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.

Conformationalequilibrium

Chemicalequilibrium

Kex

KB

Types of Chemical ExchangeIntramolecular exchange– Motions of sidechains in proteins– Helix-coil transitions of nucleic acids– Unfolding of proteins– Conformational equilibriaIntermolecular exchange– Binding of small molecules to

macromolecules– Protonation/deprotonation equilibria– Isotope exchange processes– Enzyme catalyzed reactions

A B

M+L MLBecause NMR detects the molecular motion itself, rather the numbers of molecules in different states, NMR is able to detect chemicalexchange even when the system is in equilibrium

2-state First Order Exchange

Lifetime of state A: τA = 1/k+1

Lifetime of state B: τB = 1/k-1

Use a single lifetime1/ τ =1/ τA + 1/τB

= k+1+ k-1

A Bk1

k-1

Rationale for Chemical Exchange

For slow exchange

For fast exchange

Bloch equation approach:

dMAX/dt = -(∆ωA)MAY - MAX/τA + MBX/τBdMBX/dt = -(∆ωB)MBY - MBX/τB + MAX/τA

•••

FT

FT

2-state 2nd Order Exchange

Kd = [M] [L]/[ML] = k-1/k+1

M+L MLk+1

k-1

Kd =10-3 – 10-9 Mkon = k+1 ~ 108 M-1 s-1 (diffusion-limited)k-1 ~ 10-1 – 10-5 s-1

Lifetime 1/ τ =1/ τML + 1/τl

= k-1 (1+fML/fL)fML and fL are the mole fractions of bound and free ligand, respectively

Typical Motion Time Scale for Physical Processes

SLOW FASTs ms µs ns ps fsvery slow slow fast very fast ultrafast

MACROSCOPICDIFFUSION,

FLOW

MOLECULARROTATIONS

MOLECULARVIBRATIONS

LARMORTIMESCALE

SPECTRALTIMESCALE

RELAXATIONTIMESCALE

CHEMICALEXCHANGE

3 2 1

NMR Time Scale

Time Scale Chem. Shift, δ Coupling Const., J T2 relaxation

Slow k << δΑ− δΒ k << JΑ− JΒ k << 1/ T2,Α− 1/ T2,B

Intermediate k = δΑ − δΒ k = JΑ− JΒ k = 1/ T2,Α− 1/ T2,BFast k >> δΑ − δΒ k >> JΑ− JΒ k >> 1/ T2,Α− 1/ T2,BSec-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 19F, 13C and etc.

Slow Exchange k << δA -δB

• 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/T2i,obs1/T2A,obs= 1/T2A+ 1/τA = 1/T2A+ 1/k1

1/T2B,obs= 1/T2B+ 1/τΒ = 1/T2B+ 1/k-1

line width Lw = 1/πT2 = 1/πT2 + k1/πEach resonance is broadened by ∆ Lw = k/πIncreasing temperature k increases, line width

increases

A Bk1

k-1

Slow Exchange for M+L ML• Separate resonances potentially are

observable for both the free and bound states MF, MB, LF, and LB

• 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

k1

k-1

Slow Exchange for M+L ML

• For free form 1/T2L,obs= 1/T2L+ 1/τL = 1/T2L+ k-1 fML/fL1/T2M,obs= 1/T2M+ 1/τΜ = 1/T2M+ k-1 fML/fM

For complex form1/T2ML,obs= 1/T2ML+ 1/τML = 1/T2ML+ k-1

Measurements of line width during a titration can be used to derive k-1 (koff).

k1

k-1

19F spectra of the enzyme-inhibitor complex at various mole ratio of carbonic anhydrase:inhibitor

• 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 of the complex is 2:1. No signal from the free ligand is visible until more than 2 moles of inhibitor are present.1:0.5

1:4

1:3

1:2

1:1

Free inhibitorBound ligand

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

∆δ is the chemical shift difference between the two signals in the unit of Hz.

∆δ depends on the magnetic field

kc = π ∆δ / √2 = 2.22 ∆δkc = π ∆δ / √2 = 2.22 ∆δ

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 TC

we can calculate the ∆G‡ of the process

∆G‡ = R * TC* [ 22.96 + ln ( TC / ∆δ ) ]∆G‡ = R * TC* [ 22.96 + ln ( TC / ∆δ ) ]

Fast Exchange k >> δA -δB

• A single resonance is observed, whose chemical shift is the weight average of the chemical shifts of the two individual states

δobs = fAδA +fBδB, fA + fB = 1For very fast limit

1/T2,obs= fA/T2A+ fB/T2B

For moderately fast 1/T2,obs= fA/T2A+ fB/T2B + fAfB

2 4π (∆δAB)2/ k-1

Maximal line broadening is observed when fA = fB = 0.5

A Bk1

k-1

Line Shape Simulation

Fast Exchange k >> δA -δB

For M δM,obs = fMδM +fMLδMLFor L δL.obs = fLδL +fMLδML1/T2,obs= fML/T2,ML+ fL/T2,L + fMLfL

2 4π(δML-δL)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].

