Canalization of Gene Expression and Domain Shifts in the Drosophila Blastoderm by Dynamical Attractors

The variation in the expression patterns of the gap genes in the blastoderm of the fruit fly Drosophila melanogaster reduces over time as a result of cross regulation between these genes, a fact that we have demonstrated in an accompanying article in PLoS Biology (see Manu et al., doi:10.1371/journal.pbio.1000049). This biologically essential process is an example of the phenomenon known as canalization. It has been suggested that the developmental trajectory of a wild-type organism is inherently stable, and that canalization is a manifestation of this property. Although the role of gap genes in the canalization process was established by correctly predicting the response of the system to particular perturbations, the stability of the developmental trajectory remains to be investigated. For many years, it has been speculated that stability against perturbations during development can be described by dynamical systems having attracting sets that drive reductions of volume in phase space. In this paper, we show that both the reduction in variability of gap gene expression as well as shifts in the position of posterior gap gene domains are the result of the actions of attractors in the gap gene dynamical system. Two biologically distinct dynamical regions exist in the early embryo, separated by a bifurcation at 53% egg length. In the anterior region, reduction in variation occurs because of stability induced by point attractors, while in the posterior, the stability of the developmental trajectory arises from a one-dimensional attracting manifold. This manifold also controls a previously characterized anterior shift of posterior region gap domains. Our analysis shows that the complex phenomena of canalization and pattern formation in the Drosophila blastoderm can be understood in terms of the qualitative features of the dynamical system. The result confirms the idea that attractors are important for developmental stability and shows a richer variety of dynamical attractors in developmental systems than has been previously recognized.     

Creative dynamics approach to neural intelligence

The thrust of this paper is to introduce and discuss a substantially new type of dynamical system for modelling biological behavior. The approach was motivated by an attempt to remove one of the most fundamental limitations of artificial neural networks — their rigid behavior compared with even simplest biological systems. This approach exploits a novel paradigm in nonlinear dynamics based upon the concept of terminal attractors and repellers. It was demonstrated that non-Lipschitzian dynamics based upon the failure of Lipschitz condition exhibits a new qualitative effect — a multi-choice response to periodic external excitations. Based upon this property, a substantially new class of dynamical systems — the unpredictable systems — was introduced and analyzed. These systems are represented in the form of coupled activation and learning dynamical equations whose ability to be spontaneously activated is based upon two pathological characteristics. Firstly, such systems have zero Jacobian. As a result of that, they have an infinite number of equilibrium points which occupy curves, surfaces or hypersurfaces. Secondly, at all these equilibrium points, the Lipschitz conditions fails, so the equilibrium points become terminal attractors or repellers depending upon the sign of the periodic excitation. Both of these pathological characteristics result in multi-choice response of unpredictable dynamical systems. It has been shown that the unpredictable systems can be controlled by sign strings which uniquely define the system behaviors by specifying the direction of the motions in the critical points. By changing the combinations of signs in the code strings the system can reproduce any prescribed behavior to a prescribed accuracy. That is why the unpredictable systems driven by sign strings are extremely flexible and are highly adaptable to environmental changes. It was also shown that such systems can serve as a powerful tool for temporal pattern memories and complex pattern recognition. It has been demonstrated that new architecture of neural networks based upon non-Lipschitzian dynamics can be utilized for modelling more complex patterns of behavior which can be associated with phenomenological models of creativity and neural intelligence.     

Phospholipid flip-flop modulated by transmembrane peptides WALP and melittin

Select transmembrane proteins found in biogenic membranes are known to facilitate rapid bidirectional flip-flop of lipids between the membrane leaflets, while others have no little or no effect. The particular characteristics which determine the extent to which a protein will facilitate flip-flop are still unknown. To determine if the relative polarity of the transmembrane protein segment influences its capacity for facilitation of flip-flop, we have studied lipid flip-flop dynamics for bilayers containing the peptides WALP₂₃ and melittin. WALP₂₃ is used as a model hydrophobic peptide, while melittin consists of both hydrophobic and hydrophilic residues. Sum-frequency vibrational spectroscopy (SFVS) was used to characterize the bilayers and determine the kinetics of flip-flop for the lipid component, 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC), within the mixed bilayers. The kinetic data were utilized to determine the activation thermodynamics for DSPC flip-flop in the presence of the peptides. Melittin was found to significantly reduce the free energy barrier to DSPC flip-flop when incorporated into the bilayer at 1 mol.%, while incorporation of WALP₂₃ at the same concentration led to a more modest reduction of the free energy barrier. The possible mechanisms by which these peptides facilitate flip-flop are analyzed and discussed in terms of the observed activation thermodynamics.     

