Mathematical Models of Molluscan Shell Patterns and Morphology

Elliott Cole, Gabriel Linaje-Ferrel, Salma Oraichi, Mikaela Phung

Abstract

Mollusk shells’ mesmerizing diversity of forms stems from their variations in shape, features and patterns. While these forms are highly complex and diverse, research has aimed to establish an overarching model that explains the formation of these shell characteristics. This essay explores the mathematical models that capture these natural forms and explain these recurring patterns in shells. First, a 3D mathematical model that simulates most shell forms is described. Then follow descriptions of models relating the development of three specific features of mollusk shells: bivalve interlocking, ribbing, and pigmentation. These models explain how the pattern of bivalve interlocking is the result of the mollusk growing in patterns that reduce mechanical energy, how shell ribs take shape and how reaction-diffusion patterns govern pigmentation. 

Introduction

The foundations of mathematics are intrinsically linked to shell design. Thus, a shell’s shape, the pigmented patterns on its surface, and the ribs’ arrangement all highlight mathematical notions. These concepts make it possible to optimize the survival of the mollusk in an environment that generates many evolutionary constraints such as predation. Therefore, evolutionary biology enables the shell to become an essential element for the survival of the mollusk by playing a protective role. However, all these shell components are harmoniously governed by a mathematical order. It is possible to note a vast diversity in the forms of shells. For instance, there are bivalve shells that correspond to a shell formed by two valves (Fig.1). 

Fig. 1. Bivalve mollusks are formed by two valves. Although usually connected at the hinge, the valves have been separated for a clearer view (Bunje, 2001).

In addition, there are spiral univalve shells characterized as shells that curl along a y-axis (Fig.2). 

Fig. 2. Illustration of a univalve spiral seashell (Charonia tritonis) (Adler, n.d.).

Diversity is also present within the pigmentation patterns that are present on the surface of the shells. Thus, it is possible to note parallel, perpendicular, or oblique pigmented lines (Fig.3).

Fig. 3. A representation of the pigmented pattern on the surface of the shell. The first illustration corresponds to a pattern consisting of perpendicular lines. The second illustration represents pigmented parallel lines. The third illustration depicts oblique lines (Wieschaus, 2019).  

The shell – an exoskeleton– comprises an inorganic matrix of calcium carbonate secreted by the mollusk’s mantle. The mantle (Fig. 4) corresponds to a fabric that adheres to the inner wall of the shell opening. This tissue plays a significant role with respect to mathematical application and biological phenomena within the shell.  

Fig. 4. Vertical cross-section of a bivalve’s shell and mantle (Adapted from Patrick De Deckker et al., 2016).

3D modelling of shell shapes

The basis of the mollusk shape begins with the simple spiral, which is then developed into various 3-dimensional forms. The mechanism for spiral growth begins with the fact that shells grow by adding material in the same direction. If each side grows at the same rate, this will wield a straight structure (Fig. 5A). What creates the spiral shape is a percent difference in growth rates between the two sides, resulting in a coiling that follows the direction of the smallest growth rate. A larger difference in growth rates (Fig. 5C) will result in increased coiling compared to a structure with a smaller difference in growth rates (Fig. 5B) (Picado, 2009).

Fig. 5. Two-dimensional schematic of mollusk growth. A 0% difference in growth rates (A) results in a perfectly straight shape. 10% difference in growth rates (B) results in a slight coil towards the side where less material is added. 100% difference in growth rates (C) will increase the coiling compared to the 10% difference (Adapted from Picado, 2009).

This type of spiral was named ‘logarithmic spiral’ by Jakob Bernoulli (Bernoulli, 1691), and can be described by a parametric equation, which is a type of equation that has an independent variable (a parameter) and dependent variables that are continuous functions solely dependant on that parameter (Ucal, 2022). The parametric equation is given by (see Fig 6):

where x(θ) and y(θ) are the x and y positions at a given point P, and θ is the angle between the segment OP and the x-axis. The term r(θ), representing the distance from the origin (O) to the point (P), is given by:

where A denotes the radius of the spiral when θ = 0, and ɑ denotes the angle between the tangent to the curve at point P and the radial line OP. The angle ɑ will not equate to 90°, as this would produce a circle, resulting in the formation of many concentric shells (Picado, 2009). This would create an unnecessarily thick, circular shell, which is disadvantageous for the mollusk motility.

