determination of protein structure using nuclear magnetic...

Determination of Protein Structure UsingNuclear Magnetic Resonance

FST 210, Winter, 1999

"There are only two methods that yield molecular structure to atomic resolution, onlyone yields the structure in solution"

I. Fundamentals/Overview - a Topical OutlineA. A Collection of spins in a magnetic fieldB. Effect of a radiofrequency pulseC. Fourier transform: frequency, phase, amplitude, linewidth (1D NMR)D. Generalization to 2D experiment and 2D transformsE. Through-bond connectivities using COSY and heteronuclear spectraF. Through-space connectivities using NOESYG. Internuclear distances and NOE's, dihedral angles from coupling constantsH. Structural calculations and refinement

I I . B i b l i o g r a p h yA. Covalent structure determinations of small molecules:

Silverstein & Bassler (Chem 219)

B. 2D and MQ NMR for assignmentsBax, A (1984) Two-Dimensional NMR in Liquids, Delft University Press, D.

Reidel Publishing, Dordrecht, Boston, London. Rigorous but readable.

Bax, A (1984) Bulletin of Magnetic Resonance 7, 167-183. "A Simple Description of Two-dimensional NMR Spectroscopy"

Derome, A.E. (1987) Modern NMR Techniques for Chemistry Research,, Pergammon Press, Elmsford N.Y. A very good "How-To' book.

Englander, S.W. & Wand, A.J. (1987) Biochemistry 26, 5953-5958. A brief description of assignment strategies.

Vuister, G.W., Boelens, Padilla, A. Kleywegt, G.J. & Kaptein, R. Biochemistry 29, 1829-1839, "Assignments Strategies in Homonuclear Three-Dimensional 1H NMR Spectra of Proteins".

Wüthrich, K. (1986) NMR of Proteins and Nucleic Acids, John Wiley & Sons, New York. More detailed but older review.

C. Calculation of protein & DNA structure including distance geometry, restrained molecular dynamics and energy minimization algorithms

Wüthrich, K. (1986) NMR of Proteins and Nucleic Acids, John Wiley & Sons, New York. A detailed-but-readable review.

Kaptein, R., Boelens, R., Scheek, R.M. & van Gunsteren, W.F. (1988) Biochemis t ry 27, 5389-5395. A concise (i.e., short) review.

Clore, G.M. & Gronenborn, A.M. (1987) Protein Engineering 1, 275-288.An excellent, more detailed review.

FST 210 G .M. Smi th Winter, 1999

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III. A Few Principles of NMR SpectroscopyA. Spin

Remember the Aufbau prinzip for atoms?Place electrons one-by-one in hydrogen-like orbitals...Nuclei can be described in a similar way -

Nuclear Shell TheoryNucleons (protons or neutrons) are placed in nuclear shells (like orbitals) that

have an angular momentum associated with them, usually 1/2 or 3/2.

Unfilled shells contribute the momentum of the shell times the number ofunpaired nucleons. Filled shells contribute nothing.

1 s

2p

3d

2 s

4 f

3p

5g

I

3/2 5/2

1/2 3/2

3/2

1/2

Nuclear Shells

1. Nucleons are added to the lowes t unfilled shell.2. Paired nucleons contribute no spin.3. An unpaired nucleon contributes the spin of its level.4. Protons and neutrons occupy ind ependent, identical levels.

# nucleons

4

1/2 25/2 6

42

7/2 8

64

9/2 1/2 2

7/2

2

1 0

8

Protons and Neutrons have separate-but identical shell structures and are dealtwith separately. If an atom has an odd number of protons or an odd number ofneutrons, it has a net angular momentum. The "spin" of the nucleus is the sum ofcontributions from proton levels and the neutron levels.

1H has a spin of 1/2 it has 1 proton2H has a spin of 1 both the proton and the neutron contribute3H has a spin of 1/2 the two neutrons fill the shell


3

Although nuclei are quantum mechanical particles and don't behave likemacroscopic systems, you can imagine that angular momentum behaves like "spin" i nthe following way:

When charged particles move, they create a magnetic field. An example is a nelectric current:

< < < < < < < <

> > > > > > > >electronsf low ingthru wire

>

>electrons

Magnetic fieldaround wire

>

What if the wire is curved to form a loop? And, what if the loop is very small?

electrons

f i e l d>

>

Magnetic field around a loopof electric current

Nuclei are positively charged. What happens as they "spin"? A spinning chargelooks just like a loop with a very small radius.

+

+

+

+

++

+

+

+

+ >

>

>

<<>

>

spinfie

ldfie

ld


4

The spinning charge creates a magnetic dipole. Nuclei with non-zero spin have amagnetic moment; the moment is a dipole for spin-1/2 particles, so it makes sense to saythat nuclei "spin".

