Download NMR Spectroscopy Structural Analysis

NMR Spectroscopy
in
Structural Analysis
Rainer Wechselberger
2008
ii
This reader is based on older versions which were written and maintained by a
number of people. To the best of my knowledge the following persons were involved
in the history of this document:
Rob Kaptein, Rolf Boelens, Geerten Vuister and Michael Czisch.
The current version was completely revised by me and adopted to my lecture
'NMR Spectroscopy in Structural Analysis'
I thank Hugo van Ingen and Hans Wienk for proofreading and many discussions and
suggestions.
Rainer Wechselberger, Utrecht in the summer of 2008
Please report errors in the text and/or explanations or any 'unclear' passages to:
[email protected]
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iv
I
Introduction........................................................................................................... 1
1.1
Typical applications of modern NMR ................................................................ 2
1.2
Some history of NMR ........................................................................................ 2
1.3
Aim of this course .............................................................................................. 3
1.4
General outline ................................................................................................... 4
II
Basic NMR Theory ........................................................................................... 5
III
An Ensemble of Nuclear Spins ...................................................................... 12
3.1
Ensemble of spins ............................................................................................. 12
3.2
Effect of the radio frequency (rf) field B1 ........................................................ 13
IV
Spin relaxation ................................................................................................. 16
4.1
Molecular basis of spin relaxation................................................................... 17
V
Fourier Transform NMR ............................................................................... 22
5.1
From time domain to spectrum ...................................................................... 22
5.2 Aspects of FT-NMR .............................................................................................. 27
VI
Spectrometer Hardware.................................................................................. 28
6.1
The magnet........................................................................................................ 28
6.2
The lock ............................................................................................................. 30
6.3
The shim system ................................................................................................ 31
6.4
The probe........................................................................................................... 32
6.5
The radio frequency system.............................................................................. 33
6.6
The receiver....................................................................................................... 34
VII
NMR Parameters......................................................................................... 37
7.1
Chemical shifts.................................................................................................. 37
7.1.1
Effects influencing the chemical shift ................................................................. 39
7.1.2
Protein chemical shifts ....................................................................................... 40
v
7.2
J-Coupling......................................................................................................... 41
7.2.1
VIII
Equivalent protons ............................................................................................. 43
Nuclear Overhauser Effect (NOE).............................................................. 45
8.1
Dipolar Crossrelaxation ................................................................................... 45
8.2
NOEs in biomolecules ...................................................................................... 48
IX
Relaxation Measurements............................................................................... 51
9.1
T1 relaxation measurements............................................................................. 51
9.2
9.1.1
Calculated example for an inversion-recovery experiment................................ 53
9.1.2
Applications of T1 relaxation.............................................................................. 54
T2 relaxation measurements............................................................................. 54
9.2.1
X
Applications of T2 relaxation.............................................................................. 56
Two-Dimensional NMR................................................................................... 57
10.1 The SCOTCH experiment ................................................................................ 57
10.2 2D NOE ............................................................................................................. 60
10.3 2D COSY and 2D TOCSY ................................................................................ 64
XI
The assignment problem ................................................................................. 68
11.1 Chemical shift ................................................................................................... 68
11.2 Scalar coupling ................................................................................................. 69
11.2 Signal intensities (integrals)............................................................................. 69
12.3 NOE data........................................................................................................... 69
XII Biomolecular NMR............................................................................................ 70
12.1 Peptides and proteins ........................................................................................ 70
12.1.1
Assignment of peptides and proteins ................................................................. 72
12.1.2
Secondary structural elements in peptides and proteins .................................... 76
12.2 Nucleotides and nucleic acid............................................................................ 81
12.2.1
Assignment of oligonucleotide and nucleic acid spectra................................... 87
XIII Structure determination .................................................................................. 88
vi
13.1 Sources of structural information.................................................................... 88
13.1.1
NOEs .................................................................................................................. 88
13.1.2
J-couplings ......................................................................................................... 89
13.1.3
Hydrogen bond constraints ................................................................................ 91
13.2 Structure calculations....................................................................................... 91
13.2.1
Distance-Geometry............................................................................................. 91
13.2.2
Restrained molecular dynamics ......................................................................... 94
Appendix: ................................................................................................................... 97
Appendix A:
a) Typical 1H and
13
C chemical shift of some common functional
groups and b) random coil chemical shifts for protons in amino acids. .................. 97
Appendix B:
Chemical shift distribution in 1H spectra of proteins. .................. 100
Appendix C:
Nuclear Overhauser Effect ............................................................ 101
Appendix D:
Fast exchange titration .................................................................. 104
Appendix E:
2D NOESY experiment .................................................................. 106
Appendix F:
Expected cross-peaks for COSY, TOCSY and NOESY for the
individual amino acids.............................................................................................. 108
Appendix G:
Typical chemical shift values found in nucleic acid..................... 118
Appendix H:
Typical short proton–proton distances for B-DNA....................... 119
1
I
Introduction
All spectroscopic techniques are based on the absorption of electromagnetic radiation by
molecules or atoms. This absorption is connected to transitions between states of
different energies. The nature of the electromagnetic radiation varies from hard γ-rays in
Mössbauer spectroscopy to very low energy radio frequency irradiation in NMR
spectroscopy. Other spectroscopic methods, applying electromagnetic radiation of
intermediate energy, are microwave spectroscopy (vibration and/or rotation of dipolar
groups in molecules), IR spectroscopy, where vibration states are excited, UV/vis
spectroscopy, where the electronic orbitals of atoms are involved and x-ray or atom
absorption spectroscopy involving the inner electron shells. In Nuclear Magnetic
Resonance (NMR) transitions occur between the states that nuclear spins adopt in a
magnetic field. Since the energy differences between these spin states are extremely
small, long-wavelength radio-frequency matches these differences. Accordingly, NMR is
a rather insensitive method and more sample-material is usually needed than for most
other spectroscopic methods. On the other hand, however, NMR lines are quite narrow
and therefore the resolution is usually so high that hundreds of lines can be resolved in a
single NMR spectrum. Also, the interaction between different nuclear spins is manifested
in NMR spectra, for instance, in the form of J-coupling or the nuclear Overhauser effect
(NOE). These properties have made NMR quickly an indispensable tool for structural
studies in chemistry and later also in biochemistry. NMR is the only available method to
date to determine the structure of proteins and nucleic acids in solution on an atomic
scale so that it became a well-established method in the field known as "Structural
Biology".
2
1.1
Typical applications of modern NMR
Structure elucidation
Synthetic organic chemistry (often together with MS and IR)
Natural product chemistry (identification of unknown compounds)
Study of dynamic processes
Reaction/binding kinetics
Chemical/conformational exchange
Structural studies of biomacromolecules
Proteins, protein-ligand complexes,
DNA, RNA, protein/DNA complexes,
Oligosaccharides
Drug Design
Structure Activity Relationship (SAR)
Magnetic Resonance Imaging (MRI)
MRI today is a standard diagnostic tool in medicine.
1.2
Some history of NMR
NMR was discovered in 1945 by Bloch at Stanford and Purcell at Harvard University.
For this Bloch and Purcell received the Nobel Prize for physics in 1952. Initially, it
belonged to the realm of physics but after the discovery of the chemical shift (nuclei in
different chemical surroundings have different resonance frequencies) the technique
quickly became very important as an analytical tool in chemistry. The development of
stronger magnets (maximum proton frequency is now (2006) about 1000 MHz) and of
3
multidimensional NMR methods allowed its entry in the field of biology. As a result of
its continuously increasing importance in modern chemistry, biochemistry and medicine,
two more Nobel prices for NMR followed in 1991 (Richard Ernst) and in 2002 (Kurt
Wüthrich).
1.3
Aim of this course
This course will bring the student up-to-date with the principles of modern NMR
methods and provide a basic understanding of how these methods work and how they can
be applied to derive the three-dimensional structures of biomolecules by NMR.
After a brief theoretical introduction of the basic physical principles of NMR, we will
discuss the origin of the parameters that determine the appearance of an NMR spectrum
such as chemical shift, J-coupling and line-width. Spin-relaxation (i.e. how a spin system
returns to equilibrium after excitation) is important as it determines the line widths of the
NMR signals, but also the intensity of the Nuclear Overhauser Effect, which in turn is the
major source of information for the structural analysis of biomolecules.
The modern way of recording NMR spectra is by applying short radio frequency pulses
and analyzing the response by Fourier transformation. This so-called Pulse Fourier
Transform NMR (for which R.R. Ernst received the Nobel prize for chemistry in 1991)
also allows the measurement of two-dimensional (2D) NMR spectra (and even 3D and
4D). An introduction is given to the FT-NMR technique and the principles of multidimensional NMR are reviewed. Exemplary, some basic 2D NMR experiments will be
discussed in more detail. Finally, the important process of assignment of biomolecular
NMR spectra is explained and an overview of the possibilities to extract a (3dimensional) structure out of NMR data is given.
4
1.4
General outline
I
Introduction (what is NMR and what do you study with it)
II
Theory (how does it work)
III
Ensemble of spins (from single atom to real samples)
IV
Relaxation I (after the experiment: back to equilibrium)
V
FT NMR (with a single rf-pulse to a complete spectrum)
VI
Hardware (what kind of device do you need for FT NMR)
VII
NMR parameters (what you can see in an NMR spectrum and why)
VIII
NOE (How does the Nuclear Overhauser Effect work)
IX
Relaxation II (experiments to measure relaxation properties)
X
2D NMR (how to add an extra dimension and what's the good to it)
XI
Assignment (which signal comes from which atom)
XII
Biomolecular NMR (nucleic acids and proteins, spin systems and (structural)
parameters, sequential assignment)
XIII
Structure Determination (which parameters to use, how to calculate a structure)
5
II
Basic NMR Theory
The energy states, between which transitions are observed during an NMR experiment,
are created only when a nucleus with magnetic properties is brought into an external
magnetic field. These magnetic properties of nuclei can be derived from a quantum
mechanical property, the spin angular momentum, I. Most nuclei have such a spin
angular momentum, which is represented by a corresponding spin quantum number I,
which can be integer or half-integer (I = 0, 1/2, 1, 3/2....) and we may simply speak of a
nucleus with spin I. The magnitude of the spin angular momentum is given by (2.1)
I
= h I ( I + 1)
(2.1)
where h is h/2π ( h = 6.626 · 10-34 J·s is Planck's constant). Due to its quantum mechanic
nature, any component of I along an arbitrary axis of observation, for instance the z-axis
(which is by definition the direction of the external magnetic field), is quantized:
I z = h mI
(2.2)
mI is the spin quantum number which can adopt values from -I to I in steps of 1 (a total
of 2 I +1 values). A combination of equations 2.1 and 2.2 leads to the spin-state diagrams
for nuclei with different spins (e.g. ½, 1, ³/2 ):
6
A combination of their spin angular momentum and their positive charge causes nuclei to
have a magnetic moment (compare the effect of an electric current in a circular wire).
This magnetic moment is directly proportional to the angular momentum:
μ=γ I
(2.3)
γ is called the gyro magnetic ratio. Since I is quantized, accordingly also μ is quantized
and we can express μ in terms of the spin quantum number, I or μz in terms of the
magnetic quantum number, mI:
μ = γ h I ( I + 1)
(2.4)
μz = γ h mI
(2.5)
and
Classically, if we bring a bar magnet (compass needle!) in a magnetic field, denoted B,
the magnet will tend to turn and orient itself in the field. This is a consequence of the fact
that its energy, given by
E = −μ ⋅ B
(2.6)
will then reach a minimum. For quantum mechanical objects such as nuclear spins the
situation is similar except that now only a limited set of discrete orientations (quantum
states) are available. Expression (2.6) still holds for nuclear spins. With the convention
that B lays along the z-axis we get the energy of a nuclear spin with a magnetic moment
of μ in an external magnetic field, B0 as
E = − μ z B0
(2.7)
and with Eq. 2.5 it becomes clear, that the energy of a nuclear spin depends on the
magnetic quantum number mI:
E = − γ h mI B0
(2.8)
For a spin ½ nucleus this results in two states, denoted α for mI = + ½ and β for
7
mI = - ½. As a consequence, the nuclear spins can not be found ‘turning around’ in order
to orient in an external magnetic field, but they rather can be found only in two different
orientations, corresponding to the two possible energy levels described by the two
magnetic quantum numbers:
In all forms of spectroscopy, transitions between energy levels are induced by
electromagnetic radiation of a particular frequency ν0, provided that the frequency
matches the energy difference between these energy levels:
ΔE = h ν 0
(2.9)
This is sometimes called Einstein's equation and it is a basic relation in spectroscopy.
Based on Eq. 2.8 we find that ΔE = γ h B0 = hν0, or
ν0 =
γ
B0
2π
(2.10a)
or even simpler
ω0 = γ B0
(2.10b)
8
expressed in terms of the angular frequency ω0 = 2πν0. In NMR the relation (2.10b) is
often called the "resonance condition" i.e. the condition where the frequency of the
radiation field matches the so-called Larmor frequency ωL = γB0.
For nuclei with spin I larger than ½ we have multilevel energy diagrams. However, the
selection rule ΔmI = ± 1 still holds so that we arrive at the same resonance condition.
Since the energy of the spin states depends on the strength of the external magnetic field
(equation 2.60), we can modify the figure from above and adapt it for different magnetic
fields B0:
E
E = +½γ ħB0 (β-state, m=-½)
ΔE = γ ħB
B
E = -½γ ħB0 (α-state, m=+½)
B0
The separation ΔE of the energy states α and β and thus the resonance frequency,
depends on the sort of nucleus (γ) and the strength of the external magnetic field B0 or, in
other words, on the strength of the NMR magnet (compare 2.10).
