Discrete free-boundary reaction-diffusion model of diatom pore occlusions

1CoMPLEX, University College London, Physics Building, Gower St, London, UK-WC1E 6BT, United Kingdom 2Department of Mathematics, University College London, Gower St, London, UK-WC1E 6BT, United Kingdom 3MIMS, The University of Manchester, School of Mathematics, Oxford Road, Manchester, UK-M13 9PL, United Kingdom 4Natural History Museum, Cromwell Road, London, UK-SW7 5BD, United Kingdom *Author for correspondence: l.willis@ucl.ac.uk


INTRODUCTION
Diatoms are unicellular algae, prolific in the world's waters, critical in food chains ascending to fish, and fundamental in the carbon cycle.The curious porous forms of their speciesspecific siliceous valves are a striking example of how a eukaryotic cell controls structure at a minute scale in a manner that defies synthesis in vitro.The variety of intricate patterns in valve morphogenesis is breathtaking.What physicochemical and mechanical processes can generate such variety while retaining high genotypic fidelity?In this article we briefly review current evidence on the finest structures of diatom valves, the sub-micron patterns observed in their pore occlusions.We then describe our first attempts to test a new theoretical model of pore occlusion formation.

The formation of pore occlusions is usually the last stage of valve morphogenesis
Upon cytokinesis of a diatom cell, each daughter cell initiates the construction of an organelle which houses a newly forming valve within a lipid bilayer membrane 7 nm thick called the silicalemma (Drum & Pankratz 1964, Reimann et al. 1966).Electron microscope (EM) studies have established the following common series of events in valve morphogenesis (e.g.Pickett-Heaps et al. 1990, Schmid 1994).
Pore delimitation: early formation of the larger pores -Early on the valve grows laterally, primarily from its rim, while the silicalemma expands laterally to accommodate this All rights reserved.© 2010 National Botanic Garden of Belgium and Royal Botanical Society of Belgium -ISSN 2032-3921 REGULAR PAPER Background -Diatoms are unicellular algae, prolific in nearly all aqueous environments on earth.They are encased between two siliceous valves that each feature a variety of intricately patterned speciesspecific siliceous structures.How diatoms use biological and physical processes to form these tiny detailed structures is largely unknown.This work is concerned with the smallest regular structures in diatom valves, the pore occlusions, and the processes involved in their formation.Theory and method -We developed a discrete, free-boundary, reaction-diffusion computer model to assess a new physically motivated hypothesis: pore occlusion patterns in the genus Achnanthes are simply expressions of silica diffusion and deposition within a pore covered by a membrane (silicalemma), whose deformation interacts with the growth of the pore boundary to control the silica influx.Preliminary results and discussion -Simulations generate some promising pore features such as bifurcating and curved protrusions that grow towards one another, as seen in diatom pore occlusions.But they tend to be irregular and, to date, taking into account of smoothing and regularizing effects only partially symmetrized formations.Potential future work on this point is outlined.
growth.Silica precursors, derived ultimately from soluble monosilicic acid Si(OH) 4 in the ocean (Del Amo & Brzezinski 1999), precipitate to form a delicate base-layer (a network of tessellated hexagons, or radial ribs (costae) with crossconnections, or parallel ribs (virgae) with cross-connections (vimines)), which gradually thickens and becomes more regular delimiting a regular arrangement of well-defined pores (areolae) 200-900 nm in diameter (e.g.Round et al. 1990).Soon after precipitation, the solid biogenic silica appears as a collection of loosely aggregated small nanospheres 20-50 nm diameter (Schmid 1976, Chiappino & Volcani 1977, Schmid 1979, Cox 1999: figs 5-12).Over time, the aggregations gradually condense and acquire a smooth appearance in the SEM 1 .The solid hydrated silica SiO 2 of the valve appears to be amorphous in most species even on the atomic scale (Vrieling et al. 2000(Vrieling et al. , 2003)), so crystal nucleation and directed crystal growth do not feature in morphogenesis.Although in Pinnularia viridis (Nitzsch) Ehrenb.at least, early silica deposition is directed by silica microfibrils parallel to one another and perpendicular to the valve's surface (Crawford et al. 2009).
In every diatom each pore occlusion shares a characteristic pattern that is conserved within species, yet this submicron pattern varies considerably between species; some occlusion types are restricted to particular higher taxa, e.g.hymenes are only found within the raphid diatoms.So areola type can be systematically informative, and pore occlusion 1 AFM studies have further revealed that in many species completed valves contain a variety of silica building blocks, from nanospheres 30-50 nm diameter, to smooth, texture-less, silica surfaces (Hildebrand et al. 2008).formation is somehow genetically regulated.What is the nature of this genetic regulation?As briefly discussed below, a convincing physical model of pore occlusion formation could provide some clue.
The functional role of pore occlusions is entirely unknown.In supplementary material we present a mathematical argument related to their functional role.It suggests that even if diffusion of nutrients is obstructed by solid silica components of the valve, provided that nutrients are frequently absorbed when they collide with the plasmalemma, the pores and the pore occlusions probably reduce the rate of nutrientuptake into the cell by a surprisingly small factor considerably less than 50%, even though pore occlusion openings or perforations occupy only a small fraction of the valve's surface area, usually about 5 to 20%.

