Made to Model: Mathematical Representation of Sea Sponges 

Camille Heaney, Chris Hu, Vadim Zamaruyev

Abstract

Through the observation of Porifera behaviour and the application of mathematical principles, sea sponge growth and subcellular biosilicification have been successfully modeled. Due to the prehistoric development of Porifera these processes are biologically refined, making them opportune nodes for research. The models are based on by velocity deposition, vector determination, diffusion limitations, curve normalization, and nutrient gradients. As a bisection of these mathematical processes, the design and the subsequent accuracy of these models deepen the knowledge base not just on sea sponges, but all benthic life. The benefits of these models can range from naturalist to commercial, which cements their value within the scientific community. 

Introduction

Math is the human attempt at understanding the world. Throughout the ages, the more complex human societies became, the more nuances developed in math. Counting developed for organization, equations developed for trade and transaction, and calculus developed to understand physics (FUTURISM, 2017).  

The newest application of mathematics is modeling. However, it is not that new. The first example of modeling was developed in the late 19^th century alongside algebraic and differential geometry to represent Steiner’s Roman Surface. The applications of models continued to advance through the 20^th century where attempts to represent the physical and biological world became more prevalent (Schank & Twardy, 2009). Nowadays models are built for everything from bridge collapses to neural networks. These have applications like cost reduction, safety, and even new discoveries. 

Biological models pose a special set of challenges, as biology does not often follow the most intuitive pathway and has developed without the knowledge of mathematics. As such, biological models require many mathematic components, like differential equations, probability calculations, computational geometry (geometric algorithms), and linear/non-linear algebra (spatial transformations). They also require an unexpected level of creativity. Most models are limited by the elements which rely on creativity, due to the struggle to capture nature’s “randomness”.  

Porifera are inherently asymmetrical and develop their spicules and ostia in direct response to their environmental pressures (El-Bawab, 2020), this provides many challenges in modeling. Sea sponge models can shed light on various underwater systems and processes, such as the movement of water, the arrangement of ostia, the growth of spicules, the contribution to benthic-pelagic coupling, and the aggregation of bio-silica. These could all be applied, allowing for better species identification, understanding of force withstanding properties of laminated spicules, etc. These models bring additional depth to the sea sponge way of life, especially because deep-sea environments are incredibly difficult to evaluate regularly or replicate in the lab. 

Sea sponge skeletons are made of silicon and bound by a collagen called spongin. Silicon is relatively scarce in the modern oceans as compared to Precambrian conditions. To grow and maintain their structure, Porifera take up dissolved silicon through their filtration system, which aggregates to form biogenic silica. This material is connected in specific directions depending on the external forces experienced by the sponge. Modeling both sponge growth and subcellular biosilicification allow a full picture of how this process works. This paper aims to review these growth and spicule-related models and discuss their applications. 

Modelling Growth

Computational modelling is the use of models, simplified representations of real systems and processes, to make predictions about these systems and processes. It is particularly useful in scientific studies. Comparing the output of a model to real data helps to test if our assumptions about the real world are true. Modelling several outputs for different inputs facilitates the choice of experimental conditions that are more likely to lead to new insights. Models give a glimpse of phenomena that are difficult to reproduce in the laboratory. They simulate the collision of black holes in distant space and processes happening in the depths of the ocean (Jankar et al., 2022). In the ocean, long observational experiments become prohibitively expensive. Bringing sea creatures to the surface requires special equipment and maintaining them alive requires special care. Thus, modelling is a convenient avenue for their study. The growth of sponges, a very slow process (the Demosponge Haliclona oculata grows by 1.0-1.5 cm per 10 weeks) has been extensively studied with models (Kaandorp & Kübler, 2001). 

Natural Growth of Sponges

The first step in the modelling process is to review the current knowledge on the subject. Knowing the theoretical underpinning mechanisms of the system and its empirical response to different conditions will simplify the identification of rules that should be incorporated into the model. In the case of sponges, both the macroscopic morphology and the microscopic process of growth need to be explored (Kaandorp & Kübler, 2001). 

