Doublet-split-scaling of correlation integrals in non-linear dynamics and in neurobiology

The search for the low-dimensional attractor behaviour and the dynamic self-organization of neuronal systems, from an analysis of electroencephalographic (EEG) signals, must be carried out under conditions in which the signals are not stationary for more than a few seconds. We employ a technique that we have introduced for analyzing short signals obeying a differential equation and develop it further. The technique uses the fact that in plots of slope curves of d log C(r)/d log r against log C(r), C(r)=correlation integral, for short time sequences, the dynamics may be trans-embedding-scaled, i.e. a horizontal power-law structure builds up, that is constructed from different slope curves (different embeddings), and appears at the right value of the correlation dimension, although no single slope curve exhibits scaling.Patterns of the family of slope curves are described exhibiting the doublet-split-scaling of the correlation integrals. Examples include a solution of the Mackey and Glass delay differential equation and EEG signals. The two components of a doublet differ in the dimensions of the embeddings of which they are formed, i.e. low- and high-dimensions, respectively. The advantages subsequent to recognizing trans-embedding-scaled correlation integrals and doublet-split-scaling are illustrated for EEG delta sleep signals, with emphasis on ideal doublet-split-scaling. Unambiguous evidence of attractor behaviour in delta sleep is presented.     

Time reparametrization of phase trajectories as a test for attractor-ruled dynamics: application to electroencephalographic α-waves

Time reparametrization of phase trajectories, which leaves their topological properties intact, is used to devise a means of checking the existence of an attractor. The test is applied in particular to electro-encephalographic signals, for which attractor behaviour is identified through observation of trans-embedding-scaled structures in families of slope curves of d log C(r)/d log r against log C(r), C(r) = correlation integral. The reparametrization test is quite sensitive to the artifactual attractor appearance occurring when an attention process transiently desynchronizes an -wave, as occurs when interfering with a state of relaxed wakefulness in a subject having his eyes closed. An example is given of a trans-embedding-scaled structure obeying the previously required conditions, in particular the robustness conditions, which is found invariant against reparametrization. This finding supports under the specified conditions the use of trans-embedding-scaled structures in characterizing attractors.     

Episodes of low-dimensional self-organized dynamics from electroencephalographic α-signals

 Self-organized neuronal dynamics revealed by cortical α-rhythms occur as episodes, which are rarely observed without extraction of the α-band from the other spectral components. Three episodes of an unusually long duration of 10 s, two with no signal processing after data recording at the clinic, are described and show evidence of low-dimensional α-dynamics. The evidence is gained from an analysis of scaled structures appearing in families of slope curves of the correlation integrals and is checked against time reparametrization. The data for the two unprocessed 10-s episodes are used for a test of the methodology, as well as a re-examination of the adequacy of the model of an autonomous dynamic system in steady state and of the concept of an attractor in brain dynamics investigations. Striking evidence for the model’s inadequacy is provided by the episode of subject S₁. In this example five consecutive overlapping 6-s sections do show evidence for low-dimensional dynamics, whereas the 10-s section containing those sections does not. The episode of subject 1 provides an example of α-activity which may involve self-organized dynamics extending down to low frequencies. The system (the neuronal network) showing episodes of attractor-ruled dynamics, under conditions of blurred and smoothly fading out evidence that it stays on an attractor, is designated as being ruled by a ‘shadow-attractor’. This concept is compared with that of a ‘quasi-attractor’ introduced by H. Haken in studies of physiological systems. One possible mechanism for the observed episodes is proposed, based on a time-dependent number of enslaved sub-systems. Received: 28 June 1996 / Accepted in revised form: 27 May 1995     

Studies of the cyclic adenosine monophosphate chemoreceptor ofParamecium

Summary A doublet of proteins (48,000M _r) from theParamecium cell body membrane fits several criteria for the external cAMP chemoreceptor. These criteria include: (i) selective elution from a cAMP affinity column, matching a specificity that could be predicted from the behavioral response and whole-cell binding; (ii) binding to wheat germ agglutinin indicating the presence of carbohydrate moieties indicating surface exposure; and (iii) selective inhibition of the intact cells' chemoresponse to cAMP by antibodies against the doublet. Additional evidence for the existence of a receptor, in general, comes from selective elimination of the cAMP chemoresponse by photoaffinity labeling of whole cells with 8-N₃-cAMP. The doublet proteins are not identical to the regulatory subunit of a cAMP-dependent protein kinase fromParamecium, theDictyostelium cAMP chemoreceptor, or the 42–45 kDa range proteins related to the large surface glycoprotein inParamecium. The doublet proteins are not readily separable and, as inDictyostelium, may represent two different covalent modification states of the same protein. Amino acid analysis indicates that the proteins are similar, but does not distinguish between the possibilities of proteolysis and covalent modification. Once cloned, this doublet may prove to be only the fifth external, eukaryotic chemoreceptor to be identified.     

