NMR Relaxation and Molecular Dynamics

Ecole RMN Cargese Mars 2008 NMR Relaxation and Molecular Dynamics Martin Blackledge IBS Grenoble Carine van Heijenoort ICSN, CNRS Gif-sur-Yvette

Solution NMR Timescales for Biomolecular Motion ps ns µs ms s Spin relaxation Lineshape/exchange Relaxation dispersion Real-time NMR Dipolar, scalar couplings Librational motion Normal modes Rotational diffusion Enzyme catalysis Signal transduction Ligand binding Collective motions? Protein folding Hinge Bending

Solution NMR Timescales for Biomolecular Motion ps ns µs ms s Spin relaxation Provides site specific information about backbone and sidechain dynamics throughout the protein Established experimental and analytical procedures allow routine extraction of motional description, or parameterisation in terms of amplitudes (S 2 ) and characteristic timescales (τ i ) Information is limited to motions occurring faster than the rotational correlation time of the protein (τ c ~ 5-30ns)

Solution NMR Timescales for Biomolecular Motion ps ns µs ms s Relaxation dispersion Provides site specific information about conformational exchange occurring in the 50µs-1ms range Precise determination of rates of exchange. Association with temperature dependent measurements allows for thermodynamic characterisation of exchange processes Detection and characterisation of weakly populated or invisible species Structural information is limited to interpretation of chemical shift differences between exchanging conformations

Solution NMR Timescales for Biomolecular Motion ps ns µs ms s Real time Rapid data acquisition techniques allow events to be monitored in real-time Time resolution at the level of seconds Protein folding and unfolding Reaction kinetics

Solution NMR Timescales for Biomolecular Motion ps ns µs ms s Scalar, Dipolar Couplings, Chemical shift NMR dipolar couplings provide a highly detailed description of conformational disorder occurring up to the millisecond timerange Comparison with spin relaxation can reveal extent and nature of slower motions in proteins

Local and global molecular motion from spin relaxation Brief overview of the theory of spin relaxation : Important steps Interpretation of 15 N relaxation : Spectral density mapping / Modelfree analysis Description of molecular rotational diffusion tensors from 15 N relaxation Applications of spin relaxation to studies of protein dynamics

Rf rf pulse Molecular motion Local fields Spin state excitation relaxation equilibrium

Relaxation of longitudinal and transverse magnetisation states Determines delay between acquisitions Longitudinal Relaxation Determines lifetime of the signal - linewidth of the resonance Transverse Relaxation

Longitudinal Relaxation Rate equation for R 1 (R z ) dm z (t)/dt = -R 1 (M z (t) - M z0 ) M z (t) = (M z (0) - M z0 ) exp(-r 1 t) + M 0 z M z M z 0 M z (0) T 1 180 90 t Inversion Recovery t t

Transverse Relaxation M xy 0 Rate equation for R 2 (R x ) M xy M xy (τ) =M xy 0 exp(-r 2 τ) 90 x 180 y Spin Echo (CPMG) 0 T 2 τ τ τ τ

Macroscopic Description : Bloch Equations M z M xy M z 0 M z (0) M xy 0 t dm z dt dm x dt dm y dt [ ( )] $ R z 1 M z t = " M r ( t)# B r t [ ( )] $ R x 2 M x ( t) = " M r ( t)# B r t [ ] y $ R 2 M y t = " M r t ( ) # B r ( t) ( ( )$ M eq ) ( ) M z " I ˆ z ; M x " I ˆ x ; M y " I ˆ y I ˆ x ( t) = ( I ˆ x cos#t + I ˆ y sin#t)e $R 2 t I ˆ y ( t) = ( I ˆ y cos#t $ I ˆ x sin#t)e $R 2t 0 τ d2i x S z dt =? di x S y dt =?... Microscopic description?