9
Table 1 shows some properties of nuclei important for applications in organic chemistry
and biochemistry:
Table 1: Properties of selected nuclei
Isotope
Nuclear spin
I
Resonance
gyro magnetic
frequency
-1 -1
(MHz)
Natural
ratio γ [T s ]
abundance [%]
*
1
H
1/2
600.0
2.6752 · 108
99.985
2
H
1
92.1
4.1065 · 107
0.015
12
C
0
-
-
98.89
13
C
1/2
150.9
6.7266 · 107
1.11
14
N
1
43.3
1.9325 · 107
99.63
15
N
1/2
60.8
-2.7108 · 107
0.37
16
O
0
-
-
99.76
17
O
5/2
81.4
-3.6267 · 107
0.04
19
F
1/2
564.5
2.5167 · 108
100.0
31
P
1/2
242.9
1.0829 · 108
100.0
*resonance frequency at a magnetic field of 14.092T
We will focus here on spins with I = ½ because they have only two possible energy states
and accordingly give only a single spectral line and thus are the most popular spins in
high-resolution NMR. For protein studies these are 1H, 13C and 15N (note that 12C has no
magnetic moment, and 14N has a spin 1). For nucleic acids in addition 31P is an important
nucleus.
10
So far we have given a quantum mechanical treatment of the nuclear spin in a magnetic
field considering discrete energy levels. This led to the resonance condition (2.10).
Interestingly, a similar expression can be obtained from a classical description of the
effect of a magnetic field B on a spinning magnet with a magnetic moment μ that is tilted
with respect to the field direction. While a classical non-spinning bar magnet would just
orient itself in the direction of the field (compass), a spinning magnet cannot do this but
instead performs a precession about the direction of the field. In essence this is a
consequence of the conservation of angular momentum. There is a perfect analogy with
the motion of a spinning top (Dutch: "tol") in the gravity field of the earth. The top will
also undergo a precessional motion. Mathematically, the effect of the torque acting on a
spinning magnetic moment is given by the cross product
ex
dμ
= − γ B × μ = − γ Bx
dt
μ
x
ey
By
μy
ez
Bz
μz
(2.11)
where ex, ey and ez are unit vectors along the x, y and zaxis, respectively. With the usual convention that B is
along the z-axis (Bz = B0 and Bx = By = 0) the equations of
motion for the magnetic moment become
dμ x
= γ B0 μ y
dt
dμ y
= − γ B0 μ x
dt
dμ z
=0
dt
(2.12)
It can be shown by differentiation that a correct solution is given by
11
μ x (t ) = μ x (0) cos(γ B0t ) + μ y (0) sin(γ B0t )
μ y (t ) = − μ x (0) sin(γ B0t ) + μ y (0) cos(γ B0t )
2.13)
μ z (t ) = μ z (0)
This indeed describes a precession of μ about the z-axis with an angular frequency ω0
given by
ω0 = γ B0
(2.14)
an expression identical with (2.10). Thus, we have found that the quantum based
resonance frequency corresponds exactly with the equation for the classical precession of
a spinning magnet in a magnetic field.
In NMR we will often use a classical mechanics analogy for a description of nuclear
spins. Sometimes such an analogy must break down, however, since the spins really are
quantum mechanical in nature.
12
III
An Ensemble of Nuclear Spins
3.1
Ensemble of spins
In practice we are dealing with a large number of nuclear spins. In the quantummechanical picture these are distributed over the spin states according to Boltzmann's
law:
ΔE
−
nβ
= e kT
nα
where nα and nβ are the populations of the
α and β state, respectively, and k is
Boltzmann's
constant
(k=1.3806504(24)×10−23 J/K). For I = ½
this is shown on the right.
Due to the very small energy difference ΔE = γ h B0 the
populations of the α and β states are almost equal. For a field
B0 = 14 T (Tesla) (proton frequency 600 MHz) the relative
excess of α spins is only one in 104. This is one of the main
reasons for the low sensitivity of NMR spectroscopy.
In the (classical) vector model an ensemble of spins ½ can be
described as shown on the right.
The individual spin vectors all make an angle with the field
B0 and are slightly more aligned parallel to the field than
antiparallel. In equilibrium the "phases" of the individual
spins (their xy components or positions) are randomly
(3.1)
13
distributed. Therefore the resultant magnetization vector M (the sum of all spin vectors)
is aligned along B0. Like all the individual spins that make for M,
also M has to be imagined as spinning. So, if M is tipped away
from B0 (see figure on the left) it will perform a precessional
motion about B0 with frequency ω0 = γB0. Just the way the
individual spins do. This ‘tipping’ can be done by way of a radio
frequency pulse - and involves transitions between the spin states
of the nucleus!
3.2
Effect of the radio frequency (rf) field B1
In all FT-NMR spectrometers an additional field B1 can be generated perpendicular to the
static field B0. This B1 field is created by means of radio frequency pulses (it’s basically
the magnetic component of the electromagnetic radio waves). What happens during the
pulse can best be compared to what happens to the spins, when they are brought into the
magnetic field of the spectrometer. We are just dealing with an additional magnetic field,
which tries to orient the spins in a new direction. The result of this additional field is a
rotation of the magnetization vector M around the axis of the additional field. This
rotation lasts only as long, of course, as the additional field is present. In other words:
only during the duration of the radio frequency pulse. In perfect analogy to the precession
around the B0 field, the speed of this rotation (its angular frequency) can be described as
ω1 = γ B1
(3.2)
Acting on the equilibrium magnetization M, the effect of B1 is to tilt the vector away
from the z-axis. How far the magnetization is tilted depends on the sort of nucleus (γ), the
strength of the B1 field and the duration of the pulse. Illustrating the effect of a radio
frequency pulse is a bit tricky. Actually the magnetization vector is still precessing
14
around the B0 field (with ω0) and at the same time, during the rf pulse, is tilted away from
the z axis (is rotating around the x-axis with ω1)! Fortunately there is a simple way to
describe this: the rotating frame. In a frame of reference in which the x and y axes rotate
with ω0 about the z-axis, the B1 field appears stationary (right). In the rotating frame the
axes are denoted x', y' and z'. Now, at resonance, the motion of M becomes very simple:
rotating frame
laboratory frame
While in the laboratory frame the M vector performs a complex spiralling motion, in the
rotating frame it simply precesses about the direction of the stationary B1 (in the z', y'
plane) with an angular frequency ω1 = γB1. Thus, by going over to the rotating frame of
reference we do not need to bother about the precession about B0 anymore. The effect of
an rf pulse can easily be described now: During the duration of a pulse (and only that
long!) the magnetization M is rotating around the axis from which the radio frequency
pulse is applied! The tilting of the magnetization vectors just follows the equation of
motion given in 2.13. If we keep the B1 field on for a short time such that ω1t = π/2 and
then switch it off we have tipped the magnetization along the y' axis. This is called a 90°
pulse. If we keep it on twice as long M is along the negative z-axis (180° pulse):
15
90° pulse
180° pulse
Remark: The concept of the rotating frame makes the description of NMR experiments
much easier. It is so convenient that we will use the rotating frame throughout the
complete course if not explicitly stated otherwise. For simplicity, we will use the notation
x, y, z instead of x', y', z' for the rotating frame in the following.
16
IV
Spin relaxation
After a 90° pulse from the x axis M lies along the y axis. If we leave the system now
undisturbed we will reach the equilibrium state again after a while by a process called
spin relaxation. Actually two things will happen:
i) magnetization will grow along the z-axis until the equilibrium value Meq has
been restored,
ii) the Mx and My components decrease to zero (usually faster than the equilibrium
magnetization is restored!).
The characteristic times for these processes are called T1 and T2. Thus we have
T1: longitudinal or spin-lattice relaxation time (z-magnetization),
T2: transverse or spin-spin relaxation time (x,y-magnetization).
In mathematical terms this can be described as follows:
( M z − M eq )
dM z
=−
dt
T1
Mx
dM x
=−
dt
T2
dM y
My
=−
dt
T2
(4.1a)
(4.1b)
(4.1c)
The solution of Eq. 4.1a is
M z (t ) − M eq = [M z (0) − M eq ]
and of Eq. 4.1b
M x ( t ) = M x ( 0)
− t
T
e 2
− t
T
e 1
(4.2)
(4.3)
17
and similarly for My(t). Thus, all components of M return exponentially to their
equilibrium values. An important difference between T1 and T2 processes is that the
former involves changes in z-magnetization and hence, transitions between α and β spin
states that are accompanied by an exchange of energy with the "lattice" (environment). In
contrast, T2 processes involve loss of phase coherence in the x,y plane, no energy is
exchanged with the environment.
Note that spin relaxation is a random process and should not be confused with the
coherent rotations around B0 or B1 fields. For example, T1 relaxation which affects the zcomponent does not create x- or y-magnetization (cf. Eq. 4.1).
4.1
Molecular basis of spin relaxation
What causes nuclear spins to relax? The simple answer is: exchange of energy with the
environment. But… The energies of the corresponding processes must match the ΔE
between the involved energy states. In most other spectroscopic techniques, this is
achieved by collisions (with other atoms or molecules). For the small energy differences
in NMR this is not an option. We have to look for processes with comparably small
energies (i.e. comparable frequencies) as the corresponding resonance frequencies in
NMR. We can find these in the interaction of moving magnetic dipoles. The magnetic
dipoles are the NMR nuclei themselves. The movement comes from the diffusional
motion of molecules. This process contributes both to T1 and T2 relaxation. But while
with T1 relaxation energy is exchanged with other molecules (the environment, the
‘lattice’) causing transitions between α and β states, in T2 relaxation the energy is
exchanged with spins of the same molecule, leading to a small variety in the precession
frequency of otherwise identical spins. The net-magnetization vector M is ‘split up’ in
many small components rotating with different frequencies. Eventually these components
are equally distributed in the x-y plane and no measurable transversal magnetization is
18
left. One speaks about dephasing of magnetization. It is important to note that also a
static distribution of Bz fields causes different frequencies (in different locations of the
sample) and also leads to dephasing i.e. T2 relaxation.
To understand the effect of motion, it is important to consider the time-scale of it. For T1
relaxation only motions with a frequency near the Larmor frequency ω0 are effective.
After all, they have to induce transitions between spin states and therefore must have a
frequency, which coincides with the ΔE between the states (thus, the Larmor frequency).
The time-dependence of the dipolar field comes from the
rotational diffusion (the ‘thermal motion’, see figure to the left)
of the molecules which is characterized by a rotational
correlation time τc. For times smaller than τc the orientation of
a molecule has not changed much (leftmost figure below),
while for t >> τc the correlation between different orientations
is lost (rightmost figure):
For small fast tumbling molecules τc is quite short (10-11 - 10-10 s) while for large
biomolecules it is much longer (10-8 - 10-7 s). An approximate relation of τc with the
molecular volume V and the viscosity η is given by
τc =
ηV
kT
(4.6)
For macromolecules of molecular mass Mr in H2O solution at room temperature a useful
approximation is
19
τc ≈
Mr
10 −12
2.4
(4.7)
In a randomly tumbling motion many frequencies are present. The distribution of the
frequencies of the motions is represented by the spectral density function, J(ω).
J(ω)
ω
1/τc
In this distribution the frequency ω = τc-1 acts much like a cut-off of the spectral density
function, in other words: motions with frequencies ω > τc-1 are quite rare. The area under
the curves is constant. Only the shape of J(ω) differs for molecules of different size. For a
smaller molecule for instance, J(ω) would look like:
J(ω)
1/τc
ω
τc for smaller molecules is shorter and their frequency distribution extends to higher
values of ω. On the other hand, J(ω) for low values of ω is relatively small. With
increasing size of the molecules J(ω) is getting larger for slow motions but at the same
20
time the ‘cut-off’ moves to lower frequencies. In other words, slow motions (small values
of ω) are more likely to occur for (big) biomolecules than for small organic molecules.
Not very surprising indeed!
Now what does this mean for T1 relaxation? Here, the fluctuating fields have to induce
transitions between α and β states separated by the Larmor frequency ω0 = γB0.
Therefore, the efficiency of relaxation will depend on how much this frequency is present
in the distribution of frequencies of the molecule, thus on J(ω0). This is maximal for
molecules that have τc-1 = ω0, and the efficiency will drop for both larger and smaller
molecules (longer and shorter τc values). This explains the behavior of T1 versus
correlation time τc as shown in the next figure.
The minimum in T1 (most efficient relaxation) is at ω0τc = 1, which for common NMR
fields occurs for intermediate size molecules of molecular mass Mr ≈ 1000 D. In terms of
the fluctuating fields Bx(t) and By(t) a general expression for the efficiency (or rate) of
T1 relaxation is
21
1
=γ2
T1
(B
2
x
+ By2
) J (ω )
0
(4.8)
where the average of the square of Bx(t) , < Bx2 >, is a measure of the strength of the
fluctuating fields. Assuming that the components in all directions are the same ( < Bx2 >
= < By2 > = < Bz2 > = < B2 > ) Eq. 4.8 becomes
1
= 2γ 2 B 2 J (ω 0 )
T1
(4.9)
A similar expression for T2 relaxation is
1
= γ 2 B2
T2
(
J (0) + J (ω0 ) )
(4.10)
This describes the two mechanisms that contribute to T2: the static distribution of Bz
fields (no or ‘zero’ frequency), J(0), and the effect induced by Bx(t) and By(t), which is
proportional to J(ω0). For larger biomolecules the J(0) term will dominate and becomes
approximately equal to τc. Thus, for slowly tumbling molecules we have the simple
expression
1
≈ γ 2 B2 τ c
T2
(4.11)
This means that T2 becomes progressively shorter for larger molecules and explains why
T2 unlike T1 does not go through a minimum.
22
V
Fourier Transform NMR
5.1
From time domain to spectrum
Nowadays, all modern NMR spectrometers work as so-called Fourier-transform NMR
spectrometer (FT-NMR). Earlier we have seen (chapter 3.2) that the magnetic component
of an electromagnetic field (rf), applied along the x-axis of the rotating frame on
equilibrium z-magnetization, results in a precession around the x-axis when ωrf = ω0. For
the precession frequency, ω1, we found ω1 = γ B1. If we apply this rf field for a period t =
π/2ω1 we create 'pure' y-magnetization. An rf pulse of this duration was called a 90°
pulse. The rf transmitter of an NMR spectrometer is operated by a pulse-computer, which
can generate a single rf pulse or a series of rf pulses of arbitrary length, frequency, phase,
and amplitude separated by delays of adjustable length. An rf pulse of length tp excites
the frequency-range νrf - 1/(2τp) to νrf + 1/(2τp). To excite a certain range of frequencies,
tp must be adjusted to be sufficiently short. For example, a good excitation of an NMR
spectrum of 10 kHz requires τp << 200 μs. In practice, pulses of τp = 2-20 μs are used.