Fig. 6. Graph of the logarithmic spiral where r(θ) is the distance from the origin (O) to the point (P), x(θ) and y(θ) are the x and y positions, A is the radius of the spiral when θ = 0, θ is the angle between the segment OP and the x-axis and ɑ is the angle between the tangent to the curve at point P and the segment OP (adapted from Picado, 2009).

When the angle ɑ stays constant (equiangular), this results in every increment of growth along a radius to be added in a way that maintains the shape of the spiral even as size increases. This growth process is called gnomonic growth (Picado, 2009). A gnomon represents a shape that, if subtracted from its former version, would produce a picture  with a shape similar to the original one (Stasek, 1963). Most mollusks follow this pattern (Fig. 7), with few exceptions such as the Nipponite Mirabilis, Didymoceras nebrascense and Vermicularia (Fig. 8) because most shells utilize gnomonic growth that shell modelling can utilize the logarithmic spiral as a basis for 3D modeling. 

Fig. 7. 3D modelling of mollusk examples that follow a gnomonic growth pattern: Conidae (A), Nautilus (B) and Bivalve (C) (adapted from Picado, 2009).

Fig. 8. Examples of mollusks that do not follow a gnomonic growth pattern: Didymoceras (A), Nipponite Mirabilis (B) and Vermicularia (C) (adapted from Evens, 2022; Misaki & Tsujino, 2021; Rosenberg, 2021).

Once the logarithmic spiral growth is established, two more curves (Fig. 9) can be parameterized to obtain a model that describes most forms of shell growth. The first curve, the generation curve (C), describes the outline of the shell section. In other words, this represents the size and shape of the shell aperture. The shape of curve C, represented by variable r_e (see equation 3) is most often an ellipse, although it can take many different forms. An example of this is the triangular cross-section of the Japanese Wonder shell (Fig 10). The second curve (H), describes the overall shape of the shell, which takes the form of a helical spiral, spiral that grows along an axis. Another angle introduced for the three-dimensional view is β, which is the enlarging angle of the spiral H, as shown in Fig. 9 (Picado, 2009). 

Fig. 9. Graph of three-dimensional growth of the shell. The curve H describes the overall helicospiral shape of the shell, a spiral growing along the z-axis. Curve C describes the shell’s aperture shape. Angle β represents the enlarging angle of spiral H. Angle ɸ,  μ and Ω are the rotations of the growth about the y, z and x axes respectively (adapted from Picado, 2009).

Fig. 10. Radiographs of Achatina shell (A) and Japanese Wonder shell (B). The Achatina shell demonstrates the most common aperture shape, an ellipse, whereas the Japanese Wonder shell demonstrates a triangular aperture (adapted from Myers, 1997).

In order to derive the formula for the shape of the shell, both the equations of curve C and curve H must be taken into consideration. Another aspect that must be considered to generate a model that covers most shell shapes is that the generating curve can rotate about the x-axis(μ), y-axis (Ω) and z-axis (ɸ) relative to the growth direction (Fig. 9) (Picado, 2009). These angles are particularly important as they determine the shape of the growth markings. In particular, the effects of the angle µ have been identified and result in three different growth markings, as shown in Fig. 11.. Orthoclinal growth has an angle µ of 90 degrees, opisthoclinal growth has an angle µ over 90 degrees and prosoclinal growth has an angle µ under 90 degrees (Illert, 1989). 

Fig. 11. Graph of three-dimensional growth inclination. The shell has orthoclinal growth markings when the plane of the generating curve is perpendicular to the growth direction (μ = π/2), opisthoclinal marking when μ > π/2 and prosoclinal growth markings when μ < π/2 (adapted from Illert, 1989).

Once all these main elements (the spiral formula, the generating curve (C), the overall shape of the shell (H) and the variations of the aperture angles (ɸ, μ, Ω)) are assembled, they result in the following formula describing the shell model:

The full model, which was first developed by M.B. Cortie (Cortie, 1993), also considered any protrusions found on shells, such as spines or nodules (small repetitive bumps on the shell). Indeed, M.B Cortie’s full shell model takes into consideration the number of protrusions in one shell revolution, the size of each protrusion and the separation of these protrusions (for his full derivation see (Cortie, 1993)). This enabled the replication of a large variety of existing shell shapes, as seen in Fig. 12..

Fig. 12. Simulated shells with surface features: Epitonium scalare (A), Turritella (B),  Buccinum undatum (C), Tectus niloticus (D), Olividae (E), Triplofusus giganteus (F), Argonauta (G) and the bivalve Lyonsiidae (H) (adapted from Picado, 2009).

