Some views on the role of noise in “self”-organizing systems

This paper deals with information transfer from the environment and “self”-organization in open, nonlinear systems far from thermodynamic equilibrium — in the presence of either non-stationary phase jitter noise, or amplitude stationary noise. By “self”-organization we mean here the progressive formation within the system of sequential, ordered (coherent) relationships between appropriate dynamical variables-like for example, the phase differences between the oscillating components of the system. We take up (in Section II) the classical Laser as a specific example and examine in detail the influence of phase jitter noise in the mode (phase) locking process. We find—as expected—that phase fluctuations in the cavity cause degradation of the coherent behaviour (i.e. increase the entropy) of the system — which, however, levels off, or saturates with time. Further (in Section III) we examine systems where the number of self-sustained oscillating components may vary with time in such a way that the maximum entropy of the system increases faster than the overall instantaneous entropy. We put forth the hypothesis that in such cases — because of the increase of the redundancy — the system gets organized not just in spite of, but merely because of the presence of Noise. Possible applications in biological systems (especially concerning a model of cerebral organization) are briefly discussed. It is understood here, that the system has to display some preliminary dynamical structure before the organizing procedure takes over. What happens afterwards is the subject of this paper.     

On estimating the probability of aperiodic outbursts of microbial populations from their fluctuating counts

The irregular sequence of counts of a microbial population, in the absence of observable corresponding environmental changes (e.g., temperature), can be regarded as reflecting the interplay of several unknown or random factors that favor or inhibit growth. Since these factors tend to balance one another, the fluctuations usually remain within bounds, and only by a coincidence—when all or most act in unison—does an ‘outburst’ occur. This situation can be represented mathematically as a sequence of independent random variables governed by a probability distribution. The concept was applied to reported microbial counts of ground meat and wastewater. It is found that the lognormal distribution could serve as a model, and that simulations from this model are indistinguishable from actual records. The parameters of the lognormal (or other) distribution can then be used to estimate the probability of a population outburst, i.e., an increase above a given threshold. Direct estimation of the outburst probability based on frequency of occurrence is also possible, but in some situations requires an impractically large number of observations. We compare the efficiency of these two methods of estimation. Such methods enable translation of irregular records of microbial counts into actual probabilities of an outburst of a given magnitude. Thus, if the environment remains ’stable’ or in dynamic equilibrium, the fluctuations should not be regarded merely as noise, but as a source of information and an indicator of potential population outbursts even where obvious signs do not exist.     

Theory predicts a rapid transition from niche-structured to neutral biodiversity patterns across a speciation-rate gradient

A central challenge in community ecology is to predict patterns of biodiversity with mechanistic models. The neutral model of biodiversity is a simple model that appears to provide parsimonious and accurate predictions of biodiversity patterns in some ecosystems, even though it ignores processes such as species interactions and niche structure. In a recent paper, we used analytical techniques to reveal why the mean predictions of the neutral model are robust to niche structure in high diversity but not low-diversity ecosystems. In the present paper, we explore this phenomenon further by generating stochastic simulated data from a spatially implicit hybrid niche-neutral model across different speciation rates. We compare the resulting patterns of species richness and abundance with the patterns expected from a pure neutral and a pure niche model. As the speciation rate in the hybrid model increases, we observe a surprisingly rapid transition from an ecosystem in which diversity is almost entirely governed by niche structure to one in which diversity is statistically indistinguishable from that of the neutral model. Because the transition is rapid, one prediction of our abstract model is that high-diversity ecosystems such as tropical forests can be approximated by one simple model—the neutral model—whereas low-diversity ecosystems such as temperate forests can be approximated by another simple model—the niche model. Ecosystems that require the hybrid model are predicted to be rare, occurring only over a narrow range of speciation rates.     

