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PubMed1996

Direct methods in protein electron crystallography: the ab initio structure determination of two membrane protein structures in projection using maximum entropy and likelihood.

Using maximum entropy and likelihood, an ab initio phase determination was carried out in projection at ca 6-10 A resolution for two dissimilar membrane proteins: the Omp F porin from the outer membrane of E. coli (largely beta-sheet) and halorhodopsin (largely alpha-helix). Accurate phase information found for the most likely solutions enabled potential maps to be calculated that contained most of the essential structural details of these macromolecules without the need for any image-derived phases as a starting set for phase extension or the necessity to use envelopes or electron-density histograms. A comparison with earlier calculations using the Sayre-Hughes equation coupled with phase annealing and the Luzzati flatness criterion used as a figure of merit is made.

Bacteriorhodopsins↗
PubMed1999

A multisolution method of phase determination by combined maximization of entropy and likelihood. VI. The use of error-correcting codes as a source of phase permutation and their application to the phase problem in powder, electron and macromolecular crystallography.

The use of error-correcting codes as a source of efficient designs of phase permutation schemes is described. Three codes are used, all taken from the Bricogne BUSTER program [Bricogne (1993). Acta Cryst. D49, 37-60]: the Hamming [7, 4, 3], the Nordström-Robinson (16, 256, 6) and the Golay [24, 12, 8] or its punctured [23, 12, 7] form. These are used in a maximum-entropy-likelihood phasing environment to carry out phase permutation of basis-set reflections instead of the usual quadrant permutation or magic integer approaches. The use of codes in this way inevitably introduces some errors in the phase choices, but for most structures this is not significant especially when the gain in sampling efficiency is considered. For example, the Golay [24, 14, 8] allows the permutation of 24 centric phases in such a way that only 4096 phase sets are produced instead of 2(24) = 16777216, and one of these sets has, at most, only four wrong phases. The method is successfully applied to three powder diffraction data sets of increasing complexity, and with increasing degrees of overlap {Mg(3)BN(3), Sigma-2 ([Si(64)O(128)].4C(10)H(17)N) and the NU-3 zeolite}, a sparse electron diffraction data set for buckminsterfullerene, C(60), and the small protein molecule crambin at 3 Å resolution where 42 reflections are phased with a Uweighted mean phase error of 58.5 degrees.

Journal Article↗
PubMed1999

A new type of constraint in the maximum-entropy method using ambiguous phase information from anomalous-scattering powder data.

Anomalous scattering of X-rays at a synchrotron source can be used for the ab initio structure determination of unknown crystal structures using only powder diffraction data. For noncentrosymmetric crystals, the phases of structure factors can only be determined with a remaining ambiguity, when one chemical element is used as resonant scatterer. A corresponding additional constraint function has been built into an enhanced version of the program MEED, so that now all types of information gained from an anomalous-scattering powder diffraction experiment can be used in a maximum-entropy calculation of the electron-density distribution: phased reflections, unphased reflections, intensities of groups of overlapping reflections, and now also reflections with a remaining ambiguity in the phase. This is important for practical use, since a lot of information is already lost in the powder diagram compared with single-crystal datasets and it is essential to use all remaining information. The new constraint is demonstrated with the structure of Cu(5)Zn(8).

Journal Article↗
PubMed2004

Structure of incommensurate ammonium tetrafluoroberyllate studied by structure refinements and the maximum entropy method.