M+L MLk+1

k-1

NMR Titration

• Both T6(CH3) at 1.1 ppm from DNA (A) and L(CNH2) methyl from ligand (B) are in fast exchange.

• Expanded regions from 300 MHz 1HNMR spectra from complexes between L(NH2) and d(GGTAATACC)2recorded at 10 ºC (Craik)

Residues in the linker of Calmodulin underwent sequential phases of conformational change

Binding of Inhibitor to Enzyme

• NMR studies are done at mM, making it difficult to determine Kd with any accuracy if the binding is very tight (nM Kd).

HS

HI

*

*

H

H

*

*

HS

HI*

*

koff kunf

koff [protein][ligand]Kd= =

kon [protein- ligand]

koff [protein][ligand]Kd= =

kon [protein- ligand]

Measuring Binding Constant

δM,obs = fMδM +fMLδML,The total change in δ of M∆δMo = δML- δMAt [L], ∆δM = δM,obs- δMIf [M] is fixed, ∆δM = ∆δMo[L]/([L]+Kd)At 0.5 ∆δMo, Kd = [L]Similar for

H=L, Kd = Ka∆δH = ∆δHo[H]/([H]+Ka)

M+L MLk+1

k-1

A-+H+ AHk+1

k-1Inhibitor G-3-P binds to Triosephapte Isomerase

-JMB,1976 100:319

NMR studies of Ca(II) Binding to cNTnC

cNTnC is the N-terminal domain of TnCfrom cardiac muscle with a single calcium binding site II . Peak at 10.3 ppm is G70 at the calcium binding site. – Monica et al., 1997

[Ca]t/cTnC1.460

1.243

1.026

0.809

0.635

0.505

0.374

0.244

0.114

0.027

0

6.96 ppm Kd = 144 ± 27 uM

0.0 1.0 2.0 3.0 4.0 5.0 6.0

[Ca(II)], (mM)

frac

tiona

l cha

nge

of c

hem

ical

shift

s0 mM

0.140 mM

0.228 mM

0.483 mM

Monitoring Calcium-binding by NMR

CaM-CD2-III-5G

0

0.2

0.4

0.6

0.8

1

Ca2+

The Ionization of His in Oxy & DeOxyhemglobin

C2H 7.7 ppm 8.7 ppmC4H 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.

Transferred NOE (TRNOE)

bound L free L

protein

H1 H2 H1 H2

Cross-relaxation (NOE) between two protons in the bound ligandis transferred to the free molecular by chemical exchange between bound and free ligand.The negative NOEs from the bound state can be detected by carrying NOE experiments on the free ligand in slow exchange, or one the averaged signals from free and bound ligand in fast exchange.

irradiation

Transferred NOE (TRNOE)

boundH1 boundH2kmagf

kmagr

kbindfkbindr kbindfkbindr

freeH1 freeH2kmagf

kmagr

For small ligands such as short peptides, the enhancement from cross-relaxation is almost 0. On the other hand, large molecules such as receptors have very efficient magnetization transfer. If the chemical exchange rate is great enough, magnetization is transferred from the free protons via the bound state. This gives distance information of the bound state.

bound L free L

protein

H1 H2 H1 H2

Selectively deuterated dihydrofolatereductase in the presence of 2.7 molar

equivalents of trimethoprim

control

Irriadiation st 7.34 ppm

Spectrum a-b

H6 is close to H2’(or H6’) in a folded conformation of bound trimethoprim

1

H (ppm)

Superposition of 50 energy minimized structures of FaNaCh-bound FMRFa

N-terminus

C-terminal amide

Ligand binding: Isotope editingNMR techniques exist to separate a mixture of molecules with

different isotopic labels. For example, if two proteins are mixed

together and one has 15N labels and the other doesn’t, it is possible

to only observe the amide protons of one or the other (technically it

is easier to see the amides of the protein with 15N, but either is

possible). This can be extended to mixtures of pools of ligands with

proteins where either the ligand is labeled or the protein is labeled.

More complicated mixtures can be analyzed by incorporating

different types of labels (e.g. 15N on one, 13C on another)

I(ν) - the intensity at frequency νCo-constants proportional to [protein]tot

PN and PD - the populations of N and D

Dynamic NMR to Monitor Rapid Protein Folding

The parameters needed to calculate the lineshape are Co, & ν, T2N, T2D, kf and ku.The first four parameters are obtained directly from the spectra usingLorentzian line shape simulation of the J-coupled multiplets.kf and ku can be obtained independently by simulating the position and shape of a given resonance.

The intensity as a function of frequency

--J. Sandström, 1982

McConnell equation