Topics in SSNMR and Conformational Dynamics of Biopolymers: Consequences of Intermediate Exchange

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1 Topics in SSNMR and Conformational Dynamics of Biopolymers: Consequences of Intermediate Exchange A McDermott US Canada Winter School in Biomolecular SSNMR Stowe, VT January

2 Effects on NMR Spectra: Local, Angular Dynamics on Timescales from sec to ps Krushelnitsky & Reichert Prog NMR /27/10 2 DHFR Compression: Wright et al, Chem Rev 2006

3 Relaxation Due to Rapid Angular Fluctuations H = H 0 + H 1 σ t = i[ H,σ ] ( A = e ih o t ) ( Ae ih o t ) σ t = i H 1,σ [ ] 1/27/10 3

4 Generating Second Order Expressions σ t = i H 1,σ [ ] t σ ( t) = σ ( 0) i H 1 t ' [ ( ),σ ( t ' )] dt ' σ t = t H ( t 1 ), H ( 1 t ' ), σ t ' 0 0 [ [ ( ( ) σ )]] eq dt ' 1/27/10 4

5 Separation of Spatial and Spin Variables σ t = t H t 1( ), H 1 ( t ' ), σ t ' 0 [ [ ( ( ) σ )]] eq H 1 ( t) = V α F α t α ( ) dt ' 1/27/10 5

6 Decomposition of The Local Perturbation: Cartesian Rep of CSA H CSA H 1 = γi LAB σ ( t) LAB ( t) = V α F α t α B o LAB ( ) = γ I LAB R t ( ) σ PAS R 1 ( t) B o LAB I LAB H 1 CSA ( LAB LAB I x I ) y = ˆ ˆ ˆ I z LAB ( t) = γb 0 σ zz ( ) ˆ ( ) ˆ B o LAB 0 = 0 B 0 ( ) ˆ [ LAB t I z +σ LAB xz t I x +σ LAB yx t I ] y δ σ 11 σ 12 σ 13 2 (1+η) 0 0 σ σ 21 σ 22 σ 23 = σ SYM +σ ASYM δ σ PAS,SYM = 0 2 (1 η) 0 xx 0 0 = 0 σ yy 0 σ 1/27/10 31 σ 32 σ δ 0 0 σ zz 6 σ =

7 Fluctuations in the Local Field Driven by Local or Global Reorientation H 1 ( t) = V α F α t α ( ) [ H 0,V ] α = ω α V α I LAB ( LAB LAB I x I ) y = ˆ ˆ ˆ I z LAB 1/27/10 7

8 Separation of Spatial and Spin Variables σ t = t H t 1( ), H 1 ( t ' ), σ t ' 0 [ [ ( ( ) σ )]] eq H 1 ( t) = V α F α t α ( ) dt ' J αβ * ( ω) = F α ( t)f β 0 t ' ( ) ' ( iω ( t t ) ) e d t t ' ( ) ' t A rot ( t) = e i ( ω α ω β ) t J αβ ω β αβ + [[ ]V β ] t rot [ ] rot, eq ( ){ + A,V α ( ) [ A,V α ]V } β 1/27/10 8

9 A Typical Spectral Density Function F( t)f * ( t ' ) = F( 0)F * ( 0)e t / τ c J( ω) = F( 0)F * ( 0) τ c 1+ ω 2 2 τ c τc = 10ns J(ω) in ns τc = 1ns ω in GHz 1/27/10 9

10 Isotropic Expressions for CSA and Dipolar Mechanisms R 1 CSA 3 = ω 0 2 δ 2 J( ω 0 ) R 1 S, Dip R 2 CSA = γ Cγ H 3 2r IS = 1 3 ω 6 0 δ ( 4J( 0)+ 3J( ω 0 )) I S S I S ( J( ω 0 ω 0 ) + 3J( ω 0 ) + 6J( ω 0 + ω 0 )) R 2 S, Dip = γ Cγ H 3 2r IS 2 2J(0)+ 1 2 J ω 0 I S ( ω 0 ) J ω S 0 I I S ( ) + 3J( ω 0 ) + 3J( ω 0 + ω 0 ) 1/27/10 10

11 1/26/10 11

12 Protein Applications SSNMR Cole & Torchia Chemical Physics 1991 SNase ( see e.g. Reif et al) Different Sites differ in the extent of motion Although these data were not used to determine a unique motional model, the qualitative picture agrees in some measure with other experimental and computational indications of flexibility 1/26/10 12

13 References Redfield Adv. In Magn. Reson 1965 vol 1 p1 Goldman JMR Torchia and Szabo JMR Vold and Hoatson JMR (EXPRESS) 1/26/10 13

