Biomathematics is an interdisciplinary field in rapid expansion, where the availability of cheap computing power now makes quantitative analyses important and useful for biological disciplines like Medicine, Biology and Ecology. The importance of mathematical models for these disciplines lies not only in the possibility they offer to obtain numerical answers, but also in the depth of understanding that their qualitative study has for structuring further biological research.

A problem which is always present in this field is the relative difficulty, for highly skilled biological professionals, to understand the concepts, jargon and methods of mathematicians, and viceversa. This often translates in ineffective or missed cooperation.

The goal of the school was to bring young researchers from biological disciplines and from the several mathematical specialties together, describing a wide range of applications and methods and fostering an atmosphere of dialogue and exchange of problems and solutions.

The school offered seven main courses. Some were centered on biological domains and research areas, others were centered on fundamental mathematical methods for the study of biological problems, some were attempting to elucidate material directly used to connect mathematical tools and biological applications. In addition to the classroom courses, laboratory demonstrations of currently studied biological questions, where mathematical help would be welcome, were carried out at the Mario Negri Sud biological research institute.

The following is a brief description of each course: title, duration, coordinator, instructors with number of units each, and a short summary and comment.

Units: 7

Coordinators: Alberto Gandolfi, Saverio Alberti

Instructors: K.Helin (2), G.Vecchio (1), S.Alberti (3), R.Giavazzi (1)

Comment: This course considered three topics: cell cycle control, tumor cell growth and angiogenesis.

Molecular mechanisms of the cell cycle (K.Helin) had represented a focal topic of the previous School on Mathematical Cell Proliferation, Saint Flour, June 1997, and were only briefly touched upon this time. In his second lecture, K.Helin talked about the deregulation of cell cycle control in cancer.

In his first unit, S.Alberti lectured on methods used in the study of tumor cell growth, concentrating on flow cytometry. The second lecture was devoted to describing two lines of research: (1) the production of human cytotoxic T lymphocytes to specifically kill tumor cells; (2) a dynamic study of signal transduction in tumor cells using flow cytometry and using GFP (green fluorescent protein). The third lecture was about cell adhesion molecules. It has been shown that loss of adhesion molecules such as integrins is associated with malignant cell transformation. A brief discussion of apoptosis and how it can be induced when there is a loss of cell adhesion was given. G.Vecchio gave a lecture on mechanisms of activation of oncogenes. The last two units of the course were lectured by R.Giavazzi who first presented a general overview of the phenomenon of angiogenesis especially in tumors. The steps involved in angiogenesis were discussed and their possible regulation at the genetic level were pointed out briefly. In the second lecture, R.Giavazzi discussed the stages of development of anti-angiogenic drugs that inhibit matrix metalloproteases employed by tumor cells in metastasis. Some comments were made on the benefits of anti-angiogenic drug treatment complementing chemotherapy.

Units: 10

Coordinator: Michael Mackey

Instructors: J.Belair (4), M.Mackey (6)

Comment: In this course, M.Mackey and J.Belair presented a good combination of the modeling of the blood production and some blood related illnesses and model study both by mathematical and simulation techniques.

M.Mackey presented a number of examples (cyclical neutropenia, periodic immune hemolytic anemia, aplastic anemia, cyclical thrombocytopenia), following a unified scheme: a quite detailed account of the pathology, including significant symptoms associated with such or such an illness; description of an experiment which mostly consists of testing the levels of one or several constituents of blood (neutrophils, white blood cells, platelets,etc) at different time periods; analysis of the results of the experiment, hypothesis and derivation of a model.

These models are all conceptual, not intended for use by clinicians, but as a tool to understand the functioning of the blood system. As a consequence, the analysis of the models is just qualitative, which means that one looks for equilibria, viewed as the normal states, their stability, the loss of stability and the creation of stable periodic oscillations, etc. This working program was presented in detail by J.Belair.

Units: 10

Coordinator: Alberto Gandolfi

Instructors: A.Bertuzzi (2), A.De Gaetano (4), P.Ubezio (3), L.Del Vecchio (1)

Comment: This course was roughly divided into two parts: the first part, lectured by A.Bertuzzi, L.Del Vecchio and P.Ubezio introduced the use of flow cytometry to detect and quantitate cells or cell constituents. The second part by A.De Gaetano was devoted to nonlinear parameter estimation.

