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

2. Summary of crypt model In the model by van Leeuwen et 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. 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 timestep based on initial concentrations and Wnt exposure at the current timestep. Since the Wnt model incorporates the dual role of  -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 WNT SIGNALLING MODEL CELL CYCLE MODEL CELL MECHANICS MODEL Cell-cell adhesion Target protein synthesis Cell size Biochemical cues Cell neighbours Cell position Movement Proliferation/ Differentiation

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, 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, 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.