The demosponge Raspailia inaequalis has a characteristic branching morphology (Fig. 1) shared by several sponges (de Voogd, 2024). A branching morphology resembles the shape of a tree or of an electric spark. This pattern results from a linear structure repeatedly splitting into identical structures and forming a bouquet of branches. The rules deduced from observational studies of growth of R. inaequalis can serve as an empirical basis for models. R. inaequalis has a directional growth away from the substrate. Only the tips of the branches grow, and the growth rate increases with the distance away from the substrate. The tips are capable to divide into branches. Most sponges have tips that divide into two symmetrical branches, but some have tips that divide into three. Closely growing branches continue to grow. However, they never merge because they fan out. Environmental conditions affect the growth process as well. The branches grow in the plane perpendicular to the flow of the current, collisions with obstacles stop their growth, and with age, the overall morphology changes due to damage and abrasion. An ideal model should identify a set of rules that would result in a similar morphological development. 

Fig. 1. Raspalia inaequalis growing in the Goat Island Marine Reserve (New Zealand). The specimens are the branching orange-red sponges. (Kaandorp & Kübler, 2001). 

The microscopic process of growth of demosponge Haliclona oculata (Fig. 2) inspired several models. H. oculata has a radiate accretive growth. It grows by the deposition of layers of spicules at its tips. The layers are deposited longitudinally and cemented by other spicules laterally. The spicules within the layers form a tessellation of pentagons and hexagons (Fig. 3). Moreover, different morphologies develop under different conditions. In still waters, the sponge has a branching form, while in the presence of currents, its form becomes flattened (Fig. 4). According to these observations, the mechanism of growth affects morphology. Therefore, it should be considered during the design of the model (Kaandorp & Kübler, 2001). 

The growth of R. inaequalis and H. oculata, while simple, displays a high degree of complexity when environmental influences are considered. Some trade-offs are necessary during the design to focus on the most important observations. A good starting point is to identify the environmental variables and the specific characteristics of the sponge that cause variation in growth forms. The relation between these parameters may be unknown, but guessing a rule is like making a hypothesis. After the simulation is completed, the output can be compared to observations to verify if the rule is true. If it is true, new rules can be added. As new rules are proved true, the model develops. Thus, a model develops from rules, just as a theory develops from hypotheses. 

Fig. 2. Picture of Haliclona oculata (Mayhew, 2006). 

Fig. 3. Diagram of the growth of the sponge Haliclona oculata. Lines in the cross-section (right) represent the growth axes of the spicules. The finer-scale structure of the layers (left) is a tessellation of pentagons and hexagons. In the case of H. oculata, the structure forms a three-dimensional lattice. (Kaandorp & Kübler, 2001). 

Fig. 4 Growth forms of Haliclona oculata. A) A specimen from a still water lake has a branching form. B) A specimen from a lake with currents has a flattened sheet-like form. C) A specimen from the same lake as B) displays an intermediate form. [Adapted from Kaandorp & Kübler, 2001]. 

Focusing on Parameters

There are two types of parameters to consider when modelling the growth of sponges: environmental parameters and biological parameters. However, among all these factors, food availability is the most important. Without food a sponge cannot grow. Hence, most parameters are closely linked to food availability. 

Important environmental factors are the distribution of nutrients and the type of flow around the organism. The distribution of nutrients is expressed as their concentration in the environment and flow is the velocity of the current. These factors are combined in Peclet’s number, a dimensionless quantity that reflects whether current () or diffusion () contributes more to the distribution of nutrients (Eqn. 1). Low Peclet’s values describe environments where nutrients diffuse passively, and high Peclet’s values describe environments where nutrients are transported by currents. However, the number does not tell the direction of the current (Kaandorp & Kübler, 2001). 

(1)  

Important biological factors are the structure of the aquiferous system and the secretion of growth inhibitors. (Read more about allelopathy and inhibition of growth in the chemistry paper). The aquiferous system is a network of pores (ostia), inner chambers, and openings (oscula) where food is filtered. Water enters the ostia, travels through the chambers where it is filtered, and is expelled through the osculum. An efficient filtering system has many ostia and few oscula. The efficiency determines the amount of food a sponge can filter. In a model, the characteristics of the aquiferous system are often summarized by a constant or threshold. The distribution of the inhibitor can be recorded as a concentration, just as the distribution of nutrients (Kaandorp & Kübler, 2001). 

Models provide the advantage of quantifying biological systems. Nevertheless, important qualitative features and processes are lost during the description. Take the structure of the aquiferous system, for example. Models of sponge growth translate biological processes into computational and mathematical rules that are equivalent – but do not correspond to – the actual features and processes within the organism (Kaandorp, 1991). 