14 Combined Effects of Precession, Chemical Exchange, and Relaxation M 1 t M 2 = iω + k + R 1 2 k M 1 k iω + k + R t 2 M 2 m (t) = Ue Dt U 1 m (0) 1/26/10 14

15 Chemical Exchange Intermediate Exchange Timescale C 1 k21 k 12 C2 [ K eq = C ] 2 [ C ] = k 12 1 k 21 1/26/10 15

16 Effect of Precession and Relaxation on Magnetization M 1 t M 2 = iω R M 1 0 iω t 2 R 2 M 2 M i M x i + im y i M 1 t = ( iω 1 + R 2 )M 1 M i (t) = M i (0)e ( iω i +R 2 )t M(t) = i M i (t) Matrix is diagonal, two separable, solvable equations. 1/25/10 A: Def 16

17 Effects of Exchange on Chemical Concentrations C 1 k21 k 12 C 2 C 1 t = k 12 C 1 + k 21 C 2 C 2 t = +k 12 C 1 k 21 C 2 C 1 t C 2 = k 12 k 21 C 1 k t 12 k 21 C 2 1/25/10 17

18 Effects of Chemical Exchange on Magnetization M 1 t M 2 = k 12 k 21 M 1 k t 12 k 21 M 2 1/26/10 18

19 Solving for Concentration over Time by Matrix Methods c t = K c c = C 1 ;K = k 12 k 21 ; C 2 k 12 k 21 c (t) = e Kt c (0) D = U 1 KU K = UDU 1 e Kt = Ue Dt U 1 c (t) = Ue Dt U 1 c (0) 1/25/10 19

20 Ku i = λ i u i ; u i = u i1 D = λ λ 2 U = K = a c u 11 u 12 u 21 u 22 b d K λi = a λ c λ = a + d 2 ± u i2 b d λ = a λ ( a + d) 2 4( ad bc) 2 Eigenvalue Approach to Diagonalization ( )( d λ) bc = 0 1/25/10 20 A: Meaning of λ

21 Let s Look at Three Cases λ = a+ d 2 ± a+ d X = a c K' = a c K = a c ( ) 2 4( ad bc) 2 b = 0 1 d 1 0 b d = k k k k b = k k d k 12 k 21 1/25/10 21

22 Try to Solve a Meaningless Example λ = a + d ± 2 X = a b = 0 1 c d 1 0 λ = 0 ± ( a + d) 2 4( ad bc) ( 0) 2 4( 0 +1) 2 2 = 1 1/25/10 22

23 Solve the Eigenvalue Problem for Symmetric Two Site Exchange ( a + d) 2 4( ad bc) λ = a + d ± 2 2 K'= a b c d = k k k k λ = 2k 2 ± 2k D = 2k ( ) 2 4 k 2 k 2 2 ( ) = k ± k 1/25/10 23

24 Solve for the Transformation Matrices U and U -1 Ku i = λ i u i k k u k k 1 = 2ku 1 ku 11 ku 12 = 2ku 11 ; ku 11 + ku 12 = 2ku 12 ;u 11 = u 12 k k k u k 2 = 0k u 2 ku 21 ku 22 = 0ku 21 ; ku 21 + ku 22 = 0ku 22 ;u 21 = u 22 u U = 11 u 21 = 1 1 u 12 u U = a c b ;U 1 = 1 d U d c b ;U 1 = a /25/10 24

25 Homework: Confirm that this expression yields the inverse generally U = a b c d U 1 = 1 d b U c a Homework : Confirm in this specific case that this is the correct transformation matrix K = k k k k U = Homework: Write equations for the concentration over time and check their validity 1/25/10 25

26 Arbitrary Two-Site Exchange Can Also be Solved ( a + d) 2 4( ad bc) λ = a + d ± 2 2 K = a b = k 12 k 21 c d k 12 k 21 λ = k k λ = k ± k k k 12 + k 21 ± ( k 12 k 21 ) 2 4 k 12 k 21 k 12 k 21 2 ( ) See Bernasconi 1976 Relaxation Kinetics Academic Press 1/25/10 26

27 Note that K is not symmetric; is it always diagonalizable for any dimension? Connected with microscopic reversibility, a similarity transformation (U) symmetrizes K (to form K ), assuring us that it is always diagonalizable. Exercise: Show that if U is defined as below, it renders the matrix of rate constants symmetric. A real symmetric matrix is always diagonalizable. C n k nm = C m k mn U ij = P i δ ij K ' ij = K ij K ji = P j P i K ij 1/25/10 27