Δνrf
νrf
τp
νrf-½τp
νrf
νrf+½τp
A radiofrequency pulse of duration τp (left) and the corresponding excitation
profile (right). Details in the text above.
What will happen after a single rf pulse of 90° along the x-axis? Earlier we have seen that
the magnetization vector has been rotated and now is oriented along the y-axis (phase
coherence of the spins). The receiver is tuned to the frequency ωrf. If the resonance
frequency of spin j, denoted by ωj, is equal to ωrf the magnetization will remain along the
23
y-axis (of the frame rotating at speed ωrf). The magnitude of the magnetization will be
decreased by T2 relaxation. In case spin j has a resonance frequency ωj, which is different
from ωrf, then the magnetization of spin j will precess in the frame rotating at ωrf with the
difference
frequency
Ω = ωj- ωrf.
Needless to say, also in this case the
magnetization
x component
decays
due
to
relaxation processes. The in the x-y
plane rotating magnetization vector
induces a current when it passes the
receiver coil. This is the actual signal
recorded by the receiver. In order to
y component
be able to distinguish between
positive and negative frequencies
(vectors
y
y
x
are
rotating
clockwise or counterclockwise with
y
x
which
x
the same speed), both the x- and the
y-component
magnetization
of
the
are
rotating
recorded
simultaneously. The corresponding signal induced in the receiver coil has the shape of a
decaying harmonics (sine and cosine waves) and thus is called the free-induction decay
(FID). For a number of different spins j (for example of the Hα, the Hβ, and the HN), each
with their own equilibrium magnetization Mjeq, frequency ωj, and relaxation time T2j, the
FID consists of the sum of all magnetizations:
− t
T
e 2
[
]
M y (t ) = ∑ M j (0) cos ω ′j t
j
[
M x (t ) = ∑ M j (0) sin ω ′j t
j
− t
T
e 2
]
(5.1)
24
where │Mj(0) │ = │Mjeq│ in the case of a 90° excitation pulse. The following figure
illustrates the difference in appearance of an FID containing only a single frequency and
an FID containing multiple frequencies:
The FT NMR signal (the FID) is recorded in the time domain. The signal is digitized by
an analog-to-digital (ADC) converter and stored in the memory of a computer. The
resonance frequencies, ωj', are extracted by Fourier analysis. The Fourier transformation
(Eq. 5.2) transforms the signals f(t) from the time domain to the frequency domain, g(ω):
g (ω ) =
∞
∫ f (t ) e
−iω t
dt
(5.2)
−∞
where the complex signal f(t) is defined as
f (t ) = M y (t ) + i M x (t ) = ∑ M eq
j e
i ω ′jt
− t
T
e 2
.
(5.3)
j
Fourier transformation results in a sum of complex frequency signals. The real part of
25
which describes an absorption signal:
Re[ g (ω ) ] = ∑ M eq
j
j
T2 j
1 + T2 j (ω − ω ′j ) 2
2
(5.4)
The imaginary part describes a dispersion signal:
Im[ g (ω ) ] = ∑ M
j
T2 j (ω − ω ′j )
2
1 + T2 j (ω − ω ′j ) 2
2
eq
j
(5.5)
The resonance line shape is a so called “Lorentz line shape”. Usually we are only
interested in the real part of g(ω), the absorption response, because the flanks of
absorptive line drops with ω−2 , whereas the dispersive line goes with ω−1. This means
that the absorptive line is much narrower.
From eq. 5.4 we can calculate the
relationship between T2 and the line
width at half-height, Δν1/2:
0.5 =
1
1 + T π 2 Δν 122
2
2
(5.7)
26
which can be rewritten as
Δν 12 =
1
π T2
(5.8).
Some important fourier pairs are shown here. Basically these are the ones responsible for
the shapes (or their distortion) of most lines observed on a FT-NMR spectrometer
f(t)
g(w)
I
1
0
-1
FT
t
M
1
0
-1
FT
t
I
FT
t
27
5.2 Aspects of FT-NMR
It might be that the signal-to-noise ratio is not good enough after a single scan. By coadding n successive NMR measurements the signal, S, increases by a factor n. The noise,
N, however, fortunately increases with
n only. This is due to the random nature of
noise. Hence the signal-to-noise ratio, S/N, improves as
S
~
N
n
(5.9)
FT NMR is a very flexible technique. A large number of different experiments can be
done with the FT technique, each aiming at different parameters of the molecules in study
to be extracted, such as relaxation measurements (discussed in chapter 6), multidimensional NMR (discussed in chapter 8), heteronuclear NMR, etc.
28
VI
Spectrometer Hardware
6.1
The magnet
The magnet is the core of the NMR
spectrometer.
Nowadays
mainly
persistent superconducting coils are used
to generate the high magnetic fields
necessary for high resolution NMR
(permanent magnets are only used e.g. in
food sciences or on older, lower field
NMR imaging systems). The coils consist
of
NbTi
(NbTa)3Sn)
(or
NbTi-Nb3Sn,
wires
which
NbTiare
superconducting at 4 K ( -269 °C).
Several layers of coils generate a higher
and higher field towards the innermost
section where the final magnetic field
strength is reached. In this section the
field must be extremely homogeneous over a volume of some cubic centimeters,
otherwise the resonance frequencies would vary at different locations in the sample
leading to broad and unsymmetric lines. The coil wires are also the most expensive part
of the spectrometer. To reach higher field strength, larger (and more complicated
designed) coils have to be used. This makes most of the price difference between e.g. a
700 MHz and a 900 MHz machine, which is approx. a factor of 4 (in 2006). The
necessary low temperature for superconductivity is reached by submerging the coils into
a dewar containing liquid helium (at –269 °C). This inner dewar is surrounded by a
second, outer dewar containing liquid nitrogen (-200 °C). In the course of time, both
29
helium and nitrogen evaporate – therefore the ‘magnet’ (the dewars actually!) has to be
refilled periodically (typically weekly for N2 and monthly for He).
The strength of a magnetic field is normally given in Tesla or Gauss (1 G = 10-4 T). The
strength of an NMR magnet is often described by the corresponding resonance frequency
of hydrogen atoms (‘proton-frequency’). A field of 14 T corresponds to a field strength of
600 MHz. Today (2006) the typical field strength used in biological applications are 700
MHz and 800 MHz. The highest currently available field is about 1000 MHz. The highest
field available in the Utrecht NMR laboratory is a 900 MHz spectrometer equipped with
a cryogenic probe.
Tesla
2.3
8.4
11.7
14.1
16.5
17.6
21.1
MHz
100
360
500
600
700
750
900
The need for higher and higher fields is explained by the gain in resolution and in
sensitivity. The sensitivity of an NMR experiment is usually described by the signal-tonoise ratio S/N:
S/N ~ N γ5/2 B03/2 (NS)1/2 T2/T
(6.1)
where N is the number of spins (concentration of the sample), γ the gyro magnetic ratio
of the nucleus, B0 the field strength, NS the number of individual scans per experiment,
T2 is the relaxation time and T the temperature. How a bigger field affects the S/N and
the resolution (which follows a linear dependence on B0 ) is shown in the following table:
Table 2
B0 (T)
11.7
14.1
16.5
17.6
21.1
ν (MHz)
500
600
700
750
900
S/N
1.0
1.3
1.7
1.8
2.4
resolution
1.0
1.2
1.4
1.5
1.8
30
The Biomolecular NMR laboratory at Utrecht University is housing a 360 MHz, two 500
MHz, two 600 MHz, a 700 MHz, a 750 MHz and one 900 MHz high-resolution NMR
spectrometer, one of the 600 MHz spectrometer and the 900 MHz spectrometer are
equipped with cryogenic probe systems for additional sensitivity.
6.2
The lock
Even in a very well designed magnet the magnetic field is not perfectly stable over the
long time a measurement can take (up to one week). Small deviations of the main
31
magnetic field can be compensated by the ‘lock system’ by applying correction currents
in a coil which is part of the room temperature shim system (see below). The lock system
exploits the NMR phenomenon itself: A reference NMR experiment is continuously
performed on a nucleus different from the one being studied. In most biological
experiments deuterium (2H) is used for this purpose. The deuterium spectrum is
continuously acquired and the frequency of the single deuterium line is observed.
Whenever this frequency shifts, small correction currents are applied to the lock coil to
compensate for this change therefore slightly increasing or decreasing the total magnetic
field. In most biological applications deuterium is introduced by dissolving the sample in
a mixture of 5-10% D2O in H2O.
6.3
The shim system
As stated above the magnetic field experienced by the sample must be very stable and
also very homogeneous to keep the lines as narrow as possible. Since the homogeneity is
not only a matter of the coil design, but is also influenced by the sample itself (filling
height, quality of tube etc.), for each individual sample additional field corrections have
to be applied. This is achieved by a number of correction coils (the shim system) in which
adjustable currents produce field gradients which can compensate field inhomogeneities.
There are two sorts of shim coils: superconducting ('cryo-shims') and room temperature
coils. The currents through the superconducting shim coils are usually only once adjusted
during the installation procedure of the magnet. The room temperature coils are the ones
used by the user for 'shimming' each individual sample. In practice the optimization of
the field homogeneity exploits the ‘lock’ experiment. The D2O in our sample gives a
single line in the NMR spectrum. The integral of this line is constant (as it only depends
on the number of nuclei in our sample, which is constant) but the height of the line is not:
The narrower the line the higher its maximum. The NMR operator can now manually
adjust the different currents in the different shim coils to optimize this value. This was
and still is a very time consuming procedure which requires some experience. Fortunately
32
a very fast automatic shimming method is available nowadays which employs pulsed
field gradients (PFGs) which reaches very good results within minutes. This so-called
'gradient shimming' is available on all of our machines (except the 360).
6.4
The probe
The probe (or probehead) is in many ways the most critical component in the
spectrometer. It has two main functions:
a)
to convert the radio frequency power from the amplifiers into oscillating magnetic
fields (B1 fields) and to apply these fields to the sample.
b)
To convert the oscillating magnetic fields generated by the precessing nuclear
spins of the sample into a detectable electric signal that can be recorded in the
receiver.
Both points can be achieved by a parallel tuned circuit having a coil surrounding the
sample. The tuning is dependent from the sample (on position, volume, solvent, ionic
strength). The coil has to be carefully tuned to the frequency of the nucleus of interest
(remember: the frequency of the B1 field should match the Larmor frequency). This
adjustment of the circuit is important in several ways: First, we want to transmit the
maximum possible B1 field strength to the sample. This ensures that our pulses are as
short as possible, and therefore ensures a good excitation bandwidth. Second, since NMR
is a very weak phenomenon, we do not want to loose any signal coming from the sample
by picking up only a fraction of the oscillating magnetization.
There is a variety of NMR probes. For 1H spectroscopy typically probes are used which
can hold sample tubes of 5mm diameter (with a sample volume of ~500 μl). Beside the
proton channel there is another coil for the lock system tuned on 2H (sometimes a single
double tuned coil is used for both frequencies). The same setup can be found in probes
for 3mm and 10mm tubes. The sensitivity of the probes is still improvable as reflected by
the increased sensitivity over the past 10 years (nearly a factor of 2). This shows how
33
critical coil design is for NMR purposes. In a quite recent development, cryogenic probe
systems were introduced, which consist of a probe which can be cooled with cold helium
in order to reduce the amount of electronic noise in the receiver coils (and preamplifiers)
to a minimum. The technological challenge of such a system, among others, is the fact
that the temperature of the sample, of course, must still be adjustable to as high as about
80ºC without heating the cold part of the probe. Since, on the other hand, the coils of the
probe are supposed to be as near as possible to the sample, the difficulties of designing
such a system are obvious. The gains in sensitivity with the installation of such a system
to an existing spectrometer are remarkable and can be more than a factor of 2.2 (compare
the relative sensitivities at different fields in chapter 6.1).
6.5
The radio frequency system
The rf system mainly consists of pulse generation units, the actual transmitters and
subsequent power amplifiers. It is generating the excitation pulses at the frequency of the
nucleus of interest. On modern spectrometers the frequency can be set with a precision of
0.1 Hz across a band many megahertz in width. Since we may want to apply RF pulses
out of several directions in the rotating frame, the phases of the RF waves also must be
adjustable (typically to 0.5 degree). These settings are under extremely fast computer
control with setting times of only some microseconds. The power amplifiers boost the
transmitter output to high levels (from several tens up to hundreds of watts). This assures
that short, non selective pulses can be applied to the sample.
34
6.6
The receiver
The final stage in an NMR experiments is the detection of the precessing magnetization
(x- and y-components) in the sample. As stated earlier the same coil is used for this
purpose as for excitation. This means that directly before the data acquisition the
transmitter system has to be blanked and the receiver has to be opened (this ensures that
no strong RF pulses are applied while the sensitive receiver system is on). The detection
of the high frequency signal (MHz) is quite involved. First, analog filters are applied to
the signal to reduce it to the relevant frequency range. Then the weak signal is amplified.
The incoming signal is now mixed during several stages with reference frequencies. This
mixing reduces the frequency of the signal from several MHz to the audio range (kHz).
Finally, the signal is digitized in real-time and stored. For digitization the following
relation is important: If we want to detect a certain spectral width SW we have to digitize
the signal with a time dw ('dwell time') or faster ('Nyquist theorem'):
dw =
1
2 SW
(6.2)
An example (see the following figure): Assume we digitize our FID every 5ms,
corresponding to a spectral width of 100 Hz. A frequency of 100 Hz will be sampled
twice per period (solid line), which is enough to characterize this frequency. On the other
hand, a resonance precessing with 120 Hz (dashed line) is sampled less than twice per
period, thus the frequency can not be
distinguished from a slower frequency
(80 Hz in this case, dotted line). This
means that both frequencies would
give a signal at the same position!
35
All further operations after the digitization and storing of the signal are performed in the
data processing system (the computer workstation). This includes application of window
functions, zero-filling, Fourier transformation, phase corrections, baseline corrections,
integrations in several dimensions as well as displaying and plotting the final spectrum.
36
The components of a spectrometer at a glance:
RF generator (radio sender): creates the rf signal with a frequency of less than 100MHz
up to about 900MHz.
Pulse generator: creates rf pulses of a duration of several µs up to several seconds.
