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1 NMR Spectroscopy Principles and Applications to Proteins Patrik Lundström © 2013
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NMR Spectroscopy Principles and Applications
to Proteins
Patrik Lundström © 2013
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Preface This text is intended as a complement to the book NMR spectroscopy by P.J. Hore used
for the course Biomolecular Structure Analysis (TFKE35) at Linköping University. As the name suggests, the main focus is protein applications. However, basic principles, valid for many areas of nuclear magnetic resonance spectroscopy, are covered as well. To avoid confusion the structure of the chapters and the notation closely follows Hore. Differences are clearly pointed out. After each chapter there are approximately ten problems to solve. The very last chapter does not describe a certain principle but is focused on applications and aims to put the previous chapters into context. By necessity the nature of the various chapters differs greatly. Some require a physical presentation and include a large number of equations while others are more descriptive. Regardless, I always point to applications and usually use systems that I work on myself to provide examples.
I have chosen not to include any references since this would make the text less readable. The interested reader is instead referred to the following textbooks that cover the subject in more depth than is possible here. Spin Dynamics by Malcolm Levitt is a very pedagogical introduction to both basic and advanced NMR theory and applications. The same is true for Understanding NMR Spectroscopy by James Keeler. A book that focuses on protein NMR but also covers a great deal of NMR theory is Protein NMR Spectroscopy – Principles and Practice by John Cavanagh, Wayne Fairbrother, Arthur G. Palmer III, Mark Rance and Nick Skelton.
I am indebted to several persons for either making suggestions for improvements of the text or sharing experimental data that are used to illustrate principles. I would especially like to thank Alexandra Ahlner, Cecilia Andresen, Annica Blissing, Lisa Henriksson, Markus Niklasson and Maria Sunnerhagen.
Patrik Lundström, 2013
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Table of Contents Preface ............................................................................................................................................. 2
1. The NMR Phenomenon ............................................................................................................... 7
1.1 Angular momentum and spin ................................................................................................ 7
1.2. Nuclear magnetic moments and their interaction with magnetic fields ............................... 9
1.3. The Boltzmann distribution ................................................................................................ 12
1.4. NMR spectroscopy ............................................................................................................. 13
1.5 Synopsis .............................................................................................................................. 17
1.6 Problems .............................................................................................................................. 18
2. The Chemical Shift .................................................................................................................... 19
2.1 Shielding .............................................................................................................................. 19
2.2 The chemical shift .......................................................................................................... 20
2.2.1. Chemical shift referencing .......................................................................................... 20
2.2 The origin of chemical shifts ............................................................................................... 23
2.3.1 Local effects caused by electronegativity of neighbouring atoms ............................... 23
2.3.2 Local effects caused by different hybridization ........................................................... 24
2.3.3 Dihedral angles ............................................................................................................. 25
2.3.4 Ring currents ................................................................................................................ 27
2.3.5 Hydrogen bonds ........................................................................................................... 27
2.3.6 Charges ......................................................................................................................... 28
2.4 Summary of contributions to chemical shifts in proteins .................................................... 28
2.5 Synopsis .............................................................................................................................. 30
2.6 Problems .............................................................................................................................. 31
3. Scalar Couplings ....................................................................................................................... 32
3.1 Spin couplings ..................................................................................................................... 32
3.2 The dipolar coupling ........................................................................................................... 32
3.3 The origin of scalar couplings ............................................................................................. 33
3.4 Multiplet structure ............................................................................................................... 33
3.5 Use of scalar couplings ....................................................................................................... 37
3.5.1 Structure determination of organic molecules.............................................................. 37
3.5.2 Scalar couplings as reporters of dihedral angles .......................................................... 37
3.6 Decoupling .......................................................................................................................... 38
3.7 The residual dipolar coupling .............................................................................................. 39
3.8 Synopsis .............................................................................................................................. 40
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3.9 Problems .............................................................................................................................. 41
4. The Vector Model ..................................................................................................................... 43
4.1 Radio frequency pulses ....................................................................................................... 43
4.2 Radio frequency pulses ....................................................................................................... 43
4.3 Periods of free precession ................................................................................................... 47
4.4 The spin echo ...................................................................................................................... 49
4.5 Notation for pulse sequences ............................................................................................... 50
4.6 Decoupling revisited ........................................................................................................... 51
4.7 Synopsis .............................................................................................................................. 53
4.8 Problems .............................................................................................................................. 54
5. Chemical exchange ................................................................................................................... 56
5.1 Symmetric slow exchange ................................................................................................... 56
5.2. Fast symmetrical exchange ................................................................................................ 58
5.3. Asymmetric exchange ........................................................................................................ 59
5.4. Methods used for studying chemical exchange .................................................................. 60
5.4.1. Exchange spectroscopy ............................................................................................... 60
5.4.2. Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion ....................................... 61
5.4.3. When to run EXSY and when to run CPMG experiments .......................................... 63
5.5 Synopsis .............................................................................................................................. 64
5.6 Problems .............................................................................................................................. 65
6. Relaxation .................................................................................................................................. 66
6.1 The origins of relaxation ..................................................................................................... 66
6.2 Interpretation of T1 and T2................................................................................................... 68
6.3 The nuclear Overhauser effect ............................................................................................ 68
6.4 Effect of internal motions .................................................................................................... 69
6.4.1 The model-free formalism ............................................................................................ 69
6.4.2 Spectral density mapping ............................................................................................. 71
6.5 Relaxation mechanisms ....................................................................................................... 71
6.5.1 Chemical shift anisotropy ............................................................................................. 71
6.5.2 Quadrupolar coupling ................................................................................................... 71
6.5.3 Paramagnetic relaxation enhancement ......................................................................... 72
6.6 Synopsis .............................................................................................................................. 73
6.7 Problems .............................................................................................................................. 74
7. Multi-dimensional NMR ........................................................................................................... 75
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7.1 Principles of two-dimensional NMR spectroscopy ............................................................. 75
7.1.1 Building blocks of two-dimensional experiments ........................................................ 75
7.1.2 Evolution in the indirect dimension ............................................................................. 77
7.2 Examples of homonuclear two-dimensional NMR experiments ........................................ 80
7.2.1 COSY ........................................................................................................................... 80
7.2.2 TOCSY ......................................................................................................................... 80
7.2.3 NOESY ......................................................................................................................... 81
7.3 Heteronuclear two-dimensional experiments ...................................................................... 82
7.3.1 Heteronuclear single quantum correlation spectroscopy.............................................. 83
7.4.1 TROSY ......................................................................................................................... 84
7.4 Additional concerns for three-dimensional experiments..................................................... 86
7.4.1 Sensitivity ..................................................................................................................... 86
7.4.2 Experimental time ........................................................................................................ 88
7.5 Synopsis .............................................................................................................................. 90
7.6 Problems .............................................................................................................................. 91
8. Characterization of proteins using NMR spectroscopy ............................................................. 92
8.1 Requirements for protein NMR spectroscopy ..................................................................... 92
8.1.1 Instrumentation ............................................................................................................. 92
8.1.1 Protein samples ............................................................................................................ 93
8.2 Setting up NMR experiments .............................................................................................. 94
8.2.1 Setting the temperature ................................................................................................. 94
8.2.2 Locking ......................................................................................................................... 94
8.2.3 Shimming ..................................................................................................................... 95
8.2.4 Tuning and matching .................................................................................................... 95
8.2.5 Calibration of frequencies and pulse widths ................................................................ 95
8.3 Check of protein integrity, the HSQC experiment .............................................................. 96
8.4 Resonance assignments ....................................................................................................... 97
8.5 Structure calculation ............................................................................................................ 98
8.5.1 Restraints that can be measured by NMR .................................................................... 98
8.5.2 NOE driven methods .................................................................................................... 99
8.5.3 Chemical shift driven methods ................................................................................... 100
8.6 Interactions ........................................................................................................................ 100
8.6.1 Calculation of dissociation constants from chemical shifts ....................................... 101
8.7 Dynamics ........................................................................................................................... 102
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8.7.1 Fast dynamics ............................................................................................................. 102
8.7.2 Large scale dynamics ................................................................................................. 103
8.7.3 Slow events ................................................................................................................ 103
8.8 Synopsis ............................................................................................................................ 104
8.9 Problems ............................................................................................................................ 105
9. Answers to selected problems ................................................................................................. 106
10. Collection of tables, formulas and physical constants .......................................................... 111
Spin angular momentum ..................................................................................................... 111
Nuclear magnetic moment .................................................................................................. 111
Spin energy in magnetic fields ............................................................................................ 111
Boltzmann distribution ........................................................................................................ 111
Nuclear magnetic resonance ................................................................................................ 111
Chemical exchange ............................................................................................................. 112
Nuclear spin relaxation ........................................................................................................ 112
Physical constants ............................................................................................................... 112
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1. The NMR Phenomenon
1.1 Angular momentum and spin NMR is short for nuclear magnetic resonance and we will start by explaining what
these words mean, why the phenomenon exists and how it can be used in spectroscopy. NMR is a nuclear phenomenon and to understand how it might work, it is instructive to get a picture of how a nucleus can be magnetic in classical terms. If we consider the nucleus it is composed of protons and neutrons. The protons carry the charge +1 unit charge whereas the neutrons are uncharged. The nucleus is so immensely small that we never will see it but we know some things about its properties. One is that it behaves as if it is spinning around in some sense of the word. A spinning object in the classical world would carry an angular momentum equal to
𝐈 = 𝐫 × 𝐩 1.1 I is a vector quantity with a magnitude and a direction or equivalently three components, 𝐼𝑥, 𝐼𝑦 and 𝐼𝑧. Its direction is orthogonal to both that of the position, r, and linear momentum, p, as illustrated in Figure 1.1. To distinguish vectors scalar quantities, bold font will be used for the former and italics for the latter.
Figure 1.1. A particle is rotating about an axis so that its position and momentum vectors change. This generates an angular momentum vector that is orthogonal to both the position and the momentum vectors.
Next we make the transition to angular momentum in quantum mechanics and exemplify with the hydrogen atom. Using a slightly oversimplified picture we can view this as an electron orbiting the nucleus. The electron will thus have an angular momentum by the same principle as shown in Figure 1.1. A question that requires quantum mechanics to settle is why the energy of the electron only can assume certain fixed values. One way of explaining this was proposed by Niels Bohr in the early days of quantum mechanics is that the electron only orbits the nucleus at discrete values. In 1927 Erwin Schrödinger showed that an electron orbiting a nucleus can be described as a standing wave, represented by a wave function. A standing wave puts strict demands on the mathematical form of the wave function, 𝜓(𝑥). One demand is that the function
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must be continuous. This means that for a particle moving on a circle, the wave function must have the same value at the end of the circle as at the beginning, i.e. 𝜓(2𝜋) = 𝜓(0). It is clear that this requirement is equivalent to an integer number of wave lengths in the interval 0 − 2𝜋. Only wave functions with N = 1, 2, 3, 4,…, n wave lengths in the interval are thus acceptable. The wave function with one wave length corresponds to the lowest energy level, the one with two wave lengths corresponds to the second lowest and so on. The ‘quantum’ in quantum mechanics results from the property that for certain quantities only discrete values are allowed, due to the fact that an integral number of wave lengths must be fitted in a certain interval. There are also cases where half-integral quantum numbers are allowed. A classical analogy of this is a violin string that is suspended at both ends, which corresponds to N = ½. The string can be ‘excited’ by gently touching the string at the mid-point to get N = 1, a third of the length to get 𝑁 = 3 2⁄ etc.
Figure 1.2. Examples of acceptable wave functions for a particle traveling on a circle. The respective wave functions correspond to one, two and three wavelengths in the range 0 − 2𝜋. The number N is called a quantum number and in this case it can only take integer values.
The total angular momentum of the electron may have one additional component. Imagine that the electron could spin around its axis as it orbits the nucleus; much like the earth is spinning around its axis as it revolves around the sun. In that case this rotation, which we will call spin, would also generate angular momentum and in fact also a magnetic moment. It is a property of nature that the electron and also many types of nuclei have non-zero spin that leads to a magnetic moment. For technical reasons the quantum number describing spin angular momentum can also take half integer values. The nuclear spin is the interesting property in NMR.
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The hydrogen atom can also be used to illustrate another important phenomenon The electron orbiting the nucleus can be viewed as an electric current. From high school physics we recall that a current gives rise to a magnetic field and the electron can be described as a magnetic dipole, i.e. possessing both a south and a north pole.
1.2. Nuclear magnetic moments and their interaction with magnetic fields It turns out that also the particles of the nucleus have orbital momentum and possess spin.
Typically the net angular momentum of a nucleus is simply called the nuclear spin. Since the nucleus is charged it also manifests as a magnetic dipole.
Figure 1.3. A nucleus, in this case 3He, is spinning and since it is charged it is generating a magnetic moment. Note that this is a simplified picture of the origin of nuclear magnetism and as well as of nuclear structure.
An important difference compared with the macroscopic world is that the nuclear spin is quantized in units of ℏ 2⁄ . The magnitude of the nuclear magnetic moment is given by
|𝐈| = [𝑰(𝑰 + 𝟏)]𝟏/𝟐ℏ I = 0, ½, 1, 1½ , 2, … 1.2 Note the somewhat confusing notation. We denote the angular momentum itself 𝐈 and the associated quantum number 𝐼. The constant ℏ in the equation is the Planck constant divided by 2π and its value is 1.055⋅10-34 J⋅s.
We say that I is quantized with quantum number I. The value of I is specific for each different nucleus. For the ground state of for example proton (1H), 13C and 15N it is equal to I=½. Without introducing more advanced theory of nuclear structure, we cannot explain the exact value of I for a particular nucleus but we can nevertheless state some important results. The first thing to realize is that both protons and neutrons form pairs with vanishing spin. Thus, the total nuclear spin is completely determined by the spin of any unpaired protons and neutron and in particular if both the numbers of protons and neutrons in the nucleus are even, I=0. Other results are summarized in Table 1.1.
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Table 1.1. Rules for spin angular momentum1 No. protons No. neutrons Quantum number I Even Even 0 Even Odd half integral Odd Even half integral Odd Odd integral ≠ 0 1These results can be derived from the nuclear shell model, which in fact can be used to predict the exact value of the nuclear spin for most isotopes.
The magnetic dipole moment of a nucleus is proportional to its spin angular momentum. 𝛍 = 𝜸𝐈 1.3
where the constant γ is specific for each nucleus and may be positive or negative. For proton γ = 26.75×107 T-1s-1. There is no scientific theory at present that is able to predict γ for nuclei. It has to be measured. Various nuclear properties for selected isotopes are shown in Table 1.2. Table 1.2. NMR parameters for selected nuclei. Isotope I No. protons No. neutrons γ (T-1s-1) 1H ½ 1 (odd) 0 26.752⋅107 2H 1 1 (odd) 1 (odd) 4.107⋅107 12C 0 6 (even) 6 (even) n.a. 13C ½ 6 (even) 7 (odd) 6.728⋅107 15N ½ 7 (odd) 8 (even) –2.713⋅107 16O 0 8 (even) 8 (even) n.a. 17O 5/2 8 (even) 9 (odd) –3.626⋅107
19F ½ 9 (odd) 10 (even) 25.18⋅107
Not only the magnitude of the angular momentum is quantized but so is its component along a selected axis, normally taken to be the z-axis. The angular momentum along this axis is given by
𝑰𝒛 = ℏ𝒎 m = -I, -I+1, …, I-1, I 1.4
It is clear that the allowed values of the quantum number m depend on the quantum number I and the number of values of m are equal to 2I+1. A proton can thus have two different values for m, namely m=±½. The corresponding nuclear magnetic moment along the z-axis is accordingly
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µ𝒛 = 𝜸ℏ𝒎 1.5
For a nucleus with I=½, m can only take the values ±½ and the magnetic moment along
the associated direction is thus µ𝑧 = ±𝛾ℏ𝑚. These two different states can thus be thought of as if the magnet is pointing upwards or downwards and we commonly label these states ‘spin up’ and ‘spin down’.
Figure 1.4. Illustration of ‘spin up’ and ‘spin down’ magnetic moments. The magnitude of the magnetic moment (the length of the µ arrows) are the same in the two cases but the direction is opposite in the two cases. In classical terms this would correspond to a bar magnet pointing up and down. You may wonder what the corresponding expressions for 𝐼𝑥 and 𝐼𝑦 or µ𝑥 and µ𝑦 are. The answer is that if we know Iz and/or µz we cannot know. As a consequence of Heisenberg’s uncertainty relation the maximum information that can be known at the same time about nuclear spin is the magnitude of the spin (|𝐈|) and its component along one and only one selected axis, for instance 𝐼𝑧. Precise knowledge of the values of the other components then eludes us.
The nuclear magnets will interact with external magnetic fields according to
𝐸 = −𝛍 ∙ 𝐁 = −𝛾𝐈 ∙ 𝐁 1.6
Both the nuclear magnetic moment and the external field are vectors so the scalar product
involves summing the product of their respective components. However, frequently the external field only has a component along the z-axis, usually called B0, and the interaction energy is then given by
𝐸 = −𝛾ℏ𝑚𝐵0 1.7
We see that the interaction energy will depend on m and that it will be lower if the nuclear magnetic moment is aligned with the external field instead of being opposite to it. We
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also see that the interaction energy is a linear function of 𝐵0, the higher it is, the larger the separation between energy levels with different values of 𝑚 as illustrated in Figure 4.
Figure 1.5. Nuclear energy as a function of the strength of an external magnetic field for two different nuclei, both with 𝛾 > 0. A) For a nucleus 𝐼 = ½ so that 𝑚 = −½, ½. B) For a nucleus with spin 𝐼 = 1 so that 𝑚 = −1, 0,1. Note that the slopes of the energies depend on the magnetogyric ratio and that for nuclei with negative magnetogyric ratio, the smallest (most negative) value of the quantum number m corresponds to the lowest energy level.
Not only does an external field lead to that equilibrium population will be shifted but the external field will also cause the spins to precess about the external field with a frequency given by:
ν 0 =γB0
2π 1.8
in units of Hz. The precession frequency is called the Larmor frequency and we see that it scales linearly with the applied field and that it is different for different nuclei.
1.3. The Boltzmann distribution In absence of an external field there is no preference for ‘spin up’ or ‘spin down’ so that
exactly 50% will be in the respective states. Intuitively it is clear that in the presence of an external field, more nuclei will in the lower state but just how large is the preference for the low energy state? The Boltzmann distribution provides the answer and describes the population of a particular energy level in terms of the energy of that level, 𝑝𝑖 and the energies of all possible levels, 𝑝𝑘.
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𝑝𝑖 =
𝑒−𝐸𝑖/𝑘𝐵𝑇
∑ 𝑒−𝐸𝑘/𝑘𝐵𝑇𝑘 1.9
The parameters T and kB = 1.38⋅10-23 J/K are the temperature and the Boltzmann
constant, respectively. It is often convenient to reformulate the equation to calculate the ratio of the populations of two different states in terms of their energy separation, Δ𝐸𝑖𝑗.
𝑝𝑖𝑝𝑗
=𝑒−𝐸𝑖/𝑘𝐵𝑇
𝑒−𝐸𝑗/𝑘𝐵𝑇= 𝑒−Δ𝐸𝑖𝑗/𝑘𝐵𝑇 1.10
and in all but the most extreme NMR applications the following approximation holds.
𝑝𝑖𝑝𝑗≈ 1 − Δ𝐸𝑖𝑗/𝑘𝐵𝑇
1.11
It is instructive to study this equation to see what affects the relative populations. If Δ𝐸𝑖𝑗 = 0, which is the case for a nuclear spin in the absence of an external magnetic field, the right hand side is equal to one, meaning that there are an equal number of nuclei in the two levels. As the energy difference builds up increasingly more spins will be in the lower state. This is what happens when we go to higher magnetic field strengths as in Figure 1.5. You should also note the influence of the temperature. If Δ𝐸𝑖𝑗 ≠ 0 and the temperature is 0 K, exactly all spins will be in the lower energy state and conversely at infinite temperature the populations will be equal regardless of the energy separation. There are thus two ways of increasing spin polarization (population difference). We can either increase the external magnetic field or we could lower the temperature. In practice only the former way is exploited in NMR.
