MOLEC STRU DYNAM NMR
MOLEC STRU DYNAM NMR BCH 6745
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This 32 page Class Notes was uploaded by Kavon Huel on Friday September 18, 2015. The Class Notes belongs to BCH 6745 at University of Florida taught by Arthur Edison in Fall. Since its upload, it has received 6 views. For similar materials see /class/206960/bch-6745-university-of-florida in Biochemistry and Molecular Biology at University of Florida.
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Date Created: 09/18/15
Today s Lecture 10 Mon Oct 23 Relaxation Measurements a Correlation functions b Causes of relaxation c NMR measurements of correlation functions d Example of relaxation measurements Consider random field fluctuations along the X axis 0 AF Lets monitor two of these spins for awhile Spin 1 Spin 2 B I B X X 4 4 t t Assuming these are completely random Bx 0 How can we describe the magnitude of these fluctuations Bx 0 0 1330 gt 0 Spin 1 Spin 2 2 How does the value of l3X at one point correlate with its value at another point in time Short spacing Long spacing Let s formalize this with an autocorrelation function G0 Bx th t 7 0 fast fluctuations a slow fluctuations Gk t G0 330 Go ltBftgtejlrlrc The correlation time 1 is the time constant for the exponential decay of the function How does this translate into frequencies The spectral density function Ja ZTGU 6Xp icoz39 For a randomly fluctuating field along the xaxis 2 71 2 Tc Ja 2ltBx gtAco0 TC 2ltBxgt1 021 This is a Lorentzian lineshape the natural linewidth seen in an NMR spectrum Let s graph this fast fluctuations broad spectral density function Jm r l 7500 D 500 1000 t mZn MHZ slow fluctuations narrow spectral rc 2 ns density function B X Jw 71000 7500 0 500 000 t 0327 MHZ What about less random events 02 A correlation function can describe the motion ofa molecule or part ofa molecule The correlation function above is for the isotropic dif ision of a rigid rotor Again the correlation time 15 is the time constant for the exponential decay of the function Iris approximately the amount of time the molecule takes to rotate l radian Notice that short correlation times cause the correlation function to decay rapidly and long times cause the function to decay more slowly The correlation time depends primarily on molecular size and shape as Well as solvent viscosity temperature A 5 Edison University ofFlonda Spectral Density function Note this looks very similar to the case of random fluctuations 250 Frequency MHz The spectral density function Jw is the Fourier transform of the correlation function Just as rapidly relaxing time domain signals give rise to broad lines short correlationtimes have a broad spectral density function This makes sense molecules that tumble very rapidly can sample a Wide range of frequencies Molecules that tumble slowly and have very long correlation times only sample lower frequencies S Edison Univ aslty of Florlda Spectral Density function describe internal motion For example the popular LipariSzabo model free spectral density function is I is the correlation time for internal E o o lt 2 V o E e Note this combines random fluctuations with more specific motions completely exible molecule and relaxation mechanisms A S Edison Univ aslty of Florlda Speci c sources of relaxation Dipoledipole relaxation Magma Local Dipolar Field Tumbling CSA relaxation APPLIED FIELD l INDUCED FIELD For spin 12 dipoledipole gt CSA For spin gt12 quadrupole gtgt dipoledipole gt CSA Where is this going The goal in protein dynamics is to know the spectral density function That is dif cult Several different methods have been derived to model the motions of proteins and represent the models as different spectral density functions with different parameters that are t to experimental data One approachthat is quite straightforward but requires a lot of experimental data has been developedby Peng and Wagner This is called spectral density mapping and it essentially involves experimentally measuring the spectral density