Alexander N. Gorban

Method of invariant manifold for chemical kinetics

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of the MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

Invariant manifolds for physical and chemical kinetics

Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grads Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

Principal Manifolds for Data Visualization and Dimension Reduction

Although linear principal component analysis (PCA) originates from the work of Sylvester [67] and Pearson [51], the development of nonlinear counterparts has only received attention from the 1980s. Work on nonlinear PCA, or NLPCA, can be divided into the utilization of autoassociative neural networks, principal curves and manifolds, kernel approaches or the combination of these approaches. This article reviews existing algorithmic work, shows how a given data set can be examined to determine whether a conceptually more demanding NLPCA model is required and lists developments of NLPCA algorithms. Finally, the paper outlines problem areas and challenges that require future work to mature the NLPCA research field.

Limits of the turbine efficiency for free fluid flow

An accurate estimate of the theoretical power limit of turbines in free fluid flows is important because of growing interest in the development of wind power and zero-head water power resources. The latter includes the huge kinetic energy of ocean currents, tidal streams, and rivers without dams. Knowledge of turbine efficiency limits helps to optimize design of hydro and wind power farms. An explicitly solvable new mathematical model for estimating the maximum efficiency of turbines in a free (nonducted) fluid is presented. This result can be used for hydropower turbines where construction of dams is impossible (in oceans) or undesirable (in rivers), as well as for wind power farms. The model deals with a finite two-dimensional, partially penetrable plate in an incompressible fluid. It is nearly ideal for two-dimensional propellers and less suitable for three-dimensional crossflow Darrieus and helical turbines. The most interesting finding of our analysis is that the maximum efficiency of the plane propeller is about 30 percent for free fluids. This is in a sharp contrast to the 60 percent given by the Betz limit, commonly used now for decades. It is shown that the Betz overestimate results from neglecting the curvature of the fluid streams. We also show that the three-dimensional helical turbine is more efficient than the two-dimensional propeller, at least in water applications. Moreover, well-documented tests have shown that the helical turbine has an efficiency of 35 percent, making it preferable for use in free water currents.

Constructive methods of invariant manifolds for kinetic problems

The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the di6erential equation for a manifold immersed inthe phase space ( the invariance equation). The equationof motionfor immersed man ifolds is obtained (the 1lm extension of the dynamics). Invariant manifolds are 8xed points for this equation, and slow invariant manifolds are Lyapunov stable 8xed points, thus slowness is presented as stability. A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions. The following examples of applications are presented: nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn ∼ 1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of exten sionof list of variables) to gainmore accuracy indescriptionof highly n equilibrium =ows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data;

Invariant grids for reaction kinetics

In this paper, we construct low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A grid-based version of MIM is developed (the method of invariant grids). We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grids nodes correction. The edges between the nodes appear only in the calculation of the tangent spaces. This fact determines high computational efficiency of the method of invariant grids. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis–Menten mechanism), and the hydrogen oxidation. The algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

Kinetic signatures of microRNA modes of action.

MicroRNAs (miRNAs) are key regulators of all important biological processes, including development, differentiation, and cancer. Although remarkable progress has been made in deciphering the mechanisms used by miRNAs to regulate translation, many contradictory findings have been published that stimulate active debate in this field. Here we contribute to this discussion in three ways. First, based on a comprehensive analysis of the existing literature, we hypothesize a model in which all proposed mechanisms of microRNA action coexist, and where the apparent mechanism that is detected in a given experiment is determined by the relative values of the intrinsic characteristics of the target mRNAs and associated biological processes. Among several coexisting miRNA mechanisms, the one that will effectively be measurable is that which acts on or changes the sensitive parameters of the translation process. Second, we have created a mathematical model that combines nine known mechanisms of miRNA action and estimated the model parameters from the literature. Third, based on the mathematical modeling, we have developed a computational tool for discriminating among different possible individual mechanisms of miRNA action based on translation kinetics data that can be experimentally measured (kinetic signatures). To confirm the discriminatory power of these kinetic signatures and to test our hypothesis, we have performed several computational experiments with the model in which we simulated the coexistence of several miRNA action mechanisms in the context of variable parameter values of the translation.

A random six-phase switch regulates pneumococcal virulence via global epigenetic changes

Streptococcus pneumoniae (the pneumococcus) is the world’s foremost bacterial pathogen in both morbidity and mortality. Switching between phenotypic forms (or ‘phases’) that favour asymptomatic carriage or invasive disease was first reported in 1933. Here, we show that the underlying mechanism for such phase variation consists of genetic rearrangements in a Type I restriction-modification system (SpnD39III). The rearrangements generate six alternative specificities with distinct methylation patterns, as defined by single-molecule, real-time (SMRT) methylomics. The SpnD39III variants have distinct gene expression profiles. We demonstrate distinct virulence in experimental infection and in vivo selection for switching between SpnD39III variants. SpnD39III is ubiquitous in pneumococci, indicating an essential role in its biology. Future studies must recognize the potential for switching between these heretofore undetectable, differentiated pneumococcal subpopulations in vitro and in vivo. Similar systems exist in other bacterial genera, indicating the potential for broad exploitation of epigenetic gene regulation.

Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

Model reduction and coarse-graining are important in many areas of science and engineering. How does a system with many degrees of freedom become one with fewer? How can a reversible micro-description be adapted to the dissipative macroscopic model? These crucial questions, as well as many other related problems, are discussed in this book. Specific areas of study include dynamical systems, non-equilibrium statistical mechanics, kinetic theory, hydrodynamics and mechanics of continuous media, (bio)chemical kinetics, nonlinear dynamics, nonlinear control, nonlinear estimation, and particulate systems from various branches of engineering. The generic nature and the power of the pertinent conceptual, analytical and computational frameworks helps eliminate some of the traditional language barriers, which often unnecessarily impede scientific progress and the interaction of researchers between disciplines such as physics, chemistry, biology, applied mathematics and engineering. All contributions are authored by experts, whose specialities span a wide range of fields within science and engineering

PRINCIPAL MANIFOLDS AND GRAPHS IN PRACTICE: FROM MOLECULAR BIOLOGY TO DYNAMICAL SYSTEMS

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonens self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.