+

+

+

+

++

+

+

+

+ >

spin

B. Spins in a Magnetic Field

We'll talk only about nuclei with spins of 1/2, although it is possible to do NMR o nany nucleus with non-zero spin.Common spin-1/2 nuclei are 1H, 3H, 13C, 15N, 19F & 31P

1. Quantum mechanical consequences of spin angular momentum:

a. Orientat ionIn a field with which they interact, angular momenta become oriented with

respect to the field. A particle has [2*S + 1] possible orientations, so S=1/2 particleshave two possible orientations, one with and one against the field.

The spin dipole vector does not point exactly with and against the field, but ist ipped away from the field direction.

54.7o

H z

The magnetic field exerts a force on the dipole (-µµµµ x H ), which causes the spin vectors t oprecess around the direction of the field.

H

The spin precesses at an angular frequency (radians per second) ofω = γH


5

w h e r e H is the strength of the field and γ is a constant called the magnetogyric ratio,which differs for each nucleus. Since a "cycle" is 2π radians, this frequency in cyclesper second (Hz) is

ν = γH/2π

This frequency is generally between 10MHz and 800MHz, although it depends on thestrength of the field and on γ.

When several spins see a magnetic field, they orient themselves and precessaround the field direction. We'll forget that the spins may be located on d i f f e r e n tmolecules; we're only interested in their orientation and motion. So we'll draw all of thespins' dipoles at the origin of an x,y,z coordinate system with the magnetic f ieldpointing in the z-direction. As the spin-1/2 particles precess, they sweep out two cones:

H

Field direction andindividual spins

Summation of spinvectors into a resultant.

z

You can picture the sum of all spin vectors as a single vector. It will point in the samedirection as the lower-energy states (we'll call it "up"), because there are more of these(see below), and it will point along the direction of the magnetic field, because theindividual spins are randomly distributed around the cone and their x,y-componentscancel out. This magnetization is macroscopic, so it is not strictly quantized; it does havea specific orientation and magnitude at thermal equilibrium.

H

b. EnergyIf you know the frequency of an electromagnetic wave, you can calculate its

energy using Planck's constant, h:

E = hν = γhH/2π = γ h/ H


6

c. Boltzman Distr ibutionIf the energies of the two states are different, their populations must also be

different unless the temperature is infinite. At room temperature (which is less thaninfinite), there are more nuclei with the lower-energy orientation than with thehigher-energy orientation. Boltzman's equation gives the ratio:

N u p p e rN lower

= e-∆E/kT

So, there is a slight excess of spins in the lower energy direction and the restcancel each other (add up all the spin vectors and see). So, instead of a double cone, wecan reduce the descriptio nof spins to a single vector, the "magnetization".

C. Absorpt ion/Nutat ion

1. Response to a radiofrequency pulseSuppose we give a very strong pulse of radiofrequency energy H1 in t h e

direction perpendicular to the static field Ho. We can arrange to have this secondaryfield move around Ho in the x,y-plane at the same frequency as that at which the spinsare precessing. If we let our coordinate system rotate at this frequency also, thesecondary field would appear to us (and to the spins) to be standing still (imaginejumping on a merry-go-round so that the horses will seem to stand still). The vectorrepresenting the sum of spin vectors will precess about H just as it does about any staticfield. It will do so at an angular frequency of ω = γ H radians per second. If we leave thepulse on for t seconds, the resultant will have tipped γ H*t radians. If we adjust the timeand the power (H) so that γ H*t = π/2 radians (= 90°), the spins will have tipped by 90°.This is known as a 90° pulse. Then the resultant precesses about Ho, which is the onlyapplied field remaining. In the rotating frame we can't see precession about Ho, but thespectrometer, which is in the laboratory frame, can. It sees a group of magnetic dipolesprecessing in space, which creates a fluctuating magnetic field - a radio signal. Youcan use something like an FM radio receiver to pick up the radio signal generated bythese little magnets waving around in space. This signal is called a free induction decay(FID). To obtain a spectrum, you can perform a Fourier analysis of the FID and find thefrequencies of all the spins (see below).

2. Response to a weaker RF fieldIf the secondary field (H1 ) is weak, the precession about it is slower than

transverse relaxation (see below), so instead of being tipped, the resultant just shrinksalong the z-axis. This effect is the basis of old-fashion continuous wave (CW) NMR. InCW spectroscopy, the instrument measures the absorption of the radiofrequencyenergy, analagous to a UV/Vis spectrophotometer. Spins "flop" from down to up a n d"flip" up to down with the same probability. It is only the fact that there are more spinsin the lower energy state that leads to net absorption of energy. The process continuesuntil the numbers of up-spins and down-spins are equal. Then no further netabsorption occurs, and the spins are "saturated". At saturation, the vector representingthe sum of spin vectors shrinks to zero. Another way to obtain a spectrum would be tomeasure the amount of rf absorbed by the spins while you sweep either the field or therf frequency. Whenever ν = γH/2π, absorption occurs.

