher frequency when spin A is in the β-state. In the NMR spectrum we observe a doublet (two lines with equal intensity) centered on ν b, the resonance frequency of spin B without J-interaction. Also for spin A a doublet is observed, centered around ν a. The size of the coupling is J AB (the

46 41 distance between the components of the doublets) and is called the coupling constant. It has the dimension Hz. If there is a third J-coupled spin C the pattern splits again: 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. If the

47 42 spins B and C are magnetically equivalent the doublet of doublet collapses into a triplet (three-lines with intensity ratio 1:2:1). Three equivalent neighbours (e.g. the three protons of a methyl group) result in a quartet (four lines with intensities 1:3:3:1). As you can see the intensity ratios follow the famous Pascal triangle 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 two protons under consideration. Nuclei having this type of equivalence resonate at the same frequency. For magnetic equivalence the nuclei must be a) chemical equivalent and b) must 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 line. Due to the high symmetry of the molecule all six protons are magnetically equivalent, thus showing only one frequency and no coupling with each other. In proteins magnetic equivalence due to symmetry is rare because of the high complexity of the biomolecules. But also here some 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 e.g. methyl groups or fast rotating aromatic rings.

48 43 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 γ values of the nuclei involved. Proton-proton couplings in biomolecules are 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: the J-coupling between the amide proton and the directly bond 15 N nucleus, 1 J NH is ca. 92 Hz. The onebond 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 α and 15 N over three bonds in a H α C(C=O) 15 N fragment.

49 44 VIII Nuclear Overhauser Effect (NOE) The observable intensity of the 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 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). It is the result of a relaxation process, caused by a dipoledipole 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 spin system with two spins A and B which are dipolar coupled (spatial proximity!). In the steady-state NOE experiment one resonance is selectively saturated by RF irradiation (let's say spin B). This disturbs its equilibrium magnetization, therefore spin B tries to re-establish it by exchanging magnetization with its environment. This can either be the lattice or another nucleus close-by. The NOE between the saturated spin B and another spin A is defined by the relative change in the intensity of spin A: eq ( M a M a ) NOE = 1 + η with η = (8.1) eq M a

50 45 The energy level diagram for this two-spin system is sketched on the right. Each of the spins can undergo transitions between its α and β 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. How this can be understood is schematically shown in the following figure: A A = 1.5 Δ 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 = Δ A = A 0 = Δ B = 0 A A = 0.5 Δ The most left diagram represents the situation at equilibrium. A 0 and B 0 are 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

51 46 (diagram in the middle) and finally after cross-relaxation (diagrams on the right). Saturation of the transitions of nucleus B leads to the situation in the middle. As a consequence of the saturation, the population differences for the spins-states of nucleus B disappear. Now the cross-relaxation comes 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. The relative population difference for the states of the A nucleus has increased. Consequently also the observed signal for A will be increased. 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 the A nucleus has decreased and consequently also the observed signal for A will be decreased. One can easily imagine a situation, where the two cross-relaxation mechanisms just cancel each other. Indeed a zero-crossing of the NOE is observed when (in water at room-temperature): ωτ c 1.18 η fast tumbling ω o τ slow tumbling

52 47 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 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 probably have a chance to pick up NOEs now. The other parameter which we probably can change is ω 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 C. 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, σ, in a biomolecule is τ c σ ~ (8.2) r 6 where τ c is the rotational correlation time of the proton-proton vector, and r the distance between the protons. Since T 1 values are uniform we can write for the NOE τ c η ~ (8.3) r 6

53 48 Therefore, the measured NOE can be converted to a distance. We can calibrate the NOE by comparing it with a NOE of a fixed distance r 6 ref = ref r 6 η η τ τ c ref c (8.4) If we now assume that there is no internal mobility (a rigid molecule), τ c will be uniform in the entire molecule. We can now directly calculate the distance η ref r = 6 (8.5) r ref η Eq. 8.5 provides the basis of the most important effect for structure determination by high-resolution NMR spectroscopy: the extraction of NOE-based distances between proton pairs. An example: In a protein we observe the following NOEs between the aromatic ring protons and some other protons. The distance between C δ H and C ε H is fixed at 2.45 Å. NOE intensity distance Tyr C δ H - Tyr C ε H Tyr C ε H - Val C α H r = 2.45 (0.15 / 0.02) 1/6 Tyr C ε H - Asp C α H r = 2.45 (0.15 / 0.01) 1/6 Internal motions are often fast (ωτ c,intern < 1) and contribute predominantly to W 2.