Incommensurately modulated ammonium tetrafluoroberyllate (AFB) occurs in a narrow temperature interval between the paraelectric room-temperature phase with space group Pnma (Ti = 178 K) and the ferroelectric low-temperature phase with space group Pna2(1) (Tc = 173 K). The structure is determined from accurate single-crystal X-ray diffraction data collected with synchrotron radiation at 175 K. The superspace group of the structure is Pnma(alpha00)0ss with alpha = 0.4796 (4). Both structure refinements and the maximum entropy method lead to the same structure model, which involves only single harmonic modulations. The building units of the structure are BeF4(2-) and NH4+ complex ions with approximately tetrahedral point symmetry. They are relatively rigid and the modulations consist mainly of translations of the tetrahedra and their rotations around a fixed axis. The modulation is related to changes in the network of the hydrogen bonds. The low-temperature superstructure can be described as a commensurately modulated structure with the same superspace symmetry. The first harmonic modulations of the low-temperature and incommensurate phases are related by a scale factor with a value of approximately two. In addition, the low-temperature phase exhibits a second harmonic modulation that is responsible for shifts along c and the ferroelectricity in this phase. The experimental data of the incommensurate phase do not contain any evidence for the presence of a second harmonic in the modulation functions. This suggests that the development of the second harmonic, i.e. the development of the spontaneous polarization, is responsible for the lock-in transition.

Journal Article↗
PubMed1995

Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method.

The electron-density distribution (EDD) of metallic beryllium has been derived from the structure factors of Larsen & Hansen [(1984). Acta Cryst. B40, 169-179] using the maximum entropy method (MEM). Subsequent topological analysis reveals non-nuclear maxima (NNM) in the EDD. Plots of the gradient field of the electron density illustrates this finding. A possible critical-point network for the hexagonal close-packed (h.c.p.) structure of beryllium is suggested. It is thus demonstrated that it is possible to obtain detailed topological information about the electron density in metallic beryllium without the use of a structural model. In order to test the findings of the MEM, the same set of structure factors were analysed using the multipole refinement method (MRM). Use of the MRM also reveals NNM. The results of the two different approaches to electron-density analysis are contrasted and discussed. Expressed within the framework of the theory of atoms in molecules, our results suggest that the h.c.p. structure of beryllium has no Be atoms directly bonded to other Be atoms. The structure is held together through a three-dimensional network of bonds between the NNM and Be atoms as well as between different NNM. The topological analysis thus reveals that the beryllium structure has important interactions connecting Be atoms of different basal plane layers. The breaking of these interactions when forming a surface may explain the abnormally large expansion of the inter-layer distance in the beryllium surface structure.

Beryllium↗
PubMed1993

Programs for phasing by entropy maximization as implemented in Xtal3.2: a crystallographic software system.

Xtal3.2, a crystallographic software package, is an international development project involving about 40 researchers over a full spectrum of crystallographic interests. This development has been supported by many national and international agencies and commercial institutions since the first version in 1983. The 1992 release, Xtal3.2, contains software for 95 different calculations. These range from the processing of raw diffraction data to interactive molecular graphics, atomic charge estimation, electronic publication preparation, and the structure solution and refinement of small and large molecules. Tests of the Xtal programs for phase determination and phase refinement by the application of 'maximum entropy' are presented.

Journal Article↗
PubMed1993

Construction of maximum-entropy density maps, and their use in phase determination and extension.

Methods for constructing everywhere-positive electron-density maps with Fourier amplitudes matching those for arbitrarily large sets of observed data, utilizing dual-function methods for maximization of entropy, are described. Possible strategies for utilizing these maps for the determination and extension of phases in macromolecular structure determination are suggested, and problems are discussed.

Journal Article↗
PubMed1989

A method for recovering physiological components from dynamic radionuclide images using the maximum entropy principle: a numerical investigation.

A method based on the maximum entropy principle is described for recovering physiological components from a sequence of dynamic radionuclide images. The method does not require any assumption about the shapes of the component curves. It is, however, assumed that the components are nonnegative. The method requires normalization of component curves. The sum of squares of each component curve to be estimated is normalized so as to be unity. A first-pass cardiac study and a hepatic study are selected for investigation of applicability of the method. The method is investigated numerically using computer-generated time-activity curves with six components for the cardiac study and seven components for the hepatic study. Within the regions studied, the results have demonstrated that the method may recover the component time-activity curves and the corresponding images of physiological components from a sequence of dynamic radionuclide images. The ability and limitations of the method to recover physiological components are discussed and potential applications are suggested.