Applications of shell modeling

Mollusc shells protect soft-bodied mollusks from predators and must be very strong to do so. This makes mollusk shells particularly interesting for engineers, who are constantly drawing from nature to inspire novel practices that revolutionize how humans accomplish certain tasks. However, the microstructure of molluscan shells, nacre, gets the majority of the attention when it comes to learning from the mollusk. In reality, it is not just material alone that confers strength to an object. Consider a knife, for example. It is strong, hard, and difficult to break or deform. However, a longer sword made of the same metal would perhaps be easier to warp or bend, as more internal torque can be applied to a particular point in the sword by virtue of it being longer. For this reason, there should also be interest in which features of the shell’s shape can be adopted by engineers for design purposes.

In 2012, Faghih Shojaei and their team used a mathematical model to predict gnomonic shell growth, which is very similar (though slightly different) to the one presented above, and paired it with an in silico mechanical stress prediction software to see how accurately stress distributions and failure could be predicted in various shells. Shell shapes (finite element models each consisting of around 10,000 shell elements) were imported into the software ANSYS 11.0 and were subjected to various loadings (Fig. 13.) (Shojaei et al., 2012). These were run in parallel with identical loading experiments in the real world to compare simulated stress distribution and failure locations to empirical results (Fig. 14.) (Shojaei et al., 2012).

Fig. 13. General in silico compressive test set up for a variable shell type. Movement in the z and x directions was restricted and was subject to a displacement of 2.8mm in the y direction. The virtual ‘jaw’ depicted in c) applies the compressive force (Faghih Shojaei et al., 2012).

Fig. 14. The parallel compressive experimental set up used in the real world. The moving jaw compresses the shell until failure (Faghih Shojaei et al., 2012).

The shell of Otala lactea was the main subject of these comparative tests. They are comparative in the sense that the objective is to determine whether computational programs can accurately predict the stress distribution on the shells. The results can be visualized in figures 15 and 16.

Fig. 15. Fractures sustained by the actual shells (a,b), and the locations of fracture as predicted by ANSYS 11.0 (c,d). All trials were run at first and lowest stage of load intensity (Faghih Shojaei et al., 2012).

Fig. 16. Fractures sustained by the actual shells (a,b), and the locations of fracture as predicted by ANSYS 11.0 (c,d). All trials were run at first and lowest stage of load intensity (Faghih Shojaei et al., 2012).

As can be seen, the simulated locations of high stress intensity align precisely with the observed behaviour in the real shells (Faghih Shojaei et al., 2012). The results are very promising and indicate that these types of computer software can be used to accurately model such stress distributions in mollusk shells. Why is this of relevance? Coming full circle, mollusk shells are incredibly durable and represent a facet of design patiently waiting for engineers to tap into. This study sets a precedent for analyzing and further understanding how the structure of a molluscan shell contributes to its strength, and how these design principles can be incorporated into design and engineering (Faghih Shojaei et al., 2012).

Mechanical and mathematical model of bivalve interlocking

The early sections of this paper have mainly covered the numerous patterns displayed in coiling mollusks. However, bivalve geometry has also been a subject of research among mathematicians. The particular model presented here pertains to bivalve interlocking (Fig. 17.), which refers to how the valves of bivalve mollusks (the class of mollusks including clams, mussels, etc.) fit together and leave no gap between them throughout the mollusks’ lifetime, even when damaged (Moulton et al., 2019). If interlocking patterns are maintained despite the influence of external stresses, then the formation of these matching patterns cannot be purely genetic, but must also be partially determined by mechanical forces as well (Moulton et al., 2019).

Fig. 17.:Despite having widely different patterns, bivalve mollusks manage to keep their valves interlocked with little gaps in between (Moulton et al., 2019).

Fig. 18. An open mussel gives a clear view of its two valves (Burton as cited in Gosling, 2015).

Bivalves are mollusks with two shell valves bound together by a hinge, as seen in Fig. 18.. The shell serves as protection against predators and also as a barrier to keep irritants such as sand and rocks from reaching the mollusk’s soft body (Gosling, 2015). Both valves must interlock perfectly to accomplish this goal, leaving no space between their edges. 

For two valve edges to align and interlock perfectly, the pattern at the valve edges must have a specific set of characteristics. First, the valve edges must be coplanar as seen in Fig. 19., with their sinusoidal pattern lying on the same plane. Imagine matching puzzle pieces: bend two connected pieces at their joint and they no longer align. 

