Pocket Guide to Biomolecular NMR

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Your rating has been recorded. Write a review Rate this item: Preview this item Preview this item. Pocket guide to biomolecular NMR Author: Heidelberg ; New York: English View all editions and formats Summary: Steering clear of quantum mechanics and product operators, "Pocket Guide to Biomolecular NMR" uses intuitive, concrete analogies to explain the theory required to understand NMR studies on the structure and dynamics of biological macromolecules.

For example, instead of explaining nuclear spin with angular momentum equations or Hamiltonians, the books describes nuclei as "bells" in a choir, ringing at specific frequencies depending on the atom type and their surrounding electromagnetic environment. This simple bell analogy, which is employed throughout the book, has never been used to explain NMR and makes it surprisingly easy to learn complex, bewildering NMR concepts, such as dipole-dipole coupling and CPMG pulse sequences. In this case, when you hit the nitrogen, the spring will oscillate quickly, and the energy moves back-and-forth between the atoms at a high frequency.

Notice that the units are frequency—Hertz. Because the J-coupling constant also known as the scalar coupling constant tells us how fast energy oscillates between the two connected atoms, just 2. If one nucleus is ringing strongly, the other nucleus is hardly affected. Although they are not directly bonded, they are connected via three bonds: This is called a three-bond coupling denoted by a 3 J coupling constant. Before moving on, we want to emphasize one key point about Jcoupling: It involves energy or magnetization transfer between two nuclei via the electrons in a covalent bond s.

Therefore, J-coupling occurs only between nuclei that are connected by a small number of covalent bonds. If the connection involves more than three bonds, the J-coupling constant is negligible, and the atoms are, for all practical purposes, not J-coupled. Karplus Equation The Karplus equation describes the relationship between the dihedral angle and the vicinal coupling constant.

This was first applied to the configurations of ethane derivatives but has been expanded for use in a number of systems, including peptides. Two-Dimensional Spectra Okay, when two atoms are connected by a covalent bond, energy or more specifically magnetization transfers between the two nuclei via the bonded electrons.

What good is that? For starters, we use this 28 2 Bonded Bells and Two-Dimensional Spectra phenomenon to begin matching up specific atoms in the molecule with the ringing frequencies found in the spectra. To characterize molecular structures and dynamics by NMR, the first step is to determine the ringing frequency or chemical shift for each atom in the protein. This is called chemical shift assignments, and it can be quite tricky.

If you are interested only in the biophysical properties of the backbone atoms of a protein, then you need only to determine the chemical shifts for the backbone atoms.

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But if you want an NMR image of the entire protein, you need to assign the ringing frequency for almost every atom in the protein, which is easily over a atoms. For chemical shift assignments of proteins, NMR spectroscopists almost invariably start with the backbone amide protons 1 HN —the hydrogens attached to the nitrogens in the peptide bond Fig. Notice the limits of the x-axis: This is the frequency range for virtually all amide protons in all proteins.

In general, each residue of calmodulin with the exception of the N-terminal residue and prolines has one amide proton, so there is approximately one peak for each residue of the protein. To do this, we must spread the peaks out. In other words, we need to increase the resolution of the spectrum. How could we do this?

Pocket Guide To Biomolecular Nmr

For starters, we could make the peaks narrower i. Another option is to run the experiment at a stronger magnetic field—say, for example, MHz versus MHz. Why would that 2. Two-Dimensional Spectra 29 Fig. See the end of Chap. In the end, the trick that works best is to add another dimension. Why does this increase our resolution? Think about driving to the mountains. When you are far away on level terrain, all the peaks flatten into one plane like a vertical pancake Fig. The problem is that you can see vertically, but not horizontally.

Now think about looking at these mountains from an airplane far above: Adding the extra dimension separates and spreads the peaks out. In NMR, we add extra dimensions by transferring magnetization between coupled atoms, such as J-coupled nuclei. Two-Dimensional Spectra 31 Fig. Ring the nitrogen atom. Record its electromagnetic signal. Let the nitrogen ring the attached hydrogen via J-coupling. Record the electromagnetic signal of the hydrogen. The result is a two-dimensional map of the NMR spectrum Fig. The ringing frequencies or chemical shifts of the amide hydrogens are along the xaxis and the ringing frequencies for the attached nitrogens are along the y-axis Fig.

With the extra elbow-room of an additional dimension, many peaks are now completely separated and distinct, making the spectrum easy to analyze and significantly more useful compare Fig. You can read this twodimensional spectrum in the same manner that you read a map on a longitude—latitude grid. Say your favorite place to ski is Mount Rose. How would you find Mount Rose on a map if you know it is located at Mount Rose is where the two lines intersect.