Parenthetical comment: In optical spectroscopy, the energy differences betweenlevels (hence, the populations) are much more different than for NMR, and relaxationtimes are usually shorter. Nonetheless, saturation can be observed in many compounds.


7

D. RelaxationAfter a pulse or absorption of a CW wave, the Boltzman distribution (thermal

equilibrium) is not satisfied. The spins have to give off energy in order to return to theequilibrium value. This process is called relaxation.

The energy is dissipated by passing it to other spins (the "lattice") and it occurswith a rate constant of 1/T1. T1 is called the longitudinal relaxation time or spin latticerelaxation time.

Also, you may note that, after a pulse, the spins are not randomly distributedaround their cones. Several processes, including those that contribute to T1, c a u s erandomization of orientation, so that the individual spins fan out in the x-y plane. Thisprocess is called transverse relaxation or spin-spin relaxation and its time constant iscalled T2.

It is important to note that the fact that the spins redistribute themselves aroundthe cone means that they must have slightly different precession frequencies. So, in aplot of absorption versus frequency, the resonances are not spikes. They are broadishlines. Each line is a distribution of frequencies. If the frequencies differ a lot, the linesare very broad. This also means that T2 must be short. So, T2 is related to line width. Infact, the width (in Hz at half height) = 1/πT 2.

There are about five distinctly different mechanisms that cause spin relaxation,all of which depend on the motion of the molecule in which the spins reside.

In genera l, molecules that

tumble rapidly have long T1's & long T2's (10-12s ) ( i nfact, usually T1 = T2); l ong T 2means narrow lines.tumble slowly have shorter T1's and T2's; (10-9-10-8s)

T2 ≤ T1tumble very slowly have long T1's but short T2's (protein l i n e sare very broad. complexes, ge ls ,sol ids)

What is the difference between the CW experiment and the Pulse NMR exper iment?It has to do with T2. If the precession of the resultant about the oscillating frequencyoccurs rapidly, such as during a high-power pulse, then the resultant tips into the x,y-plane. If the precession occurs slowly, T2 processes keep randomizing the spins aroundthe cone, so the resultant never tips, it just shrinks along the z-axis. It's the samephenomenon, but whether the precession of the resultant is faster or slower than T2determines whether you get tipping or saturation.

E. Spectral Interpretat ion

1. Chemical shifts.What makes NMR useful is that not all protons resonate at exactly the same

combination of field and frequency, nor do all carbons... We have said that they precessat a frequency of γH, where H is the applied field. Of course, they actually precess aboutthe to ta l field that they experience. The strong applied magnetic field i nduces magneticfields in molecules. So, the nuclei precess about the local, effective magnetic field Heff† ,not just the applied field, Ho. But, the induced field is proportional to the applied field,

† Well, this is an approximation, too. Nearly every substance (including solvents usedin making NMR samples) has a non-zero magnetic susceptibility, which means that


8

Hinduced = −σ Ho,

where sigma is the sh ie ld ing parameter; the negative sign is by convention. So the total(effective) field is

Heff = Hinduced + Ho = −σHo + Ho = (1− σ)Ho

The actual precession frequency is therefore

ω = γ (1− σ)Ho

The effect is called "chemical shift", and, since the absolute size of the shift σH o d e p e n d s

of the field strength H o, it is usually expressed as a "fraction", σH o Ho

, which is in the parts-

per-million range. So the chemical shift in ppm is the same at any field strength, but infrequency units (Hz), it's not. The chemical shift between two resonances is thereforethe difference between their σ ' s .

The shielding comes from two sources:

a. electronic shieldingCurrents are induced in electrons surrounding the nucleus and participating in

bonds with it. These currents give rise to an induced field (− σH o) and alter the chemicalshift. This effect depends strongly on the hybridization of the atom and the p a r t i a lpopulation of antibonding orbitals, etc. Hydrogen only bonds with its 1s orbital andnever hybridizes (its p orbitals are too high in energy and are never populated), so itschemical shift range is small (~10ppm). Carbon, nitrogen and phosphorous have muchmore variety in their electronic structures and have chemical shift ranges of hundredsof ppm. (However, most phosphorous compounds in biochemistry are phosphates, whichhave essentially the same structure, so the range of shifts actually observed is only ~30p p m ) .

b. other local fieldsAromatic rings and unpaired electrons (e.g., on metal ions) are sources of strong

induced fields. The actual strength of the interaction depends on the distance betweenthe nucleus and the ring or radical and the precise orientation. All nuclei (isotopes) areaffected similarly by these "ring current" and "pseudocontact" interactions. Anunpaired electron that actually resides on the nucleus under observation is said tocouple to the nuclear spin and causes a large "hyperfine" or "contact" shift.