Evidence pertaining to pore occlusion formation
The form of biogenic silica precursors in the silicalemma is unknown -Inside the cell but outside the silicalemma, silica concentrations range from 19 to 340 mM depending on species (for review, see Martin-Jézéquel et al. 2000).Soluble silica saturates at a much lower concentration of 2mM for pHs below 9 (Iler 1979), so the cell is presumably packaging the silica with some stabilizing organic agent, otherwise silica would rapidly polymerize to form large colloids (Iler 1979) that, according to Martin-Jézéquel et al. (2000), are disruptive to the cell membrane2 .Transportation of silica from within the cell to the silicalemma is thought to occur either by simple diffusion of organic ionophore-mediated mono-or polysilicic acid (Bhattacharyya & Volcani 1983), or actively by transport vesicles containing organic stabilizing agents (Schmid & Schulz 1979).But, how the cell maintains precondensed silica in a supersaturated state and in what form biogenic silica precursors arrive at and diffuse within the silicalemma remains a mystery.Silica precursors within the silicalemma could be monosilicic acid (≈ 0.3 nm diameter), polymerized silicic acid (0.5-5 nm diameter, Iler 1979), or larger organo-silica complexes.
EM images suggest the valve pattern is moulded by a finely orchestrated set of components in the cytoplasm -SEM and TEM observations of intracellular events during valve morphogenesis led Schmid (1994 and references therein) to postulate that the primary features of the valve, such as pores, are moulded by cell-orchestrated components in the cytoplasm.According to Schmid (1994) approximately 200-1000 nm diameter vesicles are sandwiched tightly between the silicalemma and the plasmalemma in a manner that interferes with the growth of the silicalemma: they border the rim of the advancing silicalemma so as to mould the pores.Pore completion is initiated by the detachment of vesicles from the plasmalemma.Schmid called these vesicles 'spacer vesicles' and had earlier postulated that they contain biogenic silica precursors for silica deposition (Schmid & Schulz 1979, Sullivan 1986).For Coscinodiscus wailesii Gran & Angst, Schmid reported that endoplasmic reticulum vesicles appear to mould the cribra.Monolayer lipid vesicles can be as small as 20 nm in diameter.
What is missing from this description is a physical explanation for the robust symmetric patterning of the valves.It seems likely that to generate this patterning efficiently, the cell has adopted and tuned some general physico-chemical principles.
An organic matrix may template silica deposition -The organelle of valve morphogenesis has not been isolated, and little is known of its contents.However, we do know that the silicalemma contains an aqueous mixture of silica precursors, free H+ ions (as the intralumenal pH is acidic, Vrieling et al. 1999), and a variety of organic components that have been discovered embedded within the siliceous valve.
The organic components include silaffins and silacidin peptides (Nakajima & Volcani 1969, Nakajima & Volcani 1970, Kröger et al. 1999, Poulsen & Kröger 2004, Wenz et al. 2008), and amphiphilic species-specific long-chain polyamines (Kröger et al. 2000, Sumper et al. 2005, Sumper & Brunner 2006, Sumper & Lehmann 2006).These interact with one another and the silica precursors and, it is postulated, organize themselves spontaneously into an energetically favourable two-phase template for silica deposition (of the pores at least; Sumper 2002, Lenoci & Camp 2008).One phase contains organic matter preventing silica deposition, the other phase is an aqueous template for silica deposition.Indeed, pioneering studies show that solutions of long-chain polyamines, silaffin peptides, and monosilicic acid precipitate in a disordered, yet approximately hexagonal pattern in vitro, reminiscent of the base-layer in some species.However, there is no direct evidence for templating by organic molecules at the tiny-scale of the pore occlusions, or that such a process could form the other, more intricate, structures of the valve.
The composition of the silicalemma may affect valve morphogenesis -Lipids and proteins often move fluidly on membranes, transiently self-organising into clusters, so it is feasible that deformations of the silicalemma cause the redistribution of its transport proteins in a manner that contributes to valve morphogenesis.Silicalemma deformations may be purely mechanical or due to the recruitment of new lipids.Grachev et al. (2008) postulated that the movement of aquaporins (membrane-bound proteins that mediate the transport of water; Kozono et al. 2002, de Groot & Grubmüller 2005) on the silicalemma gives rise to patterned precipitation of silica.Other candidates for membrane-bound proteins that contribute to valve morphogenesis are silica uptake transporters: a repertoire of monosilicic acid uptake transporters reside on the plasmalemma of the diatom cell, and are likely also to reside on the silicalemma since, upon completion of the valve, the silicalemma is presumed to fuse with the plasmalemma (for review, see Pickett-Heaps et al. 1990).Whether silicon transporters modulate the intralumenal silica concentration of the silicalemma relative to the cell is unknown.