Analogously, you can find the cross-peak for threonine on the two-dimensional NMR map in Fig. Say the amide hydrogen for threonine rings at 8. To find the cross-peak of threonine in the two-dimensional spectrum, simply draw a vertical line through 8. Like Mount Rose, threonine is where the two lines intersect. Since this peak is now distinct from all the others, we can use it as a probe to characterize the structure and dynamics of residue 29 in calmodulin.

You can probably see now how this two-dimensional spectrum Fig. In this case, we would need to add another dimension to spread these peaks out further. Two-Dimensional Spectra Mathematical Sidebar 2. To answer this question, we need to learn more about why atoms ring in the first place. Atoms carry a few intrinsic properties that dictate how they respond to forces in nature. First, atoms possess mass. The more mass an atom carries, the more it responds to gravitational fields.

Atoms also have a charge: The charge on the atom tells us how the atom responds in an electric field. Sometimes we can even feel the charge of atoms. The third property of atoms is spin. The spin tells us how an atom responds to a magnetic field. Like charge, spin comes in multiple flavors: For example, 12 C has a spin of zero. Nuclei with even mass numbers and odd number of protons have integral spin numbers. For example, 14 N has a spin of one. Nuclei with odd mass numbers have half-integral spins: Thus, all those 12 C atoms in proteins are not very useful for structure determination by NMR.

Thus, 14 N atoms in proteins are also not useful for NMR studies. To learn why 2. Read them, understand them, remember them: The x-axis gives the ringing frequency or chemical shift of each protons attached to a nitrogen atom. The y-axis gives the ringing frequency of each nitrogen atom attached to a proton. A cross-peak appears in the center of the spectrum wherever a hydrogen is bonded to the nitrogen. The total number of peaks is approximately equal to the total number of residues in the protein: Peaks should be relatively well separated with minimal overlap Fig.

Otherwise 2 Specifically, the number of peaks in a 1 H—15 N HSQC spectrum of a protein is given by the number of residues plus two times the number of glutamines and asparagines minus the number of prolines minus one for the N-terminal residue, which has a rapidly exchanging NH3 group instead of an NH group. As this list demonstrates, the 1 H—15 N HSQC provides substantial information about the state of the protein even before we assign the chemical shifts to specific atoms.

In fact, crystallographers and other biophysicists often use the 1 H—15 N HSQC to check if their proteins are folded and well-behaved before they set up crystallization screens or perform other time-consuming experiments. The sidechains of glutamine and asparagine residues have amide groups with two hydrogens attached to one nitrogen.

With this knowledge, which peaks in Fig. Explain why you have selected these peaks. Creating Two-Dimensional Spectra 37 2. Do you ever listen to talk radio or sport broadcasts on AM radio? Although its super old school, AM radio has an endearing quality that takes you back in time. How does AM radio work? In a nutshell, the AM radio in your car converts an electromagnetic wave into sound waves through your speakers Fig. Each AM radio station in an area is assigned a specific frequency at which to broadcast its signal.

On the other hand, WTEM, which plays local news and talk shows, broadcasts its signal on an electromagnetic wave with a frequency of kHz Fig. How does each station tell your car radio to play the right note? Creating Two-Dimensional Spectra 39 versus 2. You can easily see these hidden notes in Fig. The resulting sine curves have the exact frequencies of the sound waves played by the radio stations.

This is no coincidence! And, we run the experiment multiple times. In a two-dimensional NMR experiment, we start the nitrogen atom ringing; listen to its signal for a brief period; and then let it transfer magnetization to the bonded hydrogen via J-coupling Fig. For example, during the first experiment, we let the nitrogen ring for the exact time that it takes for its electromagnetic wave to make one full cycle.

In the second experiment, we let the nitrogen ring for a shorter amount of time, so that its signal is closer to zero. This time the amount of energy transferred to the hydrogen is significantly less than in the first experiment experiment 2 in Fig. In the third experiment, we let the nitrogen ring so that its signal is almost zero when it transfers energy to the hydrogen. If we reduce the time period that the nitrogen rings even further experiment 4 in Fig. The length of time that the nitrogen rings is called the t1 evolution period.

As we gradually shorten the t1 period, you can see on the righthand side of Fig. In other words, the hydrogen is broadcasting the chemical shift of its neighboring nitrogen by amplitude modulation, similar to an AM radio station. First, we perform a Fourier Transform on the hydrogen signal in each experiment. Notice that the resulting peak is at the same frequency in all experiments the hydrogen frequency but the height of the peaks varies Fig. The curve created by these oscillating peaks encodes the frequency of the directly bonded nitrogen.