28 Combined Effects of Precession, Chemical Exchange, and Relaxation M 1 t M 2 = iω + k + R 1 2 k M 1 k iω + k + R t 2 M 2 m (t) = Ue Dt U 1 m (0) 1/25/10 28

29 K + R + il = iω + R + k 2 k λ = ( R 2 + k) ± k 2 Ω 2 U = a c b ;U 1 ij = d a = i(ω + Ω 2 k 2 ) 1 ad bc k iω + R 2 + k d c b a b = i(ω Ω 2 k 2 ) c = d = k ( )( d b) M(t) = M 1 (t) + M 2 (t) = a + c ad bc ( )( a c) e λ1t + b + d ad bc e λ 2t 1/25/10 29 A: Meaning of Eigenval, Im vre

30 Amplitudes in More Detail: Slow Exchange: k<<ω ( )( d b) M(t) = M 1 (t) + M 2 (t) = a + c ad bc λ 1,2 = ( R 2 + k) ± k 2 Ω 2 R 2 + k ± iω a = i(ω + Ω 2 k 2 ) 2iΩ b = i(ω Ω 2 k 2 ) 0 c = d = k ( )( a c) e λ 1 t + b + d ad bc e λ 2 t M(t) = M 1 (t) + M 2 (t) = e λ 1 t + e λ 2 t = e iωt e (R 2 +k )t + e iωt e (R 2 +k)t S(ω) = k + R 2 ( R 2 + k) 2 + Ω ω ( ) + k + R 2 2 ( R 2 + k) 2 + ( Ω + ω) 2 1/25/10 30 A: Meaning of Ampl Re vs Im

31 Amplitudes in More Detail: Fast Limit: k>>ω ( )( d b) M(t) = M 1 (t) + M 2 (t) = a + c ad bc λ 1,2 = ( R 2 + k) ± k 2 = ( R 2 + 2k),R 2 a = i(ω + Ω 2 k 2 ) k ( )( a c) e λ1t + b + d ad bc e λ 2t b = i(ω c = d = k Ω 2 k 2 ) k M(t) = M 1 (t) + M 2 (t) 0e λ 1 t + 2e λ 2 t 2e R 2 t S(ω) 2R 2 R ( ω) 2 1/25/10 31

32 S fast (ω) 2R 2 2 R 2 + ( ω) 2 S slow (ω) = k + R 2 R 2 + k + ( ) 2 + ( Ω ω) 2 k + R 2 R 2 + k ( ) 2 + ( Ω+ω) 2 1/26/10 32

33 Differential Equation Describing Precession, Chemical Exchange, and Relaxation k 12 C 1 C2 k 21 K eq = C 2 [ ] [ ] = k 12 C 1 k 21 M 1 t M 2 = iω k R k 21 M 1 k t 12 iω 2 k 21 R 2 M 2 S fast (ω) 2R 2 R ( ω) 2 S slow (ω) = k + R 2 ( R 2 + k) 2 + Ω ω ( ) + k + R 2 2 ( R 2 + k) 2 + ( Ω + ω) 2

34 Triosephosphate Isomerase: Lid Opening ~ Turnover 1/25/10 Sharon Rozovsky et al, JMB

35 Slow Exchange Limit Contributions to Linewidth K + R + il = iω + R 2 + k k λ = ( R 2 + k) ± k 2 Ω 2 k << Ω λ R 2 + k ± iω k iω + R 2 + k 1/25/10 35

36 Fast Limit Contributions to Linewidth K + R + il = iω + R + k 2 k λ = ( R 2 + k) ± k 2 Ω 2 k >> Ω λ ( R 2 + 2k), R 2 ( ) x <<1; 1+ x 1+ x k iω + R 2 + k λ ( R 2 + 2k), ( R 2 + Ω2 ) 2k 1/25/10 36

37 Exchange Broadening Interference with Isotropic Chemical Shift Evolution Slow Limit scales with k or 1/τ c Fast Limit scales with B 2 /k or B 2 τ c Maximal effect (coalescence, critical point) when k~δω iso /sqrt(2) to k~δω iso The ability to probe this broadening allows one to study rare events (poorly populated species) 1/26/10 37

38 Determining the Rate Constant Sometimes Rex can be extracted from the Spectrum. More practically Rex can be obtained the decay of echo refocused magnetization. Either a pi train, or a spin lock refocuses the chemical shift, and provides experimentally controlled variables, such as a field strength, frequency. 1/26/10 38

39 Chemical Exchange is Apparent in Echo Detected NMR Spectra

40 Determining the Rate Constant Suppose a correlation time of 1 ms. Compare an echo with a 10 ms delay, to one with a 100us delay done 100 times. Not the same! The rare chemical event causes dephasing only for the 100us element it occurs in, but does not accumulate over the whole 10 ms period. 1/26/10 40