Fig. 19. Non-coplanar valves leave gaps between their edges (A) while coplanar valves do not (B) (Moulton et al., 2019).

However, to interlock, the patterns of the edges must also follow sinusoidal patterns that are completely in phase. That is, the patterns of their edges must be complementary to each other, just as puzzle pieces must bear matching patterns to connect properly. To explain how mollusks grow valves with matching edge shapes, a mathematical model of pattern formation will now be presented.

The curve of the mantle edge r(s) is described by a function of two horizontal and vertical coordinates, x and y. Each coordinate is itself dependent on s, the arc length along the edge. Therefore, the centre of the mantle edge curve is given by a function in the following vector form:

The mathematical model’s goal is then to find functions of x and y that best represent the patterns of bivalves in nature. 

Moulton et al. define x(s) as:

Initially, before any stresses are applied, the centre of the mantle edge has a y(s) value of 0, and r takes the form of

As seen in Fig. 20, such a curve would appear completely flat along the entire valve edge. 

Fig. 20. If there is no stress applied on the shell, the valve edges are flat (Moulton et al. 2019).

Growth of the mollusk mantle can cause internal stress in the shell (Chirat et al., 2013), while outside compressive forces are sources of external stress. Once these stresses are in play, the mantle edge deforms vertically (Chirat et al. 2013; Moulton et al., 2019), and the height of the vertical deformation is no longer a constant value of 0 as in Equation 6:

The function of y(s) is approximately the solution of the following 4^th order differential equation: 

with n being an unknown compressive force on the shell and being the growth rate of the mantle edge. Because n is unknown, there are two unknowns in this single equation, ys and n, and the equation consequently has an infinite number of solutions. However, Moulton et al.’s model determines that the mantle’s preferred deformation patterns tend to have smaller values of n. These smaller values in turn increase the likelihood of imaginary roots to the differential equation’s characteristic equation, making the solution for y(s) sinusoidal. 

Make note that Equation 7 maps the center of the mantle edges, not their point of contact. But since the focus of this paper is the interlocking of the valves’ mantle edges, adjusted equations are presented to account for half of the mantle’s width. The equation for the valve of each mantle edge’s (labelled top and bottom) point of contact is as follows:

with d12 being half the width of the mantle edge. A graph of a typical sinusoidal valve edge pattern is in Fig. 21, while a 3-D model of a typical pattern is in Fig. 22.

Fig. 21. Heights of vertical deformation of shell valve center (solid lines) and valve edge points of contact (dotted lines) as a function of the arc length (Moulton et al., 2019).

Fig. 22. 3-D model of interlocking valve edge patterns (Moulton et al., 2019).

To maximize stability, the form of the mantle edge’s curve will take the form of the configuration with the least mechanical energy. The mechanical energy stored in the two valves comes from three different sources: the bending of both valves, the resistance from their attachment to the shell edge, and the contact between both shell edges. 

Since the valve edge patterns are sinusoidal, they can be either out of phase, not interlocking, or in phase and interlocking. Perfectly out phase bivalve edges have increased forces of contact, and therefore have increased energy. As seen in Fig. 23, the energy of the out of phase pair of valves (long dashed line) is significantly higher than the energy of the in-phase pair of valves (solid line). In fact, the in-phase pair of valves’ energy is so low that it is of equal energy as a single valve (dotted line).

Fig. 23. Comparison of energy stored in a single valve edge (solid line, no diagram), two in-phase valve edges (dotted line, bottom diagram), and two out of phase valve edges (long dashed line, top diagram) (Moulton et al., 2019).

In phase valves are preferred in nature due to this lower energy, which explains why bivalve mollusks have perfectly interlocking valve edges.

Mechanical and mathematical model of shell ribbing

Ribbing resulting from feedback systems and mechanical oscillation

Commarginal ribbing (Fig. 24) is a classification of ribbing in mollusk shells which occurs parallel or almost parallel to the growth edge of the shell (Erlich et al., 2018). Many different models have been proposed for how this ribbing arises, but it is clear that there must be some sort of oscillatory relationship which can be used to model such ribbing patterns.

Fig. 24. Commarginal ribs vs antimarginal ribs in the shells Promicroceras planicosta (a) and Tridacna squamosa (b). The scale bars each represent 10mm (Erlich et al., 2018).

