Surface properties of cholesterol-containing membranes detected by Prodan fluorescence

The fluorescent membrane probe 6-propionyl-2-dimethylaminonaphthalene (Prodan) displays a high sensitivity to the polarity and packing properties of lipid membrane. Contrary to 6-lauroyl-2-dimethylaminonaphthalene (Laurdan), Prodan can also monitor the properties of the membrane surface, i.e., the polar-head pretransition. In bilayers composed of coexisting gel and liquid-crystalline phases, Prodan shows a preferential partitioning in the latter, so that the detected membrane properties mainly belong to fluid domains. In the presence of cholesterol, the packing properties of the gel phase phospholipids are modified in such a way that Prodan can penetrate and label the membrane. Although Prodan labeling of the gel phase is a function of cholesterol concentration, 3 mol percent cholesterol is sufficient for a 60% Prodan labeling with respect to the maximum labeling reached at 15 mol percent cholesterol. We present steady-state and dynamical fluorescence measurements of Prodan in bilayers in the presence of cholesterol. Our results fit the liquid-ordered/liquid-disordered phase model for cholesterol-containing membranes and show that the presence of cholesterol, in addition to modification to the phase state of the hydrophobic portion of the bilayer, strongly affects the packing and the polarity of the membrane hydrophobic-hydrophilic interface.     

On the stock estimation for some fishery systems

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called “observer” can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation of the unmeasurable stock state x(t). Moreover the convergence speed of towards x(t) can be chosen.     

Representational switching by dynamical reorganization of attractor structure in a network model of the prefrontal cortex

The prefrontal cortex (PFC) plays a crucial role in flexible cognitive behavior by representing task relevant information with its working memory. The working memory with sustained neural activity is described as a neural dynamical system composed of multiple attractors, each attractor of which corresponds to an active state of a cell assembly, representing a fragment of information. Recent studies have revealed that the PFC not only represents multiple sets of information but also switches multiple representations and transforms a set of information to another set depending on a given task context. This representational switching between different sets of information is possibly generated endogenously by flexible network dynamics but details of underlying mechanisms are unclear. Here we propose a dynamically reorganizable attractor network model based on certain internal changes in synaptic connectivity, or short-term plasticity. We construct a network model based on a spiking neuron model with dynamical synapses, which can qualitatively reproduce experimentally demonstrated representational switching in the PFC when a monkey was performing a goal-oriented action-planning task. The model holds multiple sets of information that are required for action planning before and after representational switching by reconfiguration of functional cell assemblies. Furthermore, we analyzed population dynamics of this model with a mean field model and show that the changes in cell assemblies' configuration correspond to those in attractor structure that can be viewed as a bifurcation process of the dynamical system. This dynamical reorganization of a neural network could be a key to uncovering the mechanism of flexible information processing in the PFC.     

Predicting coexistence of plants subject to a tolerance-competition trade-off

Ecological trade-offs between species are often invoked to explain species coexistence in ecological communities. However, few mathematical models have been proposed for which coexistence conditions can be characterized explicitly in terms of a trade-off. Here we present a model of a plant community which allows such a characterization. In the model plant species compete for sites where each site has a fixed stress condition. Species differ both in stress tolerance and competitive ability. Stress tolerance is quantified as the fraction of sites with stress conditions low enough to allow establishment. Competitive ability is quantified as the propensity to win the competition for empty sites. We derive the deterministic, discrete-time dynamical system for the species abundances. We prove the conditions under which plant species can coexist in a stable equilibrium. We show that the coexistence conditions can be characterized graphically, clearly illustrating the trade-off between stress tolerance and competitive ability. We compare our model with a recently proposed, continuous-time dynamical system for a tolerance-fecundity trade-off in plant communities, and we show that this model is a special case of the continuous-time version of our model.     

Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

A recurring problem in population biology - as well as other stochastic dynamical systems in biology, the physical and social sciences - is the distinction between the ‘true’ dynamics of a system and observational noise: i.e. can we from present data reliably infer e.g. biological mechanisms, or are signals swamped by noise.Here, we approach this problem using the canonical model for simple systems that exhibit complex behaviour, the logistic map. At each time-point noise is added, which allows us to study the long-term behaviour of a system which exhibits both non-linear dynamics and intrinsic noise.We show that the interplay between deterministic non-linear dynamics and simple Gaussian noise results in a perplexingly simple system when viewed statistically.In particular we show that for the case of Gaussian noise it is possible to derive at very reliable approximations for the time until the system has reached an absorbing state. This generic model allows us, for example, to study the life-time of molecular species involved in noisy feedback loops.     