Optimal firing in sparsely-connected low-activity attractor networks

We examine the performance of Hebbian-like attractor neural networks, recalling stored memory patterns from their distorted versions. Searching for an activation (firing-rate) function that maximizes the performance in sparsely connected low-activity networks, we show that the optimal activation function is a threshold-sigmoid of the neuron's input field. This function is shown to be in close correspondence with the dependence of the firing rate of cortical neurons on their integrated input current, as described by neurophysiological recordings and conduction-based models. It also accounts for the decreasing-density shape of firing rates that has been reported in the literature. Received:9 December 1994 / Accepted in revised form: 9 January 1996     

The estimation of the Kolmogorov entropy from a time series and its limitations when performed on EEG

A method to estimate a lower bound of the Kolmogorov entropy—the so calledK ₂-entropy—from a time series is presented which avoids use of the generalized correlation integral. The influence of the norm is studied. The method is demonstrated on some standard examples. The entropy of the attractor apparent in the EEG of the foetal sheep is estimated and the results are compared with results obtained from synthesized data featuring some basic properties of EEG. This gives an insight into the limitations of the procedure.     

Wave separation versus bandpass filtering: a comparative non-linear analysis of brain α-EEG signals with and without psychotropic drug treatment

Attempts to demonstrate low-dimensional attractor behaviour in the analysis of electroencephalographic (EEG) signals meet with difficulties that in part stem from a departure from single-system dynamics. In order to address this problem, the -waves can be extracted by digital filtering or by wave separation; these two techniques are compared in order to specify the conditions in which finite impulse response (FIR) bandpass filters can be used. The comparison was made using 18 EEG records of 3 min duration under resting conditions (6 subjects, 3 records per subject: prior to apomorphine administration, then 90 min and 150 min post-treatment). No presence of low-dimensional dynamic episodes in -signals was observed without digital processing. Sixty 5 s sections showing attractor behaviour were found after filtering and twenty five 5 s sections after wave separation. The mean correlation dimension was calculated for each experimental condition and for 4 subjects, in order to observe the temporal profile of the drug. When attractors were found after wave separation, bandpass filtering then also showed attractor behaviour, with the same temporal profile. However, the reverse is not true: attractors were found after bandpass filtering that were not present after wave separation; in this case the results deserve confirmation, although the temporal profiles for all cases in which attractors were found after filtering remained comparable.     

Yeast-derived recombinant human insulin-like growth factor I: Production, purification, and structural characterization

Recombinant human insulin-like growth factor I (IGF-I) is efficiently expressed and secreted fromSaccharomyces cerevisiae using a yeast -factor leader to direct secretion. However, approximately 10–20% of the IGF-I was in a monomeric form, the remaining materials being disulfide-linked aggregates. When the purified material was subjected to reverse-phase high-performance liquid chromatography (rp-HPLC), it gave two doublet peaks, I and II. Upon reduction, doublet peaks I and II converged to one doublet peak. This suggests that peaks I and II result from different disulfide structures, and the doublet feature of each peak results from other causes. Different disulfide structures between peaks I and II were also suggested from the near UV circular dichroism of these proteins. Only the peak II was biologically active, indicating that peak II has the correct disulfide structure. Concanavalin A affinity chromatography of the purified peak II doublet showed binding of the subpeak with an earlier rp-HPLC retention time, indicating that it was glycosylated. Sequence analysis of tryptic peptides suggested that Thr²⁹ was the site of glycosylation. Site-directed mutagenesis was used to convert Thr²⁹ to Asn²⁹. This substitution reduced, but did not eliminate IGF-I glycosylation, suggesting additional glycosylation sites. The site of carbohydrate addition was consistent with the model that O-glycosylations occur on hydroxyl amino acids near proline residues in -turns.     

