1 Survey on the modeling of cells, tissues and organisms

Since its foundation in 1991, the European Society for Mathematical and Theoretical Biology has been organizing triennial conferences in different European countries bringing together people who work on different aspects of mathematics but who share an enthusiasm for exchanging ideas and learning about research in this increasingly interdisciplinary field. In particular, the last conference, which was held in 2008 in Edinburgh (ECMTB’08: European Conference on Mathematical and Theoretical Biology), attracted almost 500 delegates from across the world, with increased participation by young researchers and students. Throughout the meeting there was a marked intensity to the discussions, not only during presentations and minisymposia, but also during poster sessions and breaks. Listening to some of these exchanges one could feel, perhaps for the first time, that we were witnessing the emergence of a new generation of scientists trained in project-oriented thinking, and comfortable crossing the traditional subject boundaries between philosophy, biology, mathematics, physics and other natural or social sciences.

During the first decade of the twenty-first “century of biology”, the blossoming of molecular biology and information technology, with their fast and highly accurate quantitative methods, have generated a wealth of new experimental results. We may term this work `bio-analysis’ because it involves the analytic investigation of genomic structures, metabolic processes and signalling pathways. Much of this new knowledge remains to be interpreted. This will involve identifying their causal physico-chemical mechanisms and explaining the functional roles that they play within a particular organism. Therefore, it seems that the time has now come for `bio-synthesis’, by which we mean the integration of results and methods from different spatial, temporal and organizational scales that relate to intracellular events, cell behavior, tissues, organisms and populations. The synthesis of data from a variety of space and time-scales is driving the development of new biological hypotheses, new classes of mathematical models and new numerical methods for their simulation.

In many respects the current situation resembles what happened in mathematical and theoretical biology between 1920 and 1935. Fisher and Haldane published their texts on population genetics, Lotka published his seminal book on the ‘Elements of Physical Biology’ (1925), and von Bertalanffy’s first volume of his ‘Theoretical Biology’ appeared in 1932. Most importantly, in 1935—exactly 75 years ago—a group of European biologists, mathematicians, physicists and philosophers took the initiative by founding the first journal in this area, the `Acta Biotheoretica’ at the old University of Leiden (see Hemmerich and Reydon 2005). We congratulate the current editors of this journal and the entire community on this scientific birthday, and we feel honoured to be given the opportunity to publish our special issue on this occasion.

Nowadays `synthesis’ seems to be acquired in all current ‘construction areas’ of modern biosciences: On the experimental side, the new field of `Synthetic Biology’ aims to design and construct ‘living’ cells by chemically synthesizing their genome including the functional units of their organismic life; similarly, though by using more physical techniques, the wider field of `Artificial Life’ tries to design and build ‘biorobots’ simulating life on the behavioral level by motor-sensory control. In these and other cases, the long-term success of such practical attempts at bio-synthesis will rely heavily on the ability to understand (and eventually reproduce) the biological principles and concepts of living processes from which nature has evolved. By interdependent coincidence, in Theoretical and Mathematical Biology we observe—after a longer period of fruitful `model-analysis’—an increasing tendency towards `model-synthesis’, that is trying to capture the enormous complexity of biological structures and processes. Clearly, a combined theoretical and quantitative approach will be needed to develop new models of networks and/or hierarchies of sub-models, to implement and investigate the corresponding multi-scale and multi-level mathematical systems (processes, equations or simulation algorithms) and, finally, to compare the results of the model simulations with experimental or observational data integrated from different space and time-scales.

Many of the themes and topics that were addressed during the successful and highly stimulating Edinburgh Conference, reflect this new trend for ‘synthetic’ modeling in Mathematical and Theoretical Biology. For example, the plenary talk by Ellen Baake focused on an evolutionary topic, namely Ancestral processes for mutation-selection models of population genetics. She considered both deterministic and stochastic models that describe the genetics of populations under the joint action of mutation and selection. The ancestral processes are shown to play a key role in understanding the interplay between mutation and selection.

Neil Ferguson discussed Planning for novel infectious disease outbreaks: the role of modelling. Epidemic modelling has grown in prominence as a tool to assist public health professionals and policy-makers to plan for and respond to outbreaks of human and animal diseases. Recent examples include Foot and Mouth Disease in livestock, SARS in humans, and planning for a possible human pandemic caused by H5N1 avian flu. Ferguson introduced the varied roles of modelling in outbreak response and discussed how intrinsic stochasticity together with uncertainties about disease biology, mechanisms of transmission and the population impact of controls limit our ability to predict detailed patterns of epidemic spread.

