Biology and Mathematics of Cell Proliferation (Saint-Flour, June 2-13, 1997)

   The subject of the school will be biological aspects and mathematical methods of population dynamics in connection with cell and molecular biology.
The following are the main classes. The number in parentheses is approximately the number of lectures for each class:

1. Biological mechanisms of cell proliferation (Bela Novak and John Tyson) (6)
2. Cell-Cycle and Cancer (Zvia Agur) (3)
3. Modeling of cell population dynamics (Glenn Webb) (6)
4. Branching processes in cell and molecular biology (Marek Kimmel and Ziad Taib) (9)
5. Mathematical analysis of cell population models (Ovide Arino and Eva Sanchez) (6)
6. Estimation of parameters of cell kinetics (Alessandro Bertuzzi and Alberto Gandolfi) (6)
7. Tumor growth and spatial growth (Mark Chaplain and Philippe Tracqui) (6)

Additional lectures, workshops and informal discussions will be organized in the afternoon and in the evening. The language of the school is English. Documentation will be considered essential. Each lecturer will be required to prepare lecture notes to be handed to the school participants.
Duration of the school is two weeks, including 10 days of classes, Monday through Friday.
The number of participants is limited to 50. PhD students and postdoctoral fellows are encouraged to apply.
The organizers may obtain funding for partial financial support for a limited number of school participants.

Saint-Flour, a lively small town in the pictoresque Massif Central region, is the old capital of the Haute Auvergne.

The school will take place in the Grand Seminary of St.-Flour which recently has been converted into a well equipped conference center accomodating up to 100 people. The French Society for Theoretical Biology has held its three recent annual meetings at the Grand Seminary.

1. Biological backgrounds I.:
Physiology of the cell cycle: the cell cycle, coordination of growth and division, size control, the role of nucleo-cytoplasmic ratio, early size control models, checkpoints: START, entry into- and exit from mitosis.
2. Biological backgrounds, II:
Genetics and molecular biology of the cell cycle, cell cycle mutants, cyclins and cyclin-dependent kinases, the cell cycle engine.
3. Modelling mitotic control:
Kinetic models describing M-phase Promoting Factor regulation in early embryos and growing cells, simple (2D) and detailed (multi D) models explaining experiments by cell free extracts and by fission mutants.
4. Modelling START in budding yeast:
The transcriptional activation at START, transcription factors and G1 cyclins, the relation of G1 and G2 cyclins.
5. Modelling START transition in fission yeast:
The role of cyclin dependent kinase inhibitor (rum1) in ordering S- and M-phases in the cycle, kinetic models describing endo-reduplication and mitotic cycles.
6. Modeling the meta-phase / anaphase transition:
Kinetic models of cyclin ubiquitination, hysteresis in the ubiquitination system.

1. A minimalistic model for the cell-cycle and its implications.
2. The role of time-delays and slow processes in determining the stability of the cell-cycle.
3. Can analysis of the nonlinear properties of the model serve for understanding major transition points in development? Can our conclusions be applied to the control of cancer?
4. Insights from models and experiments in cancer/AIDS chemotherapy:
the role of G0.
5. How to define the global properties of cell-cycle control?

1. Cell population models as dynamical systems:
Cell population processes can be modelled as differential equations, difference equations or functional equations in population state spaces. The population behavior can be described by analysis or computer simulation of the solutions of the equations. An example of such a population process is the loss of telomeres on the ends of chromosomes as cells divide. Telomere loss is believed to have an important role in cell senescence and the distinction of tumor and normal cells.
2. Cell population models with quiescent subpopulations:
Quiescent cells are an important consideration in many cell population processes, especially tumors. The transit of proliferating cells to quiescence or the recruitment of quiescent cells to cycling can be modelled as dynamic processes which effect the stabilization of the cell population.
3. Cell population models of tumor growth:
A tumor can be viewed as a population of cells. The growth of the tumor is a dynamic process in which proliferating and quiescent classes of cells interact. The typical Gompertzian growth of tumors can be modelled by nonlinear differential equations describing the interactions of proliferating and quiescent tumor cells.
4.Models of tumor chemotherapy:
Tumor chemotherapy is a population loss process that can be described qualitatively by population models. Chemotherapy typically effects only proliferating cells and thus the recruitment of quiescent cells to cycling during therapy is of essential importance in treatment.
5. Cell population models of the HIV infected immune system:
The HIV infected immune system can be viewed as a system of differential equations for the populations of uninfected CD4 T-cells, the infected CD4 T-cells, and the virus. The progress of AIDS can be described by the interactions of these populations.
6.Models of AIDS chemotherapy:
AZT or other chemotherapy treatment of AIDS can be described qualitatively by models of the HIV infected human immune system. The inclusion of drug resistant viral strains in these models is of essential consideration in designing treatment strategies.
7.The blood production system:
The blood production system can be viewed as a population process which is controlled by nonlinear feedback mechanisms. The normal blood production system is stable under perturbations, but the abnormal system (such as aplastic anemia) is not. Aplastic anemia can be modelled as a population process that is unstable as a result of a defective starting condition.