To get this frequency, we perform the second Fourier Transform on the curve created by the oscillating hydrogen peak Fig. A portly soprano belts out a high note at the climax of an opera and BAM! Crystal chandeliers explode and champagne flutes shatter. Is this an operatic myth, or it is physically possible for the human voice to shatter glass?

They hired Jamie Vendura, a voice coach and rock star, to attempt breaking a crystal wine glass. Although it took more than 20 attempts, Vendura could indeed shatter crystal by singing a note with a frequency matching the vibrational frequency of the glass Fig. Further, when his voice was amplified through a speaker, the glass broke immediately every time.

When the frequency of the wave is near the natural frequency at which the glass vibrates, energy builds up and eventually cracks the glass Fig. We then use these distance measurements between many pairs 43 M. Dipole-Dipole Coupling In the previous chapter, we learned that a ringing nucleus transfers its excess energy to nearby atoms that are connected by one or at most a few up to three covalent bonds. This useful phenomenon, called J-coupling, can be used to tell us which atoms are bonded to each other and helps us determine the chemical shift or ringing frequency for those atoms.

Therefore, to obtain a high-resolution image of a protein, such as calmodulin, we need a tool for measuring the distance between atoms that are not attached by bonds. So, hydrogen atoms need to be quite close to share magnetization. This is because the energy or intensity of our voice spreads out and weakens as you move away from the energy source. If you stand two feet from someone, their voice is four times softer than if you are only one feet from them Fig.

Thus, the closer Vendura gets to the glass, the more energy transfers from his larynx to the molecules in the wine glass, and the faster he can break the glass. Energy transfer via dipole-dipole coupling also depends greatly on the distance between the two atoms. The closer two atoms are in space, the more likely the atoms will exchange magnetization by dipoledipole coupling. If sound waves behaved this way, then standing two foot from someone would be 64 times softer than standing one feet from them Fig.

We would need a hearing aid at even the loudest rock concerts. Thus, energy transfer by dipole-dipole coupling is much more sensitive to distance than sound transfer—the chance of a ringing atom sharing energy with its neighbor drops off very quickly as you move away from the high-energy atom Fig. Although dipole-dipole coupling and sound have similar distance dependencies, the two energy transfer mechanisms are significantly different. For starters, if you rub your finger around the rim of crystal wine glass, the glass hums at almost a single frequency.

Does this same idea hold true for dipole-dipole coupling between atoms? However, if we ring the HN Fig. The second major difference between energy exchange by dipoledipole coupling and breaking glass with your voice is the time it takes to transfer the energy. When Vendura begins singing, it takes time for the sound waves to travel through the air and hit the atoms in the wine glass. This time is very small, less than a millisecond, but there is a moment where you can detect the sounds waves before they start rattling the atoms.

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In contrast, energy transfer by dipole-dipole coupling is nonradiative. The transfer is instantaneous, and there is no detectable electromagnetic wave between the two atoms. Or, think of two billiard balls colliding on a pool table. When the moving ball hits a still ball, almost all the kinetic energy is immediately transferred to the second ball—the same is true for two atoms in dipole-dipole coupling.

This distance dependence makes dipole-dipole coupling one of the most powerful techniques scientists have for measuring how far apart two atoms are in space. Dipole-Dipole Coupling Dipolar relaxation occurs between two spins: This proportionality also explains why dipole-dipole coupling is much more efficient for protons than for 13 C or 15 N atoms, which have much smaller gyromagnetic ratios than protons.

Overhauser presented his unique idea of energy transfer to the American Physics Society in , the top physicists in the crowd, such as Felix Bloch Physics Nobel Prize , Edward M. Ramsey Physics Noble Prize were taken aback and quite skeptical. Ramsey even wrote Dr. Overhauser a letter a few months after the meeting Overhauser, You may recall that at the Washington Meeting of the Physical Society, when you presented your paper on nuclear alignment, Bloch, Rabi, Purcell, and myself all said that we found it difficult to believe your conclusions and suspected that some fundamental fallacy would turn up in your argument.

Subsequent to my coming to Brookhaven from Harvard for the summer, I have had occasion to see the manuscript of your paper. After considerable effort in trying to find the fallacy in your argument, I finally concluded that there was no fundamental fallacy to be found. Indeed, my feeling is that this provides a most intriguing and interesting technique.

Overhauser predicted this extraordinary energy transfer mechanism, physicists went hunting for it. Sixty years later, the NOE, or energy transfer by dipole-dipole coupling, is the most important technique for determining the distance between protons in solution. Still today, the NOE provides the mainstay for three-dimensional structure determination of proteins in solution.

