NMR Theory and Techniques for Studying Molecular Dynamics

Transcription

1 NMR Theory and Techniques for Studying Molecular Dynamics Mei Hong, Department of Chemistry, Iowa State University Motivations: Molecular dynamics cause structural changes and heterogeneity. Molecular motion can average spectral lineshapes, reduce intensities, and affect NMR relaxation properties. Molecular motions are abundant in proteins and integral to their function. 1. Timescale and amplitude of motion from NMR 2. Fast motion: average tensor & symmetry 3. Experiments for measuring amplitudes of fast motion 4. Order tensor and order parameter 5. Experiments for measuring slow motions 6. 2 H quadrupolar NMR 7. Determining motional rates from relaxation NMR 8. Practical aspects of protein dynamics study by SSNMR 1

2 Timescales and Amplitudes of Motion from NMR Timescale: rate k, correlation time τ c ~ 1/k, unit s -1 reflects the stochastic nature of motion. Rate frequency. Correlation function C(t): provides a smooth curve for the random motion. C t ( ) ~ f ( 0 ) f( t) KSR & Spiess. Amplitude: motional geometry, number of sites, and their relative orientation. Rotation vs. translation: here we only discuss rotation, or reorientation. Diffusive motion: infinite number of sites, infinitesimal step size. e.g. isotropic diffusion, on a cone and in a cone. Discrete motion: e.g. methyl 3-site jump, phenylene ring 2-site jump. 2

3 Motional Regimes in NMR Fast motion: k >> δ, which is the NMR interaction strength. Amplitude and geometry information obtainable from spectral line narrowing (e.g WISE, DIPSHIFT, LG-CP, CSA recoupling, 2 H quad echo); Timescale from relaxation times (T 1, T 1ρ ); More geometry information from spectrally resolved T 1 and T 2 in 2 H spectra. Slow motion: k << δ. Exchange NMR: 2D exchange, stimulated echo, CODEX Amplitude from the distance from diagonal in the 2D spectra, or Nt r dependence in CODEX. Timescale from the t m dependence of the exchange intensity Geometry from the final value in CODEX or the off-diagonal pattern. Slow to intermediate motion: k < δ. Echo experiments (Hahn echo or quadrupolar echo) T 2 minimum, T 1ρ minimum. Intermediate motion: k ~ δ. Amplitude and timescale from 2 H lineshape and T 1ρ relaxation times. Interference with 1 H decoupling. 3

4 Motional Regimes in NMR Example: equal-population 2-site exchange: τ c >> 1 δ A δ B slow τ c 1 δ A δ B intermediate τ c << 1 fast δ A δ B Harris. Fast motion: frequency view, order tensor S and order parameter. All other regimes: time domain view essential. Slow motion: occurs during the mixing time, is directly monitored. Intermediate motion: explicit time-domain calculation. 4

5 (Fast) Motional Averaging of NMR Frequencies ω( θ, φ) = δ 1 2 3cos2 θ 1 ηsin 2 θ cos2φ NMR frequency is orientation-dependent: where (φ, θ) are the powder angles of B 0 in the tensor PAS. e.g: uniaxial interaction (η=0): ( ) Assuming fast motion among N sites with occupation probability p j, then ω = N p j ω j j=1 The average of a second rank tensor is still a second-rank tensor. So the average tensor (Σ 1, Σ 2, Σ 3 ) has δ, η and powder angles (θ a, φ a ) w.r.t. B 0. The averaged NMR frequency is: ω ( θ a,φ a ) = δ 1 ( 2 3cos2 θ a 1 η sin 2 θ a cos 2φ a ) δ, η depend on motional geometry and symmetry. δ, η of the averaged tensor usually differ from the δ, η of the original PAS. δ of an averaged dipolar coupling tensor can contain sign information. η of an averaged dipolar coupling tensor is generally not uniaxial. 5

