INADEQUATE. Christopher Anand (with Alex Bain, Sean Watson, Anuroop Sharma)

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1 INADEQUATE Christopher Anand (with Alex Bain, Sean Watson, Anuroop Sharma)

2 Small Molecules O 6 O 5 4 O O O O O 2 1 O O O 6 biological products O possible medical applications want to know structure look at C-C bonds, double-quantum structure

3 Single Quantum Single Quantum = Bloch Equations Complex Form

4 Step 1 measure C spectrum x !4000!3000!2000! Frequency (Hz)

5 Double Quantum two nuclei, arrange by coherence level

6 Measure 2Quantum create 2Q spin let it evolve (delay 1) put it back into 1Q measure (readout 2)

7 Isolated Signal Exercise in Modern NMR Pulse Sequence Design: INADEQUATE CR AXEL MEISSNER, OLE WINNECHE SØRENSEN

8 Isolated Signal readout = directly acquired Exercise in Modern NMR Pulse Sequence Design: INADEQUATE CR AXEL MEISSNER, OLE WINNECHE SØRENSEN

9 Isolated Signal readout = directly acquired delay = indirectly acquired Exercise in Modern NMR Pulse Sequence Design: INADEQUATE CR AXEL MEISSNER, OLE WINNECHE SØRENSEN

10 Problem: Noise

11 Problem: Noise

12 this structure is shown in Eq. 2 1 F 1 (khz) 3 0 Box the Signal we use the notation S ij to refe centred 4 at the location (ω, ω i a bond of i to j. 0 s 1 s 2 s F 2 (khz) s 7 s 8 s s 10

13 min m S 2 (2) + λ 1 δ x S 2 (3) + λ 2 (1 p ij ) 2 S ij + S ji 2 (4) ij + λ 3 ( S ij 2 S ji 2 ) 2 (5) ij + µ 1 i + µ 2 ij 2 j=i p ij 4 (6) p ij (7) s.t. p ij 0 (8) p ij 1 (9) δ x S 2 = i s.t. s i and s i+1 are horizontally adjacent in the same box (s i s i+1 ) 2,

14 Too Hard Non-quadratic + bi-quadratic terms takes to long to solve solve alternately for S and p (Gauss-Seidel)

15 Solve for S min S m S 2 + λ 1 δ x S 2 + λ 2 (1 p ij ) 2 S ij + S ji 2 + λ 3 ( S ij 2 S ji 2 ) 2 ij ij

16 Solve for p min p ij (1 p ij ) 2 S ij + S ji 2 + µ 1 ij + µ 2 s.t. p ij 0 p ij 1 ij p ij i 2 j=i p ij 4

17 Result F 1 (khz) F 2 (khz)

18 Without Regularization 4 G1 G5 F2 F5 F3 F4 G3 G2 G4 F6 F1 G6 3 2 F 1 (khz) F (khz) 2

19 Results > 4X reduction in scan time compared to skilled interpretation

20 Problems Can t see past O, N, etc. Still takes too long

21 Solutions Can t see past O, N, etc. go Multi-Nuclear Still takes too long

22 Solutions Can t see past O, N, etc. go Multi-Nuclear Still takes too long Optimize delay times (k-space sampling)

23 Protein NMR Know DNA Sequences Defines Strings of Amino Acids Missing Info: Protein Structure only works if folded Protein Function interaction = wiggling 20 Anand - INADEQUATE - MITACS-Field s 2011

24 2-d NMR... R C H O C N H R C H O C N H R C H O C N H... 2 frequencies couple H spins and N spins transfer spin state H-N-H H N phase variation proportional to delay delay (indirect) data collection window 21 Anand - INADEQUATE - MITACS-Field s 2011

25 2-d C-H 10 c) MAGNETIC RESONANCE IN CHEMISTRY Magn. Reson. Chem. 2007; 45: 2 4 Published Kupčes, Freeman online in Wiley InterScience Anand - INADEQUATE - MITACS-Field s 2011

