Mathematical biology From individual cell behavior to biological growth and form

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1 Mathematical biology From individual cell behavior to biological growth and form Lecture 8: Multiscale models Roeland Merks (1,2) (1) Centrum Wiskunde & Informatica, Amsterdam (2) Mathematical Institute, Leiden University

2 Examination Studiegids: Practicum assignment, 30% Final product, 30% Written exam, 40% - date will follow soon Seminars: prepare a short talk (15 mins + 5 mins discussion) on two papers Compare the two papers 18th November and 25th November

3 Paper seminars Two papers on a topic of your choice Prepare short seminar of ±15 mins, plus 5 minutes discussion What question do the papers study? What was the hypothesis? Did the papers propose alternative ideas? Did they work equally well? How was the hypothesis translated into a mathematical model?

4 Possible topics Mechanisms of phyllotaxis (upstream auxin transport vs. canalization) Gastrulation Blood vessel growth (angiogenesis) Reaction-diffusion models E.g. of finger prints Spiral waves in Dictyostelium discoideum Theory of cellular automata Morphogenesis due to differential growth

5 Mini-projects Small/tiny research project Typically based on a simulation you have seen during the computer labs Two guided afternoons and individual work Work in teams of two Work in three lab sessions: Nov. 18, Nov. 25, Dec. 2 Final presentation - December 9th ±30 mins. including discussion Introduce the problem, existing results, your research question, and your new results. Discuss the biological and mathematical relevance. What can biologists learn from your model?

6 Mini-projects Small research project (II) Final report: In the form of a paper: Introduction/Methods/Results/Discussion/Future work Figures, analysis of your model Size: around 8 pages Deadline: 31 January 2013 It s okay to choose the same topic for your seminar and research project

7 Potential topics Pattern formation (e.g. leaf venation, fur patterns, shell patterns) Theory of 1D/2D cellular automata Plant morphogenesis: interaction between reactiondiffusion and auxin pumping Blood vessel growth: extend models with additional cell types; angiogenesis in stromal tissues Branching growth (DLA): modify models with additional particle types, ballistic particle motion, aggregation probability: effect on? Models of tumor invasion Theory of cell sorting D f

8 Multiscale modeling Biological development: 1. Genes -> cell behavior 2. cell behavior->tissue shapes and patterns 3. tissue patterns-> genes and cell behavior How to model step 3? Multiscale modeling required Required levels: Gene regulatory networks Cell behavior Cell-cell signaling

9 Example of a multiscale model Dictyostelium discoideum

10 Dictyostelium discoideum Amoeba live independently When hungry, secrete camp pulse Also secrete camp if they sense camp Refractory period after camp secretion Chemotaxis against camp gradients Cells form Excitable medium (spiral waves,...) Aggregation, slug formation, culmination

11 Phenomena at all scales... Molecular level: camp sensing, camp secretion mechanism of cell motility mechanism of chemotaxis Cellular level Cell trajectories; cell velocity Cell-cell adhesion Tissue scale: aggregation slug motion

12 camp signaling in cell populations Full model would have many equations So population of m cells: m x n equations Models become complex: Expensive calculations Practically impossible to understand Can we simplify the single-cell level? Bottom line is: if cell senses camp, it secretes camp Excitable system minimal model, Fitzhugh-Nagumo

13 Simplified excitable model: Fitzhugh-Nagumo c t = D 2 c f (c) r inside amoebae c t = D 2 c d c (c c 0 ) outside amoebae r = ε(c)(kc r) t f (c) = C 1 c ε(c) = ε 1 for c < c 1 Null isoclines inside amoebae f (c) = C 2 c + a τ ε(c) = ε 2 for c 1 c c 2 f (c) = C 3 (c 1) ε(c) = ε 3 for c > c 2 Marée et al. JTB 1999

14 Model simulation (Savill and Hogeweg 1997, J. Theor. Biol. 184, 229)

15 Model simulation (Savill and Hogeweg 1997, J. Theor. Biol. 184, 229)

16 Model Simulation (Savill and Hogeweg 1997, J. Theor. Biol. 184, 229)

17 Slug moves to warm spot Warm Wave speeds up with temperature temperature Path of slug in absence of gradient Cold Marée and Hogeweg, J. Theor. Biol Pushes autocycling cell to left

18 Slug behavior is caused by the underlying genetics, but not necessarily explained by it

19 Heart modeling Sasha Panfilov - Ghent University

20 Multiscale models of tumor progression What makes cancer cells metastatic? Clonal evolution of tumor cells (Nowell, 1976) Cancer cells compete for glucose, oxygen, space What happens in harsh microenvironment? Stronger selections pressure? Relapses after cancer treatment? Model by Anderson et al., 2006 Tumor cell properties (adhesion, growth rate) affect tumor morphology

21 Tumor evolution Anderson et al. Cell 2006 Unbiased motility Motility to higher concentrations of ECM (haptotaxis)

22 Tumor evolution Anderson et al. Cell 2006 Tumor cells move randomly Tumor cells move up ECM gradients Cells produce matrix-degrading enzymes (MDE) E.g. MMPs Tumor cells consume oxygen Vasculature delivers oxygen: proportional to ECM

23 Tumor evolution Anderson et al. Cell Equations MMPs

24 Tumor evolution Anderson et al. Cell 2006 Cell-based model: possible to give each cell different properties Evolve cellular properties Exp. I: Vogelgram accumulation of mutations Fearon and Vogelstein, 1990

25 Phenotype I: Proliferative, adhesive, consumes little oxygen, produces few MMPs Phenotype IV: Slow proliferation, not adhesive, consumes lots of oxygen, produces lots of MMPs

26 Anderson et al. Cell 2006 Homogenous ECM

27 Anderson et al. Cell 2006 Homogenous ECM

28 Anderson et al. Cell 2006 Bumpy ECM

29 Anderson et al. Cell 2006 Grainy ECM

30 Progressive mutation Selection for aggressive phenotype IV (blue) Harsh environment (bumpy or grainy ECM) Fingering boundary In reality: tumors are genetically heterogeneous Alternative test for random mutation scheme; Cells jump between any of 100 parameter sets no progressive evolution possible

31 100 phenotypes Any combination of traits possible Selection rounds of oxygen limitation and oxygen excess

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33 Selection for aggressive phenotypes Selected clones: no cell-cell adhesion low oxygen consumption high proliferation high haptotaxis Stronger selection in harsh environments

34 Model predictions Tumor microenvironment guides tumor shape Homogeneous ECM: smooth tumor boundary Grainy or bumpy ECM: fingering tumor boundary Harsh environment, e.g. hypoxia: Selection for more aggressive phenotypes Consequences for treatments? Do harsh treatments induce aggressive treatments? Relapse after therapies Some authors suggest to treat cancer as chronic disease : non-resistant cells outcompete resistant ones E.g. Gatenby and others

35 Conclusions Space matters for tumor growth! Tumors grow at edges: no exponential growth Cell-based models suggest causes of invasion / fingering growth Clonal selection in harsh micro-environments (Anderson et al.) Cancer Stem Cells (Sottoriva et al.) Further reading: e.g. Gatenby See reader