You can think about the NOE as a measuring stick Fig. Second, in practice this NOE stick is best interpreted in terms of broad ranges: In other words, distance measurements by the NOE are typically only semiquantitative; it usually gives us only a rough estimate of the maximum distance between two protons. If we start ringing the HN hydrogen Fig. What would we see? Before we answer that, we need to learn one more idea about NMR spectroscopy of big, chunky molecules, such as proteins and oligonucleotides. Atoms ring incredibly softly, or more accurately the electromagnetic wave they create is extremely weak.

The only way we can detect their ringing is to have a huge quantity of atoms all ringing together. Most NMR samples contain about molecules hundreds of micromolar in 0. When we ring the HN hydrogen in the valine— threonine peptide, all HN atoms start ringing together, like a bell choir with millions of tiny bells. What happens during this delay? Most HN atoms in the sample will keep all their excess energy and continue to ring. But some HN atoms will hand off their extra magnetization to a nearby hydrogen by dipole-dipole coupling.

This extra energy allows the neighboring atom to start ringing and produce an NMR signal Fig. These nascent signals from nearby atoms create cross-peaks in the NMR spectrum. And, the relative peak height tells us approximately how close they are to the HN atom that gave them the energy. Remember that the cross-peak height depends directly on the strength of the NMR signal or how many atoms are ringing. We can use the NOE to create a similar type of two-dimensional spectrum Fig.

Ring all the amide hydrogens. Record the chemical shifts. Allow hydrogens to transfer energy to neighboring hydrogens via the NOE. Record the NMR signal of all ringing hydrogens. We read the 2D 3. Notice how the size of the cross-peaks tells us the distance between the two hydrogens.

Notice that this peak is quite large. Where does this super peak come from? In the last section we learned that ringing atoms transfer only part of their excess energy; actually most of their energy they keep for themselves during the NOE mixing time. Unfortunately, for most proteins, including average size proteins like calmodulin, the 2D-NOE spectrum is usually too crowded and overlapped to be useful for structure determination. How can we fix that? You got it—add another dimension. Two-dimensional experiments similar to these two types are all you basically need to solve the structure of small protein or DNA molecule that is less than 5—10 kDa.

But these experiments need to be stepped up a notch to be useful for larger biomolecules. Most notably, three-dimensional experiments typically start by ringing hydrogen atoms because hydrogens have the largest gyromagnetic ratio. This then requires an extra step in the experiment to record the nitrogen chemical shifts and return back to amide hydrogens before the NOE transfer.

Start ringing the nitrogen atoms and record their chemical shift. Let the amide hydrogens ring a bit and record their chemical shifts. Let the amide hydrogens transfer magnetization to nearby hydrogens via dipole-dipole coupling the NOE part. Record the NMR signals of the protons. Each side of the cube or axis gives the chemical shift for one of the three atoms recorded in the experiment: The easiest way to analyze this spectrum is to divide it up into two parts: Each cross-peak in the 2D strip represents a hydrogen near HN.

Although residue 99 is very far from residue in the sequence of calmodulin, in the threedimensional structure these two residues are quite close to each other 3. All the cross-peaks for the other amide hydrogens at 8. Actually, they are in strips located at different points along the y-axis in the HSQC plane or on different x—z planes. Thank goodness for 3D spectra! And if we need even more resolution we can simply extend the number of dimensions to four.

What about the hydrogens attached to carbon atoms? To get distance information for these protons, all we need to do is ring the carbon atoms instead of the nitrogen atoms at the beginning of the experiment Fig. And, how did we know how to label all those cross-peaks in Fig. For example, the residue calmodulin protein has 1, hydrogens, carbons, and nitrogens. How on Earth do we figure out all these chemical shifts? But for all molecules, the idea is the same. We string together different types of HSQC experiments and record the NMR signal for specific atoms as we hop through the bonds of the molecule via J-coupling.

The result is a 3D spectrum telling you which chemical shifts are connected by covalent bonds. This creates a cube of data, like Fig. But why does this 2D strip have two cross-peaks—a strong one and a weaker one at 71 and 55 ppm, respectively? In general, this is true for each residue in the protein: As you can see from Fig. What coupling constants would be used in this experiment? A Quick Review We have made it! We now have all the ideas, experiments, and tools we need to start building a three-dimensional model of a protein. A Quick Review 67 very softly. Also, the calmodulin needs to have all its 14 N atoms replaced with 15 N atoms and its 12 C atoms replaced with 13 C atoms see Mathematical Sidebar 2.

Remember you want all the cross-peaks in the 1 H—15 N HSQC to have comparable intensities and be reasonably well dispersed. In addition, the number of peaks should be approximately the same as the number of residues in the protein. Now we run 3D-heteronuclear J-coupling experiments to start determining the ringing frequency for each hydrogen, carbon, and nitrogen in calmodulin. Now we use the chemical shift table to assign the cross-peaks in the 3D 13 C- and 15 N-separated NOE experiments, like the strips shown in Fig.

