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From excitability to pearls: nacre as a complex system
The story begins with the discovery of microscopic spiral and target patterns on the surface of nacre, the iridescent material that forms the inside of many mollusc shells and the outside of their pearls, and which is now a greatly studied biomineral for its exceptional material properties that make the formation of artificial nacre a leading challenge in biomaterials today. These patterns in nacre arise from the same mechanism that Burton, Cabrera, and Frank (BCF) predicted for the growth of a crystal, except that here we are dealing not with a crystal but with a liquid crystal. The BCF growth mechanism is an example of an excitable medium, a type of system that is very common in biology - other examples of excitable systems are the behaviour of action potentials in neurons, forest fires, and cardiac tissue - based upon a fixed point surrounded by a limit cycle that is 'excitable' when the former is stable and the latter unstable, and a relaxation oscillator in the reverse case. Spatially coupled excitable elements form an excitable medium, whose spatiotemporal dynamics gives rise to spirals and target patterns. In the case of nacre, this excitable medium is the first element of self-organized, self-assembled system involving liquid crystallization, solidification, and mineralization that literally sets in stone this complex system.
Friday, February 10, 2012
Julyan Cartwright,
CSIC
Instituto Andaluz de Ciencias de la Tierra,
Consejo Superior de Investigaciones Cientificas-Universidad de Granada
From excitability to pearls:
nacre as a complex system
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Shells are structures with a first-order adaptive value and which constitute an essential part of the basic plan of the different groups
Nacre is exclusive to Mollusca: It is present (starred) in gastropods, bivalves, cephalopods (Nautilus and Spirula), and monoplacophorans
Nautilus
gastropod
monoplacophoran
bivalve
polyplacophoran scaphopod
* * *
*
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© Nature Publishing Group1966
Koji Wada, 1966
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Nacreous pearls
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Pinctada martensiiPteria avicula
grow
th fr
onts
Spiral, target and digitiform patterns in Pterioida (Bivalvia)
Pteria aviculaPteria avicula
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Where do the patterns come from?
What do the patterns tell us about the formation mechanisms of
nacre?
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+ =Text
Nacre for physicists and mathematicians
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Shells are multilayer organic-mineral biocomposites:two or three layers with different microstructures per shell
ScaphopodBivalve
Crossed lamellar
Nacre
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During biomineralization, calcite or aragonite crystals organize themselves into microstructures
Foliated (Anomia) Prismatic (Pinna) Fibrous (Propeamussium)
Homogeneous (Entodesma) Nacre (Calliostoma)
Crossed lamellar (Fragum)Prismatic (Lamprotula)
CALCITE
ARAGONITE
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Some microstructures can be assimilated to inorganic precipitates…
foliated (Ostrea edulis) calcitic prismatic (Pinna nobilis)homogeneous (Entodesma)
calcite grown with lysozymesparry calcite
marine aragonitic cement
BIOMINERALS
INORGANIC
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… but not nacre, with its characteristic brick and mortar arrangementCalliostoma zizyphinum
Gibbula pennanti
Pinctada martensii
from Mayer, G. 2005. Science 310:1144-1147
Atrina pectinata
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nacreprismatic layer
periostracum extrapallial spacemantle cells with microvellosities
interlamellar membraneintertabular membranearagonite tablet
mantle 100 µm
10 µm
20 µm
Hierarchical structure of nacre
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Relationship of nacre tablets to organic membranes
interlamellar membranes
interlamellar membrane
pericrystalline membrane
*original material from Dr. Hiroshi Nakahara, kindly ceded by Dr. Mitsuo Kakei (Meikai University)
Pinctada radiata
Haliotis gigantea
interlamellar membrane
pericrystalline membrane
pericrystalline membrane
interlamellar membranes
mantle
Clanculus jussieui
