Effects of Chemical Exchange on NMR Spectra Chemical exchange refers to any process 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
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 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
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
Rationale for Chemical Exchange For slow exchange For fast exchange FT FT Bloch equation approach: dm AX /dt = -( ω A )M AY -M AX /τ A + M BX /τ B dm BX /dt = -( ω B )M BY -M BX /τ B + M AX /τ A
2-state 2nd Order Exchange k +1 M+L ML k -1 K d = [M] [L]/[ML] = k -1 /k +1 K d =10-3 10-9 M k on = k +1 ~ 10 8 M -1 s -1 (diffusion-limited) k -1 ~ 10-1 10-5 s -1 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, respectively
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 3 2 1 RELAXATION TIMESCALE SPECTRAL TIMESCALE LARMOR TIMESCALE MOLECULAR VIBRATIONS
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 -1 0 1000 0 12 1-20 NMR time-scale refers to the chemical shift timescale. The range of the rate can be studied 0.05-5000 s -1 for H can be extended to faster rate using 19 F, 13 C and etc.
Slow Exchange k << δ A -δ B A B k1 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/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 Lw = k/π Increasing temperature k increases, line width increases
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 intensity as the excess of free ligand increases. Obtain spectra over a range of [L]/[M] ratios from 1 to 10
Slow Exchange for M+L ML 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 ). k 1 k- 1
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 = c π δ δ // 2 2 = 2.22 2.22 δ δ δ is the chemical shift difference between the two signals in the unit of Hz. δ depends on the magnetic field
Coalescence Temperature Since the rate depends on the G of the inversion, and the G is affected by T, higher temperature will make things go faster. Tc is the temperature at which fast and slow exchange meet. T>Tc, fast exchange T<Tc, slow exchange T we can calculate the G of the process G G = R ** T C * C *[[ 22.96 22.96 + ln ln (( T C / C / δ δ ))]] T C
Fast Exchange k >> δ A -δ B A B k1 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 f B, 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 ( δ AB ) 2 / k -1 Maximal line broadening is observed when f A = f B = 0.5
Fast Exchange k >> δ A -δ 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 2,ML + f L /T 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 constant for the complex can be obtained by measuring the chemical shift of the ligand resonance at a series of [L].
Measuring Binding Constant k+1 M+L ML k-1 δ M,obs = f M δ M +f ML δ ML, The total change in δ of M δ Mo = δ ML - δ M At [L], δ M = δ M,obs - δ M If [M] is fixed, δ M= δ Mo [L]/([L]+K d ) At 0.5 δ Mo, K d = [L] Similar for A - +H + k+1 k-1 AH H=L, K d = K a δ H= δ Ho [H]/([H]+K a ) Inhibitor G-3-P binds to Triosephapte Isomerase -JMB,1976 100:319
The Ionization of His in Oxy & DeOxyhemglobin C2H 7.7 ppm 8.7 ppm C4H 7.0 ppm 7.4 ppm Bohr effect: Hb takes up H + on releasing O2 The pka of His-β-146 changes by more than 1pH unit upon ionization due to the stabilization of the charged form by Asp94 in the deoxy structure.
Identifying & Designing Ca(II)-binding Proteins H 2 O 90 2.4 Å Ca 90 90 Mg 72 2.1 Å H 2 O Anchor Anchor Ca Mg Calcium geometry Magnesium geometry Yang et al. (2002) Proteins, 47, 344.
The Average Structure of CD2-6D15 D17 N60 D62 D15 W32 N60 D62 D15 D17
Calcium Titration of CD2-6D15 Using HSQC 0.3 T75 D71 T86 I18 T5 M46 H12 D62 0.2 I97 T79 T37 D17 A40 Chemical shift change (ppm) 0.1 0 S = S K 1 d1 [Ca] + [Ca] + S K 2 d2 [Ca] + [Ca] I63 N90 I18 A54-0.1 D62 D62-0.2 N60 0 20 40 60 80 100 120 [Ca] (mm) D17 D15 D17 Yang et al. (2005) JACS, 127, 2085-93 CD2-6D15
Manganese Relaxation of CD2-6D15 D26 D71 T86 T5 H12 0.64 0.56 A40 0.48 D62 log ( i -1 ) 0.4 0.32 D17 0.24 0.16 N90 I18 A54 Cyan: without Mn 2+ Purple: with Mn 2+ 0.08 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 log (r) Yang et al. (2005) JACS, 127, 2085-93 Mildvan, AS, and Cohn M, 1970, Adv. Enzymol. Relat. Areas Mol. Biol., 1970, 33, 1-70
Dynamic Properties of Ca.CD2 800 400 0 τ e (ps) T83 L89 V80 I18 400 800 M46 12 8 4 R ex (s -1 ) T37 S52 0 4 8 12 S 2 0.9 0.8 0.7 0.6 A B C C' C'' D E F G 0 20 40 60 80 100 Residue Yang et al., Biochemistry, 2005 In Press