6 Motional Geometry: Symmetry Considerations δ, η of the averaged tensor may be obtained from the symmetry of the motion: Tetrahedral jumps: δ = 0 Isotropic diffusion: δ = 0 Uniaxial rotation: η = 0 Discrete jumps over N 3 sites with C N symmetry: η = 0 For uniaxial rotation or jumps with C N symmetry (N 3), the unique axis of the average tensor is the symmetry axis: N σ PAS σ D = R 1 α P,β P, j 360 σ PAS R α P,β P, j 360 j=1 N N average σ D D: director How to calculate δ? - δ is the frequency obtained when B 0 is along the Z D axis. At this orientation, the frequency can be calculated by the equation without motion : δ = 1 δ ( 2 3cos2 β P 1 ηsin 2 β P cos2α P ) KSR and Spiess. 6

7 Symmetry of the Average (Sum) Tensor Consider equal-occupancy 2-site jumps between two uniaxial tensors (e.g. 13 C- 1 H dipolar or 2 H quadrupolar couplings). Average (or sum) tensor: Σ = σ A + σ B - must be invariant under the rotation. ( ) 2 = ( σ B + σ A ) 2 For two uniaxial tensors, the three principal axes are: Normal of the AOB plane, Σ 3 Bisector of the angle AOB, Σ 1 Normal of the bisector, in the AOB plane, Σ 2 σ A σ B Σ 3 axis: 90, 90 Σ 1 axis: β/2, β/2 Σ 2 axis: 90 +β/2, 90 -β/2 1,2,3 convention : left to right, or large to small frequencies : ω 1 > ω 2 > ω 3 β < 90 and β > 90 have different axes labels. Once the three principal axes directions are fixed, the three principal values are: ω n = 1 δ ( 2 3cos2 Θ n 1), Θ n is the direction angle between z PAS and Σ 1, Σ 2, Σ 3 7

8 Phenylene Ring Flip: Motionally Averaged Lineshape Consider 2 H or C-H dipolar spectra (η=0 PAS): Reorientation angle β = 120 ω n = 1 δ ( 2 3cos2 Θ n 1) Θ 1 = 30 Θ 2 = 60 Θ 3 = 90 ω 1 = 5 8 δ ω 2 = 1 δ 8 ω 3 = 1 δ 2 δ = 5 8 δ η = 0.6 KSR & Spiess. 8

9 Lineshapes of Two Other Common Motions Methyl 3-site jumps: β = : δ = δ 3 Equal-population trans-gauche isomerization: β = slow intermediate fast CD 3 jump trans-gauche isomerization k 10 4 s s s -1 Palmer et al, JPC (1996), 100,

10 Orientation and Magnitude of the Difference Tensor ( ) Difference tensor: Δ = σ A σ B = σ B σ A - must have sign inversion under switch of the two individual tensors. - relevant in exchange experiments. For two uniaxial tensors, the three orthogonal axes of the difference tensor are: Normal of the AOB plane, Δ 2 In the AOB plane, 45 angles from the bisector, Δ 3 and Δ 1 σ A Δ 1 axis: 45 -β/2, σ B 45 +β/2 Δ 2 axis: 90, 90 Δ 3 axis: 45 +β/2, 45 -β/2 ω Δ n = 1 δ ( 2 3cos2 Θ A,n 1) 1 δ ( 2 3cos2 Θ B,n 1) 10

11 Original DIPSHIFT Fast Motion Experiments: 1.DIPSHIFT ω exp = δ XH k homo Doubled DIPSHIFT ω exp = 2 δ XH k homo Munowitz et al, JACS, 103, 2529 (1981). Hong et al, JMR, 129,85 (1997). a separated-local field (SLF) technique: X magnetization evolves under X-H dipolar coupling. constant time t 1 : sampling occurs within one rotor period. 1 H homonuclear decoupling: e.g. MREV-8 for ν r <5 khz, FSLG for ν r ~ 5-15 khz, DUBMO, etc. X isotropic shift in ω 2 gives site resolution. t Φ( t 1 ) = 1 t 0 dtω( t) = δ 1 0 dt[ C 1 cos( ω r t + γ) + C 2 cos( 2ω r t + 2γ) ], δ µ 0 4π γ H γ x h 2 r 3 HX,and C 1,C 2 are functions of powder angles α, β, γ 2x Φ CH t ( t1 ) = 1 τ dtω( t 0 ) r dtω( t t1 ) t 1 [ τ r t ( )] ( ) ( ) ( ) ( t 1 ) = dtω t dtω t dtω t t 1 = 2 dtω t 0 ( ) and the asymmetry parameter η 1x = 2Φ CH doubling the phase allows higher ν r to be used. 11