RF amplifier: amplifies the pulse signal up to several 100 Watts.
Magnet: generates the B0-field (from about 1T up to about 21T).
Probe: holds the sample and houses the send and receive coils.
RF amplifier: amplifies the received signal from the probe.
Detector: Subtracts the base frequency from the signal, resulting in an audio frequency
(up to several KHz), containing only the differences of the resonance frequencies from
the base frequency.
AF amplifier: amplifies the audio signal.
ADC: Analog-to-digital converter.
Computer: controls all the other electronic parts, receives, stores and processes the NMR
signal.
37
VII
NMR Parameters
7.1
Chemical shifts
We have seen that the resonance frequency of a nucleus depends on its gyro magnetic
ratio γ and the magnetic field Bz. If all nuclei of the same kind (e.g. protons) would have
an identical Larmor frequency then NMR would not be a very useful technique for
studying biomolecules – we would observe just one line per sort of nucleus. Fortunately,
this is not the case since in practice different spins, even from the same sort, have a
slightly different Larmor frequency. This is because not all nuclear spins experiences the
same effective static magnetic field Beff. Instead they experience the superposition of the
external field Bz and a local field Bloc. The static field Bz induces currents in the electron
clouds surrounding each nuclear spin. These induced
currents result in local magnetic fields. The induced
current will counteract its cause (‘Lenz law’,
electromagnetism), thus the induced field will be
opposed to Bz. The nuclear spins will be ‘shielded’
from the external field. The strength of this shielding
depends on the electron density around each
individual nucleus and the strength of the static field.
Bloc = −σ Bz
(7.1)
where σ is a quantity expressing the amount of shielding. The net field experienced by
the spin becomes
Beff = B z + Bloc = B z (1 − σ )
and the new Larmor frequency is given by (compare Eq. 2.10a)
(7.2)
38
ν = γ (1 − σ )
Bz
2π
(7.3)
The shielding σ is different for different types of nuclei in a molecule because the
electron density around a nucleus is very sensitive to the chemical environment of the
nucleus (e.g. chemical bonds and neighbours). The amount of shielding is usually given
as a dimensionless parameter δ, the chemical shift, which expresses the difference in
NMR resonance frequency with respect to a reference signal
δ = 10 6 ⋅
(ν − ν ref )
ν ref
= 10 6 ⋅
σ ref − σ
≈ 10 6 ⋅ (σ ref − σ )
1 − σ ref
(7.4)
since σ ref << 1.
For the reference of the δ-scale the single line of the methyl-protons of Si(CH3)4 (TMS,
Tetramethylsilane) can be used (δ ≡ 0). For biomolecules, slightly different compounds
(e.g. TSP, (CH3)3SiCD2CD2CO2Na, Sodium-salt of trimethylsilyl-propionic acid) are
used since TMS is not soluble in water; sometimes the water resonance itself is taken.
The latter one is temperature dependent:
δ ( H 2 O ) = 7.83 −
T [ Kelvin ]
96.9
(7.5)
The dimensionless δ-scale (ppm) has the important advantage over a frequency scale
(Hz) that the chemical shift values become independent of the magnetic field Bz.
39
7.1.1 Effects influencing the chemical shift
As was mentioned above, the s-orbital electrons generate a field opposing the static field,
a shielding effect. The p-orbital and other orbital electrons with zero electron-density at
the nucleus result in a weak field reinforcing the static field. The contributions of these
two effects are of well known magnitude for the different functional groups. The table in
appendix A gives the common chemical shift values for a number of different functional
groups. A small value of δ corresponds to a shielded proton, or a low resonance
frequency.
In addition to the “constant” effects of s- and p-orbitals to the chemical shift, there are
also variable contributions resulting from the local surrounding of the nucleus (e.g.
solvent effects or interaction with other parts of the same molecule) and the local
conformation.
Aromatic and carbonyl groups have an extensive conjugated π-electron system
comprising delocalized molecular orbitals. Also in these systems the Bz-field induces
currents, the so called ring-currents, resulting in quite large magnetic moments. Their
effect on δ of a particular nucleus depends strongly on the distance and orientation with
respect to the aromatic system: above and below the aromatic ring-system an opposing
field is generated and Δδ is negative (‘upfield shift’). In the plane of the ring-system a
reinforcing field is generated and Δδ is positive (downfield shift). These effects can be as
large as -2 to 2 ppm and all nearby protons are affected. The effect decreases with 1/r6.
Paramagnetic groups, like Fe3+ in the heme-system of haemoglobin, can have a
pronounced effect on chemical shifts. An unpaired electron influences the proton
chemical shift through spatial interactions, the electron magnetic moment, and by direct
electron-proton hyperfine interaction.
Electric fields resulting from charged groups or electric dipoles polarize the electron
clouds and thus influence the chemical shifts.
H-bond formation strongly influences the chemical shift. The proton in an X-H···Y
hydrogen bond is very little shielded and thus has a large δ value. For example, the imino
40
protons in the Watson-Crick hydrogen-bonded bases of a B-DNA fragment resonate at
14-15 ppm, whereas the non hydrogen-bonded protons resonate at 10-11 ppm.
7.1.2 Protein chemical shifts
From section 7.1.1 it will be clear that also the 1H spectra of proteins will have a 'general'
appearance. An example 1H spectrum of a protein is given below:
Indicated are the typical regions where the different proton resonances are found. It is
even possible to split out the different chemical shift ranges on a per-residue basis. This is
shown in Appendix B.
Although the theory of chemical shifts is well known, it is quite complicated in practice
to accurately predict the chemical shifts in proteins. Partially this is the result of the
41
inaccuracy of the protein structures and the internal mobility. On the other hand the range
of proton chemical shifts is fairly limited (ca. 12 ppm) and the exact geometry is
relatively important.
The range of 13C chemical shifts is much larger (ca. 200 ppm) and the effects of the exact
geometry are less important. 13C chemical shifts are therefore easier to predict and can be
used in a more straightforward fashion for interpretation of spectra.
7.2
J-Coupling
J-coupling is the interaction between two spins transferred through the electrons of the
chemical bonds between them. The resonance frequency of a spin A depends on the spinstate of a second spin B and vice versa. If spin A is in the α state spin B will resonate at
slightly lower frequency whereas it will resonate at slightly higher frequency when spin
A is in the β-state. In the NMR spectrum we observe a doublet (two lines with equal
intensity) centered on νb, the resonance frequency of spin B without J-interaction. Also
for spin A a doublet is observed, centered around νa . The size of the coupling is JAB (the
42
distance between the components of the doublets) and is called the coupling constant. It
has the dimension Hz.
If there is a third J-coupled spin C the pattern splits again: the result is a doublet of
doublets. This means that the coupling of the third spin is independent of the coupling
between the first two spins. It just introduces another splitting on the first splitting. If the
spins B an C are magnetically equivalent the doublet of doublet collapses into a triplet
(three-lines with intensity ratio 1:2:1) Three equivalent neighbours (e.g. the three protons
43
of a methyl group) result in a quartet (four lines with intensities 1:3:3:1). As you can see
the intensity ratios follow the famous Pascal triangle.
7.2.1 Equivalent protons
In general equivalent protons are protons which are chemically and magnetically
equivalent. Chemical equivalence means that there is a symmetry axis in the molecule for
the two protons under consideration. Nuclei having this type of equivalence resonate at
the same frequency. For magnetic equivalence the nuclei must be a) chemical equivalent
and b) must experience exactly the same J-coupling with all other nuclei in the molecule.
Magnetically equivalent nuclei are a very special case in the coupling network: They do
not couple with each other. This explains why for instance the benzene spectrum shows
only one line. Due to the high symmetry of the molecule all six protons are magnetically
equivalent, thus showing only one frequency and no coupling with each other.
In proteins magnetic equivalence due to symmetry is rare because of the high complexity
of the biomolecules. But also here some protons can be equivalent. If a group of atoms
rotates fast enough they become magnetically equivalent as a result to dynamic
averaging. This is the case for e.g. methyl groups or fast rotating aromatic rings.
The magnitude of the J-coupling depends on the number of intervening chemical bonds,
44
the type of chemical bonds, the local geometry of the molecule, and on the γ values of the
nuclei involved. Proton-proton couplings in biomolecules are observed for protons
separated by two- or three chemical bonds. The magnitude of these proton-proton Jcouplings is relatively small, typically 2-14 Hz. Often the patterns resulting from these
2
JHH (two bonds) and 3JHH (three bonds) couplings are not resolved because of the large
line width in biomolecules.
The magnitude of heteronuclear one-bond couplings is much larger: the J-coupling
between the amide proton and the directly bond 15N nucleus, 1JNH is ca. 92 Hz. The onebond coupling between a proton and its directly attached 13C nucleus, 1JCH is ca. 140 Hz.
J-couplings involving
15
N or
13
C nuclei also exist between nuclei two- or three bonds
apart. For example, there is an interaction between Hα and
HαC(C=O)15N -fragment.
15
N over three bonds in a
45
VIII Nuclear Overhauser Effect (NOE)
The observable intensity of the signal of a nucleus depends on the intensities of other
nuclei when they are in close spatial proximity. If, for instance, two protons are situated
at a distance of less than 5 Å and the signal of one of them is saturated by selective
irradiation, the other signal will change in intensity. This effect is called the nuclear
Overhauser effect (NOE). It is the result of a relaxation process, caused by a dipoledipole interaction (dipolar coupling)between the two nuclei. We are talking here about
cross-relaxation, because the population of the spin-states of one nucleus depends on the
population of the spin-states of another one.
8.1
Dipolar Crossrelaxation
Let us consider a spin system with two spins
A and B which are dipolar coupled (spatial
proximity!).
In
the
steady-state
NOE
experiment one resonance is selectively
saturated by rf irradiation (let's say spin B).
This disturbs its equilibrium magnetization,
therefore spin B tries to re-establish it by
exchanging
magnetization
with
its
environment. This can either be the lattice or
another nucleus close-by. The NOE between
the saturated spin B and another spin A is
defined by the relative change in the intensity
of spin A:
46
NOE = 1 + η with η =
( M a − M aeq )
M aeq
(8.1)
The energy level diagram for this twospin system is sketched on the right. Each
of the spins can undergo transitions
between its α and β state resulting in the
resonance lines which are observed. The
rates of these transitions are W1a and W1b
for the transitions of the spin A and B,
resp. Dipolar interactions between spin A and B introduces two more possible transitions:
W0 and W2. These transitions involve simultaneous changes in the spin states of both the
A and the B spin.
How this can be understood is schematically shown in the following figure:
A = 1.5 Δ
A
W2 > W0
W1A
W1B
A
W0
W1B
W1A
A
small molecules
W2
A
W0 > W2
A
large molecules
A0 = B0 = Δ
A = A0 = Δ
A
B=0
A = 0.5 Δ
47
The most left diagram represents the situation at equilibrium. A0 and B0 are the relative
population differences in equilibrium (corresponding to WA and WB). A and B are the
relative differences in population of the spin-states of the A and B nucleus after saturation
(diagram in the middle) and finally after cross-relaxation (diagrams on the right).
Saturation of the transitions of nucleus B leads to the situation in the middle. As a
consequence of the saturation, the population differences for the spins-states of nucleus B
disappear. Now the cross-relaxation comes into effect. Two different cases are
distinguished here: For small molecules, W2 dominates over W0 and the result is shown
in the upper right diagram. The relative population difference for the states of the A
nucleus has increased. Consequently also the observed signal for A will be increased. For
large molecules, W0 dominates over W2 and the result is depicted in the lower right
diagram. The relative population difference for the states of the A nucleus has decreased
and consequently also the observed signal for A will be decreased. One can easily
imagine a situation, where the two cross-relaxation mechanisms just cancel each other.
Indeed a zero-crossing of the NOE is observed when ωτc ≈ 1.18 (solution in water at
room-temperature):
0.5
η
0.0
-0.5
-1.0
0.01
0.1
fast
tumbling
1.0
ωoτ
10
100
slow
tumbling
48
Suppose for a particular molecule with a particular size on a particular NMR
spectrometer the 'observed' NOE turns out to be zero. Are there any options to still
observe NOEs within this molecule? Well, obviously we can do nothing about the size of
the molecule, but the tumbling speed of the molecule, of course, depends on the viscosity
of the solvent. The viscosity is usually very sensitive to changes in the temperature. So,
when we change the temperature of the sample, the tumbling speed will change and we
probably have a chance to pick up NOEs now. The other parameter which we probably
can change is ω0, which depends on the spectrometer frequency. So we could just repeat
the experiment on a spectrometer with a different field and chances are high that we
moved away from the zero-crossing situation. A more formal derivation of the NOE is
given in appendix C.
8.2
NOEs in biomolecules
T1 relaxation times are rather uniform for biomolecules. In contrast, cross-relaxation rates
vary a lot since the strength of the dipole field is strongly dependent on the distance
between the protons. A good approximation for the cross-relaxation rate, σ, in a
biomolecule is
σ~
τc
r6
(8.2)
where τc is the rotational correlation time of the proton-proton vector, and r the distance
between the protons.
Since T1 values are uniform we can write for the NOE
η~
τc
r6
(8.3)
49
Therefore, the measured NOE can be converted to a distance. We can calibrate the NOE
by comparing it with a NOE of a fixed distance
6
η rref τ c
=
⋅
η ref r 6 τ cref
(8.4)
If we now assume that there is no internal mobility (a rigid molecule), τc will be uniform
in the entire molecule. We can now directly calculate the distance
r = rref
6
η ref
η
(8.5)
Eq. 8.5 provides the basis of the most important effect for structure determination by
high-resolution NMR spectroscopy: the extraction of NOE-based distances between
proton pairs.
An example: In a protein we observe the following NOEs between the aromatic ring
protons and some other protons. The distance between CδH and CεH is fixed at 2.45Å.
NOE
intensity
distance
Tyr CδH - Tyr CεH
0.15
2.45
Tyr CεH - Val CαH
0.02
3.43
r = 2.45 · (0.15 / 0.02)1/6
Tyr CεH - Asp CαH
0.01
3.85
r = 2.45 · (0.15 / 0.01)1/6
Internal motions are often fast (ωτc,intern < 1) and contribute predominantly to W2.