Coexistence of Tonic Spiking Oscillations in a Leech Neuron Model

The leech neuron model studied here has a remarkable dynamical plasticity. It exhibits a wide range of activities including various types of tonic spiking and bursting. In this study we apply methods of the qualitative theory of dynamical systems and the bifurcation theory to analyze the dynamics of the leech neuron model with emphasis on tonic spiking regimes. We show that the model can demonstrate bi-stability, such that two modes of tonic spiking coexist. Under a certain parameter regime, both tonic spiking modes are represented by the periodic attractors. As a bifurcation parameter is varied, one of the attractors becomes chaotic through a cascade of period-doubling bifurcations, while the other remains periodic. Thus, the system can demonstrate co-existence of a periodic tonic spiking with either periodic or chaotic tonic spiking. Pontryagins averaging technique is used to locate the periodic orbits in the phase space.     

A multivibrating switching network in homogeneous kinetics

On the basis of an abstract, simple bistable reaction system (‘homogeneous Eccles-Jordan trigger’) used as anRS flip-flop, an abstract homogeneousastable flip-flop is devised. It can be run also as amonostable flip-flop and as aT flip-flop. The qualitative behavior of the three systems can be understood, in the limiting case, with the aid of Poincaré's notion of bifurcation of steady states. The reaction system is proposed as a paradigm for a specific class of ‘decomposable’ chemical and dynamical systems (so-called DC-type dynamical automata). Two possible biological applications are mentioned.     

A Cayley tree immune network model with antibody dynamics

A Cayley tree model of idiotypic networks that includes both B cell and antibody dynamics is formulated and analysed. As in models with B cells only, localized states exist in the network with limited numbers of activated clones surrounded by virgin or near-virgin clones. The existence and stability of these localized network states are explored as a function of model parameters. As in previous models that have included antibody, the stability of immune and tolerant localized states are shown to depend on the ratio of antibody to B cell lifetimes as well as the rate of antibody complex removal. As model parameters are varied, localized steady-states can break down via two routes: dynamically, into chaotic attractors, or structurally into percolation attractors. For a given set of parameters percolation and chaotic attractors can coexist with localized attractors, and thus there do not exist clear cut boundaries in parameter space that separate regions of localized attractors from regions of percolation and chaotic attractors. Stable limit cycles, which are frequent in the two-clone antibody B cell (AB) model, are only observed in highly connected networks. Also found in highly connected networks are localized chaotic attractors. As in experiments by Lundkvistet al. (1989.Proc. natn. Acad. Sci. U.S.A. 86, 5074–5078), injection ofAb ₁ antibodies into a system operating in the chaotic regime can cause a cessation of fluctuations ofAb ₁ andAb ₂ antibodies, a phenomenon already observed in the two-clone AB model. Interestingly, chaotic fluctuations continue at higher levels of the tree, a phenomenon observed by Lundkvistet al. but not accounted for previously.     

Complex behaviours of AB model describing idiotypic network

A simple chemical model of the idiotypic network of immune systems, namely the AB model, has been developed by De Boeret al. The complexity of the system, such as the steady states, periodic oscillations and chaotic motions, has been examined by the authors mentioned above. In the present paper, the periodic motions and chaotic behaviours exhibited by the system are intuitively described. To clarify in which parameter domains concerned the system exhibits periodic oscillations and in which parameter domains the system demonstrates chaotic behaviours the Lyapounov exponent is explored. To characterize the strangeness of the attractors, the fractal dimension problem is worked out.     

