Invited ReviewModeling, inference and optimization of regulatory networks based on time series data
Introduction
Like very few other disciplines only, the modeling and prediction of genetic 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 Weber et al. (2008b), the important role played by the environment is introduced into the biological context and connected with employing the theories of optimization and dynamical systems.
According to the classical understanding of stability from science, technology and medicine (cf. Guckenheimer and Holmes, 1997), which is mostly positive in terms of some local order, a coming to a rest (recovering) or as the robust behavior of a small or large bio-system against attacks (e.g., by epidemics), there is also the negative meaning of inflexibility. In fact, in medicine, an organism, living being or bio-system which is unable to adapt to a changing environment is seriously threatened by infections, radiation and other kinds of attacks. What is more, thirdly, a stability analysis on our various expression levels can also lead to an acceptance or a rejection (and following improvement) for the model. If a component of the model behaves unbounded, then this is in contradiction with the natural-technical limitation of the gene- (environment-) expression levels which lie in trusted regions of bounded intervals (see Weber et al., 2008b).
Further, Weber et al. (2008a) survey and closer explain recent advances in understanding the mathematical foundations and interdisciplinary implications of the newly introduced gene-environment networks. They integrate the important theme of environmental protection by joint international projects into the our context of networks and their dynamics. As an example of environmental protection, they study CO2 emissions, their implications for global warming by greenhouse effect, the reduction of both and the joint implementation requested for this purpose by Kyoto protocol (cf. Kyoto, 1997a, Kyoto, 1997b, Kyoto, 1997c). Moreover, in the survey of Weber et al. (2009a), the important theme carbon dioxide emission reduction is integrated into the context of the networks and their dynamics.
In Defterli et al. (2010), time-discrete gene-environment and eco-finance networks are investigated as a subclass of target-environment regulatory systems. A different time-discretization scheme is derived for a set of dynamical models representing these networks. Rarefication of such kind of regulatory network is studied together with the corresponding mixed-integer regression problem, and a further relaxation could be obtained by means of continuous optimization. Therefore, the paper of Defterli et al. (2010) extends the existing mathematical toolbox by introducing and applying a new method of time discretization into the study and discussing its potential of improvement. In addition, a method of problem regularization, called rarefication, is recalled and applied and integrated into this research. With this, it is aimed to reach an improved modeling and prediction of our networks, and for a better service in the real-world application areas like health care, environmental protection and sustainable development, economy and society, financial sector, and the living conditions of the people (Pickl and Weber, 2001, Pickl and Weber, 2002, Gökmen et al., 2004, Taylan et al., 2007, Weber et al., 2008a, Weber et al., 2009a).
In Operational Research, eco-finance networks under uncertainty became introduced in Weber and Uğur (2007). We can find the closely related eco-finance networks presented in Kropat et al. (2008) and all these networks are regarded and analyzed as regulatory networks. Compared with the more classical genetic networks (which now appear as a subclass), the new nodes are environmental items such as, e.g., poison in soil, groundwater, air or food, emissions, radiation, but also the state of development of the financial markets, the welfare and living conditions, temperature (concerning, e.g., global warming) and, finally, education and campaigns for a healthy lifestyle. Environmental items themselves and how they exercise effects – often in mutually catalyzing or multiplicative ways, are becoming very important, the more so as we are living in a time of globalization, of rapid information exchange, of mobility and multicausalities in all kinds of bio- and social systems, communities and societies as well (see Alparslan Gök and Weber, 2010).
Clustering, but also classification, provide an insight into the structure of the data and allow to identify groups of model items which are considered to jointly act on other clusters of items of the regulatory model. The uncertain states of these clusters are represented by ellipsoids, and ellipsoidal calculus is applied to model the dynamics of the system (Kropat et al., 2010a, Kropat et al., 2010b).
As we explained, the ellipsoid-valued model groups the usually big set of items, in the game theoretical context we say: players or actors, into clusters. Let us think of the family of nations which signed the Treaty of Kyoto (Kyoto, 1997a, Kyoto, 1997b, Kyoto, 1997c) or the participants in a worldwide auction (cf. Córdoba Bueno, 2006), etc.
