Session 6: simulating crypt homeostasis in Chaste

1. Session 6: simulating crypt homeostasis in Chaste Cell-based Chaste workshop Thursday 5th January 2012

2. Summary of crypt model • In the model by van Leeuwenet al. (2009), every virtual cell carries a continuum cell-cycle control model that is coupled to an intracellular Wnt signalling network. • Given a certain Wnt stimulus, the Wnt model determines availability of key components of the cell-cycle model, which in turn defines whether a cell is ready to divide or differentiate. • Spatial variations in the extracellular Wnt signal translate into position-dependent cell proliferation and differentiation rates. • As Wnt signalling is allowed to interfere with cell–cell junctions, variable adhesion can occur within our in silico crypts. • A mechanical model, describing the attachment of cells to the underlying substrate and the attractive/repulsive forces between cells, determines cell migration.

3. Model geometry • For simplicity we 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. • 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.

4. Mechanical model

5. Mechanical model • We determine cell movement by balancing forces exerted on an individual cell by its neighbours with a drag force: • When a cell divides, a new cell is placed a smaller fixed distance away in a random direction. • The rest length between the two daughter cells increases linearly over the course of an hour to the mature cell rest length (to emulate the mitosis phase of the cell-cycle).

6. Wnt signalling model • We impose a steady linear Wnt profile up the crypt. • To characterise each cell’s Wnt response, we use simple ODE model of the Wnt-dependent progress through the cell cycle, based on the cell-cycle model developed by Swat et al. (2004). • We solve the system of ODEs numerically for each cell to calculate concentrations at the next timestepbased on initial concentrations and Wnt exposure at the current timestep. • Since the Wnt model incorporates the dual role of b-catenin in Wnt signal transduction and cell-cell adhesion, we can quantify the levels of adhesion and transcription complexes for each cell.

7. Wnt-dependent cell-cycle model • The level of transcription complexes and target-protein synthesis rates are used to link the output of the Wnt signalling model to a recent ODE model of the cell-cycle. • According to the resulting model, cells exposed to a strong Wnt signal progress more quickly through the cell cycle than cells exposed to low Wnt.

8. Wnt-dependent cell-cycle model • Hence, inclusion of a spatially varying Wnt signal into our multiscale model gives rise to cell cycles whose duration is position-dependent. • Due to the cell-cycle model’s bistability, there is a threshold Wnt level below which the G1/S checkpoint can never be passed: such cells are considered differentiated. • De-differentiation may occur.

9. Putting it all together 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

10. Implementation • Create a suitable mesh using CylindricalHoneycombMeshGenerator • Use GetCylindricalMesh()to generate a Cylindrical2dMesh • Use GetCellLocationIndices() to store which nodes correspond to ‘real’ cells • Create a vector of cells using CryptCellsGenerator • This class is templated over cell cycle model • Use Generate() to populate a vector of cells • Create a MeshBasedCellPopulationWithGhostNodes • Set up a WntConcentration singleton • Call SetType(), SetCellPopulation() and SetCryptLength()

11. Implementation • Set up a CryptSimulation2d object using the cell population • Call SetOutputDirectory() and SetEndTime() • Create a force object to simulate cell mechanics • E.g. MAKE_PTR(GeneralisedLinearSpringForce<2>, p_force) • Call AddForce() on the CryptSimulation2d object • Create a cell killer object to simulate sloughing at the top of the crypt • E.g. MAKE_PTR_ARGS(SloughingCellKiller<2>, p_killer, (&population, height)) • Call AddCellKiller() on the CryptSimulation2d object • Call Solve() on the CryptSimulation2d object

12. Implementation • Documentation and further details of the class hierarchy are available on the wiki. • You will find the tutorials for this session here: • UserTutorials/RunningMeshBasedCryptSimulations • UserTutorials/RunningVertexBasedCryptSimulations • These will guide you through the implementation of various crypt models. • Further exercises are also suggested for those who are interested.