Return to Eden: How biologically relevant can on- lattice models really be?
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Transcript of Return to Eden: How biologically relevant can on- lattice models really be?
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Return to Eden:How biologically relevant can on-lattice
models really be?
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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.
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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).
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Cellular automaton• Pattern formation.• Neural networks.• Population biology.• Tumour growth.
See Ermentrout, G.B. and Edelstein-Keshet, L., Journal of Theoretical Biology 1993
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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.
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Position jump models• Development• Pattern formation• Animal Movement• Aggregation
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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).
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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.
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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?
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Case Study 1:Position jump modelling of cell migration:
MovementT+T-
= A cell
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Signal Sensing
= A cell
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Some definitions
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Probability master equation
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Equivalent to PDE
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Local Signal Sensing
Cell Density ProfilesIndividual model – Blue histograms.PDE – Red curve.
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Growth
= A cell
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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.
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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.
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Incremental Domain Growth
= A cell
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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
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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.
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Higher Dimensions
Local sensing on a 50X50 square lattice
PDE solution surface
Individual based model – Square grid histogram
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Triangular Lattice
PDE solution surface
Individual based model – Traingle grid histogram
Diffusion on a triangular lattice
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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?
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Case Study 2:The Eden model
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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:
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Eden A
• A cell is added to any of the sites which neighbour the surface equiprobably.
# surface neighbouring sites = 12
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Example Eden A cluster
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Eden B
• A cell is added to any of the edges of the surface equiprobably.
# surface edges = 14
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Example Eden B cluster
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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
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Example Eden C cluster
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Real Tumour Slices
Images Courtesy of Kasia Bloch (Gray Institute for Radiation Oncology and Biology and the Centre for Mathematical Biology)
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Important properties•Growth rate•Morphology• Surface thickness•Genus (Holiness)
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Number of holes vs time
Eden A Eden B Eden C
All values are averaged over 50 repeats
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Surface scaling
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Surface scaling
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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
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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
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Eden universality
• In strip geometry Eden is in KPZ.• But not so in radial clusters? • Why not?
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Anisotropy
• Axial anisotropy cause problems.Eden A Eden B Eden C
The three Eden models average over 50 repeats
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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,
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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.
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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.
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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?
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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?