12 DIPSHIFT Time and Frequency Signals Original DIPSHIFT t 1 modulation expt. spectrum simulated spectrum FT time (ms) dipolar coupling (khz) dipolar coupling (khz) Doubled DIPSHIFT FT time (ms) dipolar coupling (khz) dipolar coupling (khz) Stick spectrum obtained by concatenating f(t 1,ω r ) over nτ r, followed by FT. Simulation can be in either frequency or time domain. Couplings can in principle be amplified by 2, 4 2n times by more π pulses. Hong et al, JMR, 129,85 (1997). 12

13 Manifestation of Motion from DIPSHIFT Data Cady et al, JACS, 129, 5719 (2007). 13

14 Fast Motion Experiments: 2.WISE e.g: hydration dynamics of elastin (VPGVG) n t 1 dimension: 1 H- 1 H dipolar coupling > 1 H-X dipolar coupling >> 1 H CSA. Suitable for ν r < 10 khz. Qualitative and quick assessment of mobility. In proteins, rigid-limit FWHM are khz for CH groups, khz for CH 2 groups. e.g: mobility gradient in poly(n-butyl methacrylate). Clauss et al, Macromolecules, 25, 5208 (1992). Yao et al, MRC, 42, 267 (2004). KSR and Mao, JACS, 124, (2002). 14

15 Fast Motion Experiments: 3.LG-CP simple to use: increment contact time to obtain t 1 dipolar modulation. 35 pulse prepares ρ 0 along the tilted 1 H effective field. 1 H homonuclear decoupling by magic-angle spin lock, removes H-H coupling, reveals X-H dipolar oscillation. suitable under fast MAS (ν r >~10 khz). symmetrized ω 1 and site resolution in ω 2. ω IS,LG = δ cosθ m = δ Van Rossum et al, JACS, 122, 3465 (2000). Hong et al, JPC, 106, 7355 (2002). Rigid limit CH: ~13 khz 15

16 - in the doubly rotating frame : H = ω 1I I x + ω 1S S x + Δω II z + ω IS ( t)i z S z rf part where ω IS t I S hetero. dipolar ( ) = 2δ C 1 cos( ω r t + γ) + C 2 cos( 2ω r t + 2γ) ( ) 3I z i I z j ( ) + ω II t I i I j I-I homo. dipolar γ I γ S h 2 - In the tilted frame defined by the pulses, R = e iθ mi y e iπ 2 S y, H T = ω eff,i I z + ω 1S S z + ω IS( t) ( sinθ m I x S x cosθ m I z S x ), rf part LG-CP Average Hamiltonian [ ], and δ = µ 0 - In the interaction frame defined by the rf pulses, H IS T = e ih rf t H IS e ih rf t under the n=±1 matching condition: ω eff,i ω 1S = ±ω r the time-independent average I-S dipolar coupling is: T H IS = 1 2 δsinθ m I 23 x where the 0 - quantum 2 - spin operators are : ( ) Ix ( 23) S x + I y S y, I y Iy ( 23) S x I x S y, I z 1 I x 23 ( ) C1 cosγ ( ) I y 23 ( ) C1 sinγ ( ) ( ) 2 I z S z 4π r IS 3 = 1 2 δsinθ m C 1 I 23 x ( ) 23 cosγ Iy ( ) sinγ 16 while H T II = 0.

17 LG-CP: Evolution of ρ Under the ZQ Hamiltonian The averaged I-S dipolar Hamiltonian can be written as a scalar product between the ZQ spin operator and an effective tilted LG field: T H IS = 1 2 δsinθ m C 1 I ( 23) 23 x, Iy ( ) 23, Iz ( ) cosγ sinγ = 1 2 δsinθ m C ( 1 I 23 ) BIS,LG In the tilted frame, ρ 0 I z = I z 14 ρ t ( ) I z 14 ( ) 23 + Iz ( ) ( ) 23 + Iz ( ) cos ωis,lg t where ω IS,LG = δ 2 sinθ m C 1 ( ) = I z ω IS,LG ( ) + S z 2 1+ cosω IS,LG t ( ) 2 1 cosω IS,LG t Powder average: C 1 = 3 4 sin2θ m sin2β ij, ω IS,LG β ij ( ) = δ 2 sinθ m 3 4 sin2θ m sin2β ij = 1 2 δcosθ m sin2β ij LG CP splitting= δcosθ m = δ 17