Mathematical models of predator/prey/plant interactions in a patch environment

Several tritrophic systems are characterized by local over-exploitation of the food source. Interactions between predatory mites, spider mites and their host plants are an example of such systems: either the spider mites over-exploit local patches of host plants or the spider mites are exterminated by predatory mites. It is often stated that modelling the overall population dynamics of such systems in a realistic way would soon lead to an unmanageable edifice. We advocate, however, the use of physiologically structured population models as a both general and formal mathematical framework. The advantage is that analytically tractable models may be obtained from the complex ‘master’ model by time-scale arguments or special choices of model ingredients. In this way a network of models can be derived, each concentrating on a particular aspect, all inadequate to cover the entire spectrum, but together (we hope) providing a coherent set of insights the relative importance of which can be assessed by computer experiments on the ‘master’ model. In this paper a rather realistic model of predator/prey interactions in an ensemble of host-plant patches is presented and, as an example of our approach, some special cases are derived from that model. Their analysis provided some first, useful insights. It is shown that prolonged duration of the prey-dispersal phase and prey dispersal from predator (-invaded prey) patches may result in a stable steady state, whereas a humped plant-production function may — under certain conditions — result in two stable steady states.     

Two-step concerted mechanism for alkane hydroxylation on the ferryl active site of methane monooxygenase

 A two-step concerted mechanism for the conversion of methane to methanol catalyzed by soluble methane monooxygenase (sMMO) is discussed. We propose that the enzymatic reaction mechanism is essentially the same as that of the gas-phase methane-methanol conversion by the bare FeO⁺ complex. In the initial stage of our mechanism, the ferryl (Fe—O) "iron" active site of intermediate Q and substrate methane come into contact to form the initial Q (CH₄) complex with an OFe—CH₄ bond. The C—H bonds of methane are significantly weakened by the formation of a five-coordinate carbon species, through orbital interactions between a C _3v - or D _2d -distorted methane and the Fe—O active site. The important transition state for an H atom abstraction exhibits a four-centered structure. The generated intermediate involves an HO—Fe—CH₃ moiety, and it is then converted into the final product complex including methanol as a ligand through a methyl migration that occurs via a three-centered transition state. The two-step concerted mechanism is consistent with recent experiments on regioselectivity of enzyme-catalyzed alkane hydroxylations. Received: 15 September 1997 / Accepted: 20 December 1997     

Noise-induced transition in excitable neuron models

 We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model – as has been done previously – our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles. Received: 19 September 2000 / Accepted in revised form: 4 March 2001     

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

Stable oscillations in mathematical models of biological control systems

Summary Oscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.     

Predicting the effect of climate change on temperate shallow lakes with the ecosystem model PCLake

Global average surface temperatures are expected to rise by about 1.4–5.8°C from the present until the year 2100. This temperature increase will affect all ecosystems on earth. For shallow lakes—which can be either in a clear water or a turbid state—this climate change will expectedly negatively affect water transparency though the prediction is far from conclusive and experimental investigations elucidating the potential climatic effects on shallow lakes are still rare. The aim of this study was to further shape and sharpens hypotheses on the impact of climate change on shallow lakes by applying an existing and well-calibrated ecosystem model, PCLake. We focused on asymptotic model behaviour for a range of temperature and loading scenarios in a factorial design. We conclude that climate change will likely lead to decreased critical nutrient loadings. Combined with an expected increase in the external nutrient loading, this will increase the probability of a shift from a clear to a turbid state. As the model predicts a higher summer chlorophyll-a concentration, a stronger dominance of cyanobacteria during summer and a reduced zooplankton abundance due to climate change, the turbid state itself is likely to become even more severe.     

1/f-Noise of Open Bacterial Porin Channels

General diffusion pores and specific porin channels from outer membranes of gram-negative bacteria were reconstituted into lipid bilayer membranes. The current noise of the channels was investigated for the different porins in the open state and in the ligand-induced closed state using fast Fourier transformation. The open channel noise exhibited 1/f-noise for frequencies up to 200 Hz. The 1/f-noise was investigated using the Hooge formula (Hooge, Phys. Lett. 29A: 139–140 (1969)), and the Hooge parameter α was calculated for all bacterial porins used in this study. The 1/f-noise was in part caused by slow inactivation and activation of porin channels. However, when care was taken that during the noise measurement no opening or closing of porin channels occurred, the Hooge Parameter α was a meaningful number for a given channel. A linear relationship was observed between α and the single-channel conductance, g, of the different porins. This linear relation between single-channel conductance and the Hooge parameter α could be qualitatively explained by assuming that the passing of an ion through a bacterial porin channel is—to a certain extent—influenced by nonlinear effects between channel wall and passing ion. Received: 8 May 1996/Revised: 27 January 1997     

