Numerical Methods for Pulse Sequence Optimisation

Transcription

1 Numerical Methods for Pulse Sequence Optimisation Lyndon Emsley, Laboratoire de Chimie, Ecole Normale Supérieure de Lyon, & Institut Universitaire de France ECOLE NORMALE SUPERIEURE DE LYON IUF

2 Dances with Spins: Whispered Messages from the Picometer World tb u tb u Zr tb u O O S i O S i O O O O H 2, 15! C 3 C H 3 tbu H Zr H O O H S i O S i O O O O C H 4 /C 2 H 6 H Zr O S i O O O O O H H S i O H H Zr O O H S i O S i O O O O H z 1 nano meter ys296

3 Dances With Spins designing effective Hamiltonians H = H z + H Q + H D + H cs + H J + H ext We can add rotations controlled by the experimentalist. If properly designed these rotations can selectively cancel out parts of the Hamiltonian. Rotations in Laboratory Space (magic angle spinning) Rotations in Spin Space (radiofrequency pulses) RF phase (!) 2" " -" Spatial rotation: 35 rev/second Spin rotation: 15 rev/second edumbo " 4" 6" RF periods

4 The Dance of the Spins adapting the effective Hamiltonian to our needs 1 H CP no decoupling SPINAL-64 TPPM 13 C CP R2 9 4 R2 9 4! increment interval detect H = H D H = H J H = H cs to select only the pair of spins we are interested in to measure the J coupling to detect a resolved spectrum J. Am. Chem. Soc. 128, 3878 (26).

5 Structure of the Retinylidene Ligand in Rhodopsin The Whispered Message: J couplings reveal no conjugation defects around the isomerization site. They do detect the expected penetration of the PSB positive charge into the retinylidene chain 1 H 13 C no decoupling CP CP R2 9 4 R2 9 4 SPINAL-64! TPPM can be related to the mechanism of photoisomerization? 75 7 all- interval -ret (soln) ret-psb (soln) rho (solid) Lys C9=C1 C1-C11 C11=C12 C12-C13 C13=C14 C14-C15 Atom pair Levitt and coworkers, J. Am. Chem. Soc. 128, 3878 (26).

6 Where is the Problem in the Design of Effective Hamiltonians or Transformations? free induction decay spectrum FT Luckily we have an analytical and reversible transform that links the fid with the spectrum...

7 Where is the Problem in the Design of Effective Hamiltonians or Transformations? time domain pulse frequency domain response Black Box (Bloch Equations) time domain manipulations Average Hamiltonian (decoupling, mixing...) Black Box (The denisty matrix description of NMR) Proton Chemical Shift (ppm)!2... more generally there is no analytical solution to the equation relating the pulse sequence to the spin system response. This is the general question of how does a radiofrequency field (light) interact with the nuclear spin system (matter)?

8 In general the forward transformation has no analytical solution There are three main strategies for the forward calculation: 1. Find an analytical solution for a simplified case (rare, always difficult) time domain manipulations 2. Use approximate methods: Average Hamiltonian Theory, Static Perturbation Theory, Floquet Theory 3. Solve the problem numerically d dt i H, H H D H cs H J H ext Average Hamiltonian (decoupling, mixing...)

9 Numerical Methods for Pulse Sequence Optimisation choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Has been used to generate pulses and sequences for: numerous MRI applications robust selective excitation and inversion (G3, BURP...) heteronuclear J decoupling (GARP,...) homonuclear dipolar decoupling in solids (DUMBO) coherence transfer in liquids and solids (Optimal Control) Key authors: Morris, Warren, Freeman, Emsley, Kupce, Glaser

10 Example: Broadband J Decoupling choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: One isolated spin, Bloch Equations, resonance offset or B1 misset Excitation Function: Pulse flip angles in periodic decoupling sequence. Target: Broadband (low power) J decoupling Optimisation Routine: Gradient descent Waltz+ PAR-75 GARP Shaka, Barker & Freeman, JMR 64, 547 (1985) 1 4 t2 i-1 -I -2 ABIB;

11 Example: Selective Excitation in Liquids choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: One isolated spin, Bloch Equations, resonance offset Excitation Function: Amplitude modulation described by a truncated Fourier series. Target: In phase full excitation in bandwidth. No excitation out of bandwidth. Optimisation Routine: Simulated annealing resonance offset Geen, Wimperis & Freeman, JMR 85, 62 (1989) Geen & Freeman, JMR 93, 93 (1991)