Anodonta cygnea
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Relationship between the interlamellar membranes and intertabular matrices
from Nakahara, H. 1991. In: Suga, H & Nakahara, H. (eds.) Mechanisms and Phylogeny of Mineralization in Biological Systems, 343-350. Springer-Verlag, Tokyo
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Ultrastructure of organic membranes
Pinctada radiata
*original material from Dr. Hiroshi Nakahara, kindly ceded by Dr. Mitsuo Kakei (Meikai University)
Levi-Kalisman et al. 2001. J. Struct. Biol. 135:8-17
Gibbula umbilicalisAnodonta cygnea
silk-fibroin
β-chitin fibres
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Terraced growth in bivalves
Atrina pectinata Pinctada radiata
from Nakahara, H. 1991. In: Suga, H & Nakahara, H. (eds.) Mechanisms and Phylogeny of Mineralization in Biological Systems, 343-350. Springer-Verlag, Tokyo]
Anodonta cygnea
Anodonta cygnea
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The extrapallial space; secretion of interlamellar membranes in Pinctada radiata
*original material from Dr. Hiroshi Nakahara, kindly ceded by Dr. Mitsuo Kakei (Meikai University)
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Proposed formation process of β-chitin crystallites by assimilation to the known process for cellulose crystallites
from Levi-Kalisman et al. 2001. J. Struct. Biol. 135:8-17
from Doblin, M.S. et al. 2002. Plant Cell Physiol. 43:1407-1420
Hypothesized mode of formation of β-chitin crystallites from polymer chains extruded via rosette-like structures
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Basic components of nacre and their sequence of formation
Silk-fibroin
Mineralization(aragonite)
protein coating
arrangement of β-chitincrystallites
β-chitin polymer chains(formed extrapallialy)
Form
atio
n of
inte
rlam
ella
r mem
bran
es Formation of interlam
ellar spaces
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As we shall see, molluscs make their brick walls in a rather different way than
we do...
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self-assembly of rod-shaped elements:
liquid crystals
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Liquid crystals
liquid crystal cholesteric(chiral nematic) mesophase
synthetic liquid crystal and associated defects
http://www.lcd.kent.edu/images/
cholesteric phase
nematic phase
isotropic phase
E
100,000 E
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shown how different sizes of construction units can be system-atically combined to render progressively higher levels of structuralhierarchy and incorporating different length scales. The figurecombines contributions of several groups. It also combines alreadyverified results with an outlook for still anticipated results. Forclarity, we present the complete scheme already at the start.
Hierarchical self-assembly in block-copolymer/amphiphile complexes
It is well established that various self-assembled structures areallowed by physically bonding oligomeric repulsive side chains tohomopolymers: ionic interactions are used in polyelectrolyte/surfactant complexes in the solid state as shown byAntonietti et al.,characteristic examples being polyacrylate or polystyrene sulfonatewith cationic surfactants.57,58 Further examples59 are provided bysurfactant complexes of poly(ethyleneimine),60 poly(4-vinyl-pyridine) and poly(2-vinylpyridine),61 poly(aniline),62 poly(2,5-pyridinediyl),63 poly(L-lysine),64 or cationic starch.65 Often theself-assembled structures are lamellar but cylindrical and morecomplicated phases have been reported as well.66 Hydrogenbonding allows non-charged self-assembled structures, as has beenshown using e.g. poly(4-vinylpyridine)/alkylphenols by Ikkala tenBrinke et al.,67–69 poly(ethyleneoxide)/dodecylbenzene sulfonic acidby Chen et al.,70 and poly(vinylphenol)/aminic amphiphiles byAkiyama et al.71 Due to its weakness, hydrogen bonding allowsadditional freedom to control the strength of bonding e.g. bytemperature. Also coordination is very useful for self-assembly,72,73
as it allows the tuning of the self-assembly using both the ligandsand the counter-ions.74 Fig. 2 shows a concept where four alkylchains are bonded to each repeat unit of poly(4-vinylpyridine), i.e.P4VP, where two octyl chains are due to the ligands, and one
dodecyl tail in each of the dodecylsulfonate counter-ions. The sidechain crowding leads to cylindrical self-assembly.