1. Mathematical modelling, an introduction

• 1.1. The branching property


• 1.2. The type space


• 1.3. Backward and forward approach


• 1.4. Varying environments


• 1.5. General processes

2. Biological background

• 2.1 DNA (coding and noncoding, replication, organization)


• 2.2 Cell cycle (phases, biochemistry, chemotherapy)


• 2.3 Mutations (point mutations, fluctuation test, clonal resistance)


• 2.4 Gene amplification (drug resistance, unstable, stable)


• 2.5 Cancer and cell aging (chromosome structure, telomeres)


• 2.6 Transformations of repetitive DNA (recombination, slippage, dynamic mutations).

3. The Galton-Watson process

• 3.1 Construction, functional equation and elementary properties


• 3.2 Application: cell cycle model with death and quiescence


• 3.3 Extinction and criticality


• 3.4 Asymptotic properties


• 3.5 Application: gene amplification


• 3.6 Application: iterated Galton-Watson process and expansion of DNA repeats.

4. The age-dependent process

• 4.1 The Markovian case


• 4.2 Backward differential equation and elementary properties


• 4.3 Application: clonal resistance theory


• 4.4 The Bellman-Harris process


• 4.5 Integral equations for the PGF and basic properties


• 4.6 Application: analysis of the stathmokinetic experiment

5. The general age-dependent branching process

• 5.1 Renewal theory and asymptotics of the moments


• 5.2 Asymptotic properties in the supercritical case


• 5.3 Application: Stable population theory


• 5.4 Application: The neutral theory of molecular evolution

6. Countable type processes

• 6.1 Application: two-stage mutations and fluctuation analysis


• 6.2 Stochastic model of the cell cycle with chemotherapy


• 6.3 Cell surface aggregation phenomena


• 6.4 Galton-Watson process with a countable type space


• 6.5 Stable gene amplification


• 6.6 Branching within branching


• 6.7 Bellman-Harris process with countable type spaces


• 6.8 Telomere loss


• 6.9 Branching random walks with absorbing barrier


• 6.10 Application: unstable gene amplification.

7. Branching processes with abstract type spaces

• 7.1 Markov renewal theory and asymptotics of the moments
• 7.2 Asymptotic properties in the supercritical case
• 7.3 Application: The mutation-selection balance
• 7.4 Application: Some properties of the Bell-Anderson model
• 7.5 Application: Budding yeast populations
• 7.6 Application: The evolution of transposable elements
• 7.7 Galton-Watson process with continuous type space
• 7.8 A model of unequal division and growth regulation of mammalian cells

1. A survey of mathematical models of cell population dynamics:
Through the survey, some of the main issues of cell population dynamics will be reviewed. The strong development of linear structured models during the 1980-90 decade will be more particularly focused on.
2. Linear age and size structured models:
Asynchronicity is the main topic of this lecture. Its significance and conditions implying it will be thoroughly reviewed.
3. Nonlinear models:
Density-dependence produces limitations in cell proliferation. This may lead to either a homeostatic distribution with an equal balance of incoming and outgoing cells, or non constant equilibria, ranging from periodic to aperiodic or chaotic behavior. The lecture will present two directions that have been followed recently in the study of such models: i) the development of a theory of nonlinear cell population models; ii) uncoupling of nonlinear (population) effects and linear (individual growth) effects.
4. Comparison of models:
The lecture will present some of the attempts made by several researchers during the last fifteen years for introducing some order amongst the various models which have been proposed:from phenomenological to mechanistic, from deterministic to stochastic, from the level of the populations to the level of the cells down to the level of genes and biochemical substances. This latter aspect will be expounded in the next lecture.
5. Models incorporating mechanisms of molecular biology:
This lecture focuses on models derived from the study of biochemical factors of the cell cycle. It covers the two views under which this study has been made: first, researchers concentrated on the evolution of the species, independent on the cell population. Models derived according to this view are essentially either simulation models or ordinary differential equations, possibly with a large number of variables. The other view combines the biochemical and the population approaches. It is presented in its biological context in the course on "Biological mechanisms of cell proliferation" where a model will be built. The lecture will mainly address some of the mathematical issues of the model, both its qualitative features and its comparisons with other models.

1. Reaction-diffusion models and tumour growth:
(a) dynamical analysis of brain tumour growth (P. Tracqui)
2. Reaction-diffusion models and tumour growth:
(b) patterns formation at the tumour surface (M. Chaplain)
3. Modelling angiogenesis during tumour development (M. Chaplain)
4. Modelling mechano-cellular interactions at the tumour border (P. Tracqui and M. Chaplain)
5. Modelling the immune response to cancer (M.Chaplain)