1.4. NMR spectroscopy In all areas of spectroscopy electromagnetic radiation is used to induce transitions between the energy levels associated with different quantum numbers. In NMR spectroscopy the transitions we study are between energy levels corresponding to different values of the quantum number m (not between levels of different values of the quantum number I!). Two necessary conditions for electromagnetic radiation to induce NMR transitions are
1. The frequency of the electromagnetic field must match the Larmor frequency.
2. ∆𝒎 = ±𝟏.
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If the Larmor frequency of a particular nucleus is 600 MHz only radiation of that particular frequency is effective at inducing transitions between energy levels and it is thus impossible to use electromagnetic radiation to, for instance, change the m quantum number from –1 to 1 even if the radiation has the correct frequency. This second requirement is called a selection rule. In NMR spectroscopy, the sample is placed in a magnetic field that typically is in the range 1–20 T. This leads to the low energy levels being more populated than the higher ones according to the Boltzmann distribution and thus to a net magnetization directed along the external magnetic field. Radio frequency radiation is used to induce transitions between energy levels. There are two principally different ways to record an NMR spectrum of a molecule that contain nuclei with various resonance frequencies. The first one is to continuously change the irradiation frequency (or equivalently to keep the irradiation frequency fixed and vary the magnetic field strength) and look for a response at particular frequencies (magnetic field strengths). The other way is to hit apply a short pulse of radiation containing a wide range of frequencies at once and figure out the resonance frequencies by Fourier transformation. An analogous example of these different methods of recording a spectrum is shown in Figure 1.6A-B. It is clear that the second method is more time-efficient. A representation of the actual NMR experiment for this is illustrated in Figure 1.6C.
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Figure 1.6. A-B) Two different ways of finding out which frequencies a church bell resonates at. The same principle applies to ways of finding out the Larmor frequencies in NMR. A) A loudspeaker is used to systematically expose the bell to different frequencies, one at a time. A microphone is used to record the sound. If the bell is resonant with a particular frequency, this leads to a change in intensity. B) The bell is struck with a hammer and the response is recorded by the microphone. The different frequencies of the signal can be recovered by the mathematical operation Fourier transformation. C) Illustration of the actual NMR experiment performed using the principle of (B). The net nuclear magnetization originally directed along the z-axis is rotated down to the xy-plane by means of a pulse of radio frequency radiation. The Larmor frequencies of the nuclei can be detected as an induced voltage in a coil.1) Magnetization is along the z-axis. 2) A radio-frequency pulse has flipped the magnetization onto the x-axis and it is rotating in the transverse plane which 3) induces a current with the precession frequency in a coil. 4) The frequencies of the current are recovered by Fourier transformation resulting in the NMR spectrum.
The actual NMR signal is recorded as the induced voltage in a coil by the rotating magnetization. Since a sample typically contains magnetization from several nuclei rotating at slightly different frequencies (explained in Chapter 2), the signal looks complicated. However, Fourier transformation enables recovery of the various frequencies and hence, this is the most common representation of NMR data (Figure 1.6C). For proteins there are hundreds of signals and to be able to interpret the data, the spectrum is normally recorded in two dimensions (explained in Chapter 7). A very common two-dimensional experiment is the HSQC (heteronuclear single quantum correlation spectroscopy) that contains one signal for every proton attached to 15N. The spectrum thus contains approximately as many peaks as there are residues in the protein and is often used as the fingerprint of the protein. An HSQC spectrum of the protein thiopurine methyl transferase is displayed in Figure 1.7.
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Figure 1.7. HSQC spectrum of the 28 kDa protein thiopurine methyl transferase (TPMT). The protein comprises 245 residues and there is thus approximately this number of peaks in the spectrum.
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1.5 Synopsis In this chapter you have learnt how to:
• Determine whether a nucleus is NMR active or not and if so it is a spin I=half-integer or
spin I=integer nucleus
• Construct the various spin states (allowed values of the m quantum number) from the I quantum number.
• Calculate the magnetic moment from the gyromagnetic ratio and the spin angular momentum
• Plot the different energies that results when a nucleus is interacting with an external magnetic field
• Calculate the relative populations of various spin states from the Boltzmann distribution
• Calculate the resonance frequency
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1.6 Problems 1.1 Which of the nuclei 1H, 2H, 12C, 13C, 14C, 14N, 15N, 16O, 17O and 19F are NMR active and
how can you tell? Which ones can definitely not be spin I=½ particles and why?
1.2 What is the magnitude of the spin angular momentum and its projection(s) onto the z-axis for a spin I=½ particle?
1.3 What is the magnitude of the spin angular momentum and its projection(s) onto the z-axis for a spin I=2 particle?
1.4 Calculate the magnitude of the magnetic moment of a) a proton, b) a deuteron and c) a 15N nucleus!
1.5 Calculate the possible interaction energies for the particles in 1.4 with a magnetic field of 14.1 T!
1.6 Calculate the relative populations for the ‘spin up’ and ‘spin down’ states for 1H and 15N in a 14.1 T field at 25°C!
1.7 a) Calculate the resonance frequency (unit: MHz) for a 1H nucleus in a magnetic field of
11.7 T! b) Does it make sense to refer to an NMR spectrometer operating at 11.7 T as a 500
MHz spectrometer? Does it make sense to refer to an NMR spectrometer operating at 11.7 T as a 500 MHz spectrometer?
c) What is the 13C resonance frequency at the same magnetic field?
1.8 Calculate the difference in populations between the two states for the two nuclei in 1.6! Hint: The total population is 1.
1.9 Calculate the relative populations of yourself standing on the ground and hovering 8848 m above it, respectively! The potential energy is given by E= m⋅g⋅h, where m is your mass, h is the height above the ground and g is a constant that is equal to 10 m⋅s-2. Repeat the calculation for an O2 molecule of mass 2.6576⋅10-26 kg and explain why it is hard to climb Mount Everest and difficult to breathe once you get there!
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2. The Chemical Shift
2.1 Shielding As we saw in the previous chapter a nucleus in a magnetic field resonates at a frequency
𝜈0 =𝛾𝐵02𝜋
2.1 as a result of the coupling between the magnetic field and the magnetic moment of the nucleus, the Zeeman coupling. However, this is strictly speaking only true for an isolated nucleus and not for a nucleus in an atom or a molecule. It is found that the true resonance frequency is
𝜈 =
𝛾𝐵0(1 − 𝜎)2𝜋
2.2
where σ is called the shielding constant. That is, the magnetic field a nucleus experiences is not B0 but a field different from it by a factor (1 − 𝜎). As we will see, the shielding constant is different for nuclei at different positions in a molecule and we thus get a spectrum with different resonances. The opportunity to get a unique signal for different positions in nuclei is the main reason for the importance of NMR spectroscopy in chemistry.
Shielding is caused by the electrons. It turns out that electrons, too, are influenced by the static field B0 and start to circulate about it. When doing so an induced magnetic field that counteracts B0 is induced since circulating charge equals an electric current and current induces a magnetic field.
Figure 2.1. An electron circulating the static magnetic field will induce a (much smaller) counteracting magnetic field, Bind, that is proportional to B0 with proportionality constant σ. The sum of these fields (right) is the magnetic field experienced by a nucleus and its strength is B0(1-σ).
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The induced magnetic field, Bind, is proportional to B0. The constant of proportionality is called the shielding constant and is denoted 𝜎. This means that the magnetic field a nucleus is experiencing, B, is proportional to B0 with proportionality constant (1 − 𝜎). If 𝜎 > 0 this means that the resonance frequency is lower than for a bare nucleus. This is always true in an atom but for a molecule it may be the other way, since the electrons might reinforce or reduce the static field depending on the molecular details.
2.2 The chemical shift It turns out that it is impractical to specify the frequencies for different nuclei in a
molecule directly. Two reasons are that frequencies will be different depending on which spectrometer that was used and that frequencies for various nuclei in the same molecule are so similar that it is hard to get a feel for how much they differ. For instance, the frequencies for a 1Hα and a 1Hmethyl might be 600.008851 MHz and 600.006589 MHz. These figures are very similar and we might even be tempted round both of to 600 MHz and (incorrectly) conclude that they are the same. For this reason we define a quantity called the chemical shift
𝛿 =𝜈 − 𝜈𝑟𝑒𝑓𝜈𝑟𝑒𝑓
× 106 ppm 2.3
where νref is some reference frequency and the factor 106 is used to make the chemical shifts be in the range from approximately zero to a few hundred. It is easy to verify that δ is independent of B0 and you should also be able to show that the chemical shift of the reference frequency νref = 0 ppm. If we have νref = 600.005989 in the above example the 1Hα and a 1Hmethyl chemical shifts would be 4.77 ppm and 1.00 ppm respectively. These are much easier figures to grasp and, even more importantly, they are independent of B0 and thus which NMR spectrometer that was used to record the data.
2.2.1. Chemical shift referencing The frequency νref has no physical significance but it is important for scientists in the
same field to use the same definition so that chemical shifts can be compared. In organic chemistry the molecule TMS (tetramethylsilane), which have well shielded nuclei in its four equivalent methyl groups, is used. That the reference molecule has more shielded nuclei than most other nuclei in the sample is convenient because then its signals will not overlap the others. For protein NMR spectroscopy, TMS cannot be used since it is insoluble in water which is the solvent we want to use. Instead we use DSS (4,4-dimethyl-4-silapentane-1-sulfonic acid) for most nuclei. The reference frequencies for the most common NMR active nuclei in proteins are defined according to Table 2.1 and the structures of TMS and DSS are shown in Figure 2.2.
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Table 2.1. Chemical shift referencing of NMR signals in proteins. Nucleus Reference frequency 1H The methyl protons of DSS 2H The methyl deuterons of DSS 13C The methyl carbons of DSS 15N Liquid ammonia at 25°C.
Figure 2.2. The structures of A) tetramethylsilane (TMS) and B) 4,4-dimethyl-4-silapentane-1-sulfonic acid (DSS). These molecules are routinely used to reference the signals in organic and protein applications, respectively.
For historical reasons there are several odd things with the definitions of shielding and chemical shifts. One is that the more shielded a nucleus is, i.e. higher value of σ, the lower is the chemical shift. Another is that NMR spectra are displayed with the chemical shift axis decreasing from left to right!
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Figure 2.3. The 1H NMR spectrum of ethanol. The chemical scale is increasing from right to left and it is noteworthy that the shielding decreases in that direction. The signal at 0 ppm is TMS. The reason for the splitting of the signals at 1.1 ppm and 3.6 ppm will be explained in the next chapter.
The range of the chemical shift varies greatly for different nuclei. For the most important nuclei in proteins the ranges presented in Table 2.2 are approximately valid. It is instructive to note the range of 1H on the one hand and those of 13C and 15N on the other hand. An intuitive explanation for the differences is that while 1H is surrounded by only one electron, 13C and 15N are surrounded by six and seven electrons, respectively.
Table 2.2 Approximate chemical shift ranges for selected nuclei in proteins. Nucleus Lowest chemical shift Highest chemical shift 1H 0 ppm1 12 ppm 13C 0 ppm2 180 ppm2 15N 100 ppm 140 ppm 1Chemical shifts below 0 ppm are not impossible. Such values simply mean that a particular nucleus is more well shielded than the reference nucleus. 2The chemical shifts for different 13C positions will be described later.
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2.2 The origin of chemical shifts There are two main contributions to shielding so that the total shielding can be written
𝜎 = 𝜎𝑑𝑖𝑎 + 𝜎𝑝𝑎𝑟𝑎 2.4
where the σdia is the diamagnetic and σpara is the paramagnetic contribution. The diamagnetic contribution is caused electrons circulating perpendicular to the B0 field giving rise to an induced field that counteracts B0. The paramagnetic contribution is caused by the fact that the B0 field mixes the electronic ground and excited states. The effect is that the B0 field is reinforced, and often significantly so, rather than reduced and the paramagnetic contribution thus leads to deshielding. The paramagnetic effect is much larger than the diamagnetic one. However, it only exists in molecules with unpaired electrons and will thus not be discussed further. The diamagnetic shielding can in turn be decomposed into different terms.
𝜎𝑑𝑖𝑎 = 𝜎𝑙𝑜𝑐𝑎𝑙 + 𝜎𝑑𝑖ℎ𝑒𝑑𝑟𝑎𝑙 + 𝜎𝑟𝑖𝑛𝑔𝑐𝑢𝑟𝑟𝑒𝑛𝑡 + 𝜎ℎ𝑏𝑜𝑛𝑑 + 𝜎𝑐ℎ𝑎𝑟𝑔𝑒 … 2.5 where 𝜎𝑙𝑜𝑐𝑎𝑙 depends on the the nature of the atoms within a few covalent bonds, 𝜎𝑑𝑖ℎ𝑒𝑑𝑟𝑎𝑙 depends on bond angles, 𝜎𝑟𝑖𝑛𝑔𝑐𝑢𝑟𝑟𝑒𝑛𝑡 is due to aromatic rings, 𝜎ℎ𝑏𝑜𝑛𝑑 is the contribution due to hydrogen bonds and 𝜎𝑐ℎ𝑎𝑟𝑔𝑒 is the contribution due to close by charges. In general the chemical shift thus has many contributions and it seems hard to deconvolute them. As we shall see however, quite often only a few effects dominate and in that case one can predict the chemical shift from molecular structure or alternatively predict molecular structure from chemical shifts.
2.3.1 Local effects caused by electronegativity of neighbouring atoms Since circulating electrons cause shielding it is not surprising that the distribution of
electrons around a nucleus impacts the shielding. Imagine for instance a molecule AX and you want to know the shielding of nucleus A. If X is very electronegative so that the electrons in the covalent bond largely are localized close to X, A will be less well shielded and have a higher chemical shift than if X were less electronegative. It should be pointed out that A does not need to be directly bonded to X for this to be true. For example, the proton chemical shifts of the methyl halides increase in the following order
CH3I < CH3Br < CH3Cl < CH3F 2.6
In this example the atom responsible for changing the shielding is two covalent bonds away. We can also see this for the amino acid residue valine (CH3)γ
2-CHβ-CHα-C(O)OHNH2. The proton chemical shifts increase in the following order
Hγ < Hβ < Hα < HN 2.7
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The carbon chemical shifts are affected in the same way
Cγ < Cβ < Cα < CO 2.8 Because of differences in electronegativity between the same position in different amino acids it is often quite easy to identify the identity of an amino acid from its NMR spectrum. For proteins the chemical shifts caused by local effects are usually called random coil chemical shifts and are defined as the chemical shifts for amino acids in short unstructured peptides.
2.3.2 Local effects caused by different hybridization sp3 hybridization In the absence of electron withdrawing groups, protons attached to sp3 hybridized carbon atoms typically resonate between 0 - 2 ppm. It matters whether it is attached to a primary, secondary or tertiary carbon and the order of the chemical shifts is
1H-C°1 < 1H-C°2 < 1H-C°3 sp2 hybridization Protons attached to sp2 hybridized carbon atoms resonate at higher chemical shifts than aliphatic protons attached to sp3 hybridized carbon atoms. The shift from TMS is dependent on the type of sp2 hybridized carbon atom. Protons attached to carbon-carbon double bonds resonate between 4.5 - 7 ppm. The sp2 hybridized carbon atom of the double bond has increased s-character, and is therefore more electronegative than an sp3 hybridized carbon atom. For protons in aromatic rings also ring current effects must be taken into account, which will be discussed later. This results in chemical shifts between 6 - 8 ppm. Aldehyde protons resonate between 9 - 10 ppm. This further downfield shift is due to the additional effect of the nearby electron withdrawing oxygen atom.
sp hybridization Acetylenic protons resonate between 2 - 3 ppm due to the anisotropy of the carbon-carbon triple bond. Hybridization in proteins In proteins, almost all protons are attached to sp3 hybridized carbons and nitrogens with the exception of those in aromatic rings. The hybridization of a particular carbon or nitrogen is therefore in general not important in explaining or predicting proton chemical shifts in proteins. In interpretation of NMR data for organic molecules it is however very important.
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2.3.3 Dihedral angles Since the chemical shift depends on the electron configuration it also matters how the
atoms are bonded, i.e. the angles different groups make with respect to each other. We call angles specified by the orientation of four different atoms or groups with respect to each other dihedral angles (or dihedrals). For proteins the most important of these are called ϕ (phi) and ψ (psi) and they specify if a region of the protein is in an α-helix, a β-strand or any other allowed conformation.
Figure 2.4. A) Definition of peptide dihedral angles. Shown are the angles ϕ (phi) and ψ (psi), ω (omega) and χ1 (chi one). Note that omega really is locked in the trans position since it is a partial double bond for all residue types except proline that also can be cis. Certain combinations of phi and psi corresponds to helical and others to extended structure as shown in B).
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Since certain chemical shifts are sensitive to the backbone dihedral angles, they can be used to predict secondary structure. Especially the 13Cα, 13Cβ, 13CO and 1Hα chemical shifts are reporters of secondary structure as the following table shows. The typical way to estimate secondary structure is calculate a weighted score from how much the observed chemical shifts deviate from the random coil values. Table 2.1. Average values of selected 13C chemical shifts (ppm) in different secondary structure elements.
Residue 13Cα 13Cβ 13CO coil strand helix coil strand helix coil strand helix Ala (A) 52.84 51.53 54.83 19.06 21.14 18.26 177.67 176.09 179.40 Cys (C) 57.53 56.88 61.31 29.35 30.16 27.75 174.93 173.57 176.16 Asp (D) 54.18 53.87 56.70 40.85 42.30 40.51 176.31 175.54 178.08 Glu (E) 56.87 55.52 59.11 30.20 32.01 29.37 176.43 175.35 178.61 Phe (F) 57.98 56.65 60.81 39.45 41.54 38.78 175.59 174.25 177.13 Gly (G) 45.51 45.22 46.91 -- -- -- 173.89 172.55 175.51 His (H) 55.86 55.09 59.04 29.97 31.85 29.54 174.83 174.17 176.98 Ile (I) 61.03 60.05 64.57 38.65 39.86 37.60 175.57 174.86 177.72 Lys (K) 56.69 55.40 58.93 32.79 34.63 32.27 176.34 175.31 178.40 Leu (L) 54.92 54.08 57.52 42.38 43.79 41.65 176.89 175.67 178.53 Met (M) 55.67 54.58 58.09 33.36 35.05 32.27 175.35 174.83 177.95 Asn (N) 53.23 52.74 55.45 38.55 40.12 38.61 175.08 174.64 176.91 Pro (P) 63.47 62.64 65.49 31.94 32.27 31.46 176.89 176.18 178.34 Gln (Q) 56.12 54.83 58.47 29.14 31.28 28.51 175.90 174.88 177.97 Arg (R) 56.42 55.14 58.93 30.66 32.19 30.14 176.02 175.14 178.26 Ser (S) 58.38 57.54 60.88 64.03 65.16 63.08 174.49 173.55 175.94 Thr (T) 61.64 61.06 65.61 70.12 70.75 68.88 174.70 173.66 175.92 Val (V) 62.06 60.83 66.16 32.71 33.91 31.49 175.66 174.80 177.65 Trp (W) 57.78 56.41 60.01 29.67 31.50 29.30 176.15 175.41 178.05 Tyr (Y) 57.97 56.83 60.98 38.95 40.97 38.25 175.39 174.54 177.36
An additional dihedral angle that may be important, especially for the chemical shifts of 15N and 1HN, is χ1 which corresponds to rotations about the Cα-Cβ bond vector (Figure 2.4A). If all dihedral angles are known, not only the secondary structure but also the tertiary structure is known. Thus it should be, and it is, possible to use the chemical shifts to calculate the three-dimensional structures of proteins. In Figure 2.5 the result of such a structure calculation is shown. The structure agrees extremely well with the one calculated using a more traditional approach.
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Figure 2.5. Structure of a thioredoxin domain from the protein Grx3 calculated using chemical shifts as the input. A) Secondary structure calculated with the program SSP. Values of one indicates a fully formed helix while values of minus one indicate a fully formed strand. B) Three-dimensional (tertiary) structure calculated with the program CS23D.
2.3.4 Ring currents For aromatic molecules some electrons are delocalized across the entire aromatic ring.
They will then give rise to large currents and will greatly influence the chemical shifts of nuclei that sit in the vicinity of these aromatic rings. It should be pointed out that these nuclei need only to sit close in space to the aromatic rings, not be covalently bonded to them. As a rule of thumb nuclei that sit in the same plane as the aromatic ring will get an increased chemical shift whereas nuclei above or below the plane will get their chemical shift reduced. Ring currents mostly affect nuclei with high magnetogyric ratios such as proton. The presence of aromatic residues in a protein will be very efficient at spreading out the different signals across the spectral range. The aromatic residues in proteins are phenylalanine, tyrosine, tryptophan and histidine.