Designing a Computer Model

Computational modelling is very diverse. Most sponge growth models belong to a type called cellular automata. (Famous examples of cellular automata are the self-replicating automaton developed by John von Neumann and The Game of Life developed by John Horton Conway.) A cellular automaton is a two- or three-dimensional lattice of cells that take different states. The state of each cell is determined by the state of the surrounding cells during the previous step. This relation is reflected by a rule. For example, if you have four or less active neighbours, you remain active. If you have more than four active neighbours, you become unactive. A cellular automaton can have more than two states. However, to model the growth of a sponge, two states are sufficient. A cell can either be filled by the sponge or remain empty. The difficult part is determining the rules that govern the evolution of these states based on environmental and biological factors (Kaandorp, 1991). 

To begin exploring models, the organisation of the underlying code needs to be understood, especially where certain rules and variables are more likely to be found. Most codes have two major parts: processing steps and post-processing steps. 

Processing steps are concerned with unrestricted growth. The simulation begins with a seed – an initial arrangement of cell states within the automaton, or a structure that will subsequently change (Kaandorp, 1991). During these steps, food availability is of outmost importance. The different parts of the seed will grow according to the local availability of nutrients, which is affected by the flow of currents and the structure of the seed. This is a good place for a rule relating these variables (Kaandorp & Kübler, 2001).  

Post-processing steps ascertain that the growth of the sponge is realistic. No branches should intersect. There may be a maximal distance that prevents branches from coming closer (Kaandorp, 1991). Such restrictions depend on the presence and diffusion of growth inhibitors (Kaandorp & Kübler, 2001). 

After each round of growth, the distributions of nutrients and inhibitor need to be recalculated. This relies on another cellular automaton simulation. Therefore, there is an interplay between two automata: one that models growth and another that model diffusion (Kaandorp & Kübler, 2001). 

A Model of Sponge Growth

As sessile filter-feeding organisms, sponges grow to optimize their interactions with their environment. While their growth may seem random, they grow in a specific way that can be modeled. Their asymmetric morphology depends on the environment, nutrient availability, and the flow of water during tides. Each sponge within a species grows into a unique shape (Kaandorp & Kübler, 2001). 

Accretive growth with a deposition velocity and contact with the environment 

The first model to discuss models sponge growth through deposition velocity and the environmental contact. As discussed earlier, the sponge Haliclona Oculata forms a branched plane perpendicular to the governing flow direction. The ends of the sponge are flattened which leads to anisotropic deposition of material. Anisotropic means that the deposition of the materials on the surface of the sponge is not uniform in all directions. This introduces the anisotropic growth model. The growth model is the velocity at which particles will be deposited onto the surface of the sponge for growth (Kaandorp & Kübler, 2001). 

(2a)  

(2b)  

(2c)  

The growth velocity depends on two angles and . To break this down, angle is the angle between the growth direction of the sponge (growth axis) and the direction the surface of the sponge is facing. Angle depends on the flow direction in a specific three-dimensional orientation. when assuming the flow is parallel with the yz-plane meaning there is no flow in the direction of the x-axis. is the angle that describes the orientation of the sponge’s surface according to the flow direction. When the surface normal vector (vector pointing directly out of the sponge surface) is projected onto the xz-plane, is the angle between the projection and the yx-plane (Fig. 5). when = 0 the sponge will grow the fastest because the surface of the sponge is directly aligned with the flow and growth direction. As increases the growth will slow. The maximum of is π/2; once the maximum is achieved the widening of the growth is at its minimum. The model used the variables w and η to control how the shape will widen or narrow. The constant w controls the width of the sponge as it grows. The parameter η controls the shape to prevent sharp edges. By varying η there can be smoother transitions to new branches. Each vertex has a specific direction of growth. This can depend on the angle to create a smooth continuous shape. The Model predicts the natural growth of the sponge. The growth distributions represent how sponges grow in directions to optimize their exposure to the surfaces and flow of water (Kaandorp & Kübler, 2001). 

Fig. 5. The vector determination of angle ϐ (Kaandorp & Kübler, 2001). 