41 Fast Limit Contributions to Linewidth During Pulsing: Two Site T1ρ, see Woessner 1961 J Chem Phys; fast limit 1 R ex = p 1 p 2 ΔΩ 2 /k 1+ ω e 2 k 2 CPMG: Allerhand and Gutowsky 1965 J Chem Phys; fast limit approx R ex = τ ex τ CPMG ξ sinh 1 sinh τ CPMGξ τ 2 ex ξ = k 2 4 p 1 p 2 ( ΔΩ) 2 1/26/10 See Palmer and Massi Chem Rev 2006 for discussion of other kinetic schemes 41

42 Determining the Rate Constant Note, generally no unique solution to obtain Δω p 1 p 2 and k ex =k f +k r from R ex See for example Loria s tutorial from ENC 09 for solution NMR discussion R 2 is predicted from known fast motions, or measured in a control experiment, and R ex is the extra contribution R 1ρ = R 1 cos 2 θ + R 2 sin 2 θ + R ex sin 2 θ R ex : Chemical Exchange via Isotropic Shift θ : Effective Field Angle R 1, R 2 : in the absence of exchange 1/26/10 42

43 Important Alternatives to N-site Hopping Models Smoluchowski equation for continuous diffusion in a potential, U. Potentially more direct connection to molecular interpretation. 1D example below includes diffusion and convection Finite difference numerical approaches used See work of Robert Vold: applications to SSNMR ( ) = D 2 R φ φ kt U' φ ( ) φ + 1 kt U'' ( φ ) 1/25/10 43

44 Distributed Dynamics" P(t)dt = 1 2kΓ(1+1/β) exp( t k β )dt Kaplan and Garroway showed (JMR 1982) that homogeneous and inhomogeneous distributions or rates can be distinguished based on NMR lineshapes. 1/25/10 44

45 1/25/10 45

46 Intermediate Exchange Lineshapes H H P( t) dt = 1 k dt 1/25/10 46

47 DNA Dynamics at a Methylation Site [5 -(dgatagcgctatc)-3 ] DNA dodecamer Dynamics Apparent in SSNMR Lineshape Not Seen in Solution NMR Relaxation Miller, Drobny, Varani et al JACS /25/10 47

48 Chemical Exchange During MAS ( ) = ω iso + δ C 1 cos( γ + ω r t) + C 2 cos( 2γ + 2ω r t) + S 1 sin( γ + ω r t) + S 2 cos( 2γ + 2ω r t) ω t [ ] C 1 = 2 sin( 2β) 1+ η cos( 2α ) 2 3 C 2 = 1 2 sin2 β η ( 3 1+ cos2 β)cos 2α S 1 = η 2 3 sin β ( )sin 2α ( ) S 2 = η 3 cos ( β )sin( 2α) ( ) δ η is the tensor width : σ zz - σ iso is the asymmetry : (σ xx - σ yy ) / δ R Z (α)r Y (β)r Z (γ) transforms to the Rotor Fixed Frame at t=0 from the Laboratory Fixed Frame 1/25/10 48

49 Tensor Orientation Change : Static See Solum Zilm Michl Grant /25/10 49

50 Tensor Orientation Change : Slow MAS See 1/25/10 Frydman et al 1990 and Schmidt et al

51 MAS Lineshapes: Sensitivity to Near Intermediate Exchange Conditions 1/25/10 51 Kristensen et al JMR

52 Exchange Linewidths f(ω r ) Figure 2 A Schmidt and S Vega J Chem Phys 1987 Linewidths for a system Undergoing 2 site flips: Floquet Treatment vs. Perturbation Approx. R ex δ η2 3 τ c 1+ 4ω r 2 τ c 2 ( ) + 2τ ( c 1+ ω 2 2 r τ ) c Suwelack Rothwell and Waugh 80 1/25/10 52

53 Analytic Prediction of Spectra Shin et al CPL 1998 M (t) = M (0)exp(iωt + Kt) Tr M (t) = M (0)exp(i ω(t' )dt'+kt) 0 1/26/10 53

54 ω t Fast MAS vs. Liquid ( ) = ω iso + δ C 1 cos( γ + ω r t) + C 2 cos( 2γ + 2ω r t) + S 1 sin( γ + ω r t) + S 2 cos( 2γ + 2ω r t) ω r >> δ ( ) ω iso ω t [ ] Simulated using SPINEVOLUTION (Veshtort and Griffin) (Meier and Earl, 1985 JACS; Lyerla Yannoni Fyfe 1982 Acc Chem Res 1/25/10 54

