On optimization,dynamics and uncertainty: A tutorial for gene-environment networks

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; to fully understand its foundations requires a mathematical study. This paper surveys and mathematically expands recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetic. Given the data from DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. Finally, we analyze the topological landscape of gene-environment networks in terms of structural stability. This pioneering work is practically motivated and theoretically elaborated; it is directed towards contributing to applications concerning better health care, progress in medicine, a better education and more healthy living conditions.     

Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation

This mathematical contribution is addressed towards the wide interface of life and human sciences that exists between biological and environmental information. Like very few other disciplines only, the modeling and prediction of genetical data is requesting mathematics nowadays to deeply understand its foundations. This need is even forced by the rapid changes in a world of globalization. Such a study has to include aspects of stability and tractability; the still existing limitations of modern technology in terms of measurement errors and uncertainty have to be taken into account. In this paper, the important role played by the environment is rigorously introduced into the biological context and connected with employing the theories of optimization and dynamical systems. Especially, a matrix-vector and interval concept and algebra are used; some special attention is paid to splines. From data got by DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations. This is done by Chebychev approximation and semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. This is used for testing and maybe improving the goodness of the achieved model. We analyze the topological landscape of gene-environment networks in terms of structural stability which we characterize. This pioneering practically motivated and theoretically elaborated work is devoted to a contribution to better health care, progress in medicine, better education, and to recommending more healthy living conditions. The present paper mainly bases on the authors’ and their coauthors’ contributions of the last few years, it critically discusses structural frontiers and future challenges, while respecting related research contributions, giving access and referring to alternative concepts that exist in the literature.      

On generalized semi-infinite optimization of genetic networks

Since some years, the emerging area of computational biology is looking for its mathematical foundations. Based on modern contributions given to this area, our paper approaches modeling and prediction of gene-expression patterns by optimization theory, with a special emphasis on generalized semi-infinite optimization. Based on experimental data, nonlinear ordinary differential equations are obtained by the optimization of least-squares errors. The genetic process can be investigated by a time-discretization and a utilization of a combinatorial algorithm to detect the stability regions. We represent the dynamical systems by means of matrices which allow biological-medical interpretations, and by genetic or new gene-environment networks. For evaluating these networks we optimize them under constraints imposed. For controlling the connectedness structure of the network, we introduce GSIP into this modern application field which can lead to important services in medicine and biotechnology, including energy production and material science.      

Guaranteed nonlinear parameter estimation in knowledge-based models

Knowledge-based models are ubiquitous in pure and applied sciences. They often involve unknown parameters to be estimated from experimental data. This is usually much more difficult than for black-box models, only intended to mimic a given input–output behavior. The output of knowledge-based models is almost always nonlinear in their parameters, so that linear least squares cannot be used, and analytical solutions for the model equations are seldom available. Moreover, since the parameters have some physical meaning, it is not enough to find some numerical values of these quantities that are such that the model fits the data reasonably well. One would like, for instance, to make sure that the parameters to be estimated are identifiable. If this is not the case, all equivalent solutions should be provided. The uncertainty in the parameters resulting from the measurement noise and approximate nature of the model should also be characterized. This paper describes how guaranteed methods based on interval analysis may contribute to these tasks. Examples in linear and nonlinear compartmental modeling, widely used in biology, are provided.     

Stochastic processes adapted by neural networks with application to climate, energy, and finance

Local climate parameters may naturally effect the price of many commodities and their derivatives. Therefore we propose a joint framework for stochastic modeling of climate and commodity prices. In our setting, a stable Levy process is drift augmented to a generalized SDE. The related nonlinear function on the state space typically exhibits deterministic chaos. Additionally, a neural network adapts the parameters of the stable process such that the latter produces increasingly optimal differences between simulated output and observed data. Thus we propose a novel method of “intelligent” calibration of the stochastic process, using learning neural networks in order to dynamically adapt the parameters of the stochastic model.     

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.     

Modeling, inference and optimization of regulatory networks based on time series data

In this survey paper, we present advances achieved during the last years in the development and use of OR, in particular, optimization methods in the new gene-environment and eco-finance networks, based on usually finite data series, with an emphasis on uncertainty in them and in the interactions of the model items. Indeed, our networks represent models in the form of time-continuous and time-discrete dynamics, whose unknown parameters we estimate under constraints on complexity and regularization by various kinds of optimization techniques, ranging from linear, mixed-integer, spline, semi-infinite and robust optimization to conic, e.g., semi-definite programming. We present different kinds of uncertainties and a new time-discretization technique, address aspects of data preprocessing and of stability, related aspects from game theory and financial mathematics, we work out structural frontiers and discuss chances for future research and OR application in our real world.     

