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?
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