Spiral and target patterns in bivalve nacre manifest a natural excitable medium from layer growth of a biological liquid crystal

Liquid-Crystal Layer Growth Model of Nacre

Layer or tangential growth of a solid crystal, analyzed by Burton, Cabrera, and Frank (12), involves 2 processes: on one hand, the addition of material at the edges of 2-dimensional islands on the surface of the crystal together with the occasional nucleation of a fresh island atop an earlier one and, on the other, the continual accretion of material at the edges of a helical ramp on the surface of the crystal, termed a screw dislocation (13). [Tangential growth as a whole was examined by Burton, Cabrera, and Frank (12); Frank's contribution was the analysis of screw dislocations (14)]. These 2 processes produce characteristic target and spiral patterns seen at the molecular scale in solid crystals (15). These same patterns are visible in the conspicuous features of the surface of nacre at the mesoscale in Fig. 1. In that figure, the patterns are arrangements of aragonite tablets in mineralized nacre. However, as we see in Fig. 3A, in growing nacre, interlamellar membranes extend ahead of these mineralization fronts; the latter follow the geometry laid down by the former. This is evidenced by the forms of tablets that grow at the centers of the screw dislocations in the membranes, which mold themselves to the spiral, as shown in Fig. 1F. Thus, the spirals and target patterns are in fact laid down in the interlamellar membranes, and we must establish that growth of an interlamellar membrane as a liquid crystal layer by layer can give rise to this geometry of spirals and target patterns.

We formulate a minimal model containing the fundamental physics of liquid crystallization. Growth proceeds by the incorporation of individual growth units, which are the chitin crystallites that compose the liquid crystal. Because this is an essentially discrete process, we employ a discrete coupled map lattice, or lattice Boltzmann model, with a square lattice of discrete space and time but continuous state variables, that puts nucleation and growth on an equal footing. The surface is divided into a randomized grid of cells, as introduced by Markus and Hess (16), to avoid anisotropic growth. Each cell has a growth height H(i, j), initially zero, updated at the end of each iteration of the growth. The surface is scanned on each iteration, and in each cell, there may occur either nucleation, growth, or neither as a function of the relative height of the nearest neighbors. The neighbors to each cell are assigned by proximity, defining a circle of radius R around the cell.

The essential physics we wish to capture in our model is that a fresh growth unit is incorporated into the growing surface most probably at a site where it is more strongly bound by virtue of having most neighbors. Such sites are so-called kinks: concavities at the base of steps on the surface. Thus, there is no probability of a growth unit settling above a step, because it will slide down to occupy a more favorable site below; there is a rather low probability of a growth unit settling on a flat surface and a very high probability of it occupying a site at the base of a step. This physics we can model by setting a low probability of addition of material, which is termed nucleation, at any point over the surface that is not elevated above its neighbors, together with a certainty of the addition of material, which is termed growth, at the kinks of steps.

The condition for nucleation is that the cell should not stick out far above its neighbors, that the surface be sufficiently smooth. The cell in question should have height comparable with that of its neighbors with a difference not more than Δh_N. If so, there is a small probability P_N of nucleating a fresh growth surface. Nucleation is a fundamentally stochastic process, and this is reflected in the model. If nucleation takes place, and a new growth unit is incorporated, the growth surface increases in height by 1 plus or minus a small random factor α: The condition for growth is that the cell must be at the edge of a growth front; i.e., must have neighbors higher by at least Δh_G. If so, a growth unit is incorporated: The growth height is increased by the mean of the higher neighbors ±α, and a growth front extends: where (k, l) are the coordinates for each higher neighbor, and n is the total number of higher neighbors, which depends on R. This part of the model is deterministic except for the parameter α, which adds a certain amount of noise so that the growth surfaces are not completely flat and defect free.

In this way, as a growth front grows out, its height may vary, and if it encounters another front of different height, following the rules above, defects may be spontaneously incorporated into the growth. Two growth fronts that meet may join seamlessly with each other, terminating growth; they may completely overlap each other, forming an edge dislocation; or they may overlap along part of the edge and join along the rest, nucleating a spiral defect as growth continues. All of these behaviors are comparable in the model—Fig. 4—with nacre itself (Fig. 1).

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Fig. 4.

