Mathematics and cells

One of the leading mathematicians of the late 19th century, Henri Poincaré, once wrote "Mathematics is the art of giving the same name to different things." This process of abstraction is an essential part of mathematical activity and leads to concepts applicable to many different fields. One such concept is that of a "network" – simply a set of nodes or components connected by lines or arrows, which represent a relationship between the nodes. The mathematical theory of networks (often called "graph theory") has many applications – fields of study include social networks, information networks (e.g. the World Wide Web), technological networks (electric power grid, the Internet), etc. In recent years, complex networks – those with a large number of nodes – have become a focus of interest. Such networks are also almost always dynamic, meaning that they change over time.

The new field of "systems biology" uses the concept of networks in its attempt to understand the structure and dynamics of biological systems. In particular, a cell, the basic unit of life, is modeled as a complex, dynamic network – with millions of molecules as nodes and various interactions between these as the "arrows" (or connections). This model has proven to be useful, allowing insight into biological processes and the potential to address diseases, most of which are intimately related with malfunctioning of parts of this complex network. Systems biology is in a sense an extension of molecular biology; the latter has successfully studied in the last 50 years (particularly after the discovery of the double-helix structure of DNA in 1953 by James Watson and Francis Crick) the various "components" or biomolecules. While seemingly daunting in their complexity (see Fig. 1), cellular networks exhibit several essential kinds of structure and dynamics, allowing the use of more mathematical concepts and methods in their study.

Biological networks have been analyzed in terms of their statistical properties and show a number of surprising characteristics. The two important ones are:

• "Small worlds," i.e., there are short paths between any two nodes despite the large number of nodes. A subnetwork of protein interactions in yeast, for example, has 2,115 nodes but an average path length of only 6.8 (Newman 2002).

• In many cases, "scale-free networks," i.e. the probability of a node having n connections is simply n to a constant power. This in particular implies that there are only a few highly interconnected nodes and many poorly interconnected nodes, making them robust against random errors but vulnerable to targeted attacks.

One of the really surprising discoveries in the past five years was that most complex networks share these properties, irregardless of whether they are social networks, technological ones like the Internet or informational ones like a citation network. The study of complex dynamic networks is a very active field with researchers from many disciplines.

Another approach to coping with the complexity of these networks is the familiar "divide et impera": divide into parts and study the resulting subnetworks separately. One way of dividing is inspired by viewing cells as information processing systems: the genome (the set of all genes) handles the information storage; the proteome (set of all proteins) are responsible for information processing; and the metabolome (collection of all metabolites) finally for the execution phase. The resulting important subnetworks are the gene regulatory networks (interactions between genes and proteins regulating gene expression), the protein interaction networks (including signal transduction networks) and the metabolic networks. New experimental techniques from the corresponding "omics" (genomics, proteomics, metabolomics) are delivering large data sets which will contribute to detailing structure and dynamics of these networks. The second approach utilizes what Albert Barabasi (a physicist) and Zoltan Oltvai (a pathologist) have aptly termed "Life’s Complexity Pyramid" where a cell network has a hierarchical structure of (at least) four levels: the system level (the whole cell network), the functional module level (subnetworks with a clear biological function, e.g. cell movement, cell division), the level of building blocks or recurring patterns (called "network motifs"), and then the well-known components or nodes. While the bacterium E. coli has an average of 2,500 active proteins, the module involving movement toward a chemical gradient (chemotaxis) involves only six to seven proteins. Research on motifs (which are small networks – usually up to four nodes – and fulfill an information processing function, not a biological one) has been particularly active.

Depending on the data available in publications and increasingly in well-structured, publicly accessible databases, several classes of mathematical models can be constructed. When sufficient dynamic or time-course data are available, non-linear kinetic models may be pursued, particularly in the form of differential equations. However, computational requirements are still an issue with this approach, so that today only small-scale (up to 10-15 nodes) models (for functional modules) are available. If only a finite number of values are needed, a discrete, qualitative model can be built (for only two values, this is the familiar Boolean or "on/off" model). One should note, however, that fairly complex phenomena like the body segmentation of fruit flies have been successfully described by Boolean models. Finally, if only the underlying stoichiometry of the biochemical reactions (essentially the "bare" network) is known, one can apply an approach based on finding a feasible set of steady-state solutions based on physico-chemical and other constraints. This approach uses only mathematical methods from linear algebra and convex analysis. Such efforts have already resulted in an "in silico" model for E. coli, which incorporates about 25 percent of its 4,393 genes. However, this is just a very modest step toward the vision of a "Virtual Cell."

The complexity of biological networks mandates a very close cooperation between experimenters and modelers – the rigor and passion with which this is done is characteristic for the new field of systems biology. The formation of joint-experimenter-modeler (JEM) research teams is particularly effective in this regard. The broad multi-disciplinarity needed to understand biological networks has also led to leading institutions like MIT and the Swiss Federal Institute of Technology in Zürich (ETZ Zürich) to experiment with new forms of research and development. These initiatives – CSBi (Computational and Systems Biology Initiative) at MIT and BEST (BioEngineering, bioSystems, bioTechnology) at ETZ Zürich are based on the concept of "community of practice," which has already found wide acceptance in many industrial organizations. At UP Diliman, researchers at several departments and institutes at the College of Science and the College of Engineering started the "Mathematical Life Sciences Initiative" (MLSI) last year with a series of joint-experimenter-modeler projects and a similar vision to establish a vibrant community of practice.

Systems Biology – the experimental and computational study of biological networks – is emerging as the basis for future advances in biomedicine and drug development. Research institutions and industrial companies worldwide recognize this potential and have begun to invest in the field.

For me personally, it is gratifying to see large parts of what was called "pure mathematics" when I was doing my doctoral work in the seventies at Bonn University, now finding application in the study of biological networks. Examples come as parts of algebraic geometry and topology. It is particularly amusing for me to note that in my thesis, I studied mathematical objects called "cell complexes" and now over 25 years later, I look at understanding complex cells with related methods.

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References:

(Jeong et al 2001) H. Jeong et al: Lethality and centrality in protein networks, Nature 411 (May 2001)

(Newman 2003) M.E.J. Newman: The structure and function of complex networks, arXiv:cond-mat 0303516 V1, March 25, 2003

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After finishing high school (as valedictorian) and the first two years of college (as university scholar) at the Ateneo de Manila, Eduardo Mendoza completed his B.S. Math (magna cum laude) at the University of Heidelberg. As a DAAD scholar, he finished his M.S. and Ph.D. in Math at the University of Bonn under Guenter Harder. While an assistant professor of Mathematics at Wuppertal University, he developed an interest in computer networks and decided to move to the IT industry in 1980 to pursue this interest.

Over a period of 22 years, he worked as a software developer, project manager, department head and director of consulting services for various companies, including Siemens, Scientific Control Systems (BP Group), Softlab (BMW Group) and Microsoft. At Microsoft, he received the President’s Award in 1997 for his achievements as director of Microsoft Consulting Services in Germany. From 1998 to 2002, he was responsible for Microsoft Consulting Services in EMEA (Europe, Middle East and Africa).

Beginning October 2002, he went on "early retirement" to pursue his interests in systems biology and long-standing plans to contribute to science and technology education in the Philippines. He is currently a research scientist at the Physics Department and the Center for NanoScience of the Ludwig-Maximilians-University in Munich, Germany and an adjunct professor of Mathematics at UP Diliman. He coordinates both the Mathematical Life Sciences Initiative at UPD (www.engg.upd.edu.ph/~compbio) and the Munich Systems Biology Forum (www.sysbio-muenchen.de). He can be reached at [email protected] or [email protected].