Stability and phase transitions in a mathematical model of Duchenne muscular dystrophy

We present a mathematical model to investigate the role of the immune system in the Duchenne muscular dystrophy disease. It is based on the assumption that the immune system contributes to the tissue damage and indeed its interaction with the muscle tissue after an initial endogenous damage can be described as a predator-prey system showing typical oscillations. In this article we investigate the dynamical properties of the system. We find that, for a biologically relevant parameters range, it shows two phase transitions between qualitative different behaviors corresponding to complete recover or to a state where muscle regeneration and degeneration coexist.     

The color phi phenomenon: Not so special,after all?

We show how anomalous time reversal of stimuli and their associated responses can exist in very small connectionist models. These networks are built from dynamical toy model neurons which adhere to a minimal set of biologically plausible properties. The appearance of a “ghost” response, temporally and spatially located in between responses caused by actual stimuli, as in the phi phenomenon, is demonstrated in a similar small network, where it is caused by priming and long-distance feedforward paths. We then demonstrate that the color phi phenomenon can be present in an echo state network, a recurrent neural network, without explicitly training for the presence of the effect, such that it emerges as an artifact of the dynamical processing. Our results suggest that the color phi phenomenon might simply be a feature of the inherent dynamical and nonlinear sensory processing in the brain and in and of itself is not related to consciousness.     

GDSCalc: A Web-Based Application for Evaluating Discrete Graph Dynamical Systems

Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interactions between agents as edges. Agents are in one of a finite set of states at each discrete time step and are assigned functions that describe how their states change based on neighborhood relations. Full characterization of state transitions of one system can give insights into fundamental behaviors of other dynamical systems. In this paper, we describe a discrete graph dynamical systems (GDSs) application called GDSCalc for computing and characterizing system dynamics. It is an open access system that is used through a web interface. We provide an overview of GDS theory. This theory is the basis of the web application; i.e., an understanding of GDS provides an understanding of the software features, while abstracting away implementation details. We present a set of illustrative examples to demonstrate its use in education and research. Finally, we compare GDSCalc with other discrete dynamical system software tools. Our perspective is that no single software tool will perform all computations that may be required by all users; tools typically have particular features that are more suitable for some tasks. We situate GDSCalc within this space of software tools.     

On the Stock Estimation for a Harvested Fish Population

We consider a stage-structured model of a harvested fish population and we are interested in the problem of estimating the unknown stock state for each class. The model used in this work to describe the dynamical evolution of the population is a discrete time system including a nonlinear recruitment relationship. To estimate the stock state, we build an observer for the considered fish model. This observer is an auxiliary dynamical system that uses the catch data over each time interval and gives a dynamical estimate of the stock state for each stage class. The observer works well even if the recruitment function in the considered model is not well known. The same problem for an age-structured model has been addressed in a previous work (Ngom et al., Math. Biosci. Eng. 5(2):337–354, 2008).     

A dynamical system for biological development: The case of Caenorhabditis elegans

We show how a simple nonlinear dynamical system (the discrete quadratic iteration on the unit segment) can be the basis for modelling the embryogenesis process. Such an approach, even though being crude, can nevertheless prove to be useful when looking with the two main involved processes: (i) on one hand the cell proliferation under successive divisions; (ii) on the other hand, the differentiation between cell lineages. We illustrate this new approach in the case of Caenrhabditis elegans by looking at the early stages of embryogenesis, up to several hundreds of cells ("lima bean" larval stage). We show how the many results that have been obtained by several groups can be interpreted in terms of values for the parameters controlling the dynamical system. Furthermore, we can extend the model to the cases of genetic mutations. More precisely the teratogenetic and lethal effects are associated with abnormal variation of teh control parameters with time.     

Dynamic properties of cardiovascular systems.

A recently derived mathematical model of an isolated heart is extended here to a closed-loop cardiovascular system. Taking the end-diastolic volume as state variable, the authors show that the closed-loop cardiovascular system can be described by a one-dimensional nonlinear discrete dynamical system that depends on parameters describing the systolic and diastolic properties of the heart, heart rate, total peripheral resistance, and arterial capacitance. Studies of this model show that the system possesses a rich spectrum of dynamical behavior, from stable points through stable cycles to a "chaotic" behavior. It is shown that such an analysis of dynamic behavior yields those domains in the parameter space that correspond to a normal and abnormal beating heart, when the heart ejects time-invariant and time-variant (periodic or aperiodic) stable stroke volumes, respectively. Determination of such domains may lead to better understanding of the specific pathologic mechanism involved in the evolution of an abnormal beating heart.     