Proton spectra of simple molecules that have neither unpaired electrons n o raromatic rings can be understood in terms of electron withdrawing power of adjacentsubstituents (i.e., the electrons that participate in currents that "shield" the nucleus aredrawn away so that the nucleus is "deshielded"). In general, protons nearelectronegative atoms resonate at higher frequency (or lower field becuse ofdeshielding) than those that are bonded to less electronegative atoms. So, the NMRspectrum of ethanol dissolved in water would look something like this:

when it is placed in a magnetic field, a secondary, induced field is created. So the fieldinside the sample is different from that outside the sample. We will ignore this field,the "bulk susceptibility", by assuming that every molecule in the sample feels thesame induced field. The shielding parameter σ for a spin comes from susceptibilitieswith in the same molecule.


9

H O2 HO CH CH2 3

2. Peak areas.The area under each peak is proportional to the amount of each type of proton:

within a molecule, the area reflects the structural formula (CH2 compared to CH3;comparing two molecules, the areas represent relative concentration (2H's of watercompared to the CH2 of ethanol).

3. Spin-spin spl i t t ing.A phenomenon called spin coupling also provides structural information. The CH3

group of ethanol is next to two protons, either of which could be spin up or spin d o w n( ms = +/- 1/2). If one of the adjacent CH2 protons is spin up, the energy of the CH3protons is different from what it would be if the CH2 proton were spin down. Since theC H 2 has two spins, they could be arranged in four ways: both up, both down, first up-second down and first down-second up. Both up and both down are the lowest andhighest energy situations; the second two arrangements are of equal energy butbetween the energies of the first two:

Energy

So the peak for the CH3 is split into three (a "triplet") because it is adjacent to twoother protons. The center member of the triplet is twice as large as the either of theouter two, because there are two combinations of adjacent spin that produce it.

The CH2 protons are next to three protons. Three protons can be arranged in e igh tways: three up, two up-one down (in three ways), two down-one up (in three ways), andthree down. Both of the middle arrangements are three times as probable as the outertwo as shown in the figure:


1 0

Energy

The OH of the ethanol exchanges on and off the molecule so rapidly that it doesn'tnotice how the adjacent protons are arranged. Each molecule it visits is different, so theeffect is just a blur.

Two spins that have the same chemical shift do not split each other, so the H2Oprotons are a singlet, the CH3 protons don't do any splitting among themselves... so t h espectrum of ethanol looks like the following sketch:

0PPM

F. Getting a spectrum from an FIDA typical 1-dimensional NMR spectrum is shown below. The experimental p ro toco l

is to wait for a brief period of time to let the spins reach equi l ibr ium, give aradiofrequency pulse to tip the spins into the x,y-plane, then measure the signal theprecessing spins produce:


1 1

Delay(constant)

Pulse

Measure signal

The free induction decay (in blue) is the sum of signals from each spin type. Thesignal from each kind of spin is a sine wave (the magnetic moment precesses, wavesaround in space, at its Larmor frequency) that decays exponentially (from T2relaxation). The FID is a plot of amplitude versus t ime. A spectrum is a plot of amplitudeversus f requency. So, what we really want to know is what frequencies and how much o feach are present in the FID.

In second quarter calculus, everyone learns that you can approximate any f unc t i onby adding up a bunch of sine and cosine waves:

f(x) = ao2 + ∑

n=1

∞( anc o s n x + bns i n n x )

If you added enough waves of the right frequency and amplitude, you can match anyfunction using this Fourier series. Suppose x is time (t) and f(t) is our collection of

exponentially-decreasing sine waves. We ought to be able to find the frequency (n = ω2π )

and amplitude (a's & b's) of each sine wave in the FID. A procedure called Fourieranalysis allows us to do this, to pick out the frequencies and their amplitudes in anyfunction, signal, vibration, etc. The procedure uses a Fourier transform, which togetherwith its inverse, transforms back and forth between time domain (f, the FID) andfrequency domain (S, the spectrum). Since the trigonometric sine and cosine functionsare equivalent to the exponential (via the Euler-Lagrange equation), the Fouriertransform comes in two forms, a complex exponential form

S(ω) = 1

2√ π ∫-∞

+∞f ( t ) e- iω t d t

and a pair of sine and cosine transforms

S(ω)c = ∫0

πf(t) [cos ωt] dt and S(ω)s = ∫

0

πf ( t ) [ s in ω t ] d t

Either form works and both give two "parts" which are 90° out of phase with one another(like a sine and cosine). One is usually called the "real" and the other, the " i m a gi n a r y "part. We use only the real part (the cosine transform), but both are always calculated.Actually, the forms given above yield the magnitude of the component that has afrequency of ω. To get a complete spectrum, which is a lorentzian line centered at ωo, it isconvenient to use (ω − ωo) as the variable (see below).

