http://www.cs.ox.ac.uk/chaste/

Cell-Based Modelling in Chaste

James Osborne

Timetable

• 9:30am Lecture and Demo

• 11am Practical

• 4pm Retrospective

• See https://chaste.cs.ox.ac.uk

Overview of talk

• Intro to Crypt biology

• Individual based models of tissues

• Crypt simulations

• Running Cell-Based Chaste simulations

• Demo

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

Crypts of Lieberkuhn

• 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

Paul Appleton, Dundee

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

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

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

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

Individual based models of tissues

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

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)

Cell centre models

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

Vertex models

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

Cellular Potts

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

Modelling tissue-level processes

• Geometrical constraints

• Imposed gradients

• Field equations:

• nutrient or inhibitor diffusion,

• cells as sinks/sources,

• on a growing domain

Putting it all together

Multiple possible models at each scale

Simulations of the Crypt

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

Cell movement and proliferation

WNT SIGNALLING

MODEL

WNT SIGNALLING

MODEL

CELL CYCLE MODEL

CELL CYCLE MODEL

CELL MECHANICS

MODEL

CELL MECHANICS

MODEL

Cell-celladhesion

Target protein synthesis

Cell size

Biochemical cues

Cell neighbours

Cell position

MovementProliferation/

Differentiation

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

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

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

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

Cell-Based Chaste

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

Code structure

Cell Cell PopulatioPopulatio

nn

MesMeshh

SimulationSimulation

BCSBCS Cell Cell KillersKillers

CellCell Sub-cellular Sub-cellular modelmodel

CellCell Sub-cellular Sub-cellular modelmodel

CellCell Sub-cellular Sub-cellular model model

External External FactorsFactors

• 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”

Forces/ Forces/ Update Update Rules Rules

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

Defining simulations

(1) Mesh (2) Cells(3) Cell Population (4) Simulation(5) Force/update rule(6) - Cell Killer (7) - Cell population BCS(8) (Run simulation)

Cell Cell PopulatioPopulatio

nn

MeshMesh

SimulationSimulation

CellsCells

Forces/ Forces/ Update Update Rules Rules

Cell Cell KillersKillersBCSBCS

Note: Doesn’t include PDEs – see advanced tutorials

Chaste++

Cell Cell PopulatioPopulatio

nn

MeshMesh

SimulationSimulation

CellsCells

Forces Forces

Cell Cell KillersKillers

(BCS)(BCS)

Example simulation

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