To estimate the contact with the environment, the anisotropic growth function can be combined with the additional component (low_norm_curve, av_norm_curv). The function uses the radius of convergence of each point on the surface. The radius of curvature (rad_curv) measures the amount a surface is curved at certain points. At the convex points, the radius of curvature is positive and at concave points, the rad_curv is set to zero. The radius of curvature estimates the amount that the surface is exposed. The greater the rad_curv the more it is exposed to the environment. The radius of curvature is important for the uptake of nutrients. The amount of nutrients or particles that a sponge has directly depends on the balance between the volume and the surface area. The size of the surface area describes the amount of area a sponge has that can absorb nutrients and the volume is the space that needs to be provided with the nutrients. Therefore, the ratio between the surface area and the volume shows how effectively a sponge can absorb nutrients (Kaandorp, 1995; Kaandorp & Kübler, 2001).  

In the model, the curvature is measured in six to eight directions allowing for a more accurate assessment of the sponge shape. All the radii of curvature are normalized with the equation 3. The lowest value and average values of the normalized radii of curvature are combined to form

(3)  

is a measurement of the amount of the sponge that has contact with the environment at each point (Kaandorp & Kübler, 2001). 

(3)  

(4)  

Finally, the H₂(𝑙𝑜𝑤_𝑛𝑜𝑟𝑚_𝑐𝑢𝑟𝑣, 𝑎𝑣_𝑛𝑜𝑟𝑚_𝑐𝑢𝑟𝑣)low_norm_curv, av_norm_curv)is combined with s, the size of the edges of the triangles on the sphere that starts the growth, and the anisotropic growth model. The result is a growth model simulation effect of the contact with the environment and deposition velocity. 

(5)  

With the growth function formulated the growth can be simulated. The simulation starts with a sphere of triangles put together to form a pattern of hexagons and pentagons (Fig. 6a). The growth function (Eq. 5) is the guide for how the layers grow on top of the sphere (Kaandorp & Kübler, 2001).  

In the first stage, the shape is flattened as new layers grow outward (Fig. 6b). In the next stage (Fig. 6c and d) some areas of the growth become flatter than the rest creating a minimum of curvature (Kaandorp & Kübler, 2001). This causes the growth to separate into two branches. As the branches grow the triangles will form to keep the surface tessellated. New layers of triangles were added on top of the other ones. The tangential elements are the edges of the triangles organized in layers. The longitudinal are the elements between the old and new growth (Fig. 7) (Kaandorp, 1995). As the triangles get too large, they will separate into smaller triangles. Triangles that are too small are removed.  Additionally, the growth model is made to prevent the branches from intersecting so if a part is too close to another the growth will stop (Kaandorp & Kübler, 2001). 

Fig. 6. The four stages of the geometrical growth simulation of a sponge starting with a sphere (Kaandorp & Kübler, 2001). 

Fig. 7. The growth of new triangles on top of the previous triangles. The original triangles are shaded in Gray (Kaandorp, 1995). 

This model shows how the sponges optimize the amount of nutrients they can absorb for the surface area to volume ratio. The growth considers the flow of the current, environment, and radius of curvature as the sponge grows (Kaandorp & Kübler, 2001).  

Nutrient and hydrodynamic accretive growth model 

This next model considers nutrient availability and the influence of hydrodynamics. The model assumes that the sponge’s energy source is only suspension feeding and that the food particles are dispersed by diffusion and hydrodynamics. Additionally, the model assumes that there is negligible transport of nutrients through the tissue and the local velocity is related to the amount of local food absorbed. To simulate tidal movement the simulation uses bidirectional flow (Kaandorp & Kübler, 2001). 

Two growth models can be considered for nutrient-driven accretive growth. The first type only considers local availability of nutrients included. K(c) is the local nutrient gradient along the sponge’s surface mean normal vector. 

(6)  

Equation 6 is the growth model of a sponge where s is the triangle side length the k(c) is the nutrient gradient. 

The second growth function considered the contact with the environment using the previously described equations 3 and 4. 

(7)  

The simulation adjusts the diffusion coefficient but keeps the velocity constant. These are used to calculate Peclet’s number which is the distribution of nutrients. Peclet (Pe) number as described earlier is the distribution of nutrients depending on the flow of water and the diffusion. Varying Peclet’s number affects the growth of the sponge. In addition, the simulation keeps the Reynolds number low and constant to be in a “creeping flow” regime (Kaandorp & Kübler, 2001). 

