Coherent Control of Two Nuclear Spins Using the Anisotropic Hyperfine Interaction

Yingjie Zhang, Colm A. Ryan, Raymond Laflamme, and Jonathan Baugh

Phys. Rev. Lett. 107, 170503 (2011)

Published October 17, 2011 | PDF (free)

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Figure 1 (a) The magnetic dipole field produced by an electron spin. Note that the magnetic field experienced by the nuclear spin has a transverse component along the x -axis, even though the electron spin is aligned with the external field along the -axis. Therefore, flipping the electron spin would lead to changes in both the magnitude and the direction of the magnetic field experienced by the nuclear spin. This is the essence of the anisotropic hyperfine interaction (AHF). (b) A malonic acid molecule. The red spheres are oxygen nuclei, the larger dark gray spheres are carbon nuclei, and the smaller light gray spheres are protons. One proton has been knocked off by x-ray irradiation, to leave a radical electron localized at the central carbon-13 nucleus. AHF is used in Ref. [7] to couple the nuclear spins of the carbon-13 and the proton via the electron spin.

Many nuclei have a magnetic moment that arises from their intrinsic spin. In an external magnetic field, these nuclei tend to align their moment with the field, just as a compass points to the North Pole in the Earth’s magnetic field. In many material systems, the nuclear spins are extremely well isolated from their environments, so that the dynamics of an individual nuclear spin is completely quantum coherent and governed by the Schrödinger equation of nonrelativistic quantum mechanics [1]. This high degree of quantum coherence makes some of the nuclear spins enticing candidates for bits for a quantum-mechanical computer [2]. However, the fact that these nuclear spins enjoy a quantum coherent “quiet life” also means that they are hard to access with external means, making them difficult to manipulate rapidly.

One of the established means to initialize, manipulate, and readout individual nuclear spins in a nanostructure is via an electron spin [2]. If an electron has a finite probability to be right on top of a nuclear spin, they interact through the so-called contact hyperfine interaction [1]. Otherwise they would still interact with each other through magnetic dipolar interaction, which decays as an inverse cubic function of the electron-nucleus distance, and is, in general, anisotropic (thus the name anisotropic hyperfine interaction), as shown in Fig.1(a). Electron spins are much easier and faster to control than nuclear spins because they interact strongly with both electric and magnetic fields, and their energies are in the microwave frequency regime. Indeed, in recent years there has been tremendous experimental progress in controlling and reading out single electron-spin states in a variety of systems, such as double dots [3, 4], nitrogen-vacancy (NV) centers in diamond [5], and phosphorus (P ) donors [6]. Therefore, controlling nuclear spins via their hyperfine interaction with electron spins is becoming experimentally feasible.

In a paper appearing in Physical Review Letters, Yingjie Zhang and colleagues from the University of Waterloo, Canada, have demonstrated how two nuclear spins can be coupled by an electron via anisotropic hyperfine interaction in a solid-state environment [7]. The material system studied is an x-ray-irradiated crystal of malonic acid molecules, in which a radical electron is hyperfine coupled to a 13 C nucleus and a proton, as shown in Fig.1(b). By applying various optimized electron-spin resonance (ESR) pulses in the microwave frequency range, Zhang et al. were able to create quantum coherence between the two nuclear spins, starting from an initial nuclear-spin state that is completely thermal and uncorrelated. The key for this experiment is the existence of the anisotropic component of the hyperfine interaction (AHF), which drives a nuclear spin to precess differently, depending on the electron-spin orientation, therefore allowing conditional operations on the nuclear spin. When such operations are applied to both nuclear spins, they acquire quantum-mechanical correlations between themselves from the common electron spin. In other words, the two nuclear spins can now get into lockstep in their precession—even though they do not talk to each other directly—because of the “directions” they receive from the ESR pulses via the common electron.

