Return to Eden:
How biologically relevant can onlattice 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 individualbased 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:
Movement
T-
= A cell
T+
Signal Sensing
= A cell
Some definitions
Probability master equation
Equivalent to PDE
Local Signal Sensing
Cell Density Profiles
Individual model – Blue histograms.
PDE – Red curve.
Growth
= A cell
Exponential Domain Growth
Cell Density Profiles
Individual model – Blue histograms.
PDE – Red curve.
Domain length – Green star.
Domain Growth
PDE – Red.
Average stochastic- Green.
Individual Stochastic – Black.
Density Dependent Domain Growth
Cell Density Profiles
Individual model – Blue histograms.
PDE – Red curve.
Domain length – Green star.
Domain Growth
PDE – Red.
Average stochastic- Green.
Individual Stochastic – Black.
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
Cell Density Profiles
Individual model – Blue histograms.
PDE – Red curve.
Domain length – Green star.
Domain Growth
PDE – Red.
Average stochastic- Green.
Individual Stochastic – Black.
Higher Dimensions
PDE solution surface
Individual based model –
Square grid histogram
Local sensing on a 50X50 square lattice
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
All values are averaged over 50 repeats
Eden C
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 growth
Biophys. 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
The three Eden models average over 50 repeats
Eden C
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 clusters
Journal 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 mitosis
Math. Biosci., Elsevier, 1999.
** Drasdo, D.
Coarse graining in simulated cell populations
Advances 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?