THEORY AND METHOD
A host of biological factors may influence pore occlusion formation.The only factors that are almost certainly present as pores are completed are the solid silica base-layer of the valve, biogenic silica precursors, the aqueous medium that surrounds them, and the deformable silicalemma.Our starting point was to consider possible physico-chemical relationships between these factors.We describe a theoretical process that is being assessed by a computer model.

Hypothesis: pore occlusions are simply expressions of silica diffusion and deposition within a deformable silicalemma that couples silica deposition to silica influx
The central idea is that there is a natural coupling between the locations where biogenic silica precursors enter the forming pore occlusion, and the locations on the 'pore boundary' where inward growth takes place.The silicalemma can be envisaged as a deformable membrane sandwiching a solid periodic framework, which represents a base-layer at pore delimitation stage of valve morphogenesis.The mechanical and chemical configuration of the deformable membrane influences where biogenic silica precursors enter the forming pore.(This could happen in several ways; for example, if silicon transporters span the silicalemma, they may move to the flattest region of a bendy membrane where there is least curvature, or their conformations may alter in response to membrane tension.)Points on the pore boundary close to regions where the diffusing biogenic silica precursors entered the pore will tend to accumulate precipitating silica more rapidly; inward growth will occur more rapidly here.Inward growth of the pore boundary alters the framework which the membrane sandwiches, which, in turn, alters the configuration of the membrane, and this may alter the region where silica enters the pore (fig.2; silica entry is called 'silica influx'; the points where silica enters are called the 'sources' of the silica).In this way, silica deposition and silica influx are coupled via deformations of the silicalemma.