54 49 Therefore, the NOE in mobile sub-domains will be reduced in intensity since σ = W2 W 0 (8.6) In the case of internal mobility we can not give the exact value of the distance but only an upper limit. This is because of r η τ = ref c r 6 ref ref (8.7) η τ c τ c and 1 ref τ c (8.8)

55 50 IX Relaxation Measurements 9.1 T 1 relaxation measurements The so-called inversion-recovery pulse sequence, 180 -τ-90 -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 (τ=0) = -M eq. The magnetization will return to its equilibrium value ('recover') with the relaxation time T 1, and after time τ we have: M z τ T ( τ ) = M eq e (9.1) This time dependency is shown in the figure on the right. We can measure the value of M z (τ) by applying a 90 detection pulse, which will rotate the z- component into the xy-plane. The FID is recorded and the spectrum extracted by a Fourier transformation. For τ < ln(2) T 1 the spectrum is inverted. For longer values of τ, M z (τ) has recovered to positive values and a positive signal is recorded. By repeating the experiment with increasing values of τ, 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 ω j and relaxation time T 1j has the starting magnetization:

56 51 M ( 0) = (9.2) j zj M eq Therefore, we can determine the individual T 1j by monitoring the intensities of the individual resonances j at ω j in the spectrum as a function of τ. The different time points τ of the inversion recovery sequence are measured one after another. In case more than one FID is recorded per value of τ, e.g. for S/N 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. 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.

57 Calculated example for an inversion-recovery experiment We follow the intensity of one resonance and detect signal intensities between τ = 0.001s and τ = 3s. Note that the inversion was incomplete as a result of an imperfect 180 pulse ( M ( 0) = ). Therefore we have to use the more general Eq. 4.2 in place of Eq. 9.1: j zj M eq M z ( τ ) M eq = τ T [ M (0) M ] e 1 z eq τ (s) M z (τ) ΔM(τ)=M z (τ)-m eq ln{δm(τ)/δm(0)} M z (0) M eq Since ΔM ( τ ) τ ln = ΔM (0) T1 (9.3) T 1 can be determined by linear regression analysis: Στ = 1.64; Σln(...) = -9.88; and thus T 1 = Στ / Σln(...) = s.

58 Applications of T 1 relaxation T 1 tells us how long we have to wait until the equilibrium magnetization is restored. This is important information for the setup of FT NMR experiments, where a number of experiments are repeated and the results added up in the computer. It depends on T 1 how fast we can repeat our experiments. 9.2 T 2 relaxation measurements The value of T 2 could in principle be extracted from the envelope of the FID or from the line width at half-height (Eq. 5.8). However, T 2 values obtained this way also depend strongly on the static field inhomogeneities (the shimming ). If the main magnetic field is not homogeneous over the whole sample, spins at different locations in the sample tube experience a slightly different field. This results in slightly different resonance frequencies. The total peak, which is the sum over all individual spin contributions in the sample, will be broadened due to this effect. The apparent relaxation time is T * 2 which is faster than T 2 due to spin-spin relaxation. Since only T 2 depends on the physical properties of the molecule, this is what we are actually interested in. Obviously, the contribution of a 'bad' shimming to the relaxation of our molecule is less interesting for us! '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 the 90 x-pulse. The macroscopic vector M y can be considered to consist of a sum of

59 54 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 ω 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 ω 0 and some will rotate slower, depending on the exact position in the sample. In the rotating frame (ω 0 ), the fast and slow components will start to precess in the xy-plane with frequency ( = γ ΔB (9.4) γ Bz M j ) ω 0 j This frequency is different at different locations. Hence, the transversal magnetization M y will dephase due to the inhomogeneous field. It can be shown that the spin-echo sequence eliminates the dephasing that results from these static field inhomogeneities. In order to explain how this works, we actually should consider the precession of each of the individual components M j, but fortunately the principle can be shown by picking only two spins, rotating with different speed: a slower black and a faster white one.