The current experiment points to a new way to enable nuclear-spin qubits in nanostructured systems, going beyond the complete reliance on the contact hyperfine interaction in past nuclear-spin qubit proposals [2]. Indeed, the presence of AHF allows complete control in the three-spin system, making malonic acid crystal at low temperature an interesting test bed of nuclear-spin qubit manipulations in a solid-state environment. In addition, with AHF one uses only microwave pulses (instead of both microwave and radio-frequency pulses) to control the nuclear spins, which speeds up the nuclear-spin manipulations significantly. In the current room-temperature experiment, nuclear-spin entanglement cannot be measured because they are pushed only slightly away from their initial thermal state. However, if the experiment is done at low temperature, the same experimental procedure would have led to nuclear-spin entanglement, which is the hallmark of quantum-mechanical correlation. Measurement of such correlation can be done using the microwave toolkit developed in the current experiment as well.

The future of the technique described in this paper seems bright. While malonic acid molecules may only be used to demonstrate the principle of nuclear-spin control and coupling, the same type of interactions are present in other more scalable systems. Among the most promising qubit candidates are the phosphorus donor nuclear spins in silicon (Si ) [2] and NV centers in diamond [5]. The former takes advantage of the strong contact hyperfine interaction between the phosphorus nuclear spin and the localized donor electron spin, and is set in silicon, the dominant semiconductor of today. The latter uses the spin states of a strongly localized defect in diamond, has extremely good coherence properties even at room temperature, and can be manipulated optically. Like malonic acid molecules, both these spin qubit systems have hyperfine interactions that are strongly anisotropic, which has been clearly revealed in spin-echo experiments [8, 9]. Furthermore, an experiment has already been done to perform error correction encoding between an NV-center electron spin and an environmental 13 C nuclear spin, and to couple two nuclear spins through the NV-center electron, helped by both microwave and radio-frequency pulses and optical readout [10]. Using the technique developed in the current paper, these and other experiments in and NV centers can potentially be done with less complexity, more efficiency, and faster speed, giving these materials systems a distinct advantage in terms of maneuverability.

From a broader perspective, AHF is already being used in biology and chemistry for the purpose of single nucleus recognition [1]. An all-microwave approach that can controllably couple multiple nuclear spins would enhance the toolkit of all the electron- and nuclear-spin spectroscopists. It is not clear whether such a new tool could be useful to applications such as magnetic resonance imaging, but that should hardly be an impediment to further explorations.

References

1. A. Schweiger and G. Jeschke, Principles of Pulse Electron Paramagnetic Resonance (Oxford University Press, New York, 2001).
2. B. E. Kane, Nature 393, 133 (1998).
3. J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005).
4. M. Pioro-Ladriere, T. Obata, Y. Tokura, Y. S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Nature Phys. 4, 776 (2008).
5. R. Hanson and D. D. Awschalom, Nature 453, 1043 (2008).
6. A. Morello, J. J. Pla, F. A. Zwanenburg, K. W. Chan, K. Y. Tan, H. Huebl, M. Mottonen, C. D. Nugroho, C. Y, Yang, J. A. van Donkelaar, A. D. C. Alves, D. N. Jamieson, C. C. Escott, L. C. L. Hollenberg, R. G. Clark, and A. S. Dzurak, Nature 467, 687 (2010).
7. Y. Zhang, C. A. Ryan, R. Laflamme, and J. Baugh, Phys. Rev. Lett. 107, 170503 (2011).
8. W. M. Witzel, X. Hu, and S. Das Sarma, Phys. Rev. B 76, 035212 (2007); E. Abe, A. M. Tyryshkin, S. Tojo, J. J. L. Morton, W. M. Witzel, A. Fujimoto, J. W. Ager, E. E. Haller, J. Isoya, S. A. Lyon, M. L. W. Thewalt, and K. M. Itoh, Phys. Rev. B 82, 121201R (2010).
9. L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, Science 314, 281 (2006).
10. L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, and M. D. Lukin, Science 326, 267 (2009).

