SLIDE 1 Relaxation Multidimensional Spectroscopy Summary
Biophysical Chemistry: NMR Spectroscopy
Relaxation & Multidimensional Spectrocopy Lieven Buts
Vrije Universiteit Brussel
9th December 2011
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 2 Relaxation Multidimensional Spectroscopy Summary
Outline
1
Relaxation Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
2
Multidimensional Spectroscopy Principles
3
Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 3 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Outline
1
Relaxation Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
2
Multidimensional Spectroscopy Principles
3
Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 4 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Establishment of Thermal Equilibrium
As previously mentioned, the equilibrium distribution of a collection of spins at a given temperature can be calculated using the Boltzmann distribution. However, when a sample is placed in the magnetic field for the first time or has been brought out of equilibrium by an RF pulse, it takes a certain amount of time for the equilibrium to (re)establish itself. From
- bservations it is found that when the spins are only being
affected by the external field B0, the population difference and the associated bulk magnetisation increase exponentially: ∆n(t) = ∆neq(1 − exp(− t T1 )) Mz = (1 − exp(− t T1 ))Mz,eq where T1 is a characteristic time constant.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 5
Rotation of a Tetrahedral Molecule (1)
Individual molecules rotate in a chaotic pattern. The time needed for a molecule to rotate over an average angle 1rad (≈ 57.3◦) is the rotation correlation time τc. At any give time, the molecule is rotating about a certain axis at a certain speed. From this a rotation frequency ω can be extrapolated, even when the molecule is frequently perturbed by further collisions, and rarely completes a full rotation.
SLIDE 6
Rotation of a Tetrahedral Molecule (2)
Through the exchange of thermal energy the molecules attain a certain distribution of rotational frequencies, which can be described by the spectral density function J(ω). A complete analysis results in the following analytic expression: J(ω) = 2τ c 1 + ω2τ 2
c
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 5 10 15 20 25 30 35 40 J(x, 0.2) J(x, 0.4) J(x, 0.8)
SLIDE 7 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Effect of Molecular Rotation on Relaxation
Combined with this rotational motion, the dipolar field of every spin gives rise to a small, randomly fluctuating magnetic field Brandom. The efficiency of the relaxation process is determined by the average strength of this fluctuating field (
random
fraction of the molecules that are rotating at the Larmor resonance frequency, given by J(ω0). 1 T1 = γ2 B2
random
4π)2 γ22τc r6 J(ω0)
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 8 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Rotational Motion in General
J(ω0) = τc when τc =
1 ω0 , which
leads also the mimimum possible value for T1 and the most efficient relaxation. When ω0τc << 1 (quickly tumbling molecules), J(ω0) ≈ 2τc, and faster rotation will lead to slower relaxation. When ω0τc >> 1 (slowly tumbling molecules), J(ω0) ≈
2 ω2
0τc , and
slower rotation will lead to slower relaxation.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 9 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Rotational Correlation Times
For a roughly spherical molecule in aqeuous solution at room, temperature, there is a rule of thumb that τc, expressed in picoseconds, is approximately equal to the molecular weight, expressed in g/mol. A disaccharide with M = 360 g
mol therefore has a τc of around
360 ps, and will be on the left-hand side of the previous graph in a 500MHz instrument. A small protein with M = 20000 g
mol has a τc of 20 ns, and will be
- n the right-hand side of the graph in the same conditions.
Since J(ω0) is fairly small even in the optimal case, longitudinal relaxation is always a fairly slow process by general spectroscopic standards.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 10 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Practical Implications
When the sample is first placed in the magnet, we need to wait long enough for the maximal bulk magnetisation to be established, if we want the best attainable signal after the excitation pulse. Similarly, a waiting (recycling) period is required before every repetition of the pulse/observe cycle to allow the system to return to equilibrium and to be sure that each repetition starts from the same initial conditions. It is therefore very useful to have a good idea of the magnitude
- f T1 for the system being studied, and an experiment was
designed to measure this quantity.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 11
The Inversion Recovery-Experiment
SLIDE 12 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
The Inversion Recovery Experiment
Mz(τ) = M0(1 − exp(− τ T1 ))
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 13 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Outline
1
Relaxation Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
2
Multidimensional Spectroscopy Principles
3
Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 14 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Principle
The mutual perturbation of spins as they rotate past each other at the right frequency has two distinct effects. First, the cα and cβ components are redistributed, with a slight preference for the lower-energy α state. This is the basis of the aforementioned longitudinal relaxation (also known as spin-lattice relaxation) and leads to the establishment of the thermal equilibrium. Second, the x/y orientations of the spins are modified at random, which causes them to slowly get out of rotational
- synchronisation. This leads to a reduction of the observable
transverse signal, and is therefore known as transverse relaxation.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 15 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Transverse Relaxation
The loss of phase coherence due to transverse relaxation can also be described as an intrinsic exponential fading of the
- bserved signal, with a distinct time constant T2 = 1
λ.
Mx = exp(− t T2 )M0 cos(Ω0t) = e−λtM0 cos(Ω0t) My = exp(− t T2 )M0 sin(Ω0t) = e−λtM0 sin(Ω0t) Unfortunately, there are other phenomena, such as chemical exchange and technical imperfections in the B0 field which have a similar line broadening effect. For this reason, an experiment was designed to allow the transverse relaxation to be quantified directly, removing the contributions from other effects.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 16
The Spin Echo Experiment
SLIDE 17 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
The Spin Echo Experiment
The inversion pulse in the middle of the 2τ interval causes all systematic contributions to dephasing to be compensated, leaving only the intrinsic and random dephasing due to transverse relaxation. In this way, the true value of T2 can be determined by fitting the experimental data to I(2τ) = I(0) exp(−2τ/T2).