Local variability and base sequence effects in DNA crystal structures

The importance and usefulness of local doublet parameters in understanding sequence dependent effects has been described for A- and B-DNA oligonucleotide crystal structures. Each of the two sets of local parameters described by us in the NUPARM algorithm, namely the local doublet parameters, calculated with reference to the mean z-axis, and the local helical parameters, calculated with reference to the local helix axis, is sufficient to describe the oligonucleotide structures, with the local helical parameters giving a slightly magnified picture of the variations in the structures. The values of local doublet parameters calculated by NUPARM algorithm are similar to those calculated by NEWHELIX90 program, only if the oligonucleotide fragment is not too distorted. The mean values obtained using all the available data for B-DNA crystals are not significantly different from those obtained when a limited data set is used, consisting only of structures with a data resolution of better than 2.4 A and without any bound drug molecule. Thus the variation observed in the oligonucleotide crystals appears to be independent of the quality of their crystallinity. No strong correlation is seen between any pair of local doublet parameters but the local helical parameters are interrelated by geometric relationships. An interesting feature that emerges from this analysis is that the local rise along the z-axis is highly correlated with the difference in the buckle values of the two basepairs in the doublet, as suggested earlier for the dodecamer structures (Bansal and Bhattacharyya, in Structure & Methods: DNA & RNA, Vol. 3 (Eds., R.H. Sarma and M.H. Sarma), pp. 139-153 (1990)). In fact the local rise values become almost constant for both A- and B-forms, if a correction is applied for the buckling of the basepairs. In B-DNA the AA, AT, TA and GA basepair sequences generally have a smaller local rise (3.25 A) compared to the other sequences (3.4 A) and this seems to be an intrinsic feature of basepair stacking interaction and not related to any other local doublet parameter. The roll angles in B-DNA oligonucleotides have small values (less than +/- 8 degrees), while mean local twist varies from 24 degrees to 45 degrees. The CA/TG doublet sequences show two types of preferred geometries, one with positive roll, small positive slide and reduced twist and another with negative roll, large positive slide and increased twist.(ABSTRACT TRUNCATED AT 400 WORDS)     

Direct nuclear Overhauser effect refinement of crambin from two-dimensional nmr data using a slow-cooling annealing protocol

The solution structure of crambin has been refined using a direct nuclear Overhauser effect (NOE) simulation approach (DINOSAUR) following a slow-cooling simulated annealing protocol starting from eight previously obtained nmr and the x-ray structures of crambin. Theoretical NOE intensities calculated with inclusion of local motions were directly compared to the experimental nmr data and forces were derived using a simple first-order approximation for the calculation of the NOE gradient. A dynamic assignment procedure was applied for the peaks involving unassigned diastereotopic proton pairs or equivalent aromatic protons. With this approach, R factors could be minimized in a reasonable simulation time to low values (around 0.26) while deviations from ideal bond lengths and angles are still acceptable. The improvement in R factors is accompanied by an improvement of the precision of the structures, the rms deviations (rmsd; from the average) calculated on the ensemble of nine structures decreasing from 0.65 to 0.55 Å for backbone atoms and from 1.0 to 0.85 Å for all heavy atoms. The solution structure is significantly different from the x-ray structure with rmsd for all atoms of 1.35 Å compared to 0.85 Å between solution structures. The largest differences are found for residues Thr-21 and Pro-22 in the loop region between the two α-helices and for the side chain of Tyr-29. © 1994 John Wiley & Sons, Inc.     

A geometric study of the amino acid sequence of class I HLA molecules

 HLA class I alleles are studied by representing them in a metric space where each dimension corresponds to each one of the amino acid positions. Their similarity in reference to their ability to present peptides to T cells is then evaluated by calculating the correlation matrix between the amino-acid-composition tables (or binding affinity tables) for the sets of peptides presented by each allele. This correlation matrix is considered an empirical similarity matrix between HLA alleles, and is modeled in terms of possible structures defined in the metric space of HLA class I amino acid sequences. These geometric structures are adequate models of the peptide-binding data currently available. The following clusters of HLA class I molecules are identified in reference to their ability to present peptides: Cluster I) HLA-A3/ HLA-A11/ HLA-A31/ HLA-A33/ HLA-A68; Cluster II) HLA-B35/ HLA-B51/ HLA-B53/ HLA-B54/ HLA-B7; and Cluster III) HLA-A29/ HLA-B61/HLA-B44; the last cluster showing possible similarities between alleles from different loci. In modeling these natural clusters, the geometric structures with more predictive power confirm the importance of those positions in the peptide-binding groove, particularly those in the B pocket. In addition, other positions (46, 79, 113, 131, 144, and 177) appeared to bear some relevance in determining which peptides can be presented by which HLA alleles. Received: 20 January 1998 / Revised: 30 March 1998     

Operation Everest II: An indication of deterministic chaos in human heart rate variability at simulated extreme altitude