Transition Frequencies for Heteronuclear system I-S ω H =2π.600MHz ; τ L (H)=265ps ω N =2π.60MHz ; τ L (N)=2.65ns S βα ω S ββ ω I αβ E I ω I αα ω S Local fields fluctuating at the transition frequencies of the spin system can induce relaxation to equilibrium

Relaxation rates can be described in terms of the motion of the relaxation-active interactions 15 N relaxation (spin 1/2) relaxation active mechanisms are essentially : Dipole-dipole (DD) I S DD Spin I experiences a distance and orientation dependent local field due to the magnetic moment of the nearby spin S Source of fluctuating fields

Relaxation rates can be described in terms of the motion of the relaxation-active interactions 15 N relaxation (spin 1/2) relaxation active mechanisms are essentially : Dipole-dipole (DD) + CSA DD Anisotropic electronic environment - chemical shift anisotropy Source of fluctuating fields Assumed axially symmetric and coaxial with NH

Transition Frequencies for Heteronuclear system I-S ω H =2π.600MHz ; τ L (H)=265ps ω N =2π.60MHz ; τ L (N)=2.65ns S βα ω S ββ ω I αβ E I ω I αα ω S Local motion of the neighbouring spin (and CSA) at the transition frequencies of the spin system can induce relaxation to equilibrium

Transition Frequencies for Heteronuclear system I-S 4 ββ E W 2 S W 2 I βα 3 W 0 2 αβ W 2 I spin magnetization : population W 1 I W 1 S difference between two I spin transitions (1-3), (2-4) 1 αα W - rate constants for transitions

Transition Frequencies for Heteronuclear system I-S Rate equations for magnetizations : R 1I W 2 S 4 ββ E W 2 I σ IS (I)Solomon equations βα 3 W 0 2 αβ W 2 W 1 I W 1 S R 1I - Auto relaxation rate constant 1 αα σ IS - Cross relaxation rate constant : rate of transfer of magnetization from S spin to I spin

Relaxation of longitudinal and transverse magnetisation states Longitudinal Relaxation Transverse Relaxation

How can we describe of these fluctuating local fields? Fast fluctuations Correlation function G(" ) = B loc ( t)b loc t +" ( ) # 0 Slow fluctuations Simple model : 2 G(" ) = B loc e #" " c Normalisation : c(0) = 1 " >> " c(" ) #### c $ 0 Local fluctuating fields described using a correlation function : measure of the rate of random fluctuations of the local field

How can we describe of these fluctuating Local Fields? Fast fluctuations Slow fluctuations Spectral density function P(") = 1 +& 2# ' G($ )e%i"$ d$ %& J (") = 1 2# +& ' %& c($ )e %i"$ d$ Simple model : # c J iso (") = 2 5 1+ " 2 2 # c Broad frequency distribution : Spectral density function Narrow spectral density function Local fluctuating fields described using a spectral density function : measure of the contribution of motion at each frequency. Area under the curve remains constant

Relaxation rates - Transitions induced by random fields H random (t) Spin-state variables : Available frequencies of motion that can be sampled Available transitions defined by the spin system Time-dependent geometric variables: Molecular motion J(ω) ω Random fields created by molecular motion - Allow transitions to occur

Relaxation of longitudinal and transverse magnetisation states Act on longitudinal spin states : Transitions at Larmor frequencies Longitudinal Relaxation Non-adiabatic Processes Non-secular terms ΔB i xy (t) Affect transverse spin states : Local dephasing : non-reversible! ΔB iz (t) Transverse Relaxation Non-adiabatic and adiabatic processes Secular and non-secular terms Act on transverse spin states ΔB i xy (t)

Ecole RMN Cargese Mars 2008 Identification of secular and non-secular terms...has never been easy

Relaxation of longitudinal and transverse magnetisation states Additional transverse relaxation mechanisms : R 2 R 1

Characteristic NMR timescales ps ns µs ms s Spin relaxation Lineshape/exchange Relaxation dispersion Real-time NMR Dipolar, scalar couplings 1/2πω 0 Larmor frequency 1/Δν Chemical shift coalescence T 1 relaxation Librational motion Molecular rotation Collective motions Protein folding Hinge Bending

How do we derive expressions for relaxation rate constants? Relating the motion to the rate constant W ij = f{a ij, Y, J(ω ij )} spin Interaction J(ω) Motion DD ω CSA

Liouville von Neumann Equation dσ(t)/dt = -i[h (t),σ(t)] Evolution of spin system Hamiltonians : Scalar coupling Static field B 1 field Dipolar coupling

Master Equation of Relaxation dσ(t)/dt = -i[h random (t),σ(t)] dσ(t)/dt ~ ~ = -i[h random (t),σ ~ (t)] Transformation to interaction frame (see Beat Meier s course) Evolution of spin system Stochastic Hamiltonian