It should be noted that in the related paper, Mechanical overview of mollusk shell formation and structure, a similar model is presented for model rib formation as the one detailed below. The model discussed in Mechanical overview of mollusks shell formation and structure outlines a more complex and recent model which accounts for more variable parameters, and thereby allows for fewer assumptions. The reader is encouraged to explore that model to deepen their understanding of the rib formation process. This precursor model is included here because the relative simplicity makes it slightly more digestible and encourages the reader to truly understand how the mechanical analysis informs the equations which define the model. In conjunction with the model presented in Mechanical overview of mollusks shell formation and structure, the reader can obtain a more historically complete picture of how shell researchers have built off of each other’s ideas, and progressively turned simple introductory models into more complex and, consequently, more realistic models. Finally, this model includes additional simulations and visuals, which will be addressed.

A model for such a formation can be created when one considers that exterior stresses have very little to do with the growth edge of the shell and will not influence the shell’s growth pattern significantly (Hammer, 2000). When considering commarginal ribbing, one is most concerned with the direction of shell growth, as that is what really must oscillate to generate ribbing patterns. The real culprits which most directly influence the direction of shell growth include the tension and compression of the mollusk’s soft body, as well as the elasticity of the shell’s most outer edge, also called the periostracum (Erlich et al., 2016). In fact, it is a constant regulatory mechanical feedback system between these two types of stresses, which is believed to generate commarginal ribbing (Erlich et al., 2016). However, the key requirement is that the feedback system is not fast enough to stabilize and balance the two stresses: the growth direction continuously overcompensates to reduce stress, thereby generating stress of the opposite type for which it will adjust and overcompensate again, and again and again etc. (Hammer, 2000). To investigate, one can examine a simplified model which treats the tube expansion as approximately 0. That is, the model is accurate to a certain degree for shells with tubes of constant widths, or for very slowly expanding shell tubes (Hammer, 2000).

Growth occurs at the growth surface, which refers to the opening of the shell. To motivate more complex models, a simplistic elastic circular ring represents the growth surface (Erlich et al., 2018). The elasticity comes from the rigidity of the periostracum, as well as the stretching and compression of the attached mantle (which by Newton’s 3^rd law exert the same forces back on the growth surface), both of which encourage the return of the ring to its resting state (Hammer, 2000). Hammer develops the following variables for their model: the growth rate of the shell, c_r, is treated as a constant. Therefore, the rate of change of opening is drdt=crsin⁡(φ)≈cr , where the angle of growth relative to the central axis of the tube () varies between -2φ≤2 . Taking the derivative gives d2rdt2=crdφdt  which encourages us to consider the time dependence of . The solution is to consider that the angle is proportional to the quantity R-rR , where R is the radius of the soft mantle, such that when the mantle is squished (R>r and thus R-rR<0), then dφdt is positive and the growth angle opens up (Hammer, 2000). The reverse is also true. This gives us a final differential equation, equation 11:

With a solution, equation set 12, which depends on the initial conditions at t = 0, is:

To avoid complexity, the derivation of the more complicated extension of this model is not outlined, though it should be noted that it arises by following a similar process, but by considering the radius of the shell and that of the mantle as functions of time. That is, it applies to shells with expanding tubes (Hammer, 2000). The derivation yields the equation set 13, which consists of 2 ordinary differential equations: 

 Hammer then defines the following new variables:

r(t), for the shell’s actual radius, R(t), for the ideal uncompressed or stretched radius of the mantle, c, for the general growth constant of the shell, c, for the flexibility of the periostracum, l(t), for the length of the shell, L(t), for the length of the soft parts of the shell, k, for the ratio L(t)/R(t) and finally a variable D, which takes the maximum value between either 0 or Lt-l/L(t) (Hammer, 2000). The variables can be visualized in figure 25. 

Fig. 25. Schematic representation of the variables used in equation set 13 (Hammer, 2000).

Solving equation set 13 yields some representative figures which are very useful for visualizing the predictions of the model (Fig. 26 and Fig. 27).  

Fig. 27. A plot of the radius versus length of the same solution presented in figure 26. This plot allows the actual physical shape of the shell to be visualized, by spinning the curve in 3D space around the line radius = 0. The dotted line is included to help visualize this rotation by π radians (Hammer, 2000).

Again, recent models are more complicated and comprehensive, and address complications such as antimarginal ribbing and mechanical feedback from mollusk growth. This model is certainly more digestible from a mathematical and mechanistic standpoint.