18 LG-CP for Studying Membrane Protein Motion e.g. colicin Ia channel domain, soluble > membrane-bound state Improvements of LG-CP #1: Constant-time version removes T 1ρ effect in ω 1. lower sensitivity. Hong et al, JPC, 106, 7355 (2002); Yao et al, MRC,42, 267 (2004). 18

19 Improvements of LG-CP #2: PILGRIM LG-CP Variants If at the end of the first CP period the S-spin has the same magnetization as the I-spin, then ( ) = I z S z = 2I z (23), ρ 0 compared to original LG - CP ρ 0 2 fold sensitivity enhancement ρ( t) = I z cosω IS,LG t S z cosω IS,LG t ( ) = I z (14) + I z (23), e.g. Gln: Improvements of LG-CP #3: 3D LG-CP Hong et al, JPC, 106, 7355 (2002). Lorieau and McDermott, JACS,128, (2006). 19

20 Motionally Averaged CSA Lineshape Main advantage over dipolar coupling: - Spectra give asymmetry of the motion: e.g. uniaxial motion > η = 0 - Can be recoupled by SUPER for isolated labels. f(t) ω( t) = C 1 cos ω r t + C 2 cos2ω r t + S 1 sinω r t + S 2 sin2ω r t, static lineshape : ω C 1 + C 2 MAS phase under π pulses : τ Φ = r / 2 ω( t)f( t)dt τ r / 2 choice of pulse positions : to make Φ C 1 + C 2 quasi static lineshape f( τ r 2 t) = f( τ r 2+ t), even function; τ r / 2 τ f( t)cosω r t dt = r / 2 f( t)cos2ω r t dt 0 0 Under 2π pulses: f(t) Tycko et al, JMR, 85, 265 (1989). Liu et al, JMR, 155, 15 (2002). τ r 0 ( ) SUPER dt f t = 1 2 τ r 0 ( ) CRAMA dt f t scaling factor χ is half that of the original expt. 20

21 SUPER Lineshape of a Membrane Peptide e.g. M2 peptide of influenza A virus, L40 Cα mobile immobile Cady et al, JACS, 129, 5719 (2007) 21

22 Motionally Averaged CSA: U- 13 C Labeled Proteins ω 1 =4.283 ω r χ CSA = χ CSA = 0.05 χ iso = τ r 0.499τ r 0.501τ r 0.967τ r Example: U- 13 C, 15 N-I3,N9-labeled penetratin in DMPC/DMPG, 303 K. Chan & Tycko, JCP, 118, 8378, (2003). Su et al, unpublished data. 22

23 NMR Theory and Techniques for Studying Molecular Dynamics 1. Timescale and Amplitude of Motion from NMR 2. Fast motion: average tensor and symmetry consideration 3. Experiments for measuring amplitudes of fast motion 4. Order tensor and order parameter 5. Experiments for measuring slow motions 6. 2 H quadrupolar NMR 7. Determining motional rates from relaxation NMR 8. Practical aspects of protein dynamics study by SSNMR 23

24 Derivation of Order Tensor - Uniaxial Motion Info from motionally averaged spectra usually expressed as order parameter S, which is the simplified version of order tensor S. Consider a molecule undergoing uniaxial rotation around a director Z D. The measured NMR frequency is the z-component of the interaction σ in the lab frame. It can be obtained from two coordinate transformations: σ M Θ ij uniaxial motion σ D σ D Φ ij σ L Since <σ D > is axially symmetric, the only relevant frequency in the σ M > σ D transformation is the frequency along Z D : M M M σ σ12 σ13 11 cos Θ 1z D σ zz M ZD σ M M ZD = M M M ( cosθ1z cosθ 2z cosθ 3z ) σ σ22 σ23 21 cosθ 2z M M M σ σ32 σ33 cosθ 3z 31 M averaging = cosθ iz σ ij cos Θjz, i,j D σ zz = cos M Θiz cosθ jz σ ij = cosθ iz cosθ jz 1 2 δ M ij σ ij + 1 i,j i,j δ ij σ M ij i,j δ σ D zz σiso = 2 3 S M σij M i,j i,j S ij M=σ PAS 24