Figure 2.6. Illustration of the effect of ring currents on chemical shifts. Circulating electrons give rise to a magnetic field as shown by the dashed lines. This magnetic field leads to decreased shielding (higher chemical shift) outside the ring and to increased shielding (lower chemical shift) inside or slightly above or below the center of the aromatic ring.
2.3.5 Hydrogen bonds The chemical shifts of nuclei that participate in hydrogen bonds or are bonded to such
nuclei will be affected by hydrogen bonds. It is not hard to trace this to the fact that the electron distribution will be altered by the hydrogen bond. Specifically, electrons will be removed which leads to less shielding. In general, the chemical shifts will thus get higher for protons
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participating in hydrogen bonds. It is also true that the chemical shift will increase with increasing hydrogen bond strength. The chemical shifts of amide protons in β-sheets and particularly in anti-parallel β-sheets will thus on average be higher than for amide protons in α-helices.
2.3.6 Charges Electrical charges perturb the electron distribution in their vicinity, which has an effect
on the chemical shift. It is intuitively clear that positive charges lead to decreased shielding and negative charges to increased shielding.
2.4 Summary of contributions to chemical shifts in proteins If we sum up the important effects for different nuclei in proteins the following picture
emerges. Note that the ‘local effects’ have been incorporated into residue type and nucleus.
Table 2.2. Important contributions to the chemical shift for different nuclei in proteins. Nucleus Residue type ϕ/ψ1 χ1,2 Ring current
effects Hydrogen bonds
13Cα 13Cβ 13CO 1Hα 1HN 15N 1Back bone ϕ and ψ dihedral angles. 2Side chain dihedral angle χ1. We see that the chemical shifts for many types of nuclei depend on many parameters in a complex way. However, especially the chemical shifts of 13Cα and 13Cβ depend largely on the dihedral angles phi and psi. They can thus be used to predict secondary structure.
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Figure 2.7. Proton chemical shifts of the protein ubiquitin and an illustration of typical chemical shift ranges for protons at various positions. The main features of this spectrum can be explained by local and ring current effects. Note that it is perfectly allowed, albeit rare, for chemical shifts to have negative values as shown in the figure.
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2.5 Synopsis In this chapter you have learnt how to: • Understand why not all nuclei (of the same species) in a molecule have exactly the same
resonance frequencies
• Calculate the chemical shift from the resonance frequencies of the nucleus of interest and a reference nucleus
• Understand why it is convenient to use chemical shifts rather than resonance frequencies when visualizing NMR spectra
• Understand how the electronegativity of neighboring groups affect the chemical shifts and why
• Explain the ring current effect on chemical shifts and understand that it may lead to increased or decreased chemicals depending on geometry
• Order the different 1H or 13C nuclei of an amino acid according to increasing chemical shifts if you are presented with its structure
• Explain why hydrogen bonds alter chemical shifts
• Know that the chemical shifts of certain nuclei are sensitive to dihedral angles
• Understand that the secondary chemical shift (measured chemical shift minus its random coil value) for certain nuclei is a sensitive indicator of secondary structure
• Know that the chemical shifts can be used for three dimensional structure calculations and roughly how this is done
• Explain the HSQC spectra of proteins of various sizes and in various conditions
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2.6 Problems 2.1 Verify that the chemical shift is independent of the strength of the external magnetic field! 2.2 Why is DSS rather than TMS used for chemical shift referencing in protein NMR? 2.3 Which of the amino acid residues alanine or aspartate is likely to have larger chemical
shifts for Cβ and why? 2.4 Discuss the differences between diamagnetic and paramagnetic shielding! Which one is
usually the most important for proteins and why? 2.5 Consider a pentapeptide a) on its own and b) when it binds a protein tightly. Which factors
are important for the chemical shifts of the peptide in the two cases? Is there a difference for different nuclei?
2.6 Consider a proton in a protein resonating at a frequency of 600.0006 MHz. What is its
chemical shift if the protons of DSS are resonating at a frequency of 599.999988 MHz? 2.7 Describe the ring current effect on chemical shifts! What would likely be the effect on the
chemical shifts if an aromatic ring is introduced into the interior of a protein that previously did not contain one assuming that the structure does not change?
2.8 Antiparallel β-sheets in general have higher chemical shifts for 1HN (amide proton) than α-
helices. What might be the reason? 2.9 What are random coil shifts and secondary chemical shifts? 2.10 Describe how measured chemical shifts and random coil values can be used to calculate
secondary structure in proteins! 2.11 Which one of Hβ and Hδ is likely to have the higher chemical shift for the amino acid
phenylalanine? Why? 2.12 Order the 1H chemical shifts of the following groups from lowest to highest chemical shift:
amide, methylene, methyl, α-proton, aromatic proton. Explain the ordering! 2.13 In general the chemical shift of 13Cα is higher than that of the 13Cβ. This is however not
true for the amino acid residues Ser and Thr. Explain why!
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3. Scalar Couplings
3.1 Spin couplings A nuclear spin coupling is an interaction that modifies the nuclear spin energy. The importance of the various couplings is given by how much the energy is modified. The coupling energy is typically reported in frequency units, which is fine since we learnt in Chapter 1 that they are proportional to each other, 𝐸 = ℎ𝜈. Hence, the most important such coupling is the Zeeman coupling, which is the interaction with an external magnetic field. Second in turn is the quadrupolar coupling. However, it is only present for nuclei with 𝐼 ≥ 1 and will not be considered further. Another potentially important coupling, although it is averaged to zero in solution, is the dipole-dipole coupling, which is the direct interaction between two nuclear magnetic moments. The chemical shift coupling, which is the coupling to an induced magnetic field, is of course also important as was explained in the previous chapter. The topic for most of this chapter is the scalar coupling. Although it is very weak, it survives in solution and has important consequences for the spectral appearance and in addition provides a means for magnetization transfer. Table 3.1 Summary of various spin couplings Coupling Origin Strength Zeeman coupling interaction with an external magnetic field MHz Quadrupolar coupling a coupling of electrical origin MHz Dipole-dipole coupling (direct) interaction with another nuclear spin kHz Chemical shift coupling interaction with an induced field Hz-kHz Scalar coupling indirect interaction with another nuclear spin Hz
3.2 The dipolar coupling We have covered several different mechanisms for nuclei having different chemical shifts
but a fair question is whether the spin state, i.e. ‘up’ or ‘down’, of nearby nuclei affects the chemical shift in liquid samples. After all, they are magnets that should be able to reduce or augment the experienced magnetic field. The answer is yes but we need to qualify it somewhat. Although the nucleus senses the magnetic field from nearby nuclei, this interaction, the (direct) dipole-dipole coupling, is dependent on the angle between the internuclear vector and the external magnetic field according to the following equation
𝐷 =
𝜇04𝜋ℏ𝛾𝐴𝛾𝑋𝑟𝑖𝑗3
(3cos2𝜃 − 1) 3.1
It is intuitively easy to picture that this effect averages to zero if the molecule, and thus
the internuclear vector rotates and rapidly samples all different orientations and a straightforward calculation verifies this. For a molecule in solution, the dipole-dipole coupling thus does not
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affect the chemical shifts (although it has other effects as we shall see in Chapter 6).
Figure 3.1. The direct dipole-dipole coupling between two nuclei, black and red arrows, depends on the angle the internuclear vector makes with the external field B0, indicated by a gray arrow. For a rapidly rotating molecule the dipole-dipole averages to zero.
It however turns out that electrons in covalent bonds can mediate a small part of the dipolar coupling if no more than three bonds separate the nuclei. The mediated coupling is usually quite small but, importantly, it is not dependent on orientation and therefore will not average to zero even for rapidly rotating molecules. This coupling is called the scalar coupling or J-coupling (or sometimes spin-spin coupling or indirect dipolar coupling) and significantly affects the spectral appearance by splitting signals into multiplets.
3.3 The origin of scalar couplings It is important to realize that although the scalar coupling is due to spin states of nuclei,
the effect is indirect since it is mediated by electrons. This is the reason why the scalar coupling only is important if the nuclei are separated by less than about four covalent bonds. The physics needed to explain the scalar coupling is complicated since it involves calculation of wave functions for ground states as well as excited states, which is beyond the scope of this text. An intuitive, although admittedly oversimplified, way of explaining how it might work runs as follows. Consider the simple molecule 1H19F. The molecule contains two atoms that are held together by a shared pair of electrons. Just like the nuclei 1H and 19F, electrons are spin I=½ particles and Pauli’s exclusion principle states that in an electron-pair, one of the electrons must be ‘spin up’ and the other ‘spin down’. Now consider the case when the 19F nucleus is ‘spin up’. It is then energetically favorable for the ‘spin down’ electron to be close to it. Consequently, the ‘spin up’ electron will be closer to the proton. In this way the proton senses that the 19F nucleus is ‘spin up’ and its resonance frequency, and hence chemical shift, will be modified. In the same way the 19F nucleus senses the spin state of the proton.
3.4 Multiplet structure To see how the NMR spectrum changes when the nucleus is a scalar coupled to another
one we consider a simple molecule of two magnetic nuclei, AX, where X but, not necessarily A, is a spin I=½ nucleus and show the energy level diagram for A if a) X was not present and b) if
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X is ‘spin up’ and c) if X is ‘spin down’. The Boltzmann distribution dictates that in all but the most extreme conditions about 50% of the X nuclei are ‘spin up’ and about 50% are ‘spin down’ with only slightly more spins will be ‘spin up’. When discussing the scalar coupling there is no need to consider this small difference.
Figure 3.2. Energy levels for spin A (green arrows) if it is isolated (left) and scalar coupled to the spin I=½ nucleus X (red arrows) that is parallel (middle) and antiparallel (right) with spin A in its lower level. Note that since the energy difference can be either increased or decreased due to the scalar coupling, the resonance frequency of A will be either higher or lower.
If the X spin is parallel to the A spin, its (the A nucleus) energy increases and if they are anti-parallel it decreases. Since the resonance frequency is proportional to the energy gap between levels, it follows that the proton will have different chemical shifts depending on whether the 19F is ‘spin up’ or ‘spin down’: The signal is split into a doublet with identical intensities, each of which is half of the signal if the coupling was not present.
Figure 3.3. Spectral appearance in the absence and presence of scalar coupling for an AX system. The resonance frequency increases from right to left. We are looking at the spectrum for A and the red arrows indicate the spin state of nucleus X. Note the spectrum of X would also contain a doublet split by the same amount (in Hz), it would however be centered around a different chemical shift.
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What about if a nucleus A is scalar coupled to more than one nucleus X? For the case of
two of these, AX2, we can have the following arrangements for the orientation of the spins of the X nuclei: ↑↑, ↑↓, ↓↑ and ↓↓. The resonance frequency for nucleus A would thus be smallest if both X spins are ‘up’; if one is up and the other is down (‘up-down’ or ‘down-up’ the frequency would be unaffected; and if both X spins were ‘down’, the resonance frequency of A would be largest. We thus get the spectrum in Figure 3.4 for A.
Figure 3.4. Spectral appearance of spin A in an AX2 system. The resonance frequency of A in the absence of scalar coupling is indicated in the figure. The orientation of the X spins are shown with green and red arrows.
The signal is split into a triplet. The central line will have twice the intensity of the outer lines because it corresponds to both the ↑↓ and ↓↑ configurations of the X spins. It is easy to generalize the argument to an AX3 system and see that the spectrum would comprise a quartet with intensities 1:3:3:1. The general multiplet pattern for the spectrum of A in the molecule AXn, where X is a spin I=½ nucleus is given by Pascal’s triangle (Figure 3.5).
11 1
1 2 11 3 3 1
1 4 6 4 1
Figure 3.5. The first five lines of Pascal’s triangle. It describes the multiplet pattern for the spectrum of A in a weakly coupled AXn system where X is a spin I=½ nucleus and n ranges from zero and upwards. The first line represents a singlet which is (of course) the spectrum in the absence of scalar couplings. The second line represents doublet and so on. For the triplet, Pascal’s triangle shows that the central line has twice the intensity of the outer lines. A new line in Pascal’s triangle is has one more entry than the last line and is obtained by continuing the diagonals with ‘one’ and adding the entries to the left and to the right for other entries. We can now write the general expression for the resonance frequency in the presence of scalar couplings as
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𝜈𝐴 =𝛾𝐴𝐵0(1 − 𝜎𝐴)
2𝜋− 𝐽𝐴𝑋𝑚𝑋
𝑋≠𝐴
3.2
where 𝐽𝐴𝑋 is the coupling constant (unit: Hz) and 𝑚𝑋 is the quantum number associated with the spin state for nucleus X, i.e. ±½ for a spin I=½ nucleus. It should be noted that 𝐽𝐴𝑋 can be both positive and negative, with positive values meaning that an anti-parallel arrangement of the spins A and X are energetically favored (as in the examples above) while a negative value means that the parallel configuration is stabilized. You should also note that a nucleus may be coupled to several nuclei with different coupling constants. If nucleus A is coupled to X, of course X is coupled to A with the same coupling constant and its lines will also be split into a multiplet structure, albeit centered on a different chemical shift. SOME THING ABOUT AMX SPIN SYSTEMS AND I>½ NUCLEI The way of constructing the spectrum from the energy level diagram holds when a nucleus is coupled to one or several nuclei that are spin I>½, only in this case more spin states than up and down have to be considered.
If two nuclei have the same chemical shifts due to molecular symmetry they are termed chemically equivalent. If they in addition have identical couplings to all other nuclei in the molecule they are termed magnetically equivalent. Such nuclei do not have scalar couplings to each other. Accordingly, there should be some difference in the appearance of a spectrum as the chemical shift between J-coupled nuclei decreases. If they have vastly different chemical shifts, the multiplet structure described above emerges and the nuclei are said to be weakly coupled. If the chemical shifts are similar a different, more complicated, coupling pattern is observed and the spins are strongly coupled. We will speak more about the strong coupling when discussing TOCSY experiments. Nuclei that belong to the same scalar coupled network are referred to as a spin system. Different letters are used to describe different members of a spin system. Nuclei that are weakly coupled are named by letters far apart in the alphabet, like AX or AMX. If they are strongly coupled letters close apart, like AB, are used. If two nuclei are chemically, but not magnetically, equivalent then one of them is primed, e.g. AA´. Finally, magnetically equivalent nuclei are named by same symbol, AX2.
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3.5 Use of scalar couplings Scalar couplings can be exploited to gain information about molecular structure and to
provide a means for magnetization transfer. Some examples of applications include: • Determination of the structure of small organic molecules. Characteristic multiplet patterns
show how different groups are connected. • Determination of dihedral angles for protein back bone and side-chains that can aid structure
calculations. • Transfer of magnetization. If two nuclei are coupled it is possible to transfer magnetization
between the two.
3.5.1 Structure determination of organic molecules A common application in NMR spectroscopy is to verify the identity of organic molecules. This is achieved by recording an NMR spectra and measuring positions and intensities of the various signals. Tables of approximate chemical shifts for various groups allow their identification of different groups and multiplet structures due to scalar couplings to connect these groups correctly. As a very basic example, consider the proton NMR spectrum in Figure 2.2. It consists of three signals, a singlet of relative intensity 1 at 4.7 ppm, a quartet of relative intensity 2 at 3.5 ppm and a triplet of relative intensity 3 at 1.1 ppm. From the relative intensities one concludes that the signals represent 1, 2 and 3 protons respectively and from the chemical shifts it is likely that they correspond to a proton bound to oxygen, protons in a methylene group and a methyl group, respectively. Finally, the signal that manifests as a quartet (the methylene group) must be within a few covalent bonds from three equivalent protons, i.e. a methyl group, and the one that manifest as a triplet must be within a few covalent bonds from two equivalent protons, i.e. a methylene group. Taken together this means that molecule must be ethanol. For more complex molecules a more careful analysis is needed but the procedure is the same.
3.5.2 Scalar couplings as reporters of dihedral angles The size of the scalar coupling does not only depend on which nuclei that are coupled but
also on other things, among them their geometrical arrangement. For proteins it turns out that the size of certain scalar couplings are very sensitive to certain dihedral angles. The dependence takes the following form (Karplus relation):
𝐽 = 𝐴 + 𝐵cos𝜃 + 𝐶cos2𝜃 3.3
For instance the size of scalar coupling between 1HN and 1Hα, 3JHNHα, depends on the dihedral angle φ that defines rotations around the N-Cα bond and thus determines secondary structure. If we look at the φ-dependence of this coupling, the following picture emerges:
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Figure 3.6. A) The φ dihedral angle dependence of the 3JHNHα coupling constant. Note that in general we have multiple solutions for the same coupling. However, if we look at very large (~10 Hz) and very small (~3 Hz) couplings and consider the allowed regions of the Ramachandran plot, we know that we must be in the β-strand and α-helix regions respectively. The third shaded area at 60° is not common. B) The Ramachandran plot that shows the allowed combinations of φ and ψ. The most favorable regions are colored red and the additionally allowed regions are colored yellow.
By measuring this coupling constant we thus get information of the secondary structure. This information is routinely used in NMR structure calculations.
3.6 Decoupling Although the spectral appearance because of the scalar coupling is very important since it
reports on molecular structure, it has however two negative consequences. The first one is that it reduces sensitivity, since the components of a doublet only have half the intensity of a singlet and the situation for other multiplets is even worse. The second negative consequence is spectral crowding since you will get more NMR signals than there are NMR active nuclei in the molecule. It is therefore often desirable to record NMR experiments in a fashion so that the effect of the scalar coupling is not visible.
To appreciate how this achieved, consider a molecule AX where the spins A and X have vastly different resonance frequencies so that they can be manipulated separately by RF-pulses. Imagine that we want to record the spectrum of A but only want to see a singlet. If we just apply a pulse to A, we would get a doublet due the scalar coupling. Imagine now that we record the experiment slightly differently. Immediately after the pulse on A and as soon as we start collecting the signal we start applying many 180° pulses (explained in Chapter 4) on X, one after the other. This means that the X spins that originally were ‘up’ will be ‘down’ after the first 180° pulse. It turns out that if we repeat the 180° pulses sufficiently rapidly, the A spin will simply see the average of ‘up’ and ‘down’ of X, i.e. it is like of X was not there at all and the signal will be collapsed into a singlet. The benefits are better signal to noise ratio and a simplified spectrum.
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Figure 3.7. Spectrum of nucleus A in an AX molecule where X is a spin I=½ particle in the absence (left) and presence (right) of decoupling.
The process is called decoupling and is routinely used if possible unless of course is the information content carried by the scalar coupling is desired. In practice the process is not achieved by application of repletion of simple 180° pulses but rather by more complicated sequences that emulate their action but that perform better in terms of decoupling of a wider band of frequencies at a reduced power of pulses. There are a myriad of those sequences. One of the more commonly used sequences that uses a combination of 90°, 180° and 270° pulses is called WALTZ (wideband alternating-phase low-power technique for zero-residual-splitting).
3.7 The residual dipolar coupling As stated above, the dipolar coupling does not lead to splitting for molecules in solution. One can however imagine situations where one molecular orientation is slightly favored. If so the dipolar coupling will not be exactly averaged to zero but will be scaled down. This is called the residual dipolar coupling (RDC). Since the favored direction of vectors between dipolar coupled nuclei will be different, the RDCs will also be different. By measuring RDCs it is thus possible to get information about relative orientations for different pairs of nuclei. This information can be used in protein structure calculations.
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3.8 Synopsis In this chapter you have learnt to:
• Understand why the dipolar coupling, although strong, does not lead to splitting in NMR
spectra
• Understand that the scalar coupling does lead to splitting in NMR spectra and the condition required for two nuclei being scalar coupled
• Describe how the energy levels are modified in the presence of the scalar coupling
• Be able to construct Pascal’s triangle and from it explain intensity ratios between multiplet components
• Draw the NMR spectra of A (and X) in molecules of the type AX, AX2, AX3, AX4… if A and X are spin I=½ nuclei
• Draw the NMR spectra of A in molecules of the type AMX, AMX2, AM2X2, AMX3… if A and X are spin I=½ nuclei
• Draw the NMR spectra of A in simple molecules where A is scalar coupled to a nuclei that are spin I≥1
• Name three different applications of the scalar coupling
• Describe how the scalar coupling can be used in structure calculations
• Understand why and how decoupling is used
• Understand the difference between the weak and the strong scalar coupling
• Define chemically and magnetically equivalent nuclei
• Define a spin system
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3.9 Problems 3.1 Derive the multiplet patterns of a doublet, a triplet and a quartet by considering the
resonance frequencies of A in an AXn molecule (X is a spin I=½ nucleus, A and X are weakly coupled)!