The course on flow cytometry included two aspects: 1) a description of the basic concepts of cytometry, the experimental apparatus as well as the biological substances that are to be measured; 2) theoretical aspects involved in cytometry, notably, the concept of asynchronous exponential growth, and mathematical formulae relating measured quantities (cell cycle phase percentages, mean red fluorescence) to kinetic parameters (population doubling time, mean phase duration's, potential doubling time, relative movement). Also discussed were applications of cytometry to cancer treatment: cell cycle perturbations induced by anticancer drugs, the discrimination of cytotoxic vs. cytostatic effects of drugs. As an extension of this course, one of the sessions at the Mario Negri-Sud Institute was devoted to a visit of a flow cytometry laboratory and several experiments were presented as demonstrations of the techniques used in cytometry.

The course on parameter estimation moved from a geometric presentation of the expectation surface in case space to the problems which may arise when this surface is curved. The assessment of the degree of curvature, that is of the degree of nonlinearity, allows the experimenter to make an informed decision on the reliability of the usual linear approximations. Measures of intrinsic curvature (non-modifiable) and parameter effects curvature (correctable by reparametrization) were discussed. The additional topic of population estimation by means of near-maximum likelihood methods was presented in the last lecture, with an illustration of its important practical application in reducing necessary sample sizes in biological experimentation.

Units: 6

Coordinator: Jany Vassy

Instructors: J.Vassy (1), D.Schoevaert (1), S.Lelievre (2), M.Beil (1), S.Portet (1)

Comment: This course was corally undertaken by a group of researchers and students originally from the same laboratory, Hopital Saint-Louis, Paris.

Several aspects of the cytoskeleton and the cellular matrix were introduced, as well as techniques of image analysis for exploring the cytoskeleton and simulations of the cytoskeletal formation. The course provided several arguments to motivate the subject: one idea discussed in the lectures by J.Vassy was that the cytoskeleton could link the cell environment to the nuclear machinery. The prospect presented by S.Lelievre of the reconstruction of tissues and some spectacular novelties along this line added a futuristic touch in biotechnology. The course was then concerned with two types of issues:

1) biological issues raised by the cell skeleton, its description, its role and its functioning, and more generally, the cell matrix, the extracellular matrix, were treated by J. Vassy and S.Lelievre; in particular, the role of cellular structures in the regulation of gene expression was discussed.

2) modeling issues were presented by D. Schoevaert, M.Beil, and S.Portet, with a special emphasis on the methods. The approach followed in the modelling of the geometry and movements of small filaments was explained by D.Schovaert with an example, the movement of spermatozoa. The problem arising from the gap between data collection, by electronic microscopy techniques, which is 2D, and the geometry of cell filaments, which is 3D, was pointed out, and approaches currently undertaken to both identify filaments from microscopic data and model the building-up of new filaments were discussed by M.Beil and S.Portet.

Units: 10

Coordinator: Eva Sanchez

Instructors: E.Sanchez (5), G.Webb (3), T.Kostova (2)

Comment: This is a classical although indispensable part of the mathematical theory of cell population dynamics.

The linear theory as well as well-known examples were presented by E.Sanchez. The material covered by E.Sanchez included elements of the theory of positive operators needed in the proof of asynchronous exponential growth, semigroup theory, and the mathematical treatment of a general class of cell population models.

A variety of new examples with different nonlinearities were introduced by G.Webb who discussed dynamical features of such equations: stability, the onset of instability, bifurcation and chaotic behavior were exhibited in examples. One of the examples, "Helicobacter pylora", a bacteria which colonizes the human stomach, was treated in more detail, as an example of a novel class of models of structured population dynamics, featuring genetic aspects.

T.Kostova gave an introduction to the numerical analysis of equations of age and size structured populations, after she presented some arguments justifying the interest of the subject "Why do we solve such numerical problems?". Numerical schemes for solving age and size structured problems were discussed, especially with regards the order of the methods.

Units: 10

Coordinator: Z.Agur

Instructors: Z.Agur (5), A.Swierniak (5)

Comment: This course was equally divided into a part on cancer cell modeling and chemotherapy treatment using the cell cycle clock lectured by Z.Agur and a part on optimal control theory applied to chemotherapy treatment lectured by A.Swierniak.