Discrete free-boundary reaction-diffusion model of pore occlusion formation
Mathematically this process could be modeled as a freeboundary reaction-diffusion continuum process (Appendix).This continuum model should give smooth inward growth of the pore boundary, yet as a model of the formation of pore occlusions it would be close to the limit of applicability: the tiny scale of pore occlusions components (approximately 5-40 nm) approaches that of the diffusing silica particles themselves (i.e. if the polymerized silica is 1-5 nm in diameter) and we cannot necessarily assume that the random jostling of particles due to inter-particle collisions 'averages out' at our tiny scale of observation as we would in a continuum model.Our model should consider the interactions between silica precursors and other molecules of the aqueous mixture within the pore.So we used a discrete computer model that features wandering silica precursors subject to random jostling.
In the computer model, jostling biogenic silica precursors are represented by random walkers on a square lattice of l × l sites.l is a dimensionless parameter related to the physical quantity L nm, the length of major axis of a delimited pore, and the unphysical quantity h nm, the resolution scale of observations, by l = L/h.Initially the pore boundary is a closed curve of lattice sites that matches the shape of the incipient delimited pores of the base-layer: the lattice approximation to an ellipse, hexagon, or rectangle.Random walkers, interior to the boundary, step to neighbouring non-boundary sites at rate D 0 /h 2 without any particular bias.This represents diffusion of silica precursors with diffusivity D 0 , where the physical quantity D 0 will depend on the size of the silica precursor and the viscosity of the aqueous mixture.Random walkers at sites adjacent to the boundary undergo precipitation reactions at rate d B k/h, after which they can no longer diffuse.Once the volume of precipitated random walkers is sufficient to fill the volume of a particular lattice site, the boundary grows inward by this 1 lattice site.This represents deposition of silica pre-cursors that precipitate at reaction rate k when they are within a binding distance d B of the boundary.Upon deposition, a silica precursor advances the boundary inward by its precipitated volume v B .Physical quantities d B and k will depend on the type of chemical reaction, the structures of the reacting molecules, and the ionic composition of the aqueous mixture.
As described above, in order to simulate the possible effect of a deformable silicalemma upon silica influx, at any moment the shape of the pore boundary prescribes the location of the silica sources where random walkers enter the lattice and start their walk: the silica sources are always located at points in the pore interior that are farthest away from the boundary 3 .Importantly, this causes the sources of silica to move away from boundary protrusions and, in the absence of randomness (or the analogous continuum model), would give boundary growth with symmetry-planes that match the symmetry-planes of the delimited pore.The silica influx S 0 , the number of new silica precursors that enter the pore per unit time, is a physical quantity that will depend on the mode of arrival of silica at the silicalemma.Physical assumptions -Physical assumptions of this model are: the 3D dynamics within the pore are independent of the coordinate perpendicular to the plane of the base-layer, and can be approximated in 2D; dynamics can be represented on a square, anisotropic lattice; ions within the pore screen electrostatic interactions between silica precursors, and screen silica-precipitation reactions at the boundary unless precursors are within a given binding distance; silica influx is constant in time; the precipitation reaction at the boundary is independent of boundary curvature; and the only effect of the internal aqueous mixture is to agitate thermally the silica precursors (e.g.effects of volume-occupying inert polymers or vesicles are ignored).

