Modeling mechanical energy storage in springs based on carbon nanotubes.

A modeling study of the potential for storing energy in the elastic deformation of springs comprised of carbon nanotubes (CNTs) is presented. Analytic models were generated to estimate the ideal achievable energy density in CNTs subject to axial tension, compression, bending and torsion, taking into account limiting mechanisms such as the strength of individual CNTs, the onset of buckling, and the packing density limitations of CNT groupings. The stored energy density in CNT springs is predicted to be highest under tensile loading, with maximum values more than three orders of magnitude greater than the energy density of steel springs, and approximately eight times greater than the energy density of lithium-ion batteries. Densely packed bundles of precisely aligned, small diameter single-walled carbon nanotubes are identified as the best structure for high performance springs. The conceptual design and modeling of a portable electric power source that stores energy in a CNT spring are presented as tools for studying the potential performance of a system for generating electricity from the CNTs' stored mechanical energy.

Principles of control for decoherence-free subsystems.

Decoherence-free subsystems (DFSs) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem theoretically exist, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFSs. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two physical qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.

Benchmarking quantum control methods on a 12-qubit system.

In this Letter, we present an experimental benchmark of operational control methods in quantum information processors extended up to 12 qubits. We implement universal control of this large Hilbert space using two complementary approaches and discuss their accuracy and scalability. Despite decoherence, we were able to reach a 12-coherence state (or a 12-qubit pseudopure cat state) and decode it into an 11 qubit plus one qutrit pseudopure state using liquid state nuclear magnetic resonance quantum information processors.

Incoherent noise and quantum information processing.

Incoherence in the controlled Hamiltonian is an important limitation on the precision of coherent control in quantum information processing. Incoherence can typically be modeled as a distribution of unitary processes arising from slowly varying experimental parameters. We show how it introduces artifacts in quantum process tomography and we explain how the resulting estimate of the superoperator may not be completely positive. We then go on to attack the inverse problem of extracting an effective distribution of unitaries that characterizes the incoherence via a perturbation theory analysis of the superoperator eigenvalue spectra.

NMR analog of the quantum disentanglement eraser.

We report the implementation of a three-spin quantum disentanglement eraser on a liquid-state NMR quantum information processor. A key feature of this experiment was its use of pulsed magnetic field gradients to mimic projective measurements. This ability is an important step towards the development of an experimentally controllable system which can simulate any quantum dynamics, both coherent and decoherent.

Construction and implementation of NMR quantum logic gates for two spin systems.

The implementation of small prototype quantum computers has been studied through ensemble quantum computing via NMR measurements. In such laboratory studies it is convenient to have access to a wide array of logic gates. Here a systematic approach to reduce the logic gate to an NMR pulse sequence is introduced. This approach views the truth table for a quantum logic operation as a permutation matrix that corresponds to a propagator for an NMR transition. This propagator is then used as the starting point for the derivation of a pulse sequence. Pulse sequences for all the permutations of a four level system are reported along with implementations of representative examples on a two spin-1/2 system, 13C-labeled chloroform.

Ensemble quantum computing by NMR spectroscopy.

A quantum computer (QC) can operate in parallel on all its possible inputs at once, but the amount of information that can be extracted from the result is limited by the phenomenon of wave function collapse. We present a new computational model, which differs from a QC only in that the result of a measurement is the expectation value of the observable, rather than a random eigenvalue thereof. Such an expectation value QC can solve nondeterministic polynomial-time complete problems in polynomial time. This observation is significant precisely because the computational model can be realized, to a certain extent, by NMR spectroscopy on macroscopic ensembles of quantum spins, namely molecules in a test tube. This is made possible by identifying a manifold of statistical spin states, called pseudo-pure states, the mathematical description of which is isomorphic to that of an isolated spin system. The result is a novel NMR computer that can be programmed much like a QC, but in other respects more closely resembles a DNA computer. Most notably, when applied to intractable combinatorial problems, an NMR computer can use an amount of sample, rather than time, which grows exponentially with the size of the problem. Although NMR computers will be limited by current technology to exhaustive searches over only 15 to 20 bits, searches over as much as 50 bits are in principle possible, and more advanced algorithms could greatly extend the range of applicability of such machines.

A robust method for estimating cross-relaxation rates from simultaneous fits to build-up and decay curves.