60 55 At a point (a) in the sequence we have pure y-magnetization for all individual spins. At point (b) some phase coherence is lost because each spin has precessed with its own frequency. The white one a bit faster, the black one a bit slower. After the delay τ/2 a 180 pulse from the y-direction is applied (point (c)). This pulse will invert the x- component of m j, but will not affect the y-components. During the second delay τ/2, the vectors (M j) precesses again with their individual frequency γδb j, the white one still a bit faster and the black one still a bit slower and still in the same direction as before the 180 pulse. Consequently, in point (d) both magnetization vectors have returned to the y- axis, independent of the static inhomogeneity, creating a so called 'spin-echo'. Thus, we have eliminated the effect of field inhomogeneity! By repeating the experiment for several values of τ 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 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 the rotational correlation time τ c of the molecule. Thus, analogously to the usage of T 1, we can use T 2 to determine motional parameters.

61 56 X Two-Dimensional NMR In the seventies the development of two-dimensional (2D) NMR has revolutionized NMR spectroscopy and has made the 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. Clearly, this 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 method we will discuss first a very simple 2D NMR experiment The SCOTCH experiment SCOTCH stands for spin coherence transfer in (photo) chemical reactions. If one has, for instance, a photochemical reaction A h ν B (10.1) a proton which resonates in molecule A at ω A will resonate at ω B in molecule B. The SCOTCH experiment correlates the resonance frequencies in A and B for each particular proton. In other words it enables us to find ω B in B that corresponds to ω A of the same proton in A. The pulse sequence is as

62 57 shown above. The 90 pulse creates xy- magnetization which precesses with ω A during the so-called evolution period t 1. The light pulse changes the precession frequency to ω B with which it is detected as an FID during the detection period t 2. The trick is now to increment t 1 in a regular fashion and collect a large number (typically, say, 500) FIDs belonging to different t 1 values. Thus one records a data set S(t 1, t 2 ) depending on both t 1 and t 2 (the 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, t 2 ) S( t1, F2 ) S( F1, F2 ) FIDs interferog ram 2D spectrum (10.2) The effect of incrementing t 1 is to "sample" the frequencies present during the evolution period. How this works out in practice is illustrated in Figure 10.1 (next page). After each t 1 time the signal acquired a different phase. The FIDs recorded that way have all the same frequency (ω B ), but their phase (i.e. how far the magnetization vector rotated in the xy-plane) depends on the evolution time t 1. After the first Fourier Transformation this leads to a peak at the position ω B in the F 2 -dimension with an intensity that oscillates in the t 1 direction with the frequency ω A. Looking along the t1-axis of the interferogram the signal actually looks like an FID oscillating with the frequency ω A. Hence, after double Fourier Transformation this gives a peak at (ω A, ω B ) (ω A in the F 1 - and ω B in the F 2 - dimension). This is called a cross-peak, in contrast to peaks on the diagonal which are called diagonal peaks. So, indeed, the experiment gives the connection (a spectroscopist says: 'correlation') between the resonance frequencies in A and B. In this case this is rather trivial but if there are many nuclei this method can prove very useful. All 2D NMR experiments adhere to the following scheme:

63 58 In the example above the preparation period would include a relaxation delay and the 90 pulse. The mixing period is just the light pulse. Fig. 10.1: The SCOTCH 2D NMR experiment. On the left side the sampling of the ω 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 ω B in F 2 with intensities dependent on t 1 oscillating with ω A. The second Fourier transform (t 1 F 1 ) leads to a 2D spectrum with a single cross-peak at (ω A, ω B ). The representation is as a contour plot.

64 D NOE 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. The evolution time t 1 is incremented and mixing consists of two 90 pulses separated by a constant mixing time τ m. During τ m magnetization between neighbouring spins is exchanged via cross-relaxation (see Chapter 8). For biomolecules σ AB W 0 and therefore the flip-flop transitions (αβ βα) are dominant for the NOE effect. 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, and that the carrier frequency is chosen at the Larmor frequency of spin A, ω RF = ω 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 lie along the y-axis in the rotating frame. During the evolution time t 1 the A-vector precessing at ω RF will stay the same (at least for short times when relaxation can be neglected), while the B vector precesses with a frequency ω B ω 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. 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 xyplane during t 1. This is called frequency labeling and is a common feature of the evolution period (t 1 period) of 2D NMR experiments. Focussing now on these z- components that correspond to populations of energy levels, we know that W 0 transitions during the mixing time τ m will tend to equalize populations of αβ and βα states (because

65 60 at equilibrium they are equal). Thus, the z-components of the A and B magnetizations also will become more equal (d). Finally, the third 90 pulse flips the vectors in the xyplane where the signal can be observed (e). Fig Vector diagram of the 2D NOESY experiment. Now let us see how this leads to cross-peaks in the 2D NOE spectrum. In Fig we shall look at the magnetization vectors at time d in Fig (after the mixing time τ m ). Let us first consider a trivial case where no transfer occurs, for instance because the spins are too far apart, and then the more interesting case where cross-relaxation occurs between A and B.

