Slow symmetric exchange ϕ A k k B t A B There are three things you should notice compared with the Figure on the previous slide: 1) The lines are broader, 2) the intensities are reduced and 3) the peaks are somewhat closer together than in the absence of chemical exchange If the molecule can jump back and forth between the two sites and given that rate of the jumps are slow (compared to the difference in resonance frequency between A and B) the magnetization will get defocused due to the jumps but there will be two distinct resonance frequencies. The defocusing will cause the signals to move slightly closer to each other and importantly the peaks will be broadened by an amount LW, where LW means line width. The broadening is described as: ΔLL = k π
Intermediate symmetric exchange ϕ A B t If the exchange gets into an intermediate regime, the two peaks corresponding to A and B will just about fuse to a very broad peak. Often line-broadeing is so severe in this case that the peak cannot be detected above the noise. The equations needed to describe intermediate chemical exchange line broadening are complicated and will not be given here.
Intermediate symmetric exchange Note that peak broadening and reduction in intensity is most severe for intermediate exchange. If the exchange is slower or faster, the peak(s) sharpen up.
Fast symmetric exchange Equations needed for describing line broadening due to fast exchange δν = ν A - ν B Difference in resonance frequencies between sites A and B LW 2 πδν = 2k Line broadening If the jumps between sites A and B are very rapid compared to the chemical shift difference between A and B (in frequency units), the average magnetization will end up with a phase that is the average of A and B. Correspondingly, the observed chemical shift will be in the middle of A and B. Since the jumps between sites are random processes there will be a spread around the average phase. We thus get line broadening also in this case.
Exchange regimes When we say that exchange is fast, slow or intermediate we are never talking in absolute terms but are always comparing the exchange rate (k) with the difference in resonance frequencies between A and B (δν). We call the time-scale set by resonance frequencies the chemical shift time-scale. k << δν k δν k >> δν slow exchange intermediate exchange fast exchange Condition for coalescence of peaks (when there only is a single peak at the average peak position) k πδν 2
Asymmetric exchange A k A k B B If the populations are skewed (p A p B ) the line broadening due to exchange is different. We see that the line broadening for slow exchange must be described by two equations and that the populations of A and B enters the expression for fast exchange. Slow asymmetric exchange LW = A LW = B k A π kb π Fast asymmetric exchange LW 2 4πδν p = k + k A A B p B
Asymmetric exchange for different rate regimes
Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion The effective transverse relaxation rate (the topic for next lecture) is monitored as a function of the repetition rate of 180 pulses during a constant time relaxation delay. 800 MHz 600 MHz The resulting profile is fitted to the Bloch-McConnell equations. visible ground state invisible excited state Information content Thermodynamics: p B A k AB k BA B Kinetics: k ex =k AB +k BA Structure: Δϖ AB
Applications of CPMG relaxation Identification of presence of exchange processes and their localization to specific parts of the protein. Determination of rate constants for interconversion between states. Determination of excited state populations and chemical shifts. Some applications: Protein folding Ligand binding Allosteric regulation Enzymatic catalysis
Transverse relaxation Magnetization in transverse plane Resultant magnetization in transverse plane M x time Since the resultant gets smaller the more de-phased the magnetization is, the signal will decay as a function of time. The correct expression for time evolution of transverse magnetization is M x = M 0 e t/t 2ccc (2πππ).
Longitudinal relaxation If a sample is placed in a magnetic field it will take a certain time for the new equilibrium to be established. We can write this as M z t = M z,ee (1 e t/t 1) where M z (t) is the z-component of the magnetization and M z,eq is its equilibrium value. In fact this expression is not limited to the case when the sample is moved to a different magnetic field but also to situations when the equilibrium has been perturbed by other means, for example by the application of pulses of radio-frequency irradiation.
Consequences of different values of T 1 and T 2 T 2 is related to for how long the NMR signal can be observed. It is also related to the line width of the NMR signal. The shorter T 2 is, the wider and less intense the line in the spectrum. The sensitivity is thus poor for molecules with short values of T 2. LL = 1 πt 2 T 1 is related to the time it takes the spins to reach equilibrium populations. The value therefore directs how long one has to wait before repeating the experiment. To reach equilibrium one has to wait 5 T 1 but in practice one waits shorter since the sensitivity per unit time thus increases.
The origin of relaxation Since a randomly fluctuating magnetic field causes relaxation we now turn to what causes the randomly fluctuating magnetic field. The answer is that it the stochastic modulation of various spin couplings due to molecular reorientation. The scalar coupling is the same in all directions and hence does not cause relaxation. However, the direct dipolar coupling is orientation dependent and is often the most important relaxation mechanism. It may seem odd that a coupling that is averaged to zero has any influence at all. However, relaxation depends on the square of the coupling and this is not averaged to zero.