The simulated growth begins with a spherical object on a substrate plane. The growth is modeled on a lattice grid. The grid pattern for the growth depends on the nutrient distribution and flow. The flow direction is parallel to the xz-plane (Fig. 8). The flow moves from left to right then alternates and moves from right to left. Tracer particles are released to show the nutrients. The particles are absorbed by certain areas on the sponge’s surface, and these represent where the sponge would have ostia. The sponge model grows based on the growth function and then after each iteration, the sponge growth is updated to form a shape (Fig. 8) (Kaandorp & Kübler, 2001). 

Fig. 8. The simulated growth of a sponge in a lattice grid (Kaandorp & Kübler, 2001). 

When the Peclet number is changed the morphology of the sponge changes. Figure d shows the xz-plane cross-section of two simulations of sponges with different Peclet numbers showing the concentration of nutrients. In Figure 9, black represents a high concentration of nutrients, and white means depleted nutrients. On the left, the Pe is 0 and, on the right, the Pe is 3. The morphology of the sponge with a lower Peclet value is much thinner as opposed to the sponge with a higher Peclet number because of the lower nutrient circulation (Kaandorp & Kübler, 2001). 

Fig. 9. The cross-section of the sponge simulation in different concentrations and flows of nutrients. Left is Pe=0 and right is Pe=3 (Kaandorp & Kübler, 2001). 

Fig. 10. The variation of Pe number from lower nutrient diffusion to higher nutrient diffusion shows the different morphologies of the same sponge (Kaandorp & Kübler, 2001). 

The varying Pe number and the contact with the environment affect the growth of the sponge. Figure 10 shows how as the pe increases there is a higher nutrient circulation. This leads to more nutrients at the tips upstream of the current and depletion of nutrients downstream. The growth forms in the direction of the current and there is a smoother form. (Fig. 10f-i). When the pe is low the nutrient circulation is low. This leads to sponges with higher radial symmetry, branching, and fractality (Fig. 10 a-d). When looking at Figure 4 the models generally match the natural morphology of Haliclona oculate (Kaandorp, 1991; Kaandorp & Kübler, 2001). 

Conclusion on Modelling 

The chaotic and intricate morphology of Sponges appears to be incompressible. Yet modeling can replicate the natural growth of asymmetric creatures. The rules can be created upon close examination of the environment in which sponges grow in. The rules form a simulation that produces a sponge that resembles the natural growth of sponges proving the accuracy of simulations and reflecting our understanding of the growth of sponges. As models become better, they could be used to assist species identification given that the morphology of sessile marine organisms depends on the environment (Kaandorp, 1991; Kaandorp & Kübler, 2001). 

Fractal Modelling of Biosilicification 

Sponges have mineral-based skeletal lattices in which calcium carbonate or silica is grouped into spicules, which are long, rod-like structures. Demosponges, the largest class in the Porifera phylum, have skeletons composed of a multitude of spicule forms, each comprising of a variety of specific types (fig. 11). The formation of these shapes spans a variety of complexities, but this discussion will prioritize the modelling of the monaxonal style, as this simple shape, a straight spicule with a rounded end and a pointed end, proves to be an adequate starting point to model structures with increasing complexity (Łukowiak et al., 2022). 

Fig. 11. A variety of Porifera spicule shapes, grouped and distinguished by their forms: as suggested by their names, monoaxon spicules form around a single straight or slightly curved axis, whilst trianxons and tetraxons have three and four radiating axes, respectively. The monoaxon style is shown on the far left as a long needle-like structure with a curved end diametrical from a sharp one.  (Zoology Talks) 

Spicules of Suberites domuncula  

Suberites domuncula is a demosponge with fascinating characteristic behaviour, notably attaching itself to the shells of gastropods and hermit crabs (fig. 12). However, what is of interest is their composition, as their spicule skeleton is comprised mostly of monoaxon styles (Müller et al., 2006). 

A cross section-analysis reveals additional structural complexities, such as a central canal filled with organic matter, which hints at the presence of a scaffold in the formation process of the spicule (fig. 13). Indeed, biosilica growth is a hierarchical process in which silica or silica aggregates are deposited onto an organic template. Spicule growth starts inside silicasome vesicles within sclerocyte cells, where silintaphin protein interacts with the enzyme silicatein (fig. 14) to initiate the bio-condensation of the axial filament that serves as the organic template for biosilica deposition (Javaheri et al., 2013).  