Therefore, the NOE in mobile sub-domains will be reduced in intensity since
50
σ = W0 − W2
(8.6)
In the case of internal mobility we can not give the exact value of the distance but only an
upper limit. This is because of
r = rref ⋅ 6
and
τc
≤1
τ cref
η ref τ c
⋅
η τ cref
(8.7)
(8.8)
51
IX
Relaxation Measurements
9.1
T1 relaxation measurements
The so-called inversion-recovery pulse
sequence, 180°-τ-90°-acquisition, can
be used for measuring the longitudinal
relaxation time T1. At the start of the
sequence, the equilibrium magnetization
is inverted by a 180° pulse, after which the magnetization
Mz(0) = -Mzeq . The
magnetization will return to its equilibrium value ('recover') with the relaxation time T1,
and after time τ we have:
M z (τ ) = M
eq
z
−τ ⎤
⎡
T
⎢1 − 2e 1⎥
⎢
⎥
⎣
⎦
(9.1)
This time dependency is shown in the
figure on the right. We can measure the
value of Mz(τ) by applying a 90° detection
pulse, which will rotate the z-component
into the xy-plane. The FID is recorded and
the spectrum extracted by a Fourier
transformation. For τ < ln(2) T1 the
spectrum is inverted. For longer values of τ, Mz(τ) has recovered to positive values and a
positive signal is recorded. By repeating the experiment with increasing values of τ, the
relaxation behavior can be determined and T1 extracted.
The T1 analysis is not limited to a molecule with a single resonance. In a molecule with
more spins, each of the individual spins, j, with frequency ωj and relaxation time T1j has
52
the starting magnetization:
M zj (0) = − M zjeq
(9.2)
Therefore, we can determine the individual T1j by monitoring the intensities of the
individual resonances j at ωj in the spectrum as a function of τ.
The different time points τ of the inversion recovery sequence are measured one after
another. In case more than one FID is recorded per value of τ, e.g. for S/N improvement
or in multi-dimensional experiments (discussed in chapter 8), care has to be taken that
saturation is avoided and enough time is allowed between the individual experiments for
the magnetization to relax completely. The repetition rate of the experiment should be
adjusted so that at least a time of 5·T1 is waited before the experiment is repeated.
53
9.1.1 Calculated example for an inversion-recovery experiment
We follow the intensity of one resonance and take the measurements at τ = 0.001s and
τ = 3s as start and end points. Note that the inversion was incomplete (as a result of an
imperfect 180° pulse). Therefore we have to use the more general Eq. 4.2 in place of Eq.
9.1:
“Mzeq”
Since
τ
[
]
τ (s)
Mz(τ)
ΔM(τ)=Mz(τ)-Mzeq
ln{ΔM(τ)/ΔM(0)}
0.001
-12.0
-27.0
-
0.01
-9.4
-25.4
-0.06
0.03
-7.1
-22.1
-0.20
0.1
1.1
-13.9
-0.66
0.2
7.9
-7.1
-1.36
0.3
11.3
-3.7
-2.01
1.0
14.9
-0.1
-5.59
3.0
15.0
0
M z (τ ) − M eq = M z (0) − M eq e
“Mz (0)”
−
T1
⎧ ΔM (τ ) ⎫
τ
ln ⎨
⎬=−
T1
⎩ ΔM ( 0) ⎭
T1 can be determined by linear regression analysis:
Στ = 1.64; Σln(...) = -9.88;
and thus
T1 = Στ / Σln(...) = 0.165 s.
(9.3)
54
9.1.2 Applications of T1 relaxation
T1 tells us how long we have to wait until the equilibrium magnetization is restored. This
is important information for the setup of FT NMR experiments, where a number of
experiments are repeated and the results added up in the computer. It depends on T1 how
fast we can repeat our experiments.
9.2
T2 relaxation measurements
The value of T2 could in principle be extracted from the envelope of the FID or from the
line width at half-height (Eq. 6.8). However, T2 values obtained this way also depend
strongly on the static field inhomogeneities (the “shimming”). If the main magnetic field
is not homogeneous over the whole sample, spins at different locations in the sample tube
experience a slightly different field. This results in slightly different resonance
frequencies. The total peak, which is the sum over all individual spin contributions in the
sample, will be broadened due to this effect. The apparent relaxation time is T2* which is
faster than T2. Since only T2
depends on the physical properties
of the molecule, this is what we are
actually interested in. Obviously,
the contribution of a 'bad' shimming
to the relaxation of our molecule is
less interesting for us! 'Pure' T2
times can be determined by the so
called 'spin-echo' pulse sequence,
shown on the right.
The equilibrium magnetization, Mzeq, is transferred into y-magnetization, My, by the
90°x-pulse. The macroscopic vector My can be considered to consist of a sum of
macroscopic magnetizations Mj at j different positions in the sample. Now we look at the
55
rotating frame (which rotates with the average Larmor frequency ω0 of all the spins j): As
a result of an inhomogeneous field, we will find that some of the Mj components will
rotate faster than ω0 and some will rotate slower, depending on the exact position in the
sample. In the rotating frame (ω0), the fast and slow components will start to precess in
the xy-plane with frequency
γ B z ( M j ) − ω 0 = γ ΔB j
(9.11)
This frequency is different at different locations. Hence, the transversal magnetization My
will dephase due to the inhomogeneous field.
It can be shown that the spin-echo sequence eliminates the dephasing that results from
these static field inhomogeneities. In order to explain how this works, we actually should
consider the precession of each of the individual components Mj, but fortunately the
principle can be shown by picking only two spins, rotating with different speed: a slower
black and a faster white one.
At a point (a) in the sequence we have pure y magnetization for all individual spins. At
56
point (b) some phase coherence is lost because each spin has precessed with its own
frequency. The white one a bit faster, the black one a bit slower. After the delay τ/2 a
180° pulse from the y direction is applied (point (c)). This pulse will invert the x-
component of mj, but will not affect the y-components. During the second delay τ/2, the
vectors (Mj) precesses again with their individual frequency γΔBj, the white one still abit
faster and the black one still abit slower and still in the same direction as before the
180° pulse. Censequently, in point (d) both magnetization vectors have returned to the y
axis, independent of the static inhomogeneity, creating a so called 'spin-echo'. Thus, we
have eliminated the effect of field inhomogeneity! By repeating the experiment for
several values of τ in the range 0 - 4·T2 we can determine T2 from the decay of the
intensities of the resonances in the spectrum, which now results purely from the
relaxation by random fluctuating fields and is independent from any static field
inhomogeneity.
9.2.1 Applications of T2 relaxation
In section 6.1 we have seen that the T1 and T2 values depend on the motional behavior of
the dipole-dipole vector and thus the rotational correlation time τc of the molecule. Thus,
analogously to the usage of T1, we can use T2 to determine motional parameters.
57
X
Two-Dimensional NMR
In the seventies the development of two-dimensional (2D) NMR has revolutionized
NMR spectroscopy and has made the structural studies of biomolecules possible. The
basic idea is to spread the spectral information in a plane defined by two frequency axes
rather than linearly in a conventional one-dimensional spectrum. Clearly, this provides a
large increase in spectral resolution. Also, in a 2D NMR experiment interactions between
many spins in a molecule (whether it be J-coupling or NOE-type interactions) can be
measured simultaneously. This represents an enormous time-saving for large
biomolecules. To illustrate the method we will discuss first a very simple 2D NMR
experiment.
10.1
The SCOTCH experiment
SCOTCH stands for spin coherence transfer in (photo) chemical reactions. If one has, for
instance, a photochemical reaction
ν
A ⎯h⎯→
B
(10.1)
a proton which resonates in molecule
A at ωA will resonate at ωB in
molecule B. The SCOTCH experiment
correlates the resonance frequencies in
A and B for each particular proton. In
other words it enables us to find ωB in
B that corresponds to ωA of the same
proton in A. The pulse sequence is as shown above. The 90° pulse creates xy
58
magnetization which precesses with ωA during the so-called evolution period t1. The light
pulse changes the precession frequency to ωB with which it is detected as an FID during
the detection period t2. The trick is now to increment t1 in a regular fashion and collect a
large number (typically, say, 500) FIDs belonging to different t1 values. Thus one records
a data set S(t1, t2) depending on both t1 and t2 (the FIDs). A first Fourier transformation
t 2 → F2 leads to a so-called interferogram S (t1, F2) and after a second Fourier
transformation t1 → F1 we arrive at the 2D spectrum:
S (t1 , t 2 ) ⎯FT
⎯
⎯→ S (t1 , F2 ) ⎯FT
⎯
⎯→ S ( F1 , F2 )
(10.2)
FIDs
interferog ram
2D spectrum
The effect of incrementing t1 is to "sample" the frequencies present during the evolution
period. How this works out in practice is illustrated in Figure 10.1 (next page). After each
t1 time the signal acquired a different phase. The FIDs recorded that way have all the
same frequency (ωB), but their phase (i.e. how far the magnetization vector rotated in the
xy-plane) depends on the evolution time t1. After the first Fourier transformation this
leads to a peak at the position ωB in the F2-dimension with an intensity that oscillates in
the t1 direction with the frequency ωA. Looking along the t1 axis of the interferogram the
signal actually looks like an FID oscillating with the frequency ωA. Hence, after double
Fourier transformation this gives a peak at (ωA, ωB) (ωA in the F1- and ωB in the F2dimension). This is called a cross-peak, in contrast to peaks on the diagonal which are
called diagonal peaks. So, indeed, the experiment gives the connection (a spectroscopist
says: 'correlation') between the resonance frequencies in A and B. In this case this is
rather trivial but if there are many nuclei this method can prove very useful.
All 2D NMR experiments adhere to the following scheme:
59
In the example above the preparation period would include a relaxation delay and the 90°
pulse. The mixing period is just the light pulse.
Fig.10.1: The SCOTCH 2D NMR experiment. On the left side the sampling of the ωA frequency is shown
at different values of t1. After Fourier Transformation of the FIDs (t2 → F2) the interferogram S(t1, F2)
consists of lines at ωB in F2 with intensities dependent on t1 oscillating with ωA. The second Fourier
transform (t1 → F1) leads to a 2D spectrum with a single cross-peak at (ωA, ωB). The representation is as a
contour plot.
60
10.2
2D NOE
2D NOE or NOESY is one of the
most
important
2D
NMR
experiments, because it measures
all short inter-proton distances, for
instance in a protein, in a single
experiment. The first 90° pulse belongs to the preparation period. The evolution time t1 is
incremented and mixing consists of two 90° pulses separated by a constant mixing time
τm. During τm magnetization between neighbouring spins is exchanged via cross-
relaxation (see section 8). For biomolecules σAB ≈ W0 and therefore the flip-flop
transitions (αβ → βα) are dominant for the NOE effect. We will now look at the 2D
NOE experiment in more detail. Let us assume that there are two spins, A and B, within
NOE distance, and that the carrier frequency is chosen at the Larmor frequency of spin
A, ωrf = ωA. The vector diagrams at various times in the 2D NOE pulse sequence then
look as shown in Figure 10.2.
After the first 90° pulse the magnetization vectors lie along the y-axis in the rotating
frame. During the evolution time t1 the A-vector precessing at ωrf will stay the same (at
least for short times when relaxation can be neglected), while the B vector precesses with
a frequency ωB – ωrf. The second 90° pulse tips the A-vector to the negative z-axis and
the B-vector into the xz-plane. The z-component of B is shorter than that of A. We see
that the variable t1 time acts to create z-components with different magnitudes depending
on the different Larmor frequencies (i.e. how far a particular vector did rotate in the x-y
plane during t1. This is called frequency labeling and is a common feature of the
evolution period (t1 period) of 2D NMR experiments. Focussing now on these z-
components that correspond to populations of energy levels, we know that W0 transitions
during the mixing time τm will tend to equalize populations of αβ and βα states (because
at equilibrium they are equal). Thus, the z-components of the A and B magnetizations
also will become more equal. Finally, the third 90° pulse flips the vectors in the xy-plane
where the signal can be observed.
61
Fig. 10.2. Vector diagram of the 2D NOESY experiment.
Now let us see how this leads to cross-peaks in the 2D NOE spectrum. In Fig. 10.3 we
shall look at the magnetization vectors at time d in Fig. 10.2 (after the mixing time τm).
Let us first consider a trivial case where no transfer occurs, for instance because the spins
are too far apart, and then the more interesting case where cross-relaxation occurs
between A and B.
62
The vectors are depicted for various evolution times t1 chosen such that the B-vector has
rotated through 0, 90°, 180°, 270°, and 360°. If no mixing occurs the vectors precess at
their own frequencies ωA and ωB during t1 and continue to do this during the detection
period t2. Thus, this leads to a 2D spectrum after double Fourier transformation with only
diagonal peaks at (ωA, ωA) and (ωB, ωB).
Fig. 10.3. Vector representation of spin A (solid vector) and spin B (dotted vector) at time point (d) in Fig.
10.2 for various values of t1.When no mixing occurs during τm the 2D NOE spectrum consists only of
diagonal peaks. In the case of magnetization transfer during τm cross peaks arise at (ωA, ωB) and (ωB, ωA).
In contrast, when mixing occurs in τm the equalizing effect of the W0 transitions causes
the A-vector to borrow intensity from B and vice versa. Thus, the A-vector is now
modulated with ωB for the different values of t1! Since the vector will continue to precess
at ωA in t2 this will lead to peaks both at (ωA, ωA) and (ωB, ωA) in the 2D spectrum (F1,
F2). These are the diagonal peak that we have seen before at (ωA, ωA) and an off-diagonal
cross-peak at (ωB, ωA) at the upper left half of the spectrum. In the same way, the
modulation of the B-vector with ωA leads to a symmetry related cross-peak at (ωA, ωB)
63
below the diagonal. Because mixing is reversible 2D NOE spectra are always
symmetrical. As all proton pairs within 5Å will give rise to cross peaks with intensities
inversely proportional to r6, the 2D NOE spectrum provides a map of all short protonproton distances in a biomolecule.
64
10.3
2D COSY and 2D TOCSY
In another important class of 2D NMR
experiments the magnetization transfer in
the mixing period takes place via the Jcoupling. The simplest is the COSY
(correlated spectroscopy) with the pulse
sequence shown at the right. Here the mixing period is just the second 90° pulse, which
transfers magnetization between A and B spins whenever there is a J-coupling between
them. This results in the 2D spectrum as shown in Fig. 10.4.
