From excitability to pearls: nacre as a complex system€¦ · From excitability to pearls: nacre as a complex system The story begins with the discovery of microscopic spiral and

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


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

* * *

*



© Nature Publishing Group1966

Koji Wada, 1966


Nacreous pearls


Pinctada martensiiPteria avicula

grow

th fr

onts

Spiral, target and digitiform patterns in Pterioida (Bivalvia)

Pteria aviculaPteria avicula


Where do the patterns come from?

What do the patterns tell us about the formation mechanisms of

nacre?


+ =Text

Nacre for physicists and mathematicians


Shells are multilayer organic-mineral biocomposites:two or three layers with different microstructures per shell

ScaphopodBivalve

Crossed lamellar

Nacre


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


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


… 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


nacreprismatic layer

periostracum extrapallial spacemantle cells with microvellosities

interlamellar membraneintertabular membranearagonite tablet

mantle 100 µm

10 µm

20 µm

Hierarchical structure of nacre


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



interlamellar membranes

mantle

Clanculus jussieui

Anodonta cygnea


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


Ultrastructure of organic membranes

Pinctada radiata


Levi-Kalisman et al. 2001. J. Struct. Biol. 135:8-17

Gibbula umbilicalisAnodonta cygnea

silk-fibroin

β-chitin fibres


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


The extrapallial space; secretion of interlamellar membranes in Pinctada radiata



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


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


As we shall see, molluscs make their brick walls in a rather different way than

we do...


self-assembly of rod-shaped elements:

liquid crystals


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


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.


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)


Chitin arrangement in nacre is a partially-ordered liquid crystal

Gibbula pennanti

logjam analogy

cholesteric phaseisotropic phase

days/months


Interlamellar membranes are felt-like fibrous composites

Gibbula pennanti

Anodonta cygnea

protein

β-chitin


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


Hierarchical construction of nacre


Excitable Media


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


Anexample

of an excitable element...


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/??


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/??


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.


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




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


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


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


model of liquid-crystal growth of the interlamellar membranes

SpiralsTarget patterns Double spirals


(c)

(d)(f)

(a) (e)

(b)


© Nature Publishing Group1966

Koji Wada, 1966


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’


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...


Chromosome 3 of the human genome

...but as we know more genetics, the importance of

epigenetic factors in development becomes

clearer...


...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


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.


Documents

From excitability to pearls: nacre as a complex system€¦ · From excitability to pearls: nacre as a complex system The story begins with the discovery of microscopic spiral and