PRELIMINARY RESULTS AND DISCUSSION
In all simulations, the initial shape of the pore boundary was chosen to match the delimited pores of Achnanthes coarctata (fig.3): it is an ellipse with aspect ratio 4:3.Initially there are no silica precursors within the pore (this would be the 3 In the computer model this works as follows: at each lattice site in the pore interior, the smallest distance between that site and the boundary is calculated; then silica precursors enter at the sites where this distance is a maximum.If there are multiple sites, then silica precursors enter at each site with equal probability.case if a spacer vesicle had been blocking the entry of silica).As pore occlusion patterns do not depend on time-scale or length-scale per se, only relative lengths and times need to be considered: the average time 1/(D 0 /L 2 ) for a silica precursor to diffuse the initial diameter of the pore, compared with the average time 1/S 0 between entries of new silica precursors and the average time 1/k for a precipitation reaction; the initial diameter of the pore L, compared with the binding distance d B and the incremental distance moved by the boundary upon silica deposition v B 1/2 (square root in 2D).These ratios give dimensionless parameters governing pore occlusion patterns: S 0 L 2 /D 0 ('dimensionless silica influx'), d B k L/D 0 ('dimensionless precipitation reaction rate'), and v B / L 2 ('dimensionless volume of precipitated silica').
Parameters vary in the figures as indicated in table 1. Perturbations produce a few notable effects, outlined below.Reducing the unphysical parameter l ('number of lattice sites along each axis') improves computational efficiency.Ideally, perturbing l would not change statistical features of dynamics, but changes may originate from the anisotropy of the square lattice, or from the discretization of boundary movement.The effect of perturbing l is shown in fig.4; in general, for d B k L/D 0 » 1, reducing the ratio l/(L/√v B ) strengthens the effect of anisotropy (compare left column with right column in fig.4A & B).

Slow silica influx S 0 L 2 /D 0 ≈ 1 symmetrizes pore boundary growth
As the dimensionless silica influx decreases to approximately one, dynamics of boundary growth becomes maximally sensitive to the silica deposition-silica influx coupling: as S 0 L 2 / D 0 decreases, eventually the average time between the entry of a silica particle into the pore and its deposition is shorter than the average time between entries of different silica particles, and so every time silica enters the pore, boundary growth causes the sources of silica to move in response 4 .This symmetrizes pore boundary growth (compare fig. 4A & C); 4 Decreasing S 0 L 2 /D 0 far below one means that the pore is empty for long periods of time, and so further decreases in S 0 L 2 /D 0 hardly alter statistical features of boundary growth, except for the time-scale.
protrusions that form early on have less of a growth advantage, and the more uniform growth of several, more equally spaced protrusions is apparent, rather than growth of just 1-4 protrusions near the minor axis of the ellipse (fig.4C).Rapid silica influx (S 0 L 2 /D 0 » 1 -fig.4D) concentrates silica densely near silica sources, forming one connected silica blob.The entire blob is rapidly invaded by the boundary when a protrusion reaches it.Our choice for how silica sources move in response to boundary growth seems to prevent opposing protrusions from meeting and joining unless silica influx is rapid.

Rapid precipitation reactions d B k L/D 0 » 1 encourage growth of protrusions
When the dimensionless precipitation reaction rate is much greater than one, silica precursors aggregate to the boundary once they reach it: dynamics is 'diffusion-limited'.The resulting boundary structures, familiar from models of diffusion-limited-aggregation (Witten & Sander 1983), are too irregular to look like pore occlusions (fig.4A-D).Reducing the dimensionless reaction rate to approach one (fig.4E), silica is more uniformly distributed within the pore, so different sites on the pore boundary grow inward at more similar rates; this smoothes inward protruding bumps and fattens protruding fingers, but also reduces the effect of silica depositionsilica influx coupling (fig.4E & F).If measured values of d B , k, and D 0 in Achnanthes imply d B k L/D 0 is close to one (L ≈ 200 nm is known), then boundary dynamics is described as 'reaction-limited', and in this case our model cannot adequately explain pore occlusion formation.

The lower bound on the volume v B /L 2 of precipitated silica seems to necessitate an additional physical explanation for boundary smoothness
The value of l (number of lattice sites along each axis) sets the resolution scale for observations.If l = 1, nothing is seen at all.If l = L/√v B , dynamics are observed at the scale of individual precipitating silica particles, and boundary growth is bound to look somewhat jagged, as particles arrive at the boundary by diffusive random jostling.The only chance of seeing a regular, smooth boundary growth in this model is to observe dynamics at a resolution scale determined by l where 1 « l « L/√v B .As the initial diameter of the pore L ≈ 200 nm, the lower bound on the diameter of precipitated silica precursors of ≈ 0.2 nm (diameter of SiO 2 ) sets the maximum value of L/√v B ≈ 200/0.2= 1000 and therefore the maximum range for l.For L/√v B = 1000 and l = 300 « L/√v B , in the diffusion-limited case, pore boundary growth is still random and jagged (results not shown).This and further simulations indicate that, in the diffusion-limited case, there is no physically plausible resolution scale that could account for smooth pore formation, so the model seems to require an additional physical explanation for this smoothness.