Hans Othmer addressed Multiscale analysis in biology—Paradigms and problems. New techniques in cell and molecular biology have produced huge advances in our understanding of signal transduction and cellular response in many systems, and this has led to better cell-level models for problems ranging from biofilm formation to embryonic development. However, many problems involve very large numbers of cells, and detailed cell-based descriptions are computationally prohibitive at present. Thus rational techniques for incorporating cell-level knowledge into macroscopic equations are needed for these problems. Othmer discussed several examples that arise in the context of cell motility and pattern formation.

In his talk on Multiscale models of cell and tissue dynamics Luigi Preziosi showed how multiphase and continuum models could be used to understand different aspects of solid tumour growth, particularly how the behaviour of tumour cells is influenced by the combination of host cells, extracellular matrix and extracellular liquid that define their environment. Cells proliferate, reorganise and deform while binding to each other (and the extracellular matrix) via adhesion molecules. Preziosi presented a test case to illustrate how tumour growth can be influenced by mechanical stress, focusing on how and where it can generate plastic reorganisation of the cells and how it can lead to compression of the surrounding tissue.

In a lively presentation, James Sneyd considered Calcium, smooth muscle and asthma. He explained that, as far as we know, the only known function of the contraction of airway smooth muscle is to cause, at least in part, asthma. He introduced experimental data which suggests that the contraction of airway smooth muscle may be controlled by the frequency of oscillation of intracellular free calcium. Sneyd then developed a mathematical model to test this hypothesis and, in so doing, demonstrated how mathematical modelling and nonlinear dynamical systems can act as a bridge to link those experimental results and biological hypotheses.

John Tyson introduced Stochastic modeling of the cell division cycle. The eukaryotic cell cycle is controlled by a complex network of interacting proteins and genes. A large number of deterministic models (nonlinear ODEs) of this network successfully account for many average properties of populations of proliferating cells. Recent data on the molecular composition of individual cells reveal that more accurate stochastic models of the molecular events are needed to simulate the dynamics of individual cells. Tyson reviewed progress on attempts to formulate stochastic models of yeast cell cycle control that are realistic, accurate and informative. In addition to molecular noise due to finite numbers of reactant molecules in a yeast cell, he also considered random effects due to imprecise division and binomial distribution of protein molecules between daughter cells.

As one of the two prize winners of the “Reinhart-Heinrich Doctoral Thesis Award 2007”, Antonion Politi delivered a special plenary lecture on Systems biology perspectives on calcium signaling and DNA repair, for an extended summary see Alt (2008).

Several of the 40 minisymposia that were organized at ECMTB08 addressed questions of direct relevance to medicine and healthcare. Thus, of the eight review articles selected for this volume, five deal with case studies, where mathematical modeling is used to design, improve or optimize specific medical therapies such as dopamine treatment (1), prediction of cancer growth (2), cancer radiotherapy (3), bone regeneration (4) or blood gas exchange (5). Further articles reveal how complex mathematical models of infection spread and autoimmune response may be used to explain the transition from healthy control to disease dynamics (6) and (7).

These and other applications of mathematical modeling to medicine, environmental protection and ecological management, are necessarily based on a thorough and systematic investigation of the systems that underpin natural life and which can be observed by isolated cells and their interactions with cell populations. Here we uncover a surprising variety of mechanisms, strategies and functional controls that have been developed by elementary organisms during their evolution.

New approaches to model synthesis clearly require new ideas in theoretical thinking. Here ‘nature’ often serves as a good teacher. During the last decade there has been a significant increase in mathematical model development. This may involve adapting or combining existing models from physics or chemistry in order to incorporate biological features, as in (2), or the creation of genuinely new, biologically-inspired model systems—an example is the application of general nonlinear operator systems from renewal theory to structured population dynamics, as it is used in (8).

In the following we briefly summarize the articles selected for this special issue.

  1. (1)

    Intracellular regulatory systems for the secretion (of hormones or neurotransmitters) are realized by multi-step pathways that involve the synthesis, mobilization, docking and exocytosis of membrane vesicles, and range from the molecular to the organ level. At each stage of this complex chain of events, different chemical, mechanical and electro-physiological mechanisms shape the response. The first article by Tsaneva-Atanasova, Osinga, Tabak and Pederson is a minisymposium review of Modeling mechanisms of cell secretion which focusses on three different aspects of this chain: regulated granule formation, vesicle trafficking and bursting patterns in pituitary or endocrinic cells. In order to understand the regulation process at the system level, the different sub-models could be combined to produce an integrated theory of cell secretion.