66 61 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 ω A and ω 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 only diagonal peaks at (ω A, ω A ) and (ω B, ω B ). Fig Vector representation of spin A (solid vector) and spin B (dotted vector) at time point (d) in Fig for various values of t 1.When no mixing occurs during τ m the 2D NOE spectrum consists only of diagonal peaks. In the case of magnetization transfer during τ m cross peaks arise at (ω A, ω B ) and (ω B, ω A ). In contrast, when mixing occurs in τ m the equalizing effect of the W 0 transitions causes the A-vector to borrow intensity from B and vice versa. Thus, the A-vector is now modulated with ω B for the different values of t 1! Since the vector will continue to precess at ω A in t 2 this will lead to peaks both at (ω A, ω A ) and (ω B, ω A ) in the 2D spectrum (F 1, F 2 ). These are the diagonal peak that we have seen before at (ω A, ω A ) and an off-diagonal cross-peak at (ω B, ω A ) at the upper left half of the spectrum. In the same way, the

67 62 modulation of the B-vector with ω A leads to a symmetry related cross-peak at (ω A, ω B ) below the diagonal. Because mixing is reversible 2D NOE spectra are always symmetrical. As all proton pairs within 5 Å will give rise to cross peaks with intensities inversely proportional to r 6, the 2D NOE spectrum provides a map of all short protonproton distances in a biomolecule. A more mathematical description of the NOESY experiment is seen in Appendix D D COSY and 2D TOCSY In another important class of 2D NMR experiments the magnetization transfer in the mixing period takes place via the J-coupling. The simplest is the COSY (correlated spectroscopy) with the pulse sequence shown at the upper right. Here the mixing period is just the second 90 pulse, which transfers magnetization between A and B spins whenever there is a J-coupling between them. This results in the 2D spectrum as shown in Fig Fig. 10.4: COSY spectrum of two J-coupled nuclear spins A and B. The sign of the cross-peaks is indicated. The regular 1D spectrum is drawn above. It can be seen that the cross-peaks reflect the fine structure of the 1D spectrum (doublets in this case) and therefore can be used to measure J-couplings. Note the typical negativepositive sign pattern in the cross-peaks (but not in the diagonal peaks).

68 63 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. Such low-resolution COSY spectra are shown below for the two amino acids alanine and valine: As measurable J-couplings only arise between nuclei separated by less than four chemical bonds, the connectivity patterns can be easily predicted. We note that in a protein a network of J-coupled protons does not extend beyond an amino acid residue because the CαH of residue i is separated from the NH of residue i+1 by four chemical bonds. Each of the amino acids forms a separate spin-system. A COSY spectrum is a valuable tool for the identification of the types of amino acids. Another important J-coupling based 2D experiment is the TOCSY which stands for total correlation spectroscopy. The pulse sequence is shown below. The mixing period here consists of a complicated pulse train. This has the effect of transferring magnetization through a whole network of J- coupled spins.

69 64 For instance, in a molecular fragment we have non-zero J-couplings 3 J AB and 3 J BC but 4 J AC is close to zero. During the TOCSY mixing period A-magnetization is transferred to B via J AB and then to C via J BC in multiple transfer steps. Hence a cross-peak will arise between A and 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 Fig. 10.5: Fig. 10.5: COSY and TOCSY spectra of a three proton spin-system where J AB and J BC are non-zero and J AC =0. Because of the symmetry only the part above the diagonal has been drawn. It should be clear from this that a TOCSY spectrum always contains the COSY as a subspectrum. Although the information content of COSY and TOCSY spectra is in principal the same, in complicated spectra with a lot of overlap the TOCSY spectrum is still useful.

70 65 This is because if B and C in the example of Fig 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 difficult for B and C. We leave the sketch of the TOCSY spectrum of alanine and valine as an exercise (compare Fig and 10.5). 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.

