Geometry is Key: Mathematical Modeling of Whiskers and Antennae

Table of Contents

Abstract

Rat whiskers can be  modeled by Euler spirals,  curves with linear change in curvature. One hypothesis explaining vibrissae shape is that the linear growth of rat whiskers creates linear curvature. Another hypothesis is that the Euler spiral is an optimal shape to satisfy its sensory needs. The patterning of the mystacial pads on which those whiskers grow can also be modeled mathematically by Reaction-Diffusion systems. Furthermore, the Barrel Cortex of rodents, to which whiskers are linked, can also be modeled by this Reaction-Diffusion. Through this modeling, we can better understand how the patterns arise. However, mathematics can elucidate more than the “how” of structures; it can also explain the “why.” The physiology of insect antennae impacts their sensory performance. The varying lengths and angles between segments of insect antennae can change both the antennae’s range and the quality of the tactile information. When investigating olfaction, the number of sensory hairs and the leakiness of the structure can modify odorant capture efficiency. Moths often possess antennae composed of simple filaments. By modeling the effects of scales on the passage of different particles in the airflow, the experimental results indicate that the scales on the filamentous antennae enhance their efficacy.

Introduction

Whiskers and antennae are crucial sensory appendages for animals and insects to perceive their environment. For this reason, they have evolved to be relatively optimal and elegant in shape. However, their arrangement, structure, and function vary widely due to environmental and evolutionary factors. The geometry and mechanism of whiskers and antennae can be mathematically analyzed concerning the function or growth of animals/insects. By modeling the curvature and array of whiskers, the sensory mechanism of whiskers regarding a rat’s ability to detect the surrounding environment can be explored. Effective insect communication requires the antennae, which function as the olfactory sensory organs, to interpret information effectively. Applying mathematical models demonstrates how geometry optimizes the efficacy of odor detection and, consequently, the sensing efficiency that contributes to the survival of a species.

The Shape of Whiskers

Rats are colorblind and have poor vision and thus must use other senses, such as smell and whisker detection, to perceive their environment. Rat whiskers (vibrissae) sense by whisking, the process of periodically moving the snout to gather information about an object’s shape, size, and texture (Sofroniew & Svoboda, 2015). When in contact with an object, the object applies pressure on the whiskers, which is transmitted into reaction forces at the follicle’s base (Starostin et al., 2020a). The brain then processes reaction forces and vibration frequencies to allow rats to interpret their surroundings. 

An essential component of sensing is the whiskers’ shape change when in contact with an object. The shape of whiskers, notably their curvatures, affects the manner in which whiskers bend during whisking (Starostin et al., 2020a). Thus, the study of whisker geometry can provide further insight into whiskers’ sensory mechanisms. One important finding by Starostin et al. (2020a) is that the centerline of rat whiskers (Fig. 1) can be modeled using a universal curve known as the Euler Spiral.

Fig. 1 Two-dimensional scan of a rat vibrissa. Notice how the left (proximal) end has a smaller curvature than the right (distal) end [Adapted from Starostin et al., 2020].

What is the Euler Spiral?

The Euler spiral (or Cornu spiral) was first proposed by James Bernoulli when trying to solve problems relating to elasticity. It was named after Leonhard Euler, who solved Bernoulli’s elasticity problem. The Euler Spiral has various applications in physics and engineering. For instance, Augustin Fresnel found that the spiral describes the diffraction of light from the edge of a half-plane (Cornu Spiral), and Arthur Talbot invented the Cornu spiral (independently of Euler and Bernoulli) to construct smoother railway tracks for train turns.

The key aspect of the Euler spiral (Fig. 2) is that its curvature is a linear function of its length. That is, the spiral gets more “curvy” as it approaches the endpoints in the first and third quadrants and gets less “curvy” when it approaches the origin. (To visualize the concept of increasing curvature, please see this animation: Euler Spiral (Clothoid) Animation)

Many different equations for plotting the Euler spiral exist. For instance, on the complex plane, it can be plotted using the equation (Cornu Spiral):

Where i is the imaginary axis that lies vertically, and S(t) and C(t) are the Fresnel integrals used in describing diffraction patterns (Tatum, 2021):

This method of describing Euler spirals on the complex plane is most useful when dealing with the physics of light (Tatum, 2021). After transforming Equation (1) into the real plane using arc-length parametrization, the Cesàro equation is obtained:

Where s is the scaled arc length, k is the curvature, and A and B are constants called Cesàro coefficients. This Cesaro equation highlights the linear aspect of its curvature and is used to model rat whiskers (Starostin et al., 2020a).