Deriving expressions for relaxation rate constants d dt " t d dt " t ( ) = #i ( ) = # [ H r ( t), " ( t )] τ c <<t<<t relax % [ H r ( t), [ H r ( t # $ ), " ( t ) #" éq ]] 0 & d$ Spin state evolves under the action of the stochastic Hamiltonian H r (t) Stochastic Hamiltonian causes relaxation if : (a) double commutator does not vanish, (b) motion exist at characteristic frequencies H r " ( t) = F q q d dt " = # 1 2 % p,q ( ) t ( ) ( ) A q ( ) A p #q ( q) J q $ p ( ) ( q [,[ A ) p,(" # " )]] éq A (q) are spin operators describing the nature of the interaction F (q) (t) are spatial functions describing the reorientation of relaxation active interactions Motion (spatial function) can be described in terms of the spectral density function J(ω)

Deriving expressions for relaxation rate constants Calculation of relaxation rate constants : Decomposition of relaxation d dt " = # 1 2 Hamiltonian for each mechanism (DD, CSA) Calculation of double commutators : Determine efficiency of relaxation due to each term % p,q ( ) A p #q ( q) J q $ p ( ) ( q [,[ A ) p,(" # " )]] éq E IS = µ 0 4"r IS 3 Dipolar interaction (DD) $ r % µ I r µ S # 3 & r ( µ I r ) r µ S r r IS 2 r IS { } H DD = d I r S r " 3I # S # d = µ 0$ I $ S h 4%r IS 3 * H DD = d 1" 3cos 2 $ + # % &, ( ) r IS ' ( ) ( ) I z S z " 1 4 I + S " + I " S + " 3 2 sin# cos#em i- ( I ± S z + I z S ± ) " 3 4 sin2 #e m2i- I ± S ±. / 0 ( ) ' ( )

d dt " = # 1 2 Deriving expressions for relaxation rate constants Calculation of double commutators ( ) A q #q ( )[ A p [ ( )]] J q $ ( q ) ( ) ( q % p, A ) p, " # " éq F 0 p,q H r t " ( ) = F q q ( ) t ( ) = 1" 3 cos 2 # F 1) = sin # cos#e " i$ F = sin # cos#e i$ ( ) = sin 2 #e "2i$ F = sin 2 #e 2i$ F 2 A ( 0) ( 0) ( = A 1 + 0) ( A2 + A 0) Heavy. 3 ( 0) d A 1 = 2 " 2I z S z A ( q) ( 0) d A 2 = # 4 I + S # ( 0) d A 3 = # 4 I ( 0) #S + = A * 2 ( ) ( t) = e ih 0 t ( A q ) "ih e t 0 q = A p H 0 = " I I z + " S S z ( 0) ( " 1 = 0 " 0) ( 2 = " 0) I # " S " 3 = " S # " I ( 1) ( " 1 = " 1) I " 2 = " S " ( 2 ) = " I + " S $ p ( ) t ( ) e i# p q A ( 1) ( 1) ( = A 1 + 1) A2 = A (#1) ( 1) 3d A 1 = # 4 " 2I (#1) 3d +S z A 1 = # 4 " 2I #S z ( 1) 3d A 2 = # 4 " 2I (#1) 3d z S + A 2 = # 4 " 2I z S # A ( 2) = A (#2) A ( 2) = # 3d 4 I +S + A (#2) = # 3d 4 I #S #

Deriving expressions for relaxation rate constants Most commonly measured 15 N relaxation rates : This formalism can be used generally to develop expressions for relaxation rates of different spin states avec d 2 = " µ 0 Dipolar terms : $ # 4! % + - -, - -. R 1 R 1 2 & I 2& S 2h 2 r IS 6 : ( S z ) = d 2 { ( ) +6J (' I +' S )} 4 J ( ' I (' S )+ 3J ' S ( I z ) = d 2 { ( ) + 6J (' I +' S )} 4 J ( ' (' I S)+ 3J ' I R( I z ) S z ) = * SI = * IS = d 2 R 2 ( S x,y ) = d 2 R 2 ( I x,y ) = d 2 R zz ( ) = 3d 2 2I z S z { ( ) ( J (' I (' S )} 4 6J ' I +' S { ( ) +6J (' I ) +6J (' I +' S )} 8 4J( 0)+ J ( ' I (' S )+ 3J ' S { ( ) +6J (' S ) +6J (' I + ' S )} 8 4J( 0)+ J ( ' I (' S )+ 3J ' I 4 R 2z ( 2I z S x,y ) = d 2 R 2z { J (' S )+ J (' I )}+ / HH { 8 4J( 0)+ J ( ' I (' S )+3J(' S )+ 6J (' I +' S )}+ / HH ( 2I x,y S z ) = d 2 8 4J( 0)+ J ( ' I (' S )+3J ' I R 0Q ( I + S ( ;I ( S + ) = d 2 R 2Q ( I + S + ;I ( S ( ) = d 2 { ( ) +6J (' I +' S )} { 8 2J ( ' (' I S)+ 3J (' S ) +3J (' I )}+ / HH { 8 3J ( ' S) +3J (' I )+12J(' I +' S )}+ / HH