71 66 XI The assignment problem The interpretation of an NMR spectrum always starts with the identification of resonance frequencies and their corresponding nuclei in the molecule. This so-called assignment of resonances constitutes an essential step in the structure determination process by high resolution NMR spectroscopy that always precedes 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), it 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 lines become broader and the overlap becomes increasingly severe. We focus here on the basic principles of assignment, i.e. which parameters of the NMR spectrum can be used. We will come back to the special case of the assignment of spectra of biomacromolecules in the next chapter Chemical shift The chemical shift of signals gives a first indication of the surrounding of the corresponding nucleus in a molecule. We saw before (chap ) on the example of a protein spectra, that proton chemical shifts usually are grouped according to the environment in which they are located in the molecule. In our example those were the amide and aromatic protons in the left half of the spectrum, the H α protons just right of the water signal (center of the spectrum), protons of aliphatic side chains still to the right of them 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

72 67 and 13 C chemical shifts, which can help to identify the origin of particular resonances in NMR spectra (see appendix A) Scalar coupling From chapter 7.2 we know that we can identify coupled nuclei with help of their coupling constant. If we look, for instance, at the signals of the methyl group and the CH 2 group of ethanol, we find them split into multiplets due to scalar coupling. The number of multiplet components gives us an indication of what the neighbouring group looks like (i.e. 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 signal of the CH 2 group is a quartet due to the coupling with the three equivalent protons of the neighboring methyl group). The distance of the components of these multiplets (the coupling constant) is exactly the same in the triplet and in the quartet. 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. The integral of the proton signal of a methyl group is just 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 (basic) structural features of the molecule we are looking at. It is quite straight

73 68 forward for example to identify neighbouring protons in an aromatic ring system, because their distance from each other is quite short and well known (~2.45 Å). Once we know one of them, we can relatively easily identify the others on hand of a NOE spectrum. Especially in biomacromolecules, where the spin systems of the individual residues cannot be connected by scalar coupling experiments, NOE data is often the only outcome to still get a sequential assignment.

74 69 XII Biomolecular NMR Some of the most important sorts of biomolecules are nucleotides and amino acids. Although also lipids and carbohydrates play important roles in biological processes, in this course we will focus on the first two groups. Nucleic acid is the bearer of the genetic information and is involved in protein synthesis where it acts as a template containing the sequential information for all proteins occurring in organisms. Each three consecutive nucleotides in a gene code for a particular amino acid in a protein. In addition there are control regions (stop codons and sequences where proteins involved in transcription and transcription regulation bind). The role of proteins is very diverse. We know them for example as enzymes and regulators, as building material of cells and their compounds, and as carrier of information within and between cells. Obviously both nucleic acids and proteins play a major role in the function of all living organisms and accordingly also most defective disorders of them can be traced back to the malfunction of proteins or to defects in nucleic acid. This makes these molecules to very popular subjects of study in a variety of research disciplines. Their structural and functional understanding is supposed to give insight into how and why they work and what probably goes wrong in the case of diseases. Knowing the exact composition and function of a particular virus, e.g. can lead to the development of anti-viral drugs. The exact knowledge of the structure and function of a particular enzyme can lead to the development of e.g. inhibitors which can deactivate the enzyme when needed. We will have a close look here at the structural properties of nucleotides and peptides and how their different spin systems translate into different features in NMR spectra Peptides and proteins Peptides and proteins are mainly build from twenty different naturally occurring amino acids. They all share the same basic structure and only differ in their side chain R:

75 70 H α Amino group H 2 N C α C R Side chain O OH Carboxyl group In peptides, amino acids are linked via a so-called peptide bond. The amino group of one residue is connected with the carboxyl group of another: H O H H 2 N C α C N C α C R H R O OH Note that the peptide bond is planar due to its partial double bond character! Amino acids are usually referred to with either a one-letter or a three-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

76 71 Amino acids can be classified by the character of their side chain as: aliphatic (A, V, L, I, (G), aromatic/ring (F, Y, W, H, P), carboxylic (D, E, N, Q), sulfur/hydroxy containing (C, M, S, T, Y) and charged (K, R). The chemical formulas of the natural occuring amino acids together with their COSY, TOCSY and NOESY spectra are shown in appendix E Assignment of peptides and proteins A strategy based upon homonuclear 2D experiments (COSY, TOCSY, and NOESY) was developed in the 80 s (K. Wüthrich, 1986). This approach is discussed in this chapter. A more recent approach employs uniformly 15 N and 15 N, 13 C labeled proteins. The strategy uses so-called triple-resonance experiments (involving 1 H, 15 N and 13 C) to transfer magnetization through the polypeptide chain employing the large one-bond homo- and heteronuclear J-couplings. For larger proteins several patterns corresponding to residues of a certain type are present in a COSY, e.g. several alanines give rise to similar patterns. How can we decide which of the alanines in the protein sequence corresponds to a particular pattern? The solution 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 sequence) with 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.