Modeling immune reactivity in secondary lymphoid organs

Models of the dynamical interactions important in generating immune reactivity have generally assumed that the immune system is a single well-stirred compartment. Here we explicitly take into account the compartmentalized nature of the immune system and show that qualitative conclusions, such as the stability of the immune steady state, depend on architectural details. We examine a simple model idiotypic network involving only two types of B cells and antibody molecules. We show, for model parameters used by De Boeret al. (1990,Chem. Eng. Sci. 45, 2375–2382), that the immune steady state is unstable in a one compartmental model but stable in a two compartment model that contains both a lymphoid organ, such as the spleen, and the circulatory system. This work was performed under the auspices of the U.S. Department of Energy.     

Dynamical phase coexistence: A simple solution to the “savanna problem”

We introduce the concept of dynamical phase coexistence to provide a simple solution for a long-standing problem in theoretical ecology, the so-called “savanna problem”. The challenge is to understand why in savanna ecosystems trees and grasses coexist in a robust way with large spatiotemporal variability. We propose a simple model, a variant of the contact process (CP), which includes two key extra features: varying external (environmental/rainfall) conditions and tree age. The system fluctuates locally between a woodland and a grassland phase, corresponding to the active and absorbing phases of the underlying pure contact process. This leads to a highly variable stable phase characterized by patches of the woodland and grassland phases coexisting dynamically. We show that the mean time to tree extinction under this model increases as a power-law of system size and can be of the order of 10,000,000 years in even moderately sized savannas. Finally, we demonstrate that while local interactions among trees may influence tree spatial distribution and the order of the transition between woodland and grassland phases, they do not affect dynamical coexistence. We expect dynamical coexistence to be relevant in other contexts in physics, biology or the social sciences.     

Invasion of an intermediate predator: the dynamics of fish populations in the mathematical model of a trophic chain (as applied to the Syamozero lake)

We present a mathematical model of the dynamics of a spatially heterogeneous predator-prey population system. A prototype of the model system is the Syamozero lake fish community. We study the impact of the invader, an intermediate predator, on the dynamics of the fish community. We show that the invasion can lead to the appearance of chaotic oscillations in the population density. We show also that different dynamical regimes resulting from the invasion, i.e., stationary, non-chaotic oscillatory and chaotic ones, can coexist. The "choice" of a specific regime therewith depends on the initial invader density. Our analysis of solutions of the mathematical models shows that the successful invasion of the alien species takes place solely in the absence of the competition between the invaders and the native species.     

Irregular dynamics in up and down cortical states

Complex coherent dynamics is present in a wide variety of neural systems. A typical example is the voltage transitions between up and down states observed in cortical areas in the brain. In this work, we study this phenomenon via a biologically motivated stochastic model of up and down transitions. The model is constituted by a simple bistable rate dynamics, where the synaptic current is modulated by short-term synaptic processes which introduce stochasticity and temporal correlations. A complete analysis of our model, both with mean-field approaches and numerical simulations, shows the appearance of complex transitions between high (up) and low (down) neural activity states, driven by the synaptic noise, with permanence times in the up state distributed according to a power-law. We show that the experimentally observed large fluctuation in up and down permanence times can be explained as the result of sufficiently noisy dynamical synapses with sufficiently large recovery times. Static synapses cannot account for this behavior, nor can dynamical synapses in the absence of noise.     

Bifurcation analysis of a neural network model

This paper describes the analysis of the well known neural network model by Wilson and Cowan. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inhibitory populations of neurons. We analyze the dependence of the model's behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied. In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.     

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.     

Dynamics of neutral biodiversity

Hubbell's neutral model has become a major paradigm in ecology. Whereas the steady-state structure is well understood, results about the dynamical aspects of the model are scarce. Here we derive dynamical equations for the Simpson diversity index. Both mean and variance of the diversity are proven to satisfy stable linear system dynamics. We show that in the stationary limit we indeed recover previous results, and we supplement this with numerical simulations to validate the dynamical part of our analytical computations. These findings are especially relevant for experiments in microbial ecology, where the Simpson diversity index can be accurately measured as a function of time.