12 Example: Selective Inversion in Liquids choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: One isolated spin, Bloch Equations, resonance offset Excitation Function: Amplitude modulation described by a sum of Gaussians. Target: Full inversion in bandwidth. No effect out of bandwidth. Optimisation Routine: Gradient descent Emsley & Bodenhausen, CPL 165, 469 (199) Emsley & Bodenhausen, JMR 97, 135 (1992) resonance offset

13 More Challenging Examples? choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: Two J coupled spins, rapid relaxation Excitation Function: Flip angles and delays in a multi-pulse sequence Target: Maximum coherence transfer CRIPT TROPIC Optimisation Routine: Optimal control Frueh et al. J. Biomol. NMR 32, 23 (25)

14 More Challenging Examples? choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: One spin, with quadrupolar coupling, under MAS in a powder Excitation Function: Individual time steps in an amplitude and phase modulated continuous irradiation sequence. Target: Maximum excitation efficiency Enhanced MQ MAS experiments Optimisation Routine: Optimal control Vosegaard et al., JACS 127, (25) + many other examples for optimal control from Glaser, Khaneja and coworkers

15 More Challenging Examples? choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: Two dipolar coupled spins. Homonuclear Dipolar Decoupling Excitation Function: Continuous phase modulation defined by a truncated Fourier series Target: Maximise the chemical shift, remove the dipolar coupling Optimisation Routine: Gradient descent Sakellariou et al., CPL 319, 253 (2) t q Dipolar coupling d 12 /khz 3 6 n BLEW a n cos n c t 8 b n sin n c t 1H / khz q radio-frequency phase ( ) 2 DUMBO-1 DUMBO-1 64 steps 5 ns per step time / s Dipolar coupling d 12 /khz H / khz

16 The Solution to All Your Problems? choice of function choice of model choice of target generation of random starting point computer model calculates response comparison with target response good enough? yes experimental test on spectrometer good enough? yes accept new sequence choice of optimisation routine no no optimisation routine calculates next try Model Spin System: Two dipolar coupled spins. The weakness of numerical optimisation in that the computer is that the solution you obtain, by whatever method, is only as good as the accuracy with which the model system you set up actually reproduces experiment! Out of many results, the 5-TR pulse sequence shown in Figure 1a yielded the highest 3Q coherence excitation.

17 radio-frequency phase ($) 2" " DUMBO-1 What Next? 64 steps 5 ns per step #" time / µs MAS dimension 139 Hz 133 Hz.26 ppm.32%! 4 2 ppm CRAMPS dimension DUMBO-1 was developed using a computer simulation approach. It works very well. But still not good enough for chemistry...!! How can we do better than.32% residual?

18 How Can We Improve the Accuracy of the Predicted Spin System Response? We need < 1% accuracy......we routinely adjust the B homogeniety to << 1% accuracy by shimming directly on the NMR signal response. We would never imagine shimming by first calculating the field map, to predict the best shim values, and then try them!!

19 The edumbo Approach experimental Decoupling Uses Mind Boggling Optimization 6 Use the NMR signal to find high-performance pulse sequences through!( t) = # iterative optimisation: ( a n cos( autoshimming n" c t) + b n sin( n" c tsequences. )) n= choice of function choice of model sample generation of random starting point spectrometer acquires spectrum comparison with ideal response good enough? yes accept new decoupling sequence no optimisation routine calculates next try no need Has for been computer used to modelling generate of pulses the spin and system sequences (to <1% for: accuracy)! heteronuclear dipolar decoupling under MAS (CM) homonuclear dipolar decoulping under MAS (edumbo) increased carbon-13 transverse dephasing times T 2 (TDOP) Chem. Phys. Lett. 376, 259 (23) increased sensitivty in solid-state 1 H- 13 C INEPT & HSQC

20 The edumbo Approach to Heteronuclear Dipolar Decoupling in Solids choice of function generation of random starting point choice of model sample spectrometer acquires spectrum comparison with ideal response good enough? no yes accept new decoupling sequence Model sample: [2-13 C] Glycine Optimisation Method: simplex Quality Factor: intensity of the carbon-13 CPMAS peak optimisation routine calculates next try Decoupling Chem. Phys. Lett. 376, 3-4 (23). 13C spectra of Glycine 15 khz 1H irradiation khz MAS 115.9% 116.4% 11.% 48.6 Hz 48.5 Hz 52.2 Hz 8.6 Hz 53.9% ppm CW etppm XiX CM Phase (t) a -a c time Acquisition Cosine Modulation (CM) (t) = a cos (t/ c ) where c and a are variables for optimisation.

