INADEQUATE Christopher Anand (with Alex Bain, Sean Watson, Anuroop Sharma)
Small Molecules O 6 O 5 4 O 1 2 3 O O O O 2 1 O 5 3 4 O O 6 biological products O possible medical applications want to know structure look at C-C bonds, double-quantum structure
Single Quantum Single Quantum = Bloch Equations Complex Form
Step 1 measure C spectrum x 10 6 14 12 10 8 6 4 2 0!4000!3000!2000!1000 0 1000 2000 3000 4000 Frequency (Hz)
Double Quantum two nuclei, arrange by coherence level
Measure 2Quantum create 2Q spin let it evolve (delay 1) put it back into 1Q measure (readout 2)
Isolated Signal Exercise in Modern NMR Pulse Sequence Design: INADEQUATE CR AXEL MEISSNER, OLE WINNECHE SØRENSEN
Isolated Signal readout = directly acquired Exercise in Modern NMR Pulse Sequence Design: INADEQUATE CR AXEL MEISSNER, OLE WINNECHE SØRENSEN
Isolated Signal readout = directly acquired delay = indirectly acquired Exercise in Modern NMR Pulse Sequence Design: INADEQUATE CR AXEL MEISSNER, OLE WINNECHE SØRENSEN
Problem: Noise 8000 6000 4000 2000 0 2000 4000 6000 8000 4000 3000 2000 1000 0 1000 2000 3000 4000
Problem: Noise 8000 6000 4000 2000 0 2000 4000 6000 8000 4000 3000 2000 1000 0 1000 2000 3000 4000
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 3 0 0 0 0 0 0 0 0-1 -2-3 -2-1 0 1 2 F 2 (khz) s 7 s 8 s 9 0 0 s 10
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,
Too Hard Non-quadratic + bi-quadratic terms takes to long to solve solve alternately for S and p (Gauss-Seidel)
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
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
Result 4 3 2 F 1 (khz) 1 0-1 -2-3 -2-1 0 1 2 F 2 (khz)
Without Regularization 4 G1 G5 F2 F5 F3 F4 G3 G2 G4 F6 F1 G6 3 2 F 1 (khz) 1 0-1 -2-3 -2-1 0 1 2 F (khz) 2
Results > 4X reduction in scan time compared to skilled interpretation
Problems Can t see past O, N, etc. Still takes too long
Solutions Can t see past O, N, etc. go Multi-Nuclear Still takes too long
Solutions Can t see past O, N, etc. go Multi-Nuclear Still takes too long Optimize delay times (k-space sampling)
Protein NMR Know DNA Sequences Defines Strings of Amino Acids Missing Info: Protein Structure only works if folded Protein Function interaction = wiggling 20 http://en.wikipedia.org/wiki/protien Anand - INADEQUATE - MITACS-Field s 2011
2-d NMR... R C H O C N H R C H O C N H R C H O C N H... pulse @ 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
2-d C-H 10 c) 20 30 40 50 60 70 80 MAGNETIC RESONANCE IN CHEMISTRY Magn. Reson. Chem. 2007; 45: 2 4 Published Kupčes, Freeman online in Wiley InterScience 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 22 Anand - INADEQUATE - MITACS-Field s 2011
Linear Forward Problem 23 Anand - INADEQUATE - MITACS-Field s 2011
Linear Forward Problem f(k i ) = m f(x j )e 1k i,x j, j=1 23 Anand - INADEQUATE - MITACS-Field s 2011
Linear Forward Problem f(k i ) = m j=1 f(x j )e 1k i,x j, f(k 1 )... = S f(x 1 )... + 1... f(k n ) f(x m ) n 23 Anand - INADEQUATE - MITACS-Field s 2011
Linear Forward Problem f(k i ) = m j=1 f(x j )e 1k i,x j, f(k 1 )... = S f(x 1 )... + 1... f(k n ) f(x m ) n S i,j = e 1k i,x j. 23 Anand - INADEQUATE - MITACS-Field s 2011
Linear Forward Problem f(k i ) = m j=1 f(x j )e 1k i,x j, f(k 1 )... = S f(x 1 )... + 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
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
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
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
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
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
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
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
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
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 1 7+2 26 0.99 2 38+2 119 0.72 greedy random < 80% efficient 2x more samples with greedy random 3 26+2 83 0.82 4 15+2 50 0.87 5 22+2 71 0.80 6 2+2 11 1.00 28 Figure 3. Efficiencies found by greedy random + trustanand method - INADEQUATE for all planes - MITACS-Field s clustered from 20113d
Better in 5D 6D Higher Dimensions 3D 4D 29 17 peaks with full frequency information Efficiency increases with dimension (34 samples) fewer samples required in higher dimensions Anand - INADEQUATE - MITACS-Field s 2011
To Do Celebrate 4X faster experiments Design mixed C-O-N experiments Make delay optimization numerically robust 30 Anand - INADEQUATE - MITACS-Field s 2011
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