Robustness in biological regulatory network III: Application to genetic networks controlling the morphogenesis

This Note deals with the mathematical notions of entropy and stability rate in interaction graphs of genetic networks, in the particular context of the genetic threshold Boolean random regulatory networks (getBrens). It is proved that in certain circumstances of particular connectance, the entropy of the invariant measure of the dynamical system can be considered both as a complexity and a stability index, by exploiting the link between these two notions, fundamental to characterize the resistance of a biological system against endogenous or exogenous perturbations, as in the case of the n-switches. Examples of biological networks are then given showing the practical interest of the mathematical notions of complexity and stability in the case of the control of the morphogenesis.     

Entropy embedding and fluctuation analysis in genomic manifolds

The complexity of typically high-dimensional genomic data requires computational work prone to integrate different biological information sources through efficient model solutions. Usually, one step involves dimensionality reduction (DR), which requires projecting the input data onto a low dimensional subspace, and often leads to an embedding. Thus, DR should be able to filter out the uninformative dimensions and recover the original variables. This step is of crucial relevance for any reverse engineering and statistical inference attempt to reconstruct the dynamics underlying the biological systems under study, i.e. the interactions between its genes or proteins. DR has become almost a standard practice just following the pre-processing steps typically applied to the experimental measurements (mining, normalization, etc.). In this work, the data for the analysis reflect expression values of genes whose dynamics are affected by perturbation experiments. In particular, the aims are to monitor the response of genes involved in a certain pathway, and then to isolate their biological variability from any possible external influence. Last, it is of interest to control the stability of the system; with this regard, we look at dynamical aspects of data through embedding theory and entropy fluctuation analysis. We demonstrate that a redundant biological system can in principle be reduced to a minimal number of almost independent components. In particular, such structures detect the higher-order statistical dependencies in the training data in addition to the correlations. Two popular DR techniques are compared in relation to their ability to extract the most salient features, allow gene selection, and minimize the various interferences due to algorithmic approximation errors and variable noise covers.     

Computing the complexity for Schelling segregation models

The Schelling segregation models are “agent based” population models, where individual members of the population (agents) interact directly with other agents and move in space and time. In this note we study one-dimensional Schelling population models as finite dynamical systems. We define a natural notion of entropy which measures the complexity of the family of these dynamical systems. The entropy counts the asymptotic growth rate of the number of limit states. We find formulas and deduce precise asymptotics for the number of limit states, which enable us to explicitly compute the entropy.     

Polynomial entropies and integrable Hamiltonian systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.     

Complex evolution of a multi-particle system

This paper studies a discrete dynamical system of interacting particles that evolve by interacting among them. The computational model is an abstraction of the natural world, and real systems can range from the huge cosmological scale down to the scale of biological cell, or even molecules. Different conditions for the system evolution are tested. The emerging patterns are analysed by means of fractal dimension and entropy measures. It is observed that the population of particles evolves towards geometrical objects with a fractal nature. Moreover, the time signature of the entropy can be interpreted at the light of complex dynamical systems.     

Maps serving the combined coupling for use in environmental models and their behaviour in the presence of dynamical noise

Many physical, biological as well as the environmental problems, can be described by the dynamics of driven coupled oscillators. In order to study their behaviour as a function of coupling strength and nonlinearity, we considered dynamics of two maps serving the combined coupling (diffusive and linear) in the above fields. Firstly, we have considered a logistic difference equation on extended domain that is a part of the maps, that is discussed using its bifurcation diagram, Lyapunov exponent, sample as well as the permutation entropy. Secondly we have performed the dynamical analysis of the coupled maps using Lyapunov exponent and cross sample entropy in dependence on two coupling parameters. Further, we investigated how dynamical noise can affects the structure of their bifurcation diagrams. It was done (i) by the noise entering in two specific ways, that disturbs either the logistic parameter on extended domain or (ii) by an additive “shock” to the state variables. Finally, we demonstrated the effect of forcing by parametric noise, introduced in all maps’ parameter, on Lyapunov exponent of coupled maps.     

Stability Analysis for Discrete Biological Models Using Algebraic Methods

This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Gr?bner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.     

Entropy transmission in C1-dynamical systems

Three entropies of a state in C¹-dynamical systems are introduced and their relations and dynamical properties are studied. The entropy (information) transmission under a channel between two dynamical systems is considered. We find a condition under which our entropy becomes a dynamical invariant between two systems.     

Dynamics of quantum entropy maximization for finite-level systems

The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.     

Upper bound of time derivative of entropy for a dynamical system driven by quasimonochromatic noise

The quasimonochromatic noise (QMN) is the “truly colored” noise, and in this paper the upper bound of time derivative of entropy for a dynamical system driven by QMN is studied. The dimension of Fokker–Planck equation is reduced by the way of linear transformation. The exact time dependence of the upper bound for the rate of entropy change is calculated based on the definition of Shannon’s information entropy and the Schwartz inequality principle. The relationship between the properties of QMN and dissipative parameters and their effect on the upper bound for the rate of entropy change is also discussed.     