25 Definition of Order Tensor The general Saupe matrix (order tensor) S is defined as: S ij 1 2 3cosΘ icosθ j δ ij For uniaxial motion, the diagonalized σ D is : σ D = D where σ // = σ zz, σ = 3 2 σ iso 1 2 σ // σ σ σ //, δ = σ D zz σiso = 2 3 S i,j σ ij i,j is true in any common frame of S and σ. Further transforming <σ D > to the lab frame gives the standard expression for the measured NMR frequency with orientation dependence: σ L zz σiso = cosφ iz σ D σiso cosφ ii iz = 1 i 2 δ cos2 Φ xz + cos 2 Φ yz + δ cos 2 Φ zz = 1 2 δ 1 cos2 Φ zz + δ cos 2 Φ zz = δ 1 2 3cos2 Φ zz 1 Φ zz : polar angle of the director in the lab frame. 25

26 Properties of the Order Tensor The order tensor is traceless: The order tensor is symmetric: S ii = 3 cos 2 Θ 2 i 3 = 0 2 i i S ij = S ji (cosθ i cosθ j = cosθ j cosθ i ) Thus, S tensor has 5 independent elements, and can always be diagonalized to give its own principal axis system. The order parameter along a bond is the projection of the order tensor onto that vector. δ = 2 σ,pas S 3 ii PAS σii = 2 i 3 δ S σ,pas zz δ 2 S σ,pas xx + σ,pas Syy = δ S σ,pas zz = δ 1 2 3cos2 θ PD 1, where θ PD = Θ 3Z σ,pas Thus, the order parameter along a tensor s principal axis, S zz, is directly measurable as the ratio δ δ. Since ω 0 oriented = 1, bond order parameters 2 δ 3cos2 θ PD 1 + ω iso contain the same orientation information that is sought after in alignedmembrane experiments. 26

27 Uniaxially Averaged CSA Lineshapes of β-sheet Peptides 13 CO 15 N Hong and Doherty, Chem. Phys. Lett, 432, 296 (2006). 27

28 Order Tensor and Order Parameter of Limiting Cases Determining a traceless and symmetric S tensor requires 5 independent NMR couplings; Symmetry considerations may reduce the number of unknowns. If the molecule is rigid (i.e. no segmental motion), then there is a single S tensor for the whole molecule (all segments have the same order). If the rigid molecule rotates about a single director axis, then the S tensor is uniaxial, i.e. η S = 0, with the unique axis along the director, and PAS S zz = 1 (complete order). If the rigid molecule rotates about its own molecular axis and an external director axis, then the S tensor is uniaxial along the molecular axis, PAS S zz = Smol = 1 2 3cos2 θ MD 1 In this case, S along a bond, S bond is related to S mol by: S bond ( θ PD ) δ δ = 1 2 3cos2 θ PM 1 S mol S mol can be smaller than 1 due to tilt of the molecular axis from Z D or wobbling of the molecular axis. Example: - Cholesterol rings in the lipid membrane; - Lipid chains have additional internal segmental motions. 28

29 Slow Motion: 2D Exchange NMR 2D spectrum S(ω 1, ω 2 ) is a joint probability: intensity distribution: geometry of motion geometry encoded in a single angle β for a uniaxial interaction t m dependence > correlation time. S (ω 1,ω 2 ;β) for an η=0 tensor KSR and Spiess. 29

30 2D Exchange of Lipid Membranes e.g. lateral diffusion of lipids over the curved surface of liposomes. Tensor PAS: uniaxial due to lipid axial rotation; Reorientation-angle distribution R(β; t m ). ( ) S ω 1,ω 2 ; t m 90 = dβr( β; t m )S ω 1,ω 2 ;β 0 ( ) t m =0, no exchange. t m dependence gives info on the lateral diffusion τ c = r 2 /6D L, which reveals the vesicle size r. Marasinghe et al, JPC B, 109, (2005). 30