• Molecular Description

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  Classification:	Proton Transport
  Structure Weight:	54340.28

  Molecule:	Bacteriorhodopsin
  Polymer:	1	Type:	protein	Length:	249
  Chains:	A, B
  Mutation:	D128A
  UniProtKB:	P02945

1. Department of Chemistry and Biochemistry, UCLA-DOE Center for Genomics and Proteomics, Molecular Biology Institute,
2. Department of Physiology, and,
3. The NPI-Semel Institute, Pasarow Mass Spec Laboratory, University of California, Los Angeles, California 90095, USA
4. Department of Medicine and Biomedical Sciences Graduate Program, University of California, San Diego, La Jolla, California 92093-0656, USA

Correspondence to: James U. Bowie¹ Correspondence and requests for materials should be addressed to J.U.B. (Email: bowie@mbi.ucla.edu).

The few evaluations of hydrogen-bond contributions in membrane proteins have tested the effect of single point mutants on either the free energy of unfolding or the free energy of dissociation^{4, 8, 9}. However, these measurements combine hydrogen-bond contributions with desolvation and many other factors¹⁰, so the hydrogen-bond contribution cannot necessarily be extracted without the incorporation of correction factors¹¹ that are particularly uncertain for membrane proteins.

The energetic complexities of single side-chain alterations can be illustrated by mutations in bacteriorhodopsin residues T90 and D115 that make two hydrogen bonds near the centre of the membrane (Fig. 1). We eliminated the hydrogen bonds by making T90A and D115A mutations and measured the change in the free energy of unfolding with an SDS unfolding assay⁹. The T90A mutation decreases stability by 1.3 ± 0.1 kcal mol^-1, whereas the D115A mutant increases stability by 0.5 ± 0.1 kcal mol^-1. The large variation suggests that hydrogen-bonding alone does not dominate the stability effects, and other energetic contributions must be accounted for. Below we present evidence that a principal factor is changes in solvation free energy in the unfolded protein.

Figure 1: Double-mutant cycles for hydrogen-bonding interactions in bacteriorhodopsin.

For each cycle shown, the difference in free energies of unfolding (black number by the arrow) was measured for the pair of proteins connected by the arrow. Free energies of unfolding are compared at an SDS concentration at which the wild-type protein (WT) is 50% unfolded to minimize extrapolations needed. Errors are s.d. for three separate measurements. Next to each double-mutant cycle is a close-up view of the relevant hydrogen bond shown as blue dotted line between the altered side chains along with the heavy atom donor–acceptor distance. Donor and acceptor residues are labelled in green and blue, respectively. Donor–acceptor distinction in the two strongest interactions was arbitrary. On the basis of hydrogen-bonding patterns and nearest neighbours, it seems that all the potentially charged residues are the neutral species. The inset (bottom right) shows the location of each interaction in the context of the protein (PDB ID 1C3W). The planes of green dots indicate the estimated position of the edge of the hydrocarbon region of the bilayer as defined previously²⁸. Any interaction mediated by the residues that contain at least one atom in the hydrocarbon region is mapped with the red line, and the interaction in the lipid/water interface region is mapped with a blue line.

High resolution image and legend (507K)

To examine the effects of the T90A and D115A mutations on the folded state of bacteriorhodopsin, we solved the structures of the D115A mutant and a T90A/D115A double mutant (T90A proved too unstable to crystallize). We were unable to detect any structural changes in the mutant proteins that would obviously explain the contrasting energetic consequences, beyond the loss of density around the deleted side chains (see Fig. 2a).

Figure 2: Characterization of the T90A, D115A and T90A/D115A mutants.