Our model of liquid-crystal layer growth reproduces the features seen in micrographs of nacre growth and shows—what cannot be seen in nacre samples—how they evolve in time. (A) Target patterns—see Fig. 1B—after (i) 87 and (ii) 93 iterations; parameters α = 0.08, Δh_N = 0.1, P_N = 10-3, Δh_G = 0.8, R = 10. (B) Colliding growth fronts and spiral—see Fig. 1C—after (i) 300 and (ii) 310 iterations; parameters α = 0.06, Δh_N = 0.06, P_N = 5 × 10⁻⁶, Δh_G = 0.6, R = 4. (C) Paired spirals—see Fig. 1 D and E—after (i) 230 and (ii) 250 iterations; parameters α = 0.08, Δh_N = 0.09, P_N = 10-4, Δh_G = 0.8, R = 5. The color scheme is shown alongside each subfigure; lighter is higher. These 3 cases can be found in supporting information (SI) Movie S1, Movie S2, and Movie S3.

In our model the set of parameters comprises:

• α Stochastic term, typically ±0.05–0.1, that varies the height of the growth cell on nucleation and growth, otherwise renormalized to be unity. This term is responsible for defects. Physically this growth-height variation is related to the fact that the growth units in the case of nacre are nonidentical; they are rods of differing lengths. An additional factor in the appearance of height variations leading to defects in all layer growth is the presence of impurities.

• Δh_N Threshold of elevation with respect to the neighbors below which a cell's neighborhood is considered sufficiently flat for nucleation of a fresh layer to occur. A larger value of Δh_N implies that more nucleation will take place, typically set similar in magnitude to α.

• P_N Probability of nucleation occurring in a given cell in which, according to Δh_N, it may occur; typical value 10⁻³ to 10⁻⁶. P_N in combination with Δh_N determine the frequency of nucleation. If P_N is zero, screw dislocations leading to spiral growth are the only growth mechanism; if P_N is large, the surface is rough.

• Δh_G Threshold of difference in height with the neighbors for there to be an active neighbor. There must be at least 1 active neighbor for a site to be a kink and so to have growth in a cell. Physically, this parameter in combination with α determines the frequency of screw dislocations. Typical values to produce spiral growth are 0.4–0.8. If Δh_G is large, spiral growth is suppressed and growth proceeds by layers. If Δh_G is small, growth fronts that encounter one another cross to produce edge dislocations.

• R Radius of the circle of neighbors, typically set between 3 and 10. The model begins with a randomized grid (16): A point is set at random within each cell at the beginning of the simulation. Subsequently the neighbors of the cell are sought by tracing a circle of radius R about this point. Owing to the initial random positioning of the center of the circle, this procedure produces a varying number of neighbors per cell. As with α, the randomization introduced with the grid here is related to the nonidentical nature of the chitin crystallites that are the growth units for the nacre. A larger radius R implies wider terraces between growth steps. This is physically related to the surface diffusion of the growth units. If we look at nacre from different species, we indeed find that some have steeper, narrower terraces, and others have shallower, wider ones.

In Fig. 4, we display the evolution in time of 3 different scenarios in our model, to be compared with the various phenomena in nacre of Fig. 1. In each case, the simulation began at time 0 with a flat surface of zero height. In Fig. 4A, we show the evolution of target patterns; we see the system after 87 and 93 iterations. Each terrace expands laterally, and, once the uppermost terrace has expanded sufficiently, a fresh terrace nucleates on it. This is similar to what is seen in Fig. 1B in nacre and corresponds to island growth, the first process of Burton–Cabrera–Frank (12). In Fig. 4B, we show, after 300 and 310 iterations, colliding growth fronts and an evolving spiral similar to those seen in nacre in Fig. 1C. As the spiral rotates and grows out, it meets other growth fronts coming from the top right. The fronts of the same height annihilate each other by joining seamlessly as they meet. Last, in Fig. 4C, we show, after 230 and 250 iterations, paired spirals similar to those in the nacre of Fig. 1 D and E. Fig. 4 B and C corresponds to spiral growth, which is the second process of Burton–Cabrera–Frank. Those in Fig. 4B are single screw dislocations, whereas the paired spirals in Fig. 4C are Frank–Read sources (26).

These time evolutions can be seen in nacre samples by looking down through the nacre layers, as the third dimension of space—into the material—is equivalent to time. However, although that procedure takes us back in time, it does not indicate which events are coeval, as the model does. From our model, we can see that Wada (1) was correct in noting the parallel between nacre growth and the Burton–Cabrera– Frank (12) dynamics [in fact, Wada (1) singled out the spiral growth mechanism that was Frank's contribution (14)] of layer growth, albeit, for a liquid crystal rather than the solid crystal Wada had envisaged.