15 N relaxation rates R 1, R 2 and noe R 1 = d 2 /4{J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N )} + c{j(ω N )} Non-secular terms R 2 = d 2 /4{4J(0) + J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N ) + 6J(ω H ) } + c{3j(ω N ) + 4J(0)} (I z S z ) = d 2 /4{- J(ω I - ω S ) + 6J(ω I + ω S )} Secular terms d = (µ o /4π) (γ I γ S ћ) /r 3 c = (Δσ) ( γ S B o )2 / 3 For two covalently bound spins, r ij can be considered fixed - J(ω) provides reorientational information

R 1 = d 2 /4{J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N )} + c{j(ω N )} R 2 = d 2 /4{4J(0) + J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N ) + 6J(ω H ) } + c{3j(ω N ) + 4J(0)} Dependence of relaxation rates on characteristic correlation time of the motion ωτ c = 1

R 1 = d 2 /4{J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N )} + c{j(ω N )} R 2 = d 2 /4{4J(0) + J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N ) + 6J(ω H ) } + c{3j(ω N ) + 4J(0)} Fast motion or Extreme narrowing limit : ωτ c << 1 R 2 R 1 ( τ c ) ωτ c = 1

R 1 = d 2 /4{J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N )} + c{j(ω N )} R 2 = d 2 /4{4J(0) + J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N ) + 6J(ω H ) } + c{3j(ω N ) + 4J(0)} Slow motion limit : ωτ c >> 1 ωτ c = 1

Nuclear Overhauser enhancement (noe) η=1+γ I σ IS /γ S ρ S ; σ IS = d 2 /4{-J(ω I - ω S ) + 6J(ω I + ω S )}

Nuclear Overhauser enhancement 1 H- 1 H (noe) σ IS = (µ o /8π) 2 (γ I γ S ћ) 2 {-J(0) + 6J(2ω H )}/r 6 Principal conformational constraint for solution structure determination Often assume a common dynamic contribution

Determination of Macromolecular Structure in the Solution State Interproton Distances from NOE Measurement

Determination of Macromolecular Structure in the Solution State

Relaxation rates - Transitions induced by random fields H random (t) Spin-state variables : Available frequencies of motion that can be sampled Available transitions defined by the spin system Time-dependent geometric variables: Molecular motion J(ω) ω Random fields created by molecular motion - Allow transitions to occur

15 N relaxation A probe for local molecular flexibility Different dependence on ω of R 1, R 2 and noe samples the spectral density function at different frequencies defined by the spin system For 15 N relaxation, dominant relaxation mechanisms are the dipoledipole and csa interactions. Often considered to be coaxial Measures dynamics at each amide site in the peptide chain 15 N relaxation allows the measurement of angular reorientation, defined by angular time correlation or spectral density functions

Sampling of the spectral density function - Heteronuclear two spin system J(ω) 0 ω x ω H -ω x ω H ω H +ω x R 1 (X) R 2 (X) R 2 H z X x R 2 H z X z R 1 (H) noeh-x

Sampling of the spectral density function - Heteronuclear two spin system Approximation : J(ω) J(0) > J(ω N ) >> J(ω H -ω N )~J(ω H )~J(ω H +ω N ) 0 ω x ω H -ω x ω H ω H +ω x R 1 (X) R 2 (X) R 2 H z X x R 2 H z X z R 1 (H) noeh-x

Analysis of relaxation data using Reduced spectral density mapping Relaxation rates Spectral density mapping