Heart↗
PubMed1991

Frequency analysis of the electrocardiogram with maximum entropy method for identification of patients with sustained ventricular tachycardia.

Late potentials in the terminal phase of the QRS-complex during sinus rhythm have been proposed to identify a subgroup of patients with myocardial infarction at risk of ventricular tachycardia (VT). Frequency analysis of the ECG with Fourier transform (FFT) has been applied for detection of these microvolt level signals, but is limited by poor frequency resolution of short data segments and spectral leakage. We therefore developed frequency analysis using the maximum entropy method (MEM) based on an autoregressive (AR) model. Orthogonal electrocardiograms were recorded from the body surface of patients with and without VT, and healthy persons after low noise, high-gain amplification. Multiple 40 ms segments (time intervals 2 ms, AR-parameters tapered) were analyzed (spectrotemporal mapping): low-frequency components were eliminated by building difference spectra with optimal high order and fixed low order. The MEM-spectra revealed high frequency components (40-200 Hz) in the terminal phase of the QRS-complex and in the ST-section in 26/38 patients with VT, but only in 2/20 without VT and in 1/20 healthy persons (p less than 0.05). Unlike FFT, MEM allowed localization of late potentials by the analysis of short data segments. Thus, MEM offers promise for noninvasive identification of patients with sustained VT after myocardial infarction and detailed analysis of late potentials.

Algorithms↗
PubMed2004

Biomagnetic source detection by maximum entropy and graphical models.

This article presents a new approach for detecting active sources in the cortex from magnetic field measurements on the scalp in magnetoencephalography (MEG). The solution of this ill-posed inverse problem is addressed within the framework of maximum entropy on the mean (MEM) principle introduced by Clarke and Janday. The main ingredient of this regularization technique is a reference probability measure on the random variables of interest. These variables are the intensity of current sources distributed on the cortical surface for which this measure encompasses all available prior information that could help to regularize the inverse problem. This measure introduces hidden Markov random variables associated with the activation state of predefined cortical regions. MEM approach is applied within this particular probabilistic framework and simulations show that the present methodology leads to a practical detection of cerebral activity from MEG data.

Algorithms↗
PubMed2006

Monotonicity of approximate entropy during transition from awareness to unresponsiveness due to propofol anesthetic induction.

The ability to monitor the physiological effects of sedative medication accurately is of interest in clinical practice. During the anesthetic agent driven transition to unresponsiveness, nonstationary changes such as signal amplitude variations appear in electroencephalography. In this paper, it is studied whether the application of the approximate entropy (ApEn) method to electroencephalographic (EEG) signal produces a monotonic response curve during the transition from awareness to unresponsiveness. Data from fourteen patients, undergoing propofol anesthetic induction were studied. To optimize the ApEn performance, different parameter choices were carefully evaluated. It was assumed with our protocol, that the level of anesthesia changes monotonically with the elapsed induction time. The monotonicity of the ApEn change was assessed with the prediction probability statistic (PK). The monotonicity of the ApEn time-series depends on the parameters employed in the algorithm and the varying signal amplitude. Depending on the parameter values, the median PK value ranged from 0.886 to 0.527. Thus, a good directionality and concordance was observed, but the nonstationarity of the signal affected the results. In conclusion, EEG-based ApEn measure shows a nonlinear response during propofol induction. With a judicious choice of parameters, a monotonic response is confirmed using PK statistic.

Adult↗
PubMed2005

EEG spectral entropy, heart rate, photoplethysmography and motor responses to skin incision during sevoflurane anaesthesia.