It has been shown that fluctuation of human heartbeat intervals (heart rate variability, HRV) reflects variations in autonomic nervous system activity. We studied HRV at simulated altitudes of over 6000 m from Holter electrocardiograms recorded during the Operation Everest II study (Houston et al. 1987). Stationary, 30-min segments of HRV data from six subjects at sea level and over 6000m were supplied to (1) spectral analysis to evaluate sympathetic and parasympathetic nervous system (SNS, PNS) activity, (2) the analysis of Poincaré section of the phase space trajectory reconstructed on a delayed coordinate system to evaluate whether there was fluctuation with deterministic dynamics, (3) the estimation of the correlation dimension to evaluate a static property of putative attractors, and (4) the analysis of nonlinear predictability of HRV time series which could reflect a dynamic property of the attractor. Unlike HRV at sea level, the recordings at over 6000 m showed a strong periodicity (period of about 20 s) with small cycle-to-cycle perturbation. When this perturbation was expressed on a Poincaré section, it seemed to be likely that the perturbation itself obeyed a deterministic law. The correlation dimensions of these recordings showed low dimensional values (3.5 ± 0.4, mean±SD), whereas those of the isospectral surrogates showed significantly (P < 0.05) higher values (5.3 ±0.5) with embedding dimensions of 5.6 ± 0.9. At over 6000 m, the correlation coefficients between the observed and the predicted time series with the prediction time of < 4 beats were significantly (P < 0.01) higher than those for the surrogate data, whereas there was no significant difference in the nonlinear predictability between the observed and the surrogate data at sea level. The results of the spectral analyses showed that, at over 6000 m, there was hardly any power > 0.15 Hz in the HRV spectra possibly due to PNS withdrawal. Hence, these deterministic and/or chaotic dynamics might be mediated by variations in SNS activity at over 6000 m. Present address and address for correspondence: Laboratory for Exercise Physiology and Biomechanics, Faculty of Education, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan     

Multipole electrostatic potential derived atomic charges in NDDO-methods with <Emphasis Type="Italic">spd</Emphasis>-basis sets

The recently introduced multipole approach for computing the molecular electrostatic potential (MEP) within the semiempirical neglect of diatomic differential overlap (NDDO) framework [Horn AHC, Lin Jr-H., Clark T (2005) Theor Chem Acc 114:159–168] has been used to obtain atomic charges of nearly ab initio quality by scaling the semiempirical MEP. The parameterization set comprised a total of 797 compounds and included not only the newly parameterized AM1* elements Al, Si, P, S, Cl, Ti, Zr, and Mo but also the standard AM1 elements H, C, N, O and F. For comparison, the ZDO-approximated MEP was also calculated analytically in the spd-basis. For the AM1*-optimized structures, single-point calculations at the B3LYP, HF and MP2 levels with the 6-31G(d) and LanL2DZP basis sets were performed to obtain the MEP. The regression analysis of all 12 combinations of semiempirical and ab initio MEP data yielded correlation coefficients of at least 0.99 in all cases. Scaling the analytical and multipole-derived semiempirical MEP by the regression coefficients yielded mean unsigned errors below 2.6 and 1.9 kcal mol⁻¹, respectively. Subsequently, for 22 drug molecules from the World Drug Index, atomic charges were computed according to the RESP procedure using XX/6-31G(d) (XX=B3LYP, HF, MP2) and scaled AM1* multipole MEP; the correlation coefficients obtained are 0.83, 0.85 and 0.83, respectively. Figure: Schematic representation of the atomic charge generation: The molecular electrostatic potential (MEP) is calculated using the AM1* Hamiltonian; then the semiempirical MEP is scaled to DFT or ab initio level, and atomic charges are generated subsequently by the restraint electrostatic potential (RESP) fit method. Electronic supplementary material Supplementary material is available in the online version of this article at and is accessible to authorized users. Proceedings of “Modeling Interactions in Biomolecules II”, Prague, September 5th–9th, 2005.     

Competition for Binding Between Veratridine and KIFMK: An Open Channel Blocking Peptide of the RIIA Sodium Channel

Veratridine, an alkaloid isolated from the rhizome of V. album, binds and slows the inactivation of the brain sodium channels. The synthetic pentapeptide KIFMK causes a voltage- and use-dependent open-channel block of the RIIA (rat brain type IIA) sodium channel (Eaholtz, Scheuer & Catterall, 1994). Our studies on the RIIA sodium channel expressed in CHO cells reveal that the fraction of veratridine modified sodium channels decreases linearly with increasing KIFMK concentration. However, the time constant for dissociation of veratridine from the channel remains unchanged in the presence of a high concentration of KIFMK, as opposed to that in the presence of QX314 where the dissociation appears to be more complex. These data are consistent with mutually exclusive binding of the open channel blocking peptide and veratridine to the brain sodium channel. Received: 19 November 1996/Revised: 31 July 1997     

When and where was the most recent common ancestor?