77 72 Step1: Spin system identification The identification of spins belonging to the same spinsystem can be performed on the basis of the COSY and the TOCSY experiment (see the example of an Ala-Ala peptide fragment on the right). In both spectra, protons belonging to a certain amino acid can be identified. A summary of all the expected patterns for the different amino acid types is given in the appendix E. Some of the amino acids have a very typical pattern, for instance Gly where the side chain just consists of two H α, or prolines where no H N is present. Some other amino acids share a common pattern, e.g. the so-called AMX spin systems where AMX represents the H α and the two H β protons of the 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 no J-couplings between H β and protons in the ring are observable (4 bonds involved). 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. Again, these cross peaks are included in appendix E. For bigger proteins the overlap makes it harder to identify such patterns. Also the lines are broader in general which is due to the shorter T 2 relaxation times. This 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 spectrum. In order to make a so-called sequential assignment, i.e. correlating the COSY patterns to individual amino acids in the primary sequence, we have to correlate the COSY pattern of the alanine to the COSY pattern of its sequential neighbour.

78 73 Unfortunately, when using only proton NMR, no 1 H- 1 H J-couplings of appreciable size exist over the peptide bond since the shortest connection of two protons in neighbouring residues involves four bonds. Consequently, the individual amino acid residues form isolated spin systems. Fortunately we can employ another mechanism of magnetization transfer. The short sequential distances between consecutive residues result in cross peaks in the NOESY spectrum. The figure on the right 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 cross peak between the H α 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 αn. Similar, the distances between H β or H N of spin i to the neighbour H N of spin i+1 are represented by d βn and d NN. On the other hand the tertiary structure of the protein will also lead to intense signals from non-sequential cross peaks. 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 (mostly X-ray) structures. Table 12.1 shows for example that 98% of all d αn distances shorter than 2.4 Å correspond to sequential distances. Naturally, the score drops with increasing distance limit. Similar values are obtained for the d NN and d βn distances. This means that for two residues i and i+1 d αn, d NN and d βn cross peaks most likely result from sequential NOEs. The probability for identifying a sequential connection increases dramatically when simultaneously two (or more) short distances can be found. Table 12.1 shows that if simultaneously two NOEs are found between two amino acids, they most likely result from sequential residues.

79 74 Table 12.1 Distance (Å) j i = 1 (%) d αn (i,j) d NN (i,j) d βn (i,j) d αn (i,j) 3.6 && d NN (i,j) d αn (i,j) 3.6 && d βn (i,j) d NN (i,j) 3.0 && d βn (i,j) Step 3: Matching to the sequence The next step is to locate this fragment of two (or more) residues in the protein 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 (or TOCSY) spectra, a given di-peptide fragment has a probability of uniqueness of 56%, the tri-peptide and tetra-peptide fragments of 95% and 99%, respectively. As expected, increasing the length of the fragment increases the uniqueness. A tetra-peptide fragment is usually sufficiently unique

80 75 to allow for identification in the polypeptide chain Secondary structural elements in peptides and proteins The intensities of sequential NOEs contain some information on the secondary structure because they depend on the local conformation of the polypeptide backbone. Take, for instance, the distance between an H α proton of a residue i to the H N proton of the following residue (i+1). In an extended strand this distance is short (2.2 Å) whereas it is 3.5 Å in a helical conformation. This information together with analysis of other characteristic medium range NOEs (between residues which are less than 4 positions apart in the sequence) is sufficient to specify the secondary structure element. An overview of important sequential- and medium-range proton-proton distances is given in the Figure Fig. 12.1: Characteristic sequential and medium range NOE connectivities. Recognition of secondary structural elements, i.e. α-helices, β-sheets, and turns,