The bioenergetic coordination of a complex biological system is revealed by its adaptation to changing environmental conditions

The properties of the phosphate uptake system of the cyanobacterium Anacystis nidulans have been studied during the transition from a phosphate-deficient non-growing state to a non-deficient growing state. In the phosphate-deficient state the high affinity phosphate transport system in the cell membrane is extremely adaptive. As a result of these adaptive features the phosphate transport system cannot be described by determinate, fixed parameters, because the transport system is influenced by the measurement of the uptake process itself. When the growing state has been initiated by a persisting phosphate pulse, the transport system rapidly loses its adaptive features and can then be characterized by determinate parameters that remain unchanged for a long period of time, even if no uptake occurs in that time. Depending on the amount of phosphate stored during a pulse the cell makes a choice between slow or fast growth. In the latter case the light harvesting and energy converting machinery is completely reorganized before growth commences. Thereby the components of this machinery conform to each other and to the stable properties of the phosphate transport system. It is suggested that the mutual adjustment of these adaptive energy converting subunits is guided by attractors that function as the final cause for the development of the whole system.An application of this model to an analysis of the selforganization of aquatic ecosystems is discussed.     

Transbilayer (flip-flop) lipid motion and lipid scrambling in membranes

This paper reviews the current knowledge on the various mechanisms for transbilayer, or flip-flop, lipid motion in model and cell membranes, enzyme-assisted lipid transfer by flippases, floppases and scramblases is briefly discussed, while non-catalyzed lipid flip-flop is reviewed in more detail. Transbilayer lipid motion may occur as a result of the insertion of foreign molecules (detergents, lipids, or even proteins) in one of the membrane leaflets. It may also be the result of the enzymatic generation of lipids, e.g. diacylglycerol or ceramide, at one side of the membrane. Transbilayer motion rates decrease in the order diacylglycerol ? ceramide ? phospholipids. Ceramide, but not diacylglycerol, can induce transbilayer motion of other lipids, and bilayer scrambling. Transbilayer lipid diffusion and bilayer scrambling are defined as two conceptually and mechanistically different processes. The mechanism of scrambling appears to be related to local instabilities caused by the non-lamellar ceramide molecule, or by other molecules that exhibit a relatively slow flip-flop rate, when asymmetrically inserted or generated in one of the monolayers in a cell or model membrane.     

Phospholipid flip-out controls the cell cycle of Escherichia coli

Phospholipids are the principal constituents of biological membranes. In Escherichia coli, phospholipids are involved in the metabolism of other envelope constituents such as lipoprotein, lipopolysaccharide, certain envelope proteins and peptidoglycan. They are also involved in the regulation of the cell cycle. DNAA, the key protein in the initiation of chromosome replication, is activated by acidic phospholipids only when these are in fluid bilayers, whilst interruptions of phospholipid synthesis inhibit both the initiation of chromosome replication and cell division. The transmembrane movement or flip-flop of phospholipids from one monolayer to the other requires the passage of the polar head group through the hydrophobic core of the bilayer. Hence, in many systems, flip-flop is a slow process with half-time of days. Flip-flop accompanies the formation of non-bilayer structure. Such structures form under certain conditions of packing density and composition and have been observed both in vitro and in vivo. In bacteria, flip-flop appears to be extremely rapid, with half-times as fast as 3 min being observed. However, such rapid flip-flop may not be characteristic of all phospholipids. The asymmetrical distribution of phosphatidylethanolamine in the plasma membrane of Bacillus megaterium has been attributed to the existence of two classes of this phospholipid. In E. coli, studies of the metabolic turnover of phosphatidylserine, phosphatidylglycerol and phosphatidic acid also reveal the existence of distinct classes of these phospholipids. In this article I propose that, in E. coli, a class of phospholipids does indeed escape the rapid flip-flop mechanism; this class probably includes a subpopulation of the acidic phospholipids. Therefore during the cell cycle these phospholipids accumulate in the inner monolayer of the cytoplasmic membrane and so cause an increase in its packing density; at a critical density, phospholipids "flip out" from the inner to the outer monolayer. This flip-out occurs once per cycle and initiates cell cycle events.     

Dynamics of diluted attractor neural networks with delays

We present an analysis of the attractors of a deterministic dynamics in formal neural networks characterized by binary threshold units and a nonsymmetric connectivity. It is shown that in these networks a stored pattern or a pattern sequence is represented by a cloud of attractors rather than by a single attractor. Dilution, which we describe by a power-law scaling, and delayed couplings are shown to equip this type of network with a dynamic behaviour that is interesting enough for simplified models of biological motor systems. Received: 27 November 1992/Accepted in revised form: 22 September 1993     

On Ebenman's model for the dynamics of a population with competing juveniles and adults

A general version of a model of Ebenman for the dynamics of a population consisting of competing juveniles and adults is analyzed using methods of bifurcation theory. A very general existence results is obtained for non-trivial equilibria and non-negative synchronous two-cycles that bifurcate simultaneously at the critical valuer=1 of the inherent net reproductive rater. Stability is studied in this general setting near the bifurcation point and conditions are derived that determine which of these two bifurcating branches is the stable branch. These general results are supplemented by numerical studies of the asymptotic dynamics over wider parameter ranges where various other bifurcations and stable attractors are found. The implications of these results are discussed with respect to the effects on stability that age class competition within a population can have and whether such competition is stabilizing or destabilizing. Supported by National Science Foundation Grant No. DMS-8714810.     