BACKGROUND: Analgesia is a part of balanced anaesthesia, but direct indicators of nociception do not exist. We examined the relationship between motor reactions and physiological variables during skin incision in sevoflurane anaesthesia and hypothesized that nociception could be detected and graded by significant changes in these variables. METHODS: Thirty-one women scheduled for abdominal hysterectomy participated in the study. Anaesthesia was induced with fentanyl (1 microg kg(-1)), propofol (1 mg kg(-1)) and sevoflurane. Skin incision was performed 14 min after induction during 1.6% end-tidal sevoflurane anaesthesia without neuromuscular blockade. Electrocardiography (ECG), photoplethysmography (PPG) and electroencephalography (EEG) were registered, and a range of variables was computed from these signals. The postincision values, normalized with respect to their preincision values, of movers vs. non-movers were compared. The variables showing significant differences between movers and non-movers were used to develop a logistic regression equation for the classification of patients into movers or non-movers. RESULTS: Twenty-six patients were eligible for analysis, and 12 (46%) displayed a motor reaction to skin incision (movers). Many ECG, PPG and EEG-related variables showed significant differences between the pre- and postincision periods. The best classification performance, assessed by leave-one-out cross-validation, between movers and non-movers was achieved with the combination of response entropy of EEG, RR-interval and PPG notch amplitude. The corresponding equation yielded 96% correct classification with 90% sensitivity and 100% specificity. The classification performance of any single variable alone was considerably worse. CONCLUSION: Combination of information from different sources may be required for monitoring the adequacy of analgesia during anaesthesia.

Adult↗
PubMed2001

Maximum entropy approach for modeling random uncertainties in transient elastodynamics.

A new approach is presented for analyzing random uncertainties in dynamical systems. This approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The information used does not require the description of the local parameters of the mechanical model. The probability model is deduced from the use of the entropy optimization principle, whose available information is constituted of the algebraic properties related to the generalized mass, damping, and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. The fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed. Finally, an example is presented.

Journal Article↗
PubMed1993

A model of auditory pattern analysis based on component-relative-entropy.

Recently results from detection and information theory have been used to predict performance of human observers in experiments involving the discriminability of changes in individual components of unfamiliar or uncertain tone patterns [R. A. Lutfi, J. Acoust. Soc. Am. 91, 3391-3401 (1992)]. In this paper, the theoretical analysis is expanded to yield a general rule of pattern analysis that accounts for nearly all of the major results of such experiments conducted over the last 18 years. The rule can be stated as follows: Component discriminability in an unfamiliar tone pattern (profile or sequence) is a linearly increasing function of the component's relative entropy (CoRE) in the pattern. Ancillary: The threshold for detection of a change decreases by one order of magnitude (factor of 10) for each one bit increase in CoRE. This precise relation of threshold to CoRE is demonstrated repeatedly in a review of past studies, and is shown to account for the relative effects of a variety of important variables and their interactions. Such variables include the total duration of the pattern; the number of tones in the pattern; the number of tones subject to change; the relative level and duration of individual tones in the pattern; the relative variability and the physical dimension of the tones' parameters that are subject to change; the number and position of targets in the pattern; the psychophysical procedure used (method of adjustment versus same-different), the type of discrimination task (frequency versus intensity discrimination), and the manner of presentation of the tones (simultaneous or sequential). The CoRE rule seems to reflect a general property of auditory analysis wherein the perception of patterns in an ensemble is dominated by those features that dominate the variance of the ensemble.

Auditory Perception↗
PubMed1994

Effect of component-relative-entropy on the discrimination of simultaneous tone complexes.

Results are reported from a number of experiments investigating the effects of stimulus uncertainty on the discrimination of simultaneous multitone complexes. With few exceptions the data were well described by a model in which average threshold for the detection of a change in one or more components varies with the components' relative entropy (CoRE) in the complex [R.A. Lutfi, J. Acoust. Soc. Am. 94, 748-758 (1993)]. In different experiments, thresholds for the detection of a change in target level were found to be linearly related to CoRE for manipulations in the number of targets, the number of context tones, and the relative degree of level perturbation on tones from trial to trial. The slopes of the functions did not change with measures taken at different points on the psychometric function, and in most cases were close to the theoretical value of b = -1.0, as predicted by strict application of the model. Also as predicted (but for the exception of one listener) increasing the level of the target beyond that of the context had little or no effect on threshold. A correlational analysis of the trial-by-trial data revealed a tendency of listeners to give negative weight to frequencies immediately above the target, and to adjust weights in response to changes in the relative degree of level perturbation. Though inconsistent with the basic premise of the model, the variation in weights was not so great as to materially affect the predictions for mean thresholds.