In one of her lectures, Z.Agur described the regulation of the cell-cycle clock, using a theoretical model for the embryonic cell-cycle, where the clock is modeled as a single limit cycle. It was suggested that the apparent wealth of behavior of morphogenetic tissues is in correspondence with the similar richness of behavior of the model, which, in some cases, can exhibit a "srange attractor". In the other lectures, she dealt with mathematical models used in designing protocols for the treatment of cancers, illustrating by examples how it is possible to control host toxicity of phase-specific drugs by connecting drug scheduling to the cell cycle duration.

The optimization part of the course, lectured by A.Swierniak, served the purpose of an introduction to optimal control theory and its applications in various problems of cell population dynamics. It dealt with the dynamic programming technique applied respectively to discrete - time and continuous-time optimal control problems and elaborated the maximum principle. It was then explained how to use these two most important control optimization techniques to solve the linear quadratic problem, both for finite and infinite control time interval. Finally, an introduction to stochastic optimization problems was made, the possibility of tackling such problems by means of the previously described techniques was discussed, and some applications of optimal control theory to cancer chemotherapy scheduling were presented.

Units: 14

Coordinator: Marek Kimmel

Instructors: M.Kimmel (6), R.Chakraborty (4), W.Amos (4)

Comment: This course introduced very recent developments on population genetics, with several applications, for example, to the history of growth and migrations of modern humans, or cancer induced by somatic mutations.

An illuminating insight into the biology of genetics was given by W.Amos, as part of the course. The main object of study in this course was the genome, and its various units and subunits: genes, DNA, satellite DNA, mini- and microsatellites, mitochondrial DNA, and some of their applications in apparently unrelated areas.

The effect of the expansion of DNA triplet repeats in the development of such diseases as the fragile X syndrome and Huntington's diseases, the concept of genetic drift and the neutral theory of evolution, cancer cell genetics and carcinogenesis, were presented by R.Chakraborty.

Considerations on the history of growth and migrations of modern humans were presented by M.Kimmel, exploring organically, in a series of lectures, the mathematical concept of coalescence.Several aspects of population genetics, as well as branching processes, were presented in detail in the examples treated.

Some of the tools of probability theory used in the course were presented in more detail in extra classes held by collaborators of Marek Kimmel.

As a complement to the courses, laboratory demonstrations were shown at the Mario Negri-Sud Institution during three afternoons. These activities were introduced by members of the Research staff of the Institutewho, occasionally, gave lectures on aspects of cell biology which had not been dealt with in the courses. Below is the list of presentations given at the Mario Negri Sud Institute:

- S. Alberti: Experimental techniques in oncology and molecular biology: transfection.

- S. Caltagirone, A. Di Castelnuovo: Cancer and chemotherapy in animal models.

- M. Di Mascio: Electrophysiology and microdialysis in the study of the CNS.

- C. Dossi: Production of transgenic mice by pronuclear microinjection of foreign DNA.

- M.Pessia: Potassium channels studied in Xenopus Laevis oocytes.

- A. Sala: Regulation of cell-cycle, oncogenes and anti-oncogenes.

- P. Ubezio, N. Martelli: Measurement of cellular DNA content by flow citometry.

- R. Weigert: Electron microscopy in cell biology.

In addition to the courses, 40' lectures were held throughout the duration of the school. The talks complement the courses, either at the level of mathematical techniques or results, or by showing further topics not covered by the regular courses.

__General interest:__

- E. Beretta: Sufficient conditions for non-existence of periodic solutions in some classes of delay differential equations.

- D. Morale: From a stochastic system of interacting individuals to a non linear diffusion continuous aggregation model.

- W.P. Coleman: The Conditional Neyman-Pearson Lemma and model choice.

__More on cell migration:__

- T. Hoefer: Chemotaxis and aggregation in the cellular slime mould. (2 lectures)

- L. Preziosi: Free boundary problems for solid tumor evolution.

__More on age and size structured populations:__

- M. Iannelli: Modelling age-structured populations.

- R. Rudnicki: Markov semigroups and their applications to population dynamics.

