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
Relaxation Multidimensional Spectroscopy Summary Outline Relaxation 1 Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect Multidimensional Spectroscopy 2 Principles Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect Outline Relaxation 1 Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect Multidimensional Spectroscopy 2 Principles Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary 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 observations it is found that when the spins are only being affected by the external field � B 0 , the population difference and the associated bulk magnetisation increase exponentially: ∆ n ( t ) = ∆ n eq ( 1 − exp ( − t )) T 1 M z = ( 1 − exp ( − t )) M z , eq T 1 where T 1 is a characteristic time constant. Lieven Buts Biophysical Chemistry: NMR Spectroscopy
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 τ c . 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.
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: 2 τ c J ( ω ) = 1 + ω 2 τ 2 c 1.6 J(x, 0.2) 1.4 J(x, 0.4) 1.2 J(x, 0.8) 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 30 35 40
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary 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 B random . The efficiency of the relaxation process is determined by the average strength of this fluctuating field ( B 2 ), and by the � � random fraction of the molecules that are rotating at the Larmor resonance frequency, given by J ( ω 0 ) . 4 π ) 2 γ 2 � 2 τ c 1 J ( ω 0 ) = ( µ 0 = γ 2 � B 2 � J ( ω 0 ) random r 6 T 1 Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect Rotational Motion in General J ( ω 0 ) = τ c when τ c = ω 0 , which 1 leads also the mimimum possible value for T 1 and the most efficient relaxation. When ω 0 τ c << 1 (quickly tumbling molecules), J ( ω 0 ) ≈ 2 τ c , and faster rotation will lead to slower relaxation. When ω 0 τ c >> 1 (slowly tumbling molecules), J ( ω 0 ) ≈ 0 τ c , and 2 ω 2 slower rotation will lead to slower relaxation. Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary 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 τ c , expressed in picoseconds, is approximately equal to the molecular weight, expressed in g/mol. A disaccharide with M = 360 g mol therefore has a τ c 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 τ c of 20 ns, and will be on the right-hand side of the graph in the same conditions. Since J ( ω 0 ) 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
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary 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 of T 1 for the system being studied, and an experiment was designed to measure this quantity. Lieven Buts Biophysical Chemistry: NMR Spectroscopy
The Inversion Recovery -Experiment
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect The Inversion Recovery Experiment M z ( τ ) = M 0 ( 1 − exp ( − τ )) T 1 Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect Outline Relaxation 1 Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect Multidimensional Spectroscopy 2 Principles Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary 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 c α and c β 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
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary 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 observed signal, with a distinct time constant T 2 = 1 λ . M x = exp ( − t ) M 0 cos (Ω 0 t ) = e − λ t M 0 cos (Ω 0 t ) T 2 M y = exp ( − t ) M 0 sin (Ω 0 t ) = e − λ t M 0 sin (Ω 0 t ) T 2 Unfortunately, there are other phenomena, such as chemical exchange and technical imperfections in the � B 0 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
The Spin Echo Experiment
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect The Spin Echo Experiment The inversion pulse in the middle of the 2 τ 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 T 2 can be determined by fitting the experimental data to I ( 2 τ ) = I ( 0 ) exp ( − 2 τ/ T 2 ) . Lieven Buts Biophysical Chemistry: NMR Spectroscopy
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect Implications For quickly rotating molecules (short τ c ), T 2 more or less matches T 1 . For slowly rotating molecules however, T 2 continues to drop with increasing τ c , 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
Relaxation Longitudinal Relaxation Multidimensional Spectroscopy Transverse Relaxation Summary The Nuclear Overhauser Effect Outline Relaxation 1 Longitudinal Relaxation Transverse Relaxation The Nuclear Overhauser Effect Multidimensional Spectroscopy 2 Principles Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy
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