3.2 Construct Pascal’s triangle! 3.3 A proton in a protein has a shielding constant of -1.0×10-5. The shielding constant of the
reference nucleus is 0.
a) What is the chemical shift of the proton in the absence of scalar couplings?
b) Now consider the case where this proton is scalar coupled to an 15N nucleus with coupling constant 92 Hz. What are the chemical shifts of the two components of the doublet at a static magnetic field of 11.7 T?
c) Repeat your calculation from (b) at a static magnetic field of 18.8 T!
3.4 Sketch the NMR spectrum of nucleus A in the molecule AX2! X is a spin I=1 nucleus and the coupling constant is JAX. Indicate the relative intensities of the various components and their separation in Hz.
3.5 Sketch the NMR spectrum of nucleus A in the molecule AM2X3 if a) JAX >> JAM
b) JAM >> JAX! All nuclei are spin I=½ particles.
3.6 Why does the direct dipole-dipole coupling not cause splitting in NMR spectra of liquid
samples? 3.7 What is the condition for the validity of the weak coupling limit? 3.8 Describe chemical and magnetic equivalence! 3.9 Describe how scalar couplings can be used in protein structure calculations! 3.10 Consider the molecules 12C1H4, 13C1H4, 12C2H4 and 13C2H4 and sketch the 1H, 2H and 13C
NMR spectra in all applicable cases! The carbon and hydrogen nuclei are coupled if possible. Why did I not ask you to sketch the 12C NMR spectra?
3.11 Solve the structures of the organic molecules shown on the next page with the aid of their
1H NMR spectra! The numbers within brackets indicate relative intensities of the signals.
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4. The Vector Model In the vector model magnetization is considered as a vector that can rotate about various axes. Although it is a simplified model that for example is not applicable for interacting nuclei, it nevertheless can be used to evaluate the result of many NMR epxeriments. Since an NMR experiment consists of radio frequency pulses and delays, the description will start with the effect of these on the magnetization and will conclude with the result of sequences of pulses and delays.
4.1 Radio frequency pulses At equilibrium, the magnetization vector will point along the external field, taken to be
the z-axis. This is so because at thermal equilibrium more nuclear spins are pointing up than pointing down and the magnetization vector can be viewed as the sum of all magnetic moments. Since the NMR signal results from the magnetization moving around the z-axis we first need to bring it down to the transverse plane, or at least away from the z-axis, so that we can detect it. We do this by applying pulses of radiation that are resonant with the precession frequencies. The frequencies of these pulses are typically 10-1000 MHz (depending on nucleus and field strength of the magnet), i.e. in the radio frequency region of the electromagnetic spectrum. We thus call these pulses radio frequency pulses or RF-pulses for short.
The angle of which an RF-pulse rotates the magnetization vector, i.e. the flip-angle, depends on its strength, denoted by γB1, and for how long it is applied. In NMR spectroscopy the most commonly flip-angles are 90° and 180°. The flip-angle is given by
𝛽 = 𝛾𝐵1𝑡𝑝 4.1 where tp is the duration of the pulse. The typical length of a 90° pulse is 5-50 μs, and accordingly a 180° pulse is twice as long. These durations are so short that we often may regard the pulse to be able to instantly flip the magnetization from one direction to another. We then do not care about what happens to the magnetization during the pulse.
4.2 Radio frequency pulses Pulses are applied along a certain axis, typically x, y (or -x or -y). What this means is that
they rotate the magnetization about this axis. We will now look at some examples of this. First we consider 90° pulses applied to equilibrium magnetization along the z-axis, also called longitudinal magnetization.
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Figure 4.1 The result of a 90°y pulse on magnetization along the positive z-axis is to bring it to the positive x-axis.
The figure shows what happens if magnetization along the z-axis is turned 90° around the y-axis, that is after application of a 90°y pulse. The result is magnetization along the x-axis. If we instead apply a 90°x the magnetization now is aligned along the negative y-axis.
Figure 4.2 The result of a 90°x pulse on magnetization along the positive z-axis is to bring it to the negative y-axis. You should be able to evaluate what happens if 90°-x and 90°-y pulses are applied to longitudinal magnetization.
We now turn to the effect of 90° pulses to transverse magnetization, that is magnetization in the xy-plane, and first consider a 90°y pulse applied to x-magnetization. The result is that rotation 90° around the y-axis yields a magnetization vector pointing along the negative z-axis.
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Figure 4.3. The result of a 90°y pulse on magnetization along the positive x-axis is to bring it to the negative z-axis. What about a 90°y pulse applied to y-magnetization? The result is shown in Figure 4.4.
Figure 4.4 A 90°y pulse on magnetization along the positive y-axis does not lead to any observable consequences.
The result is that nothing that we can detect happens. This is exactly for the same reasons as we cannot detect equilibrium magnetization precessing about the z-axis. We can summarize this into a rule: Rotation of magnetization around the same axis as it is aligned is not detectable. You should now be able to evaluate
𝑀𝑥90°𝑦⎯ ? 𝑀𝑦
90°𝑥⎯ ? 𝑀−𝑥90°𝑥⎯ ?
where Mi is magnetization aligned along the i-axis. If you can do this, you essentially know all there is to know about 90° pulses.
We now consider 180° pulses and we are going to make it a little more complicated by
also considering magnetization that is aligned somewhere in between two axes. We start simple however. The effect of a 180°y pulse on z-magnetization is of course that it is inverted.
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Figure 4.5. The result of a 180°y pulse on magnetization along the positive z-axis is to bring it to the negative z-axis.
You should be able to show that the same result is obtained also for a 180°x pulse. For
180° pulses applied to transverse magnetization it is enough to consider the xy-plane since a 180° pulse never can tilt the magnetization away from this plane.
Figure 4.6. The result of various 180° pulses on different kinds of transverse magnetization. Only the x- and y-axes are shown since a 180° pulse cannot tilt the magnetization out of the transverse plane.
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Since we already have established that all rotations about the axis where the magnetization is aligned are undetectable, there is no need to show the effect of a 180°x pulse on x-magnetization and similar. It is however useful to show the effect of a 180° pulse on magnetization not aligned along a particular axis. We thus consider magnetization that is somewhere in between the x- and y-axes.
Figure 4.7. The result of various 180° pulses on yet different kinds of transverse magnetization.
Hopefully you will not find anything strange with these rotations. We have done exactly as before, rotated the magnetization 180° around certain axes. As we will see in a minute, 180° flips of magnetization at arbitrary positions in the transverse plane is an important building block in many NMR experiments.
4.3 Periods of free precession When the magnetization vector is turned into the transverse plane by means of a 90°
pulse we can detect that it is precessing around the z-axis with frequency ν since a current with this frequency is induced in a coil but what happens in detail and how do we describe this mathematically? Imagine that we have applied a 90°y pulse to equilibrium magnetization and thus turned it along the x-axis. The magnetization vector will immediately start to rotate around the z-axis. If we wait for a while the magnetization will be directed along the y-axis, if we wait
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some more along the negative x-axis and if we wait yet some more it will be back along the x-axis. It is very easy to show that the x- and y-components of the magnetization are (if they are along x at t=0) are given by:
𝑀𝑥 = 𝑀0cos(2𝜋𝜈𝑡) 4.1a
𝑀𝑦 = 𝑀0sin(2𝜋𝜈𝑡) 4.1b
For now we ignore the fact that the signal decays over time but will consider that in Chapter 6. We can make a plot of for instance the x-component and get a cosine function or the y-component and get a sine function. Using this we can also see how the free induction decay (FID) (represented by the x-component) looks if we have two nuclei with different precession frequencies (chemical shifts).
Figure 4.8. The free induction decay, FID, for A) three nuclei with different resonance frequencies and B) their sum, i.e. the observed NMR signal.
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4.4 The spin echo The FID looks fairly complex already in the case with only three nuclei with different
resonance frequencies. An FID from a large molecule would comprise a large number of overlaid cosine signals and look even more complicated. In a protein there would for example typically be hundreds of overlaid signals. All frequencies can of course be recovered by Fourier transforming the signal to switch the representation from the time domain to the frequency domain but a useful way of picturing what happens with the magnetization as a function of time is to plot the angle it makes with the x-axis. This angle, also called the phase, is given by
ϕ(𝑡) = 2π⋅ν⋅t 4.2 We would thus get the following plot where both nuclei from the previous example are plotted in the same chart.
Figure 4.9. The phase of the magnetization with respect to the x-axis after application of a 90°y pulse and precession for a period t=τ. To the left the magnetization is shown as rotating vectors and to the right the phase with respect to the x-axis is shown as a function of time.
This is a convenient way to picture what happens if we do something to the magnetization, e.g. apply a 180° pulse when it is in the transverse plane. As you should be able to see, one way of describing a 180°x pulse is that it inverts the angle ϕ. Let us apply such a pulse at t = τ and evaluate the magnetization at time t =2τ.
Figure 4.10. The phase of the magnetization, i.e. its angle with the respect to the x-axis is inverted after application of a 180° (x) pulse at t=τ. After an additional period τ the magnetization will be realigned with the x-axis regardless of resonance frequency.
The effect is clearly that the magnetization is at the same location at t=2τ as it was at t=0, regardless of resonance frequency, i.e. chemical shift. For this reason the sequence τ-180°-τ is
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called a spin echo or a Hahn echo after the scientist Erwin Hahn who first described it. The spin echo is useful if you need to bring back the magnetization to where it was originally, regardless of chemical shift.
4.5 Notation for pulse sequences A pulse sequence is typically represented horizontal lines representing delays and boxes representing pulses. At the end of the pulse sequence a representation of the FID drawn to show when the signal is detected. To distinguish 90° and 180° pulses narrow rectangles are used for the former and wide rectangles are used for the latter. It is of course also necessary to specify the phase of the pulses and that is done by writing x, y etc. above the pulse. Complicated combinations of pulses, for example decoupling sequences, are typically shown as very wide rectangles with the name of the sequence inside the box. With this notation the pulse-acquire and spin echo experiments are written as shown in Figure 4.11.
Figure 4.11. Pulse schemes for two experiments that yield nearly identical spectra. The difference is that the one in (B) will be of lower intensity. How much lower depends on the length of the delay τ. A) Pulse-acquire experiment. B) Spin echo experiment.
For nuclei that have very different resonance frequencies, such as those of different isotopes it is possible to manipulate the nuclei separately. To show this it is then also necessary to specify on which pulse a given pulse acts. Two examples involving the scalar coupled nuclei I and S are shown in Figure 4.12.
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Figure 4.12. Two variations of the spin echo experiment for scalar coupled nuclei. A) The application of simultaneous 180° pulses on nuclei I and S refocuses the chemical shift of I at 2τ while the scalar coupling evolves for the entire period 2τ. If τ is set to 1/4JIS the spectrum will comprise a doublet where the two components of the doublets have opposite signs. B) By omitting the 180° pulse on nucleus S at τ, the I chemical shift as well as the scalar coupling is refocused. With the decoupling, here WALTZ-16, the expected doublet is collapsed into a singlet of twice the intensity as the components of the doublet.
4.6 Decoupling revisited There are many applications where several spin echoes are combined. One is decoupling that already have been introduced, in which many pulses are applied on a nucleus that is scalar coupled to the nucleus of interest. As shown in Figure 4.13 a 180° pulse in this case exchanges the components of a multiplet, a doublet in this case. If no pulses are applied there will be two distinct phases at all times. If one pulse is applied in the middle of the delay, the phase will be refocused to a position with intermediary phase at the end of the delay but at no other time points. If more pulses are applied there will be refocusing also at other time points and if they are applied continuously there will refocusing at all times, i.e. it is as if the scalar coupling did not exist, we have achieved decoupling.
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Figure 4.13. Illustration of how decoupling can be achieved. Application of A) zero, B) one, C) four and D) hundreds or thousands of 180 ° pulses on the scalar coupled nucleus.
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4.7 Synopsis In this chapter you have learnt to:
• Explain the concept of net magnetization and in what direction it points at equilibrium
• Know the concept of the rotating frame
• Understand why we cannot detect the NMR signal at equilibrium
• Explain how different 90° and 180° pulses rotate the magnetization
• Explain what happens the magnetization when it is in the xy-plane in three different
ways: 1) rotating arrows, 2) cosine functions, 3) phase with the respect to the x-axis as a function of time
• Understand and make sketches of the magnetization in pulse sequences of the types 90°y - τ - 180°x - τ and 90°y - τ - 180°y - τ, where τ is a delay
• Be able to understand and write simple pulse sequences in standard notation
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4.8 Problems 4.1 Describe the rotating-frame and the effective field! (Both using words and equations) 4.2 Two nuclei are precessing with frequencies of 1 Hz and -3 Hz in the rotating frame. At
time t=0 they are aligned with the x-axis
a) Sketch their x-components as a function of time! b) Sketch their y-components as a function of time!
4.3 Evaluate the result of a 90°x pulse to magnetization aligned with
a) the z-axis b) the x-axis c) the y-axis d) the negative z-axis e) the negative x-axis f) the negative y-axis
4.4 Repeat what you did in 4.3 but for a 90°-y pulse! 4.5 Repeat what you did in 4.3 but for a 180°y pulse! 4.6 If magnetization in the transverse plane makes the angle -30° with the x-axis. Where is it
after the
a) application of a 180°x pulse? b) application of a 180°y pulse?
4.7 Two nuclei have the resonance frequencies 1 Hz and 2 Hz in the rotating frame. At time
t=0 they are aligned with the x-axis.
a) sketch their phases with respect to that axis as a function of time! Also sketch the x-components of their magnetizations!
b) sketch their phases with respect to the x-axis at time t=2τ if a 180°x pulse is applied at time t=τ!
c) the same as in (b) but with a 180°y pulse 4.8 Describe the spin echo and what it is used for!
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4.9 Here you will evaluate three examples of the spin echo. Consider an AX spin system where nuclei A and X can be manipulated separately by RF-pulses. They could for instance be a proton and an 15N nucleus, respectively. The resonance frequency of spin A is νA ± ½JAX Hz in the presence of the coupling to X. The magnetization for nucleus A is originally aligned with the x-axis. Sketch its phase(s) with respect to the x-axis for 0 ≤ t ≤ 2τ if
a) a 180°x pulse is applied to only A at time t=τ! b) a 180°x pulse is applied to both A and X at time t=τ! c) a 180°x pulse is applied to only X at time t=τ! d) Summarize the results in a-c using words!
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5. Chemical exchange Molecules are not static. Chemical bonds can break and form and the structure can also
fluctuate on different time-scales. This is especially true for large complicated molecules and for proteins, dynamics it is often a prerequisite for proper function. An enzyme would for instance hardly work if all residues were completely fixed with respect to each other and binding of ligands would also not be feasible if the protein was not somewhat plastic. protein folding is also an example of dynamics. In NMR we refer to dynamics as chemical exchange regardless of whether bonds are broken and formed or the molecule merely changes its structure. I shall now give a brief summary of how chemical exchange affects the NMR signals and spectra.
5.1 Symmetric slow exchange We start by considering a molecule that can be in two different conformations, A and B,
with equal probability so that the populations pA = pB = 0.5. A certain nucleus, say X, has the chemical shift δXA when the protein is in state A and chemical shift δXB when it is in state B. If the magnetization was along the x-axis at t=0 and the jumps between states are very rare the angle it would make with the x-axis at later time points would be
Figure 5.1. The phase of the magnetization for nucleus X when it is states A and B, respectively, if there is no interconversion of the two states.
We have one chemical shift for the nucleus when it is in state A and another when it is in state B. Now assume that there is a small chance that the molecule would change state from A to B or vice versa during the course of the experiment. It should be noted that these jumps happen at random time points and only the average rate of jumps per unit time, which we denote k, can be known. The above graph now changes to
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Figure 5.2. The phase of the magnetization for nucleus X when it is states A and B, respectively, if there is slow interconversion of the two states.
Note that in reality there can be more than one jump during the time course. The important thing though is how the spectrum changes. It is clear that some of the molecules that originally were in state A are in state B at some point during the experiment and vice versa. This means that the two NMR lines will be somewhat closer to each other than if there was no exchange. You see this by taking the average of the trajectories close to A and B respectively and see that they are neither at A or B. The other thing is that the spreading out of magnetization is equivalent with the lines get broader and less intense. If we define ∆LW as the extra line-broadening due to exchange, it is related to the rate constant of the jumps, k, as
Δ𝐿𝑊 =
𝑘𝜋
5.1
This equation is valid for slow exchange and the populations are equal, i.e. slow symmetric exchange.
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5.2. Fast symmetrical exchange In the other extreme, when jumps are very frequent we get the following graph
Figure 5.3. The phase of the magnetization for nucleus X when it is states A and B, respectively, is shown with thick lines. If there is rapid stochastic interconversion of the two states, the phase will be centered at the population averaged phase of A and B. Three trajectories are shown in thin lines.
It should be obvious that the angle the magnetization makes with the x-axis is centered on the average of A and B. We thus only see the average chemical shift. Another difference for fast exchange is that the line will sharpen up (get less broad) when the exchange gets faster. To calculate the exchange broadening in the case of fast exchange, we also need to define the difference in resonance frequencies for states A and B
δν = δXA − δX𝐵 5.2 The exchange broadening for fast symmetrical exchange is then given by
Δ𝐿𝑊 =
𝜋𝛿𝜈2
2𝑘 5.3
What about the intermediate situation? This occurs when the rate constant k is of the
same order of magnitude as the δν. The two peaks coalesce into one very broad peak when
𝑘 =
𝜋𝛿𝜈√2
5.4
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The peak(s) will never get broader than at this rate and we call this regime intermediate exchange. Any slower or faster exchange and the peak(s) will sharpen up. The detailed expression for the line broadening for intermediate exchange is very complicated and will not be given here. When defining what is fast exchange etc. we always compare the rate constant with δν and thus have
𝑘 ≪ 𝛿𝜈 slow exchange 𝑘 ≈ 𝛿𝜈 intermediate exchange 𝑘 ≫ 𝛿𝜈 fast exchange
The difference in resonance frequencies for states A and B, δν, thus defines the chemical
shift time-scale. If the entire molecule experiences a coordinated exchange process with a common rate constant, different nuclei can thus be in different exchange regimes depending on how much their resonance frequencies change when the molecule jumps to the other state.
It should also be noted that if either the exchange rate or the difference in resonance frequencies is modified the exchange regime can be changed. As an example, consider raising the sample temperature. At low temperatures there is often a high barrier against jumps between sites so that exchange is negligible. As the temperature is raised, the rate constant gets increasingly larger and accordingly the exchange would be classified as first slow then intermediate and finally fast. An example of instead changing the difference in resonance frequencies is to record the experiment at different magnetic fields. Chemical exchange that is fast at a field of 7 T can be intermediate or slow at a field of 18.8 T.
5.3. Asymmetric exchange If the populations for states A and B, pA and pB, respectively, are not equal we have
asymmetric exchange. To describe it we need to introduce the rate constants kA and kB that refer to the rate of jumps out of state A and B respectively. For asymmetric exchange they are not the same. For slow exchange the extra line broadening due to exchange is then given by
Δ𝐿𝑊𝐴 =
𝑘𝐴𝜋
5.5a
Δ𝐿𝑊𝐵 =𝑘𝐵𝜋
5.5b
You should be able to see that the less intense of the two peaks will get more broadened
and often it will be so broad that it will not be seen above the noise in the spectrum. That you only see one peak is thus not clear-cut proof of intermediate to fast chemical exchange. If the populations are unequal, the expression for fast exchange also changes to
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Δ𝐿𝑊 =4𝜋𝛿𝜈2𝑝𝐴𝑝𝐵𝑘𝐴 + 𝑘𝐵
5.6
Often we write it in the following form
Δ𝐿𝑊 =
4𝜋𝛿𝜈2𝑝𝐴𝑝𝐵𝑘𝑒𝑥
5.7
where we have introduced the exchange rate 𝑘𝑒𝑥 = 𝑘𝐴 + 𝑘𝐵.