The role of noise in a predator–prey model with Allee effect

The existence and implications of alternative stable states in ecological systems have been investigated extensively within deterministic models. However, it is known that natural systems are undeniably subject to random fluctuations, arising from either environmental variability or internal effects. Thus, in this paper, we study the role of noise on the pattern formation of a spatial predator–prey model with Allee effect. The obtained results show that the spatially extended system exhibits rich dynamic behavior. More specifically, the stationary pattern can be induced to be a stable target wave when the noise intensity is small. As the noise intensity is increased, patchy invasion emerges. These results indicate that the dynamic behavior of predator–prey models may be partly due to stochastic factors instead of deterministic factors, which may also help us to understand the effects arising from the undeniable susceptibility to random fluctuations of real ecosystems.     

Interaction of poly(l-lysines) with negatively charged membranes: an FT-IR and DSC study

The influence of the binding of poly(l-lysine) (PLL) to negatively charged membranes containing phosphatidylglycerols (PG) was studied by DSC and FT-IR spectroscopy. We found a general increase in the main transition temperature as well as increase in hydrophobic order of the membrane upon PLL binding. Furthermore we observed stronger binding of hydration water to the lipid head groups after PLL binding. The secondary structure of the PLL after binding was studied by FT-IR spectroscopy. We found that PLL binds in an α-helical conformation to negatively charged DPPG membranes or membranes with DPPG-rich domains. Moreover we proved that PLL binding induces domain formation in the gel state of mixed DPPC/DPPG or DMPC/DPPG membranes as well as lipid remixing in the liquid–crystalline state. We studied these effects as a function of PLL chain length and found a significant dependence of the secondary structure, phase transition temperature and domain formation capacity on PLL chain length and also a correlation between the peptide secondary structure and the phase transition temperature of the membrane. We present a system in which the membrane phase transition triggers a highly cooperative secondary structure transition of the membrane-bound peptide from α-helix to random coil. Dedicated to Prof. K. Arnold on the occasion of his 65th birthday.     

Metastable states, phase transitions, and persistent neural activity

A new mathematical model to describe the spiking rate of a neural population is derived, which considers both the mean and the variance of the activity. Bifurcation analysis identifies a critical interval of parameter values in which the standard bistability regime coexists with an additional third attractor corresponding to the metastable state of bounded mean activity and high variance. To understand the structure of spatio-temporal activity in the metastable state, we study a simple discrete-time model of binary elements with random noise locally coupled on the grid, which produces rich dynamics including metastability. A critical value of the noise amplitude is identified; in the vicinity of this value the system is flexible and can easily generate transitions between UP and DOWN metastable states, either autonomously or in response to a control process. These metastable states and phase transitions provide a proper basis for the modelling of persistent neural activity reported in many experimental studies.     

Effects of Intrinsic and Extrinsic Noise Can Accelerate Juxtacrine Pattern Formation

Epithelial pattern formation is an important phenomenon that, for example, has roles in embryogenesis, development and wound-healing. The ligand Epithelial Growth Factor (EGF) and its receptor EGF-R, constitute a system that forms lateral induction patterns by juxtacrine signalling—binding of membrane-bound ligands to receptors on neighbouring cells. Owen et al. developed a generic ordinary differential equation model of juxtacrine lateral induction that exhibits stable patterning under some conditions. The model predicts relatively slow pattern formation. We examine here the effects of both intrinsic and extrinsic cellular noise arising from the stochastic treatment of this model, and show that this noise could have an accelerating effect on the patterning process.     

