cellular automata & molluscan shells

Cellular Automata

& Molluscan Shells

By Andrew Bateman and Ryan Langendorf

Cellular Automata

Wolfram class I:

Wolfram class II:

Wolfram class III:

Wolfram class IV:

Where Did That Shell Come From?

•The outer edge of the mantle lays down calcium carbonate crystals in a protein matrix.

•The periostracum is the outer, organic layer that both protects the shell and gives it its pattern.

Shell Patterns: What Do We Know?

• Not much!

• Evolutionary advantage?– Cone shells have vibrant patterns to warn of

their poison– Ermentrout, Campbell, and Oster say none

• Pigments get permanently laid down over time in a synchronized manner along the leading edge– There is likely interaction between the cells

laying down the pigments

Why Bother With Cellular Automata?

• The mathematician’s answer:They look right.

• The (mathematical) biologist’s answer:

Local Effects of activation and inhibition dominate pigment, and thus pattern,

production.

Activation & Inhibition

Kusch & Markus Propose The Meaning of (Marine)

Life

What Makes It Tick?

random activation and expression of geneproduction of the inhibitoractivation when lots of activated cells in the neighbourhood

quantity of inhibitor in the neighbourhooddeactivation when lots of inhibition

decay of the inhibitor

BiologyMath

What can such a simple model produce?

Strengths & limitations• Strengths:

– The patterns resemble those on the shells – Biology:

•Activation/inhibition is taken into account•All shells can be generated from the same

set of rules– In real life all the shells are made in a

similar fashion• Limitations:

– Patterns differ in details and regularity– Tenuous biological connection

•Scale?•Why use specific parameters?•How derive the specific rules?

Our Improvement: Multiple Genes

Biology Of Our Model

• There are two types of patterns on some shells.

• This indicates there might be multiple genes involved in the creation of the patterns.

• Activation and inhibition is still assumed to be the mechanism behind the production of the patterns.

Playing GodRefresher:

•Activation is randomly triggered and then spreads. •As it spreads inhibitor builds up. •Once the inhibitor reaches a threshold level deactivation occurs.•The inhibitor then decreases.

Our Twist:•If a cell in deactivated, there is a lot of activated cells around it, and there is a lot of inhibitor around it, then a second gene is activated.•The background color produced while this second gene is active is different.•The inhibitor decreases over time.•Once the inhibitor drops below a threshold level the gene is deactivated and pigment production reverts to its previous state.

One Gene

Actual Pattern

Two Genes

Asynchronous

Are Kusch, Markus, And We God?

•If all shells are created in similar ways, why do some versions of the model require the inhibitor to decay linearly and others for it to decay exponentially?

•Is gene activation random?

•How is a neighbourhood’s effect on a cell evaluated?

•Is it realistic to have only inhibitor toggling a gene on and off?

•When a new gene is expressed, is color the only thing changed? Should the pattern differ as well?

Real Life??

• The patterns generated with two genes were more realistic, but still different from the actual ones.

• Our multiple gene model is an extension of one we deem questionable in its biological groundings.

• Multiple genes?

In an abalone one color is exclusively associated with a specific gene. Perhaps the colors on cone shells are similarly controlled, and thus further genetic research is warranted in species displayingsuch patterns.

A New Kind Of Science?

• If there are multiple genes at work, how do they interact, if at all?

• Diffusion equations?

• Neural models?

• A new style of art?

“Everything which is computable can be computedwith… [a] cellular automaton”

- W. Poundstone

“As regards cellular automata models, they make no connection with any of the underlying biological processes”

- J.D. Murray

de Vries, G, et al. A Course in Mathematical BiologyMurray, J.D. Mathematical BiologyKusch, I. and M. Markus. “Mollusc Shell Pigmentation: Cellular Automaton Simulations and Evidence for Undecidability”http://www.stephenwolfram.com/publications/articles/ca/84-universality/9/text.htmlhttp://mathworld.wolfram.com/ElementaryCellularAutomaton.htmlhttp://math.hws.edu/xJava/CA/http://www.weichtiere.at/english/gastropoda/terrestrial/escargot/shell.htmlhttp://www.sealifegifts.net/nautical_decorations.htmlhttp://cephalopodia.blogspot.com/2007/02/five-deadly-animals-that-may-save-your.htmlhttp://www.biochemistry.unimelb.edu.au/research/res_livett.htmhttp://www.scuba-equipment-usa.com/marine/JUN05/Textile_Cone_Shell(Conus_textile).htmlhttp://en.wikipedia.org/wiki/Asynchronous_Cellular_Automatonhttp://online.sfsu.edu/~psych200/unit5/52.htmhttp://www.art.com/asp/sp-asp/_/pd--13060293/sp--A/Jaguar_CloseUp_of_Fur_Pattern_Pantanal_Brazil.htm

Made Possible By:A sincere thanks to Mark and Tomas, without whom this project would not have been realized.