K ex. Conformational equilibrium. equilibrium K B

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2 Effects of Chemical Exchange on NMR Spectra Chemical exchange refers to any yprocess in which a nucleus exchanges between two or more environments in which its NMR parameters (e.g. chemical shift, scalar coupling, or relaxation) differ. DNMR deals with the effects in a broad sense of chemical exchange processes on NMR spectra; and conversely with the information about the changes in the environment of magnetic nuclei that can be derived from observation of NMR spectra. K ex Conformational equilibrium K B Chemical equilibrium

3 Types of Chemical Exchange Intramolecular exchange Motions of sidechains in proteins A B Helix-coil transitions of nucleic acids Unfolding of proteins Conformational equilibria Intermolecular exchange Binding of small molecules l to macromolecules Protonation/deprotonation equilibria Isotope exchange processes Enzyme catalyzed reactions M+L ML Because NMR detects the molecular motion itself, rather the numbers of molecules in different states, NMR is able to detect chemical exchange even when the system is in equilibrium

4 2-state First Order Exchange A B k1 k -1 Lifetime of state A: τ A = 1/k +1 Lifetime of state B: τ B = 1/k -1 Use a single lifetime 1/ τ =1/ τ A + 1/τ B = k +1 + k -1

5 Rationale for Chemical Exchange For slow exchange For fast exchange FT FT Bloch equation approach: dm AX /dt = -(Δω A )M AY -M AX /τ A + M BX /τ B dm BX /dt = -(Δω B )M BY -M BX /τ B + M AX /τ A

6 2-state 2nd Order Exchange M+L k +1 M+L ML K d = [M] [L]/[ML] = k -1 /k +1 k -1 K -3-9 d =10 10 M k on = k +1 ~ 10 8 M -1 s -1 (diffusion-limited) k ~ s Lifetime 1/ τ =1/ τ ML + 1/τ l = k -1 (1+f ML /f L ) f ML and f L are the mole fractions of bound and free ligand, ML L g respectively

7 Typical Motion Time Scale for Physical Processes SLOW very slow slow fast very fast ultrafast s ms μs ns ps fs FAST MACROSCOPIC DIFFUSION, FLOW CHEMICAL EXCHANGE MOLECULAR ROTATIONS RELAXATION SPECTRAL LARMOR TIMESCALE TIMESCALE TIMESCALE MOLECULAR VIBRATIONS

8 NMR Time Scale Time Scale Chem. Shift, δ Coupling Const., J T2 relaxation Slow k << δ Α δ Β k << J Α J Β k << 1/ T 2, Α 1/ T 2, B Intermediate k = δ Α δ Β k = J Α J Β k = 1/ T 2, Α 1/ T 2, B Fast k >> δ Α δ Β k>>j Α J Β k>>1/t 2, Α 1/ T 2, B Sec NMR time-scale refers to the chemical shift timescale. The range of the rate can be studied s -1 for H can be extended to faster rate using 19 F, 13 C and etc.

9 k1 Slow Exchange k << δ -δ A B A δ B k -1 Separate lines are observed for each state. The exchange rate can be readily measured from the line widths of the resonances Like the apparent spin-spin relaxation rates, 1/T 2i,obs 1/T 2A,obs = 1/T 2A +1/ 1/τ A = 1/T 2A +1/k 1 1/T 2B,obs = 1/T 2B + 1/τ Β = 1/T 2B + 1/k -1 line width Lw = 1/πT 2 = 1/πT 2 +k 1 /π Each resonance is broadened by Δ Lw = k/π Increasing temperature increases k, line width increases

10 Slow Exchange for M+L ML k 1 k- 1 Separate resonances potentially are observable for both the free and bound states M F,M B,L F, and L B The addition of a ligand to a solution of a protein can be used to determine the stoichiometry of the complex. Once a stoichiometric mole ratio is achieved, peaks from free ligand appear with increasing i intensity it as the excess of free ligand increases. Obtain spectra over a range of [L]/[M] ratios from 1 to 10

11 Slow Exchange for M+L ML k 1 k- 1 For free form 1/T 2L,obs = 1/T 2L + 1/τ L = 1/T 2L + k -1 f ML /f L 1/T 2M,obs = 1/T 2M + 1/τ Μ = 1/T 2M + k -1 f ML /f M For complex form 1/T 2ML,obs = 1/T 2ML + 1/τ ML = 1/T 2ML + k -1 Measurements of line width during a titration can be used to derive k -1 (k off ).

12 19 F spectra of the enzyme-inhibitor complex at various mole ratio of carbonic anhydrase:inhibitor Free inhibitor 1:4 1:3 1:2 1:1 1:0.5 Bound ligand At -6 ppm the broadened peak for the bound ligand is in slow exchange with the peak from free ligand at 0 ppm. The stoichiometry t of the complex is 2:1. No signal from the free ligand is visible until more than 2 moles of inhibitor are present.

