Mathematical modeling of biological systems

B RIEFINGS IN BIOINF ORMATICS . VOL 14. NO 4. 411^ 422 Advance Access published on 14 October 2012 doi:10.1093/bib/bbs061 Mathematical modeling of biological systems Santo Motta and Francesco Pappalardo Submitted: 30th April 2012; Received (in revised form) : 23rd July 2012 Abstract Keywords: mathematical biology; computational models; systems biology INTRODUCTION Revolutions in biotechnology and information technology have produced enormous amounts of data and are accelerating the process of knowledge discovery of biological systems. These advances are changing the way biomedical research, development and applications are conducted. Clinical data complements biological data, enabling detailed descriptions of both healthy and diseased states, as well as disease progression and response to therapies. The availability of data representing various biological states, processes and their time dependencies enables the study of biological systems at various levels of organization, from molecules to organism and even up to the population level [3–5]. Multiple sources of data support a rapidly growing body of biomedical knowledge, however, our ability to analyze and interpret this data lags far behind data generation and storage capacity. Mathematical and computational models are increasingly used to help interpret biomedical data produced by high-throughput genomics and proteomics projects. The application of advanced computer models enabling the simulation of complex biological processes generates hypotheses and suggests experiments. Computational models are set to exploit the wealth of data stored on biomedical databases through text mining and knowledge discovery approaches. Modeling is the human activity consisting of representing, manipulating and communicating real-world daily life objects. As one can easily realize, there are many ways to observe an object or, equivalently, there are many different observers for the same object. Any observer has ‘different views’ of the same object, i.e. ‘there is no omniscient observer with special access to the truth’. Each different observer collects data and generates hypothesis that are consistent with the data. This logical process is called ‘abduction’. Abduction is not infallible, though; with respect to a scientific unknown, we are all blind. A system is a collection of interrelated objects. For example, a biological system could be a collection of different cellular compartments (e.g. cell types) specialized for a specific biological function (e.g. white and red blood cells have very different commitments). An object is some elemental unit upon which observation can be made but whose internal structure is either unknown or does not exist. The choice of the elemental unit defines the representation scale of the system. A model is a description of a system in Corresponding author. Santo Motta, Department of Mathematics and Computer Science, University of Catania, V.le A. Doria, 6, 95125 Catania, Italy. Tel.: þ39 095 7383073. Fax: þ39 095 330094. Email.: Santo Motta, Laurea in Physics (University of Catania,1970) and MSc in Applied Mathematics (University of London, Queen Mary College, 1971). Associate professor of Mathematical Physics (Faculty of Pharmacy, Department of Mathematics and Computer Science, University of Catania). His present scientific interests are BioMaths, BioComputing and BioInformatics. Francesco Pappalardo is a researcher at the University of Catania. He was a visiting research scientist at the Dana-Farber Cancer Institute in Boston and at the Molecular Immunogenetics Labs, IMGT in Montpellier. His major research area is computational biology. ß The Author 2012. Published by Oxford University Press. For Permissions, please email: Mathematical and computational models are increasingly used to help interpret biomedical data produced by high-throughput genomics and proteomics projects. The application of advanced computer models enabling the simulation of complex biological processes generates hypotheses and suggests experiments. Appropriately interfaced with biomedical databases, models are necessary for rapid access to, and sharing of knowledge through data mining and knowledge discovery approaches. 412 Motta and Pappalardo This article is organized as follows: in the next two sections (Models of Systems and The Modeling Process), we describe the types of models and the modeling processes in scientific investigations in a general context; then in the next section (Models in Biology: Scales and Complexity) we go more specific and talk about models in biology and medicine; few examples of models are briefly shown in the section ‘Tools and Applications’; finally we draw our ‘Conclusions’ in the last section. MODELS OF SYSTEMS Not all scientific models are expressed in a precise, numerical and quantitative way. Actually, one can identify four different types of models: verbal models, conceptual or diagrammatic models, physical models and formal models. In this article, we focus mainly on diagrammatic and formal models and we concentrate on the model building process. Verbal models In verbal models the system is described in words. These models, based on observations, usually evidence in a simple way the objects and relations among the objects in the system. A verbal model is a rough and sometime ambiguous qualitative representation of the knowledge of the system. These kinds of models are used in the first approach to the analysis of biological system. Conceptual or diagrammatic models. In conceptual or diagrammatic models the system is described by a graphical representation of the objects and the relationships describing the underlying dynamical processes. To develop these kind of models the understanding of the available data needs to be sufficient to have a detailed (even if not exhaustive) idea of the objects (or entities) and relations. A conceptual model (CM) represents ‘concepts’ (objects or entities) and relationships between them. In computer science, CM are also referred to as domain models. A CM is expressly independent from the design and free from implementation concerns. The aim of a CM is to convey the meaning of terms and concepts used by ‘domain experts’ to rationalize the problem and to find relationships among the different concepts. The CM aims to clarify the meaning of the usually ambiguous terms to minimize as much as possible problems arising from terms of constitutive objects and the relationships among them, where the description itself is, in general, decodable or interpretable by humans. Generally speaking, a system is an unknown ‘black box’ (S) which, under a specific external stimulus (input E) produces a response (output R) [19]. Using this general definition, one can identify three primary scientific uses of models [12]: (i) synthesis or knowledge discovery; to use the knowledge of inputs E and outputs R to infer system characteristics; (ii) analysis and predic (...truncated)