Hydrophobic free energy eigenfunctions of pore, channel, and transporter proteins contain beta-burst patterns.

Hydropathy plots are often used in place of missing physical data to model transmembrane proteins that are difficult to crystallize. The sequential maxima of their graphs approximate the number and locations of transmembrane segments, but potentially useful additional information about sequential hydrophobic variation is lost in this smoothing procedure. To explore a broader range of hydrophobic variations without loss of the transmembrane segment-relevant sequential maxima, we utilize a sequence of linear decompositions and transformations of the n-length hydrophobic free energy sequences, Hi, i = 1...n, of proteins. Constructions of hydrophobic free energy eigenfunctions, psil, from M-lagged, M x M autocovariance matrices, CM, were followed by their all-poles, maximum entropy power spectral, Somega(psil), and Mexican Hat wavelet, Wa,b(psil), transformations. These procedures yielded graphs indicative of inverse frequencies, omega-1, and sequence locations of hydrophobic modes suggestive of secondary and supersecondary protein structures. The graphs of these computations discriminated between Greek Key, Jelly Role, and Up and Down categories of antiparallel beta-barrel proteins. With these methods, examples of porins, connexins, hexose transporters, nuclear membrane proteins, and potassium but not sodium channels appear to belong to the Up and Down antiparallel beta-barrel variety.

Mode matches in hydrophobic free energy eigenfunctions predict peptide-protein interactions.

The dominant statistical hydrophobic free energy inverse frequencies amino acid wavelengths as hydrophobic modes, of neurotensin (NT), cholescystokinin (CCK), the human dopamine D2 receptor [(DA)D2], and the human dopamine transporter (DAT) were determined using orthogonal decomposition of the autocovariance matrices of their amino acid sequences as hydrophobic free energy equivalents in kcal/mol. The leading eigenvalues-associated eigenvectors were convolved with the original series to construct eigenfunctions. Eigenfunctions were further analyzed using discrete trigonometric wavelet and all poles, maximum entropy power spectral transformations. This yielded clean representations of the dominant hydrophobic free energy modes, most of which are otherwise lost in the smoothing of hydropathy plots or contaminated by end effects and multimodality in conventional Fourier transformations. Mode matches were found between NT and (DA)D2 and between CCK and DAT, but not the converse. These mode matches successfully predicted the nonlinear kinetic interactions of NT-(DA)D2 in contrast with CCK-(DA) D2 on 3H-spiperone binding to (DA) D2, and by CCK-DAT but not NT-DAT on [N-methyl-3H]-WIN 35,428 binding to DAT in (DA)D2 and DAT cDNA stably transfected cell lines without known NT or CCK receptors. Computation of the dominant modes of hydrophobic free energy eigenfunctions may help predict functionally relevant peptide-membrane protein interactions, even across neurotransmitter families.

The development of nuchal atonia associated with active (REM) sleep in fetal sheep: presence of recurrent fractal organization.

The behavioral state of active or rapid eye movement sleep (REMS) is dominant during fetal life and may play an important role in brain development. One marker of this state in fetal sheep is neck nuchal muscle atonia (NA). We observed burst within burst NA patterns suggestive of recurrent fractal organization in continuous 13 day in utero recordings of NA during the third trimester. Consistent with fractal renewal processes, the cumulative mean and standard deviation (SD) diverged over this time and the tail of NA distributions fit a stable Lévy law with exponents that remained invariant over the periods of development examined. The Hurst exponent, a measure of self-affine fractals, indicated that long-range correlations among NA intervals were present throughout development. A conserved complex fractal structure is apparent in NA which may help elucidate ambiguities in defining fetal states as well as some unique properties of fetal REMS.

Mode matches and their locations in the hydrophobic free energy sequences of peptide ligands and their receptor eigenfunctions.

Patterns in sequences of amino acid hydrophobic free energies predict secondary structures in proteins. In protein folding, matches in hydrophobic free energy statistical wavelengths appear to contribute to selective aggregation of secondary structures in "hydrophobic zippers." In a similar setting, the use of Fourier analysis to characterize the dominant statistical wavelengths of peptide ligands' and receptor proteins' hydrophobic modes to predict such matches has been limited by the aliasing and end effects of short peptide lengths, as well as the broad-band, mode multiplicity of many of their frequency (power) spectra. In addition, the sequence locations of the matching modes are lost in this transformation. We make new use of three techniques to address these difficulties: (i) eigenfunction construction from the linear decomposition of the lagged covariance matrices of the ligands and receptors as hydrophobic free energy sequences; (ii) maximum entropy, complex poles power spectra, which select the dominant modes of the hydrophobic free energy sequences or their eigenfunctions; and (iii) discrete, best bases, trigonometric wavelet transformations, which confirm the dominant spectral frequencies of the eigenfunctions and locate them as (absolute valued) moduli in the peptide or receptor sequence. The leading eigenfunction of the covariance matrix of a transmembrane receptor sequence locates the same transmembrane segments seen in n-block-averaged hydropathy plots while leaving the remaining hydrophobic modes unsmoothed and available for further analyses as secondary eigenfunctions. In these receptor eigenfunctions, we find a set of statistical wavelength matches between peptide ligands and their G-protein and tyrosine kinase coupled receptors, ranging across examples from 13.10 amino acids in acid fibroblast growth factor to 2.18 residues in corticotropin releasing factor. We find that the wavelet-located receptor modes in the extracellular loops are compatible with studies of receptor chimeric exchanges and point mutations. A nonbinding corticotropin-releasing factor receptor mutant is shown to have lost the signatory mode common to the normal receptor and its ligand. Hydrophobic free energy eigenfunctions and their transformations offer new quantitative physical homologies in database searches for peptide-receptor matches.