Homochirality and the Need for Energy

The mechanisms for explaining how a stable asymmetric chemical system can be formed from a symmetric chemical system, in the absence of any asymmetric influence other than statistical fluctuations, have been developed during the last decades, focusing on the non-linear kinetic aspects. Besides the absolute necessity of self-amplification processes, the importance of energetic aspects is often underestimated. Going down to the most fundamental aspects, the distinction between a single object—that can be intrinsically asymmetric—and a collection of objects—whose racemic state is the more stable one—must be emphasized. A system of strongly interacting objects can be described as one single object retaining its individuality and a single asymmetry; weakly or non-interacting objects keep their own individuality, and are prone to racemize towards the equilibrium state. In the presence of energy fluxes, systems can be maintained in an asymmetric non-equilibrium steady-state. Such dynamical systems can retain their asymmetry for times longer than their racemization time.     

A neural mechanism of hyperaccurate detection of phase advance and delay in the jamming avoidance response of weakly electric fish

 The weakly electric fish Eigenmannia can detect the phase difference between a jamming signal and its own signal down to 1 s. To clarify the neuronal mechanism of this hyperaccurate detection of phase difference, we present a neural network model of the torus of the midbrain which plays an essential role in the detection of phase advances and delays. The small-cell model functions as a coincidence detector and can discriminate a time difference of more than 100 s. The torus model consists of laminae 6 and 8. The model of lamina 6 is made with multiple encoding units, each of which consists of a single linear array of small cells and a single giant cell. The encoding unit encodes the phase difference into its spatio-temporal firing pattern. The spatially random distribution of small cells in each encoding unit improves the encoding ability of phase modulation. The neurons in lamina 8 can discriminate the phase advance and delay of jamming electric organ discharges (EODs) compared with the phase of the fish's own EOD by integrating simultaneously the outputs from multiple encoding units in lamina 6. The discrimination accuracy of the feature-detection neurons is of the order of 1 s. The neuronal mechanism generating this hyperacuity arises from the spatial feature of the system that the innervation sites of small cells in different encoding units are distributed randomly and differently on the dendrites of single feature-detection neurons. The mechanism is similar to that of noise-enhanced information transmission. Received: 10 July 2000 / Accepted in revised form: 19 January 2001     

A Riemannian manifold analysis of endothelial cell monolayer impedance parameter precision

Endothelial cell adhesion and barrier function play a critical role in many biological and pathophysiological processes. The decomposition of endothelial cell adhesion and barrier function into cell–cell and cell–matrix components using frequency dependent cellular micro-impedance measurements has, therefore, received widespread application. Few if any studies, however, have examined the precision of these model parameters. This study presents a parameter sensitivity analysis of a representative cellular barrier function model using a concise geometric formulation that includes instrumental data acquisition settings. Both model state dependence and instrumental noise distributions are accounted for within the framework of Riemannian manifold theory. Experimentally acquired microimpedance measurements of attached endothelial cells define the model state domain, while experimentally measured noise statistics define the data space Riemannian metric based on the Fisher information matrix. The results of this analysis show that the sensitivity of cell–cell and cell–matrix impedance components are highly model state dependent and several well defined regions of low precision exist. The results of this study further indicate that membrane resistive components can significantly reduce the precision of the remaining parameters in these models. This work was supported by a National Science Foundation CAREER Award (AE), BES-0238905, and in part by the American Heart Association under Grant 0265029B (AE).     

The enhancement of stability of p53 in MTBP induced p53–MDM2 regulatory network

We have modeled an MTBP-MDM2–p53 regulatory network by integrating p53–MDM2 autoregulatory model (Proctor and Gray, 2008) with the effect of a cellular protein MTBP (MDM2 binding protein) which is allowed to bind with MDM2 (Brady et al., 2005). We study this model to investigate the activation of p53 and MDM2 steady state levels induced by MTBP protein under different stress conditions. Our simulation results in three approaches namely deterministic, Chemical Langevin equation and stochastic simulation of Master equation show a clear transition from damped limit cycle oscillation to fixed point oscillation during a certain time period with constant stress condition in the cell. This transition is the signature of transition of p53 and MDM2 levels from activated state to stabilized steady state levels. We present various phase diagrams to show the transition between unstable and stable states of p53 and MDM2 concentration levels and also their possible relations among critical value of the parameters at which the respective protein level reach stable steady states. In the stochastic approach, the dynamics of the proteins become noise induced process depending on the system size. We found that this noise enhances the stability of the p53 steady state level.