Fig. 2 The Euler Spiral. The limit points of the curve are marked by crosses at the points 2π4,2π4 and -2π4,-2π4 (Euler Spiral).

Mathematical modeling of Rat Vibrissae Curvature

Starostin et al. (2020a) noticed that rat whiskers have a noticeable change in curvature, with some whiskers having an inflection point but no more than a maximum of one inflection point. They modeled the shape of each whisker from a data set of 523 rat vibrissae and found that the set of whiskers from a rat will span a large portion of the Euler spiral (Fig. 3). 

Starostin et al.’s mathematical modeling of the whiskers falls primarily on the following two equations:

Where C represents the curvature function scaled over some constant (A/2) and α is a parameter that depends on the angle between the x-axis and the tangent to the centerline. Larger values of the constant A represent more changes to the curvature of the shape. These arc-length parametric equations describe the curve as coordinates on an x-y plane. In addition, each whisker is individually rescaled without affecting their shapes to facilitate comparison with other whiskers.

Fig. 3 Visualization of rat whiskers forming the Euler spiral from the collection of 516 whiskers from 15 rats stacked. Notice how, for each color (representing a different animal), the whiskers combine to form a large portion of the Euler spiral [Adapted from Starostin et al., 2020a].

Hypotheses for Vibrissae Shape

The modeling of rat vibrissae by the Euler curve with linear curvature is likely a representation of linear laws that govern rat whisker growth. Presumably, rat whiskers grow by the same amount every day, with variations depending on season and nutrition (Starostin et al., 2020a). This linear growth with time translates to a linear curvature because newly born cells on one side of the whisker follicle occupy more volume than on the opposite side (Starostin et al., 2020a). Since whiskers grow linearly in length, and the whiskers get more “curvy” by a linear factor at the follicle site, then the whiskers will have a linear curvature. Fig. 4 shows the growth rates across time on a section of vibrissae.

Fig. 4 A magnified cross-sectional piece of a whisker. Its width is 2w, with the centerline represented by a grey dashed line. s, s+, s− are the arc lengths, and the array of arrows indicates the growth rates across the section. t shows the direction of accumulation of vibrissae cells with time.(Starostin et al., 2020b).

Additionally, the linear curvature of whiskers can be explained by sensory needs. The tactile sensory surface (Fig. 5), also called the shroud, is the surface area covered by the whisker tips. This virtual area represents the space where the vibrissae set can sense objects. The whisker tips are oriented at around a 45-degree angle to the shroud surface (Starostin et al., 2020a). Having the whisker tips almost perpendicular to the sensory surface is likely an evolutionary result of optimizing the largest surface that can be perceived with a given set of vibrissae. To achieve this, a whisker whose base is at a specific position on the mystacial pad must have a corresponding specific position at the shroud surface (Starostin et al., 2020a). These conditions, also known as the two-point G Hermite interpolation, are generally satisfied with a shape with linear curvature. In contrast, a shape with constant curvature generally does not satisfy these conditions (Starostin et al., 2020a). Thus, the reasons behind whiskers’ curvature are presumably related to their growth patterns and sensory functions.

Fig. 5 Modeling of the arrangement of whiskers on a rat’s right mystacial pad. A) The blue ball represents the base of the rat whisker on the mystacial pad, while the pink ball represents the tips of the whiskers. B) The grid surface models the shroud, which is described as the surface spanned by the whisker tips. The blue arrows are the arrows tangent to the whisker tips, and the red arrows are normal to the ellipsoid surface (Starostin et al., 2020a).

Arrangement of Whiskers: Reaction-diffusion

Simple Mystacial Pad modeling with Reaction-Diffusion

Though nature evolves towards greater efficiency, there is usually great complexity hidden in the apparent simplicity. 

The whiskers of mice are one such example. The pattern of follicle locations on the mystacial pad is almost grid-like, made up of vertical and horizontal lines, and follows a pattern (Fig. 6). The whiskers, as mentioned, have intrinsic curvatures that fit the Euler spiral. As Krause et al. (2021) describe, the periodic patterning of hair follicles is a “structurally homogeneous field”. But this pattern can result from nonequilibrium thermodynamic processes and can potentially be modeled as a stationary Turing pattern, which is characteristic of a reaction-diffusion system. This modeling can be simplified by assuming a Neumann boundary condition (Fig. 7), which is a type of boundary that limits the patterning, and it says that “the normal derivative at a boundary to be zero or a constant” (Venkateshan & Swaminathan, 2014). 