This paper introduces a novel computational method for estimating relaxation rates among pairs of spin orders. This method simultaneously estimates all the auto- and cross-relaxation rates from the same measurements, and avoids the ill-conditioning problems associated with multiexponential fits. The method models the relaxation dynamics by a system of linear differential equations, and assumes that measurements of the spin orders have been made at an equally spaced sequence of time points. It computes a nonlinear least-squares fit of the exponential of the rate matrix at the shortest time point to these measurements. Preliminary estimates of the exponential matrix and initial spin orders from which to start the computations are obtained by solving simpler linear-least-squares problems. The performance of the method on simulated 2 x 2 test problems indicates that when measurements at eight or more equally spaced times spanning the maximum and inflection points of the build-up curves are available, the relative errors in the rates are usually less than the relative errors in the measurements. The method is further demonstrated by applying it to the problem of determining the cross correlation-induced cross-relaxation rates between the in-phase and antiphase coherence of the amide groups in the 15N-labeled protein oxidized flavodoxin. Finally, the possibility of extending the method to other kinds of relaxation measurements and larger spin systems is discussed.

An evaluation of least-squares fits to COSY spectra as a means of estimating proton-proton coupling constants. I. Simulated test problems.

A computational method is described that takes an initial estimate of the chemical shifts, line widths and scalar coupling constants for the protons in a molecule, and refines this estimate so as to improve the least-squares fit between an experimental COSY spectrum and the spectrum simulated from these parameters in the weak-coupling approximation. In order to evaluate the potential of such refinements for estimating these parameters from COSY experiments, the method has been applied to a large number of sample problems which were themselves simulated from standard conformations of the amino acids, along with 25 near-native conformations of the protein bovine pancreatic trypsin inhibitor. The results of this evaluation show that: (i) if the chemical shifts are known to within ca. 0.01 ppm and no noise or artifacts are present in the data, the method is capable of recovering the correct coupling constants, starting from essentially arbitrary values, to within 0.1 Hz in almost all cases. (ii) Although the precision of these estimates of the coupling constants is degraded by the limited resolution, noise and artifacts present in most experimental spectra, the large majority of coupling constants can still be recovered to within 1.0 Hz; the local minimum problem is not made significantly worse by such defects in the data. (iii) The method assigns an 'effective' line width to all the resonances, and in the process can resolve overlapping cross peaks. (iv) The method is not capable of determining the chemical shifts a priori, due to the presence of numerous local minima in the least-squares residual as a function of these parameters.

An evaluation of least-squares fits to COSY spectra as a means of estimating proton-proton coupling constants. II. Applications to polypeptides.

A new computational method for simultaneously estimating all the proton-proton coupling constants in a molecule from COSY spectra [Yang, J.-X. and Havel, T.F. (1994) J. Biomol. NMR, 4, 807-826] is applied to experimental data from two polypeptides. The first of these is a cyclic hexapeptide denoted as VDA (-D-Ala1-Phe2-Trp3-Lys(Z)4-Val5-Phe6-), in deuterated DMSO, while the second is a 39-residue protein, called decorsin, in aqueous solution. The effect of different data processing strategies and different initial parameter values on the accuracy of the coupling constants was explored. In the case of VDA, most of the coupling constants did not depend strongly on the initial values chosen for the optimization or on how the data were processed. This, together with our previous experience using simulated data, implies strongly that these values are accurate estimates of the coupling constants. They also differ by an average of only 0.36 Hz from the values of the 14 coupling constants that could be measured independently by established methods. In the case of decorsin, many of the coupling constants exhibited a moderate dependence on their initial values and a strong dependence on how the data were processed. With the most successful data processing strategy, the amide-alpha coupling constants differed by an average of 1.11 Hz from the 21 values that could be measured by established methods, while two thirds of the three-bond coupling constants fell within 1.0 Hz of the ranges obtained by applying the Karplus relation to an independently computed ensemble of distance geometry structures. The averages of the coupling constants over multiple optimizations using random initial values were computed in order to obtain the best possible estimates of the coupling constants. Most clearly incorrect averages can be identified by large standard deviations in the coupling constants or the associated line widths and chemical shifts, and can be explained by strong coupling and/or overlap with the water signal, the diagonal peaks or other cross peaks.

Matrix decompositions of two-dimensional nuclear magnetic resonance spectra.