Lastly we convert the NOE assignments into a list of approximate inter-proton distance restraints for the protein. For an average size protein like calmodulin, the 3D 13 C- and 15 Nseparated NOE experiments should supply 1,—2, distance restraints, such as the ones shown in Table 3. Building a Three-Dimensional Structure Now we can start building the protein! At this point the hard work is basically behind us.

After some heavy computations, poof! Out pops a bunch of three-dimensional structures of calmodulin Fig. But why are there multiple structures? Which one is right? And, how did the computer calculate these structures in the first place? There are several approaches, but one method implemented in Xplor-NIH is known as simulated annealing. This process is a bit like cooking dry spaghetti. You start off with a stiff rod of pasta and throw it into a boiling hot water Fig.

When the pasta warms ups and adsorbs the water, it loosens up and starts wiggling around. The string of pasta samples many different conformations as it tumbles around in the hot water. What happens if we repeat the experiment with a fresh strand of spaghetti? Because the initial conditions at the moment the pasta hits the water are slightly different in each case, and thus, the end result will be different. We take our protein in this case calmodulin and adjust all the backbone torsion angles to make an extended conformation Fig. For the protein, this wiggling involves rotating torsion angles in both the backbone and the sidechains.

At the beginning when the water is very hot, the algorithm allows the protein to move around randomly like a boiling noodle. Specifically, the software program ensures the following: If the protein starts folding into a configuration where one of these three requirements is violated, the software program will nudge the protein back into a conformation that preserves these properties. Then we run simulated annealing many times to fold the protein into a three-dimensional structure that satisfies the laws of protein chemistry and the NOE distant restraints. The random element arises from the fact that the initial velocities for the atoms are assigned random values at the start of the calculation.

Will the final structure look like the first one or will it form a completely different conformation like the spaghetti did on its second experiment? NOEs between atoms that are far apart in the amino acid sequence tell us what regions of the protein are near each other in the folded state.

These types NOE restraints provide key folding instructions for the software program. If we have enough of these restraints, then the algorithm will guide the protein into the same overall conformation almost every time we perform the structure calculation Fig. Therefore, we keep adding more and more distance restraints to our NOE list until almost all of the calculations create a bundle of structures that have approximately the same fold.

When the RMSD for backbone atoms is References and Further Reading 73 these additional types of restraints help improve the precision of the structure calculations. Methods Mol Biol J Mol Biol Overhauser A In: Chapter 4 Silencing of the Bells: Relaxation Theory Part One You know, what these people do is really very clever. They put little spies into the molecules and send radio signals to them, and they have to radio back what they are seeing. Say it slowly, and the word itself conjures calming images—a hammock in the sun, a cool breeze at the beach, or a mug of hot chocolate on a snowy day.

No matter how you like to wind down after a tough week or stressful semester, relaxation is about releasing pent up energy and returning to a lower energy state that you think of as normal. But to reach a state of zen takes time. Most of us need at least a few days to release pent up energy and completely relax. A great way to accelerate the relaxation process is to exercise. Go for a run, hit the gym, even go dancing! Exercise is a second type of 75 M. Relaxation Theory Part One relaxation that we humans need. Like stressed out humans, ringing nuclei also have extra energy.

And, they have multiple ways to release the excess energy, defocus their signal, and relax back to the low-energy state. No matter the route, the result is the same—the nuclei return to where they started before the NMR experiment, and our precious NMR signal vanishes. In many cases, this relaxation is a nuisance because it leaves us with less than ideal NMR data. But for large macromolecules, like proteins and nucleic acids, the relaxation is a blessing in disguise.

So turn on the Norah Jones, grab a cool beverage, and get ready to learn about how atoms kick back and relax. The amplitude of the wave slowly decreases until the signal withers into a straight line Fig. Two Paths to Relax 77 B Fig. The bells are very tiny so each one only whispers the note Fig. This note is our NMR signal Fig. So what stops the choir? What causes the note at 9. Individual bells stop ringing! Every nucleus contributes to the NMR signal Fig.

Atoms with large T1 values can play long, drawn out notes that last more than a second Fig. But atoms with short T1 values play short eighth notes that last only a fraction of a second Fig. R1 tells us how quickly the bells shut off. Okay, so the amplitude of the NMR signal decreases as individual nuclei in the sample stop ringing. That makes good sense, but can you think of another mechanism that would decrease the intensity of the electromagnetic wave?