If our FID were a single sine wave that went on forever, the Fourier transformwould produce a frequency-domain spectrum consisting of a δ function (a spike) at thefrequency of the sine wave, say ωo. If the FID were composed of several sine waves, the


1 2

FT would yield a series of spikes of relative height equal to the relative amplitudes of thesine waves.

But the FID does not go on forever. It decays exponentially with a rate constant of 1T2

.

The Fourier transform of an exponential decay is a lorentzian lineshape, so each spectralline would not be a spike, but a lorentzian line centered at its own chemical shift ωo:

S(ω) = T2

1 + (ω -ω o)2 T22 .

For several lines, the spectrum would be the simple sum of several equations like thatabove. Clearly the width of the line depends on T2. It is easy to show that the width at half

height measured in Hz (ω/2π) is 1

πT2.

Now that you know how the FT works, you don't need to do it anymore. You can pickout all of the important information by eye. For instance, a FID that has the equation

f(t) = So e {

- tT2

} cos

ω2π t

will give a spectrum "S" that is a lorentzian line centered at ω, with a height p ropor t iona l

to "So" and with a width (in Hz) of 1

πT2.

The phase of the line, i.e., the fact that it is a peak instead of a valley or somethingwith a positive a n d a negative lobe, is arbitrary. Only relative phases are reallymeaningful. But they are meaningful, so we'd better make provisions for phase . A moregeneral description of the wave (FID) resulting from a single set of spins including thephase is

f(t) = So e - tT2 [ ]c o s

ω2π t + b

where b expresses the relative phase of the peak.

G. Two-Dimensional NMR spectroscopyAll two-dimensional NMR experiments are done in approximately the same way.

Suppose we take a series of FID's with additional pulses and some delays. One of the delayswe'll call t2. During this delay, we'll digitize the signal, so our FID will be f (t2). We'll callthe other delay t1 and have it occur before we digitize the signal (which is why wenumbered them in this order). We'll do at least one pulse before the t2 delay to perturbthe spins from equilibrium. During the t2 delay, the spins move around (e.g., precess)according to the various interactions they experience. These interactions include theinteraction with the static field Ho, spin coupling, etc. There will also be a constant delayat the beginning to let the spins return to equilibrium before we start the cycle again.The figure shows only two values of t1, but we could obtain 256 or 512 spectra withprogressively longer values of t1.


1 3

t 2t prep t 1(evolution)

Delay(constant)

Pulse(s)Measure signal

∆ t 1

t 1(evolution)

t prep

Suppose also that, for some reason, either the amplitude So or the phase b of the signalsarising from some nuclei vary periodically with t1. (We'll see some examples of how thismight happen, in class). We could take a large number of FID's, varying that delay timesystematically. Specifically, let's take 16 or 32 scans at a fixed value of t1, save this as onespectrum, then increment t1 and take another spectrum... When we transformed the F I D ' s(with respect to t2 ), we would see that the amplitudes or phases of some peaks variedsinusoidally. How would you find the frequency of the variation? Take a Fouriertransform in the other dimension, t1. The result would be a 2D NMR spectrum, with f2(which came from transform of the t2 time domain) as one frequency dimension and f1(which came from the transform of t1) as the new dimension.

Exactly what information is contained spectrum depends on the sequence ofpulses and delays (the "pulse sequence") used to acquire the spectra. (There are alsoways to filter out peaks that you are not interested in that would otherwise complicatethe spectrum or obscure the peaks of interest.) The most common types of 2D-NMRexperiments are those that allow us to determine which peaks are J-coupled to eachother (and therefore, bonded to each other or to the same group) and those that allow usto determine which peaks arise from spins that are near one another in space (i.e., < 5Åapart). These 2D spectra can be homonuclear (both axes correspond to 1H spectra of thesample), or heteronuclear (one axis is proton shift, the other, X-nucleus shift).

The homonuclear shift correlation spectrum can be depicted as a three-dimensional surface of intensity plotted versus the chemical shift axes, as shown below.The normal one-dimensional spectrum is found along the diagonal or as the p ro jec t i onon either axis. The additional resolution and information over the one-dimensionalexperiment is obtained from the off-diagonal peaks which are present only if the twodiagonal peaks are spin coupled. The information is most easily extracted from c o n t o u rplots, such as the one shown below on the right.


1 4

Three-dimensional plot (left) and contour plot (right) of the aromatic region of aproton COSY (a "magnitude COSY") spectrum of the protein ubiquitin.