Pigmentation pattern formation in space and time

Fundamental principles of mathematics govern the pigmentation process on a shell’s surface. Thus, many evolutionary constraints have led to the complexity of pigmentation. Indeed, biological motifs present a wide variety of pigmentation patterns (Fig. 28). From an evolutionary perspective, this diversity implies a lack of selectivity of a particular pigmentation model (Fowler et al., 1992). The formation of patterns is closely related to the presence of the pigment during the process of shell growth.

Fig. 28 The diversity of pigmented shapes and patterns present on the surface of mollusk shells (Oster, 2009).

The mathematical reaction-diffusion model, developed by Meinhardt and Klinger, visualizes the pigmentation process on the surface of shellfish (Meinhardt, 2012). Indeed, this model describes the formation of complex patterns concretely through reaction-diffusion. Thus, reaction-diffusion corresponds to a mathematical model that makes it possible to highlight the evolution of the concentration of a substance in a defined space. In the context of mollusks, this substance corresponds to pigmentation. By being distributed in a confined space, the substance is subjected to two distinct phenomena: local chemical reactions and the diffusion process (Yan, 1993). It is important to note that diffusion is a phenomenon that involves a displacement of the substance in order to establish a balance for concentrations. The substance moves from one place of high concentration to another medium of low concentration.

The activator is a substance that has control over the deposition of pigmentation on the surface of the shell. The way pigments are secreted strongly resembles an oscillatory model (Fowler et al., 1992). Thus, each pigment secretion is considered a new wave in this oscillation. Therefore, the secretion of pigments is caused by the presence of pigment cells at the edge of the mantle. The mantle corresponds to a tissue that adheres to the inner wall of the shell opening (Vittadello et al., 2021). In addition, the activator stimulates pigmentation production through a positive feedback mechanism, an autocatalytic mechanism. The main objective of positive feedback is to amplify the effects of disturbances exerted on the system, i.e. the shellfish as a whole. In addition, it is essential to note that the enzyme that allows the activator’s production is called the substrate (Ball, 2012).

In order to form a defined pattern on the shell, it is necessary to inhibit the production of pigmentation secreted by the neighbouring activator. It is important to note that the inhibition phenomenon is initiated by the mollusk’s neural network (Meinhardt, 2012). Since the diffusion speed of the inhibitor is faster than the diffusion speed of the activator, this makes it possible to form stable patterns over time. The antagonistic reaction must therefore occur quickly. This process makes it possible to limit the spread of the secretion of the activator (Fig. 29).

Fig. 29. Illustration of the activator-inhibitor mechanism within the formation of pigment motifs on the surface of shells (Oster, 2009).

The premises of this model were developed by Alan Turing. Indeed, shellfish includes Turing structures due to the presence of the activator and the inhibitor, both morphogens (Ball, 2012). Initially, these morphogens are in balance because there is no pattern on the shell. Then, the equilibrium is compromised due to instability caused by the difference in diffusion rate between the activator and the inhibitor (Vittadello et al., 2021). This phenomenon leads to the formation of pigmented patterns (Fig. 30).

Fig. 30 Diagram of the activator-inhibitor mechanism. The green line corresponds to the activator, and the red line represents the inhibitor. Figure b highlights the stage of equilibrium maintained between the two morphogens. Then, step c demonstrates that the autocatalytic reaction generates the formation of a progressive activator wave. Illustrations d and e show inhibitor stimulation that limits the spread of the activator (Fowler et al., 1992). 

Thus, the formation of pigment patterns (Fig. 31) is characterized as an antagonistic interaction due to the activation of pigmentation production and inhibition. The production of pigmentation occurs in a confined space. In contrast, inhibition occurs simultaneously at other autocatalytic sites.

 Fig. 31 Example of a Turing structure developed using the Gierer-Meinhardt system (Vittadello et al., 2021). 

It is essential to subdivide the pigmentation process into two mathematical models: the substrate-activator model and the inhibitor-activator model (Fowler et al., 1992). These models are therefore highlighted through non-linear coupled differential equations. Although the direct approach of these models is mathematical, these models offer a biochemical explanation for the formation of pigment patterns.

The substrate-activator model

The principle of inhibition is fundamental in the pigmentation process on the shells’ surface. In effect, the inhibitory action is due to the depletion of the substrate, which is essential for the production of the activator (Fowler et al., 1992). These partial differential equations highlight this relationship between the existence of inhibition in the system and the amount of substrate present.