All of these waves sum together to create the NMR signal that we detect in the spectrometer. When these waves all have exactly the same frequency and start ringing together, they amplify each other, producing one unified wave that is much stronger than an individual wave Fig. Instead of reinforcing or augmenting each other, the waves begin to cancel each other out! This is called dephasing, and it diminishes the NMR signal because the positive parts of the waves equal the negative parts of the wave and they sum to zero Fig. Hydrogens with large T2 values can hold the same name note for an extended period of time Fig.

T2 is also called the transverse relaxation time and spin-spin relaxation time Table 4. More detailed description Alternate names The time it takes for the nuclei to return their magnetic needles to the z-axis. The time it takes for NMR the signal to decay due changes in the ringing frequency dephasing 1.

R2 tells us the rate of dephasing. Ticket to the Protein Prom Relaxation is our ticket to the biomolecular dance. The decaying NMR signal possesses surprisingly rich information about the internal motion of macromolecules and provides a rare glimpse of how biomolecules function during catalysis and signaling. How does the decay in the NMR signal Fig. To answer this, we need to go back to Chap. However, the exact ringing frequency or chemical shift of a specific hydrogen depends not only on the external field but also on local magnetic fields created by the surrounding electrons and other atoms remember that a magnetic field is created by moving electrons.

For example, the HN atom in Fig. But when the 82 4 Silencing of the Bells: These electrons significantly reduce the total magnetic field felt by the HN , causing its ringing frequency to shift almost 1 ppm or Hz at As we just learned, altering the ringing frequency speeds up T2 relaxation, making our signal decay faster Fig.

So, there we have it—a connection between relaxation and the dynamics of macromolecules! To understand the full story, we need to learn more of the details of our NMR experiments, especially about what makes an atom ring in the first place. Boltzmann Distribution When a sample is placed into an NMR spectrometer the individual spins align with the magnet field.

Nuclei aligned with the field are in a lower energy state than those pointing in the opposite direction of the magnetic field. This leads to two distinct states with a difference in energy given by Fig. Note that as Bo increases, so does the energy separation. This explains why the resolution of ringing frequency increases as the strength of the magnetic field increases.

How many spins populate each state? Fortunately for us, the two states are populated unequally, and this population difference is what gives rise to the intensity of our NMR signal. That is, the greater the population difference, the stronger our signal. This explains why one can quickly collect a spectrum when we detect 1 H atoms, but we need far more scans and thus increased signal averaging to get the same intensity when we detect the ringing of 13 C atoms.

Probably not, given the abundance of cheap GPS devices now available. But back in the olden days, like 4.

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Imagine sitting on a sailboat—ocean to the left, ocean to the right, ocean in the front and behind—no direction to be found. Now place a compass in your hand. The tiny little bar magnet wobbles around, and then settles itself—north! In an NMR spectrometer, we have a super-strong magnetic field pointing toward the ceiling or along the z-axis. So when we put the protein or DNA sample in the magnet, the hydrogen nuclei line up vertically like toothpicks in a box Fig.

This is how all NMR experiments begin—hydrogen nuclei standing up straight along the z-axis in response to an external magnetic field Fig. Now what happens when we apply a magnetic field perpendicular to the z-axis, say along the x-axis?

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Like miniature compasses, the nuclei 86 4 A Silencing of the Bells: During this time, the little compasses in the nuclei try to keep up and align with the rotating magnetic field. As a result, they start spinning around the z-axis, like they are sitting on top of a record player Fig. You got it—an electromagnetic field! As the nuclei revolve around the z-axis, they create a fluctuating magnetic field.

This is how the nuclei ring and create our NMR signal! Ok, this all seems reasonable: When that field starts spinning in the xy-plane, the tiny bar magnets in the nuclei align with the revolving field and start spinning too. The original field in the z-axis is incredibly strong, one of the strongest magnets in the world.

This is where NMR becomes amazing, almost magical. A remarkable phenomenon, resonance, allows relatively weak forces to have huge effects. The key to 88 4 Silencing of the Bells: Relaxation Theory Part One Fig. Think of pushing a lb NFL lineman on a playground swing Fig. To start him swinging, you could use all your strength and apply a few tremendously powerful pushes. Or, you could be more elegant and simply give the behemoth many small pushes every time the swing returns to you Fig. Although the pushes are relatively weak, their effect accumulates over time because their timing matches the natural swinging frequency.

With resonance to assist you, the huge man is flying high without much effect at all! If we apply an oscillating electromagnetic field in the xyplane with a frequency that nearly matches this preferred spinning frequency, even a relatively weak field will eventually flip the magnets down into the xy-plane and start them spinning around the z-axis 4.