Heteronuclear shift correlation experiments give the same information exceptthat the projection on one axis is a proton spectrum and on the other axis is thespectrum of the other nucleus (element) such as 1 3C, 1 5N or 3 1P, and there is nodiagonal, since the spectra of the two nuclei have no peaks in common. Cross peaksoccur at the position where a proton is coupled to the heteronucleus. This t e c h n i q u eprovides remarkable enhancement in resolution in the spectra of both nuclei bydistributing the peaks into two frequency dimensions instead of one and by editing outpeaks for nuclei that are not coupled to the other element.

The other class of two-dimensional NMR spectra, NOESY spectra, displays crosspeaks that represent nuclei that are within 5Å of each other in the three-dimensionalstructure of the protein. They need not be bonded to one another. NOESY stands fornuclear Overhauser effect (or enhancement) spectroscopy. We'l l describe themechanism of the NOE, later.

The following table lists some 2D experiments. We'll discuss some of these in class.


1 5

Summary of 2D experiments by acronym ("SY" = "spectroscopy")

NAME Freq. Domain Dimensions Special Info Obtained

COSY Correlated… Chemical Shift (δ) cross-peaks (off-diagonal) on each axis indicate spins linked by

J -coup l ing

HSC Heteronuclear Shift CorrelationProton chemical shift " " (i.e., identifiesX-nucleus chemical shift bonded groups)

HMBC Heteronuclear MultiBond Coherence Like HSC but optimized for Same as HSC small (JXH <10Hz) coupling

cons tan ts

NOESY Nuclear Overhauser Effect… cross-peaks indicate spinseach axis, chemical shift within ~4 Å of each other.(same as COSY)

TOCSY Total Correlated… see HOHAHA

HOHAHA Homonuclear Hartman-Hahn Identifies "families" of spinschemical shift coupled to one another buton each axis not to others.

INADEQUATEIncredible Natural Abundance Double Quantum Transfer Experiment

Chemical shift on one axis Same info as COSY but more sum or difference of chemical sensitive for hetero nuclei;

shifts on the other lacks diagonal peaks.

2DJ J Spectrum Decoupled spectrum on one Multiplicity and couplingJ multiplets on the other constants of each peak

Relayed COSY (RCT)Relayed Coherence Transfer… Cross-peaks indicate two spins

Same as COSY are coupled to the same spin (e.g., NH proton and βCH

are both coupled to α C H )- -in the last five years, a whole slough of NMR pulse sequences have been devised thatallow one to obtain crosspeaks among the protons, carbons and nitrogen atoms in a spin-coupled system, provided that the protein is enriched in 15N and 13C. These are mostlythree- and four-dimensional experiments; only two dimensions are analyzed at a time.These experiments are named according to the nuclei through the information (the"coherence") is transferred. A popular one is HNCA. If a nucleus is involved in thepathway but does not produce a signal, its letter appears in parentheses.


1 6

H. Making Assignments to the Protein SpectrumThis is the hard part. In general, COSY and HOHAHA experiments are used to

identify which resonances belong to the same amino acid and to what kind of amino acidthey belong. These spectra may be taken in 2H 2O, so that exchangeable protons (-NH and-OH) become substituted with 2H and simplify the spectrum. Because Cγ is not protonated,the aromatic amino acids present a problem, usually solved by NOESY spectra. Sequentialassignments can be obtained over short stretches using the COSY experiment to identifyCα H-NH pairs in the same residue and the NOESY experiment to identify Cα H-NH pairs inadjacent residues. These experiments must be performed in H2O so that the NH protonsare not lost. This method works best in α -helical stretches because of the proximity ofthe CH and NH protons. A substantial fraction of the proton resonances can be identifiedusing these methods alone. There are also homonuclear 3D-NMR experiments that can b eemployed if assignments are unusually difficult to obtain.

A substantial improvement in ease of assignment can be made if the protein canbe labeled uniformly with 1 5N, so that all nitrogens in the protein are be NMR-activenuclei; heteronuclear shift correlation experiments can then be employed. Theseexperiments, similar in concept to COSY spectra, allow the pairing of nitrogenresonances to their own NH proton and, in separate experiments, to the Cα H of the sameresidue. The heteronuclear multiple bond correlation experiment accomplishes the s a m eresult. The "relay" experiments in which the connectivity is passed through thenitrogen to the attached proton via the nitrogen's nuclear spin are used to e s t a b l i s hchemical connectivities even where the conformation is unfavorable for direct NOESYspectra. In addition, the chemical shift of the 1 5N can be used as a third dimension toseparate proton resonances in crowded regions of the spectrum (e.g., the Cα H region).

