Cell-Based Modelling in Chaste: Understanding Tissue Renewal & Wnt Signalling

1. Cell-Based Modelling in Chaste James Osborne

2. Timetable • 9:30am Lecture and Demo • 11am Practical • 4pm Retrospective • See https://chaste.cs.ox.ac.uk

3. Overview of talk • Intro to Crypt biology • Individual based models of tissues • Crypt simulations • Running Cell-Based Chaste simulations • Demo

4. Tissue renewal • Tissue renewal and regeneration are crucial for the survival and longevity of multicellular organisms • Replacement of the outer layers of the epidermis ensures maintenance of the skin’s structure and mechanical properties despite its continuous and direct exposure to a wide range of damaging factors • For a similar reason, one of the most rapidly renewing tissues in the body is the intestinal epithelium

5. Crypts of Lieberkuhn Paul Appleton, Dundee • This process is driven by stem cell proliferation within the crypts of Lieberkuhn • There are ~20 million crypts in the large intestine • About 700 cells in each and renewal every 3-5 days

6. Role of Wnt signalling • There is evidence that Wnt signalling plays a central role in maintaining the intestinal stem-cell niche and regulating normal crypt dynamics • It has been proposed that a spatial gradient of extracellular Wnt factors along the crypt axis determines position-dependent rates of cell proliferation, differentiation and death • Wnt signalling is initiated when extracellular Wnt factors bind to specific receptors on the cell surface WNT LEVEL

7. Wnt signalling in cancer • Most cancers can be initiated by a wide number of different mutations, but almost all CRCs carry activating mutations in the Wnt pathway • Over 90% of CRCs begin with either double-hits that truncate or disable APC or a single-hit in b-catenin • Both these genetic alterations result in ‘activation’ of the Wnt signalling pathway, build-up of b-catenin, and transcription of Wnt target genes • Thus the Wnt pathway plays a crucial role in the initiation of colorectal cancer • http://www.molecularmovies.com/movies/kellermcgill_clonalconveyorbelt.mov

8. Crypt modelling • Several fundamental biological questions remain to be resolved • position and number of stem cells • how different cellular processes are regulated and coordinated to maintain crypt homeostasis • As a crypt is a complex, highly-regulated system, a theoretical approach is valuable for attaining deeper understanding of its dynamics • Mathematical and computational modelling can provide insights that complement and reinforce knowledge acquired by experiments

9. Crypt modelling • Mathematical modelling has been used to investigate aspects of CRC for over half a century • Models for crypt dynamics have been used to investigate the mechanisms of • cell migration, • tissue recovery following irradiation, • malignant transformation, • stem-cell dynamics • Recent advances have led to multiscale models where we study the effect of pathway mutations on the tissue as a whole

10. Individual based models of tissues

11. Cell-level models • Explicitly consider individual cells • Track cell movement, size, shape • Influences from ‘above’ and ‘below’ • Cellular automata, cellular potts, cell-centre based models, vertex models

12. Cell centre models • Cell (centres) represented as points in space • Forces between centres: • Overdamped springs; • Hertz laws .etc. • Connectivity: • OS (node based); • Voronoi tessellation (mesh based)

13. Cell centre models

14. Vertex models • Cell represented as polygons whose vertices are free to move • Control over cell size and cell-cell adhesion/interactions • Evolution by potentials, or other force balance

15. Vertex models

16. Cellular Potts • Cells composed of a collection of lattice points • Probabilities associated with different moves • Monte Carlo simulations to update cells, multiple sweeps per timestep

17. Cellular Potts

18. sub-cellular processes: cell cycle models • Simple agent based models • Cell cycle or other metabolic pathways • Typically a system of non-linear ODEs • Coupled to extracellular concentrations • Cell division, cell size/shape, cell-stromal and cell-cell adhesion, cell fate http://teachline.ls.huji.ac.il www.biocarta.com

19. Modelling tissue-level processes • Geometrical constraints • Imposed gradients • Field equations: • nutrient or inhibitor diffusion, • cells as sinks/sources, • on a growing domain

20. Putting it all together Multiple possible models at each scale

21. Simulations of the Crypt

22. Model setup • For simplicity we first focus on an individual crypt, treating the 3D tubular crypt as a monolayer of cells lying on a cylindrical surface • We take a discrete approach, modelling each cell individually using a cell centre model • For simulation purposes, it is convenient to roll the crypt out onto a flat planar domain and impose periodic boundary conditions on the left and right sides • Impose a gradient of WNT, high at the bottom and low at the top

23. Cell movement and proliferation Target protein synthesis WNT SIGNALLING MODEL CELL CYCLE MODEL Biochemical cues Cell-cell adhesion Cell position CELL MECHANICS MODEL Cell size Cell neighbours Proliferation/ Differentiation Movement

24. Basic Crypt • The yellow cells are proliferating as they experience sufficient Wnt , the red cells are differentiated as the levels of Wnt are not high enough • A cell and its progeny is labelled in blue and you see that this cell takes over the crypt

25. Clonal expansion and niche succession • We can follow expansion of a clonal population in silico • The progeny of one stem cell can eventually take over the whole crypt • Our model predictions are consistent with experimental evidence • It has been proposed that monoclonal, mutant crypts constitute the earliest stage of colorectal adenomas • These monocryptal lesions can expand further by crypt fission

26. Introducing mutations • We can add mutant cells to the simulation to investigate the onset of CRC • Mutant cells proliferate independently of external cues and experience a higher adhesion to stromal tissue

27. 3D crypt • We can also investigate cell dynamics on different geometries for example the test tube shaped crypt • This allows us to accurately represent the number of cells at the base of the crypt

28. Cell-Based Chaste

29. Functionality Tissue level: • external factors; • PDEs: reaction diffusion, etc. • Cell level: • cell centre; • OS, • voronoi • vertex based; • cellular Potts; • cellular automata • Sub-cellular level: • rule based; • stochastic; • ODE based cell cycles models/ other networks

30. Cell Cell Cell Sub-cellular model Sub-cellular model Sub-cellular model Code structure • Open source modular code • C++ classes to represent individual elements • Simple to add new functionality, using sub-classes • Uses existing libraries: PETSc, CVODE .etc. • “CellBasedCodeStructure” External Factors Mesh Cell Population Forces/ Update Rules BCS Cell Killers Simulation

31. Setting up a simulation To specify a simulation you need to decide on the following: • Type of cell level model • interaction forces/rules • boundary forces/rules • Sub cellular model • proliferation, growth, death .etc. • Diffusible species • how these interact with cells • boundary conditions

32. Defining simulations Mesh Cells (1) Mesh (2) Cells (3) Cell Population (4) Simulation (5) Force/update rule (6) - Cell Killer (7) - Cell population BCS (8) (Run simulation) Cell Population Forces/ Update Rules BCS Cell Killers Simulation Note: Doesn’t include PDEs – see advanced tutorials

33. Chaste++ Mesh Cells Cell Population Simulation Forces Cell Killers (BCS)

34. Example simulation

35. CellBasedDemo • “UserTutorials/CellBasedDemo” • Changing cell level model • Changing CCM • Changing force • Introducing killers • Introducing boundary conditions • Cell signalling (i.e. PDEs) covered in advance tutorials