Fig.10.4: COSY spectrum of two J-coupled nuclear spins A and B. The sign of the cross-peaks is indicated.
The regular 1D spectrum is drawn above. It can be seen that the cross-peaks reflect the fine structure of the
1D spectrum (doublets in this case) and therefore can be used to measure J-couplings. Note the typical
negative-positive sign pattern in the cross-peaks (but not in the diagonal peaks).
65
COSY spectra are often recorded at low resolution so that the fine structure is not visible.
In this way they are used to trace networks of J-coupled nuclei. Such low-resolution
COSY spectra are shown below for the two amino acids alanine and valine:
As measurable J-couplings only arise between nuclei separated by less than three
chemical bonds, the connectivity patterns can be easily predicted. We note that in a
protein a network of J-coupled protons does not extend beyond an amino acid residue
because the CαH of residue i is separated from the NH of residue i+1 by four chemical
bonds. Each of the amino acids forms a separate spin-system. A COSY spectrum is a
valuable tool for the identification of the types of amino acids.
Another important J-coupling based 2D experiment is the TOCSY which stands for total
correlation
spectroscopy.
The
pulse
sequence is shown on the right. The
mixing
period
here
consists
of
a
complicated pulse train. This has the
effect
of
transferring
magnetization
through a whole network of J-coupled spins.
66
For instance, in a molecular fragment
we have non-zero J-couplings JAB and JBC but JAC is close to zero. During the TOCSY
mixing period A-magnetization is transferred to B via JAB and then to C via JBC in
multiple transfer steps. Hence a cross-peak will arise between A and C even though there
is no direct J-coupling between these spins. To illustrate this, a comparison between
COSY and TOCSY spectra for the ABC fragment is shown in Fig. 10.5:
Fig. 10.5: COSY and TOCSY spectra of a three proton spin-system where JAB and JBC are non-zero and
JAC=0. Because of the symmetry only the part above the diagonal has been drawn.
It should be clear from this that a TOCSY spectrum always contains the COSY as a subspectrum. Although the information content of COSY and TOCSY spectra is in principal
the same, in complicated spectra with a lot of overlap the TOCSY spectrum is still useful.
This is because if B and C in the example of Fig. 10.5 are in crowded spectral regions but
A is not, then the whole spin system can be observed on a vertical line at the A-position,
67
while this would be difficult for B and C. We leave the sketch of the TOCSY spectrum of
alanine and valine as an exercise (compare Fig. 10.4 and 10.5).
Finally, it should be mentioned that the TOCSY is a sub-spectrum of the NOESY for
almost all cross-peaks. We will come back to this point in the following chapter.
68
XI
The assignment problem
The interpretation of an NMR spectrum always starts with the identification of resonance
frequencies and their corresponding nuclei in the molecule. This so called ‘assignment’ of
resonances constitutes an essential step in the structure determination process by high
resolution NMR spectroscopy that always precedes the actual calculation of structures.
While the assignment of smaller organic compounds with only a few 1H nuclei can often
be solved easily by means of a single experiment (e.g. COSY), it is much more
complicated for bigger and more complex molecules like peptides, proteins and nucleic
acids. Not only does the number of resonances increase with increasing size, also the line
width increases as a consequence of the shorter T2 relaxation times. As a result the lines
become broader and the overlap becomes increasingly severe.
We focus here on the basic principles of assignment, i.e. which parameters of the NMR
spectrum can be used. We will come back to the special case of the assignment of spectra
of biomacromolecules in the next chapter.
11.1
Chemical shift
The chemical shift of signals gives a first indication of the surrounding of the
corresponding nucleus in a molecule. We saw before (chap. 7.1.2) on the example of a
protein spectra, that proton chemical shifts usually are grouped according to the
environment in which they are located in the molecule. In our example those were the
amide and aromatic protons in the left half of the spectrum, the Hα protons just right of
the water signal (center of the spectrum), protons of aliphatic side chains still to the right
of them and finally the methyl groups on the right edge of the spectrum. If you look at the
vast variety of organic molecules you can find many more functional groups and
chemical environments a nucleus can be situated in. Most of the protons belonging to
such a group share their individual ranges of chemical shifts. Tables are available for 1H
and 13C chemical shifts, which can help to identify the origin of particular resonances in
69
NMR spectra (see appendix).
11.2
Scalar coupling
From chapter 7.2 we know that we can identify coupled nuclei on hand of their coupling
constant. If we look, for instance, at the signals of the methyl group and the CH2 group of
ethanol, we find them split into multiplets due to scalar coupling. The number of
multiplet components gives us an indication of how the neighbouring group looks like
(i.e. the signal of the methyl group is a triplet due to the coupling with the two equivalent
protons of the neighbouring CH2 group. The signal of the CH2 group is a quartet due to
the coupling with the three equivalent protons of the neighboring methyl group). The
distance of the components of these multiplets (the coupling constant) is exactly the same
in the triplet and in the quartet. This helps to identify partners with a scalar coupling
between them.
11.2
Signal intensities (integrals)
The intensity, or better the integral of a signal tells us to how many equivalent nuclei a
particular signal corresponds to. The integral of the proton signal of a methyl group is just
three times as large as the integral of a single proton in the same molecule.
12.3
NOE data
NOE data can be very helpful for the assignment, especially when we already have an
idea of (basic) structural features of the molecule we are looking at. It is quite straight
forward for example to identify neighbouring protons in an aromatic ring system, because
their distance from each other is quite short and well known (~2.45Å). Once we know
one of them, we can relatively easily identify the others on hand of a NOE spectrum.
Especially in biomacromolecules, where the spin systems of the individual residues
cannot be connected by scalar coupling experiments, NOE data is often the only outcome
to still get a sequential assignment.
70
XII Biomolecular NMR
Some of the most important sorts of biomolecules are nucleotides and amino acids.
Although also lipids and carbohydrates play important roles in biological processes, in
this course we will focus on the first two groups. Nucleic acid is the bearer of the genetic
information and is involved in protein synthesis where it acts as a template containing the
sequential information for all proteins occurring in organisms. Each three consecutive
nucleotides in a gene code for a particular amino acid in a protein. In addition there are
control regions (stop codons and sequences were proteins involved in transcription and
transcription regulation bind). The role of proteins is very diverse. We know them for
example as enzymes and regulators, as building material of cells and their compounds,
and as carrier of information within and between cells. Obviously both nucleic acids and
proteins play a major role in the function of all living organisms and accordingly also
most defective disorders of them can be traced back to the malfunction of proteins or to
defects in nucleic acid. This makes these molecules to very popular subjects of study in a
variety of research disciplines. Their structural and functional understanding is supposed
to give insight into how and why they work and what probably goes wrong in the case of
diseases. Knowing the exact composition and function of a particular virus, e.g. can lead
to the development of anti-viral drugs. The exact knowledge of the structure and function
of a particular enzyme can lead to the development of e.g. inhibitors which can deactivate
the enzyme when needed.
We will have a close look here at the structural properties of nucleotides and peptides and
how their different spin systems translate into different features in NMR spectra.
12.1
Peptides and proteins
Peptides and proteins are mainly build from twenty different naturally occurring amino
acids. They all share the same basic structure and only differ in their side chain R:
Amino group
H2N
Hα
O
|
Carboxyl group
Cα C
|
OH
R
Side chain
71
In peptides, amino acids are linked via a so-called peptide bond. The amino group of one
residue is connected with the carboxyl group of another:
H2N
H
O
|
Cα C
N
|
|
R
H
H
O
|
Cα C
OH
|
R
Note that the peptide bond is planar due to its partial double bond character! Amino acids
are usually referred to with either a one-letter or a three-letter code:
Glycine
Gly
G
Histidine
His
H
Alanine
Ala
A
Proline
Pro
P
Valine
Val
V
Aspartate
Asp
D
Leucine
Leu
L
Glutamate
Glu
E
Isoleucine
Ile
I
Asparagine
Asn
N
Serine
Ser
S
Glutamine
Gln
Q
Threonine
Thr
T
Lysine
Lys
K
Phenylalanine
Phe
F
Arginine
Arg
R
Tyrosine
Tyr
Y
Cysteine
Cys
C
Tryptophane
Trp
W
Methionine
Met
M
72
Amino acids can be classified by the character of their side chain as: aliphatic (A, V, L, I,
(G)), aromatic/ring (F, Y, W, H, P), carboxylic (D, E, N, Q), sulfur/hydroxy containing
(C, M, S, T, Y) and charged (K, R). The chemical formulas of the natural occuring amino
acids together with their COSY, TOCSY and NOESY spectra are shown in appendix F.
12.1.1 Assignment of peptides and proteins
A strategy based upon homonuclear 2D experiments (COSY, TOCSY, and NOESY) was
developed in the 80’s (K. Wüthrich, 1986). This approach is discussed in this chapter.
A more recent approach employs uniformly
15
N and
15
N,13C labeled proteins. The
strategy uses so-called triple-resonance experiments (involving 1H,
15
N and
13
C) to
transfer magnetization through the polypeptide chain employing the large one-bond
homo- and heteronuclear J-couplings, this method will briefly be discussed at the end of
this chapter.
For larger proteins several patterns corresponding to residues of a certain type are present
in a COSY, e.g. several alanines give rise to similar patterns. How can we decide which
of the alanines in the protein sequence corresponds to a particular pattern?
The solution involves three steps:
1. Find patterns of coupled interconnected spins (spin-systems) belonging to amino
acid residues using COSY and TOCSY (amino acid identification).
2. Connect neighbouring spin-systems (in the sequence) with sequential NOEs.
3. Match stretches of connected spin-systems with the (known) amino acid sequence
for unique fits.
Let us focus now on the individual steps.
73
Step1: Spin system identification
The identification of spins belonging to the same spin-system can be performed on the
basis of the COSY and the TOCSY experiment (see the example of an Ala-Ala peptide
fragment on the right). In both spectra, protons
belonging to a certain amino acid can be identified. A
summary of all the expected patterns for the different
amino acid types is given in the appendix F. Some of
the amino acids have a very typical pattern, for
instance Gly where the side chain just consists of two
Hα, or prolines where no HN is present. Some other
amino acids share a common pattern, e.g. the so-called
AMX spin systems where AMX represents the Hα and the two Hβ protons of the amino
acid. To this group the following residues belong: Phe, Tyr, Trp, His, Ser, Cys, Asp and
Asn. In the COSY and TOCSY spectra of Phe no J-couplings between Hβ and protons in
the ring are observable (4 bonds involved). This gap can be closed if in addition the
NOESY spectrum is used since the distance between these protons is typically smaller
than 5 Å. Again, these cross peaks are included in appendix F.
For bigger proteins the overlap makes it harder to identify such patterns. Also the lines
are broader in general which is due to the shorter T2 relaxation times. This also results in
a reduced efficiency of the magnetization transfer during the mixing period in TOCSY
since the magnetization is transversal during this time.
Step 2: Identification of neighbouring residues
When more than, for instance, one alanine is present in the protein, it is not a priori clear
which alanine in the primary sequence corresponds to a certain alanine pattern in the 2D
COSY spectrum. In order to make a so-called sequential assignment, i.e. correlating the
74
COSY patterns to individual amino acids in the primary sequence, we have to correlate
the COSY pattern of the alanine to the COSY pattern of its sequential neighbour.
Unfortunately, when using only proton NMR, no 1H-1H J-couplings of appreciable size
exists over the peptide bond since the shortest
connection of two protons in neighbouring residues
involves four bonds. Consequently, the individual
amino acid residues form isolated spin systems.
Fortunately we can employ another mechanism of
magnetization transfer. The short sequential distances
between consecutive residues result in cross peaks in
the NOESY spectrum. The figure on the right
summarizes the sequential assignment approach: The
type of the spin system is identified using COSY and/or TOCSY experiments (bold
arrows), whereas the sequential connectivity is established by sequential NOESY cross
peaks (dotted arrows). A cross peak between the Hα proton of a spin i and the HN proton
of the neighbouring spin i+1 results from a short distance between these two residues,
often referred to as dαN. Similar, the distances between Hβ or HN of spin i to the
neighbour HN of spin i+1 are represented by dβN and dNN .
On the other hand the tertiary structure of the protein will also lead to intense signals
from non-sequential cross peaks. How can we be sure to observe a sequential peak?
The statistics of these short distances have been investigated on the basis of thousands of
known (mostly X-ray) structures. Table 12.1 shows for example that 98% of all dαN
distances shorter than 2.4 Å correspond to sequential distances. Naturally, the score drops
with increasing distance limit. Similar values are obtained for the dNN and dβN distances.
This means that for two residues i and i+1 dαN, dNN and dβN cross peaks most likely result
from sequential NOEs. The probability for identifying a sequential connection increases
dramatically when simultaneously two (or more) short distances can be found.
Table 12.1 shows that if simultaneously two NOEs are found between two amino acids,
they most likely result from sequential residues.
75
Table 12.1
Distance (Å)
j - i = 1 (%)
dαN (i,j) ≤ 2.4
98
≤ 3.0
88
≤ 3.6
72
dNN (i,j) ≤ 2.4
94
≤ 3.0
88
≤ 3.6
76
dβN (i,j) ≤ 2.4
79
≤ 3.0
76
≤ 3.6
66
dαN (i,j) ≤ 3.6 && dNN (i,j) ≤ 3.0
99
dαN (i,j) ≤ 3.6 && dβN (i,j) ≤ 3.4
95
dNN (i,j) ≤ 3.0 && dβN (i,j) ≤ 3.0
90
Step 3: Matching to the sequence
The next step is to locate this fragment of two (or more) residues in the protein sequence.
For bigger proteins there might still exist several possibilities. Naturally, we can try to
link more patterns together and try to make tri-peptide, tetra-peptide, and even bigger
fragments. The uniqueness of such di-, tri-, and tetra-peptide fragments in proteins with
less than 200 residues has also been investigated. If all residue types could be identified
unambiguously in the fragment from the COSY (or TOCSY) spectra, a given di-peptide
fragment has a probability of uniqueness of 56%, the tri-peptide and tetra-peptide
76
fragments of 95% and 99%, respectively. As expected, increasing the length of the
fragment increases the uniqueness. A tetra-peptide fragment is usually sufficiently unique
to allow for identification in the polypeptide chain.