Note that the spectral line is now seen at the population-weighted average of δXA and δXB. You should be able to convince yourself that the expressions for asymmetric exchange reduce to those for the symmetric case if you set pA = pB = 0.5. You should also see that there is a lot of information encoded in the exchange line broadening, especially for the intermediate to fast time scales. It depends on exchange rate (kinetic information), the populations (thermodynamics) and the difference in chemical shifts of the states, which can be converted to the chemical shift for the various states (structural information). There are dedicated NMR experiments where all these parameters can be determined.
5.4. Methods used for studying chemical exchange
5.4.1. Exchange spectroscopy The exchange spectroscopy (EXSY) experiment lets you measure rate constants for a
system that is in slow exchange. Essentially the experiment is an HSQC with a long delay inserted between the t1 and t2 periods. The 15N chemical shift is recorded for the two states, here called N and U, during t1. During the subsequent delay the states are partially interconverted due to exchange. Magnetization is then transferred to 1H for detection. Magnetization that was in the N state during t1 but is in the U state during t2 will manifest as a cross peak with the N 15N and U 1H chemical and vice versa. From the intensities of the cross peaks as a function of the duration of the delay the rate constants can be determined.
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Figure 5.4. The EXSY experiment for a system exchanging between the N and the U states. These could be the native and unfolded states, respectively. A) A protein is exchanging between states N and U. The populations are pN=67% and pU=33% and the exchange rate is sufficiently slow for peaks of both states to be present in the spectrum. B) First the 15N chemical shift is recorded, then a delay is inserted during which exchange takes place and finally the 1H chemical shift is recorded. This results in cross peaks between the two states. C) By recording the experiment as a function of the exchange delay, the respective rate constants can be extracted as the initial slope of buildup cross peak intensity.
5.4.2. Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion CPMG relaxation dispersion is based on the fact that if there are random jumps between
resonance frequencies a 180° pulse in the middle of a delay is not able to refocus the magnetization perfectly. However, as more of these pulses are inserted during the delay refocusing gets progressively better and accordingly the line width should change as a function the number of refocusing pulses. The method was invented by Carr and Purcell and subsequently improved on by Meiboom and Gill, hence the name.
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Figure 5.5. A) Evolution of magnetization at two sites with resonance frequencies of ωA and ωB, respectively, following a 90° pulse and a delay τ. B) In the absence of chemical exchange a 180° pulse in the middle of a delay is able to refocus the magnetization. C) In the presence of exchange (random jumps between the two resonance frequencies) the refocusing is not perfect and peak broadening results. Note that the small peak will be broadened more. By application of increasingly more 180° pulses the refocusing gets progressively better as comparison of the top and bottom panels suggests.
In the CPMG relaxation dispersion experiment the effective transverse relaxation rate, which is closely related to the line width as you will see in the next chapter, is monitored as a function of the repetition rate of 180° pulses during a constant time relaxation delay.
For a system in intermediate exchange the CPMG experiment is extremely powerful. It provides information on the rate constants for interconversion between exchanging states and also populations and chemical shifts of excited states that cannot readily be seen in the spectrum. To summarize this one gets information about kinetics, thermodynamics and structure.
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Figure 5.6. Example of a CPMG relaxation dispersion profile measured for a Glu30 13CO in an SH3 domain from the protein Abp1p at static magnetic fields of 600 MHz (red symbols) and 800 MHz (blue symbols). The lines represent the best fit to the data. R2,eff = 1/T2,eff is the effective relaxation rate for a particular value of νCPMG.
5.4.3. When to run EXSY and when to run CPMG experiments For EXSY exchange must be slow enough for observation of peaks corresponding to
both states but not so slow that the magnetization dies during the delay. In practice this means that the rate constants should be on the order of 1/s. For decent sensitivity in the extraction of both rate constants, the populations of the exchanging states should be approximately equal.
For using CPMG at its full potential (extraction of rate constants, populations and excited state chemical shifts) the exchange should be on the intermediate time-scale which often means rate constants on the order of 100-1000/s. The method works best when the populations are skewed. However, the population of the excited state must at least 0.5%. In practice, it is much more common that conditions are suitable for CPMG than EXSY. Hence, CPMG is by far the more common experiment.
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5.5 Synopsis In this chapter you have learnt to:
• Describe what chemical exchange means
• Explain the spectra for slow, intermediate and fast chemical exchange
• Know the chemical shift time scale and when exchange is slow, intermediate or fast
• Know the difference between symmetric and asymmetric chemical exchange and why
exchange is symmetric or asymmetric
• Know two methods for studying chemical exchange in proteins
• Describe the information content of a CPMG relaxation dispersion experiment
• Describe important biological processes that are coupled with protein dynamics
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5.6 Problems 5.1 Describe in words and figures the spectral appearance for slow, intermediate and fast
chemical exchange between to states! 5.2 Describe some common processes leading to chemical exchange in a) small molecules
and b) proteins! Describe why it is called chemical exchange in the first place! 5.3 What is the difference between symmetric and asymmetric exchange? 5.4 Sketch the spectra for a nucleus in chemical exchange between two sites A and B as a
function of kex = kA + kB in the range 0 – 10000 s-1 (five representative spectra)! The population of state B is 20% and the difference in resonance frequencies between A and B is 300 Hz. At approximately which frequency do the two peaks coalesce into one?
5.5 Describe the EXSY and CPMG experiments and especially their information content! 5.6 You happen to know (by some other method) that a protein is in exchange between states
A and B, that the population of state B is 2% and that the exchange rate is 300 s-1. When you run an NMR experiment you see only one peak for a particular nucleus. However the spectrum is very noisy and you estimate the noise to about 10% of the signal you see. Can you determine from the spectrum if this particular nucleus is in slow or fast exchange? If not, how can you change the experiment to find out?
5.7 A protein is exchanging between its native state (N) and a folding intermediate (I) with a
rate constant of 200 s-1. Is the exchange regime (slow, intermediate, fast) necessary the same for all nuclei in the protein the same? If not, explain why!
5.8 The resonance frequencies of the two δ protons of Phe residues in proteins are often
different. Yet one almost always only observes one signal in NMR spectra. Why? Suggest some experimental conditions where one would expect to see two signals!
5.9 You want to calculate the structure of a protein folding intermediate. By fluorescence
stopped-flow experiments you have determined the population of the intermediate to be 3% and the exchange rate 600 s-1. Since you are an experienced NMR spectroscopist you know that this means that the protein is in intermediate exchange for many nuclei. Suggest NMR experiment(s) you would need to run and then how you would proceed to complete the task!
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6. Relaxation When an NMR sample is inserted into the magnet, the emergence of the equilibrium
magnetization, i.e. population difference between nuclei of various energy levels, is not instantaneous. Similarly, if the equilibrium magnetization is disturbed by an RF-pulse, the decay of transverse magnetization and the return of longitudinal magnetization to the equilibrium value take a certain time. We introduce two time constants to describe these processes. T1 is the time constant for the return to equilibrium magnetization and T2 is the time constant for the decay of transverse magnetization. We refer to both T1 and T2 processes as relaxation.
6.1 The origins of relaxation It is important to understand that relaxation is not the result of spontaneous or stimulated
emission of a photon because in NMR that is such a rare event that it does not need to be considered. Instead, it is the result of the nuclei experiencing a randomly fluctuating magnetic field because of molecular motion. The randomly fluctuating fields can have different origins and one of the most important is due to the dipole-dipole coupling. This coupling averages to zero for randomly rotating molecules in solution but the average of its square does not (Figure 6.1). We can write
⟨Δ𝐵2⟩ ≠ 0 6.1 where
∆B2 is the square of the fluctuating field at a certain point in time. The larger ⟨Δ𝐵2⟩, the faster relaxation is. This is however not the only thing that affects the relaxation rates.
Figure 6.1. Illustration of random fluctuations of the magnetic field (green) with zero average and its square (red). The average of square of the fluctuations is clearly different from zero.
Another important aspect is how fast the molecule is tumbling and, for dipolar relaxation, the magnetogyric ratio of the nucleus in question. For the effect of the molecular tumbling on relaxation we introduce the spectral density function. For a rigid spherical molecule it can be written
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𝐽(𝜔) =25⋅
𝜏𝑐1 + 𝜔2𝜏𝑐2
6.2
where ω =2πν (2×π times the frequency) is the angular frequency and τc is the correlation time for molecular tumbling. The correlation time is approximately equal to the average time for the molecule to rotate one radian. Small molecules have small correlation times and vice versa but it also depends on the viscosity of the sample and thus on temperature and type of solvent.
The size of ⟨Δ𝐵2⟩ differs for different relaxation mechanisms. If relaxation is due to the dipolar coupling it is usually referred to as 𝑑2 and is given by
𝑑2 =(𝜇0/4𝜋)2 ℏ2𝛾𝐼2𝛾𝑆2
𝑟𝐼𝑆6 6.3
where 𝛾𝐼 is the gyromagnetic ratio for the nucleus for which relaxation is calculated, 𝛾𝑆 is the gyromagnetic ratio for the dipolar coupled nucleus, 𝑟𝐼𝑆 is the internuclear distance between the two and the physical constants have their usual meanings. This tells us that high gyromagnetic ratio nuclei are most efficient at inducing relaxation and that in order to do so the internuclear distance needs to be short.
If the dipolar coupling between nuclei I and S is the only relaxation mechanism, the
relaxation rates 1/T1 and 1/T2 for nucleus I are given by
1𝑇1
=𝑑2
4[3𝐽(𝜔𝐼) + 6𝐽(𝜔𝑆 + 𝜔𝐼) + 𝐽(𝜔𝑆 − 𝜔𝐼)] 6.3
1
𝑇2=𝑑2
8(4𝐽(0) + 3𝐽(𝜔𝐼) + 6𝐽(𝜔𝑆) + 6𝐽(𝜔𝑆 + 𝜔𝐼) + 𝐽(𝜔𝑆 − 𝜔𝐼)) 6.4
The important spectral densities are thus at resonance frequencies for nuclei I and S as
well as their sums and differences and for T2 relaxation also zero frequency. It is easily seen that T2 will decay monotonically with increasing τc, and thus with increasing molecular weight, for T2 while T1 while show a minimum and then increase with increasing τc.
It is also interesting to understand why the expressions for 1/𝑇1 and 1/𝑇2 differ. As stated above, relaxation is caused by randomly fluctuating magnetic fields due to molecular motion. Only fields fluctuating perpendicular to the magnetization of interest cause relaxation. Fluctuating fields in the transverse (xy) plane give rise to terms proportional to 𝐽(𝜔𝐼), 𝐽(𝜔𝑆), 𝐽(𝜔𝑆 + 𝜔𝐼) and 𝐽(𝜔𝑆 − 𝜔𝐼). For T1 relaxation, all magnetization will be perpendicular to the transverse plane whereas for T2 relaxation only half of the magnetization will be perpendicular to
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a particular direction in this plane. This explains the factor ½ in the expression for 1/𝑇2. Random fields along the z-axis give rise to the term proportional to 𝐽(0) and are only important for transverse relaxation.
The expressions also allows us to say some things regarding the magnitudes of 1/𝑇1 and 1/𝑇2. We first note that the total number of spectral densities in the expressions are the same in both cases. In the particular case 𝐽(0) = 𝐽(𝜔𝐼) = 𝐽(𝜔𝑆) = 𝐽(𝜔𝑆 + 𝜔𝐼) = 𝐽(𝜔𝑆 − 𝜔𝐼) it then follows that 1/𝑇1 = 1/𝑇2. This is actually true for small molecules. We also see that the minimum value for 1/𝑇2 is 1/𝑇2 = 0.5/𝑇1and this would happen in the unlikely event that the fluctuations in the z-direction vanish. In general we have 1/𝑇2 ≥ 1/𝑇1 since 𝐽(0) ≥ 𝐽(𝜔0).
For convenience, the expressions 1/𝑇1 and 1/𝑇2 are frequently referred to as 𝑅1 and 𝑅2,
respectively.
6.2 Interpretation of T1 and T2 Since NMR spectroscopy is dependent of net equilibrium magnetization at the start of the
experiment, T1 determines how long time we need to wait until we can repeat the experiment. The time 5×T1 is often taken to be the time for equilibrium magnetization to be completely reestablished but in practice one repeats the experiment faster than this and rather adds data from more experiments with somewhat lesser sensitivity rather than using data from only one or a few experiments with high sensitivity. For proteins T1 is on the order of one second and 1-3 s is the usual delay between successive experiments.
T2 determines the line width of the spectral lines. The shorter T2 is, the wider (and less intense!) line shapes since (in absence of chemical exchange) we have
𝐿𝑊 =1𝜋𝑇2
6.6
For sensitive NMR experiments it is thus desirable to have a short T1 but a long T2, which
according to the equations above often are conflicting goals. Short T2 due to large τc is what sets the size limit for protein NMR. Large proteins will simply yield so wide lines that they are not visible above the experimental noise.
6.3 The nuclear Overhauser effect It turns out that T1 relaxation and also the equilibrium magnetization for one nucleus is
dependent on the spin state, i.e. ‘up’ or ‘down’ of neighboring spins. The effect is called the nuclear Overhauser effect (NOE) and is due to dipolar cross-relaxation. Just like 1/T1 and 1/T2, dipolar cross relaxation, 𝜎𝐷𝐷, can expressed as a sum of spectral densities.
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𝜎𝐷𝐷 =𝑑2
46𝐽(𝜔𝑆 + 𝜔𝐼) − 𝐽(𝜔𝑆 + 𝜔𝐼) 6.7
The minus sign between the terms in the expression has important implications for the
manifestation of NOE for molecules of different size. It turns out that the effect can be positive, negative or vanish. We shall not explain the physics behind the NOE in more detail here but we note the NOE can be used to characterize molecular dynamics and that it can be used to transfer magnetization between nuclei in NOESY experiments (explained in Chapter 7). From Equation 6.5 it is clear that dipolar cross-relaxation and thus the NOE is strongly distance dependent. For protons the maximum internuclear distance for the appearance of the NOE is 5 Å. For most other nuclei it is significantly less. The amount of nuclear Overhauser enhancement can be reported in various ways. One common way is to define the parameter
𝜂 =𝐼𝑠𝑎𝑡 − 𝐼0
𝐼0
where 𝐼𝑠𝑎𝑡 is the intensity when the dipolar coupled nucleus is saturated and 𝐼0 is the intensity in a normal experiment. Using this definition 𝜂 can range from –1 to +0.5 if the two nuclei are of the same species. In applications involving heteronuclear dipolar coupled nuclei in proteins, usually another parameter, simply called the NOE, is commonly reported
𝑁𝑂𝐸 =𝐼𝑠𝑎𝑡𝐼0
This parameter is widely used to characterize fast internal motions (see below).
6.4 Effect of internal motions
6.4.1 The model-free formalism Molecules are not static entities that tumble as rigid bodies. Rather, their structures
fluctuate on many time-scales. It turns out that motions that are faster than overall tumbling affects the relaxation rates. While the expressions for 1/T1 and 1/T2 remain the same, a spectral density function that takes internal motions into account can be written
𝐽(𝜔) =
25⋅
𝑆2𝜏𝑐1 + 𝜔2𝜏𝑐2
+2(1 − 𝑆2)𝜏𝑒
1 + 𝜔2𝜏𝑒2 6.8
where S2 is the order parameter that describes how restricted the fast bond vector are (1 for rigid and 0 for completely flexible) and τe is approximately equal to the correlation time for these fast
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motions. For proteins τc is on the order of nanoseconds and τe is on the order of picoseconds. If we want to characterize the three parameters τc, τe and S2 we need three different experiments. The first two are measurements of T1 and T2 and the third is provided by measurement of NOE. This way of modeling molecular motion is called the model-free formalism.
Figure 6.2. Molecular motions according to the model-free formalism. The molecule is pictured as an ellipsoid, reorienting with correlation time 𝜏𝑐. Also shown are two nuclei, A and B, that are relaxed by nuclei that can diffuse freely on the perimeter of their respective cones. The nuclei A and B are typically 15N that are relaxed by their attached protons. In this case relaxation depends on global motion as well as the local motion of the attached protons.
The appearance of the spectral density should not be surprising. If the molecule has internal dynamics, these too must lead to relaxation and their time-scale, expressed as τe, and their magnitude, described by S2, must be important.
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Figure 6.3. The relaxation rates R1, R2, and the heteronuclear NOE measured for a thioredoxin domain from the protein Grx3. From these measurements the order parameter S2 was calculated. The order parameters show that the thioredoxin domain is very rigid, except for the termini and a short region around residue 40.
6.4.2 Spectral density mapping The model-free formalism is only valid when the internal motions are statistically independent from the overall tumbling. If this condition is violated one must resort to other methods for describing internal motions. One such approach is spectral density mapping, where the goal is not calculation of order parameters and correlation times for the internal motions but rather to determine the spectral density function. The method relies on the fact that the relaxation rates are linear combinations of spectral density function, implying that if enough relaxation have been measured, the spectral density functions can be determined by linear algebra methods.
6.5 Relaxation mechanisms In the explanation of the origins of relaxation the dipolar coupling was used as an
example of a spin coupling leading to relaxation. Since the dipolar coupling is not the only spin coupling there are also more relaxation mechanisms since all couplings that are modulated by molecular motions cause relaxation.
6.5.1 Chemical shift anisotropy For the most important nuclei in proteins 1H, 13C and 15N, the other important relaxation
mechanism is the chemical shift anisotropy (CSA). Contrary to what you might think, the chemical shift is not the same in all directions but depends on molecular orientation so that it changes as the molecule tumbles. The reason why we talk about a single chemical shift is that the molecule normally reorients so rapidly that we only see the average value. Just as for the dipolar coupling, ⟨𝐵0,𝐶𝑆𝐴⟩ = 0 but ⟨𝐵0,𝐶𝑆𝐴
2 ⟩ ≠ 0 and relaxation results. Especially for 15N and certain positions of 13C the chemical shift anisotropy is large and this relaxation mechanism may be as large as or larger than dipolar relaxation. Since the resonance frequency, and thus the chemical shift anisotropy in frequency units depends on the external magnetic field whereas the dipolar coupling does not it follows that the importance of the chemical shift anisotropy is especially important at high magnetic field strengths.
6.5.2 Quadrupolar coupling For nuclei with spin 𝐼 > ½ the quadrupolar coupling, which is huge compared to the
dipolar and CSA couplings, dominates the relaxation behavior to the extent that other mechanisms need not be considered. The only quadrupolar nucleus that occasionally is used in
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protein NMR spectroscopy is 2H. The origin of the quadrupolar coupling is electrical rather than magnetic and results from the non-vanishing electric quadrupole of spin 𝐼 > ½ nuclei with the electric field gradient.
6.5.3 Paramagnetic relaxation enhancement For molecules that contain unpaired electrons we must also consider parametngic relaxation enhancements (PRE). The theoretical framework for PREs is very similar to that for the normal dipolar coupling. The difference is that the dipole a nucleus interacts with this time is an electron rather than a proton and since the magnetic moment of the latter is huge so is the relaxation rate. The PRE is also proportional to the inverse sixth power of the distance between the nucleus and the proton and the PRE can thus be used as a measure of their speration. Importantly, this allows longer range distance restraints than are available for relaxation induced by the interaction with a nuclear dipole.
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6.6 Synopsis In this chapter you have learnt to:
• Understand the principal difference between T1 and T2 relaxation
• Know what causes relaxation and that relaxation is coupled to molecular motions
• Understand why (dipolar) relaxation is caused nearby nuclei
• Know that relaxation rates (1/T1 and 1/T2) can be calculated as linear combinations of spectral density functions
• Know the implications for different values of T1 and T2.
• Qualitatively describe the nuclear Overhauser effect and what it can used for
• Understand what information that can be extracted from measurements of T1, T2 and the NOE according to the model-free formalism
• Describe what can be learnt about proteins by the above measurements and the model-
free formalism
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6.7 Problems 6.1 Two of the most common nuclei in proteins are 13C and 15N and it is interesting to
calculate their respective relaxation time constants. In a particular case we have for a 13C nucleus <∆B2>=1.28⋅1010 T2 and for a 15N nucleus <∆B2>=3.10⋅109 T2. Both nuclei have S2=1. The protein is at a magnetic field of 11.7 T at 310 K and is tumbling with τc = 5 ns. Calculate 1/T2 for both nuclei and explain a consequence of the different relaxation rates for the NMR signals in the two cases!