Regulatory networks are often characterized by the presence of a large number of variables and parameters resulting in a complexity which is beyond man’s everyday perception. The development of high-throughput technologies has resulted in a generation of massive quantities of data. This technological progress has been accompanied by the development of new mathematical methods for the analysis of such highly interconnected systems that allows to gain deeper insights in the dynamic behavior of complex regulatory systems in biology, finance and engineering sciences. In Kropat et al. (2010b) a special class of so-called TE-regulatory systems (target-environment regulatory systems) is addressed.
There is a rich list of roles and performances delivered which are associated and assigned to ellipsoids. They include: (i) encompassing of objects, (ii) inner or outer approximation of shapes and bodies, of discrete or continuous kinds of sets, (iii) support for classification of objects and discrimination of different objects, (iv) defining critical points or contours which mark tails of higher dimensional and connected versions of tails that describe neighbourhoods of infinity, usually with small values of small probabilities assigned, (v) set-valued generalizations of numbers, and generalizations of balls with a reduced wealth of symmetries but still highly symmetrical, (vi) geometrical representation of linear mappings which execute certain expansions and contractions (herewith, deformation; e.g., applied to a ball) and rotations, with respect to axes in an orthogonal system of coordinates, (vi) geometrical representation of some symmetry breakings, compared with balls, (vii) geometrical representation of dependencies, especially, of variances and correlations, (viii) easy measurability and support for an approximate measuring of other sets and subsets.
The paper is organized as follows. In Section 2, we introduce and analyze time-continuous and time-discrete gene-environment networks, as a subclass of target-environment regulatory systems, without emphasizing the concept of uncertainty. A different discretization scheme is applied in the study of time-discrete networks and nonlinear mixed-integer programming is used to calculate the unknown parameters in the optimization problem. Finally, the corresponding values of the gene-expression levels at discrete time points tk (k = 0, 1, … , l) are obtained and compared with the given data to see the performance of the new method. The need and importance of studying gene-environment networks under the concept of uncertainty is explained and these networks are studied under the interval and ellipsoidal uncertainty in Section 3. Basic notions of interval calculus and ellipsoidal calculus, and interval games are also presented in this section. Section 4 gives some examples from the theory of cooperative interval games and discuss the ellipsoidal extension, then Technology-Emission-Means (TEM) model and its interval-valued version is investigated. Our remarks and future works are presented in Section 5.
Section snippets
A class of dynamical models
The subject of genetic network is one of the important and promising research areas for most of the branches of modern science. A genetic network can be expressed as a weighted directed graph containing nodes (vertices) that represent the genes, and the arcs with functional weights describing the influences of each gene onto the other genes in the network. The aim is to find and predict these interactions between genes. In the literature, many analytic and numerical methods are constructed to
Examples and selected related topics
In this section, we turn to applications and further extensions of our theory on gene-environment networks under uncertainty. We demonstrate their usefulness for modern OR-applications in the energy sector and discuss their relation to the so-called technology-emissions means model in CO2-emissions control as an example of an eco-finance network (Kropat et al., 2008, Özceylan et al., 2010). In particular, we show how methods from game theory and the concepts of cooperative interval games and
Conclusion
In the first part of the paper, we contribute to an improved modeling of gene-environment networks, including their rarefication which may be regarded as a regularization, and to the numerical solution of their dynamics. By this, we supported to a better future prediction of evolution and behavior of these kind of networks in time, with important consequences in health care, environmental protection, finance and in the field of education.
Combining our new numerical methods with those concepts
Acknowledgements
The authors cordially thank the coauthors of all the works on which this survey article bases, and they express their gratitude to EJOR for the invitation to this paper project. This work is partially supported by the Scientific and Technical Research Council of Turkey.
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Honorary positions: University of Aveiro, Portugal; University of Siegen, Germany; Universiti Teknologi Malaysia, Skudai, Malaysia.