The cool part is that once we shut off this fluctuating electromagnetic field, the nuclei continue to spin at their favored frequency Fig. Now the details of the NMR experiment become clearer. That big hammer at the beginning of the experiment is actually a fluctuating electromagnetic wave Fig. And, the 90 4 Silencing of the Bells: This last part is the key idea—weak magnetic fields can a have a significant effect on nuclei when the electromagnetic field oscillates at a frequency close to their resonant frequency.

With this idea in mind, the connection between relaxation or the decrease in signal decay and molecular motion is obvious: Molecular motions produce oscillating magnetic fields that interfere or disrupt the resonance ringing in our NMR signal. Repeat the idea if you need to because it is critical for studying the dynamics of proteins.

As we just learned, each hydrogen in a protein has a tiny magnet inside its nucleus Fig. Yes, this field is weak and affects only hydrogens in its own neighborhood. However, when the motion has a frequency close to the resonance frequency of a nearby nucleus, then BAM! You guessed it—these relatively weak magnetic fields have a significant impact on the ringing of nearby nuclei. What about all those electrons whizzing around in covalent bonds? We already know that moving electrons produce magnetic fields too. So when these bonds tumble around in solution with the rest of the protein or move during structural rearrangements, they also create fluctuating magnetic fields that can affect nearby nuclei too.

Our field provides precise instructions for the spinning nuclei: All the nuclei follow the directions and rotate around the z-axis together Fig. In contrast, the neighboring nuclei and electrons create oscillating fields that sound like random gibberish in comparison to the precise commands of our the applied pulse. These random commands disrupt and counteract the orderly ringing established by our applied electromagnetic field. This causes their electromagnetic waves in the xy-plane to start canceling each other out, and our NMR signal decreases Fig.

So to understand the internal dynamics creating these local oscillating fields, we need to characterize how the NMR signal decays. This is where T1 and T2 come to help us. Each type of relaxation reports on different kinds of motion inside a molecule. Think of them as tiny newscasters stationed inside the nucleus, transmitting information about the fluctuating fields around them by diminishing the NMR signal. Relaxation Theory Part One motion Fig. T1 Have you ever played tetherball?

You hit the ball to your right, and the ball begins to circle around the pole at a specific frequency counter-clockwise. If you give the ball a nudge to the right every time it passes by you, the angle between the ball 4. When you stop hitting it, the ball falls back to the pole in response to gravity right. What happens when you stop hitting the volleyball?

It will continue to rotate around the pole because the ball has angular momentum. But the angle between the ball and the pole will slowly decrease as gravity pulls the volleyball back down to the ground Fig. The energy that you put into the system by hitting the ball is lost as the ball falls down to the pole. Relaxation Theory Part One electromagnetic wave in the xy-plane Fig. Instead, they respond to the huge magnetic force This is T1 relaxation—the nuclei releasing their extra energy as they return their little compass needles back in alignment with the large magnetic field along the z-axis see Mathematical Sidebar 4.

Why is this important? For starters, the T1 relaxation rate tells us how long we need to wait between NMR experiments. Remember that the signal we detect in an NMR experiment is a fluctuating magnetic field in the xy-plane only Fig. We just point our electromagnetic radar at the xy-plane and disregard fluctuations along the z-axis.

T1 95 NMR signal. T1 tells us how fast this occurs: Here are few other key ideas to remember about T1 relaxation: What is your T1 value—do you like to stay for a long time in a high-energy state or do you hurry home after a stressful day of work? This is also called the equilibrium magnetization Io because at equilibrium, the spins are aligned with the field. When the applied electromagnetic field is turned off, the spins will relax back to equilibrium i.

The rate of the return to the z-axis is given by a first-order rate expression: Coherence and T2 97 above. If you measure the signal right after the applied electromagnetic pulse is turned off, the intensity along the x-axis will be large. But if you wait a period time t, a subset of the nuclei will have relaxed back to equilibrium i.

Thus, by measuring the intensity I at a number of time points after the pulse and integrating the equation above, you can determine the longitudinal relaxation time T1. Coherence and T2 Okay, so T1 tells us how quickly the nuclei return their little compass needles to the z-axis, destroying our precious signal in the xy-plane. But what about T2? How does it contribute to decay in NMR signal? To answer this question, we need to learn more about a concept called coherence. But for NMR spectroscopists and other physical chemists, coherence has a totally different meaning: Think of watching a college marching band from the top bleachers of a stadium.

Although the band contains hundreds of musicians, 3 Comparing phase coherence to walking in lock-step is the idea of Nobel Prizewinning physicist Wolfgang Ketterle. He used this analogy in when he and his laboratory created one of the first Bose-Einstein condensates—a new state of matter in which atoms line up in lock-step at super low temperatures to become a coherent wave. The cohesion arises from the marching synchronization: At any moment in time, the angle between the left leg and right leg will be exactly the same for each member of the band Fig.