I. Dihedral angles from coupling constantsOne of the prime movers in molecular dynamics has been Martin Karplus. Before

he began doing MD, he studied hyperfine coupling and gave us "the Karplus Relation",which says that the coupling constant J between vicinal protons depends on thedihedral angle between them. A pictorial representation of the relation is shown below:

4

5

6

7

8

9

10

Dihedral Angle

0 9 0 1 8 0

J

Variation of J with Dihedral Angle3H - H

The exact value of J depends on other things, so a particular system must be calibrated.But, the dihedral angle can be estimated from the value of J. The cross-peaks of phase-sensitive COSY experiments show symmetrical multiplet structure and J cam be estimateddirectly from the spectra. These peaks are antiphase and tend to cancel each other out i n


1 7

the middle so that J appears to be artificially large. J can be calculated more accuratelyby simulation.

J. Internuclear distances from NOE's

Spins that are near each other in space, but not J-coupled still influence eachother: they are dipoles and are dipolar-coupled. The dipolar coupling averages to zero insolution, but the strength of the coupling fluctuates as the molecule tumbles. Thefluctuation occurs because the dipoles try to remain oriented with or against the field,but the angle θ between the field direction and the vector connecting the dipoles mustchange as the molecule tumbles. The strength of the interaction is proportional to (1-3 c o s2θ ), so as θ varies, the field varies. This variation in the dipolar interaction is afluctuating magnetic field (like an rf pulse) and induces transitions if it occurs at atransition frequency. These transitions are the relaxation processes that lead to T1 andT2.

Apart from relaxation, dipolar coupling means that spin populations (i.e.,up/down ratios) are influenced by the populations of neighboring spins. Saturating aresonance (equalizing the number of spin-up and spin-down nuclei by irradiation)perturbs the system from the Boltzman distribution. The relaxation processes work toachieve a new steady state, which means that the spin populations of spins that you didnot irradiate also change. Therefore irradiation of one transition can alter the intensityof another peak. This effect is called the Nuclear Overhauser Effect, or NOE. It is the basisfor the NOESY spectrum.

One of the relaxation processes is the "cross relaxation" or "flip-flop" mechanism inwhich one spin goes from up to down and its neighbor goes from down to up. This isapproximately an energy-conserved transition, because the energy emitted by one spinis absorbed by the other. This process dominates dipolar relaxation in macromolecules.The rate of buildup of an NOE is therefore a direct measure of the cross-relaxation rate σ .

HH

r ?

Internuclear Distances - the NOE

proton Aproton B

macromolecule

proton Bproton A(This figure is described more fully in the lecxture)

Since the dipolar effect depends on distance, so does the rate of energy transfer. In fact,the relation holds that

σ ij ∝ τc r ij - 6


1 8

where σ i j is the cross relaxation rate between a pair of nearby protons separated bydistance r , and τ c is the correlation time for isotropic tumbling. If you know thedistance between a pair of protons, you can calibrate the system.

r i j = rcal (σc a l

σ i j)

16

The buildup rate of many NOE's can be measured at the same time by taking a NOESYspectrum with a short "mixing time". The idea is that you can measure the rate of a firstorder reaction by measuring a single time-point, providing that the time point is on thelinear portion of the reaction. Also, secondary processes can confuse matters if themixing time is too long. Note that you calculate the distance by taking the sixth root. Thisstep truncates errors; a factor-of-two error produces only a 12% error in the answer.

A NOESY spectrum taken at short (e.g., 50 ms) mixing time constitutes a matrix ofpairwise internuclear distances.

K. Secondary Structure from NMR Spectra.The effects of secondary structure can be inferred from NMR spectra at least four

ways.1) Since the formation of hydrogen bonds is involved in stabilizing both α -he l i x

and β -sheet, some indication of the presence of secondary structure can be obtainedfrom the measurement of the rate of deuterium exchange of the amide N-H protons.

2) The chemical shifts imposed by the presence of sheet or helix are slightlydifferent (H-bonding in general causes a downfield shift, but the effects of sheet andhelix are somewhat different). People often plot the difference between the observedchemical shift of a spin (e.g., Cα H) and the chemical shift of the same spin in anunstructured state versus residue number. A run of positive peaks in this plotcorresponds to β -structure, and a run of negative peaks corresponds to α -s t ructure.

3) In terms of the Ramachandran diagram, the difference between helix and sheetis in the value of the torsion angle ψ , which turns out to be harder to measure than thevalue of φ. Luckily, the value of φ for an α -helix is smaller than that for a parallel β -sheet, which is a bit smaller than that for an antiparallel β -sheet, and these can bemeasured from the values of coupling constants betrween Cα -H and N-H protons (fromthe Karplus relation). The following able can be used:

2° S t r u c t u r e φ (°) 3JCH-NH (Hz)α -he l i x -57 3.9parallel β - shee t -119 9.7antiparallel β - shee t -139 8.9

The value of these coupling constants can be calculated from phase-sensitive COSYspec t ra .

