Topics in SSNMR and Dynamics of Proteins: Consequences of Intermediate Exchange

Transcription

1 Topics in SSNMR and Dynamics of Proteins: Consequences of Intermediate Exchange A McDermott, Columbia University Winter School in Biomolecular NMR, Stowe VT January

2 Effects on NMR Spectra: Local, Angular Dynamics on Timescales from sec to ps Krushelnitsky & Reichert Prog NMR /23/08 2 DHFR Compression: Wright et al, Chem Rev 2006

3 Notional Regimes for Motion Regime Definition Timescale Consequences Fast τ c << 1/Δω ps-ns Relaxation, reduced apparent tensor breadth Intermediate Exchange τ c 1/Δω τ c > T 2 (or ZQ T 1ρ T 2 etc.) µs Lineshape, s -1 short coherence times Slow T 1 >τ c >1/Δ ms-sec Population ω exchange Ultraslow τ c > T 1 min- days Time Dep Expts Rate and Amplitude controlled by Temp, Pressure and Hydration 1/23/08 3 A: Def

4 Chemical Exchange Intermediate Exchange Timescale C 1 k21 k 12 C2 [ K eq = C ] 2 [ C ] = k 12 1 k 21 1/23/08 4

5 Differential Equation Describing Precession, Chemical Exchange, and Relaxation M 1 t M 2 = iω k R k 21 M 1 k t 12 iω 2 k 21 R 2 M 2 1/23/08 5 A: Def next

6 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. 1/23/08 A: Def 6

7 Effects of Chemical Exchange on Magnetization M 1 t M 2 = k 12 k 21 M 1 k t 12 k 21 M 2 1/23/08 7

8 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 1/23/08 8

9 Solving for Concentration over Time by Matrix Methods v c t = K v c v c = C 1 ;K = k 12 k 21 ; C 2 k 12 k 21 v c (t) = e Kt v c (0) D = U 1 KU K = UDU 1 e Kt = Ue Dt U 1 v c (t) = Ue Dt U 1 v c (0) 1/23/08 9

10 Ku v v i = λ i u i ; u v 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 1/23/08 10 A: Meaning of λ

11 Let s Look at Three Cases λ = a + d ± 2 X = a b = 0 1 c d 1 0 K'= a c K = a c ( a + d) 2 4( ad bc) 2 b d = k k k k b = k 12 k 21 d k 12 k 21 1/23/08 11

12 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 1/23/08 12

13 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 1/23/08 13

14 Solve for the Transformation Matrices U and U -1 Ku v v i = λ i u i k k u v 1 = 2ku v 1 k k ku 11 ku 12 = 2ku 11 ; ku 11 + ku 12 = 2ku 12 ;u 11 = u 12 k k k u v k 2 = 0k v 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 /23/08 14

15 Homework: Confirm that this expression yields the inverse generally U = a b c d U 1 = 1 U d c b 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 1/23/08 15

16 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 12 k 21 2 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 1/23/08 16

17 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. HOMEWORK: 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 1/23/08 17

18 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 v m (t) = Ue Dt U 1 v m (0) 1/23/08 18

19 K + R + il = iω + R 2 + k 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 λ 1 t + b + d ad bc e λ 2 t 1/23/08 19 A: Meaning of Eigenval, Im vre

20 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 λ1t + b + d ad bc e λ 2t M(t) = M 1 (t) + M 2 (t) = e λ 1t + e λ 2t = 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 1/23/08 20 A: Meaning of Ampl Re vs Im

21 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 1t + 2e λ 2t 2e R 2t S(ω) 2R 2 R ( ω) 2 1/23/08 21

22 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 1/23/08 22

23 Triosephosphate Isomerase: Lid Opening~Turnover 1/23/08 23 Sharon Rozovsky et al, JMB 2001

24 Slow Exchange 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 + k ± iω k iω + R 2 + k 1/23/08 24

25 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 1/23/08 25

26 Exchange Broadening Interference with Isotropic Chemical Shift Evolution Slow Limit scales with k or 1/τ c Fast Limit scales with 1/k or 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) 1/23/08 26