31 1D Stimulated Echo: Time-Domain Exchange 2D time signal: 1D analog of 2D exchange: t 1 = τ. Allows fast determination of τ c by varying t m, avoids multiple 2D. similar to 2D exchange, most often applied to static samples. ( ) = [ cosω( θ 1 )t 1 isinω( θ 1 )t 1 ] e iω θ 2 f t 1, t 2... denotes powder averaging. ( )t 2 = e iω ( θ 1)t 1 e iω θ 2 ( )t 2 1D time signal: t 2 =t 1 =t e. - Segments without frequency change: ω(θ 1 )=ω(θ 2 )=ω (diagonal), or t m =0 M SE t e - Segments with frequency change, M SE ( t e ) = e iω ( θ 1)t e e iω θ 2 ( ) = e iωt e e iωt e = 1 { 2 scans ( )t e = e iω [( θ 2) ω( θ 1 )]t e 0 due to destructive interference. 1D simulated echo intensity = 2D spectrum s diagonal intensity. 31

32 Stimulated Echo: An Improved Sequence Built-in z-filter eliminates T 1 effects. S 0 : first t z = 1 ms, then t m S: first t m, then t z = 1 ms Second and third 90 pulses phase-cycled together to create cos(ω 1 t)cos(ω 2 t) in one scan and sin(ω 1 t)sin(ω 2 t) in another. Every 2 scans: M SE (τ;t m ) = M 0 cos(ω 1 τ)cos(ω 2 τ) + sin(ω 1 τ)sin(ω 2 τ) = M 0 cos(ω 1 ω 2 )τ) = cos ((ω 1 ω 2 )τ) S( ω 1,ω 2 ;t m )dω 1 dω 2 Powder averaging: 2D spectrum S(ω 1,ω 2 ; t m ) is the probability of finding molecules with freq ω 1 before t m and freq ω 2 after t m. At t m =0, Rec: y y ω 1 = ω 2, cos( ω 1 ω 2 )τ = 1, M SE τ;0 ( ) = S ω 1,ω 2 ;t m ( )dω 1 dω 2 With increasing t m, ω 2 ω 1, echo intensity decreases. For symmetric n-site jumps, the stimulated echo intensity decays exponentially with a time constant τ c : M SE τ >> 1 δ ;t m = 1 n + (1 1 n ) M 0 e t m τ c Marasinghe et al, JPC B, 109, (2005). 32

33 Stimulated Echo Under MAS: CODEX Rec: y y 180 pulse train recouples X-spin CSA. 90 storage and read-out pulses are phase-cycled together; after the second recoupling period, the accumulated MAS phase for 2 scans is: where cosφ 1 cosφ 2 - sinφ 1 sinφ 2 = cos(φ 1 +Φ 2 ) = cos( Φ 2 - Φ 1 ) Φ 1 = N t r 2 t 0 ω 1 ( t)dt r ω 1 ( t )dt t = N r 2 2 t r 2 0 Φ 2 = N 2 t r 2 0 ω t 2( t)dt + r ω 2 ( t t )dt = N r 2 ω 1 ( t)dt t r 2 0 ω 2 ( t)dt No reorientation: ω 1 = ω 2, > Φ 1 +Φ 2 = 0, cos(φ 1 +Φ 2 )=1, a stimulated echo. With reorientation, ω 1 ω 2, > cos(φ 1 +Φ 2 ) < 1, echo decay. Same T 1 correction by t m /t z switch between S 0 and S. deazevedo et al, JCP, 112, 8988 (2000). 33

34 CODEX Sensitive to Small-Angle Reorientation The normalized CODEX signal is parameterized by the product of CSA and recoupling time, δnt r (analogous to REDOR). Normalized signal can be considered in terms of the difference phase: S( t m,δnt r ) S 0 ( t m,δnt r ) = cos Φ 2 Φ 1 ( ) = cos Φ Δ where Φ Δ = N 2 t r / 2 ( ω 2 ω 1 )dt = N ω Δ t 2 t r / 2 ( )dt ( ) The difference tensor of a uniaxial interaction (η=0) has the symmetry-determined orientation on the right > It can be shown that: ω Δ 22 = 0 ω Δ 33 = ω Δ 11 = 3 δ sinβ 2 i.e. η Δ = 1 ω Δ 33 ω Δ 11 = 3δ sinβ = ω 33 ω 11 2sinβ Thus, the CODEX signal scales with sinβ or β for small angles. Usual angle dependence is (3cos 2 β-1)2, which scales as β 2. 34