__More on branching processes:__

- Z. Taib: Modelling Cell Populations Using Branching processes: Theory And Specific Applications (2 lectures)

- V.Vatutin: Branching processes and genetic divergence

Contributed talks of 20' length were given by participants throughout the duration of the school:

B.Aguda: Control of the restriction point in the cell cycle

M.Alexandersson: Branching processes in cell populations

O.Angulo: The numerical integration of a model with growth-rate depending on the total population

J.Arino: A structured, discrete model of phytoplankton growth in the chemostat. Comparison with experiments

Y.Daniel: Discrete models for homeostatic systems- The bone marrow (joint work with Z.Agur)

L.G. de Pillis: Modeling Cancer Tumor Growth with Immune Resistance and an Optimal Control Approach to Treatment

R.Eidukevicius: Probability space and maximum likelihood estimation without tears

G.Funk: Modeling VSV-neutralizing antibody titers during acute and memory phase

A.Koch: Bacteria that live on other bacteria

T.Lipniacki: Non-linear waves in DNA

P.Magal: Existence of oscillating periodic solutions for a state-dependent delay differential equation (joint work with O.Arino)

J.Masel: The kinetics of prion replication

D.Morel: Computer model to simulate epithelial homeostasis and tumour genesis

N.Noykova: Sensitivity analysis and parameter estimation in a model of anaerobic waste water treatment processes with substrate inhibition (joint work with M.Gyllenberg)

A.Petrovski: Intelligent search for optimal chemotherapy treatments

R.Regoes: Evolution of virulence in a heterogeneous host population

T.Ritz: Understanding light-harvesting systems: a complex interplay between symmetries and atomic details

S.Sagitov: Multiple mergers of ancestral lines in exchangeable population models

C.Shaw: A genealogical approach to the polymerase chain reaction (joint work with M.Kimmel)

K.Takumi: A dynamical model of infection and disease by gastroinstestinal pathogens: model structure and plan of work

P.Widlak: The DNA-damage-induced cell cycle checkpoints

Building on the experience of the previous Saint Flour 1997 Summer School, the 1999 Termoli Biomathematics Summer School succeeded in creating a tightly interactive, cordial, well organized and even more didactically intense environment. Part of the success stems from few key concepts: short lectures with enough breathing space in between and plenty of support didactic material (papers, course notes, videos); high instructor to participant ratio (30 to 77) with invited instructors mingling and talking with the participants in all working and social occasions; substitution of the student/teacher opposition with the younger researcher/older researcher paradigm (e.g. by providing plenty of presentation opportunities for younger people); responsabilization of the participants by involving them in the organizational as well as in the scientific issues; last but not least, generous support from the European Commission enabling many deserving participants to contribute with their presence.

The main result of the school has been to expose young researchers, at the beginning of their scientific service to the community, to an array of different techniques, concepts, needs and philosophies. This 1999 Summer School contributed in particular to the improvement of the communication between the two biological and mathematical components by presenting real problems necessitating real solutions rather than expounding theoretical concepts with some connection with biology. This was done not only by having a substantial biological faculty, but also by physically taking young mathematicians from the classroom and bringing them to the biological research labs of the Mario Negri Institute. The usual complaint of biologists, that mathematics is too difficult, was this time accompanied by the healthy impression of many young mathematicians who found high-level biology rather hard. Another gap which was made smaller was that between the deterministic people (ODE, PDE, control) and the stochastic people (stochastic processes, SDE, statistical genetics, MonteCarlo): students of one branch had to explain their jargon, their tools and their proposed solutions to students of the other and to biologists. The atmosphere was challenging and fertile.

In the coming years the participants to the Termoli school will recognize in their own work situations some of the problems they heard discussed and will study them in greater depth, building also on the personal relationships begun at the school. Many opportunities were evaluated to exchange students and post docs among the participating laboratories, and substantial progress was made in reinforcing the identity and interrelationships of the european research network in this small ambitious field.

Towards the end of the second week, an informal meeting took place where some thirty young researchers enjoyed some lively criticism of this school’s organizational choices, teaching criteria and selection of materials. Many of them volunteered to remedy all faults by getting involved in the organization of the next school, which is expected to be just perfect.

Finally, a project of a book to be published by Wiley, Publ. in its Biomathematics Series (ed. S.Levin), based on the lecture notes of the courses, is under way. The book will be a collective work, uniting the efforts of the instructors, and will be edited by Z.Agur, an associate editor of the Wiley series.