To maintain this lockstep marching requires two conditions: First, the band members must start marching together. Second, the musicians must walk at the same pace. A slight change in stride frequency amongst the band members will quickly scramble their stepping synchrony and diminish their fluid, cohesive motion. Electromagnetic waves can also walk in lockstep, and when they do, physicists say that the waves are coherent. Like the band members, waves must maintain the same pace or frequency. But they also must start off at the same point in the wave.

Coherence and T2 99 o A. Two waves are in phase when they begin fluctuating at the same point, like the two in Fig. Otherwise the waves are out of phase Fig. Think back to our discussion in Chap. Remember the second racer needed to skate around the inner rink at the same 4 Silencing of the Bells: Relaxation Theory Part One frequency as the first skater racing in the outer rink in order for the two teammates to connect with each other when the second skater enters the race Fig.

But matching frequencies is not enough; the skaters also need to be in phase. The same goes for waves. In contrast, when the waves start with the same phase are in phase and have identical frequencies, the waves overlap perfectly and their signals sum together, creating one wave with twice the amplitude of the individual waves Fig. Physicists say that the waves are coherent or have phase coherence. You will hear these terms often in NMR, so be sure you understand them.

Coherence is a fabulous phenomenon, and it is essential for NMR. The electromagnetic wave produced by an individual nucleus is too weak to detect—it has a teeny-tiny amplitude. Look again at the coherent waves in Fig. Their amplitudes continue to add up productively as long as the waves maintain identical frequencies. If their frequencies drift off a bit Fig.

Now when we add up the waves, the total amplitude is less than the amplitude of the individual waves. The NMR signal is vanishing! NMR spectroscopists call this process dephasing or lose of phase coherence. Eventually their signals will cancel completely, and the net signal will be zero. Coherence and T2 we collect the NMR data, the frequency of individual nuclei begin to shift and encompass a broad range of frequencies Fig. Remember the NMR spectrum tells us the ringing frequency for each atom in the molecule. So if T2 relaxation causes an atom type to shift its frequency randomly as we collect the NMR data, does this mean we get more than one peak in the NMR spectrum?

Not quite; the spreading out of frequencies makes the peak broader. In fact, the quicker the waves lose coherence, the broader the peak. In a nut-shell, the spreading out of frequencies, spreads out the NMR peak! In other words, the NMR signal appears as one sine wave with one frequency that lasts forever left, Fig. If we perform a Fourier Transform Fig. Relaxation Theory Part One on this curve, what will the spectrum look like? This particular curve has one sine wave at 9. Now imagine the other extreme. Say that each of HN hydrogens rings at one of 1, different frequencies between 9.

The Fourier Transform will find all 1, frequencies and give 1, individual peaks. However, if the frequencies are too close to each other, their peaks will blend together and produce one broad plateau that encompasses all 1, frequencies. In addition, the height of the plateau at a particular frequency will tell us how well that frequency is represented in the ensemble of waves. In this example, any frequency between 9. In reality, atomic nuclei behave as a mixture of these two extremes Figs. They start off like the first case with all HN nuclei ringing together at the same frequency, and they end up like case two with each nucleus ringing at a slightly different frequency.

It will catch all these frequencies even though they shift over time. It will find the original ringing frequency 9. The result is a stretched-out peak centered around 9. The Lorentzian shape is quite beautiful. Coherence and T2 Fig. In contrast, atoms with short T2 values atoms that quickly go out of tune will have a broad peak that is more similar to the plateau Fig.

Pocket Guide to Biomolecular NMR

Short T2 values give us fat peaks, and long T2 values give us thin peaks. At that instant, all the nuclei are in phase i. However, due to small differences in their resonance frequencies, they will immediately begin to dephase, reducing the intensity of the signal that we can observe in the xyplane. As these spins continue to spread out along the xy-plane, we see an oscillation in the intensity observed along the x-axis. The rate at which the coherence is lost is given by a first-order rate expression: In particular, slight variations in the external magnetic field along the z-axis i.

A spin-echo experiment, as described in Chap. Chapter 5 Relaxation Theory Part Two: Imagine staring at a red light two feet in front of your face. The light stays red for a few seconds and then instantly changes to green Fig. What do you see? Red and green flashing lights, of course—nothing interesting about that. Okay, now imagine that the light switches from red to green extremely fast, like once every few milliseconds.

What would you see? It takes the light about ms to travel up your optic nerve to the cerebral cortex where the photons get processed into electrical signals that represent the color red in your thoughts. Because the lights are now flashing between green and red faster than your brain can process, the signal gets blended and summed up. Instead of seeing two flashing lights, you now see only one light—a yellow light Fig.

Remember that red and green lights combine together to create yellow.