81 76 constitutes an important element in the structure determination process. Most of the observable NOE cross peaks between different residues are due to this secondary structure. The α-helix, β-sheet, and turn conformation result in characteristic short distances which in turn result in characteristic NOEs in the 2D NOE spectrum. Also the 3 J HNHα coupling has traditionally been used as a marker for secondary structure as well as the presence of slowly exchanging amide protons (cf. Chapter 13). Fig shows the short distances in an anti-parallel β-sheet and in a parallel β-sheet. Fig. 12.2: Anti-parallel (top) and parallel (bottom) β-sheet. Sequential NOEs are indicated by open arrows, interstrand NOEs by solid arrows. Hydrogen bonds connecting the strands are shown by wavy lines.

82 77 The anti-parallel β-sheet is characterized by short d αn (i,i+1) and interstrand d αα (i,j) distances whereas the d αα (i,j) distance is much longer (4.8 Å) in parallel β-sheet. Also the d NN (i,j) distance in parallel β-sheet is much longer compared to the d NN (i,j) distance in anti-parallel β-sheet. In Fig short distances are shown for an α-helix. An α-helix is characterized by a close proximity of residues i and i+3 and residues i and i+4. The d αn (i,i+3) and d αn (i,i+4) NOEs are therefore clear markers for this element of secondary structure. In addition to the aforementioned short distances, the sequential d NN (i,i+1) distance is also short, and strong sequential d NN NOEs can be found in the spectrum. Fig. 12.3: α-helix. The sequential d NN is shown (2.8 Å) together with d αn (i,i+1), d αn (i,i+2) and d αn (i,i+3), Note that the side-chains are not shown. Turns are characterised by short distances between residues i and i+2; in particular the d NN (i,i+2) distance. In general, however, since there exists a large number of turns, which

83 78 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 characteristic patterns and short distances is given in Fig Fig. 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 also given. The short distances in secondary structures are also listed in the following table: Table 12.2: Secondary structure specific atomic distances (in Å) Distance α-helix helix β β p turn I a turn II a d αn d αn (i,i+2) d αn (i,i+3) d αn (i,i+4) 4.2 d NN d NN (i,i+2) d βn d αβ (i,i+3) a for turns, the two numbers apply for the distances between residues 2, 3 and 3, 4 respectively

84 Nucleotides and nucleic acid Nucleic acid is mainly build from five different nucleotides. All of them share a common general structure: They consist of a nucleobase, a pentose-sugar ring (ribose in the case of RNA or 2'-deoxy ribose in DNA) and a phosphate group which links the nucleotide units to each other. (Nucleo) Base Phosphate Sugar Fig. 12.5: Common structure of nucleotides There are two sorts of nucleobases: The purine bases adenine and guanine and the pyrimidine bases uracil (only found in RNA), thymine (only found in DNA) and cytosine. The base is connected to the sugar moiety via a glycosidic bond at the 1' carbon of the pentose ring. 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. It should be clear from this, that NMR spectra of large nucleic acids are much more complex than NMR spectra of peptides of comparable size. While peptides are build from as many as 20 different building blocks (the different 'natural' amino acids) nucleic acid is build 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. This can lead to very crowded regions in the NMR spectra and can complicate the spectral assignment considerably.

85 80 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 Fig. 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) Fig. 12.7: The pentose sugar rings of RNA and DNA

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

87 82 The figure below illustrates how the nucleotide units are linked by the phosphate groups in oligo nucleotides and in nucleic acids. If we analyze the spin systems of this molecule, 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! Fig : 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 adopts. The two strands of the helix adhere to each other by means of hydrogen bonding between bases of the adjacent stands. These so-

88 83 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 (A) always pairs with Uracil (U) in RNA and with Thymine (T) in DNA. Guanine (G) always pairs with Cytosine (C). The hydrogen bonds are shown in the figure below. Note that the G C pair is more stable than the A T pair, because G C consists of three hydrogen bonds and A T only of two. Fig : Watson-Crick base pairs. A-T on the left, G-C on the right There are two major conformation in which the double helices of DNA and RNA usually are found. The so-called B-form and the A-form (see figure 12.12). Both of them are right-handed (like ordinary corkscrews) and differ mainly in their 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 table 12.3: 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

89 84 A-form double helix (RNA) B-form double helix (DNA) Fig : The two major helical conformations Information about typical chemical shift values of the A- and B-DNA and typical short distances can be found in appendices F and G.

