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Geometry before Euclid
RIKEN / June 18, 2014
Robert Sinclair
Mathematical Biology Unit
What is this talk about?
Our (vertebrate) three-dimensional body
plans require that cells know where they
are, in three-dimensions, to make the
correct development of organs and limbs
possible.
How did life discover three dimensions?
Were body plans always three-dimensional,
or do we see a progression in the fossil
record?
Some Context
http://1.bp.blogspot.com/-J3NZOMBlvP4/Ti5Ls2yB-1I/AAAAAAAABT4/8Ks2mgGBv9s/s1600/Age+of+the+Universe.jpg
Early life...
Albani et al., Large colonial organisms with coordinated
growth in oxygenated environments 2.1 Gyr ago, Nature 466,
100–104 (1 July 2010) doi:10.1038/nature09166
Pizza Discs...
Kevin J. Peterson et al., A Fungal Analog for Newfoundland
Ediacaran Fossils? Integr. Comp. Biol., 43:127–136 (2003)
Early life...
Phase contrast image of Monosiga brevicollis, by Stephen
Fairclough, Wikimedia Commons.
What can we say about
evolution’s early
exploration of dimensionality,
from one-dimensional, linear
organisms
to truly two-dimensional
organisms?
Who got to one dimension first?
Bacteria
make
filaments
Equispaced
differentiation:
heterocysts
John C. Meeks et al., Cellular differentiation in the cyanobacterium Nostoc punctiforme,
Arch Microbiol (2002) 178 : 395–403 DOI 10.1007/s00203-002-0476-5
Horodyskia (1.5 billion years old)
Fedonkin et al., Middle Proterozoic (1.5Ga) Horodyskia moniliformis Yochelson and
Fedonkin, the Oldest Known Tissue-Grade Colonial Eukaryote,
Smithsonian Contributions to Paleobiology, No. 94 (2002)
Fractals?
http://ancientshore.files.wordpress.com/2010/02/
cpc-07-mistaken-pt-nl-fractofusus-204lr.jpg?w=600&h=379
Fractals?
Guy M. Narbonne, Modular Construction of Early Ediacaran Complex Life Forms
Science 305, 1141 (2004); DOI: 10.1126/science.1099727
Slope = 1, so one-dimensional
Slope = 2 (about), so two-dimensional
Slope = ...
Slope = ..., so it has fractional dimension
Fractofusus
misrai
http://content65.eol.org/content/2011/08/06/01/50247_orig.jpg
http://content64.eol.org/content/2011/11/01/22/05409_580_360.jpg
A New Frontier:
Two Dimensions
Christof Niehrs, On growth and form: a Cartesian
coordinate system of Wnt and BMP signaling
specifies bilaterian body axes, Development 137,
845-857 (2010) doi:10.1242/dev.039651
Dickinsonia
About 560 million years ago
Dzik & Ivantsov, 2002
Dickinsonia
A Conformal Map
Can one make a twodimensional body plan with
a one-dimensional toolkit
(a single morphogen)?
Hormogonia
Elsie Lin Campbell et al., Characteristics of Hormogonia Formation by Symbiotic Nostoc spp.
in Response to the Presence of Anthoceros punctatus or Its Extracellular Products, Applied
and Envirnomental Microbiology, Vol. 55, Jan. 1989, p. 125-131
Hapsidophyllas
Multibranched Rangeomorphs from the Ediacaran Mistaken
Point Assemblage, Newfoundland, Canada <E.L. Bamforth,
Master of Science Thesis, Queen's University, 2008>
Brigid LM Hogan, Bone morphogenetic proteins in development,
Current Opinion in Genetics & Development 1996, 6:432-438
Paleontological Journal, Vol. 38, No. 3, 2004, pp.
247–253. Translated from Paleontologicheskii Zhurnal,
No. 3, 2004, pp. 21–26.
GREGORY J. RETALLACK, Growth, decay and burial compaction of
Dickinsonia, an iconic Ediacaran fossil, Alcheringa 31, 215-240.
Can one make a twodimensional body plan with
a one-dimensional toolkit
(a single morphogen)?
YES!
Swartpuntia
Guy M. Narbonne et al., The Youngest Ediacaran Fossils from
South Africa, Journal of Paleontology, Vol. 71 (1997) 953-967.
First Impressions:
The fossil record seems to show that
while there was a progression from one
to two dimensions, the developmental
mechanisms may have remained onedimensional, until the later emergence of
true bilaterians etc.
Were these precambrian organisms
“geometric pioneers” but doomed to
failure because their developmental
mechanisms were not truly higherdimensional?
Hypotheses/Thoughts
Explore before map-making.
Don’t build without a plan.
“Foraging” before coordinate
system definition.
No complex body plan without
coordinates.
What about us?
Robin Hayman et al., Anisotropic encoding of three-dimensional space by place cells and grid
cells, Nature Neuroscience, Vol. 14, No. 9 (September 2011) 1182-1188. doi:10.1038/nn.2892
What about bats?
Michael Yartsev and Nachum Ulanovsky, Representation of Three-Dimensional Space in the
Hippocampus of Flying Bats, Science, Vol. 340 (April 2013) 367-372. doi:10.1126/science.
1235338
What about bats?
Fig. 3 of Michael Yartsev and Nachum Ulanovsky, Representation of Three-Dimensional Space in
the Hippocampus of Flying Bats, Science, Vol. 340 (April 2013) 367-372. doi:10.1126/science.
1235338
Geometry before Euclid
RIKEN / June 18, 2014
Robert Sinclair
Mathematical Biology Unit