Return to Eden:How biologically relevant can on-lattice

models really be?

Outline

• What sorts of on-lattice models are there?• What do/can we model on-lattice?• Pros.• Cons.• Two case studies– Position jump modelling of cell migration.– Models for tumour growth.

Types of on lattice model

• Cellular automaton.– Exclusion processes.– Game of life.

• Cellular Potts model.• Lattice gas automaton.– Lattice-Boltzmann.

• Ising model.• Position jump models (on lattice).

Cellular automaton• Pattern formation.• Neural networks.• Population biology.• Tumour growth.

See Ermentrout, G.B. and Edelstein-Keshet, L., Journal of Theoretical Biology 1993

Cellular Potts models• Immunology• Tumour growth

•Metastasis•Developmental biology

Cellular Potts Model of single ovarian cancer cell migrating through the mesothelial lining of the peritoneum.

Position jump models• Development• Pattern formation• Animal Movement• Aggregation

Advantages• Simple to formulate and adapt.• Easy to explain to biologists.• Can capture phenomenological details.• Mathematically and computationally

tractable.• Makes multiscale description possible

(i.e. can often derive PDEs).

Problems with on-lattice models

• Geometry - Cells aren’t squares!• Hard to convince biologists.• Changing lattices are difficult to deal

with (i.e. how to implement cell birth/death).• Inherent anisotropy.• Artificial noise effects.

What’s best for…

• …Parallelisation of code?– Can both on-lattice and off-lattice individual-based

models be parallelised equally well?• …Boundary condition implementation?– Which type of model deals best with curved

boundaries for example?

Case Study 1:Position jump modelling of cell migration:

MovementT+T-

= A cell

Signal Sensing

= A cell

Some definitions

Probability master equation

Equivalent to PDE

Local Signal Sensing

Cell Density ProfilesIndividual model – Blue histograms.PDE – Red curve.

Growth

= A cell

Exponential Domain Growth

Domain GrowthPDE – Red.Average stochastic- Green.Individual Stochastic – Black.

Cell Density ProfilesIndividual model – Blue histograms.PDE – Red curve.Domain length – Green star.

Density Dependent Domain Growth

Domain GrowthPDE – Red.Average stochastic- Green.Individual Stochastic – Black.

Cell Density ProfilesIndividual model – Blue histograms.PDE – Red curve.Domain length – Green star.

Incremental Domain Growth

= A cell

Connecting to a PDE• In order to connect the PDE with the cell

density we had to enforce a Voronoi domain partition.

Interval Centred Domain Partition

Vornoi Domain Partition

Diffusion on the Voronoi domain partition

Domain GrowthPDE – Red.Average stochastic- Green.Individual Stochastic – Black.

Cell Density ProfilesIndividual model – Blue histograms.PDE – Red curve.Domain length – Green star.

Higher Dimensions

Local sensing on a 50X50 square lattice

PDE solution surface

Individual based model – Square grid histogram

Triangular Lattice

PDE solution surface

Individual based model – Traingle grid histogram

Diffusion on a triangular lattice

Growth in two-dimensions?

• Circular or square domain to make PDEs tractable.

• Apical growth?• How much can lattice sites push each other

out of the way?• Can we make on lattice models replicate real

biological dynamics, at least qualitatively?

Case Study 2:The Eden model

The Eden model• Produces roughly circular growth (especially

for large clusters)• Start of with an initial “cell” configuration or a

single seed.• Square cells are added one at a time to the

edges of the cluster in one of three ways:

Eden A

• A cell is added to any of the sites which neighbour the surface equiprobably.

# surface neighbouring sites = 12

Example Eden A cluster

Eden B

• A cell is added to any of the edges of the surface equiprobably.

# surface edges = 14

Example Eden B cluster

Eden C

• A surface cell is chosen equiprobably and one of its edges chosen equiprobably to have a cell added to it.

# surface cells = 8

Example Eden C cluster

Real Tumour Slices

Images Courtesy of Kasia Bloch (Gray Institute for Radiation Oncology and Biology and the Centre for Mathematical Biology)

Important properties•Growth rate•Morphology• Surface thickness•Genus (Holiness)

Number of holes vs time

Eden A Eden B Eden C

All values are averaged over 50 repeats

Surface scaling

Surface scaling

Universality Classes (UC)

• By finding these coefficients we can classify these models into universality classes.

• Some well-known universality classes are:

Name Z

EW ¼ ½ 2

KPZ 1/3 ½ 3/2

MH 3/8 3/2 4

Tumour universality class

• Brú et al*. found a universality class for tumours.

• They placed tumours in the MH universality class.

*Brú, A.; Albertos, S.; Luis Subiza, J.; Garcia-Asenjo, J. & Brú, I.The universal dynamics of tumor growthBiophys. J., Elsevier, 2003, 85, 2948-2961

Eden universality

• In strip geometry Eden is in KPZ.• But not so in radial clusters? • Why not?

Anisotropy

• Axial anisotropy cause problems.Eden A Eden B Eden C

The three Eden models average over 50 repeats

Anisotropy correction

• Even model C exhibits a 2% axial anisotropy.• But Paiva & Ferreira* have found a way to

correct for this.• Once corrected and surface thickness

determined in the proper way it was found the radial Eden clusters fall into the KPZ UC.

*Paiva, L. & Ferreira Jr, S.Universality class of isotropic on-lattice Eden clustersJournal of Physics A: Mathematical and Theoretical, IOP Publishing, 2007,

Mitosis

• Off-lattice Eden model – Ho and Wang*.– Isotropic but no use to us as it’s off lattice.

• On lattice with limited pushing range – Drasdo**.– Limited range of pushing.– Anisotropic.

*Ho, P. & Wang, C.Cluster growth by mitosisMath. Biosci., Elsevier, 1999.

** Drasdo, D.Coarse graining in simulated cell populationsAdvances in Complex Systems, Singapore: World Scientific, 2005.

Adapted mitosis model• Division in 8 neighbouring directions.

• No limit as to how far we can push other cells.• Isotropic? Tentative yes.• Universality class? Too early to say.

Summary

• Lattice model examples.• Pros and cons.• Position jump case study.• Cluster growth case study.• Lattice models can be compared to real-world

phenomena (e.g. universality classes, genus).• But how realistic are they?

Discussion points

• Will on-lattice models continue to be of use in the future?

• Will on lattice models ever be as realistic as off-lattice models?

• Why use a lattice model when an off-lattice model works just as well (and vice versa)?

• Do lattice models have a role in communicating our modelling ideas to biologists?