13 Coalescence Rate For A B equal concentrations, there will be a rate of interchange where the separate lines for two species are no longer discernible The coalescence rate k c = π Δδ / 2 = 2.22 Δδ Δδ is the chemical shift difference between the two signals in the unit of Hz. Δδ depends on the magnetic field

14 Coalescence Temperature Since the rate depends on the ΔG of the inversion, and the ΔG is affected by T, higher temperature will make things go faster. Tc is the temperature t at which fast and slow exchange meet. T>Tc, fast exchange T<Tc, slow exchange T we can calculate the ΔG of the process ΔG = R * T C * [ ln ( T C / Δδ ) ] T C

15 Fast Exchange k >> δ A -δ B k1 A B k -1 A single resonance is observed, whose chemical shift is the weight average of the chemical shifts of the two individual states δ obs = f A δ A +f B δ B, f A + f B = 1 For very fast limit 1/T 2,obs = f A /T 2A + f B /T 2B For moderately fast 1/T 2,obs = f A /T 2A + f B /T 2B + f A f B2 4π (Δδ AB ) 2 / k -1 Maximal line broadening is observed when f A = f B = 0.5

16 Fast Exchange k >> δ A -δδ B k+1 M+L ML k-1 For M δ M,obs = f M δ M +f ML δ ML For L δ L.obs = f L δ L +f ML δ ML 1/T 2,obs = f ML /T 2ML 2,ML + f L /T 2L 2,L + f ML f L2 4π (δ ML -δ L ) 2 / k -1 A maximum in the line broadening of ligand or protein resonances occurs during the titration at a mole ratio of approx. ligand:protein 1:3 The dissociation i constant t for the complex can be obtained by measuring the chemical shift of the ligand resonance at a series of [L].

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24 Diffusion by Pulse Field Gradient Price et al. Concepts Magn. Reson. (1997) 9,

25 Pulse Sequence Δ I = I o exp[-(γδg) 2 (Δ + 2τ 2δ/3)D] D = kt/f γ: gyromagnetic ratio f = 6πηr D: diffusion i constant t k: Boltzman constant t Δ: interval between gradient pulse f: friction coefficient δ: PFG duration time η: solvent viscosity G: gradient strength r: hydrodynamic radius

26 N15 Edited Chou et al. J. Biomol. NMR (2004) 29,

27 Calibration of Gradient Strength 500 MHz 600 MHz y = exp(-m1* M0^2) Value Error m e e-10 1 Chisq e-05 NA 1 R NA 0.8 y = exp(-m1* M0^2) Value Error 0.8 m e e-10 Chisq NA 0.6 R NA 0.6 m1 Chisq R y = exp(-m1 * M0^2) Value Error 1.465e e NA NA gradient strength gradient strength Gzlvl1=32768 = 53.2 G/cm = 64.6 G/cm

28 Diffusion Constant is Related to Molecular Weight diameter Lee et al. Biochimica et Biophysica Acta (2002) 1598,80 87

29 Diffusion Dependent on Conformation ApoCaM r = 22.4 ± 0.3 Å NMR: 22 Å 1.2 m1 Chisq R y = exp(-m1 * M0^2) Value Error e-06 NA NA m1 Chisq R y = exp(-m1 * M0^2) Value Error e-06 NA NA Ca 2+ -CaM r = 22.8 ± 0.5 Å apocam Ca-CaM X-ray: 23 Å gradient strength

30 Binding Constant Determination D obs = X L D L + X PL D PL X PL = (D L D obs )/(D L D PL ) K a = X PL /((1 X PL )([P] 0 X PL [L] 0 )) 0.5 mm cyclohexylacetic acid in D 2 O; D = 6.85 x10 6 cm 2 s mm cyclohexylacetic acid plus 0.5 mm β-cyclodextrin in D 2 O; D = 5.39 x10 6 cm 2 s -1 K a = 1800 ± 100 M -1 Cameron et al. J. Org. Chem. (2001) 66,

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32 Limitations The transverse relaxation time constant T 2 limits the interval where diffusion i can be observed. Small diffusion i coefficients i (D < m 2 s -1 ), associated with macromolecules with masses larger than 50 kda are difficult to measure For weak binding, the change in diffusion constant is too small, and it is difficult to get the binding constant Singlet State Diffusion Spectroscopy stores the nuclear spin order as a singlet state. This state relaxes with a time constant T s that can be much longer than both T 1 and T 2 Further reading: Cavadini et al. Concepts Magn. Reson. (2008) 32A,