90 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 being that for the identification of a particular residue always the combination of COSY/TOCSY and NOESY is needed, because the bases form isolated spin systems and cannot be assigned to their sugars without making use of NOE data. Tables to be used for this purpose with common shift values and short distances in nucleic acids can be found in appendix F.

91 86 XIII Structure determination In this chapter we will discuss the method of determination of the complete 3D structure of a protein. Apart from the bond lengths and most 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 sequence) provide important constraints on the structure. 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 ~10kD protein typically between 1000 and 2000 cross peaks can be observed in a 2D NOE spectrum. We have seen in Chapter 7 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 principal 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 this high precision cannot be obtained in practice. The first is local mobility. Remember that the simple relation of Eq. 8.5 was derived on the assumption of equal τ c for the protons of reference and unknown distance. Only for very rigid proteins this assumption is really valid. Also, if there is a conformational equilibrium the r -6 dependence gives values much more weighted to the short distances in the average. Thus, the actual average distances may be longer than they appear in the case of conformational averaging. The second reason is the so-called "spin-diffusion" effect. This is the result of multiple transfer steps of magnetization A B C during

92 87 the mixing time τ m that may disturb the intensity of the NOE cross peak between A and C. Only for very short τ c can this indirect transfer path be neglected. For these reasons the NOE based distance constraints are often used in the form of 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 the three-bond coupling constant, 3 J, is related to the dihedral angle (between the two outer bonds) and therefore provides a constraint on this angle (θ). This is expressed in the Karplus relation J = Acos 2 ( θ ) + B cos( θ ) + 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 α-proton, J HNHα depends on the backbone angle φ as follows: J = 6.51cos 2 ( φ 60) 1.76 cos( φ 60) (13.1a)

93 88 The form of the Karplus relation for this case is shown in Fig Fig. 13.1: The Karplus curve for the protein backbone φ angle as described in the text. A complication is that several values of φ may belong to a particular J. Also, conformational averaging may lead to average values of J in the range 6-7 Hz. Therefore, the most reliable values for J HNHα are 9 10 Hz for extended (β-sheet) structure and 3 4 Hz for α-helical structure (Fig 12.4). Values around 6 7 Hz are often difficult to interpret. Similar Karplus curves exist for CH-CH J-couplings from which side chain χ angles can be derived Hydrogen bond constraints When a protein is dissolved in D 2 O many of the amide protons do exchange rapidly with deuterium and therefore disappear from the spectrum. However, often some NH signals remain in the spectrum for some time and exchange only slowly in time. These slowly exchanging NHs invariably are present in hydrogen-bonds such as occur in α-helices and

94 89 β-sheets. If the H-bond acceptor is known 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 these are given the distance ranges Å for the distance between NH and O and Å between N 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) with the so-called Distance-Geometry (DG) program. The resulting structures can then be refined with Molecular Dynamics (MD) calculations which include also energy terms for electrostatic interactions etc. The calculation procedure will now be briefly discussed Distance-Geometry The structure of a macromolecule containing N atoms can be perfectly described by specifying the N(N 1)/2 distances between the 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, an algorithm called Distance-Geometry (DG) that converts distances into Cartesian coordinates for a much smaller number of distances even if they are not precisely known. This algorithm has the following steps: 1. Set up distance matrices for upper bound (u) and lower bound (l) distances between all atoms in the structure. These include the so-called holonomic distances (from bond length and bond 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).

95 90 2. Smoothing By this we mean the adjustment of upper and lower bounds 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 have u u + u (13.2) ij ik 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 matrix D with random trial distances between upper and lower bounds. 4. Embedding This is the procedure of finding the best 3D structure that belongs to the distance matrix D. We will not explain this further here. 5. Optimization Because the matrix D, of course, is not perfect, it is usually necessary to regularize the structure coming from the embedding stage. Optimization involves the minimization of the coordinates against a distance error function. Because of the random nature of step 3, the repetition of steps 3 5 will produce slightly different structures. Ideally, if this is done many times, an ensemble of structures is

96 91 obtained, which samples the conformational space consistent with the experimental data. An example of such an ensemble of structures is shown in Fig Fig Structures of the protein crambin calculated by the DG procedure Restrained molecular dynamics Molecular Dynamics (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 r i 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