21 High-Resolution Proton NMR Spectroscopy: edumbo-1 Model sample: [2-13 C] L-Alanine Optimisation Method: [2-13 simplex C] L-Alanine Quality Factor: resolution of the CH Doublet 1 JCH doublet on carbon-13 1 khz 1 H-Homodecoupling q = a (I 1 + I 2 ) - b (" 1 - " 2 ) - J CH X! M 1 J CH X! dec edumbo edumbo-1 22 DUMBO khz MAS 22 khz MAS C Chemical Shift (ppm) C Chemical Shift (ppm) Chem. Phys. Lett. 398, 532 (24).

22 RF phase ($) High-Resolution Proton NMR Spectroscopy: 2% % -% edumbo % 4% 6% RF periods edumbo-1 F2: MAS - 1 H Chemical Shift (ppm) 1 H linewidths (FWHH) - #-AspAla dipeptide 1 OH #-AspAla Dipeptide 12 1 NH 8 NH Hz F1: edumbo H Chemical Shift (ppm) 6 CHAla 13 Hz (.25 ppm) CHAsp 4 CH2 2 CH khz MAS:!*(CH 3 ) " exp! cor (CH 3 ) DUMBO-1 72 Hz Hz (.3 ppm) edumbo Hz Hz (.26 ppm) 22 khz MAS:!*(CH 3 ) " exp! cor (CH 3 ) DUMBO-1 82 Hz Hz (.34 ppm) edumbo Hz Hz (.32 ppm) edumbo Hz Hz (.28 ppm)

23 Intrinsic Linewidths in Solids: Coherent Control of Transverse Dephasing Times 1 H CP 1H spin decoupling Model sample: [2-13 C] Glycine Optimisation Method: simplex Quality Factor: intensity of the signal after a 3 ms spin echo 13 C CP T 2 ' t 2 T 2 * Very small difference in unrefocused linewidths (<1 %) a.u T2 ' =24.62ms.2 T2 ' =47.8ms T2 ' =6.47ms. CW sec ecm standard Very large difference in refocused linewidths (>5 %) 1/( T 2 ') ecm (6 Hz) standard (13 Hz) J. Am. Chem. Soc. 125, (23)

24 8 Transverse Dephasing Optimised Spectroscopy Coherent Control of Transverse Dephasing Times ! 3. " carbon-13 single quantum frequency (ppm) J. Am. Chem. Soc. 125, (23) carbon-13 double quantum frequency (ppm) 13C double quantum frequency (ppm) ppm eoptimisation of the coherence lifetime essential to find the best decoupling conditions with CM or TPPM O 28 C3 C3 O C14 C17 C C9 C9 ChemPhysChem 5, 869 (24) C12 C24 C4 C24 C13 C2 C4 C1, C1, C22 C1, C2 C8 C25 C16 C25 C7 C ppm 13C single quantum frequency (ppm) C16 # c/# 1 C a (rad) C2 C26/27 C15 C26/27 C23 C29 C15 C11 C21 C19 C18 1 khz of decoupling 1 khz of MAS speed ~4 mg, 4 mm rotor 4 days, good S/N

25 "Decoherence Times": Liquids & Solids T 2 solid ~ 3 ms (1Hz) Peak height (a.u.) T 2 ' solid ~ 1 ms (3Hz) T 2 ' liquid ~ 21 ms (15Hz) (protein of 2 kd, 35C, 11.8 T) T 2 * solid = 6.6 ms (48Hz) 3 Hz 48 Hz ppm t (ms) Apparent T 2 Refocused T 2 Incoherent T 2 potentially very long, depends T 2 * solid << T 2 ' solid < T 2 solid << < << T 2 * liquid < T 2 ' liquid ~ T 2 liquid mainly on internal dynamics decreases as a function of the molecular weight Effective coherence lifetimes in solids can be longer than in liquids! SSNMR experiments can be more efficient than equivalent liquid-state experiments.

26 36 18 Direct eoptimisation of 1 H- 13 C J-INEPT edumbo steps (as implemented at 15 khz) 4 ns each, " c =2 µs! " # 2 1 H edumbo 13 C! $# " #! $# #! $#! edumbo 2! edumbo 2! " edumbo " Het. Dec. acq CH % CH & 1. CH edumbo after eopt on INEPT at 15 khz " c /2 " c CH ppm phase angle " c /2 " c Model sample: [U- 13 C] Isoleucine Optimisation Method: simplex Quality Factor: intensity of the signal after an INEPT transfer difference " c /2 " c ppm Direct experimental optimisation of the CH2 transfer efficiency increases the proton coherence lifetime, leading to a >6% increase in sensitivity. J. Am. Chem. Soc. 127, (26).