35 CODEX: Reorientation Angles and Number of Sites ε( δnt r ;β) Total signal is a weighted sum of β-dependent curves: E( t m,δnt r ) = 90 0 R ( β ) ε( δnt r ;β)dt dβ 3-site jump Jump motions: S ( t m >> τ c,δnt r >> 1) = 1 S 0 M Isotropic jump Isotropic diffusion Uniaxial rotation KSR et al, Encyclop NMR, 9, 633 (2002). 35

36 2 H NMR for Studying Protein Motion large interaction strength, khz, sensitive to motional geometry. anisotropic relaxation experiments probe ns - ms motions. requires site-specific labeling. no multiplex advantage. fast motion amplitude information replaced by C-H dipolar experiments under MAS, e.g. LG-CP. Quadrupolar echo Anisotropic T 2 Anisotropic T 1 : different relaxation rates across motionally averaged lineshape; large frequencies give faster relaxation. CD 3 rotation 2D exchange: ω 1 symmetrized version of the η=0 CSA pattern. Phe ring flip Palmer et al, JPC, 100, (1996). 36

37 NMR Theory and Techniques for Studying Molecular Dynamics 1. Timescale and Amplitude of Motion from NMR 2. Fast motion: average tensor and symmetry consideration 3. Experiments for measuring amplitudes of fast motion 4. Order tensor and order parameter 5. Experiments for measuring slow motions 6. 2 H quadrupolar NMR 7. Determining motional rates from relaxation NMR 8. Practical aspects of protein dynamics study by SSNMR 37

38 NMR Relaxation - A Primer Requirements for relaxation: a magnetic interaction. random motion at the appropriate frequency. Consider an isolated spin subject to an isotropic random field B L (t)=<b L >f(t). dρ( t) dt 0 ( ) ( ) C( τ) ( ) = e τ τ c, i.e. decays to 0, isotropic motion. The spatial part of H(t) is γb L f t so double commutation means f( t) f t τ Assume C τ = i [ h H ( t ),ρ( 0) ] h 2 In the rotating frame, H(t) is modulated by e -inω 0 t So Fourier transform C τ t [ H( t), [ H( t τ),ρ] ]dτ = 1 0 h 2 ( ) : C ( τ ) e iω 0 τ 0 dτ J ω 0 ( ) J( ω 0 ) = t [ H( t), [ H( t τ),ρ] ]dτ 0 τ c 1+ ω 0 2 τ c 2 J( ω 0 ) spectral density, power available from the fluctuations at frequency ω 0 Abragam C(τ) 38

39 Relaxation Rate, J(ω), & Correlation Time The above calculations lead to relaxation rates including terms like: T 1 1 γbl 2 τ c 1+ ω 0 2 τc 2 <γb L >: strength of the local field driving relaxation. For heteronuclear dipolar relaxation, γb L = ω IS. <γb L > 2 dependence: relaxation is second order. -1 ω 0 τ c << 1, T 1-1 τc ; ω 0 τ c ~ 1, T 1,min ω 2 ω 0 a Lorentzian function centered at ω 0 = 0 J(ω) log(t 1,T 1ρ ) T 1 Extreme narrowing limit T 1ρ log(ω) Logarithmic plot stretches small ω regime. log(τ c ) T 2 no T 2 minimum for undecoupled systems. 39 Harris.

40 T 1 and T 1ρ Relaxation Main local fields driving relaxation: dipolar couplings quadrupolar couplings CSA Inversion recovery For X-spin T 1 relaxation due to H-X dipolar coupling and X-spin CSA, R 1,X = ω 2 HX [ J( ω H ω x ) + 3J( ω x ) + 6J( ω H + ω x )] + c 2 J( ω x ), where c = Δσ ω x 3 4 T 1 relaxation is sensitive to motion near the Larmor frequencies, ~10-9 s -1 For 1 H T 1ρ relaxation under LG spin lock at ω e, driven by H-H and H-X dipolar couplings, R 1ρ,H = 1 10 δ 2 J(ω HH{ e ) + 2J(2ω e ) +6J(ω H ) + 6J( 2ω H )} δ 2 2J(ω XH e ) + 3J(ω x ) + J(ω x ω H ) + 3J(ω H ) { + 6J(ω x + ω H )} T 1ρ is sensitive to motion near ~10 5 s -1, which is common in membrane proteins. X-spin T 1ρ is usually not measured due to the need for 1 H decoupling, which causes undesirable reverse CP. Mehring. 40 Huster et al, Biochemistry, 40, 7662 (2001).