a, Omit electron density maps and overlay of refined mutant and wild-type structures for D115A (top) and T90A/D115A (bottom) mutants. The wild-type structure (PDB ID 1PY6) is shown in rust, D115A in blue and T90A/D115A in grey. The mutated side chains are shown in ball-and-stick representation and labelled. The side chains of all residues within 4 Å of T90 and D115 of the wild-type (WT) structure were eliminated during refinement for the omit map and are shown here with the exception of W182, which was left out for clarity. The electron density map is contoured at 1.0σ and 1.5σ for D115A and T90A/D115A, respectively. b, Plot of the number of hydrogens exchanged in the denatured state against time for peptides overlapping the T90A mutation (top), a region between T90A and D115A mutation (middle) and the D115A mutation (bottom). In brief, wild-type and mutant proteins were unfolded in SDS and incubated in D₂O; the exchange reaction was quenched by rapidly lowering the temperature and pH. The proteins were then digested with pepsin, distinct peptides were separated chromatographically and the change in the mass envelope was measured by electrospray ionization-mass spectroscopy. The maximum scale on the y axis is the maximum number of exchangeable backbone amide hydrogens. Error bars are s.d. estimated with results from triplicate experiments. c, A plot of average exchange rates for peptides throughout the protein (top) and a schematic illustration of the bacteriorhodopsin structure (bottom) showing the sequences covered by the deuterium exchange experiment in light red. Error bars on the x axis reflect the range of the peptic peptides, and those on the y axis are s.d. for ten simulated data sets incorporating the experimental errors observed in the exchange time courses (see Methods).

High resolution image and legend (329K)

To probe the consequences of the mutations on the unfolded state, we developed a hydrogen-exchange assay. Unfolded-state backbone hydrogens that are shielded from solvent by burial in the detergent micelle will exchange at a slower rate than backbone hydrogens exposed to the aqueous phase^{12, 13}. Figure 2b shows the detailed time course of exchange for the unfolded state of the wild-type and mutant proteins at three regions, one resolved by the peptide overlapping the site of the T90A mutation, the second overlapping a region in between the sites of the T90A and D115A mutations, and the third overlapping the site of the D115A mutation. Figure 2c summarizes the average exchange rates of peptides throughout the unfolded states.

The T90A mutation modestly slows the exchange in the vicinity of position 90, whereas D115A markedly slows exchange in the vicinity of position 115. Although the sequence effects on intrinsic exchange rates¹⁴ are uncertain in an SDS environment¹⁵, the results suggest that the polar to non-polar substitutions alter the unfolded state by increasing burial in the detergent micelle at the sites of mutation. The larger change in polarity in D115A than in T90A is consistent with the larger effect on exchange rate and probably explains the stabilizing effect of the D115A mutation. In particular, the loss of the favourable escape of D115 to solvent could increase the free energy of the unfolded state in the D115A mutant, compensating for the increased free energy of the folded state. Thus, solvation effects in the unfolded state may mask the hydrogen-bond contribution that we wish to measure.

In an effort to obtain side-chain interaction energies within the folded state, we turned to double-mutant cycle analysis. Double-mutant cycle analysis has the potential to measure the free energy of side-chain interaction directly in the context of the folded protein by cancelling out energetic perturbations in both the folded and unfolded states that are not due to the interactions between the side chains^{16, 17}. Thus, desolvation contributions and any other new interactions made in the unfolded state can be eliminated. As a result, double-mutant cycle analysis can be interpreted as reporting the contribution of the hydrogen-bonded interaction to the free energy of the folded state, not the difference in free energy between the folded and unfolded states. The unfolded state becomes simply a common reference state in which the interaction of interest is broken (see Supplementary Methods).

We were able to express and purify complete single-mutant and double-mutant sets for eight interhelical hydrogen-bonding interactions as shown in Fig. 1. Four of the hydrogen bonds are in the middle of the hydrocarbon core region of the bilayer, three are on the edge of the hydrocarbon core and one is in the interfacial region. The strongest interactions were T46–D96 and T90–D115, each contributing -1.7 ± 0.3 kcal mol^-1, and T170–S226, contributing -0.8 ± 0.3 kcal mol^-1. The strongest interactions, between T46 and D96 and between T90 and D115, both involve two hydrogen bonds, corresponding to about -0.9 kcal mol^-1 per hydrogen bond. Y185–D212 and S193–E204 make weaker, but favourable, interactions contributing -0.4 ± 0.4 and -0.5 ± 0.3 kcal mol^-1, respectively. The K30–Y43 and E9–Y79 interactions were found to make no measurable contribution to stability, and W189–Y83 was found to be slightly destabilizing, contributing +0.4 ± 0.2 kcal mol^-1.