27 Fast Limit Contributions to Linewidth T1ρ, Related to Woessner 1961 J Chem Phys; fast limit R ex = p 1 p 2 ΔΩ 2 /k 1 1+ ω e 2 k 2 CPMG: Allerhand and Gutowsky 1965 J Chem Phys; fast limit approx R ex = 1 2τ ex 1 1 τ CPMG ξ sinh 1 sinh τ CPMGξ τ 2 ex ξ = k 2 4 p 1 p 2 ( ΔΩ) 2 1/23/08 27

28 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 ) 1/23/08 28

29 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. 1/23/08 29

30 1/23/08 30

31 Intermediate Exchange Lineshapes H H P( t) dt = 1 k dt 1/23/08 31

32 Triosephosphate Isomerase Lid Motion Rate ~ Turnover

33 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 /23/08 33

34 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 1/23/08 34

35 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 1/23/08 35

36 Tensor Orientation Change : Slow MAS 1/23/08 36 See Frydman et al 1990 and Schmidt et al 1986

37 Tensor Orientation Change : Static See Solum Zilm Michl Grant /23/08 37

38 MAS Lineshapes: Sensitivity to Near Intermediate Exchange Conditions 1/23/08 38 Kristensen et al JMR

39 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 1/23/08 39

40 Exchange Linewidths f(ω r ) R ex δ 2 15 for three crystallites 1+ η2 3 τ c 1+ 4ω r 2 τ c 2 ( ) + 2τ ( c 1+ ω 2 2 r τ ) c R ex δ ( 1+ 3η) 4ω 2 r τ c R ex 3 15 δ 2 1+ η2 τ c 3 1/23/08 40

41 R ex Linewith Contributions for Interference with Chemical Shift Evolution Slow Limit scales with k or 1/τ c Fast Limit scales with 1/k or τ c Maximal effect (coalescence, critical point) when k~δω iso /sqrt(2) to k~δω iso 1/23/08 41

42 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 related to 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.) 1/23/08 42

43 More Generally, This Approach Also Applies to the Density Matrix, With Couplings, etc. v m t = il + R + K ( ) v v ρ = il + R + K t ( ) v ρ m L = [ H,ρ] v ( il(t )+R+K) v ρ (t 0 + t) = e ρ (t 0 ) 1/23/08 43

44 Interference With Decoupling; Amine Rotor R ex 4ω d 2 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 /23/08 44

45 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.) 1/23/08 45

46 R ex -like Interference with Coherent Transfer Assume Coherent Transfer of Longitudinal Magnetization (eg Iz-Sz) involves Intermediary Coherence (eg ZQC) Conformational Exchange Causes Interference with Exchange Mechanism, either because of Failed Decoupling or Fluctuations in Time Dependent Chemical Shift Could be thought of as an exchange broadening on ZQC 1/23/08 46

47 R ex -like Interference with Coherent Transfer (1) ZQ Coherence experiences fluctuations in dipolar couplings. Dephasing amplitude related to ω D, Maximal effect when k~ω 1 (2) ZQ Coherence experiences chemical shift fluctuations due to changes in orientation of CSA. Dephasing amplitude related to δ, and maximal effect when k~ω r 1/23/08 47

48 R ex Interference with Coherent Transfer Specific Example which MV set up in SPINEVOLUTION: Two Carbons during RFDR Magnetization Exchange Period Begin with magnetization on one spin only and follow Iz and Sz over time. 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). 1/23/08 48

49 Efficiency of Cross Peak Development in RFDR: Varying the Chemical Exchange Rates 1/23/08 49

50 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). There may be 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). 1/23/08 50

51 1/23/08 51

52 Simulate Tensor Orientation Change During MAS (DMS) Conditions as for Schmidt et al, : 108 degree hop, axial static tensor of width 4.6 khz; Simulation with SPINEVOLUTION. Very broad interference condition 1/23/08 52

53 Tensor Orientation Change : Slow MAS 1/23/08 53 See Frydman et al 1990 and Schmidt et al 1986