Dynamical entropy is conserved during cocaine-induced changes in fetal rat motor patterns.

Our previous studies demonstrated that the intra-cisternal (IC) administration of cocaine to fetal rats increased motor activity and decreased responsiveness to perioral stimulation. One explanation for these observations comes from the behavioral pharmacology of stimulant drugs: increased motor activity is often associated with a decrease in its variety. Previous power spectral transformation of this data suggests an alternative explanation: cocaine-induced hyperactivity fixates a new behavioral pattern with complexity equal to that of saline controls. We explore these possibilities using statistical techniques derived from studies of nonlinear dynamical systems, examining patterns of the total motor activity of the individual fetus as counts per 5 s interval on either gestational day E20 or E21 for 20 min following IC injections of saline, 2.5 or 10 mg/kg of cocaine. The results are consistent with a state in which increased spontaneous activity is associated with the emergence of a new dynamical pattern which conserves entropy and provides experimental support for a fundamental conservation-variational relation, hT approximately equal to lambda 1 x DR, that has been proven for abstract models of chaotic dynamical systems. A multivariate analysis of variance (MANOVA) followed by appropriate analyses of variance (ANOVAs) and pairwise comparisons revealed that, whereas cocaine induced increases in the total amount of motor activity, the rate of increase in the variety of new sequences in activity counts over time did not change with treatment and age conditions. This invariant is quantified by an absence of change in topological entropy, delta hT = 0. The analyses also showed that, in order to maintain hT values, compensatory changes took place in the leading Lyapounov characteristic exponent, lambda 1 (the distance between sequential values 'stretched' along the increasing amplitudes of the variations) such that delta lambda 1 > 0, and the correlation dimension, DR (the hierarchical range of possible values, 'complicated clustering') was reduced, so that delta DR < 0. Our findings are consistent with the idea that the association between cocaine-induced increases in activity and decreases in adaptive response are not due to the dynamical entropy loss of decreased behavioral variety, but are rather the result of competitive interference by a drug-induced, equally complex, new pattern of spontaneous behavior.

Nonlinear dynamical patterns as personality theory for neurobiology and psychiatry.

ADVANCES in the theory of nonlinear differential equations and their statistical representations have yielded a powerful, qualitatively descriptive yet quantitative language that captures characteristic patterns of behavior (what the psychoanalyst Roy Schafer calls "continuity, coherence, and consistency of action") that has begun to influence studies of complex systems in motion as diverse in specifics as signatory patterns of discharge of neurochemically defined single neurons and the dynamical structures characteristic of a particular composer's music. What might be called personality theories of neurobiological dynamics have arisen to replace neurobiological theories of personality. It is in this way that rigorously proven and powerful general mathematical insights have changed the face of determinism in research in brain and behavior. Two examples: (1) Very complicated looking behavior of neurobiological forced-dissipative (expanding and contracting) systems over time take place on low dimensional abstract surfaces on which only a few underlying abstract parameters control the action. (2) Independent of specific details (chemical, electrical, and/or behavioral), there exist a relatively few fundamental categories of behavior in time and transitions, among them a property called universality. Results from this new theoretical, in contrast with experimental, reductionism yield analogies with and new approaches to historically important dynamic ideas about personality and character patterns that are equally relevant to micro and macrocomplex systems such as neural membrane receptor proteins and individual personality styles. Research findings achieved over the past decade and a half in our laboratory and others in neurochemistry, neurophysiology, and animal and human behavior, as well as the results of a new demonstration experiment involving the prediction of dynamical category membership from abstract expressive motion in humans, are used to exemplify this use of a quantitative dynamic category theory across disciplinary levels in brain and behavior. Multiple measures of complexity adapted from current research in the statistical properties of chaos on unobtrusively observed and reconstructed orbits on the computer screen made by non-premorbid subjects executing content-free, computer-game-like tasks with a mouse, were used to reliably differentiate the "signatures" of two Axis II diagnoses as established using SCID-II criteria. Whereas the techniques of nonlinear systems have achieved some success in quantifying and stimulating the dynamical styles of relatively local phenomena such as the spontaneous behavior of neuronal membrane conductances, single neurons, neural networks, and field electrical events, we think that the real power of these techniques lies in their quantitative description and statistical prediction of global patterns of behavior of entire systems. For example, since the late 1970s our work has shown that these measures could be used to discriminate categories of drug action and dose when applied to patterns of rat exploratory behavior in space and time. The combination of abstract generality and quantitative precision of these methods suggests their usefulness as a cross-disciplinary language for fields like psychiatry that deal with complicated behavior of both neurobiological elements and "the whole person."