The activator, essential within the pigmentation process, has a concentration called α. The autocatalytic activator diffuses from cell to cell at a speed denoted by D_a. It is important to note that diffusion is highly dependent on position (x) and time (t), as the formation of a one-dimensional relief pattern along the growth edge corresponds to a temporal record (Yan, 1993). Then, its quantity decreases according to a speed characterized by μ. In addition, the autocatalytic aspect of the reaction is closely related to the presence of the substrate. Thus, in the case of the substrate, it is possible to note its diffusion rate within the system through the variable D_s. Indeed, its quantity decreases according to a speed that is characterized by v. In addition, the substrate is produced at a constant rate which is noted . As an autocatalytic phenomenon, small activator concentrations form in proportion to 2. This variable demonstrates that the autocatalytic phenomenon is not linear (Oster, 2009).

This production of the activator is only feasible in the presence of the substrate. Therefore, the amount of the substrate decreases when the amount of the activator increases. It is, therefore, essential to note that the variable corresponds to a simple constant of proportionality. Then, it is very likely that the autocatalysis mechanism saturates high concentrations of the activator. The control of the saturation level of the activator concentration corresponds to parameter k. Indeed, this saturation limits autocatalysis (Yan, 1993). With a high activator concentration, the autocatalytic reaction slows down due to insufficient enzymes needed during catalysis. This situation hence leads to saturation. Equations (14) and (15) highlight how the variable 0  represents a small amount of activator that is previously present to initiate the autocatalytic process (Fowler et al., 1992). The formation of parallel lines to the direction of growth of the shell is illustrated in Fig. 32. This figure clearly shows the application of equations (14) and (15).

Fig. 32 Illustration, which presents a pigmented pattern of stripes, generated through the equations related to the substrate-activator model. Here are the quantitative values of the variables that are present in equations (1) and (2): = 0.01 ± 2.5%, = 0.001, = 0.01, D 0_a = 0.002, =  0.015, = 0, D v_s =  0.4 and k = 0 (Fowler et al., 1992).

In order to form patterns, the amount of the substrate must be sufficient as this allows the maintenance of a constant production of activators. It is also necessary for the substrate to diffuse faster than the activator. Thus, the activator’s peak (a maximum) in the oscillation is formed when its concentration is high. This maximum causes substrate depression (decreasing substrate concentration) (Yan, 1993). The formation of a second maximum for the activator requires an inevitable delay so that the substrate has time to increase its concentration again. Due to this delay, it is possible to see a distance between the maximums of the activator (Fig. 33). This demonstrates that activator production is interrupted when the amount of substrate is insufficient (Fowler et al., 1992).  

Fig. 33 Diagram that illustrates the activator-substrate mechanism as a function of time (Meinhardt, 2012).

The activator-inhibitor model

Through the activator-inhibitor model, it is possible to explain the formation of oblique lines. Indeed, the presence of these characteristic lines is due to the propagation of colliding activator concentration waves (Fowler et al., 1992). It is important to note that the movement of these waves occurs along the direction of hull growth. 

This type of pigmentation is present in shellfish (Oliva’s porphyria), which is shown in Fig. 34. When concentration waves cause a collision, the following pattern is formed:

Fig. 34 Oblique pigmented patterns at the growth lines of the shell Oliva porphyria (Fowler et al., 1992).

According to an observation of the shell, it is possible to note a close correlation between the number of pigment waves and the elapsed time. Indeed, the progression of these waves is constant over time (Meinhardt, 2012). Through this observation, it is essential to introduce another mechanism with a much more global control over the process of pigment pattern formation, namely the control of the total amount of activator. When the activator concentration is insufficient, this mechanism can also generate new activator waves to initiate the formation of new patterns (Fowler et al., 1992). Thus, this mechanism is represented in the following partial differential equations.

The activator-inhibitor system is highlighted through equations (16) and (17). Like the activator-substrate model, activator production is an autocatalytic process (Fowler et al., 1992). In addition, the presence of the activator allows the catalysis of the production of the inhibitor (h). The inhibitor, an antagonist, decreases the amount of the activator. This decrease is proportional to  (1/ h+h0), which is presented in equation (16). It is important to note that the concentration of the activator and inhibitor is expressed as a function of position (x) and time (t). Then, this process is regulated by the presence of a hormone (c), which is presented in equations (16) and (17). This hormone makes it possible to control the entire amount of activators. Along the growth edge of the shell, the concentration of the hormone is considered constant, as the hormone diffuses very quickly (Gong et al., 2012).  