Fig. 1 One of the potential scenarios to construct hierarchically self-assembled polymeric structures. Construction units of different sizes allow a naturalselection of different self-assembled length scales. Structural hierarchy is shown for amphiphiles complexed with both block copolymers (Ikkala and tenBrinke et al.10,12,16) and rod-like polymers (in collaboration with Monkman and Serimaa et al.52,54). Combination of block copolymers and mesogenicoligomers has been described by Thomas and Ober et al.55 Combination of polymeric colloidal spheres andblock copolymers has been reported by Kramerand Fredrickson et al.56
Fig. 2 Coordinated comb-shaped polymeric supramoleculespoly[(4VP)Zn(2,6-bis-(n-octylaminomethyl)-pyridine)(DBS)2] and theircylindrical self-assembly as suggested by small angle X-ray scattering.The magnitude of the scattering vector is given by q ~ (4p/l)sinq where2q ~ scattering angle and l ~ 0.154 nm.74
2 1 3 2 C h e m . C o m m u n . , 2 0 0 4 , 2 1 3 1 – 2 1 3 7
Text
Hierarchical self-assembly in polymeric complexes: Towards functional materials
Olli Ikkala and Gerrit ten Brinke, ChemComm, 2131-2137, 2004.
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An example of a proposed biological liquid crystal: The twisted plywood structure of the exoskeleton in arthropods
formed with α-chitin
from Neville, A.C. 1993. Biology of fibrous composites. Cambridge Univ. Press, N.Y.Carcinus maenas (crab) Hidrocyrius (water bug) Copris (beetle)
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Chitin arrangement in nacre is a partially-ordered liquid crystal
Gibbula pennanti
logjam analogy
cholesteric phaseisotropic phase
days/months
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Interlamellar membranes are felt-like fibrous composites
Gibbula pennanti
Anodonta cygnea
protein
β-chitin
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Membrane separation with protein coating
90 nm
500 nm
*original material from Dr. Hiroshi Nakahara, kindly ceded by Dr. Mitsuo Kakei (Meikai University) Pinctada radiata
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Hierarchical construction of nacre
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Excitable Media
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Examples of excitability:
B-Z reagentNerve cellsCardiac cells, muscle cellsSlime mold (Dictystelium discoideum)CICR (Calcium Induced Calcium Release) Forest Fires
Features of Excitability:
Threshold Behavior RefractorinessRecovery
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Anexample
of an excitable element...
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University of UtahMathematical Biology
theImagine Possibilities
Two Variable Models
Following is a summary of two variable models of excitablemedia. The models described here are all of the form
dv
dt= f(v, w) + I
dw
dt= g(v, w)
Typically, v is a “fast” variable, while w is a “slow” variable.
v
w
V-(w) V0(w) V+(w)
f(u,w) = 0
g(v,w) = 0
W*
W*
Excitable Cells – p.13/??
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University of UtahMathematical Biology
theImagine Possibilities
FitzHugh-Nagumo Equations
0.20
0.15
0.10
0.05
0.00
-0.05
w
1.20.80.40.0-0.4v
dw/dt = 0
dv/dt = 0
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
1.00.80.60.40.20.0time
v(t)
w(t)
0.20
0.15
0.10
0.05
0.00
-0.05
w
0.80.40.0-0.4v
dw/dt = 0
dv/dt = 0
0.8
0.4
0.0
-0.4
3.02.52.01.51.00.50.0time
v(t)
w(t)
(Go Back)Excitable Cells – p.15/??
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3
period during which another new layer cannot be formed, because any new material will
of preference be incorporated at the growing edge nearby.
3. Coupled map lattice model
Layer growth proceeds by the incorporation of individual growth units — it is essentially
a discrete process — so we employ a discrete model. Thus we construct a minimal
coupled map lattice model of the growth dynamics (Fig. 1). The surface is divided up
into cells, which may be thought of as the size of the growth units that compose the
crystal. In order to avoid anisotropic growth, we use a randomized grid, as introduced
by Markus and Hess [8], and, defining a neighborhood radius, R, the neighbors are all
elements within this radius
|rij ! rkl|<R,
where ij and kl are the cell coordinates on the bidimensional lattice and we use the
length of one cell as the length unit. The parameter R will influence the separation
between terraces on the growing crystal surface, and so physically we can associate it to
the surface diffusion of the depositing particles. Each cell has an associated height, Hij ,
initially zero and updated at the end of each time iteration. Hij is a continuous variable
into which a random component is introduced, as we show below, since there has to
be certain variability for defects to occur. Cells can be considered to be in an excitable
state or in a refractory state. The excitable state means that the cell may nucleate a new
island or add new material at a growth front. The condition for nucleation is that the
cell in question must be on a flat surface, meaning that the height difference with its
neighboring cells must be smaller than a certain margin, !HN :
!
neighbors
!H <!HN .