12.1.2 Secondary structural elements in peptides and proteins
The intensity of the sequential NOEs contains some information on the secondary
structure because they depend on the local conformation of the polypeptide backbone.
Take, for instance, the distance between an Hα proton of a residue i to the HN proton of
the following residue (i +1). In an extended strand this distance is short (2.2 Å) whereas it
is 3.5 Å in a helical conformation. This information together with analysis of other
characteristic medium range NOEs (between residues which are less than 4 positions
apart in the sequence) is sufficient to specify the secondary structure element. An
overview of important sequential- and medium-range proton-proton distances is given in
the Figure 12.1.
Figure 12.1: Characteristic sequential and medium range NOE connectivities.
77
Recognition of secondary structural elements, i.e. α-helices, β-sheets, and turns,
constitutes an important element in the structure determination process. Most of the
observable NOE cross peaks between different residues are due to this secondary
structure.
The α-helix, β-sheet, and turn conformation result in characteristic short distances which
in turn result in characteristic NOEs in the 2D NOE spectrum. Also the 3J(HN-Hα)
coupling has traditionally been used as a marker for secondary structure as well as the
presence of slowly exchanging amide protons (cf. Chapter 10).
Fig. 12.2 shows the short distances in an anti-parallel β-sheet and in a parallel β-sheet.
78
Fig.12.2: Anti-parallel (top) and parallel (bottom) β-sheet. Sequential NOEs are indicated by open arrows,
characteristic interstrand NOEs by solid arrows. Hydrogen bonds connecting the strands are shown by
wavy lines.
The anti-parallel β-sheet is characterized by short dαN(i,i+1) and dαα(i,j) distances
whereas the dαα (i,j) distance is much longer (4.8 Å) in parallel β-sheet. Also the dNN(i,j)
distance in parallel β-sheet is much longer compared to the dNN(i,j) distance in antiparallel β-sheet. In Fig. 12.3 short distances are shown for an α-helix. An α-helix is
characterized by a close proximity of residues i and i+3 and residues i and i+4. The
dαN(i,i+3) and dαN(i,i+4) NOEs are therefore clear markers for this element of secondary
structure. In addition to the aforementioned short distances, the sequential dNN(i,i+1)
distance is also short, and strong sequential dNN NOEs can be found in the spectrum.
Fig.12.3: α-helix. The sequential dNN is shown (2.8 Å) together with dαN(i,i+1), dαN(i,i+2) and dαN(i,i+3),
Note that the side-chains are not shown.
Turns are characterised by short distances between residues i and i+2; in particular the
79
dNN(i,i+2) distance. In general, however, since there exists a large number of turns, which
all have mirror images as well, e.g. I, I’, II, II’, III, III’, the precise nature of the turn is
hard to establish from NOE and J-coupling data alone. An overview of the characteristic
patterns and short distances is given in Fig. 12.4.
Fig. 12.4: Characteristic NOEs for several secondary structure elements. The thickness of the bars reflects
the strength of the NOE. The thicker the bar the stronger the NOE (and the shorter the distance between the
protons involved). 3J coupling constants are also given.
80
The short distances in secondary sructures are also listed in the following Tabel 12.2:
Distance
α-helix
310-helix
β
βp
turn Ia
turn IIa
dαN
3.5
3.4
2.2
2.2
3.4
2.2
3.2
3.2
dαN(i,i+2)
4.4
3.8
3.6
3.3
dαN(i,i+3)
3.4
3.3
3.1-4.2
3.8-4.7
dαN(i,i+4)
4.2
dNN
2.8
2.6
4.5
2.4
2.4
3.8
4.3
2.9-4.4
3.6-4.6
3.6-4.6
3.6-4.6
dNN(i,i+2)
4.2
4.1
dβN
2.5-4.1
2.9-4.4
dαβ(i,i+3)
a
2.6
2.5-4.4
4.3
3.2-4.5
4.2
3.7-4.7
3.1-5.1
for turns, the two numbers apply for the distances between residues 2, 3 and 3, 4 respectively
81
12.2
Nucleotides and nucleic acid
Nucleic acid is mainly build from five different nucleotides. All of them share a common
general structure: They consist of a nucleobase, a pentose-sugar ring (ribose in the case of
RNA or 2'-deoxy ribose in DNA) and a phosphate group which links the nucleotide units
with each other.
(Nucleo) Base
Sugar
Phosphate
Figure 12.5: Common structure of nucleotides
There are two sorts of nucleobases: The purine bases adenine and guanine and the
pyrimidine bases uracil (only found in RNA), thymine (only found in DNA) and cytidine.
The base is connected to the sugar moiety via a glycosidic bond at the 1' carbon of the
pentose ring. In the common nucleotides the phosphate group is attached to the 5' carbon
of its sugar. A nucleotide which has no phosphate group is called a nucleoside. In
oligonucleotides and in nucleic acid, the phosphate group is linked between the 5' carbon
of one pentose and the 3' carbon of the next. It should be clear from this, that NMR
spectra of large nucleic acids are much more complex than NMR spectra of peptides of
comparable size. While peptides are build from as many as 20 different building blocks
(the different 'natural' amino acids) nucleic acid is build from only four different
nucleotides. Consequently the number of occurrences for a particular kind of nucleotide
is in average five times as high as for any particular amino acid. This can lead to very
crowded regions in the NMR spectra and can complicate the spectral assignment
considerably.
82
Figures 12.6 to 12.9 show the different nucleobases and sugars with their numbering
schemes and the eight different RNA and DNA nucleotides.
Adenine
5
6
5
1
8
6
2
7
4
4
1
3
Uracil
2
3
9
5
Thymine
Guanine
6
5
1
3
2
1
2
7
8
6
4
4
Cytosine
3
9
5
6
4
3
1
2
Figure 12.6: The five different nucleobases found in Nucleic acid
5'
5'
4'
1'
3'
2'
Ribose (RNA)
Figure 12.7: The pentose sugar rings of DNA and RNA
4'
1'
3'
2'
2'-deoxy ribose (DNA)
83
AMP
Adenosine5'-phoshate
GMP
Guanosine5'-phoshate
UMP
Uridine5'-phoshate
CMP
Cytidine5'-phoshate
Figure 12.8: Ribonucleotides (RNA)
dAMP
Adenosine5'-phoshate
dGMP
Guanosine5'-phoshate
dTMP
Thymidine5'-phoshate
dCMP
Cytidine5'-phoshate
Figure 12.9: 2'-Deoxy-ribonucleotides (DNA)
84
The figure below illustrates how the nucleotide units are linked by the phosphate groups
in oligo nucleotides and in nucleic acids. If we analyze the spin systems of this molecule,
we find that both the link
between
sugar
and
nucleobase and between
the individual nucleotide
units (via the phosphate
groups) reach further than
three bonds before the
next proton can be found.
In other words: The bases
and
the
isolated
sugars
spin
form
systems.
This is important when it
comes to think about an
assignment strategy for
these molecules!
When we look at the
structural properties of
Figure 12.8: The phosphate-sugar backbone of DNA
nucleic acid, the most
prominent feature is the double-helical conformation it adopts. The two strands of the
helix adhere to each other by means of hydrogen bonding between bases of the adjacent
stands. These so-called base-pairs were found by James Watson and Francis Crick and
accordingly named 'Watson-Crick base pairs'. Base pairs are always built from one purine
and one pyrimidine base. Adenine (A) always pairs with Uracil (U) in RNA and with
Thymine (T) in DNA. Guanine (G) always pairs with Cytosine (C). The hydrogen bonds
are shown in the figure below. Note that the G-C pair is more stable than the A-T pair,
because G-C consists of three hydrogen bonds and A-T only of two.
85
Figure 12.9: Watson-Crick base pairs. A-T on the left, G-C on the right
There are two major conformation in which the double helices of DNA and RNA usually
are found. The so-called B-form and the A-form (see figure 12.10). Both of them are
right-handed (like ordinary corkscrews) and differ mainly in their width and height of
their turns. A third conformation, the Z-form is left-handed and of minor importance. The
important parameters of the A-, B- and Z-form helix are listed in table 12.3:
Geometry attribute
A-form
B-form
Z-form
Helix sense
right-handed
right-handed
left-handed
Repeating unit
1 bp
1 bp
2 bp
Rotation/bp
33.6°
35.9°
60°/2
Mean bp/turn
10.7
10.0
12
Inclination of bp to axis
+19°
-1.2°
-9°
Rise/bp along axis
0.23 nm
0.332 nm
0.38 nm
Pitch/turn of helix
2.46 nm
3.32 nm
4.56 nm
Mean propeller twist
+18°
+16°
0°
Glycosyl angle
anti
anti
Sugar pucker
C3'-endo
C2'-endo
C: anti
G: syn
C: C2'-endo
G: C2'-exo
Diameter
26 nm
20 nm
18 nm
86
A-form double helix (RNA)
B-form Helix (DNA)
Figure 12.10: The two major helical conformations
Information about typical chemical shift values of the A- and B-DNA and typical short
distances can be found in appendix F.
87
12.2.1 Assignment of oligonucleotide and nucleic acid spectra
The assignment strategy for nucleic acid spectra is very similar to the one discussed in
chapter 13 for peptides and proteins. The major difference being that for the identification
of a particular residue always the combination of COSY/TOCSY and NOESY is needed,
because the bases form isolated spin systems and cannot be assigned to their sugars
without making use of NOE data. Tables to be used for this purpose with common shift
values and short distances in nucleic acids can be found in appendix F.
88
XIII Structure determination
In this chapter we will discuss the method of determination of the complete 3D structure
of a protein. Apart from the bond lengths and most bond angles, which are known on the
basis of the amino acid sequence, the most important source of structural information is
the NOE. In particular, the so-called "long-range" NOEs (those between protons more
than four residues apart in the sequence) provide important constraints on the structure.
We will first see which experimental NMR parameters can be translated into structural
constraints and then describe the computational structure calculation procedure.
13.1
Sources of structural information
13.1.1 NOEs
For a ~10kD protein typically between 1000 and 2000 cross peaks can be observed in a
2D NOE spectrum. We have seen in Chapter 7 how NOE intensities can be converted
into proton-proton distances with the aid of a reference distance (for instance the distance
of 2.45 Å between neighbouring protons on an aromatic ring). In principal the r-6 relation
between NOE and distance should give very precise distances. For example, if there is a
10% error on the NOE intensity this translates in only a 1.5% error in the distance!
However, there are two reasons why this high precision cannot be obtained in practice.
The first is local mobility. Remember that the simple relation of Eq. 7.21 was derived on
the assumption of equal τc for the protons of reference and unknown distance. Only for
very rigid proteins this assumption is really valid. Also, if there is a conformational
equilibrium the r-6 dependence gives values much more weighted to the short distances in
the average. Thus, the actual average distances may be longer than they appear in the
case of conformational averaging. The second reason is the so-called "spin-diffusion"
effect. This is the result of multiple transfer steps of magnetization A → B → C during
89
the mixing time τm that may disturb the intensity of the NOE cross peak between A and
C. Only for very short τc can this indirect transfer path neglected. For these reasons the
NOE based distance constraints are often used in the form of distance ranges rather then
precise distances:
strong NOE
1.8 - 2.7 Å
medium NOE
1.8 - 3.5 Å
weak NOE
1.8 - 5.0 Å
This procedure works well in practice because it turns out that for a high precision of the
structure a large number of NOE constraints is more important than precise distances.
13.1.2 J-couplings
The magnitude of the three-bond
coupling constant, 3J, is related
to the dihedral angle (between
the
two
outer
bonds)
and
therefore provides a constraint
on this angle (θ).
This is expressed in the Karplus relation
J = A cos 2 (θ ) + B cos(θ ) + C
(13.1)
where A,B and C are parameters that depend on the particular situation. For instance, the
J-coupling between the amide proton and the α-proton, JHNHα depends on the backbone
angle φ as follows:
90
J = 6.51 cos 2 (φ − 60) − 1.76 cos(φ − 60) + 1.60
(13.1)
The form of the Karplus relation for this case is shown in Fig. 13.1.
Fig. 13.1: The Karplus curve for the φ angle as described in the text.
A complication is that several values of φ may belong to a particular J. Also,
conformational averaging may lead to average values of J in the range 6-7 Hz. Therefore,
the most reliable values for JHNHα are 9-10 Hz for extended (β-sheet) structure and 3-4 Hz
for α-helical structure. Values around 6-7 Hz are often difficult to interpret. Similar
Karplus curves exist for CH-CH J-couplings from which χ angles can be derived.
91
13.1.3 Hydrogen bond constraints
When a protein is dissolved in D2O many of the amide protons do exchange rapidly with
deuterium and therefore disappear from the spectrum. However, often some NHs remain
in the spectrum for some time and exchange only slowly in time. These slowly
exchanging NHs invariably are present in hydrogen-bonds such as occur in α-helices and
β-sheets. If the H-bond acceptor is known with certainty, for instance, because we have
several short- and medium-range NOEs defining the secondary structure, we can use
distance constraints corresponding to the H-bonds. Usually these are given the distance
ranges 2.1 - 2.3 Å for the distance between NH and O and 3.1 - 4.3 Å between N and O.
13.2
Structure calculations
Several computer programs exist that are able to calculate the 3D structure of a protein
based on distance and dihedral angle constraints. Often one starts using only geometric
constraints (distances and angles) with the so-called Distance-Geometry (DG) program.
The resulting structures can then be refined with Molecular Dynamics (MD) calculations
which include also energy terms for electrostatic interactions etc. The calculation
procedure will now be briefly discussed.
13.2.1 Distance-Geometry
The structure of a macromolecule containing N atoms can be perfectly described by
specifying the N(N-1)/2 distances between the atoms (except for the chirality). Since this
is a very large number we will never have so many distance constraints in practice. There
is, however, an algorithm called Distance-Geometry (DG) that converts distances into
92
Cartesian coordinates for a much smaller number of distances even if they are not
precisely known. This algorithm has the following steps:
1. Set up distance matrices for upper bound (u) and lower bound (l) distances between all
atoms in the structure. These include the so-called holonomic distances (from bond
length and bond angles) and the NOE distance constraints. For those elements for which
no information is available the upper bound is set to a large value and the lower bound to
the sum of the van-der-Waals radii (1.8 Å).