function at select NMR frequencies This overall approach will be outlined on the next few slides Regardless of the technique one chooses for dynamics measurements the next few slides will also show the mathematical relationship between relaxation parameters such as T1 and T2 and the Jm S Edison Univ asity of Florida Measuring Dynamics Background reading Cavanagh et al Protein NMR Spectroscopy Principles and Practice Academic Press 1996 Peng and Wagner Investigation of Protein Motions via Relaxation Measurements Methods in Enzymology 239 Eds James and Oppenheimer p 563596 1994 We introduced T1 and T2 in a phenomenological sense in the Bloch equations This was back in the good old days when we only had one spin with its x y or z orientation to worry about When we got to 2 spins we had 16 terms Each of these terms has different relaxationbehavior on its own autorelaxation Terms can also relax through interactions with each other like the 1H 1H NOE cross relaxation How does relaxation occur and what information can we gain by measuring relaxation rates Some Autorelaxation Rates for an I S two spin system Table adapted from Peng and Wagner 10 Koo 1m on 1w Jm m pL pT RSSZ eg T1 of S 0 3dc d 0 6d 0 0 RSSX eg T2 of s 2d2c3 3dc2 dz 3d 3d 0 0 RIz eg Tl ofI 0 0 d 3d 6d 1 0 RIX eg TZ ofI 2d 3d dz 3d2 3d 0 1 RS212sz newl 0 3dc 0 3d 0 1 0 Etc see Peng and Wagner for more 25 and A5 are the frequency and chemical shift anisotropy CSA for spin S CSA is the difference in chemical shi when th molecule is oriented along different axes relative to BE S Edison Univ asity of Florida For example the relaxation rate for 15N along the z axis eg T1 is 2 2h2 A2602 RNNZ IWHN 7 QN 3JQ N 6Ja HN Mill 3 N Ja N My This looks much more complicated than it really is Most ofWhat you see are just constants that are known The important concept is that the relaxation rate depends on the spectral density at different frequencies RNNZ 0c a JcoHN 0N b JcoN c JcoHN 0N Where a b and c are constants A S Edison Univ asity of Florida Remember that our goal is to know the spectral density function Each of the different auto and cross relaxation terms is similar to the one for the T1 of 15N and is just different combinations of the spectral density functions at different frequencies RNNZ M a JcoHN 0N b JcoN c JcoHN 0N Peng and Wagner and others have derived equations relating the different spectral densities to relaxation rates this is just algebra not physics For example 110 7gigsRsltsRS2Izsxgti RmaIzsz 7 R 12 12d4cl S Edison Univ asity of Florida Many measured relaxation rates give rise to the spectral densities These rates can be measured and if you have enough spectrometer time at different frequencies you can directly measure the spectral density at different frequencies just using 1H and 15N For example a 1176 T 500 MHZ magnet will allow measurement of the following frequencies 0 50 MHz N at 1176 T 500 MHz 450 MHz 1H 7 15N 550 MHz 1H 15N A 176 T 750 MHZ magnet will measure 0 75 MHZ 750 MHz 675 MHZ and 825 MHz Peng and Wagner and others have derived pulse sequences to make all the necessary measurements for this Please see the original work for details A S Edison Univ asity of Florida Nature Smcmm1amplvblecular Btutugy 1 945 r 949 mm MaguusWulfrWatzVuleai r r H dim in a N n WWW u mmqu hummus W 1 M m aquot Homework 1 Calculate the dipolar term for a 13C and proton that are 1A apart 2 Calculate the breadth ofthe CSAin Hz for a 13C spin at 176 T and 117 T ifits value is 200 ppm t moving to substantially larger magnetic elds Today s Lecture 4 Wed Oct 9 Data Collection a Digitization and Spectral Width b Fourier Transform c Quadrature detection lson Unwasity ofFlonda 2006 What happens next 1 A sample With a single resonance eg water is placed in a strong static B0 magnetic eld and the sample reaches equilibrium 2 An rfB1 eld is applied at a frequency 10 Hz away from the single resonance in the sample The E1 eld is applied along the Xaxis long enough to rotate the magnetization vector from the zaxis into the Xy plane Then