27 36 18 Direct eoptimisation of 1 H- 13 C J-INEPT edumbo steps (as implemented at 15 khz) 4 ns each, " c =2 µs! " # 2 1 H edumbo 13 C! $# " #! $# #! $#! edumbo 2! edumbo 2! " edumbo " Het. Dec. acq edumbo after eopt on INEPT at 15 khz " c /2 " c J-HMQC 1. CH 2 phase angle " c /2 " c J-INEPT difference " c /2 " c ppm Proton Chemical Shift (ppm) In carbon-13 labelled compounds, the efficiency of the transfer is far superior to that of the J-HMQC sequence. J. Am. Chem. Soc. 127, (26).

28 !!!!!!!! Through-Bond INEPT Based HSQC for Proteins 1 H 13 C Solid-State Refocused 1 H- 13 C INEPT experiment!" 2 t 1 edumbo!" # $# # $# 2 #!" $# edumbo edumbo!!! " edumbo " " 2 " % % %' %' Het. Dec. t mm MAS probe 192 scans Total expt. time = 2 hours 1 khz edumbo decoupling Sample temperature = 5! C Sample cooling using nitrogen gas through the BCU-X MAS using dry air 1H Chemical Shift (ppm) khz 7 MHz C Chemical Shift (ppm) J. Am. Chem. Soc. 127, (25).

29 I don t like numerical methods, because they don t provide any understandable result. It s all too much of a black box.

30 eoptimised Heteronuclear Decoupling a ecm (a) 2 Phase (t) cw TPPM 1 H CP ecm (a, x variable) ecm (reference) acq -a Overall rotation angle c 13 C CP T2' evolution (b) (c) c / Numerical optimisation converges to a c / 1 1 very simple answer for heteronuclear decoupling (a close cousin of TPPM), and demonstrates that the parameterisation is very sensitive Can we determine why it works? a (rad) a (rad) 2 J. Chem. Phys. 121, 3165 (24)

31 Proton Decoupling: ecm & TPPM Modulation Frame HORROR Conditions From the Rotating Frame... z 1 y H rf CM c x (t) = a cos (2 c t) This point is found to be a modulation frame HORROR condition. The numerical result provided the key to understanding the mechanism & parameterisation of TPPM! c / to the Modulation Frame (defined as the frame rotating around the mean axis x at the frequency of the modulation c ) 1. 8 z y.95 6 eff H eff.9 4 eff x eff and eff are functions of a and x = c/ a (rad) 2 J. Chem. Phys. 121, 3165 (24).

32 Summary I If the approximate analytical solution is sufficiently accurate, Average Hamiltonian Theory (and cousins) is a good platform for pulse sequence development. It provides a detailed understanding. J decoupling in liquids is an excellent example of where AHT methods work very well to describe the experimental observations. time domain manipulations d dt i H, H H D H cs H J H ext Average Hamiltonian (decoupling, mixing...)

33 Summary I If approximate methods fail, numerical methods can provide better solutions. This is often the case where single spin dynamics are sufficient to describe the problem accurately. time domain manipulations Selective pulses are an excellent example of where numerical computer optimisation works well. Liquid state coherence transfer is another area where this approach works. d dt i H, Methods using computer simulations of the spin system can only be as accurate as the simulation itself. H H D H cs H J H ext Average Hamiltonian (decoupling, mixing...)

34 Summary I In many cases, especially in solid-state NMR, computer simulations do not reproduce the experimental behavior sufficiently accurately to allow useful results. In these cases, direct optimisation of the NMR signal naturally provides more accurate results, and can generate the best pulse sequences. time domain manipulations Homonuclear dipolar decoupling is an excellent example of where the experimental optimisation yields the best results. d dt i H, H H D H cs H J H ext Average Hamiltonian (decoupling, mixing...)

35 Summary II eoptimisation: An Experimental Approach to Pulse Sequence Design Weaknesses of eoptimisation: sensitivity. needs a robust experimental quality factor that can be reliably calculated automatically. in most cases, does not provide feedback for understanding. needs a robust optimisation method that can deal with noise in the quality factor.

36 Summary II eoptimisation: An Experimental Approach to Pulse Sequence Design Advantages of eoptimisation: naturally integrates all the error terms. can be adapted to almost any problem, even when neither a theoretical nor a numerical description is available. Is easy to carry out. in some cases, may provide feedback for understanding. can be combined with any robust optimisation method. as long as the model compound is valid, no need for reoptimisation for each sample. it works!

37 Anne Guido Sylvian. Nicolas M. Nicolas G. Lyndon Christophe Bénédicte Gwendal Luminita Bénédicte Nicolas G. Gaël Lyndon Sabine Anne ECOLE NORMALE SUPERIEURE DE LYON IUF