Two-dimensional NMR spectra are rectangular arrays of real numbers, which are commonly regarded as digitized images to be analyzed visually. If one treats them instead as mathematical matrices, linear algebra techniques can also be used to extract valuable information from them. This matrix approach is greatly facilitated by means of a physically significant decomposition of these spectra into a product of matrices--namely, S = PAPT. Here, P denotes a matrix whose columns contain the digitized contours of each individual peak or multiple in the one-dimensional spectrum, PT is its transpose, and A is an interaction matrix specific to the experiment in question. The practical applications of this decomposition are considered in detail for two important types of two-dimensional NMR spectra, double quantum-filtered correlated spectroscopy and nuclear Overhauser effect spectroscopy, both in the weak-coupling approximation. The elements of A are the signed intensities of the cross-peaks in a double quantum-filtered correlated spectrum, or the integrated cross-peak intensities in the case of a nuclear Overhauser effect spectrum. This decomposition not only permits these spectra to be efficiently simulated but also permits the corresponding inverse problems to be given an elegant mathematical formulation to which standard numerical methods are applicable. Finally, the extension of this decomposition to the case of strong coupling is given.

Does compactness induce secondary structure in proteins? A study of poly-alanine chains computed by distance geometry.

A few years ago, lattice model studies indicated that compactness could induce polymer chains to develop protein-like secondary structures. Subsequent off-lattice studies have found the amounts of induced structure to be relatively small. Here we use distance geometry to generate random conformations of compact poly-alanine chains of various chain lengths. The poly-alanine chains are subjected only to compactness and excluded volume constraints; no other energies or conformational propensities are included in the chain generation procedure. We find that compactness leads to considerable stabilization of secondary structure, but the absolute amount of secondary structure depends strongly on the criteria used to define helices and sheets. By loose criteria, much secondary structure arises from compactness, but by strict criteria, little does. The stabilization free energy of secondary structure provided by compactness, however, appears to be independent of criteria. Since real helices and sheets in proteins can be identified by strict criteria, we introduced small energy perturbations to compact poly-alanine chains using the AMBER force field. Small refinements produced good alpha-helices. For beta-sheets, however, larger refinements are necessary. Compactness appears to impart stability, but not much structural specificity, to secondary structures in proteins. Compactness acts more like diffusion as a force, a result of ensemble statistics, than like pair interactions such as hydrogen bonding.

SESAME: a least-squares approach to the evaluation of protein structures computed from NMR data.

A method is proposed for defining a probability distribution on an ensemble of protein conformations from a 2D NOE spectrum, while at the same time back-calculating the experimental spectrum from the ensemble. This enables one to assess the relative quality and significance of the conformations, and to test the consistency of the ensemble as a whole with the experimental spectrum. The method eliminates the need to integrate the cross-peak intensities and is surprisingly insensitive to random noise in the spectrum. In this communication, these advantages are demonstrated by applying the method to simulated data, for which the correct result is already known.

The solution structure of eglin c based on measurements of many NOEs and coupling constants and its comparison with X-ray structures.

A high-precision solution structure of the elastase inhibitor eglin c was determined by NMR and distance geometry calculations. A large set of 947 nuclear Overhauser (NOE) distance constraints was identified, 417 of which were quantified from two-dimensional NOE spectra at short mixing times. In addition, a large number of homonuclear 1H-1H and heteronuclear 1H-15N vicinal coupling constants were used, and constraints on 42 chi 1 and 38 phi angles were obtained. Structure calculations were carried out using the distance geometry program DG-II. These calculations had a high convergence rate, in that 66 out of 75 calculations converged with maximum residual NOE violations ranging from 0.17 A to 0.47 A. The spread of the structures was characterized with average root mean square deviations () between the structures and a mean structure. To calculate the unbiased toward any single structure, a new procedure was used for structure alignment. A canonical structure was calculated from the mean distances, and all structures were aligned relative to that. Furthermore, an angular order parameter S was defined and used to characterize the spread of structures in torsion angle space. To obtain an accurate estimate of the precision of the structure, the number of calculations was increased until the and the angular order parameters stabilized. This was achieved after approximately 40 calculations. The structure consists of a well-defined core whose backbone deviates from the canonical structure ca. 0.4 A, a disordered N-terminal heptapeptide whose backbone deviates by 0.8-12 A, and a proteinase-binding loop whose backbone deviates up to 3.0 A. Analysis of the angular order parameters and inspection of the structures indicates that a hinge-bending motion of the binding loop may occur in solution. Secondary structures were analyzed by comparison of dihedral angle patterns. The high precision of the structure allows one to identify subtle differences with four crystal structures of eglin c determined in complexes with proteinases.