Temporal properties of some biological systems and their fractal attractors

In this paper we analyse time series data as the growth of organisms using markers such as treerings and otolith deposits (fish). The series studied belong to two tree species (Pinus uncinata, Fagus sylvatica) and one fish species (Dicentrarchus labrax). Spectral analyses of the time series growth show that the main frequencies of fluctuation may be due to variations of the energy input. However, any causal explanation must consider the internal continuous readjustment in the system as reported by the corresponding chaotic properties of the asymptotic decay of the spectra time structure. Since the output of noisy and chaotic systems tend to show similar spectral densities, an attempt to differentiate them has been carried out. The chaotic behaviour has been characterized by the study of the attractors. The dimmensions of these multiple topologies were 3.2 and 3.4 for the tree species and 2.3 for the fish species. Therefore, we are dealing with fractal attractors and the minimum number of variables that can be used to describe the systems are 4 and 3 respectively. It is suggested that some of the variables that most influence growth are those obtained by the response functions in the case of trees.     

The rainbow bridge: Hamiltonian limits and resonance in predator-prey dynamics

 In the presence of seasonal forcing, the intricate topology of non-integrable Hamiltonian predator-prey models is shown to exercise profound effects on the dynamics and bifurcation structure of more realistic schemes which do not admit a Hamiltonian formulation. The demonstration of this fact is accomplished by writing the more general models as perturbations of a Hamiltonian limit, ℋ, in which are contained infinite numbers of periodic, quasiperiodic and chaotic motions. From ℋ, there emanates a surface, Γ, of Nejmark-Sacker bifurcations whereby the annual oscillations induced by seasonality are destabilized. Connecting Γ and ℋ is a bridge of resonance horns within which invariant motions of the Hamiltonian case persist. The boundaries of the resonance horns are curves of tangent (saddle-node) bifurcations corresponding to subharmonics of the yearly cycle. Associated with each horn is a rotation number which determines the dominant frequency, or “color”, of attractors within the horn. When viewed through the necessarily coarse filter of ecological data acquisition, and regardless of their detailed topology, these attractors are often indistinguishable from multi-annual cycles. Because the tips of the horns line up monotonically along Γ, it further follows that the distribution of observable periods in systems subject to fluctuating parameter values induced, for example, by year-to-year variations in the climate, will often exhibit a discernible central tendency. In short, the bifurcation structure is consistent with the observation of multi-annual cycles in Nature. Fundamentally, this is a consequence of the fact that the bridge between ℋ and Γ is a rainbow bridge. While the present analysis is principally concerned with the two species case (one predator and one prey), Hamiltonian limits are also observed in other ecological contexts: 2n-species (n predators, n prey) systems and periodically-forced three level food chain models. Hamiltonian limits may thus be common in models involving the destruction of one species by another. Given the oft-commented upon structural instability of Hamiltonian systems and the corresponding lack of regard in which they are held as useful caricatures of ecological interactions, the pivotal role assigned here to Hamiltonian limits constitutes a qualitative break with the conventional wisdom. Received: 2 November 1998     

Inhibition of type 2A secretory phospholipase A2 reduces death of cardiomyocytes in acute myocardial infarction