the B1 eld is turned off lson Unwasity ofFlonda 2006 Precession and relaxation A 5 Edison University ofFlonda 2006 Detection The sample rotates in the xy plane at a frequency of 10 Hz and relaxes in the xy plane T 2 and along the zaxis T 1 The rotating magnetization induces a current in the same coil Which produced the B1 eld pulse This current is ampli ed and detected in the spectrometer However the analog signal is converted to digital by sampling the data at select time points ur l 1 digital A 5 Edison University ofFlonda 2006 Time Domain Signal The time domain signal is collected point by point and can be thought of as a vector d d0 d1 dz erl with M points To make the math work the count begins at 0 For example here is a 10 Hz signal sampled for 02 seconds with 001 sec spacing between points 1 do 0 dioo d1 05 39dz O 05 01 O 15 O 2 M13 0 o o 05 etc 0 0 O 1 o o lson Unwasity ofFlonda 2006 The spectral or sweep Width is determined by the spacing between time points IMPORTANT IMPORTANT SWi IMPORTANT IMPORTANT SW is the spectral width in Hz and At is the spacing between points lson Unwasity ofFlonda 2006 What if a peak is out of the SW These two frequencies are 10 and 110 Hz sampled at Dt001 seconds The FT would have a SW of 100 Hz and the two peaks are indistinguishable This is called aliasing and can be a big experimental problem ifyou are not careful A 5 E645 b ls 2006 Fourier Transform The Fourier Transform converts the time domain signal to the frequency domain There are several types of FT and we will examine the discrete PT DPT because it works on discrete poinw like NMR data 1 N4 2m knN DFT f W dke 0 1 is the nth point in the frequency Vector Nis the number oftime data points dk iis J l k and n are integers that refer to the points in 11C and A S Edi Notes adapted from NMR Data Processing Hoch and Stem WileyLiss 1995 Nanci F1 0 dz 2006 Fourier Transform For example here is the vector d for dice 2mm N 100 time points of Cos2 Pi20 t 20 Hz frequency d1 0 309017 70 809017 70 809017 0 309017 0 309017 70 809017 70 809017 0 309017 lt0 lt5 1 lt5 00 lt5 lt50 U lt5 00 lt5 lt5 1 lt5 9 lt5 lt0 lt5 1 lt5 00 lt5 lt0 lt5 1 lt5 00 lt5 lt0 lt5 1 00000000 wwwwwwww oooooooo ltoltoltoltoltoltoltolto 00000000 0000000 0000000 wwwwwwww oooooooo ltoltoltoltoltoltoltolto 00000000 0000000 0 309017 70 809017 70 809017 0309017 70 809017 70 809017 1 1 1 1 1 0 309017 70 80901710 80901710 309017 1 1 70 8090171 1 70 8090171 11 70 8090171 11 0309017 70 809017 70 8090171 1 70 8090171 1 70 8090171 1 70 8090171 1 1 1 1 1 10 309017 70 809017 70 809017 0 309017 1 lt5 9 lt5 lt0 lt5 1 lt5 00 oo lt5 lt5 lt0 lt5 1 lt5 00 lt5 lt0 lt5 1 A S Edison Unwasity ofFlonda 2006 Fourier Transform f d e Zm39knN Here istherealpartofffor 100 n k frequency points of for the previous 1 A S Edison Unwasity ofFlonda 2006 What does it look like 0 0 see thh 0 01 see spaemg between pomts 5 Here 15 the real pan ofthe DFT ofthe 4 signal above Several Lhmgs are wonh Mug 1 There are zpeaks each 20 Hz from 2 enhee edge 2 The spectral wxdth 15 100 Hz 3 The peaks are qune sharp A 5 Mean UnlvemiyafFlnndA Z Why 2 peaks from one frequency e 5 Here is a plot of Cos2Px20t 7 n n x x n 2 2 n 3 en 5 These are indistinguishable even h che 39 7 n re ore 2 freq A A A A A A mm and cm ochquot negative e 5 e T e T VW VW A 5 Mean UnlvemiyafFlnndA Z Why 2 peaks from one frequency WM U U f U W Here rs aplot of 5rh2pr2o We can distinguish different signs in Sin functions Here rs a plot of Sm2P1720t A 5 men UnlvemiyafFlnndA Z FT of a sine function a 3975 u 5 a 25 ma quot 25 01 see spaerhg between perms h 5 u 75 of r the signal above Several thmgs are wonh hohhg 7 1 There are 2 peaks each 20 Hz from enher edge Nowthey are opposxte each 0 W 2 2 The speehal wxdth rs 100 Hz 4 3 The peaks are quite sharp n 2n n so an m A 5 men UnlvemiyafFlnndA Z SumsDifferences 0f sine and cosine FTs 5 i 3 2 PT of cosine function i 2 in en a m I i 2 A PT of s1ne function 2 4 if n in in en Sn m H II in a s leference i 2 A 5 Edison University ofFlonda 2 in en Sn m 2006 Quadrature