Sampling of the spectral density function - Heteronuclear two spin system J(ω) ω =2πf - Larmor frequency, depends on static field strength 0 ω x ω H -ω x ω H ω H +ω x R 1 (X) R 2 (X) ω/2π =f noeh-x 80MHz 800MHz 18.8 Tesla 40MHz 400MHz 9.4 Tesla

Longitudinal 15 N Relaxation : Static Field Dependence ωτ c = 1

Nuclear Overhauser enhancement (noe) Field Dependence η=1+γ I σ IS /γ S ρ S ; σ IS = d 2 /4{-J(ω I - ω S ) + 6J(ω I + ω S )}

Spectral density function : J(ω) J(w) Describes the mobility of the inter-nuclear vector in terms of the distribution of frequency components Fourier transform of the time-dependent autocorrelation function c(t ) w

Auto-correlation function : c(τ) Correlates the value of a parameters at a time t with its value at time (t+τ) c(τ) represents the probability of finding the same value of the parameter at a time τ later - memory

Auto-correlation Functions c(τ) Correlates the value of a parameter at a time t with its value at time (t+τ) τ c(τ) represents the probability of finding the same value of the parameter at a time τ later - memory c(0) = 1 Random movement - c(t ) falls off with τ c(t ) < 1

Auto-correlation Functions - Rigid and Spherical Molecule Simplest function - decreasing exponential Describes rotational movement of a rigid sphere. Only one parameter τ c characterises motion c(τ) Two spheres of different diameter : τ c =2ns and 8ns Stokes Law τ c τ c 10 ns τ

Auto-correlation Functions - Rigid and Spherical Molecule Simple function - Lorentzian : Describes rotational movement of a rigid sphere. Width at half height = 2/τ c J(ω) (ns) Two spheres of different diameter : τ c =2ns and 8ns 200 400 ω (MHz)

c(τ) In reality this function can of course be much more complex - Interactions are modulated by local motions and by the global motion of the molecule τ If these motions are not correlated, and occur on different time scales, the corrlation functions can be described by : c(τ) = c g (τ)c i (τ) and J(ω) = J g (ω) * J i (ω)

Restricted Internal Motion c g (τ)c i (τ) J g (ω) * J i (ω) τ ω τ c

Unrestricted Internal Motion c g (ω)c i (ω) J g (ω) * J i (ω) ω ω τ c

Information available from 15 N Relaxation c(τ) τ Reminder : Beyond a few τ c there is no remaining memory No direct information about slower internal motion?

Analysis of Internal Mobility using 15 N Relaxation J(ω) ω Characterisation of the motion by analytical modelling of J(ω) or c(τ) Free diffusion about an axis Restricted rotation about an axis Diffusion/rotation about an axis Diffusion on the surface of a cone Discrete n-site jumps J(ω) - can be precise, but complex: Number of dynamic parameters to be fitted can exceed the number of experimentally available relaxation rates. Differentiation?

Analysis of Movement without a Geometric Model Modelling of c(τ) (et J(ω)) using a simple mathematical function Reduces the number of degrees of freedom Allows possibility of statistical analysis of treatment Lipari-Szabo Model Internal and global motion are supposed independant Simplest model : c(τ) described by two exponential functions : c(τ) = c g (τ)c i (τ) c i (0)=1; c i ( )=S 2 For isotropic overall tumbling - c i (τ) = S 2 + (1- S 2 )exp(-τ/ τ i ) c g (τ) =(1/5) exp(-τ/ τ c )

Lipari-Szabo Model-free Approach : Auto-correlation Function τ c c i (τ) S 2 τ i I I II c i (τ) S 2 c g (τ) III τ i c i (τ) S 2 -Generalised Order Parameter describing amplitude of angular reorientation τ i τ i t II S 2 III τ c τ Internal and global motion are supposed independant τ i << τ c

Lipari-Szabo Model-free Approach : Spectral Density Function Similarly the spectral density function can be described by the sum of two Lorentzians, describing the internal and overall motion : J(ω) = (2/5) (S 2 τ c /(1+ ω 2 τ c2 ) + (1- S 2 ) τ e /(1+ ω 2 τ e2 ) ) All relaxation rates can be described in terms of this simple function τ c is common to the whole molecule {S 2, τ e } can be determined for each site from {R 1, R 2, noe}