The results of the double-mutant cycles suggest three main conclusions. First, hydrogen-bonded side-chain contributions are quite variable and depend on the characteristics and local environment of each hydrogen bond. Second, the strength of a hydrogen-bonded interaction is not strongly correlated with the location in the protein. For example, the T170–S226 interaction in the interfacial region contributes -0.8 ± 0.3 kcal mol^-1, whereas the T185–D212 interaction in the centre of the hydrocarbon core contributes only -0.4 ± 0.4 kcal mol^-1. Third, the eight hydrogen-bonding interactions studied here make a remarkably modest average contribution of only about 0.6 kcal mol^-1, which corresponds to a roughly threefold effect on an equilibrium constant at room temperature.

Protein folding experiments are complex and not all variables can be eliminated, so we sought an additional, independent evaluation of the hydrogen-bond contribution in membrane proteins. We reasoned that if hydrogen-bond strengths were low, we would see a large number of unsatisfied hydrogen-bonding groups in membrane protein structures¹⁸. To test this idea, we examined six membrane protein structures solved at 1.7 Å resolution or better. HBPLUS was used to identify the hydrogen-bonding of all polar atoms within the hydrocarbon core region of the bilayer. Any polar atoms that made no hydrogen bonds were further verified by eye.

The results are summarized in Table 1 and reveal that unsatisfied hydrogen bonds are not rare in the hydrocarbon core region (also see Supplementary Information). Of 2,892 protein donors and acceptors examined in the hydrocarbon core region, 111 have no hydrogen-bonding partner (about 4%). We believe this to be a low estimate of the number of unsatisfied hydrogen bonds because the HBPLUS criteria permit even marginal hydrogen bonds to be counted. Moreover, the crystal structure reports the predominant conformation and does not report the fraction of time for which a hydrogen bond is broken.

Table 1: Unsatisfied hydrogen-bond donors and acceptors in hydrophobic cores of membrane proteins

Full table

The hydrogen-bonded interaction strengths we measured in a membrane protein are very similar to hydrogen-bond strengths in soluble proteins measured in a variety of double-mutant cycle analyses (see Supplementary Fig. 1). Retrospectively, this finding is not unreasonable because the polarities of the interiors of soluble proteins and membrane proteins are quite similar^{19, 20}. Because folding studies in soluble proteins are well accepted, we decided to validate our findings further by comparing hydrogen-acceptor distance distributions in membrane and soluble proteins. As summarized in Fig. 3, the buried hydrogen-bond distances in the interior of soluble and membrane protein transmembrane regions are statistically indistinguishable (see Supplementary Fig. 3 for full distributions), both averaging 2.02 Å. However, the hydrogen-bond distances in surface residues are markedly different. For the transmembrane regions of membrane proteins, the hydrogen-acceptor distances on the surface are slightly shortened to 1.98 Å on average, whereas for soluble proteins the average distance lengthens to 2.08 Å. These results further validate our results, indicating similar contributions from interior hydrogen bonds in soluble and membrane proteins. It also hints that hydrogen bonds at the surface of membrane proteins may be stronger than what we have measured here for interior ones.

Figure 3: Comparison of average hydrogen-bond distances in different environments.

The arrows point towards the shorter hydrogen bonds. The P value is the probability that the distance distributions are different by random chance based on Student’s t-test. The distributions are shown in Supplementary Information.