Alternating zinc fingers in the human male associated protein ZFY: 2D NMR structure of an even finger and implications for "jumping-linker" DNA recognition.

ZFY, a sex-related Zn-finger protein encoded by the human Y chromosome, is distinguished from the general class of Zn-finger proteins by the presence of a two-finger repeat. Whereas odd-numbered domains and linkers fit a general consensus, even-numbered domains and linkers exhibit systematic differences. Because this alternation may have fundamental implications for the mechanism of protein-DNA recognition, we have undertaken biochemical and structural studies of fragments of ZFY. We describe here the solution structure of a representative nonconsensus (even-numbered) Zn finger based on 2D NMR studies of a 30-residue peptide. Structural modeling by distance geometry and simulated annealing (DG/SA) demonstrates that this peptide folds as a miniglobular domain containing a C-terminal beta--hairpin and N-terminal alpha-helix (beta beta alpha motif). These features are similar to (but not identical with) those previously described in consensus-type Zn fingers (derived from ADR1 and Xfin); the similarities suggest that even and odd ZFY domains bind DNA by a common mechanism. A model of the protein-DNA complex (designated the "jumping-linker" model) is presented and discussed in terms of the ZFY two-finger repeat. In this model every other linker is proposed to cross the minor groove by means of a putative finger/linker submotif HX4HX3-hydrophobic residue-X3. Analogous use of a hydrophobic residue in a linker that spans the minor groove has recently been described in crystallographic and 3D NMR studies of homeodomain-DNA complexes. The proposed model of ZFY is supported in part by the hydroxyl radical footprint of the TFIIIA-DNA complex [Churchill, M.E.A., Tullius, T.D., & Klug, A. (1990) Proc. Natl. Acad. Sci. U.S.A. 87, 5528-5532].

A new method for building protein conformations from sequence alignments with homologues of known structure.

We describe a largely automatic procedure for building protein structures from sequence alignments with homologues of known structure. This procedure uses simple rules by which multiple sequence alignments can be translated into distance and chirality constraints, which are then used as input for distance geometry calculations. By this means one obtains an ensemble of conformations for the unknown structure that are compatible with the rules employed, and the differences among these conformations provide an indication of the reliability of the structure prediction. The overall approach is demonstrated here by applying it to several Kazal-type trypsin inhibitors, for which experimentally determined structures are available. On the basis of our experience with these test problems, we have further predicted the conformation of the human pancreatic secretory trypsin inhibitor, for which no experimentally determined structure is presently available.

The sampling properties of some distance geometry algorithms applied to unconstrained polypeptide chains: a study of 1830 independently computed conformations.

In this paper we study the statistical geometry of ensembles of poly (L-alanine) conformations computed by several different distance geometry algorithms. Since basic theory only permits us to predict the statistical properties of such ensembles a priori when the distance constraints have a very simple form, the only constraints used for these calculations are those necessary to obtain reasonable bond lengths and angles, together with a lack of short- and long-range atomic overlaps. The geometric properties studied include the squared end-to-end distance and radius of gyration of the computed conformations, in addition to the usual rms coordinate and phi/psi angle deviations among these conformations. The distance geometry algorithms evaluated include several variations of the well-known embed algorithm, together with optimizations of the torsion angles using the ellipsoid and variable target function algorithms. The conclusions may be summarized as follows: First, the distribution with which the trial distances are chosen in most implementations of the embed algorithm is not appropriate when no long-range upper bounds on the distances are present, because it leads to unjustifiably expanded conformations. Second, chosing the trial distances independently of one another leads to a lack of variation in the degree of expansion, which in turn produces a relatively low rms square coordinate difference among the members of the ensemble. Third, when short-range steric constraints are present, torsion angle optimizations that start from conformations obtained by choosing their phi/psi angles randomly with a uniform distribution between -180 degrees and +180 degrees do not converge to conformations whose angles are uniformly distributed over the sterically allowed regions of the phi/psi plane. Finally, in an appendix we show how the sampling obtained with the embed algorithm can be substantially improved upon by the proper application of existing methodology.