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 18 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Implications
For quickly rotating molecules (short τc), T2 more or less matches T1. For slowly rotating molecules however, T2 continues to drop with increasing τc, making transverse relaxation increasingly efficient for slower molecules. Transverse relaxation is one of the main obstacles when studying large (slowly tumbling) macromolecules in solution.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 19 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Outline
1
Relaxation Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
2
Multidimensional Spectroscopy Principles
3
Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 20 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Principle
When one of a pair of nuclei is excited by strong RF radiation at its resonance frequence, this can affect the intensity of the NMR signal of the other nucleus in the pair. The effect of the first spin on the signal intensity of the second spin can be quantified by the ratio −1 < η = i − i0 i < 1 2
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 21
Energy Diagram
The Overhauser effect is caused by non-radiative transitions between the four energy levels of a coupled spin pair. Here, W1,I and W1,S ∼ J(ω0), W2,I/S ∼ J(2ω0) and W0,I/S ∼ J(0).
SLIDE 22 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Implications
All other factors being equal, the magnitude of the NOE depends only on the distance between the coupled nuclei, via a factor of r−6. Therefore the NOE can in principle be used to measure interatomic distances. Most of the time however, conformational changes and other complications make it very difficult to exploit this effect quantitatively. The NOE is therefore generally used in a quantitative way to determine which atoms are close together (less than 5 to 6 Å) in a molecular structure, and which aren’t.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 23 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Interpretation of NOEs (1)
In simple cases, NOE ratios can be interpreted
molecules are more dynamic, this becomes very difficult.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 24 Relaxation Multidimensional Spectroscopy Summary Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
Interpretation of NOEs (2)
NOEs can be very useful in determining structural details and distinguishing isomeric structures.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 25 Relaxation Multidimensional Spectroscopy Summary Principles
Outline
1
Relaxation Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect
2
Multidimensional Spectroscopy Principles
3
Summary
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 26 Relaxation Multidimensional Spectroscopy Summary Principles
Pulse Sequences
By using a well-timed sequence of radiofrequency pulses and timed intervals, the final signal can be manipulated to reflect different aspects of the interactions between spins in the molecule. The details of such experiments can be represented graphically as a so-called pulse sequence:
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 27
Frequency Labelling (1)
SLIDE 28
Frequency Labelling (2)
SLIDE 29
COSY Spectra (COrrelation SpectroscopY)
This experiment provides information about "through-bond" interactions by means of scalar couplings.
SLIDE 30 Relaxation Multidimensional Spectroscopy Summary Principles
NOESY Spectra (NOE SPectroscopy)
This experiment provides information about "through-space" interactions due to dipolar couplings and then Overhauser effect.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 31
NOESY Spectra
The NOESY spectrum has a cross-peak for every pair of protons that is in close proximity in the folded protein structure.
SLIDE 32 Relaxation Multidimensional Spectroscopy Summary Principles
Triple Resonance Spectra
When magnetically active nuclei of different element (1H, 13C,
15N, ...) are present in the system, their distinct frequency
ranges can be exploited to open up additional dimensions and distribute the signals over an even larger space. It also becomes possible to select specific atomic configurations.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 33 Relaxation Multidimensional Spectroscopy Summary
Summary (1)
In a first approximation, each spin executes a precession motion about the direction of the external field, ignoring any motions of the molecule it is part of. When two spins move in just the right way with respect to each other, their magnetic dipoles can interact and cause a non-radiative transition between the two energy levels. Since encounters of molecules rotating at the right frequencies are relatively rare, this is a fairly inefficient and slow process.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 34 Relaxation Multidimensional Spectroscopy Summary
Summary (2)
On the long term, these sporadic interactions are responsible for the relaxation of the collection of spins. In between the relaxation corrections, the spins complete billions of undisturbed Larmor precession cycles. Through the process of longitudinal relaxation (with a time constant T1), the spin ensemble gradually returns to the equilibrium distribution, with population differences compatible with the Boltzmann distribution. T1 can be experimentally determined using the inversion recovery experiment.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 35 Relaxation Multidimensional Spectroscopy Summary
Summary (3)
Through the process of transverse relaxation (with a time constant T2) the spins gradually go out of phase in the x,y plane, causing the transverse component of the magnetisation to decrease. This is an intrinsic factor contributing to the line width of NMR signals. External factors, such as inhomogeneities in the external magnetic field, also contribute to the broadening of NMR
- signals. The spin echo experiment allows us to measure
the intrinsic value of T2, correcting for these systematic errors.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 36 Relaxation Multidimensional Spectroscopy Summary
Summary (4)
The nuclear Overhauser effect (NOE) is a complex relaxation phenomenon, involving the exchange of energy between nuclei involved in dipolar coupling by virtue of being in close spatial proximity. The effect can be used to map interatomic distances in a quantitative of qualitative way. By performing a series of similar experiments with multiple RF pulses and variable waiting intervals, we can obtain multidimensional spectra.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 37 Relaxation Multidimensional Spectroscopy Summary
Summary (5)
Different pulse sequence select different modes of interaction between atoms. The resulting spectra contain peaks that depend on the correspondng structural properties of the molecule, The position of a peak in such a spectrum is the intersection of the resonance frequences of all the atoms that are correlated by the experiment, while its intensity is determined by the strength of the selected interaction.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
SLIDE 38 Relaxation Multidimensional Spectroscopy Summary
Summary (6)
By combining experiments that select atoms with specific covalent connectivity patterns (COSY, TOCSY) with experiments that depend on the spatial structure (NOESY), we can gather the necessary information to determine the structure and conformation of macromolecules.
Lieven Buts Biophysical Chemistry: NMR Spectroscopy