Extended Model-free Approach Introduction of an additional movement with intermediate correlation time τ s c i (τ) = S 2 + (1- S f 2 )exp(-t/ τ f ) + (S f 2 -S 2 )exp(-t/ τ s ) J(ω) = (2/5) (S 2 τ c /(1+ ω 2 τ c 2 ) + (S f 2 - S 2 )τ s /(1+ ω 2 τ s 2 )) ; τ f 0 tτ c τ f τ s τ f S f 2 S 2 τ s τ f τ

Chemical Shift Exchange - Contribution to R 2 Measured R 2 can contain a contribution due to rapid exchange R ex between conformers having different chemical shifts (see Carine s talk) Conformational exchange (t exch - ms) Can lead to artefacts in model-free analysis if undetected

Statistical Analysis of Heteronuclear Relaxation Data J(ω) = (2/5) (S f 2 (τ c /(1+ ω 2 τ c 2 ) + (1- S s 2 ) τ s /(1+ ω 2 τ s 2 )) Measurements - (R 1, R 2, noe) Dynamic Parameters - τ c (S f 2, S s 2, τ s, R ex )

Generalised Analysis with no Geometric Model Lipari-Szabo Approach c(t c(τ) ) (and J(ω)) J(w)) described using abstract mathematical model Internal and global motion supposed independent and exponential in time Internal motion described by an amplitude (S 2 ) and correlation time (t (τ e ) tτ c is common to the whole molecule {S 2, tτ e } can be determined for each site from {R 1, R 2, noe} All relaxation rates can be described in terms of this simple function S 2 measures thermally accessible orientations - Local contribution to entropy Model-free approach susceptible to artefacts due to Under-determination - Over-interpretation Poorly determined τ c - (using rigid regions) Local/overall motion correlated Non-globular behaviour

Statistical Analysis - Estimation of τ c J(ω) = (2/5) (S f 2 (τ c /(1+ ω 2 τ c 2 ) + (1- S s 2 ) τ s /(1+ ω 2 τ s 2 )) In limit where internal motion is considered fast - librational (restricted motion on 10s picosecond timescale) J(ω) = (2/5) (S f 2 (τ c /(1+ ω 2 τ c 2 )) Ratio R 2 /R 1 can be considered independent of {S 2,τ s } R 2 /R 1 τ c

Statistical Analysis of Heteronuclear Relaxation Data Modelled Parameters Model 1 S 2 Model 2 S 2,τ i Model 3 S 2,R ex Model 4 S 2,τ i,r ex Model 5 S f 2, S s 2, τ s Iterative modelling of relaxation data using increasingly complex J(ω) until data satisfy estimated confidence level

R 1 = d 2 /4{J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N )} + c 2 {J(ω N )} R 2 = d 2 /4{4J(0) + J(ω H - ω N ) + 3J(ω N ) + 6J(ω H ) + 6J(ω H + ω N )} + c 2 {4J(0) + 3J(ω N )} (I z S z ) = d 2 /4{- J(ω I - ω S ) + 6J(ω I + ω S )} Dependence of 15 N relaxation rates on Lipari- Szabo parameters {S 2, τ i } Rigid vector attached to a rotor with overall correlation time 10ns 500MHz spectrometer frequency

R 1 = d 2 /4{J(ω H - ω N ) + 3J(ω N ) + 6J(ω H + ω N )} + c 2 {J(ω N )} R 2 = d 2 /4{4J(0) + J(ω H - ω N ) + 3J(ω N ) + 6J(ω H ) + 6J(ω H + ω N )} + c 2 {4J(0) + 3J(ω N )} (I z S z ) = d 2 /4{- J(ω I - ω S ) + 6J(ω I + ω S )} Dependence of 15 N relaxation rates on Lipari- Szabo parameters {S 2, τ i } Isotropic global motion with overall correlation time 10ns 500MHz spectrometer frequency

Dependence of 15 N relaxation rates on Lipari- Szabo parameters {S 2, τ i } Isotropic rotor with overall correlation time 10ns 500MHz spectrometer frequency Isocontours of relaxation rate values R 1, R 2 and noe

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data 2 R 1 (s-1) 1 0 12 R 2 (s-1) 6 0 noe 0.8 0.4 0 Dependence of 15 N relaxation rates on Lipari-Szabo parameters {S 2, τ i } Isotropic rotor with overall correlation time 10.0ns