2. Smoothing
By this we mean the adjustment of upper
and
lower
bounds
using
triangular
inequalities. Consider three atoms i, j and k
with upper bounds uij, uik, and ujk, and lower
bounds lij, lik and ljk. The maximum distance
between i and j is when i,j and k are
colinear. Thus we have
uij ≤ uik + u jk
(13.2)
A similar relation exists for the lower bound lij:
lij ≥ lik − u jk
(13.3)
These relations are applied to all distances in the set until no further changes are obtained.
3. Select a matrix D with random trial distances between upper and lower bounds.
93
4. Embedding
This is the procedure of finding the best 3D structure that belongs to the distance matrix
D. We will not explain this further here.
5. Optimization
Because the matrix D, of course, is not perfect, it is usually necessary to regularize the
structure coming from the embedding stage. Optimization involves the minimization of
the coordinates against a distance error function.
Because of the random nature of step 3, the repetition of steps 3-5 will produce slightly
different structures. Idealy, if this is done many times, a family of structures is obtained,
which samples the conformational space consistent with the experimental data. An
example of such a family of structures is shown in Fig. 13.2.
Fig. 13.2. Structures of the protein crambin calculated by the DG procedure.
94
13.2.2 Restrained molecular dynamics
Molecular Dynamics (MD) programs were originally designed to simulate the dynamic
behaviour of atomic or molecular systems. It turns out that a purely classical approach
works well for this purpose. For the initial coordinates and velocities of all atoms i,
Newtons equation of motion is solved:
d 2 ri
mi 2 = Fi
dt
(13.4)
where mi is the mass, ri the position vector and Fi the force acting on atom i. These forces
are given by
Fi = −
∂V
∂ri
(13.5)
The empirical potential energy function V (also called the force-field) contains many
terms
V = Vbond length + Vbond angle + Vdihedral + Vvan der Waals + Velectrostatic + ⋅ ⋅ ⋅ ⋅
(13.6)
that will not be specified here. For a faithful simulation of the dynamics the integration
step should be rather small, say 1 fs (femto second), so that for a dynamic trajectory of
several tens of ps or a ns very many integrations have to be carried out.
For the application of this method for NMR structure refinement we just add another term
in the potential energy function that reflects the NOE distance constraints:
VNOE = k ( rij − uij ) 2
=0
= k (lij − rij )
2
if
rij > uij
if
lij ≤ rij < uij
if
rij < lij
(13.7)
95
This function looks as shown at the
right.
The effect of this term is the following:
If the distance rij in the current model is
too large (larger than the upper bound
uij) than a force is acting to decrease it.
Conversely, if rij is too small a force
will tend to increase it until rij lies
between the bounds. As this will happen for all 1000-2000 distance constraints the
structure will usually satisfy the constraints after a MD run better than before. At the
same time it will be close to a minimum with respect to the potential energy function of
Eq. 13.6. Thus, this so-called restrained MD algorithm acts as an efficient minimizer of
the energy. Of course, if the potential energy decreases the kinetic energy would increase.
To avoid this 'heating effect' the system is normally coupled to a 'thermal bath' so that
excessive kinetic energy is drained off. An illustration of this procedure is shown in
Figure 13.3. Again, a family of structures is calculated starting with a random starting
structure. It can be seen that the final cluster of structures is better defined than those
from the DG calculations. This is usually expressed in terms of the root-mean-squaredeviation (RMSD) calculated for the ensemble of structures. Typically, RMSD values for
good NMR structures are in the range 0.3 Å to 0.5Å for the backbone atoms.
96
Fig. 13.3: Calculation of the structure of dimeric interleukin-8 by a combination of Distance-Geometry and
Molecular Dynamics. The start is a random structure obtaining only the sequence information (only the
backbone is shown). 10 DG structures are calculated, which are the starting structures for the MD
calculation. After cooling down the precession of the structure is greatly increased.
97
Appendix:
Appendix A: a) Typical 1H and 13C chemical shift of some common functional groups
and b) random coil chemical shifts for protons in amino acids.
a)
98
b) Random coil 1H chemical shifts for the 20 common amino acid residues
NH
Hα
G
8.39
3.97
A
8.25
4.35
1.39
V
8.44
4.18
2.13
CH3
0.97
0.94
L
8.42
4.38
1.65
Hγ
CH3
1.64
0.94
0.90
I
8.19
4.23
1.90
CH2
CH3
CH3
1.48
0.95
0.89
T
8.24
4.35
4.22
CH3
1.23
4.66
3.22
2.99
H2,6
H3,5
H4
7.30
7.39
7.34
3.13
2.92
H2,6
H3,5
7.15
6.86
F
Y
8.23
8.18
4.60
4.44
P
Hβ
H
2.28
2.02
C
8.31
4.69
3.28
2.96
S
8.38
4.50
3.88
M
8.42
4.52
2.15
2.01
γ CH2
δCH2
2.03
3.68
3.65
1.19
NH
Hα
Hβ
8.41
4.63
3.26
3.20
H2
H4
8.12
7.14
H2
H4
H5
H6
H7
NH
7.24
7.65
7.17
7.24
7.50
10.2
γ CH2
2.31
2.28
W
8.09
4.70
3.32
2.99
D
8.41
4.76
2.84
2.75
E
8.37
4.29
2.09
1.97
N
8.75
4.75
2.83
2.75
NH2
7.59
6.91
Q
8.41
4.37
2.13
2.01
CH2
NH2
2.38
6.87
K
8.41
4.36
1.85
1.76
γ CH2
δCH2
εCH2
NH3+
1.45
1.70
3.02
7.52
R
8.27
4.38
1.89
1.79
γ CH2
δCH2
NH
1.70
3.32
7.17
For X in GGXA, pH 7, 35ºC (Bundi and Wüthrich 1979)
γ CH2
εCH3
2.64
2.13
7.59
6.62
99
Random Coil Chemical shifts (in ppm) for the 20 common amino acids in acidic 8 M urea (from Peter E.
Wright & H. Jane Dyson et. al., Journal of Biomolecular NMR, 18: 43–48, 2000).
100
Appendix B: Chemical shift distribution in 1H spectra of proteins.
101
Appendix C: Nuclear Overhauser Effect
The population change of the αα-state, d/dt nαα, after a disturbance from equilibrium can
be expressed using the W0, W1a, W1b and W2 rates. The rate equation is:
d
eq
eq
nαα = − (W1a + W1b + W2 )( nαα − n αα
) + W2 ( n ββ − n ββ
)
dt
+ W1a ( n βa − n βeqa ) + W1b ( n aβ − n aeqβ )
(C.1)
Thus, the αα-state looses magnetization (first term), but also gains some magnetization
from the ββ-, the βα− and the αβ-states.
Similar expression can be found for the time dependence of the other three states.
Now we look at the net population differences of spin A and B, na and nb, resp. These are
n a = ( nαα − n βα ) + (nαβ − n ββ )
n b = ( nαα − nαβ ) + (n βα − n ββ )
(C.2a,b)
From this we can calculated the time derivate
d
d
d
d
d
na =
nαα − n βα + nαβ − n ββ
dt
dt
dt
dt
dt
d
d
d
d
d
nb =
nαα − nαβ + n βα − n ββ
dt
dt
dt
dt
dt
(C.3a,b)
If we now introduce in Eqs. C.3 the results from Eq. C.1 (and the expressions for the
other spin states) we can derive the dependency of the population from the transition
rates:
d
na = − (W0 + 2W1a + W2 )( na − naeq ) − (W2 − W0 )( nb − nbeq )
dt
d
nb = − (W0 + 2W1b + W2 )( nb − nbeq ) − (W2 − W0 )( na − naeq )
dt
(C.4a,b)
102
The first terms of Eqs. C.4a and C.4b describe the T1 relaxation of spins A and B,
respectively. Hence we have
1
= ρ a = (W0 + 2W1a + W2 )
T1a
1
= ρ b = (W0 + 2W1b + W2 )
T1b
(C.5a,b)
The second term of Eqs. C.4 results in transfer of magnetization from A to B. We define
the cross-relaxation σ between A and B as
σ = W2 − W0
(C.6)
With this definition, from Eqs. C.4 we arrive at the famous Solomon-Bloembergen
equations:
d
n a = − ρ a ( n a − n aeq ) − σ ( n b − n beq )
dt
d
n b = − ρ b ( n b − n beq ) − σ ( n a − n aeq )
dt
(C.7a,b)
Now let us return to the steady-state NOE experiment of chapter IIX. Spin B was
selectively saturated (nb = 0). After some time the two-spin system will reach a steady
state with
d
na = 0
dt
Eq. C.7a then becomes
(C.8)
103
d
na = 0 = − ρ a ( na − naeq ) − σ ( nb − nbeq )
dt
(C.9)
so that the NOE (Eq. 8.1) becomes
η=
( na − n eq
)
a
n eq
a
eq
σ n
σ γ
=
⋅ eq =
⋅
ρa n
ρa γ
b
b
a
a
(C.10)
Here we exploited the fact that the macroscopic magnetization Ma is proportional to the
population na, and that the populations are themselves proportional to the gyromagnetic
ratio (Eqs. 2.8 and 4.1).
For identical nuclei γa = γb, and ρa = ρ, and Eq. C.10 reduces to the simple form
η=
σ
ρ
(C.11)
104
Appendix D: Fast exchange titration
To get from Eqn. 9.6 to Eqn. 9.10 the following rearrangements need to be done:
÷ν b
ν = ν a (1 − x) +ν b x
⎛ ν ⎞
ν ν ⋅x
ν
ν νa
= (1 − x) + x = a − a + x = a + x ⋅ ⎜⎜1 − a ⎟⎟
νb νb
νb νb
νb
⎝ νb ⎠
−
⎛ ν ⎞
ν νa
− = x ⋅ ⎜⎜1 − a ⎟⎟
νb νb
⎝ νb ⎠
x=
νa
νb
⎛ ν ⎞
÷ ⎜⎜1 − a ⎟⎟
⎝ νb ⎠
(ν − ν a )
(ν b − ν a )
(9.10)
Another expression for the molar ratio can be found by looking at the equation
for the equilibrium of the reaction
⎯
⎯→
←
⎯⎯
E+H+
E⋅H+
The molar ratio is simply the amount (concentration) of product molecules divided by the
complete amount of molecules in the solution, which leads us to the first part of Eqn. 9.8:
x=
[EH ]
[E ] +[EH ]
Substitution of [EH] by the expression on the left side of Eqn. 9.7 eventually brings us to
the right part of Eqn. 9.8
x=
x=
x=
[EH ]
[E ] +[EH ]
[H ][E ]/ K a
[E ] + [H ][E ]/ K a
[H ]
[H ] + K a
From this, Eqn. 9.9 can be derived by inversion,
[EH ] = [H ][E ]
Ka
÷[E ] and ⋅ K a
(9.8)
105
1 [H ] + K a
=
[H ]
x
subsequent decomposition of the fraction on the right side of the equation and some
rearrangement
K
1 [H ] K a
=
+
= 1+ a
x [H ] [H ]
[H ]
−1
K
1
1 x 1− x
−1 = a = − =
[H ] x x x
x
the logarithm of which is
log
1− x
= log K a − log[H ]
x
now, knowing that the pH (or pKa) are defined as the negative logarithm of [H] (or Ka),
we finally reach the expression from Eqn. 9.9:
log
(1 − x )
= pH − pK a
x
(9.9)
106
Appendix E: 2D NOESY experiment
In mathematical terms the 2D NOE experiment can be described as follows. During the
evolution period the transversal magnetization of nucleus B can be written as
M B = M Beq e
iω B t
1
=
M Beq [cos(ω B t1 ) + i ⋅ sin(ω B t1 )]
(E.1)
This results at time point c (Fig. 10.2) in a z-component
M B ,z = − M Beq cos(ω B t1 )
(E.2)
According to the Solomon equation (Eq.C.7) we have for MA a dependency from MB
d
M A = − ρ a ( M A − M Aeq ) − σ AB ( M B − M Beq )
dt
(E.3)
For short mixing times τ m << T1 = ρ A−1 we can neglect spin-lattice relaxation and Eq. E.3
becomes approximately
d
M A ≈ − σ AB ( M B − M Beq )
dt
(E.4)
d
M A ≈ σ AB M Beq [1 + cos(ω B t1 )]
dt
(E.5)
Using Eq. (E.2) this becomes
For short mixing times τm we can approximate this by
ΔM A
τm
= σ AB M Beq [ 1 + cos(ω B t1 )]
(E.6)
107
and thus a fraction of the A-magnetization ΔMA is modulated with ωB
ΔM A = σ AB τ m M Beq [ 1 + cos(ω B t1 )]
During the detection period this evolves with e
iω A t2
(E.7)
and after Fourier transformation we
will have a cross-peak at (ωB, ωA) in the 2D spectrum (F1, F2), correlating the protons A
and B in the 2D NOESY spectrum. The intensity of this cross-peak is
I (ω B , ω A ) ~ σ AB τ m
(E.8)
Since for biomolecules we have
σ AB ~
τc
(E.9)
6
rAB
we get for the cross-peak intensity from Eq. (E.8):
I (ω B , ω A ) ~
τ mτ c
6
rAB
(E.10)
108
Appendix F: Expected cross-peaks for COSY, TOCSY and NOESY for the individual
amino acids.
109
110
111
112
113
114
115
116
117
118
Appendix G: Typical chemical shift values found in nucleic acid
119
Appendix H: Typical short proton–proton distances for B-DNA
All distances are given in Å. Sequential distances (to its 3' neigbor) below the diagonal,
intraresidual distances above the diagonal.
H6/8
3.8
2.1
3.6
4.1
4.9
3.4
3.5
1'
3.0
2.3
3.9
3.6
4.5
3.8
4.1
2'
1.8
2.4
3.9
3.8
2''
2.7
4.0
4.9
3'
2.7
3.7
2.9
4.1
4'
2.7
2.3
4.9
5'
1.8
3.4
5''
4.4
2.3
4.9
4.2
1.8
4.3
3.3
4.8
3.5
3.8
3.2
4.5
3.6
4.0
4.7
4.1
2.3
4.4
4.8
3.8
4.0
2.1
3.6
H6/8