41 1 H Decoupled X-Spin T 2 For X-spin T 2 under a 1 H decoupling field of ω 1,H, R 2,X = γ 2 2 γx H h 2 5 r 6 J(ω 1,H ) = ω 1,H τ c << 1, R 2,X τ c ω τ 1,H c >> 1, R 2,X τ c ω 2 1,H γ H 2 γx 2 h 2 τ c 5 r 6 1+ ω 2 τc 2 1,H Solids X-spin T 2 has a minimum. Similar to 1 H T 1ρ, decoupled X-spin T 2 is sensitive to ~10 5 s -1 motions. Many membrane peptides and proteins exhibit R 2 -enhanced line broadening due to µs motion. R 2 ( s 1 ) ω 1,H =100 khz ω 1,H =50 khz Rothwell and Waugh, JCP, 74, 2721 (1981) 41

42 Definition : C( τ) = f( t) f( t τ) For example, f( t) = 1 2 3cos2 θ( t) 1 which leads to the 2 nd - order correlation function : C 2 τ More about Time Correlation Function ( ) 5 P 2 ( cosθ( t) ) P 2 ( cosθ( t τ) ) Higher order correlation functions also exist. In general, C L τ ( ) P L ( cosθ( t) ) P L ( cosθ( t τ) ) For discrete jumps, τ c (C 2 ) = τ c (C 4 ) = τ c (C 6 ) For diffusive motion, τ c from higher order C L functions differ from τ c (C 2 ). Determining τ c (C L ) for different order can distinguish jump motion and diffusive motion. 42

43 Determining τ c in Anisotropically Mobile Systems For activated processes, τ = τ 0 e E a RT 1 T 2 Varying temperature to map out τ 0 and E a. antimicrobial peptide TPF4 E a = 6.4 kcal/mol τ 0 =2 x s e.g. hexamethylbezene, 3 phases: I: CH 3 C 3 motion (< 116 K), II: C 6 + C 3 motions ( K) For anisotropic motion: C(τ) decays to a finite value, S 2 > 0. J(ω) depends on S: J( ω) = 1 S 2 τ c 1+ ω 2 τ 2 c Varying the field: ω 0 for T 1, ω 1 for T 1ρ. Relaxation time ratios at different fields depend only on τ c. Rothwell and Waugh, JCP, 74, 2721 (1981) Huster et al, Biochemistry, 40, 7662 (2001). Doherty et al, Biochemistry, Epub (2008) Colicin Ia channel domain 43

44 Practical Aspects of Protein Dynamics From SSNMR Types of Motions in Proteins Segmental methyl group rotation (CH 3 ) amine rotation (NH 3 ) phenylene ring flip (Phe, Tyr) no ring flip (Trp, His) torsional libration sidechains: mobility gradient backbone amides (NH-CO) trans-gauche isomerization large-amplitude diffusion of loops and termini Global whole-body uniaxial rotation (membrane peptides) near isotropic diffusion (e.g. elastin) 44

45 Diagnostic Methods to Identify Motion CP versus DP (direct polarization) CP: selects immobile species. DP: selects mobile species due to narrow lines, high intensity, and short 13 C T 1. heterogeneously mobile systems give different T 2 s under CP and DP. e.g. POPC/POPG membrane in the presence of an antimicrobial peptide Mani et al, Biochemistry, 43, (2004). 45

46 Diagnostic Methods to Identify Motion CP intensity loss: intermediate timescale motion interferes with: 1 H-X CP 1 H decoupling Intensity can be retrieved by varying T. 13 C CP Doherty et al, Biochemistry, 2007 (Epub ahead of print) Cady et al, JACS, 129, 5719 (2007). 46

47 Other Mobility-Tailored Techniques 3. Very large-amplitude motion: J-coupling based techniques can be used to selectively detect mobile segments. J-INEPT HCC Andronesi, JACS, 127, (2005). J-INEPT HNCACB 4. Rigid-body uniaxial rotation of membrane peptides around the bilayer normal allows orientation determination. 47