Dependence of dephasing on magnitude of random fluctuations of the magnetic field < B 2 > = 0.5 a.u. < B 2 > = 1.0 a.u. A randomly fluctuating magnetic field in the z-direction leads to dephasing of transverse magnetization. This is one cause of the decay of the NMR signal. The process is called T 2 -relaxation, or transverse relaxation. A similarly fluctuating field in the xy-plane leads to T 1 -relaxation a.k.a. longitudinal relaxation. This is related to how fast equilibrium populations are established.
The square of the random fluctuations of the experienced field can be calculated In many cases, the most important cause of the fluctuations of the magnetic field is caused by the dipolar coupling. Although < B>=0, for rapidly tumbling molecules in solution, the average of its square < B 2 > does not vanish. It can be calculated as ΔB 2 = μ 0/4π 2 ħ 2 γ I 2 γ S 2 It is proportional to the square of the gyromagnetic ratios of the coupled nuclei and inversely related to the sixth power of their separation. Dipolar relaxation is thus most important for nuclei with high gyromagnetic ratios that are close in space. r II 6
Random motion, time correlation functions and spectral density functions The level of correlation between the orientation at t=0 and later points in time can be quantified by the time correlation function. For a rigid molecule it is TTT = e t/τ c where τ c is called the correlation time. The TCF can be Fourier-transformed to see which frequencies it contains. The result is called the spectral density function. It has the following form for a rigid molecule J ω = 2τ c 1 + ω 2 τ c 2
Important frequencies in the spectral density functions It turns out that the only frequency that is important for longitudinal relaxation is the resonance frequency ω 0 = 2πν 0. For transverse relaxation ω = ω 0 is also important but even more important is ω =0. The spectral density function at these frequencies thus enter the expressions for the relaxation rates in the way shown on the next slide.
Expressions for relaxation rates 1 T 1 = ΔB 2 J ω 0 1 T 2 = 1 2 ΔB2 J 0 + J ω 0 J ω = 2τ c 1 + ω 2 τ c 2 NOTE THAT I USE A SLIGHTLY DIFFERENT NOTATION THAN HORE DOES. ω = 2ππ From the expressions it is clear that T 2 will drop monotonically as molecular motions get slower while T 1 will pass a minimum and then increase. The timeconstant τ c depends on the molecular mass, the temperature and the viscosity of the sample.
The nuclear Overhauser effect (NOE) One of the strangest but also most important manifestations of relaxation is that the spin state of one spin (S) affects the populations of another spin (I). Depending on what we do to S we can make the NMR signal for I stronger, weaker or even inverted. The NOE also provides a mean to transfer magnetization between S and I in a distance dependent manner. This forms the basis of the NOESY experiment that is used to measure internuclear distances in molecules.
The NOE is caused by simultaneous spin flips The important spectral densities are J ( 0 2ω ) J (0) flip-flip transition flip-flop transition From the expression for < B 2 > it is obvious that the NOE is most pronouned for high gyromagnetic ratio nuclei that are close in space.
Fast internal motions and the model-free formalism If the direction of a bond vector can be modulated by global tumbling as well as fast internal motions, the spectral density function is modified to J(ω) = 2S 2 τ c 1+ ω 2 τ c 2 + 2(1 S 2 )τ e 1+ ω 2 τ e 2 τ e is the effective correlation time for the fast motions and the order parameter S 2 represents the restriction of the internal motions. A value of 1 means total restriction, i.e. a rigid bondvector whereas a value of 0 means total flexibility. We see that the contribution of the internal motions to the spectral density function increases as S 2 decrease. This way of describing tumbling and internal motions is called the model-free formalism.
To determine all three model-free parameters, R 1, R 2 and NOE must be measured. The relaxation rates R 1, R 2, and the heteronuclear NOE measured for a protein. The order parameters, S 2,were calculated and the tumbling time was estimated to τ c = 8.8 ns.
Order parameters for the protein E140Q-Tr2C The topology is a four helix bundle with two calcium binding loops Do you think the order parameters make sense? S 2 τ c = 4.4 ns
R 2 I = I 0 exp( R 2 t) Intensities (A.U.) R 2 (s -1 ) Time (s)
Relaxation experiments can be used to determine whether a protein dimerizes R 2 Intensities (A.U.) R 2 (s -1 ) Time (s) [1.0 mm]
The ratio R2/R1 can be used to determine the shape of a molecule prolate oblate α The data is compatible with a prolate shaped molecule
How to measure T 1 Inversion recovery Intensity Time For short delays, the signal will be inverted (red signals). It will then increase, pass zero and continue to increase to the equilibrium value, i.e. as if the 180 and relaxation delay were omitted.
How to measure T 2 CPMG spin-echoes Intensity Time (The 90 pulse should have phase x and the 180 pulses should have phase y )
Alternative method to measure T 2 Do all CPMG spin-echoes in one go! Intensity Time
How to measure the NOE The NOE is measured by performing the two experiments in the figure. The first experiment simply comprises an 90 pulse on nucleus A, while the other also includes saturation of the attached nucleus X. Typically, A= 15 N and X= 1 H. The NOE is calculated as the ratio of the intensities of experiment 2 and experiment 1. NNN = I 2 I 1