During acute myocardial infarction (AMI), ischemia leads to necrotic areas surrounded by border zones of reversibly damaged cardiomyocytes, showing membrane flip-flop. During reperfusion type IIA secretory phopholipase A₂ (sPLA₂-IIA) induces direct cell-toxicity and facilitates binding of other inflammatory mediators on these cardiomyocytes. Therefore, we hypothesized that the specific sPLA₂-IIA-inhibitor PX-18 would reduce cardiomyocyte death and infarct size in vivo. Wistar rats were treated with PX-18 starting minutes after reperfusion, and at day 1 and 2 post AMI. After 28 days hearts were analyzed. Furthermore, the effect of PX-18 on membrane flip-flop and apoptosis was investigated in vitro. PX-18 significantly inhibited sPLA₂-IIA activity and reduced infarct size (reduction 73 ± 9%, P < 0.05), compared to the vehicle-treated group, without impairing wound healing. In vitro, PX-18 significantly reduced reversible membrane flip-flop and apoptosis in cardiomyocytes. However, no sPLA₂-IIA activity could be detected, suggesting that PX-18 also exerted a protective effect independent of sPLA₂-IIA. In conclusion, PX-18 is a potent therapeutic to reduce infarct size by inhibiting sPLA₂-IIA, and possibly also by inhibiting apoptosis of cardiomyocytes in a sPLA₂-IIA independent manner. A. van Dijk and P. A. J. Krijnen have contributed equally to the study.     

复杂性与脑功能

EEG代表了大脑活动的一种电信号,但是用它来研究脑的功能活动是非常困难的.近年来由于非线性动力学的新发展,为我们提供了从一维EEG的时间序列提取脑的多维动力系统的信息,其中一个重要的方法是测量“关联维数”,但发现EEG是非平稳的混沌态,分维的知识只能给出系统的几何特征.而非平稳性表现出的是动态特性.因此我们对EEC的“复杂性”进行了研究,并与其它已知的标准的奇异吸引子做了比较.     

Neural Avalanches at the Critical Point between Replay and Non-Replay of Spatiotemporal Patterns

We model spontaneous cortical activity with a network of coupled spiking units, in which multiple spatio-temporal patterns are stored as dynamical attractors. We introduce an order parameter, which measures the overlap (similarity) between the activity of the network and the stored patterns. We find that, depending on the excitability of the network, different working regimes are possible. For high excitability, the dynamical attractors are stable, and a collective activity that replays one of the stored patterns emerges spontaneously, while for low excitability, no replay is induced. Between these two regimes, there is a critical region in which the dynamical attractors are unstable, and intermittent short replays are induced by noise. At the critical spiking threshold, the order parameter goes from zero to one, and its fluctuations are maximized, as expected for a phase transition (and as observed in recent experimental results in the brain). Notably, in this critical region, the avalanche size and duration distributions follow power laws. Critical exponents are consistent with a scaling relationship observed recently in neural avalanches measurements. In conclusion, our simple model suggests that avalanche power laws in cortical spontaneous activity may be the effect of a network at the critical point between the replay and non-replay of spatio-temporal patterns.     

Dipole Theory of Heat Production and Absorption in Nerve Axon

Exact formulas are derived for the energy change of a dipole system with two energy states (or bands) in a changing field in two cases: (a) no dipole flip-flop and (b) dipole flip-flop caused by stimulation. Based on these formulas, the positive and negative heats are calculated. The results are in good agreement with experiment in case b but are 60-180% larger in case a. Furthermore, the theory shows that the negative heat cannot be less than the positive heat in case a but can be either way in case b, the latter result being found prevalent in experiment. It is concluded that nerve excitation is most likely to involve dipole flip-flop at the membrane surface. The theory is consistent in the interpretations and correlations of the electrical, optical, and thermal effects observed in nerve axon.     

Large species shifts triggered by small forces

Changes in species composition of communities seem to proceed gradually at first sight, but remarkably rapid shifts are known to occur. Although disrupting disturbances seem an obvious explanation for such shifts, evidence for large disturbances is not always apparent. Here we show that complex communities tend to move through occasional catastrophic shifts in response to gradual environmental change or evolution. This tendency is caused by multiple attractors that may exist in such systems. We show that alternative attractors arise robustly in randomly generated multispecies models, especially if competition is symmetrical and if interspecific competition is allowed to exceed intraspecific competition. Inclusion of predators as a second trophic level did not alter the results greatly, although it reduced the probability of alternative attractors somewhat. These results suggest that alternative attractors may commonly arise from interactions between large numbers of species. Consequently, the response of complex communities to environmental change is expected to be characterized by hysteresis and sudden shifts. Some unexplained regime shifts observed in ecosystems could be related to alternative attractors arising from complex species interactions. Additionally, our results support the idea that ancient mass extinctions may partly be due to an intrinsic loss of stability of species configurations.