6.2 Describe what the relaxation times T1 and T2 mean! 6.3 What causes nuclear spin relaxation? 6.4 A protein is tumbling with a correlation time of τc = 5 ns at a magnetic field strength of
14.1 T. One of the 15N nuclei of the protein has an order parameter of S2 = 1 while another has the order parameter S2 = 0.5 end τe = 0 ps. The average of the square of stochastic fluctuations of the magnetic field, <∆B2>, are equal in the two cases. For which nucleus does the NMR signal decay most rapidly?
6.5 Briefly describe the nuclear Overhauser effect! 6.6 State some applications of the nuclear Overhauser effect! 6.7 Show that the function σ(ω) = 6J(2ω) – J(0) is zero for a particular value of τc! What are
the implications for η (nuclear Overhauser enhancement) for small, medium size and large molecules, respectively?
6.8 How is protein dynamics modeled according to the model-free formalism? 6.9 A protein is tumbling with a correlation time of τc = 5 ns at a magnetic field strength of
18.8 T. An 15N nucleus in one of the residues has the order parameter S2 = 1. You are performing an NMR experiment that has to be repeated many times. You want to wait 5×T1 before repeating the experiment. For how long do you have to wait? What is the purpose of waiting for so long?
6.10 Show that 1/𝑇1 = 1/𝑇2 for a small molecule whereas 1/𝑇2 ≫ 1/𝑇1 for a large molecule!
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7. Multi-dimensional NMR There are several reasons doing NMR experiments in two or more dimensions. The
intuitive motivation is that the signals usually are better resolved in multidimensional experiments than they would be in a one dimensional experiment but perhaps the most important aspect is related to why they are spread out in two dimensions at all and that is because the nuclei giving rise to off-diagonal peaks must be correlated in some way. Multidimensional NMR thus provides correlations between nuclei and depending on how the experiment is carried out, different correlations can be detected. In multidimensional experiments several aspects not present in one-dimensional experiments need to be considered. In this chapter these extra concerns will be presented and in addition a number of principally different experiments in two and three dimensions will be presented and discussed.
7.1 Principles of two-dimensional NMR spectroscopy For simplicity the general ideas will be exemplified with two-dimensional experiments
but the generalization to more dimensions is straightforward and should be easy to do. In NMR textbooks a two dimensional experiment is often described by four different building blocks that will be introduced here.
7.1.1 Building blocks of two-dimensional experiments 1. Preparation
In two-dimensional experiments we want to correlate two different nuclei and we thus want to know their chemical shifts in the respective dimensions. We use the preparation period to transfer magnetization to the nucleus that is to be detected in the first dimension. A fair question is why we include this block all together. Why not just start the experiment on the nucleus of interest and let the preparation period just be a 90° pulse on that nucleus? The reason for this is sensitivity. If the nucleus of interest is not a proton but of species X we will gain a factor of γ1H/γX in sensitivity from the ratio of magnetic moments between the two nuclei and another factor of γ1H/γX because the difference in populations between ‘spin-up’ and ‘spin-down’, neglecting relaxation losses during the transfer period. Unless the nucleus is a proton we are thus well advised to include this element. The usual way preparation is performed is to transfer magnetization one or several steps with the aid of the scalar coupling. For instance, if the amide nitrogen in a protein is to be detected in the indirect dimension preparation consists of the element (90°)x,H τ 180°x,H+180°x,N τ 90°)y,H + (90°)x,N, where τ is adjusted to 1/4J. Since the nitrogen is in the transverse plane we can detect its chemical shift. 2. Evolution
During this period the chemical shift of the nucleus in the indirect dimension is detected. In its most simple form this period takes the form of a delay, sandwiched between two 90° pulses that brings the magnetization to and from the transverse plane respectively. The experiment is
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repeated with this delay successively incremented by 1/sw, where sw/2 is the highest frequency component relative to the radio-frequency carrier that is considered. In reality, a simple delay will often not suffice. Consider for instance the usual case where the nucleus of interest is coupled to another one of a different species. Unless a 180° pulse is applied on that nucleus halfway through the delay the scalar coupling will lead to doublets in that dimension which usually is not desired. In the case of scalar coupling to a nucleus of the same species this method of mitigating the effect of the scalar coupling can of course not be used. Instead one has to resort to a method called constant time evolution that will be described later. 3. Mixing
During the mixing period magnetization is transferred to the nucleus on which the signal is detected. It is thus during this period that correlations between the two nuclei are established and depending on the nature of the mixing period different correlations can be seen. For instance if it comprises the weak scalar coupling, one can see correlations between nuclei that are directly scalar coupled to each other. This is the case for instance in COSY (correlated spectroscopy) and HSQC (heteronuclear single quantum correlation spectroscopy) experiments. If it uses the strong scalar coupling one can see correlations between all atoms in the scalar coupled network such in the TOCSY (total correlation spectroscopy). A mode of mixing that does not use the scalar coupling but instead an indirect effect of the direct dipolar coupling, dipolar cross relaxation, can instead transfer magnetization through space so that correlations between atoms that are close in space can be established. This is the idea of the NOESY (nuclear Overhauser spectroscopy) experiment. 4. Detection
During the detection period the NMR signal is detected as an oscillating current. In order to determine the sense of the rotation of the magnetization also here both the cosine and the sine component needs to be detected. However, here this can be done in an electronically way, which is beyond the scope of the text, so there is no need to repeat the experiment to record each component. During detection the same complications as for the evolution persist, i.e. there will be scalar coupling evolution to coupled spins. In principle one could apply 180° pulses on the coupled nucleus between each of the sampled points but this is usually not the method of choice. Instead one resorts to a method called broadband decoupling, or decoupling for short. The idea is to apply a repeated sequence of composite 180° pulses at a much higher rate than the strength of the coupling. The effect is that the spin state of the coupled nucleus changes between ‘up’ and ‘down’ so fast that the detected nucleus only sees the average. The multiple structure that otherwise would arise is thus collapsed. For nuclei of the same species decoupling can usually not be applied and the multiplet structure persists. Sometimes, such as for NMR spectroscopy of small organic molecules, this can actually be an advantage, since the multiplet structure yields information on the structure of the molecule. Also for proteins, the splitting in the indirect dimension can sometimes be used to measure scalar couplings, such in the COSY experiment that will be described later.
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7.1.2 Evolution in the indirect dimension The specifics of the preparation and mixing periods will be covered when the various
experiments are discussed and the detection period will not be discussed further. We will instead focus on the evolution period in this section. For simplicity we first consider the case where the S nucleus is not (or at least only very weakly) scalar coupled to the I nucleus, the evolution period with flanking 90° pulses would look as in Figure 7.1.
Figure 7.1. Example of an indirect evolution period. Before the first 90° pulse magnetization is aligned with the z-axis and is then flipped to the x-axis. It will rotate around the z-axis for a period t1 and the final pulse returns what is left of magnetization along the x-axis to the z-axis.
Assuming that the magnetization after the first pulse is Sx, it will evolve as follows during the period t1
Sx Sxcos(2πνSt1) + Sysin(2πνS t1) 7.1
We thus have all we know to record the chemical shift and since we have the cosine as well as the sine component we can even record the sense of rotation, i.e. whether it is positive or negative relative to some reference frequency. However, note what happens after the next pulse which is used to rotate the magnetization back to the z-axis so that magnetization can be transferred to the I nucleus by means of the mixing period. The effect of this pulse is
Sxcos(2πνS t1) + Sysin(2πνS t1) Szcos(2πνS t1) + Sysin(2πνS t1) 7.2
We are thus only able to turn the cosine component to the z-axis and it seems like we
cannot reach our goal of detecting magnitude as well as the sign of the chemical shift. However, consider the modification to the first flanking pulse of the evolution block shown in Figure 7.2.
Figure 7.2. Example of an indirect evolution period. Before the first 90° pulse magnetization is aligned with the z-axis and is then flipped to the x-axis. It will rotate around the z-axis for a period t1 and the final pulse returns magnetization aligned with the y-axis to the z-axis. Note that for t1=0 no magnetization is aligned with the y-axis.
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Sxcos(2πνSt1) + Sysin(2πνS t1) Sxcos(2πνS t1) + Szsin(2πνS t1) 7.3
We have now instead detected the sine modulated component! Thus, in order to detect
both components in the indirect dimension one method is to record the experiment twice with different phases for one of the pulses flanking the evolution period and to store the results separately. During processing the two are treated as the cosine and sine components of a rotating, or alternatively the real and imaginary components of a complex, signal. This method of obtaining frequency discrimination or quadrature detection in the indirect dimension is called States detection, or sometimes States-Haberkorn detection, after the scientist(s) that first described the method. In fact it is more common to increment the phase of the first pulse by 90° but the net result is the same. Apparently States detection and similar schemes of obtaining quadrature detection is wasteful because as much as half the signal is discarded both when the real and the imaginary components are detected. Mark Rance and co-workers however showed that it is possible to retain both terms to gain a sensitivity enhancement of √2 for certain experiments if a more complex pulse sequence and a more elaborate phase cycling scheme is used. His method is called preservation of equivalent pathways (PEP) or simply sensitivity enhancement. Lewis Kay later modified the method to rather use pulsed field gradients with changing polarity and this is now the de facto standard way of recording most heteronuclear experiments. The details of sensitivity enhancement are beyond the scope of this text. To guarantee that all frequency components are reproduced at the correct spectral positions the signal must be sampled according to the Nyquist theorem that states that the sampling rate must be twice the period of the highest frequency that is considered. If these frequencies are in a window called sw, ranging from –sw/2 to sw/2, the sampling rate must thus be 1/sw. Frequencies outside of this window will be ’folded’ into (the wrong position of) the spectrum by a phenomenon called aliasing. Aliasing is not unique to NMR spectroscopy. You most likely already have seen one example of it in old western movies where carriage wheels apparently rotate backwards despite the fact that the carriage itself most definitely moves forward. The reason for aliasing in this case is that the sampling rate of the camera is not fast enough to correctly capture the fast rotation of the wheels. Sometimes aliasing is deliberately used in NMR to increase the spectral resolution for a given experimental time. A subtlety that needs to be taken into account when writing NMR pulse sequences and processing data is that there will be evolution also during the flanking pulses and if the evolution performed as above it can be shown that
𝑡1,𝑡𝑟𝑢𝑒 = 𝑡1 + 4 ⋅ 𝑝𝑤 𝜋⁄ 7.4
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where pw is the duration of the flanking pulse. Clearly a delay of t1,true = 0 can never be achieved this way and this point must back calculated from the other points using linear prediction. It is however not necessary to start sampling at t1,true = 0. The other option is to start at t1,true = 1/(2sw). As long as 1/2sw > 4pw/π, also the first point is correctly sampled in this case. Yet another option is to modify the evolution block to
Figure 7.3. A spin echo makes it possible to sample at t1=0 in the indirect dimension. where a fixed delay of 2τ and 180° pulse has been added. It should be easy to see that evolution during the second 90° pulse cancels that during the first pulse so that in this case t1,true = t1 and one can start sampling at time zero.
If the spin of interest, S, is scalar coupled to another spin, I, we will have scalar coupling evolution as well as chemical shift evolution during t1. This would lead to a multiplet structure in this dimension as well as to reduced sensitivity and should be avoided if possible. If we assume that I and S are of different species so that they can be manipulated separately by radio frequency pulses, we have the possibility to modify the evolution period to
Figure 7.4. Refocusing of scalar coupling to spin I in evolution period of spin S can be achieved by application of an I-selective 180° pulse in the middle of the period.
According to a result from the previous chapter the net effect of these pulses and delays
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is to refocus the scalar coupling between I and S when the chemical shift of spin S evolves.
7.2 Examples of homonuclear two-dimensional NMR experiments To exemplify the concepts discussed so far I will now describe three two-dimensional
experiments in detail. The experiments in question are COSY, TOCSY and HSQC. They all use the scalar coupling to transfer magnetization in between nuclei in the two dimensions. The first two experiments are homonuclear whereas the third is heteronuclear. The strong scalar coupling is used for the TOCSY experiment and the weak scalar coupling for the others.
7.2.1 COSY COSY is an acronym for correlated spectroscopy and was the first two-dimensional
experiment that was used to correlate to nuclei. The weak scalar coupling is used to transfer magnetization between the nuclei that are correlated. The experiment is very economical, it only employs two pulses and during t1 as well as t2 both the chemical shift and the scalar coupling evolves. The resulting spectrum thus contains doublets in both dimensions. The COSY pulse sequence is shown in Figure 7.6.
Figure 7.6. Pulse sequence for the COSY experiment. Simultaneous chemical shift evolution and magnetization occurs during t1. The signal is detected during t2.
A weakness with the COSY experiment is that it can only correlate nuclei that are directly scalar coupled. Consider for instance the case where a nucleus I is coupled to nucleus M. Nucleus M, but not nucleus I, is in turn coupled to nucleus S. One will thus not get a cross-peak between I and S. In the TOCSY experiment, described below, such cross-peaks do appear.
7.2.2 TOCSY Contrary to the case of COSY the TOCSY (total correlation spectroscopy) experiment
uses the strong scalar coupling to transfer magnetization. It turns out that this coupling can transfer magnetization between all nuclei in a spin system. Thus, cross-peaks appear also for nuclei that are not directly scalar coupled. However, all nuclei must be in the same spin system to be correlated.
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Figure 7.7. Pulse sequence for the TOCSY experiment. Magnetization transfer occurs between the last two 90° pulses. The block marked DIPSI-2 is an isotropic mixing sequence, designed to let all spins experience the strong coupling.
A very common application for the TOCSY experiment in protein NMR is to establish the identity of amino acid residues. This works well since amino acid residues with only a few exceptions comprise a spin system and often have unique chemical shift signatures (c.f. Figure 7.9).
7.2.3 NOESY NOESY (nuclear Overhauser effect spectroscopy) differs from the COSY and TOCSY
experiments in that the dipolar rather than scalar coupling is used to transfer magnetization. The dipolar coupling is a through space rather through bond coupling implying that cross peaks only appear for nuclei that are close in space. For two protons the practical upper limit for this is around 5 Å and for most other nuclei significantly shorter.
Figure 7.8. Pulse sequence for the NOESY experiment. Magnetization transfer occurs between the last two 90° pulses. Note the similarity with the TOCSY experiment! In Figure 7.9 the information content of the COSY, TOCSY and NOESY experiments is shown. A common strategy when assigning the resonances to different protein nuclei is to record these
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experiments to identify spin systems and too connect them sequentially.
Figure 7.9. The information content in COSY, TOCSY and NOESY spectra, illustrated for three residues Glu, Ala and Leu, colored red, green and orange, respectively, in a protein that not necessarily forms a contiguous tripeptide.
7.3 Heteronuclear two-dimensional experiments There is nothing that prohibits transferring magnetization between nuclei of different
species, provided that they are coupled to each other. In fact, these experiments are often easier to perform since the magnetization of nuclei resonating at vastly different frequencies can be manipulated separately. It is for instance possible to apply pulses on one nuclei at a time and to perform decoupling on a specific nucleus. Nuclei other than protons are called heteronuclei. Therefore these experiments are referred to as heteronuclear experiments.
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7.3.1 Heteronuclear single quantum correlation spectroscopy The most important heteronuclear experiment is without question the heteronuclear single quantum correlation (HSQC) experiment. It correlates a particular heteronucleus, usually 15N, with its attached proton and can be thought of as a heteronuclear version of the COSY experiment. The HSQC is also instructive to use as an example of many of the principles that apply to two-dimensional NMR spectroscopy. The flow of magnetization in the HSQC experiment is 1H 15N (t1) 1H (t2). By performing the experiment this and not for instance 15N (t1) 1H (t2) or 1H (t1) 15N (t2), sensitivity is improved since the larger polarization of proton yields a larger magnetization vector and its larger resonance frequency induces a larger current in the detection coil.
The pulse sequence of the HSQC is shown in Figure 7.10. The block between the first pair of 90° pulses transfers magnetization 1H to 15N. You should be able to convince yourself that the scalar coupling between the two nuclei is active for the entire period 2τ while the chemical shift is refocused. The next block is used to record the 15N chemical shift. We thus have it evolve for a period t1. During this period it is undesired to have any scalar coupling evolution. The pulse on proton at t1/2 is used to refocus the chemical shift. Magnetization is then transferred back to proton using a similar block as at the start of the sequence. Finally, the signal is detected during t2. Once again, it is not desired to let the scalar coupling evolve and decoupling on 15N is therefore applied.
Figure 7.10. Basic pulse sequence for the HSQC experiment. The different building blocks are highlighted in yellow. Narrow and wide rectangles depict 90° and 180° pulses, respectively. To detect the sine modulated component during t1, the phase of the pulse on 15N proceeding the evolution period should be changed from x to y.
The popularity of the HSQC is due to many reasons. Some of these are that it is a sensitive experiment that can be performed within minutes, it yields simple spectra even for large proteins and the positions of the signals are sensitive to changes in sample conditions or protein status. It can thus be used to verify that a protein is well folded that it is not degraded. It is also useful for characterization of ligand binding since it possible to measure the dissociation constant
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and to determine where binding takes place. Finally, the experiment can be used as a building block in other experiments such as relaxation experiments and three-dimensional experiments.
7.4.1 TROSY NMR spectroscopy is an inherently insensitive and for large molecules rapid transverse relaxation the sensitivity is further reduced. A way of increasing the sensitivity is thus to reduce the transverse relaxation rate. This can be done in various ways. The simplest of those is perhaps to increase the tumbling rate of the molecule by raising the temperature. In fact, protein NMR experiments are often performed at elevated (compared to physiological) temperature for this purpose. Of course a protein is only well-folded in a limited temperature range so there is a limit. However, to raise the limit one can choose a protein from a thermophilic organism. Another way is to reduce the dipolar contribution to relaxation by replacing all protons except the ones used for excitation and detection by deuterons. This is often combined with the TROSY (transverse relaxation optimized spectroscopy) approach where the slow relaxing component of doublets is selected (Figure 7.10). For small molecules this leads to a 75% reduction in sensitivity since three of the four components of the multiplet are discarded. For large molecules, on the other, it leads to increased sensitivity since then only the slowest relaxing component contributes significantly to the signal. TROSY works because of mutual cancellation of two relaxation effects, the dipolar and chemical shift anisotropy mechanisms. Since especially the latter is field dependent, TROSY works best at particular magnetic fields. For 15N-1H spin pairs optimal results are achieved at fields of 800–1000 MHz. For optimal results it is also necessary to reduce relaxation pathways that are not cancelled as much as possible. This is achieved by perdeuteration where protons that not are probed are replaced by deuterons.
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Figure 7.10. Illustration of the TROSY effect. A) In the non-decoupled spectrum the components of the doublet will be visible in both dimensions. The different components will have different line-widths that differ greatly for large proteins. B) In the decoupled spectrum the doublet is collapsed. C) In the TROSY experiment the slow-relaxing (narrow) component is selected. This leads to dramatic improvement of sensitivity and resolution for large proteins. D) TROSY spectrum of the homodimer of the S677/680A mutant of the kinase domain from EphB2 acquired at 800 MHz. The molecular weight is 70 kDa.
TROSY type experiments can be used instead of the regular HSQC as shown in Figure 7.10D but are also readily incorporated into three dimensional pulse sequences.
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7.4 Additional concerns for three-dimensional experiments Most concepts for two-dimensional experiments can be straightforwardly generalized to
three or more dimensions. The main difference for experiments that correlate three different nuclei is that there must be two evolution periods and additional mixing periods. The first three-dimensional experiments that were commonly used were the combination of simple homonuclear two-dimensional experiments and the HSQC that lead to TOCSY-HSQC and NOESY-HSQC experiments. In these experiment one first records the chemical shift of a particular proton during t1, then transfers magnetization to an amide proton using the strong scalar coupling or dipolar cross relaxation during the first mixing period. Then a normal HSQC is recorded with the 15N chemical shift recorded during t2 and the amide proton chemical shift detected during t3. By spreading out the spectrum in three dimensions, signal overlap is greatly reduced the combination of these two experiments are routinely used to assign the resonances of small proteins.