4) The periodicity of the secondary structure leads to different NOE patterns. In anα -helix, the Cα H of a residue (residue i) and the NH of the third residue (i+3) along thehelix are close to one another and give a strong NOE or NOESY crosspeak. The proximityof the Cα H to the NH of the next residue (i+1) is fairly large, and the NOE is small. In a β-sheet, the Cα H proton is close to the NH proton of the very next residue (i+1).Consequently, if a run of several residues showing strong i to i+3 "connectivity" andperhaps weak i to i+1 connectivity ( and 3JCα H-NH = ~4Hz), it is fairly certain that thoseresidues exist in an α -helix. If the connectivity is i to i+1 and coupling constants are 9-10Hz, the structure is β .


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l. Calculation of a Solution Structure from NMR Measurements and CovalentS t r u c t u r e .Two or three methods for deducing the structure of a protein from internucleardistances exist. They are called distance geometry, molecular dynamics and simulateda n n e a l i n g .Distance Geometry (programs DISGEO & DISMAN)

Input: upper and lower bound matrices U and Lwhere elements uij and lij come from

1. covalent structures (bond lengths, angle constraints)2. NOE-determined constraints3. dihedral angles from J coupling

What it does:Set up a matrix D s.t. lij ≤ dij ≤ uij .An "embedding routine" finds a "structure" consistent with D .

Output: A 3D structure (cartesian or φ/ψ coordinates)

Distance Bounds Driven Molecular Dynamics"Molecular dynamics" is a computer simulation system for modeling how proteins moveabout an energy-minimum conformation. It can also be used to adjust a structure to fitthe NMR-derived distances. In this case, it would be a distance driven, not energy driven"motion". You could start with an arbitrary structure, or use one derived from distancegeomet ry .

Input: U , L & D from DG.

What it does:Adjusts structure by pre tend ing that there is a physical force holding the distances within the distance limits.

Fi = mir̈ ,letting Fi vary with displacement from some position

Fi = ∂V/∂rib u t

V = K

∑dij>uij

( d i j

2 - ui j2 )2 + ∑

lij>dij( l i j

2 - di j2)2

Ref inement (Energy Minimization, Restrained MD)

Purpose:Simultaneous optimization with resp. to

1 ) potential energy and2 ) e x p e r i m e n t

Method: Same as MD, but use theoretical potential energies and fictitious NMR-determined "pseudopotentials"

V = Vbond + Vangle + Vdihedral + VvdW + Vcoulomb + Vdc


2 0

usually the distance constraints are expressed as

Vdc = 12 Kdc (dij -di j(NMR) )2

Often REM is done first, then Vdc is inserted.

Notes: In the calculations,there is no solvent.there are no ions present (Hofmeister).

What happens to salt bridges?What charge-charge repulsions are overemphasized?How do these problems differ from those encountered by

c r ys ta l l og raphe rs?Can NMR structures be used as trial structures to determine the phase angle for X-ray crystallography?


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SUMMARY: NMR Studies of Protein StructuresTake-Home Lessons

1. In a strong static magnetic field, the magnetic field associated with aradiofrequency pulse can cause nuclear spins to precess around the static field.The precessing spins create a radiofrequency signal.

2. The Fourier transform of this signal yields an NMR spectrum: it convertsamplitude as a function of time (i.e., a sine wave) to amplitude as a function offrequency (i.e., reports the amplitude of each sine wave in the signal).

3. A series of spectra taken after incremented delays (after a pulse or series ofpulses) can be Fourier transformed with respect to the delay time to yield a secondfrequency dimension - a 2D NMR spectrum.

4a. Off-diagonal peaks of a COSY spectrum tell which peaks arise from nuclei that arecoupled to one another. Coupling requires the existence of three or fewer bondsbetween the nuclei, such as H-C-C-H.

4b. Off-diagonal peaks of a NOESY spectrum tell which peaks arise from nuclei thatare within ~4.5Å of one another.

5. Coupling constants can be used to deduce dihedral bond angles and NOE's can beused to obtain internuclear distances.

6. Secondary structure can be detected by measuring nuclear Overhauser effects,coupling constants and chemical shifts. Isotope exchange rates can be used toconfirm the presence of secondary structure.

7. Structures can be calculated from dihedral angles and internuclear distancesusing algorithms such as Distance Geometry, Distance-Bounds-Driven MolecularDynamics and Simulated Annealing.

8. Denaturation/renaturation experiments in conjunction with isotope exchangeexperiments can give information about the sequence of events in proteinfolding or unfolding.

determination of protein structure using nuclear magnetic...

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