Mixing Closures for Conservation Laws in Stratified Flows

A closure for shocks involving the mixing of the fluids in two-layer stratified flows is proposed. The closure maximizes the rate of mixing, treating the dynamical hydraulic equations and entropy conditions as constraints. This closure may also be viewed as yielding an upper bound on the mixing rate by internal shocks. It is shown that the maximal mixing rate is accomplished by a shock moving at the fastest allowable speed against the upstream flow. Depending on whether the active constraint limiting this speed is the Lax entropy condition or the positive dissipation of energy, we distinguish precisely between internal hydraulic jumps and bores. Maximizing entrainment is shown to be equivalent to maximizing a suitable entropy associated to mixing. By using the latter, one can describe the flow globally by an optimization procedure, without treating the shocks separately. A general mathematical framework is formulated that can be applied whenever an insufficient number of conservation laws is supplemented by a maximization principle.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

Semihyperbolic mappings

Summary Semihyperbolic dynamical systems generated by Lipschitz mappings are investigated. A special form of robustness of topological entropy under perturbations of a Semihyperbolic mapping is discussed, and weakened forms of persistence and of structural stability are considered. Proofs are based on the concept of bi-shadowing, which is a stronger version of the shadowing lemma.     

Entropy increase in dynamical systems

The rate of increase of the non-equilibrium entropy introduced by Goldstein and Penrose, defined on nonstationary probability measures for an abstract dynamical system, is quantitatively related to the Kolmogorov-Sinai entropy of the system. It is shown in particular that for ergodic systems the asymptotic rate of entropy increase coincides with the Kolmogorov-Sinai entropy. Supported in part by NSF Grant No. PHY 78-03816.     

Time-reversal symmetry,Poincaré recurrence,irreversibility, and the entropic arrow of time: From mechanics to system thermodynamics

In this paper, we use a large-scale dynamical systems perspective to provide a system-theoretic foundation for thermodynamics. Specifically, using a state space formulation, we develop a nonlinear compartmental dynamical system model characterized by energy conservation laws that is consistent with basic thermodynamic principles. In addition, we establish the existence of a unique, continuously differentiable global entropy function for our large-scale dynamical system, and using Lyapunov stability theory we show that the proposed thermodynamic model has convergent trajectories to Lyapunov stable equilibria determined by the system initial energies. Finally, using the system entropy, we establish the absence of Poincaré recurrence for our thermodynamic model and develop a clear connection between irreversibility, the second law of thermodynamics, and the entropic arrow of time.     

Stability analysis of shear-thinning flow between rotating cylinders

Stability of the shear thinning Taylor–Couette flow is carried out and complete bifurcation diagram is drawn. The fluid is assumed to follow the Carreau–Bird model and mixed boundary conditions are imposed. The low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. It is observed, that the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the shear thinning effects increases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, shear-thinning Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Complete flow field together with viscosity maps are given for different scenarios in the bifurcation diagram.     

An introduction of logical entropy on sequential effect algebra

The main aim of this study was to introduce logical entropy on dynamical systems that their state spaces were sequential effect algebra. In this regard, logical partition was defined on sequential effect algebra and then based on logical partition concept, logical entropy on partitions, conditional logical entropy, and logical entropy on dynamical systems were introduced and their features were analyzed. In addition, it was proved that this entropy is an invariant object under isomorphism relation.     

Entropy increase for ℤd-actions on a Lebesgue space

Our aim is to introduce the concepts of the entropy increase and the asymptotic rate of entropy increase for a ?^d-action on a Lebesgue space and to show that for ergodic ?^d-actions the asymptotic rate of entropy increase coincides with the Conze—Katznelson—Weiss (CKW) entropy. The result is the multidimensional analogue of the Goldstein result for onedimensional dynamical systems.     

Symbolic dynamics and complexity in a physiological time series

A general approach to non-stationary data from a non-linear dynamical time series is presented. As an application, the RR intervals extracted from the 24 h electrocardiograms of 60 healthy individuals 16–64 yr of age are analyzed with the use of a sliding time window of 100 intervals. This procedure maps the original time series into a time series of the given complexity measure. The state of the system is then given by the properties of the distribution of the complexity measure. The relation of the complexity measures to the level of the catecholamine hormones in the plasma, their dependence on the age of the subject, their mutual correlation and the results of surrogate data tests are discussed. Two different approaches to analyzing complexity are used: pattern entropy as a measure of statistical order and algorithmic complexity as a measure sequential order in heart rate variability. These two complexity measures are found to reflect different aspects of the neuroregulation of the heart. Finally, in some subjects (usually younger persons) the two complexity measures depend on their age while in others (mostly older subjects) they do not – in which case the correlation between is lost.