Factors that smooth and regularize pore formation
We investigated two physically motivated effects that smooth and regularize pore formation (computer algorithms detailed in Appendix).
Effect ( 1) is 'volume-occupation' of diffusing silica, dependent on the volume v S of silica precursors in solution, it prevents silica precursors from entering or diffusing into lattice sites whose volume is already filled by silica.In the  results of fig. 4, silica precursors do not 'see one another' and the volume of silica in each lattice site is unlimited, an assumption that is invalid when silica precursors are concentrated (i.e. in the diffusion-limited case, when S 0 L 2 /D 0 » 1).
Effect (2) is 'resettling' of precipitated silica.This envisages recently deposited silica being knocked a short distance (the 'resettling length' R L ) into a more tightly packed configuration by molecules in the surrounding solution.In the computer model, silica precursors may step to neighbouring sites within a certain distance, which are also adjacent to the boundary, but less exposed to the solution in the pore lumen.
How effects (1) and ( 2) alter the model's dynamics is shown in fig. 5.These generate more promising dynamics (compare with Fig. 4): volume-occupation encourages symmetric growth (compare figs 4A & 5A); resettling over a short-range smoothes the boundary without eliminating protrusions, helping protrusions to grow with a uniform thickness (fig.5B).But to date, these smoothing and regularizing effects are not sufficient to account for the smooth, connected patterns seen in Achnanthes coarctata.We see that resettling-parameters strongly affect boundary dynamics (compare figs 4A & 5B), so, in future, smoothing algorithms should be realized by physical derivations which express smoothing-parameters in terms of potentially measurable physical quantities.

Relating physical quantities to pore occlusion type
Dimensionless combinations of physical quantities S 0 , D 0 , d B k, v B , and the size and shape of the delimited pore, govern pore occlusion formation according to our model, and may be influenced by genetic change.A convincing physical model that robustly mimicked pore occlusion formation in Achnanthes coarctata, and also reproduced the variety of pore occlusions observed in other species, could lead to predictions about how the cell physically regulates pore occlusion patterns.For example, to a very rough approximation our model implies that reducing the precipitation reaction rate k should cause a shift from pore occlusions resembling cribra, to pore occlusions resembling foriculae (compare fig. 1 with fig.4C  & 4E).Such predictions may provide clues to the nature of genetic regulation.