 The structured coalescent is investigated for single-locus, digenic samples in the diffusion limit of the unidimensional stepping-stone model for homogeneous, isotropic migration and random genetic drift. Let T denote the scaled time to the most recent common ancestor (MRCA) of the two genes, and let Z designate the scaled deviation of the position of the MRCA from the average position of the two genes. The joint probability density of T and Z is evaluated explicitly. Both the marginal and conditional distributions of T have infinite expectation, as does the marginal distribution of Z. Conditioned on T = τ, the distribution of Z is Gaussian with mean zero and variance 2τ. The main results are extended to anisotropic migration. The results establish the existence of and define in the diffusion limit a retrospective stochastic process for digenic samples in one spatial dimension. Received: 1 May 2001 / Revised version: 2 September 2001 / Published online: 8 February 2002     

A variable target intensity-restrained global optimization (VARTIGO) procedure for determining three-dimensional structures of polypeptides from NOESY data: Application to gramicidin-S

Summary A global optimization method for intensity-restrained structure refinement, based on variable target function (VTF) analysis, is illustrated using experimental data on a model peptide, gramicidin-S (GS) dissolved in DMSO. The method (referred to as VARTIGO for variable target intensity-restrained global optimization) involves minimization of a target function in which the range of NOE contacts is gradually increased in successive cycles of optimization in dihedral angle space. Several different starting conformations (including all-trans) have been tested to establish the validity of the method. Not all optimizations were successful, but these were readily identifiable from their large NOE R-factors. We also show that it is possible to simultaneously optimize the rotational correlation time along with the dihedral angles. The structural features of GS thus obtained from the successful optimizations are in excellent agreement with the available experimental data. A comparison is made with structures generated from an intensity-restrained single target function (STF) analysis. The results on GS suggest that VARTIGO refinement is capable of yielding better quality structures. Our work also underscores the need for a simultaneous analysis of different NOE R-factors in judging the quality of optimized structures. The NOESY data on GS in DMSO appear to provide evidence for the presence of two orientations for the ornithine side chain, in fast exchange. The NOESY spectra for this case were analyzed using a relaxation rate matrix which is a weighted average of the relaxation rate matrices for the individual conformations.     

Silica‐scaled chrysophytes from Abisko (Swedish Lapland)

The silica‐scaled chrysophyte flora of Swedish Lapland (near Abisko) was examined. Chrysophyte scales were found in 18 of the 32 investigated water bodies (lakes and pools). Altogether, 28 taxa from the classes Chrysophyceae and Synurophyceae were found. The most abundant species was Synura echinulata. Three species were observed for the first time in Sweden: Mallomonas maculata, M. rasilis and Spiniferomonas serrata.     

Combining geometric morphometrics,molecular phylogeny,and micropaleontology to assess evolutionary patterns in Mallomonas (Synurophyceae: Heterokontophyta)

Synurophytes, also known as scaled chrysophytes, are ecologically important algae that produce an array of siliceous structures upon which their taxonomy is based. Despite occupying a key position within the photosynthetic heterokonts, the evolutionary history of synurophytes remains poorly constrained. Here, modern and Middle Eocene siliceous scales of the morphotaxon Mallomonas insignis are used as a model to investigate synurophyte evolutionary patterns. Structural details of scale morphology were examined comparatively with scanning electron microscopy and scored for geometric morphometric analyses to assess the stability of shape characters. Although consistent size differences exist (modern scales are larger than Eocene counterparts), the populations cannot be differentiated on the basis of shape or microstructural detail, implying considerable evolutionary stasis in scale morphology. A time‐calibrated relaxed molecular clock analysis using a three‐gene concatenated data set (27 strains) suggests that the M. insignis lineage predates the available fossil record, having diverged from closest congeneric taxa in the Cretaceous (≥94 Ma). However, the molecular analysis also implies that considerable genetic variability is present within several morphotaxa of Mallomonas, implying that substantial genetic variability has arisen despite the retention of uniform scale morphologies, and resulting in the widespread occurrence of cryptic taxa. Results from the synurophyte lineage are consistent with the notion of protracted ghost ranges (>10 Ma) implied by the molecular phylogenies of other algal groups, together pointing to the paucity of the fossil record of these organisms on these timescales.