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data 2 R 1 (s-1) 1 0 12 R 2 (s-1) 6 0 noe 0.8 0.4 0 Dependence of 15 N relaxation rates on Lipari-Szabo parameters {S 2, τ i } Isotropic rotor with overall correlation time 10.0ns

Dependence of 15 N relaxation rates on Lipari-Szabo parameters {S 2, τ i } Isotropic rotor with overall correlation time 4.22ns

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data R 1 = (2.53 ± 0.06) s -1 R 2 = (5.55 ± 0.04) s -1 noe =0.650 ± 0.006 S 2 =0.81±0.02 τ i =44±11ps

R 1 = (2.29 ± 0.08) s -1 R 2 = (5.09 ± 0.4) s -1 noe =0.52 ± 0.04

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data R 1 = (2.29 ± 0.08) s -1 R 2 = (5.09 ± 0.4) s -1 noe =0.52 ± 0.04 S 2 =0.73±0.02 τ i =78±20ps

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data R 1 = (2.64 ± 0.15) s -1 R 2 = (6.35 ± 0.16) s -1 noe =0.66 ± 0.02 S 2 =0.90±0.03 τ i =?

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data from Calbindin

Lipari-Szabo Modelfree Analysis : Fitting {S 2, τ i } to relaxation data from Calbindin Apo form Calcium bound

TENSOR2 Graphical and statistical analysis of global and internal motion in proteins from 15N relaxation data

Monte Carlo based error analysis of the precision of Lipari-Szabo parameters {S 2, τ i } fitted to 15N to relaxation data using the Modelfree approach : Ca-bound Calbindin

Can Dynamics from Relaxation Lead to Thermodynamic Understanding? S 2 provides a direct measure of conformational order A component of conformational entropy can be estimated from S 2

Internal Mobility from 15 N heteronuclear relaxation rates Relaxation rates R 1, R 2 and noe sample rapid (ps-ns) dynamics 15 N relaxation samples angular reorientational properties Transformation to spectral density terms via matrix inversion Analytical geometric description of motion in general underdetermined Lipari-Szabo - Abstract mathematical model decouples internal and global motion. Internal mobility described in terms of order parameter S 2 and local correlation time τ i Quantitative measurement of extent and timescale of internal motion Provides a measure of a contribution to entropy Connection to physical model is lost in this approach : realistic models require additional probes or molecular dynamics simulations

Systems where Lipari-Szabo approach is not valid Partially folded or molten globule states Highly disordered proteins Multi-domain proteins exhibiting differential domain dynamics Multiple, complex internal motions..

Anisotropic Rotational Diffusion D xx Ω D zz D yy J g (ω)=σa j τ j /(1+ ω 2 τ 2 j ) a j = fn(ω) a j define the geometry of the relaxation mechanism relative to the rotational diffusion tensor D

Anisotropic Rotational Diffusion J g (ω)=σa j t j /(1+ ω 2 τ j 2 ) D xx 5 independent components of spectral density function D zz J(ω) D yy R 1 (N) R 2 (N) noeh-n 0 ω N ω H -ω N ω H ω H +ω N ω

Anisotropic Rotational Diffusion J g (ω)=σa j τ j /(1+ ω 2 τ 2 j ) 5 independent components of spectral density function : Relative contribution depends on orientation of relaxation mechanism within the molecular frame J(ω) R 1 (N) R 2 (N) 0 ω N

Anisotropic Rotational Diffusion D xx D zz J(ω) D yy R 1 (N) R 2 (N) 0 ω N

Characterisation of anisotropic rotational diffusion from heteronuclear relaxation D zz z D xx α,β,γ Laboratory frame D yy y x 6 parameters define the diffusion tensor components and their orientation relative to the molecular frame (D xx, D yy, D zz, α, β, γ) Requires >6 relaxation rates/orientations

Determination of hydrodynamic behaviour of proteins in solution from 15 N relaxation : TENSOR2

Internal Mobility in the Presence of Anisotropic Rotational Diffusion J (ω) S2 Σ{a j τ j /(1+ω 2 τ j2 )} + (1- S 2 ) τ e /(1+ω 2 τ e2 )

Lipari-Szabo Type Internal mobility analysis using iso- and anisotropic rotational diffusion

Molecular Dynamics Simulations and NMR Relaxation Correlation with experimental data Identification of motional models Identification of additional probes for more precise motional information