26 Linear Forward Problem 23 Anand - INADEQUATE - MITACS-Field s 2011

27 Linear Forward Problem f(k i ) = m f(x j )e 1k i,x j, j=1 23 Anand - INADEQUATE - MITACS-Field s 2011

28 Linear Forward Problem f(k i ) = m j=1 f(x j )e 1k i,x j, f(k 1 )... = S f(x 1 ) f(k n ) f(x m ) n 23 Anand - INADEQUATE - MITACS-Field s 2011

29 Linear Forward Problem f(k i ) = m j=1 f(x j )e 1k i,x j, f(k 1 )... = S f(x 1 ) f(k n ) f(x m ) n S i,j = e 1k i,x j. 23 Anand - INADEQUATE - MITACS-Field s 2011

30 Linear Forward Problem f(k i ) = m j=1 f(x j )e 1k i,x j, f(k 1 )... = S f(x 1 ) f(k n ) f(x m ) n S i,j = e 1k i,x j. is linear 23 Anand - INADEQUATE - MITACS-Field s 2011

31 Moore-Penrose Inverse f(x 1 )... = (S S) 1 S f(k 1 )... f(x m ) f(k n ) 24 Anand - INADEQUATE - MITACS-Field s 2011

32 Moore-Penrose Inverse f(x 1 )... = (S S) 1 S f(k 1 )... f(x m ) f(k n ) can invert with M-P pseudo-inverse 24 Anand - INADEQUATE - MITACS-Field s 2011

33 Moore-Penrose Inverse f(x 1 )... = (S S) 1 S f(k 1 )... f(x m ) f(k n ) can invert with M-P pseudo-inverse optimize {k} for M-P 24 Anand - INADEQUATE - MITACS-Field s 2011

34 Worst-Case Noise ~ Conditioning of n (S S) i,j = e 1k l,x j x i l=1 25 Anand - INADEQUATE - MITACS-Field s 2011

35 Worst-Case Noise ~ Conditioning of n (S S) i,j = e 1k l,x j x i l=1 expected maximum error ~ 1/ minimal eigenvalue 25 Anand - INADEQUATE - MITACS-Field s 2011

36 Worst-Case Noise ~ Conditioning of n (S S) i,j = e 1k l,x j x i l=1 expected maximum error ~ 1/ minimal eigenvalue constraining eigenvalues = Semi-Definite Programming (SDP) 25 Anand - INADEQUATE - MITACS-Field s 2011

37 Real SDP 26 min {k i } λ subject to A λi 0 satisfies (2) A 2i 1,2j 1 = n l=1 l=1 cosk l, x j x i n A 2i,2j = cosk l, x j x i n A 2i,2j 1 = sink l, x j x i l=1 n A 2i 1,2j = sink l, x j x i l=1 Anand - INADEQUATE - MITACS-Field s 2011

38 Trust Region + SDP Step ch min k λ subject to A k + α = 1...n β = 1...r (k α,β k α,β ) A k α,β λi 0, k A 2i 1,2j 1 k α,β = (sink α, x j x i )(x j,β x i,β ) A 2i,2j k α,β = (sink α, x j x i )(x j,β x i,β ) A 2i,2j 1 k α,β = (cosk α, x j x i )(x j,β x i,β ) A 2i 1,2j k α,β = (cosk α, x j x i )(x j,β x i,β ) k α,β k α,β π/2 max x j,β x i,β 27 Anand - INADEQUATE - MITACS-Field s 2011

39 Works well in 3D! Figure 2. SDP optimization = 100% efficient Trust region method does much better than random sampling, whether applied once in a final step, or after each additional random point. plane number of peaks number of samples efficiency greedy random < 80% efficient 2x more samples with greedy random Figure 3. Efficiencies found by greedy random + trustanand method - INADEQUATE for all planes - MITACS-Field s clustered from 20113d

40 Better in 5D 6D Higher Dimensions 3D 4D peaks with full frequency information Efficiency increases with dimension (34 samples) fewer samples required in higher dimensions Anand - INADEQUATE - MITACS-Field s 2011

41 To Do Celebrate 4X faster experiments Design mixed C-O-N experiments Make delay optimization numerically robust 30 Anand - INADEQUATE - MITACS-Field s 2011

42 Thanks to Adrian, Dhavide, Hongmei for the invitation MITACS and Field s the conference NSERC and IBM for funding my students for their hard work 31 Anand - INADEQUATE - MITACS-Field s 2011