CONCLUSION
This is the first computational study that aims to identify the primary processes that govern pore occlusion formation.Computer studies have previously been used to investigate the stage of pore delimitation in valve morphogenesis (Gordon & Drum 1994, Parkinson et al. 1999, Bentley et al. 2005, Lenoci & Camp 2008).The most closely related study is by Parkinson et al. who used diffusion-limited aggregation (a well-known 2D model of randomly jostling molecules which wander from infinity to an outward growing dendritic structure nucleated by some central seed -Witten & Sander 1981) to model the arrival of microtubule-guided silica precursors at the margin of an expanding silicalemma.Here random walkers wandered from a line source or a set of point sources a fixed distance beyond the rim of the incipient valve.Our model inverts this process: random walkers on the interior of a nucleating surface cause inward growth.By locating sources at points that are farthest from the inward growing pore boundary, an extra stabilizing mechanism that encourages symmetric growth was automatically included.Further, we investigated a precipitation reaction rate that controls whether the model is diffusion-limited or reaction-limited.Our simulations generated some promising dynamics, but as yet we have been unable to account for the smooth and regular deposition of randomly jostling particles of silica at such tiny scales, or for the topology of pore occlusion perforations that in our model must result from fusions of finger-like protrusions of solid silica.
Although firm conclusions require a more thorough investigation of perturbations to our model (such as investigations of other smoothing algorithms, of 3D and non-lattice based models), these preliminary results suggest that the proposed process omits some important physical effect which enables pore occlusions to form in a smooth and connected manner: our hypothesis should be revised.The following interesting observations will be used to guide these revisions.Pore occlusion patterns are often all aligned, usually in the direction of the major axis of the diatom.What causes this?Could it be an effect of anisotropic tension in the silicalemma?The size of areola when pore occlusions begin to form is remarkably uniform in most species.Is this due to the orchestrated removal of 'spacer' vesicles that prevented silica deposition, or could it be due to the destabilization of a spatially homogeneous solution within the pore as the length of the pore decreases?Finally, solid silica components and perforations of pore occlusions often have similar wave-like thicknesses.Is there a physical explanation for this?Revisions could include models of multiple chemical species within a delimited pore, possibly featuring a number of identical zwitterionic polymers that coprecipitate with silica so as to further enhance the silica influx-silica deposition coupling as proposed in this paper.Or models of polymers or tiny spacer vesicles, either within the silicalemma or pressed-up against its surface, that prevent silica deposition in the volumes they occupy (the mere presence of macromolecules can encourage nanometermicron-scale organization through what are known as excluded volume forces).
The great advantage of exploring ideas with theoretical and computational models is that all forces influencing results are automatically controlled, which is difficult to achieve in the laboratory, and the effects of select forces can be investigated relatively quickly.This is also a disadvantage: proving biological statements by theoretical models alone is rarely possible, as only select forces are included in any idealized simulation.At extreme scales, physical reasoning can be used to say 'only forces X, Y, and Z are important', and we can be confident that our model is an accurate model of the real world.But at cellular scales, it is evolutionarily inevitable that for most phenomena many different forces may be involved.Theoretical and computer models of cellular processes are then not much use on their own, but are primary tools for comparing the consequences of hypotheses with data from the laboratory.This is what we aim to do ultimately with our model of pore occlusion formation.

Figure 1 -
Figure 1 -Illustrations of types of pore occlusions.The perforations in cribra, rotae, and foriculae are approximately 40 nm across, while hymenes are distinguished by their tiny perforations 5-10 nm.Rotae have one or more bars on their axes.Foriculae have a single, simply connected perforation.

Figure 2 -
Figure2-Deposition of silica precursors alters the shape and size of the inward-growing delimited pores in the base-layer (the 'pore boundary', coloured black in the diagram) and this, in turn, alters where silica precursors enter the pore interior (the 'sources of silica', coloured blue in the diagram) so that different regions of the pore boundary thicken at different rates.The initial shape of the delimited pores may be a hexagon, or an ellipse, or a rectangle.

Figure 3 -
Figure 3 -SEM micrographs of partially formed pore occlusions of Achnanthes coarctata showing a possible time sequence 1-4 of their formation.

Figure 4 -
Figure 4 -Simulations of pore occlusions.Random walkers, coloured blue, represent biogenic silica precursors.The pore boundary is coloured white.Each row is a time series 1-4 for the parameters of table 1.In the left-hand column the scale l = 200; in the right-hand column l = 100.A: the simulation is too random and jagged compared to a pore occlusion.B: small volume of precipitated silica precursors -the smaller the ratio l/(L/√v B ), the stronger the effect of anisotropy.C: slow silica influx symmetrizes boundary growth.D: fast silica influxboundary expands as one connected blob.E: slow precipitation reaction rate smoothes boundary and eliminates protrusions.F: intermediate precipitation reaction rate.

Figure 5 -
Figure 5 -Simulations showing the effects of volume-occupation and resettling.A: volume-occupation symmetrizes boundary growth.B-D: resettling helps protrusions to grow with a uniform thickness.Parameters are in table 1.