High resolution image and legend (181K)

Many of the hydrogen-bonded residues we tested are involved in function, so it is possible that they are in a separate class from structural hydrogen-bonded side chains. However, previous work eliminating hydrogen bonds in structural residues is consistent with our findings. For example, a Gln residue that makes two hydrogen bonds across the interface of an OMPLA dimer contributes less than 1 kcal mol^-1 to dimerization (less than 0.5 kcal mol^-1 per hydrogen bond)²¹. An Asp to Ala substitution in a designed transmembrane helix oligomer decreases the free energy of association by 1.8 kcal mol^-1 (0.9 kcal mol^-1 per potential hydrogen bond)⁴. A T87A substitution in glycophorin A decreases the free energy of dimerization by 0.9 kcal mol^-1 (about 0.5 kcal mol^-1 per hydrogen bond in the dimer)²². Mutations in hydrogen-bonding polar residues in the T-cell receptor ζ-subunit dimer affected dimerization by a maximum of 7.7-fold, as indirectly measured by assembly rate, which corresponds to a maximum of about 1.2 kcal mol^-1 (0.6 kcal mol^-1 per residue in the dimer)⁸. Analysis of hydrogen-bonding mutations in bacteriorhodopsin (ref. 9) suggest a contribution of about 1 kcal mol^-1 (ref. 23). Because double-mutant cycle analysis was not employed in these cases, however, the results combine hydrogen-bond contributions with other effects that could contribute favourably or unfavourably^{10, 24}.

Although our results indicate that most hydrogen-bond interactions observed in membrane proteins make modest energetic contributions, it does not mean that polar interactions cannot be strong. It has been found¹⁶ that charge-stabilized salt-bridge interactions can contribute 5.6 kcal mol^-1, and mutations in residues that hydrogen-bond to ligands in the β-adrenergic receptor can have marked effects on ligand binding²⁵.

Why, then, are hydrogen-bond interactions not much stronger on average? It is possible that optimal geometries are difficult to achieve, that there are entropic costs to fixing hydrogen-bonded groups, and that polar groups in the protein can increase the local dielectric constant²⁶. In addition to possible physical limitations, there may also be evolutionary pressure favouring weak hydrogen bonds. Evolutionary pressure for weak hydrogen bonds could come in the form of the conformational flexibility needed for protein function⁶. Moreover, the helical distortions that are common in membrane proteins would be hard to create by random mutation if the breakage of hydrogen bonds presented a large energy barrier. It is also possible that weak hydrogen bonds are more robust evolutionarily. Strong side-chain hydrogen bonds, once established, could no longer be altered by mutation without destroying fitness. Thus, proteins that rely on a strong hydrogen bond for stability would be more likely to be lost from a population than proteins that rely on more broadly distributed stabilizing interactions. Whatever the mechanism, our results indicate that it is not difficult to make and break interactions between polar residues, enabling the structural variation and dynamic flexibility necessary to optimize membrane protein function and folding. The results also suggest that a primary mechanism for the prevalence of disease-causing substitutions of polar residues may not be inappropriate hydrogen-bond formation but, instead, alterations in bilayer partitioning⁶. However, the loss or gain of even a weak hydrogen bond could tip the balance between biological function and dysfunction.

1. White, S. H. How hydrogen bonds shape membrane protein structure. Adv. Protein Chem. 72, 157–172 (2005) | PubMed |
2. Popot, J. L. & Engelman, D. M. Helical membrane protein folding, stability, and evolution. Annu. Rev. Biochem. 69, 881–922 (2000) | Article | PubMed | ISI | ChemPort |
3. Zhou, F. X., Merianos, H. J., Brunger, A. T. & Engelman, D. M. Polar residues drive association of polyleucine transmembrane helices. Proc. Natl Acad. Sci. USA 98, 2250–2255 (2001) | Article | PubMed | ChemPort |
4. Gratkowski, H., Lear, J. D. & DeGrado, W. F. Polar side chains drive the association of model transmembrane peptides. Proc. Natl Acad. Sci. USA 98, 880–885 (2001) | Article | PubMed | ChemPort |
5. Adamian, L. & Liang, J. Interhelical hydrogen bonds and spatial motifs in membrane proteins: polar clamps and serine zippers. Proteins 47, 209–218 (2002) | Article | PubMed | ISI | ChemPort |
6. Partridge, A. W., Therien, A. G. & Deber, C. M. Polar mutations in membrane proteins as a biophysical basis for disease. Biopolymers 66, 350–358 (2002) | Article | PubMed | ChemPort |