Another example of three-dimensional experiments are the triple resonance experiments that correlate nuclei of three different species, almost always 13C, 15N and 1H. These experiments can be viewed as generalizations of the HSQC to a third dimension. The flow of magnetization is 1H13C (t1) 15N (t2) 1H (t3). There are a large number of these experiments that differ by which type(s) of 13C that are detected. Some common examples include the HNCA that correlates 15N and 1H of residue i with 13Cα of residues i and i-1, the HNCO that correlates 15N and 1H of residue i-1 with 13CO and the HNCACB that correlates 15N and 1H of residue i with 13Cα and 13Cβ of residues i and i-1. The experiments that correlate 15N and 1H of residue i with 13C of both residues i and i-1 are particularly useful for obtaining resonance assignments of large proteins.
The extra transfer and evolution periods in three dimensional experiments lead to some additional concerns that will be discussed briefly in this section.
7.4.1 Sensitivity The longer the experiment, the more magnetization will be lost due to relaxation prior to
the detection period. Three-dimensional experiments by definition include three extra blocks that contributes to longer experimental times. The obvious one is of course the extra evolution period but in addition you also need a transfer step to that nucleus and one from it. Especially if the coupling constants are small, remember a transfer step is on the order of 1/2J, the losses can be very significant and in practice make certain experiments impossible. If we for instance compare an HSQC with an HNCO which is the most sensitive triple resonance experiment, the former only has transfer periods to and from 15N using the 92 Hz coupling constant. The minimum duration of this experiment is thus 2×1/2JHN = 1/JHN = 10 ms. The HNCO also needs these transfer periods but in addition requires a transfer step to and from 13CO using the 15 Hz coupling. The duration of these extra steps are thus 2×1/2JNC = 1/JNC = 67 ms. For a small
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protein with a relaxation rate of 5 s-1, the intensity would be reduced by a factor exp(-5×0.067) = 0.72, which is very tolerable but for a larger protein with a relaxation rate of 20 s-1 the intensity would be reduced by a factor exp(-20×0.067) = 0.26, which is quite substantial. For an HNCA the transfer time required is around 100 ms so that the reduction compared to an HSQC is 0.60 and 0.14 for the two cases respectively and to make matters worse the signal is split into two peaks in the carbon dimension, making the sensitivity of the individual components less than 10% of an HSQC. Note that this only applies to the first increment in t1. At later increments the reduction will be even larger.
To not lose more signal than necessary it is imperative to reduce the duration of the pulse sequence as much as possible and one way of doing this is to use the same block for two purposes simultaneously. It turns out that it is possible to transfer magnetization and frequency label a nucleus at the same time. We consider the following block and start with a density operator equal to 2NzCz.
Figure 7.11. Combination of chemical shift evolution and magnetization transfer. The period for evolution of the scalar coupling is always 2τ whereas the time for chemical shift evolution increases from zero to maximally 2τ.
We see that the block can be described by as 1) a 90° pulse on N to create the
operator 2NxCz, 2) chemical shift evolution of this operator for a period t1, 3) scalar coupling evolution for a period τ, 4) simultaneous 180°x pulses on the N and C spins and finally 5) a 90° pulse on the C spin. If we set τ =1/(4J) we get
𝜎(𝑡) = 𝑁𝑧 cos(𝜔𝑁𝑡1) + 𝑁𝑧 sin(𝜔𝑁𝑡1) 7.5
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Apparently we have succeeded in getting complete transfer of magnetization and frequency labeling in a period τ rather than τ+t1. You should convince yourself that constant time evolution only reduces the duration of the pulse sequence if applied to the second indirect dimension. For the first indirect dimension it would rather increase by an amount τ-t1. The potential drawback with simultaneous evolution and mixing is resolution. For each increment t1 is increased by 1/sw and obviously this can only go on as long as τ-t1/2 ≥ 0 and thus one must sometimes settle for fewer increments than desired.
7.4.2 Experimental time A two dimensional experiment has to be recorded for 2×n1 times as long as a one-dimensional experiment, where n1 is the number of increments in the indirect dimension and the factor of two is due to that the cosine and sine components in the indirect dimension cannot be detected simultaneously. This means that if the one-dimensional experiment takes 8 s to perform, a two-dimensional experiment with n1=64 requires 17 minutes. Three-dimensional experiment accordingly require 2×n1×2×n2 as much time and with n1=64, n2=64 the experimental time increases to 36 h. Since sensitivity is reduced one often wants to record the basic one-dimensional experiment for longer, which means that the total time increases even more. It is thus obvious that reduction of the measurement time as much as possible is imperative. The usual way to reduce experimental time is to sacrifice resolution. That is to reduce n1 or n2 or both. If for example n1 is set to 32 and n2 to 48 the experiment that required 36 h can be recorded in less than 14 h. Of course, in doing so we get less resolution and it might be harder to interpret the spectrum. An alternative way of reducing experimental time without this but perhaps with other sacrifices is to not reduce n1 or n2 but to only record a selected number of data points and reconstructing the ones not recorded. This is referred to as non uniform sampling and using this strategy it may be feasible to record a three-dimensional spectrum in a few hours and even higher dimensional spectra in a reasonable time.
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Figure 7.11. Illustration of uniform (left) and non-uniform (right) sampling in t1 and t2 with n1 = n2 = 16. In the example of non-uniform sampling, only 25% of the data points are sampled. The experiment can thus be recorded in a quarter of the time and still yield the same resolution.
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7.5 Synopsis In this chapter you have learnt to:
• Understand why two-dimensional NMR is needed for large molecules
• Describe how a two-dimensional NMR experiment is performed
• Know what is needed to record all frequencies properly
• Describe the principles of the experiments COSY, TOCSY and NOESY
• Describe the information content in the above experiments
• Describe the principle of an HSQC experiment
• State several reasons that makes the HSQC such an important experiment
• Describe how the NMR signals corresponding to the protein backbone can be assigned to specific nuclei
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7.6 Problems 7.1 Draw a sketch that shows that high frequency signals are missed if the sampling
rate is too slow when the signal is detected! 7.2 How can one distinguish between positive and negative (relative to the spectral
midpoint) frequencies? Does it make sense to use complex numbers to describe the NMR signal?
7.3 Describe the principles of a two-dimensional experiment! 7.4 How can one work out the resonance frequencies in the indirect (t1) dimension
from a series of amplitude modulated one-dimensional spectra? 7.5 Describe COSY, TOCSY and NOESY with respect to a) mechanism of transfer of
magnetization and b) information content! 7.6 Explain why the HSQC is such a useful experiment! 7.7 Describe two principally different ways of obtaining sequence-specific resonance
assignments for a protein!
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8. Characterization of proteins using NMR spectroscopy In the previous chapters, fundamental principles of NMR have been presented
along with parameters that can be measured and what they can be used for. The purpose of this chapter is to summarize applications presented in the previous chapters. Many of the figures you already have seen will be reused but may be discussed in a different context. The chapter will start with requirements for recording NMR spectra and will continue with experiments to characterize sample integrity, structure, dynamics and interactions.
8.1 Requirements for protein NMR spectroscopy
8.1.1 Instrumentation The NMR spectrometer consists of two main parts, the magnet and the console.
The magnet is typically a superconducting electromagnet. Once it is charged, the current flows more or less forever. The magnet can thus be viewed as an electromagnet that is always on. For superconduction to be possible the magnet needs to be extremely cold. It is cooled by liquid helium to a temperature of 4.2 K (-269 °C). What you see is the ‘thermos flask’ that contains the liquid helium and surrounding liquid nitrogen that is used to reduce the boil off rate of helium. NMR experiments are very sensitive to even small vibrations. For this reason, the magnet is commonly suspended by pressurized air. A typical magnetic field for NMR is 10-20 T, which is more than 100,000 times the earth magnetic field of ~0.00005 T. At short distances, credit cards, cell phones and watches may be damaged; magnetic objects can fly towards the magnet and you are thus not allowed to bring such objects close to the magnet. An even bigger concern is if you have a pacemaker or certain types of metallic implants. You can then get seriously injured or even killed from getting too close to the magnet.
The console harbors all electronics needed to perform experiments. This includes devices for applying pulses of electromagnetic radiation (in the radio frequency range) with frequencies that match the proton, 13C, 15N and deuterium resonance frequencies. It also includes devices for receiving the NMR signal and processing it. In addition the console manages the temperature during the experiment, contains equipment for intentionally changing the magnetic field among other things.
The probe is the connection between the magnet and the console. The sample is inserted into the probe and it contains the coils that serve the dual task of applying the pulses and receiving the signal. There are two types of probes: Room temperature probes and cryogenically cooled ones. Cryogenically cooled probes have superior sensitivity but need to be cooled to 25 K to work and also need to be kept at pressures close to vacuum. The reason for the increased sensitivity of cryoprobes is that electrical noise is decreased
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at low temperatures.
Figure 8.1. Schematic view of a superconducting magnet suitable for NMR spectroscopy. In order to be super conducting the solenoid has to be cooled to approximately 4.2 K. The bulk of the magnet thus consists of a compartment to hold the liquid helium used to cool the magnet and a surrounding compartment of liquid nitrogen used to reduce the rate of the helium boil off.
In addition to a high field NMR spectrometer you will also need to have access to a laboratory with capabilities for expression and purification of isotopically enriched proteins.
8.1.1 Protein samples NMR is a notoriously insensitive technique, which puts very strict demands on
which samples that can be used. Specifically, highly concentrated protein solutions are needed. Optimally you want concentrations on the order of 1 mM, corresponding to several mg/ml. To put this into context it is instructive to compare with the requirements for other techniques. For instance in circular dichroism spectroscopy one hundred fold lower concentrations can be used and in fluorescence spectroscopy even less is enough. The protein must also be soluble for extended periods of time at these concentrations. This effectively narrows down the range of proteins that can be studies with NMR spectroscopy. Specifically, membrane proteins are not compatible with NMR spectroscopy unless special precautions are taken.
The large size of proteins effectively prohibits analysis of one dimensional NMR
spectra since the signals blend into one another. It is possible to use the proton chemical shift for both dimensions of a two dimensional experiment. In a previous chapter we have for instance presented the COSY, TOCSY and NOESY experiments. In practice, they are only feasible for the smallest proteins and even these are more easily studied if the proteins are isotopically labeled with 15N and sometimes 13C. The reason why isotopical
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labeling is so beneficial is that it opens up the possibility for other classes of experiments that both are more sensitive and more easily interpreted. Isotopic labeling is commonly achieved by overexpressing the protein in a well-defined medium that is supplemented with an isotopically labeled nitrogen and/or carbon source. A common growth medium is called M9 and comprises 6 g/l Na2HPO4, 3 g/l KH2PO4, 0.5 g/l NaCl, 1 mM MgSO4, 0.1 mM CaCl2, 0.5 g/l NH4Cl, 3 g/l glucose, vitamins and suitable antibiotics. Note that the use of M9 medium enables control of which isotopes of nitrogen and carbon that will be incorporated into the protein. Use 15NH4Cl and the protein will be 15N labeled and use [13C6]-glucose for it to be 13C labeled.
In order to only measure on the protein of interest it is of course necessary to purify it once it is expressed. Proteins are typically purified by passing them over columns that have different affinities for different proteins. Examples include immobilized metal ion affinity chromatography (IMAC), ion exchange chromatography and size exclusion chromatography (gel filtration). The purified protein is subsequently concentrated and dissolved in a suitable buffer. To be able to keep the magnetic field in the experiment constant, 5–10% D2O is typically added.
8.2 Setting up NMR experiments For NMR spectroscopy of small organic molecules, the sensitivity and resolution
is normally so good that most of the steps described below can be omitted. This is not advisable for protein applications and these steps that take from minutes to a few hours should be completed.
8.2.1 Setting the temperature The chemical shifts and the line shape of the signals are strongly dependent on
temperature. Furthermore, the structure of a protein might change with temperature. Therefore the sample temperature is set to a predefined value that is regulated to within ±0.1 °C. The samle temperature should not vary during the course of the experiment.
8.2.2 Locking In the introduction it was stated that the current that generates the magnetic field
goes on indefinitely. This is almost true but in reality it decreases slightly over time and if this drift was not compensated for, the magnetic field at the start and end of the experiment would differ and spectra of low quality would result. To remedy this we detect how much the field decays and compensate by increasing the current in a small electromagnet so that the total field is always the same. To do this we continuously perform an additional NMR experiment on deuterium and the current in the electromagnet is adjusted so that the deuterium frequency remains constant. This process
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is called locking.
8.2.3 Shimming A powerful magnetic field is not enough for a successful NMR experiment, it is
also important that the magnetic field is homogenous across the sample volume. If it was not, nuclei in one part of the tube would experience a different field than nuclei in another part of the tube with different resonance frequencies as a result. It is very hard to obtain sufficient homogeneity in the superconducting magnet and it is thus surrounded with some twenty small electromagnets with different properties and by adjusting currents in these it is possible to get a homogeneity that is better than one part in one billion. The process is called shimming.
8.2.4 Tuning and matching Just like a radio that transmits and/or receives radiofrequency radiation, the coil in
the probe must be tuned correctly for efficient application of radio frequency pulses and detection of the signal. The probe is tuned so that the resonance frequency in circuit is the same as that of the radio frequency pulses. Furthermore, the impedance in the probe coil should be adjusted so that it matches the impedance in the spectrometer electronics to maximize sensitivity. The tuning and matching procedure is performed by adjusting two capacitors mounted close to the coil.
8.2.5 Calibration of frequencies and pulse widths When performing protein NMR you typically have 1 mM protein dissolved in
water. The concentration of water is thus 55 M and since there are two protons in one water molecule the total proton concentration in water is 110 M, thus 100,000 higher than of the molecule you are interested in. If NMR experiments were set up naively, the water signal would simply drench the relevant signals and special tricks are required to filter out water as well as possible. One procedure that facilitates this is to apply all proton RF-pulses exactly at the water resonance frequency. Since this frequency may vary slightly depending on sample conditions and temperature we need to do this every time we set up an NMR experiment. The way to calibrate the water frequency is irradiate water with a long, weak pulse and then detect the remaining water signal. If the long, weak pulse is exactly on resonance, the water signal is destroyed. The experiment is thus repeated with many different values of the irradiation frequency.
The maximum NMR signal is obtained if the pulse width is exactly 90°, i.e. the magnetization vector is rotated from the z-axis to the transverse plane. Additionally, many experiments require some pulses to be exactly 180°. We thus need to calibrate pulse widths as well. Contrary to intuition it is not so easy to see the maximum signal so
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we instead set up the experiment to look for zero (or as low as possible) signal. One way of calibrating the pulse is thus to look for a 360° pulse, i.e. one that rotates the magnetization from the z-axis all the way around to the z-axis again. The pulse width of the 90° pulse is obtained by dividing the length of this pulse by four.
Figure 8.2 A) Experiment for calibration of water frequency. The water signal is destroyed completely by a long weak pulse if it is at the correct frequency. B) Experiment for calibration of pulse width. The pulse width is varied until the signal almost disappears. This means that the 360° pulse width has been found. The 90° pulse width is of course a quarter as long. The reason why the 360° pulse is calibrated is that it is easier to see a minimum than a maximum and that return to equilibrium is more rapid.
8.3 Check of protein integrity, the HSQC experiment The importance of the HSQC experiment cannot be overstated. For good reasons
it is always the first experiment that is tried for a new protein. Using the experiment it is possible to check if the protein is folded or if it forms aggregates. If it does, the sample conditions are probably not right for NMR. If the protein is unfolded, and it is predicted to not be, the buffer may have to be changed, perhaps the pH or ionic strength need to be optimized. A protein that forms aggregates may be too concentrated and dilution or making sure that the protein is not too concentrated at any stage during purification is
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necessary. Since the HSQC is sensitive enough that it can be recorded within minutes, one is usually always run before other more complicated experiments to verify that the protein is intact.
8.4 Resonance assignments The most straightforward way of assigning many of the resonances in a protein is
to record a series of experiments called triple resonance experiments. The reason for this name is that they correlate nuclei of three different species, i.e. 1H, 15N and 13C. The various experiments of this class are named after which nuclei they correlate. Examples are HNCA (correlates 1HN, 15N and 13Cα), HNCO (correlates 1HN, 15N and 13CO) and HNCACB (correlates 1HN, 15N, 13Cα, 13Cβ). Certain experiments use one or more nuclei as ‘stepping stones’ for magnetization without recording their chemical shifts. These nuclei are then written within brackets. An example is CBCA(CO)NH (correlates 1H, 15N, 13Cα, 13Cβ and the magnetization flows via 13CO).
Many of the experiments work as follows: They start with magnetization on 1HN of residue i. Magnetization is then transferred to 15N of the same residue via the 92 Hz scalar coupling between the two nuclei. Magnetization can then be transferred to selected 13C nuclei of both the same (i) and the preceding residue (i-1) at the same time. The relevant 13C chemical shifts are then recorded during the period t1. Magnetization is then first transferred back to 15N of residue i and its chemical shift is recorded during t2 and finally to 1HN where the signal is detected during t3. We will thus get 13C nuclei for two residues correlated with every 1HN/15N and, importantly, every 13C should be both correlated with 1HN/15N of residue i and residue i-1. Usually the intra-residual correlation is stronger (a more intense peak) so in general one can tell if the correlation is intra- inter-residual. There are also some experiments that only records correlations with 13C of the preceding residue.
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Figure 8.3. Sequential resonance assignments from triple resonance experiments. The red and green circles are particular correlations to for instance 13Cα or 13CO. Red circles depict correlations with a 13C nucleus of the same residue and green circles to 13C of the preceding residue. A) ‘Strips’ with 13C correlations in no particular order. Resonance assignment involves ordering the ‘strips’ as shown in B).
8.5 Structure calculation Although the most precise protein structures are calculated by x-ray
crystallography and this method in addition enables structure calculations of very large systems, structure determination by NMR spectroscopy is still competitive for a number of applications. When solving structures by NMR, one never has a fully determined system, i.e. as many measured parameters as the degrees of freedom. One thus has to settle for determining a family of structures that fulfill all restraints. Usually a structure calculation is performed many times and the result is presented as the ensemble of the ten structures with the lowest energy.
8.5.1 Restraints that can be measured by NMR Distance restraints. The nuclear Overhauser effect (NOE) is highly distance dependent and thus provides a way to measure internuclear distances for protons that are closer than approximately 5 Å. The magnitude of the NOE between nuclei I and S to a first approximation depends on 𝑟𝐼𝑆−6. Since this is only an approximation and additionally the NOE cannot be quantified very accurately it is common to divide the NOEs into three classes corresponding to strong, medium and weak and assign distances 1.8–2.7 Å, 1.8–3.3 Å and 1.8–5 Å, respectively. The reason why the shortest distance always is kept at 1.8 Å is that internal mobility may weaken the NOEs. Dihedral angle restraints. There are several parameters that are sensitive to backbone
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dihedral angles. Examples include the chemical shifts of 13Cα, 13Cβ, 13CO and 1Hα as well as the scalar coupling between 1HN and 1Hα. Relative bond angle restraints. The dipolar coupling is dependent on direction so if it could be measured for, say, 15N-1HN one would obtain information on the relative angles between all 15N-1HN bond-vectors. As was discussed in Chapter 3 the dipolar coupling vanishes for molecules in solution but it is possible to partially align molecules and thus measure a scaled down form of the dipolar coupling called the residual dipolar coupling (RDC). There are several ways to partially align molecules including adding filamentous phages, bicelles or enclosing the sample in stretched or compressed gels. If one studies Equation 3.1 it is clear that more than one angle can give rise to the same dipolar coupling. To get unambiguous results it is thus necessary to measure RDCs for several dipolar coupled nuclei or to use more than one alignment medium.
8.5.2 NOE driven methods A necessary hypothesis is that the native state of a protein can be defined as the
state of lowest energy. Calculating the structure of a protein is thus equivalent of minimizing its energy. To do this we assign potentials that describe how the energy changes when distances and angles change. For instance, dihedral angles that are not in the allowed regions of the Ramachandran plot are assigned high energies and interatomic distances that result in atoms being on top of each other are also being penalized by high energies. Energies are also assigned to the measured parameters. A distance between two nuclei of more than 5 Å should result in a large energy if an NOE between the two are observed. Formally the structure is calculated by minimizing the equation
𝜒2 = 𝑆𝑐𝑎𝑙𝑐 − 𝑆𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 8.1
Unfortunately, minimization is a very difficult computational problem. While it is easy to find local minima, it is hard to verify that such a minimum corresponds to the global minimum. One possible way of calculating a protein structure would thus be to repeat the calculation from enough starting points so that all local minima are found to see which one is deepest. Since the number of local minima are too many this is however not feasible. Instead a technique called simulated annealing is used. Here, the structure calculation is first performed at high temperature, which means that the protein can escape out of shallow minima. When the temperature is lowered the chance of escape is gradually decreased and the protein descends into a deep minimum. It is however by no means clear that this corresponds to the global minimum so the process is repeated, perhaps 100 times, and the protein structure is represented by the ensemble of the ten lowest energy structures. The quality of the structures is gauged by checking that all
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dihedral angles and nuclear distances are within permissible ranges and compatible with experimental data.
