1. Cellular Automata & Molluscan Shells
By Andrew Bateman and Ryan Langendorf

2. Cellular Automata

3. Wolfram class I:
Wolfram class II:
Wolfram class III:
Wolfram class IV:

4. 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.

5. 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

6. 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.

7. Activation & Inhibition

8. Kusch & Markus Propose The Meaning of (Marine) Life

10. What Makes It Tick? Biology Math 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

11. What can such a simple model produce?

17. Strengths & limitations
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?

18. Our Improvement: Multiple Genes

19. 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.

20. 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.

21. Two Genes
One Gene
Actual Pattern

23. Asynchronous

29. 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?

30. 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?

31. 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.

32. 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?

39. “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

40. 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”