This parameter will define the frequency of nucleations and target patterns. In
crystallization it is related with the supersaturation of the precipitating material. The
condition for growth is that the cell must be at the edge of a growth front, meaning it has
at least one neighbor with a height difference larger than a certain threshold, !HG:
!H >!HG.
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4
This parameter is responsible for the appearance of screw dislocations. Physically there
can be several causes for this kind of defects, like the flexibility of the bonds of the lattice,
the heterogeneity of growth units (for example in protein- and virus-crystal growth, and
in liquid crystallization), and the presence of impurities. If the cell does not fulfill either
of these conditions, then it is considered to be in a refractory state. The duration of this
state can be altered by increasing or decreasing the radius of the neighborhood; a larger
radius implies more neighbors and hence a longer refractory period, which is reflected
in the separation between terraces in the spiral and target patterns. However, a larger
neighbor radius would also imply a higher probability of being at the edge of a step.
This is reflected in the front thickness, meaning that the number of cells that experience
growth in each iteration is greater, giving the appearance that the front is spreading
faster. But since our time units (iterations) are completely arbitrary, the kinetics of crystal
growth are beyond the focus of our model.
If the cell is in a position to nucleate, it must first pass a probability check with
probability PN , as nucleation is a stochastic process. PN is generally small (! 1). If the
check is positive and nucleation takes place, the height of the cell is increased by 1 plus
or minus a small random factor !:
Hij(t+ 1) =Hij(t) + 1± !.
On the other hand, when growth takes place, the height of the cell is increased by the
mean height difference with its higher neighbors:
Hij(t+ 1) =Hij(t) +!
k,l
Hkl(t)"Hij(t)
n± !,
where (k, l) are the coordinates for each higher neighbor and n is the total number
of higher neighbors, which depends on R. This growth algorithm is performed
simultaneously for all cells, and the process is iterated.
4. Results
When a nucleation event occurs it automatically inhibits further nucleation in its
neighborhood and all new material will be added at the edge of the newly formed
island. If this island can grow through several time steps without varying its height
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Target patterns are growth hillocks from the 2D nucleation of interlamellar membranes
Pteria avicula
monomolecular steps
from De Yoreo & Vekilov. 2003. Rev. Mineral. Geochem. 54:57-93
calcite
canavalin
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Spiral and labyrinthine patterns are from screw dislocations of interlamellar membranes
Pteria hirundoPteria avicula
synthetic liquid crystalgraphite canavalin
canavalin
http://www.mc2.chalmesse/pl/lc/engelska/
from Mc Pherson et al. 2004. J. Synchroton Rad. 11:21-23
from Mc Pherson et al. 2000. Annual Rev. Biophys. Biomol. Struct. 29:361-410
from Rakovan & Jaszczak. 2002. Amer. Mineralog. 87:17-24
screw dislocation
Pteria avicula
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5
Table 1. Model parameters
Description Influence Physical meaning
! Stochastic term Lattice defects Impurities
!HN Height margin to consider flatenough to nucleate
Nucleations and targetpatterns
Supersaturation
!HG Height threshold to consider thebase of a step
Screw dislocations andspiral patterns
Particle shape variability
R Radius of the cell neighborhood Terrace separation Surface diffusion
much, a second island may nucleate on top of it. If this process continues periodically
it will produce a target pattern (Fig. 2a). In order to produce this pattern we set a
broad nucleation margin !HN to favor nucleation and a low stochastic term ! to avoid
rough surfaces. In crystallization these parameters would respectively translate to a high
supersaturation of the precipitating material and a low shape variability of its growth
units. Although we can currently only describe the qualitative relation between the model
and the physical parameters, understanding this behavior is necessary in order to obtain
a quantitative relation.