Detection In order to distinguish the sign of frequencies NMR signals must be collected along both thex eg cosine and y eg sine axes This is called quadrature detection The signal is treated as a real part cosine and an imaginary part sine They are both equally real and important but this allows easy mathematical manipulation from Euler s relations C0st 139Sint equot C0st iSint e39quot real imaginary real imaginary A 5 Edison University ofFlonda 2006 Quadrature Detection a asuh2m1x5m12h2m1 Here 15th DFT af39he mu pamtrumuan a n shave wnh a frequzn InexpenmemuNMR 91 measured slangbathaxes and cambmzd n Next Lecture 5 Wed Oct 11 mam upuamn um m may 391 qmlum mahuliu a RF pulses xxxwand Today s Lecture 1 Rabi Molecular Beam experiment continued from last time 2 Fri Oct 4 Behavior of nuclear spins in a magnetic eld 11 a Bloch equations b Phenomenological introduction to T1 and T2 c RF Pulses d Teach Spin apparatus A S Edison Unwasity ofFlonda 2006 I I Rabi molecular beam experiment to measure 7 Feynman Lectures on Physics B N Detector The coil produces a magnetic eld along the xaxis going into the board 0 When the frequency reaches resonance particles no longer reach the detector A S Edison Unwasity ofFlonda 2006 y Relates frequency and magnetic eld Planck s Law Important NMR equation relating frequency With magnetic eld strength lson Univasity ofFlonda 2006 Bulk Magnetization The magnetic moment u is a vector parallel to the spin angular momentum I The gyromagneto or magmetogyro ratio v is a physical constant particular to a given nucleus Unfortunately the vast majority of the magnetic moments cancel one another The Boltzmann excess in the cc state add together to create bulk angular momentum J and magnetization M lson Univasity ofFlonda 2006 Gyroscopes Classical physics tells us about the motion of a magnet in a magnetic eld Lt is the gyroscope s angular momentum r its radius from the fixed point of rotation m its mass and g the force of gravity A S Edison Unwasity ofFlonda 2006 Bloch Equations We can make the equations easier to deal With by multiplying both sides by y Multiply by v Remember that M D A S Edison Unwasity ofFlonda 2006 Bloch Equations Mo t X L What does this equation yd 39 describe After suitable choices for B this equation will predict that nuclear magnetization will precess at a frequency w0YB0 FOREVER Nothing in the equation is a restoring force to cause the magnetization to relax back to equilibrium However reallife NMR experiments relax lson Unwasity ofFlonda 2006 Bloch Equations Therefore Felix Bloch made the following modi cations to the basic equation dM Ct dt M6 YBU RMU M 01 W Empirical modi cation in which a relaxation matrix R acts on magnetization that is different from the equilibrium state M 0 lson Unwasity ofFlonda 2006 Bloch Equations This equation is easiest to understand broken into its matrix components A S Edison Univasity ofFlonda 2006 Changing the frame of reference PROBLEM The Bloch equations we have shown so far are helpful but still too complicated The problem is that as soon as magnetization is put into the xy plane it starts to precess we will see that soon at NMR equencies eg 500 MHZ Thus the actual trajectory of the motion is VERY COMPLICATED SOLUTION We will de ne a coordinate system that rotates around the zaxis at the same NMR equency This is accomplished by de ning the following wrfis the frequency of the NMR transmitter and Be is the frequency of the peak we are interested in observing If the two e the same this is called onresonance lnthis case the effective magnetic eld strength along the zaxis is 0 A S Edison Univasity ofFlonda 2006 Bloch Equations in the Rotating Frame Substituting QyBOmrf Where B0BZ and is not timedependent into the Bloch equations yields The r superscript refers to amagnetic eld in the rotating frame Now try some Mathematica simulations to see what these mean iSOn Univasity ofFlonda 2006 Next Monday s Lecture 3 Wed Oct 4 Introduction to NMR Parameters a Chemical shi iBMRB database b J couplingiKarplus