Due to the presence of a small number of progressive waves of pigment, this results in the production of a small concentration of hormone (Fowler et al., 1992). This hormone significantly promotes the disintegration of the h inhibitor. Therefore, the presence of its antagonist (the inhibitor) is compromised. This phenomenon increases the concentration of the activator in order to form new waves of pigmentation. Thus, the hormone makes it possible to introduce the principle of negative feedback because it constantly maintains the number of progressive waves.

The periodic formation of pigment patterns in space is considered a morphogenetic characteristic. Thus, inhibitor maxima are located around activator maxima (Vittadello et al., 2021). The fact that these two types of maxima do not overlap enables the formation of patterns at regular intervals (Fig.35). When the maxima of the inhibitor and those of the activator come very close, this situation forms irregularities within the pattern. Thus, the secretion of pigmentation by activators is affected by inhibiting activity.  

Fig. 35 Reaction diagram of the activator-inhibitor mechanism that illustrates the interaction of these two morphogens as a function of time.  The green line corresponds to the activator and the red line represents the inhibitor (Vittadello et al., 2021).

The elementary patterns present on the surface of shells

Due to the wide variety of pigmentation patterns, patterns can be presented as lines, stripes or spots (Fowler et al., 1992). However, it is possible to highlight elementary patterns such as parallel, perpendicular or oblique lines to the growth lines of the shell. Growth lines, therefore, correspond to transverse lines formed by the shell’s alternating periods of growth and rest (Gong et al., 2012). 

The patterns parallel to the growth lines highlight a spatial-periodic relationship because the formation of this pigment pattern is stable over time (Fig. 36). The deposition of pigmentation, which is governed by oscillatory behaviour, is the primary cause of the realization of parallel patterns (Yan, 1993). Indeed, some cells from the mantle secrete pigmentation at specific intervals. Thus, at certain intervals, these cells do not secrete pigments (Oster, 2009).

Fig. 36 Pigmented patterns parallel to shell growth lines (Fowler et al., 1992).

In the case of perpendicular lines, they are also characterized as stable patterns over time. However, the meaning of these patterns demonstrates that the pigmentation deposition also oscillates over time (Fig. 37). Thus, the main distinction in pattern formation is that inhibitory activity occurs more slowly (Yan, 1993).

Fig. 37 Pigmented patterns perpendicular to shell growth lines (Meinhardt, 2012). 

Oblique lines are formed due to activator progressive waves corresponding to pigment production (Fowler et al., 1992). The transition between an oscillatory mode and a stable pigment production mode causes branching and crossing formation (Fig.38). Thus, a pigment cell, which can produce pigment, induces a neighbouring cell to start pigment production.

Fig. 38 Oblique pigmented patterns at the growth lines of the shell Amoria undulata (Fowler et al., 1992).

Then, the formation of the checkerboard pattern results from the interaction of three parameters (Fig.39). The first parameter corresponds to the autocatalytic substance, the activator. The last two parameters correspond to the antagonists of the activator (Yan, 1993). The second parameter is an inhibitory substance that is capable of diffusing. Therefore, this substance makes it possible to form a pattern in space, while the third parameter, an inhibitory substance that is non-diffusible, generates a pattern in time.

  Fig. 39 The pigmented checkerboard patterns that are present on the surface of the shell Conus marmoreus (Fowler et al., 1992).

Conclusion

As mollusk shells grow, they adopt growing patterns that follow known mathematical patterns. While the famous logarithmic spiral in gnomonic growth is a prime example, these patterns extend beyond coiling in gastropods. From the geometrical shapes of apertures to the inclinations of the apertures, repeated patterns can be observed in the shapes of shells across all mollusk species. 

Mollusk shells have also been documented to follow mathematical patterns while under internal and external stresses. Perfectly interlocking sinusoidal patterns arise in the edges of bivalves, even when under forces of compression that cause their edge pattern to deviate from a flat one. Ribs in shells arise to relieve internal stresses caused from mantle growth. The oscillating internal stress leads to variations in the shell radius, which appear as marginal and commarginal ribs.  

Furthermore, these mathematical patterns are not limited to the shapes of the mollusk, as they can also be found on their surface in their pigmentation patterns. Using activator-inhibitor mechanisms aligning with Alan Turing’s reaction-diffusion model, shells display pigmentation lines parallel, perpendicular, or oblique to their growth lines.

Modelling the growth of mollusks is relevant in engineering to investigate the distribution of stresses on mollusks. Experiments have already been carried out to test which shell shape resists the best to compressive forces, but it is clear that more could be done to study mollusk shell resistance to other stress types.

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