Modeling the complexity of genetic networks: Understanding multigenic and pleiotropic regulation

Molecular genetics presents an increasingly complex picture of the genome and biological function. Evidence is mounting for distributed function, redundancy, and combinatorial coding in the regulation of genes. Satisfactory explanation will require the concept of a parallel processing signaling network. Here we provide an introduction to Boolean networks and their relevance to present-day experimental research. Boolean network models exhibit global complex behavior, self-organization, stability, redundancy and periodicity, properties that deeply characterize biological systems. While the life sciences must inevitably face the issue of complexity, we may well look to cybernetics for a modeling language such as Boolean networks which can manageably describe parallel processing biological systems and provide a framework for the growing accumulation of data. We finally discuss experimental strategies and database systems that will enable mapping of genetic networks. The synthesis of these approaches holds an immense potential for new discoveries on the intimate nature of genetic networks, bringing us closer to an understanding of complex molecular physiological processes like brain development, and intractable medical problems of immediate importance, such as neurodegenerative disorders, cancer, and a variety of genetic diseases.     

确定高精度参数问题

在航天器精确制导等高科技的实际问题中,必须高精度地估计模型中的大批参数,建立高精度的数学模型,考虑较简单的确定高精度参数问题:食饵-捕食者系统.对于绝大多数微分方程得不到解析解,尤其是非线性微分方程这样的情况,运用稳定性理论和常微分方程几何理论来分析该生态模型.在数据分析处理中,采用了大量优化算法,如灰色系统辩识方法,多项式曲线选阶及拟合算法,牛顿迭代法等等.最后,通过MATLAB仿真验证了本方法的可行性.     

STABILITY ANALYSIS OF YEE SCHEMES TO PML AND UPML FOR MAXWELL EQUATIONS

We utilize Fourier methods to analyze the stability of the Yee difference schemes for Berenger PML （perfectly matched layer） as well as the UPML （uniaxial perfectly matched layer） systems of two-dimensional Maxwell equations. Using a practical spectrum stability concept, we find that the two schemes are spectrum stable under the same conditions for mesh sizes. Besides, we prove that the UPML schemes with the same damping in both directions are stable. Numerical examples are given to confirm the stability analysis for the PML method.     

Switching Stability of Automotive Roll Dynamics Subject to Interval Uncertainty

In this paper we apply some recent results on the stability of switched systems with interval uncertainty for analyzing the roll stability of automotive vehicles subject to bounded parametric uncertainties and switching. The application is motivated by the fact that the roll dynamics of an automotive vehicle is affected in a significant and a nonlinear fashion due to possible sudden switches in the vehicle's center of gravity (CG) height, as well as the uncertainty in the suspension parameters. Utilizing a recent stability result, it is possible to model such parametric variations as bounded interval uncertainties for this problem, and obtain easily verifiable conditions for analyzing the stability of the resulting switched/uncertain system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A discrete generalization of the extended simplest equation method

We modified the so-called extended simplest equation method to obtain discrete traveling wave solutions for nonlinear differential-difference equations. The Wadati lattice equation is chosen to illustrate the method in detail. Further discrete soliton/periodic solutions with more arbitrary parameters, as well as discrete rational solutions, are revealed. We note that using our approach one can also find in principal highly accurate exact discrete solutions for other lattice equations arising in the applied sciences.     

Parameter estimation for forward Kolmogorov equation with application to nonlinear exchange rate dynamics

Diffusion processes are widely used for mathematical modeling in finance e.g. in modeling foreign exchange rates. Stochastic differential equations describing diffusion processes are linked directly to the forward Kolmogorov equations. In order to calibrate the models, efficient algorithms identifying the system parameters are in demand. Taking into account nonlinear effects in volatility and drift and dependence on observed economical data, which are not directly modeled, one obtains problems which cannot be treated by standard numerical methods. The coefficients are rapidly oscillatory and strong instabilities may arise. To handle these problem we develop special numerical methods, which are used to simulate the nonlinear dynamics of exchange rates depending on economic data. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling gene regulatory networks with piecewise linear differential equations

Microarray chips generate large amounts of data about a cell’s state. In our work we want to analyze these data in order to describe the regulation processes within a cell. Therefore, we build a model which is capable of capturing the most relevant regulating interactions and present an approach how to calculate the parameters for the model from time-series data. This approach uses the discrete approximation method of least squares to solve a data fitting modeling problem. Furthermore, we analyze the features of our proposed system, i.e., which kinds of dynamical behaviors the system is able to show.     

Discrete exact solutions to some nonlinear differential-difference equations via the (G′/G)-expansion method

We extended the (G′/G)-expansion method to two well-known nonlinear differential-difference equations, the discrete nonlinear Schrödinger equation and the Toda lattice equation, for constructing traveling wave solutions. Discrete soliton and periodic wave solutions with more arbitrary parameters, as well as discrete rational wave solutions, are revealed. It seems that the utilized method can provide highly accurate discrete exact solutions to NDDEs arising in applied mathematical and physical sciences.     

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

In this paper, we study the forward and the backward in time problems for a class of nonlinear diffusion equations with respect to the pseudo‐differential operator. Herein, we investigate the stability of the solution of the forward problem in relationship with parameters of the pseudo‐differential operator and initial data. Besides, as known, the backward in time problem is instability. Hence, we give a method to regularize the solution of the backward problem in the case of the parameters are perturbed.     

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.     

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.     

Delay equations with fluctuating delay related to the regenerative chatter II: analysis of reduced bifurcation equation

In this article, we study the reduced bifurcation equations of the nonlinear delay differential equations with periodic delays, which models the machine tool chatter with continuously modulated spindle speed to determine the periodic solutions and analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions close to the new stability boundary.