equation c Tl d T2 e NOE f Dipolar Couplings iSOn Univasity ofFlonda 2006 Today s Lecture 6 Mon Oct 14 Product operators II a Scalar I coupling b Multiple pulse experiments lson Univasity ofFlonda 2006 5 N Last lecture we discussed rotations of bulk magnetization Z l 0 0 0 0 0 0 0 o Cos sm 0 0 0 sm sm 0 Sin Co l 0 0 Cos Cos y Z This was the example of a rotation of X MZ around the x axis by an angle 1 The vector went toward the y axis lson Unwasity ofFlonda 2006 Now we get a bit more complicated I How many states does a spin 12 particle have I 2 don t forget 21l I How do the 2 quantum states relate to 3D Cartesian space I Different Cartesian orientations arise from a superposition of quantum mechanical states lson Univasity ofFlonda 2006 Quantum mechanics again For a proper treatment of the subject please see books by Abragam Ernst et al Goldman Farrar or Cavanagh et al The spin be in two state I The angular momentum can be represented by I lson Univasity ofFlonda 2006 Quantum Mechanics Rotations QM still involves rotations Now it is slightly more complicated so I will just present one case and let the interested students read the complete treatments RAB cos0 2 isin0 2 1 srn0 2 cos0 2 Here is the rule for an xpulse with an angle 6 After the proper manipulations and learning the rules about how todo it you get the same answer as a 3D rotation of bulkmagnetization cos8 2 isin62 l 0 cos8 2 isin8 2 isin8 2 cos62 HO 1 isin8 2 cos8 2 1 cos6 isin6 2 isin6 cos8 Z cos6 1y sin6 lson Univasity ofFlonda 2006 Why am I telling you this No this isn t a course in quantum mechanics and you are not expected to learn how to do the previous example It is very satisfying to do so I encourage you to learn it some day Here is Why you need to at least see the QM You can get the same answer using QM as Bloch equations QM is the way to understand different spatial orientations of a spin with only 2 states You can t understand the next part without QM Therefore you can either work hard to learn the real way to do it OR justbelieve a set of rules I will give you Your choice lson Univasity ofFlonda 2006 Scalar J Coupling When two spins are connectedby chemical bonds in a way that leads to splitting in the NMR signal they are said to be coupled Usually spins are coupled when they are separated by 1 2 or 3 covalent bonds Here are the different states that 2 spins I and S can have if they are NOT coupled S 1 NOT S as IX 1y IZ OLTPLED SX Sy SZ lson Univasity ofFlonda 2006 Scalar J Coupling Here are the different states that 2 spins I and S can have if they are are coupled Notice that we now need to consider all of these together rather than individually because they are coupled 16 I S states Single Quantum IX ly 12 SK Sy SZ Multiple Quantum IXSX IXSy lySX lySy Antiphasel S l S SXIZ SylZ xzyz Spin or JOrdered lZSZ Unity l for the math to work lson Univasity ofFlonda 2006 Scalar J Coupling These states each form a 4x4 matrix that is the direct product of the two corresponding 2x2 matrices For example lson Univasity ofFlorida 2006 What do you need to know 394x4 matrices such as IZSZ shown on the last slide can be made for all 16 IS states These can then be used in rotations as shown for the x pulse of a single spin The coupling operator is given by mljstzi z Where It is 314 J13 is the value of the scalar coupling between I and S I is the tim e and ZIZSz is the 4x4 matrix shown on the last slide lson Univasity ofFlorida 2006 Rules for the coupling operator It only acts on magnetization with one component in the x y plane Ix SK IXSZ Iysz SXIZ Sylz MEMORIZE THESE 1X M Ix cosnmt 2552 sinnmt 11921252 1y cosnmt ZIXSZ sin mt 71 21 539 ZIySZ 15 z Ze ZIySZ cosnmt IX s1n71 mt ZIXSZ mfst zsz 21sz c0snmt 1y sinnmt unmismimda 2006 Rules for the sign of rotation X M X assetmt 21ysz sinmSt 1y M S ZIZSZ 1y assetmt 21x52 shimmt ml tzf ZIySZ 46 ZIYSZ 0050143 Ix sinmSt y X x or y is the transverse component of the spin 21x52 ML 21st commt 1y sinmSt lson Univasity ofFlonda 2006
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