Cellular Automata & Molluscan Shells - University of Alberta

overwhelmedblueearthΤεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 3 χρόνια και 8 μήνες)

139 εμφανίσεις

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

²
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.




7KH??PDWKHPDWLFDO??ELRORJLVW∙V?DQVZHU?




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 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

decay of the inhibitor

Biology

Math

What can such a simple model
produce?

Strengths & limitations


Strengths:

²
The patterns resemble those on the shells

²
Biology:


Activation/inhibition is taken into account


All shells can be generated from the same set of rules

²
In real life all the shells are made in a similar
fashion


Limitations:

²
Patterns differ in details and regularity

²
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.

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 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

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


Made Possible By:

A sincere thanks to Mark and Tomas, without

whom this project would not have been realized.