Fig. 12.  S. domuncula attached to the mollusk shell inhabited by a hermit crab (Le Grancher, 2021). 

Fig. 13. Cross-section of a broken spicule. The central canal, which contained organic matter, has been emptied. Its presence confirms the formation pattern of the spicule, where silica particles aggregate onto a protein template (Javaheri et al., 2013). 

Fig. 14. Assembly of iron oxide (γ-Fe₂O₃) particles through silicatein and silintaphin interaction. Images captured by scanning electron micrograph. Bar 20 nm (Wiens et al., 2009). Notice the amorphous protrusions of particle aggregates. Similar behaviour occurs with other mineral particles, such as the silica that forms sponge spicules. 

Mathematical Model

To accurately represent and predict biosilicification, a spatiotemporal model is required, as the structure of the mineral is not only dependant on time, but also on the physical arrangement of particles and their interactions in three-dimensional space. Ordinary differential equations, the process habitually privileged in systems biology modelling, is insufficient as it neglects critical spatial information in particle aggregation. Therefore, the biosilicification model needs to combine the partial differential equations required to solve the diffusion equation by modelling random walks, with a particle-based model to represent the morphology of the structure (Javaheri et al., 2013). The diffusion equation solved by the PDEs is as such: 

(8)  

Where 𝜑(𝑟→, 𝑡)𝜑(r→, t) is the particle density at location and time 𝑡, 𝐷 is the diffusion coefficient, and is the vector differential operator. However, if the diffusion coefficient 𝐷 does not depend on the particle density, the equation becomes linear, simplifying the equation to: 

(9)  

Diffusion is the net movement of particles from an area of high concentration to an area of lower concentration. This net flow results from a random motion of these particles, and by discretizing space and time, eq. 9 simulates random walks, capturing particle trajectories within the sponge’s cellular environment, in what can be described to be Brownian motion.

Diffusion-Limited Aggregation Model  

The Diffusion-Limited Aggregation Model, or DLA, is a computer model used to simulate aggregation based on random walks of particles, especially in systems where diffusion is the primary source of motion. One of DLA’s key benefits in the modelling of spicule growth is its ability to produce systems with fractal properties (Meakin, 1995). Indeed, experimental methods such as x-ray scattering have determined the fractal nature of biosilica deposition around the silicatein core. X-ray scattering measures the distribution of matter at small scales by analyzing how X-rays are deflected by the sample. The resulting scattering patterns follow a power-law relationship, allowing the calculation of the fractal dimension, which quantifies the irregularity of the silica surface. For biosilica, this method has revealed a consistent fractal dimension, confirming its self-similar morphology across different length scales. As such, the first steps of biosilica formation can be described using a DLA framework. This approach provides insight into how diffusion and random particle movement shape silica morphology before post-synthesis processes, like hardening and electrostatic interactions, refine the final structure (Javaheri et al., 2013). 

Simulation

To replicate the initial stages of spicule formation, the simulation model leverages the principles of diffusion and particle aggregation within vesicles of sclerocytes. Small silica particles, found within these silicasome vesicles, are the building blocks for these spicules. The central protein filament serves as the nucleation site, which guides the particle aggregation (Javaheri et al., 2013). This simulation by Javaheri and colleagues uses a Cartesian coordinate mesh and assumes a cubic 3D space within the vesicle. Reflective boundary conditions ensure that particles remain within the vesicle by mirroring them back into the simulation box upon contact with the boundaries. 

In this space, silica particles perform random walks, slightly influenced by the electrostatic forces of the acidic proteins, until they attach onto the growing aggregate. As the aggregate grows, uneven protrusions tend to appear, as it becomes more difficult for particles following random walks to lodge themselves in crevices near the core before attaching to a protruding structure, as particles will stick to the aggregate as soon as they reach it with a probability of 1 (Javaheri et al., 2013).  