Adult↗
PubMed1985

Transition from B to Z DNA: contribution of internal fluctuations to the configurational entropy difference.

The internal motions of the double-stranded DNA oligomer (dCdG)3 (dC, deoxycytidylate; dG, deoxyguanylate) in the B and Z forms have been calculated in the harmonic approximation. A complete vibrational analysis has been made, and the resulting normal mode frequencies have been used to evaluate the vibrational entropy of B and Z DNA. The greater flexibility of the B DNA hexamer leads to an entropic stabilization relative to the stiffer Z DNA hexamer of 22 calories per mole per kelvin at 300 K. The calculated value is of the same order as that (21 to 27 calories per mole per kelvin) obtained from nuclear magnetic resonance measurements on the methylated duplexes (m5dCdG)3 and (dCdGm5dCdGdCdG). This result demonstrates the importance of internal motions, which have been neglected in earlier studies of the transition from B to Z DNA, in the stability of different nucleic acid conformers.

DNA↗
PubMed1992

Quantification of hormone pulsatility via an approximate entropy algorithm.

Approximate entropy (ApEn) is a recently developed formula to quantify the amount of regularity in data. We examine the potential applicability of ApEn to clinical endocrinology to quantify pulsatility in hormone secretion data. We evaluate the role of ApEn as a complementary statistic to widely employed pulse-detection algorithms, represented herein by ULTRA, via the analysis of two different classes of models that generate episodic data. We conclude that ApEn is able to discern subtle system changes and to provide insights separate from those given by ULTRA. ApEn evaluates subordinate as well as peak behavior and often provides a direct measure of feedback between subsystems. ApEn generally can distinguish systems given 180 data points and an intra-assay coefficient of variation of 8%. This suggests ApEn as applicable to clinical hormone secretion data within the foreseeable future. Additionally, the models analyzed and extant clinical data are both consistent with episodic, not periodic, normative physiology.

Algorithms↗
PubMed2001

Impact of pulsatility on the ensemble orderliness (approximate entropy) of neurohormone secretion.

Regular patterns of neurohormone secretion are driven by underlying pulsatile and subordinate (feedback sensitive) dynamics. Measures of time-series orderliness, e.g., the approximate entropy (ApEn) statistic (Pincus SM. Proc Natl Acad Sci 88: 2297-2301, 1991), vividly discriminate pathological and physiological patterns of hormone release. To investigate how specific pulsatility features impact regularity estimates, we have examined the sensitivity of the ApEn metric to systematic variations in the frequency, amplitude, and half-life of simulated neurohormone pulse trains (Veldhuis JD, Carlson ML, and Johnson ML. Proc Natl Acad Sci 84: 7686-7690, 1987) and compared the impact of a high vs. low baseline luteinizing hormone (LH) pattern regularity state mimicking the normal female luteal phase and the young male, respectively. Shortening the interpulse interval length elevated ApEn in both pulsatility models, thereby signifying greater ensemble series irregularity. The frequency sensitivity of ApEn was robust to several complementary renditions of ApEn and to variations in experimental uncertainty, basal (nonpulsatile) LH secretion, and secretory burst amplitude. ApEn rose with increasing hormone half-life, especially in the face of low baseline variability emulated by midluteal LH secretion profiles. High variability of secretory burst amplitude, pulse duration, or interpeak intervals increased ApEn in the more orderly femalelike construct; in the highly irregular malelike LH pulse model, these variability changes had little effect on ApEn. In summary, the ensemble regularity statistic, ApEn, quantifies unequal pattern orderliness in neurohormone pulse trains with minimal dependence on mean pulse amplitude, interpulse baseline, or (subthreshold) sample uncertainty. Thus ApEn monitors changing secretory event frequency and interpulse variability with sensitivity to starting pattern regularity, providing a mechanistic linkage between model evolution and statistical change.

Female↗