Another common pattern in both excitable media and crystal growth is the spiral
(Fig. 2b). When two growth fronts with slightly different heights collide, a portion of
one may overlap the other, and continue its growth revolving around the dislocation
center. These patterns are produced with a small active neighbor threshold !HG, which
allows growth fronts to overlap instead of annihilating each other and a larger stochastic
term ! that introduces variability in the front height. Physically these parameters would
imply a crystalline lattice with high tendency to create dislocations of whatever type,
whether through the presence of impurities or due to the size and shape variability of the
growth units.
An important aspect of the behavior of excitable media is that excitable waves cannot
pass through each other, but rather destroy each other, as the refractory period does not
permit wave propagation to continue; nor do waves reflect from boundaries. In the well-
known excitable model of forest fires, for example, a second fire cannot pass until the
vegetation burnt by the first has grown back, and excitable cardiac cells cannot fire again
until they have recovered from their earlier firing. In Fig. 3a we see a sequence of two
growth fronts colliding and annihilating each other. In this case the height of both fronts
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model of liquid-crystal growth of the interlamellar membranes
SpiralsTarget patterns Double spirals
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(c)
(d)(f)
(a) (e)
(b)
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© Nature Publishing Group1966
Koji Wada, 1966
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Pinctada martensiiPteria avicula
grow
th fr
onts
Spiral, target and digitiform patterns in Pterioida (Bivalvia)
Pteria aviculaPteria avicula
These patterns have to be traced back to the interlamellar membranes; nacre tablets are a ‘decoration’
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calcite aragonite
(Falini et al. 1996. Science 271:67-69; Belcher et al. 1996. Nature 381:56-58)
calcit
ic po
lyanio
nic pr
oteins
aragonitic polyanionic proteins
both
Crystallization in the presence of proteins extracted from the shell
from Belcher et al. 1996. Nature 381:56-58
Genetic control on calcium carbonate polymorph by soluble shell proteins...
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Chromosome 3 of the human genome
...but as we know more genetics, the importance of
epigenetic factors in development becomes
clearer...
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...and we need to complement an
understanding of the genetics and proteomics...
...with an understanding of the physical processes of self-organization and self-
assembly involved
cholesteric phase
nematic phase
isotropic phase
E
100,000 E
Biological processes take place far from equilibrium and need to be viewed in terms of non-equilibrium
thermodynamics and nonlinear dynamics Friday, February 10, 2012
Richard Feynman:
“What I want to talk about is the problem of manipulating and controlling things on a small scale.
As soon as I mention this, people tell me about miniaturization, and how far it has progressed today. They tell me about electric motors that are the size of the nail on your small finger. And there is a device on the market, they tell me, by which you can write the Lord's Prayer on the head of a pin. But that's nothing; that's the most primitive, halting step in the direction I intend to discuss. It is a staggeringly small world that is below. In the year 2000, when they look back at this age, they will wonder why it was not until the year 1960 that anybody began seriously to move in this direction.”
There's Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics
Talk given at the annual meeting of the American Physical Society at the California Institute of Technology, 29th December 1959
There's plenty of room at the bottom
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Coworkers
Antonio Checa (Granada)Bruno Escribano (Bilbao)Marthe Rousseau (Vandoeuvre les Nancy)Ignacio Sainz-Diaz (Granada)Ana Vasiliu (Bucharest)Marc-Georg Willinger (Berlin)
PublicationsJ. H. E. Cartwright & A. Checa, “The dynamics of nacre self assembly”, J. Roy. Soc. Interface 4, 491–504 , 2007.A. G. Checa, J. H.E. Cartwright, & M. Willinger, “The key role of the surface membrane in why gastropod nacre grows in towers”, Proc. Natl Acad. Sci. USA 106 38–43, 2009.J. H. E. Cartwright, A. G. Checa, B. Escribano, & C. I. Sainz-Dıaz, “Spiral and target patterns in bivalve nacre manifest a natural excitable medium from layer growth of a biological liquid crystal ”, Proc. Natl Acad. Sci. USA, 106 10499–10504, 2009. A. G. Checa, J. H.E. Cartwright, & M. Willinger, “Mineral bridges in nacre”, J. Struct. Biol. 176, 330–339,, 2011.J. H. E. Cartwright, A. G. Checa, B. Escribano, & C. I. Sainz-Dıaz,“Crystal growth as an excitable medium”, Phil. Trans. Roy. Soc A 2012.
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