  2. (2)

    The next article focuses on applications of lattice-gas celular automaton and lattice Boltzmann models for interacting cell populations. Chopard, Quared, Deutsch, Hatzikiriou and Wolf-Gladrow review a minisymposium on Lattice-gas cellular automaton models for biology: from fluids to cells, in which this powerful modeling tool and its generalizations are first used to simulate the shear flow conditions that lead to the clotting in cerebral blood vessel aneurysms. The authors then explain how the models can be adapted to describe the growth and invasion of solid tumors. Here cell division and migration depend on ‘environmental’ factors and structures, including measurable inhomogeneities in the fibrillar arrangement of the extracellular matrix.

  3. (3)

    In their article, Enderling, Chaplain and Hanhnfeldt address Quantitative modelling of tumour dynamics and radiotherapy. Cancer is a complex disease, and mathematical models have been proposed to identify key mechanisms that underlie dynamics and events at every scale of interest, and recently increasing effort is being paid to multi-scale models that bridge different scales. The ultimate goal of cancer models, however, is to gain insight into the processes that regulate carcinogenesis, and to use this insight to guide the design of new treatment protocols that reduce mortality and improve the quality of life of the patients. Treatments for cancer typically involve a combination of surgery and radiotherapy (for localized tumours) or chemotherapy (for systemic treatment of advanced cancers). Although radiation is widely used, most scheduling is based on empirical knowledge and fails to exploit the existence of sophisticated models that can predict tumor growth dynamics and responses to treatment. The authors discuss ideas for combining dynamic tumor growth models with radiation response models to simulate and understand the complexity of radiotherapy treatment and failure.

  4. (4)

    Geris, Gerisch and Schugart discuss Mathematical modeling in wound healing, bone regeneration and tissue engineering. The related fields of wound healing, bone regeneration and tissue engineering have been active areas for mathematical modeling throughout the last decade. The authors review a selection of these recent models. In wound healing, the models have focused on the inflammatory response in order to improve the healing of chronic wounds. For bone regeneration, the mathematical models have been used to design new treatment strategies for normal and impaired fracture healing. In the field of tissue engineering, mathematical models have been proposed to analyze the interplay between cells and biochemical cues from the porous scaffold in which they are embedded in order to ensure optimal nutrient transport and maximal tissue production.

  5. (5)

    Batzel and Bachar discuss the Modeling the cardiovascular-respiratory control system: data, model analysis, and parameter estimation. Several key areas in modeling the cardiovascular and respiratory control systems are reviewed and examples are given which reflect the research state of the art in these areas. Attention is given to the interrelated issues of data collection, experimental design, and model application including model development and analysis. Examples are provided of current clinical problems which can be examined via modeling, and important issues related to model adaptation to the clinical setting.

  6. (6)

    Pinto, Aguiar, Martins and Stollenwerk present aspects of Dynamics of epidemiological models, focussing on biological thresholds that appear in these dynamic models. In particular, the authors show how to compute the thresholds that determine the appearance of an epidemic disease for stochastic SIS or SIRI models and how to compare corresponding (pseudo-)steady states for different approximations.

  7. (7)

    The sibling paper on Dynamics of immunological models by Pinto, Burroughs, Ferreira, Oliveira and Stollenwerk address the same problem of computing thresholds, now for the deterministic immune dynamics of regulatory T-cell control. By using different methods they are able to predict when an effective immune response will be stimulated.

  8. (8)

    A very promising but detailed approach for validating a mathematical model against experimental data involves formulating it as an inverse problem. The last article by Doumic, Maia and Zubelli, On the calibration of a size-structured population model from experimental data, illustrates how this may be done when the data are measurements of the size distribution of E. coli cells at division under controlled conditions, with a constant doubling time. For two different simplifying assumptions, the authors are able to construct and solve the inverse problem by using a hybrid regularization model of filtering and quasi-reversibility. Possible extensions of their results by appropriate choices of the regularization parameters are also discussed.

Such a review of currently important problems in biological research and their—at least partial—clarification by mathematical modeling and analysis, reveals the fruitfully developing process of scientific adaptation between empirically based biosciences and theoretically oriented biomathematics. In particular, it shows how the interdisciplinary research in Mathematical and Theoretical Biology opens manifolds of roads towards an analytic and synthetic understanding of organismic structures, processes and functions that comprise the whole spectrum of cellular life—from genetic control and epigenetic regulation to tissue formation and organogenesis. In this way, we can be confident to continue these basic research activities in responsible contact to the issues of health care and environment protection.

Thus encouraged, we look forward to the following European Conference on Mathematical and Theoretical Biology (ECMTB’2011) that will take place in Krakow, Poland, from June 28 to July 2, 2011.