55 R ex Linewith Contributions for Interference with Chemical Shift TENSOR Refocusing Slow Limit scales with k or 1/τ c Fast Limit scales with 1/k or τ c Amplitude : crystal selection and δ/ ω r Maximal effect when k~ω r Effect gone when ω r > δ (or if k is moved out of kinetic regime by heating, cooling, drying etc.) ω r is a powerful variable! 1/26/10 55

56 MAS: R 1ρ Dispersion Curve DEC 0ms 20 Intensity intensity ms magnetization khz Freq. (khz) time (msec) Time (ms)

57 Spectral Density is Centered on Conditions involving the Rotor Lines Random Fluctuation Model - Approximate Initial Relaxation due to CSA during a Spinlock and MAS for Fast Limit Farès, Qian, and Davis, J Chem Phys 2005 Experimental Analysis Involving Dipolar Mechanisms: Methyl Rotors: Akasaka Ganapathy MacDowell Naito J Chem Phys 1983 Experimental DMS: Krushelnitsky et al SSNMR 2002

58 A Caution Regarding Interpretability Vanderhart and Garroway J Chem Phys /26/10 58

59 R 1ρ and Chemical Exchange DEC R1rho, s -1 k= 1, 1000, 30,000, >10^10 s Figure 2. (A) Simulations of DMS R 1ρ relaxation, showing the dependence of R 1ρ on the timescale of dynamics: k ex = 1 Hz (green), 1 khz (blue), 30 khz (red), inginitely fast (yellow). (B) Expansion of intermediate Spinlock exchange simulations Field, in the region khz of khz applied spin- lock power. (inset) Structure and motional mode of DMS.

60 k ex (T, ω o, ω 1, ω r ) Global Analysis R 2 (T, ω o, ω 1, ω r ) I(t, T, ω o, ω 1, ω r ) Prior 2H Analysis: Brown Vold & Hoatson 96 This Analysis (C. Quinn & McDermott 09) T 2 ~10-60 s -1

61 More Generally, This Approach Also Applies to the Density Matrix, With Couplings, etc. m t = il + R + K ( ) ρ t = ( il + R + K) ρ m L = [ H,ρ] ( il(t )+R+K) ρ (t 0 + t) = e ρ (t 0 ) See discussion by Saalwachter & Fischbach JMR /25/10 61

62 Interference With Decoupling; Amine Rotor R ex 4ω 2 d I( I +1) 15 τ c 1+ ω τ c R ex 4ω 2 d I( I +1) τ c 15 R ex 4ω 2 d I( I +1) 15 1 ω 1 2 τ c Rothwell and Waugh 1981 Long et al JACS /26/10 62

63 R ex Linewith Contributions for Interference with Proton Decoupling Slow Limit scales with k or 1/τ c Fast Limit scales with 1/k or τ c Amplitude related to ω D Maximal effect when k~ω 1 Effect gone when ω 1 >>> ω D HH (or if k is moved out of kinetic regime by heating, cooling, drying etc.) 1/25/10 63

64 R ex -like Interference with Coherent Transfer Assume Coherent Transfer of Longitudinal Magnetization (eg Iz-Sz) involves Intermediary Coherence (eg ZQC) Conformational Exchange Causes an exchange broadening on ZQC: ZQ Coherence experiences fluctuations in dipolar couplings and chemical shift fluctuations due to changes in orientation. Dephasing amplitude related to δ and maximal effect when k~ω r 1/26/10 64

65 R ex Interference with RFDR Buildup Specific Example in SPINEVOLUTION: Two Carbons during RFDR Magnetization Exchange Period Begin with magnetization on one spin only and follow Iz and Sz over time. For some simulations, one of them is also in Chemical Exchange with both CSA (δ= 10 khz) and couplings to attached protons changing. ω r moderate (16 khz). 1/26/10 65

66 Exchange Interference in MAS NMR: Dynamics in MAS NMR modulates large anisotropic interactions (1) CSA refocusing under MAS is defeated by large angle jumps if k~ω r and δ~ω r, (2) Decoupling by proton irradiation is defeated by moderate or large angle jumps when k~ω 1 and ω D ~ω 1 (3) The performance of coherent transfer sequences is degraded by large angle jumps over a similar range of conditions (provided transverse coherences are important in the spin evolution). I emphasized useful analogies between these problems and the wellunderstood effect of exchange induced interference with the evolution of the isotropic shift, and the resultant signature broadening effects (which of course also occurs for MAS NMR). 1/25/10 66

67 1/25/10 67