Three different scenarios spicule growth have been devised based on sample images depicting the growth of S. domuncula spicules: 

1. Fixed boundaries: precursor particles are randomly located within the silicalemma, the vesicle membrane of the silicasome (fig. 15) 

2. Expanding Boundaries: the silicalemma expands according to the growth of the spicule, maintaining a constant distance from the aggregate. (fig. 16) 

3. Volume-Wide Distribution: like (2), boundaries expand, but silica particles are randomly distributed within the entire volume of the vesicle. (fig. 17) 

Fig. 15. a. Aggregate after joining 2000 particles to the core in scenario (1). b. a closer view. (Javaheri et al., 2013). Particles starting from boundary-adjacent planes had ample room for diffusion, causing the formation of loosely packed aggregates with extensive branching. Indeed, the longer travel distance for each particle allows for a greater influence of random motion. The resulting structure was more diffusive, with noticeable voids and irregularities, reflecting the less constrained nature of particle movement in the simulation. 

Fig. 16. Aggregate after joining 20000 particles to the core in scenario (2). (Javaheri et al., 2013). As the boundaries expand to simulate vesicle growth, the produced aggregate is more compact and denser. This scenario better reflects biological conditions, where spatial confinement progressively influences silica aggregation, demonstrating the significance of vesicle dynamics in shaping morphology. 

Fig. 17. Aggregate after joining 20000 particles to the core in scenario (3). (Javaheri et al., 2013). This scenario results in the most compact and isotropic aggregate. The widespread availability of silica particles ensures rapid attachment near the growing aggregate, reducing voids and creating tightly packed structures. This outcome closely resembles the morphology of natural spicules. 

Fractal dimensions 

Fractal dimension analysis using the box-counting method revealed values of 2.2 for Case 2 and 2.48 for Case 3, indicative of the compactness of the aggregates. The higher fractal dimension of Case 3 nears the ideal DLA conditions for three-dimensional objects (≈ 2.5), suggesting that this setup provides the most biologically realistic model for initial spicule growth (Javaheri et al., 2013).  

However, these early-stage models represent only the initial aggregation of silica particles. The natural progression of spicule formation involves significant post-synthesis modifications that refine and harden the silica structure. For instance, biosintering, process where water molecules are released from amorphous silica, as well as surface migration, where particles within the aggregate move to lower-energy configurations, contribute to a more compact and robust structure (Javaheri et al., 2013). These processes are driven by interactions such as van der Waals forces, ionic bonds, and hydrophobic effects, particularly between silica particles and proteins like silicatein oligomers. The anisotropic surface properties of these proteins guide the transformation of fractal aggregates into filamentous and hardened morphologies, as observed in mature spicules. 

Conclusion

As some of the oldest benthic organisms, sea sponges have become the groundwork and cornerstones of many aquatic systems. They act as keystone species, nutrient sinks and cyclers, and now a piece in the ongoing puzzle of biological modeling.  

In summary, the modeling of sea sponge growth and the subcellular biosilicification process are two approaches taken by analysts to further understand Porifera.  

The asymmetry and complexities of sponge growth have been accurately human modeled by using velocity deposition, vector determination, diffusion limitations, and curve normalization. These aspects come together to build a system which grows via branching with the length regulated by contact with the environment and available nutrients. These model results are consistent with the shape of branched sea sponges, proving the logic that has developed throughout years of natural selection. 

Biosilicification modeling uses the power of ordinary differential equations, partial differential equations, and random walks to simulate the aggregation of silica particles. The model demonstrates the arrangements formed by diffusion limitation, but stops there, as biological models actively contribute to the final formation of biosilica in spicules. However, with diffusion alone, the modeled aggregation of silica accurately reflects reality within the biological world. 

Of course, these models are not perfect. An improvement in the specificity of parameters and acute modeling systems will provide increasingly more accurate designs. But perfection is not necessary for models to be relevant.  

 The role of sea sponges in their communities and the mysteries within the benthic-pelagic system can begin to be uncovered, alongside the fantastic strength of spicules, and the organisms’ ability to remain rooted despite external pressures. Other applications include the development of eco-sustainable resources, the improvement of human made aqua culture farms, and the ability to understand the development and growth of spicules. In times of record flooding and hurricanes, an analysis of growth systems and holdfasts could result in stronger underwater foundations, while the models also further an avenue for betterment of construction materials within the laminated system. Further impacts could lie withing the fields of mechanical engineering, civil engineering, and architecture.  

Ultimately, the ability to model such a complex biological system proves how far we have come from the days when counting was enough to understand our world. But it also sheds light on all that we have left to learn. For now, Porifera largely remain a mystery, but wielding the power of mathematics we can continue to discover their ancient secrets.  

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