Transcript 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?
Math
Biology
decay of the inhibitor
random activation and
expression of gene
production of the inhibitor
activation when lots of
activated cells in the
neighbourhood
quantity of inhibitor in the
neighbourhood
deactivation when lots of
inhibition
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 God
Refresher:
•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.
Two Genes
One Gene
Actual Pattern
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 displaying
such 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 computed
with… [a] cellular automaton”
- W. Poundstone
“As regards cellular automata models, they
make no connection with any of the underlying
biological processes”
- J.D. Murray
Made Possible By:
A sincere thanks to Mark and Tomas, without
whom this project would not have been realized.
de Vries, G, et al. A Course in Mathematical Biology
Murray, J.D. Mathematical Biology
Kusch, 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.html
http://mathworld.wolfram.com/ElementaryCellularAutomaton.html
http://math.hws.edu/xJava/CA/
http://www.weichtiere.at/english/gastropoda/terrestrial/escargot/shell.html
http://www.sealifegifts.net/nautical_decorations.html
http://cephalopodia.blogspot.com/2007/02/five-deadly-animals-that-may-save-your.html
http://www.biochemistry.unimelb.edu.au/research/res_livett.htm
http://www.scuba-equipment-usa.com/marine/JUN05/Textile_Cone_Shell(Conus_textile).html
http://en.wikipedia.org/wiki/Asynchronous_Cellular_Automaton
http://online.sfsu.edu/~psych200/unit5/52.htm
http://www.art.com/asp/sp-asp/_/pd--13060293/sp--A/Jaguar_CloseUp_of_Fur_Pattern_Pantanal_Brazil.htm