8.5.3 Chemical shift driven methods Since the chemical shifts depend on structure it is possible to use them to provide
restraints for structure calculations. Several different approaches exist for this, one of which is CS-ROSETTA. The CS-ROSETTA method works as follows:
1. The protein is divided into fragments of a few amino acids each. 2. The sequence and chemical shifts of these fragments are compared with ones
present in a database. 3. If the same fragment is found in the database and also the chemical shifts match,
the secondary structure of the fragment should be very similar to the one in the database.
4. The fragments, using the determined dihedral angles from the database search, are assembled.
5. An energy minimization in 1000-10000 different ways to obtain this number of protein structures is performed.
6. The 10 structures with the lowest energy are used to represent the protein structure.
If the method worked these structures should be very similar. There are other methods that also exclusively use chemical shift restraints but more resemble the traditional methods. Figure 8.4. Comparison of protein structures determined using different methods. A) The structure of a thioredoxin domain from the protein Grx3 calculated using NOEs. B) The structure of the same protein calculated using CS-ROSETTA. C) Overlay of the two structures. The pairwise root-mean square-deviation between the two structures is 0.6 Å.
8.6 Interactions A common point of interest is whether a protein is interacting with something.
This something can be various things, for instance itself (dimerization), another protein, an organic molecule or an ion. We refer to all these interactions as ligand binding and NMR is well suited to characterize these events under a large range of conditions. Schematically we write binding as
𝑃 + 𝐿 𝑃𝐿 8.2
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where 𝑃 is the protein, 𝐿 is the ligand and 𝑃𝐿 is the complex. It is important to realize that binding is a dynamic event with a binding rate and a dissociation rate. We thus always have a chemical exchange situation and different ways are used to study ligand binding depending on the exchange rate. For fast exchange the peak should move linearly towards the position of PL as more ligand is added. For slow to intermediate exchange, the peak should first broaden at P and then appear at PL where it will gradually sharpen up as more ligand is added.
8.6.1 Calculation of dissociation constants from chemical shifts As we learnt in Chapter 5, fast chemical exchange manifests as broadened peaks
at the population weighted average resonance frequencies for the two states. The latter property means that if we are in fast exchange, the peaks that experience altered chemical environment upon binding will move in the spectrum as a function of the amount of added ligand. Peak movement is an indicator of binding but it is even possible to extract the dissociation constant from fitting the movement as a function of concentration to the following expression:
𝐾𝑑 =
[𝑃][𝐿][𝑃𝐿] 8.3
𝜛 = 𝜛𝑃 + (𝜛𝑃𝐿 − 𝜛𝑃)[𝐿0] − [𝐿]
[𝑃0] 8.4
𝐿 = 14
([𝑃0] − [𝐿0] + 𝐾𝑑)2 + [𝐿0]𝐾𝑑 −12
([𝑃0] − [𝐿0] + 𝐾𝑑) 8.5
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Figure 8.4. Titration of a drug to the protein thiopurine methyltransferase. A) HSQC spectra for different concentrations of the drug are shown in different colors. Peaks that move are involved in binding. B) The concentration dependent peak movements can be used to obtain the dissociations constant.
8.7 Dynamics Although protein dynamics can be studied by other methods, NMR spectroscopy
is unique in that it allows dynamics to be probed at atomic resolution. Moreover, the NMR signal is sensitive to dynamics spanning time scales of more than twelve orders of magnitude (picoseconds to seconds or longer).
8.7.1 Fast dynamics Fast dynamics is typically defined as motions that occur on time-scales that are
equal to or faster than molecular tumbling. In practice this means that these motions are tumbling (nanosecond time-scale) and vibrations of bond vectors (picosecond time-scale). From the expressions for relaxation rates it is clear that they can modeled from correlation times for tumbling, fast motions and the amplitude of the latter motions. If we want to determine these three parameters, three different relaxation rates must be measured. Almost always, R1, R2 for 15N and the 1H-15N heteronuclear NOE are measured for this purpose.
The correlation time for tumbling can be estimated from the R2/R1 ratio for
residues that are in rigid parts of the protein. If the protein structure is known it is possible to calculate molecular tumbling accurately and it is fair to ask why relaxation experiments are needed in the first place. One reason is that it provides a convenient way of identifying dimerization.
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The internal motions are modeled in terms of order parameters and internal correlation times. Since flexibility provides an entropic contribution to free energy it is possible to use changes in order parameters as a measure of gain/loss of entropy upon events as ligand binding.
8.7.2 Large scale dynamics The correlation time for large scale motions such as those involved in folding,
ligand binding and enzymatic catalysis are typically several order of magnitude slower than tumbling and especially if they are on the millisecond time-scale they can be probed by CPMG relaxation dispersion. By this powerful technique it is possible to extract information about exchange rates as well as populations and chemical shifts of excited states.
8.7.3 Slow events For instance unfolding can be so slow that the time scale is days or even more. A
common method for analyzing dynamics that are on the order minutes to days is hydrogen exchange. For this method, a 15N labeled protein is purified and lyophilized. Immediately beofer the NMR experiment it is dissolved into D2O. This means that labile protons will be exchanged for deuterons at a rate that reflects how well protected they are. By recording HSQCs repeatedly for a day or more it is possible to extract these rates that can provide information on local and global stability.
Figure 8.5. Hydrogen exchange profiles for the protein thiopurine methyltransferase. Clearly, the amide protons for residues V86 and A80 are not equally well protected.
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8.8 Synopsis In this chapter you have learnt:
• How the NMR spectrometer works
• Requirements for a NMR protein sample
• How to produce an isotopically labeled sample
• How NMR experiments are set up
• How resonances are assigned
• How protein structures are calculated
• How NMR spectroscopy can be used to study interactions
• How NMR spectroscopy can be used to study protein dynamics
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8.9 Problems 8.1 Describe the NMR spectrometer! 8.2 State some properties a protein must have for it to be a suitable NMR sample! 8.3 Explain the following procedures, and why they are important, when setting up an
NMR experiment:
a) Locking b) Shimming c) Tuning d) Calibration of pulse widths
8.4 Describe two different approaches for obtaining sequential assignments! Also
discuss advantages and disadvantages with the two methods! 8.5 State some relevant parameters that can be measured by NMR for structure
determinations! 8.6 Discuss two methods for performing structure calculations by NMR! 8.7 Interactions can manifest differently.
a) What is the requirement for being able to follow ligand binding by changes in chemical shifts?
b) How can interactions be studies if this requirement is not fulfilled?
8.8 Why is it necessary to measure R1, R2 and NOE to perform model-free analysis of protein dynamics?
8.9 Why is CPMG relaxation dispersion MR spectroscopy such a powerful method? 8.10 State a few applications for the hydrogen exchange experiment!
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9. Answers to selected problems Lecture 1 1.10 Every nucleus except 12C and 16O is NMR active. 2H and 14N cannot be spin I=½
particles.
1.11 |𝐈| = √0.75 ∙ 1.05 ∙ 10−34 J⋅s 𝐼𝑧 = ±½ ∙ 1.05 ∙ 10−34 J⋅s
1.12 |𝐈| = √6 ∙ 1.05 ∙ 10−34 J⋅s 𝐼𝑧 = −2,−1,0,1,2 ∙ 1.05 ∙ 10−34 J⋅s (any of the values within curly brackets is allowed)
1.13 |𝛍| = 26.752 ∙ 107 ∙ 0.75½ ∙ 1.05 ∙ 10−34 J/T (proton) |𝛍| = 4.107 ∙ 107 ∙ 6½ ∙ 1.05 ∙ 10−34 J/T (deuteron) |𝛍| = 2.713 ∙ 107 ∙ 0.75½ ∙ 1.05 ∙ 10−34 J/T (15N) Note that the negative sign of γ in the last case is removed by the ‘absolute sign’.
1.14 𝐸 = −26.752 ∙ 107 ∙ −½, ½ ∙ 1.05 ∙ 10−34 ∙ 14.1 J (proton)
𝐸 = −4.107 ∙ 107 ∙ −1,0,1 ∙ 1.05 ∙ 10−34 ∙ 14.1 J (deuteron) 𝐸 = 2.713 ∙ 107 ∙ −½, ½ ∙ 1.05 ∙ 10−34 ∙ 14.1 J (15N)
1.15 𝑝𝑑𝑝𝑢
= 𝑒−26.752∙107∙14.1∙1.05∙10−34
1.38∙10−23∙298.15 = 0.99990374
1.16 a) 𝜈 = 26.752 ∙ 107 ∙ 11.7
2𝜋= 498.152 MHz
b) Yes! c) 𝜈 = 6.728 ∙ 107 ∙ 11.7
2𝜋= 125.28 MHz
1.17 𝑝𝑑 + 0.99990374𝑝𝑑 = 1 ↔ ∆𝑝 = 4.81 ∙ 10−5 (or 48 out of a million)
1.18 𝑝8848𝑚𝑝𝑔𝑟𝑜𝑢𝑛𝑑
= 𝑒−80∙10∙8848
1.38∙10−23∙298.15 = 𝑒−1.72∙1027
Not impossible, but pretty slim chance to say the least. 𝑝8848𝑚,𝑂2𝑝𝑔𝑟𝑜𝑢𝑛𝑑,𝑂2
= 𝑒−32∙1.661∙10−27∙10∙8848
1.38∙10−23∙298.15 = 𝑒−1.14 = 0.32
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Lecture 2
2.1 δ = (ν – νref)/νref⋅106 = (γB0(1 – σ)/2π – γB0(1 – σref)/2π)/ γB0(1 – σref)/2π⋅106 = (σref – σ)/(1 – σref)⋅106 ppm.
2.2 δ = (ν – νref)/ νref⋅106 = (νref – νref)/ νref⋅106 = 0. 2.3 Because TMS is insoluble in water (and proteins are not soluble in chloroform). 2.4 Aspartate. Its Cβ is closer to an electronegative group which means that it is less
well shielded. 2.5 See Hore. Paramagnetic centers are normally not present in proteins and hence if so
paramagnetic shielding does not need to be taken into account. However, if such a center is present it is extremely important since the effect is large.
2.6 See lecture notes! 2.7 (600.0006 – 599.999988)/599.999988⋅106 ppm = 1.02 ppm 2.8 Some nuclei of the protein should get higher and some lower chemical shifts.
(Nuclei in the plane of the ring get higher shifts and ones stacked above or below it get lower chemical shifts). In essence the signals are spread out more. See Hore!
2.9 Hydrogen bonds are shorter for antiparallel β-sheets than for α-helices, which leads
to less shielding and thus higher chemical shifts for 1HN in antiparallel β-sheets (given that everything else is unchanged).
2.10 Random coil shifts are the chemical shifts in short (unordered) peptides. The
chemical shifts of ends and loops in proteins are usually very close to random coil values. The secondary chemical shift is the difference between the chemical shift for a nucleus in a protein and its tabulated random coil value, δsec = δ – δr.c.. The secondary chemical shift for a subset of the nuclei is a powerful indicator of secondary structure. See lecture notes!
2.11 See above. 2.12 Hδ because of ring current effects. See Hore! 2.13 methyl < methylene < α-proton < aromatic proton < amide proton
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2.14 For these particular residues, Cβ is close to an electronegative atom (oxygen). 2.15 For Tyr 13Cε is close to an oxygen Lecture 3 3.1 Resonance frequencies will be ν = νA – ∑mXJAX where the summation is over
all nuclei X.
Doublet: one X and mX is ±½ ν = νA ±½JAX. Equal intensities Triplet: 1:2:1 intensity ratios. Centered around νA and split by JAX Hz. Quartet: 1:3:3:1 intensity ratios. Centered around νA and split by JAX Hz.
3.2 Start with the number 1 at the top row. To make a new row: Sum up the numbers
above to the left and to the right. A non-existing number counts as zero. 3.3 a) 10 ppm
a, b) First calculate νref = 11.7⋅26.752⋅107(1 – 0)/2π. Then calculate the frequency of the two components as ν± = 11.7⋅26.752⋅107(1 – (–10-5))/2π ± ½⋅92. Finally, calculate the chemical shifts as δ± = (ν± – νref)/νref ⋅106 ppm. The difference in chemical shifts might seem small but is readily visible in the spectrum. c) Just substitute 18.8 for 11.7 in the above calculations. The results of (b) and (c) will not be identical and the reason is that the coupling constant is independent of the external field. Once again, albeit small, the difference is readily visible in the spectrum.
3.4 Each X can be up (u), down (d) or zero (z) and correspond to m = 1, -1, and 0, respectively. The sum of m ranges from –2 to 2 and we have (dd) = –2, (dz, zd) = –1, (du, ud, zz) = 0, (uz, zu) = 1, (uu) = 2. We thus get five lines centered at the resonance frequency of the uncoupled nucleus and split by JAX. The relative intensities will be 1:2:3:2:1.
3.5 A quartet of triplets. 3.6 It is averaged to zero by rapid molecular motion. 3.7 JAX << δνAX
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3.8 Chemical equivalence: Two nuclei are related to each other by a symmetry
operation. They will then have identical chemical shifts. Magnetic equivalence: In addition to the above they should have identical
couplings to all other nuclei. See Hore! 3.9 Karplus relations for scalar couplings. See Hore and notes! 3.10 12C1H4 proton spectrum: a singlet (magnetic equivalence) 13C1H4 proton spectrum: a doublet (split by coupling to 13C) 13C1H4 13C spectrum: a pentet (split by coupling to 4 protons) 12C2H4 deuterium spectrum: a singlet (magnetic equivalence) 13C2H4 deuterium spectrum: a doublet (split by coupling to 13C) 13C2H4 13C spectrum: Complicated, see below! m=+1 ‘u’, m=0 ‘z’, m=-1 ‘d’ -4 dddd (one possibility) -3 dddz and permutations (four possibilities) -2 dddu and permutations (four possibilities) -2 ddzz and permutations (six possibilities) -1 dduz and permutations (twelve possibilities) -1 dzzz and permutations (four possibilities) 0 dduu and permutations (six possibilities) 0 duzz and permutations (twelve possibilities) 0 zzzz (one possibility) Do the same for the positive values (just switch the ‘u’ and ‘d’ labels) The result is that the 13C signal will be split into nine lines centered at the frequency for an uncoupled 13C nucleus and split by the coupling constant. The relative intensities will be 1:4:10:16:19:16:10:4:1. 3.11 1) 1,1-dibromethane 2) ethylacetate 3) methylethylketon (MEK) 4) t-butylacetate
5) acetaldehyde 6) ethylbensen
Lecture 4 4.1 Ω = ν – νRF ∆B = B0 – γB1/2π. See Hore! 4.2 a) cosine functions with frequencies of 1 Hz and -3 Hz [Mx = M0⋅cos(2π⋅ν⋅t)]
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b) sine functions with the same frequencies 4.3 a) –y b) x c) z d) y e) –x f) –z 4.4 Just rotate about the ‘–y’ axis instead. 4.5 Will change ‘+’ to ‘–’ for all axes except the rotation axis.
c) 4.6 a) +30° b) 210° (–150°) 4.7 a) The phase will increase linearly with time with a proportionality coefficient of
‘2π⋅ν’. d) The phase will increase linearly with time with a proportionality coefficient of
‘2π⋅ν’. e) Aligned with the x-axis. f) Aligned with the negative x-axis.
4.9 a) Aligned with the x-axis (chemical shift and scalar coupling refocused)
b) The components at ±πJAX⋅t radians with respect to the x-axis (only chemical shift refocused)
c) At 2π⋅νA⋅t radians with respect to the x-axis (only scalar coupling refocused) Lecture 6 6.1 ω0,13C = 6.728⋅107⋅11.7 s-1 = 7.872⋅108 s-1.
ω0,15N = 2.713⋅107⋅11.7 s-1 = 3.174⋅108 s-1.
J(0) = 2⋅5⋅10-9 = 1⋅10-8 s-1 J(ω0,13C) = 2⋅5⋅10-9 / (1 + (7.872⋅108⋅5⋅10-9)2) = 6.09⋅10-10 s-1 J(ω0,15N) = 2⋅5⋅10-9 / (1 + (3.174⋅108⋅5⋅10-9)2) = 2.84⋅10-9 s-1
1/T2,13C = 0.5(1⋅10-8 + 6.09⋅10-10)⋅ 1.28⋅1010 = 67.8 s-1
1/T2,15N = 0.5(1⋅10-8 + 2.8410-9)⋅ 3.10⋅109 = 19.9 s-1
6.4 The one with S2 = 1 decays more rapidly.
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10. Collection of tables, formulas and physical constants
Spin angular momentum |𝐈| = [𝐼(𝐼 + 1)]½ℏ I =0, ½, 1, 1½ , 2, … 𝐼𝑧 = 𝑚ℏ m = -I, -I+1, …, I-1, I
Nuclear magnetic moment 𝛍 = 𝛾 ⋅ 𝐈
Spin energy in magnetic fields 𝐸𝑚 = 𝜇𝑧𝐵0 = −𝑚𝛾ℏ𝐵0 Δ𝐸0 = ℏ𝛾𝐵0
Boltzmann distribution 𝑝𝑢𝑝𝑝𝑒𝑟𝑝𝑙𝑜𝑤𝑒𝑟
= 𝑒−Δ𝐸𝑘𝑏𝑇
Nuclear magnetic resonance
𝜈0 =𝛾𝐵02𝜋
𝜈 =𝛾𝐵0(1 − 𝜎)
2𝜋
𝛿 =
𝜈 − 𝜈𝑟𝑒𝑓𝜈𝑟𝑒𝑓
× 106 ppm
𝜈 =𝛾𝐵0(1 − 𝜎)
2𝜋−𝐽𝐴𝑋𝑚𝑋
𝑋
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Chemical exchange 𝐴 𝐵
Δ𝐿𝑊𝑖 =𝑘𝑖𝜋
Δ𝐿𝑊 =4𝜋(𝛿𝜈)2𝑝𝐴𝑝𝐵𝑘𝐴 + 𝑘𝐵
𝛿𝜈 = 𝜈𝐴 − 𝜈𝐵
Nuclear spin relaxation 1𝑇1
= ⟨Δ𝐵2⟩𝐽(𝜔0)
1𝑇2
=12⟨Δ𝐵2⟩𝐽(0) + 𝐽(𝜔0)
J(ω) =2τ c
1+ ω 2τ c2
𝐽(𝜔) =2𝜏𝑐
1 + 𝜔2𝜏𝑐2
𝐽(𝜔) =2𝑆2𝜏𝑐
1 + 𝜔2𝜏𝑐2+
2(1 − 𝑆2)𝜏𝑒1 + 𝜔2𝜏𝑒2
𝐿𝑊 =1𝜋𝑇2
Physical constants ℎ = 6.626 ⋅ 10−34 kg ⋅ m2 ⋅ s−1 Planck’s constant ℏ = 1.05 ⋅ 10−34 kg ⋅ m2 ⋅ s−1 Planck’s reduced constant 𝑘𝐵 = 1.38 ⋅ 10−23 J ⋅ K−1 Boltzmann’s constant 𝑅 = 8.3145 J ⋅ mol−1 ⋅ K−1 The universal gas constant 𝑁𝐴 = 6.022 mol−1 Avogadro’s constant 𝜇0 = 4𝜋 ⋅ 10−7 T ⋅ m ⋅ A−1 Permeability of free space
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Table 10.1 Magnetic properties for selected isotopes Isotope Spin quantum number (I) Gyromagnetic constant (γ) (T-1s-1) 1H 1/2 26.752⋅107 2H 1 4.107⋅107 12C 0 - 13C 1/2 6.728⋅107 15N 1/2 -2.713⋅107 16O 0 - 17O 5/2 -3.63⋅107
19F 1/2 25.18⋅107