Fig. 6 The mystacial patterning in rodents is grid-like. There are approximately 30 hair follicles, and they are placed in almost straight vertical and horizontal lines [Adapted from Lucianna et al., 2016].

Fig. 7 Steady state solutions to Reaction-Diffusion. The x-axis is space, and the y-axis is the concentration of some chemical u. The concentrations, and resulting patterns, vary based on location. (a) All the periodic solutions. (b) The solutions for a specific boundary type, the Neumann boundary condition, a subset of the periodic solutions [Adapted from Krause et al., 2021].

Reaction-Diffusion: Effect in Patterning

The Reaction-Diffusion model can describe patterns as “spots, stripes, and maze on the surface of an animal coat through chemical interaction among cells” (Fig. 8) (Wakamiya et al., 2011). Although the pattern of follicle localization is not listed above, it has similar periodic patterning and can likely also be described by this model. In Reaction-Diffusion, the initial parameters are important and can determine the final pattern. The vibrissae placode, the structure that gives rise to rat whiskers, that forms at different times—the initial conditions—seem to result in different patterns in the end (Ahn et al., 2013). This is because the timing of the formation will influence the diffusion and bring about different patterns (Fig. 9). 

Fig. 8 A figure that illustrates the principle in the model of Reaction-Diffusion; different activators and inhibitors undergo reactions that, along with different diffusions, influence the final pattern, including spots, stripes, and maze (Wakamiya et al., 2011).

Fig. 9 There is delay in placode formation in a mutant mouse, and the end pattern is different. A and A’ are normal, and B and B’ are the mutants that have delayed placode formation and have abnormal patterning. In this case, there is a protein that is removed, which is the primary reason for this abnormal development, but the temporal effect on patterning can still be seen [Adapted from Ahn et al., 2013] .

The time variable is not the only factor that can influence Reaction-Diffusion systems. Different protein interactions can also result in different patterns. For example, Wise, a potential ligand for Lrp4, a protein expressed in placodes of skin appendages, causes a reduction in the number of hair follicles for non-LRP4 mutants (Ahn et al., 2013). This reduction in vibrissal follicles results in a different pattern (Fig. 10). In fact, it may be a type nu reaction-diffusion pattern, as classified by “Pearson’s Classification (Extended) of Gray-Scott System Parameter Values” (Fig. 11) (Pearson’s Classification (Extended) of Gray-Scott System Parameter Values at MROB, n.d.). 

Fig. 10 After treatment with Wise, the control sees a decrease in the number of follicles (A to A’), but the mutant is not affected (B to B’). This shows that molecular interactions (reactions) influence the final pattern. The pattern, especially in B and B’, is similar to a nu type pattern [Adapted from Ahn et al., 2013].

Fig. 11 Progression of the ν-type pattern as time increases. The solitons, represented by green dots, uniformly cover the space as time goes on. Each frame is double the time (Pearson’s Classification (Extended) of Gray-Scott System Parameter Values at MROB, n.d.).

The type nu pattern in modeling Reaction-diffusion systems depends on the starting pattern, which then shrinks and splits into solitons, which are solitary and localized waves (Pearson’s Classification (Extended) of Gray-Scott System Parameter Values at MROB, n.d.). These solitons drift apart and spread in space. Many other types of reaction-diffusion result in different patterns, some of which could possibly model the rat vibrissae formation pattern, like type zeta and lambda, but nothing that visually corresponds as well to Fig. 10 as the type nu pattern. These other patterns are covered briefly in Fig. 12.

Fig. 12 A: type lambda, which models mitosis; the solitons arrange into a hexagon grid and stop moving. B: type zeta, which is similar to lambda, but has spatial-temporal chaos and mitosis, as well as a little die-off. (Pearson’s Classification (Extended) of Gray-Scott System Parameter Values at MROB, n.d.)

Similar to the apparent simplicity of rat whisker localization patterns, reaction-diffusion modeling is “simple,” in that it is governed by just two reactions (Reaction-Diffusion by the Gray-Scott Model: Pearson’s Parameterization at MROB, n.d.).

where U is an activator species and V is an inhibitor species, and 

where V is the same as before, and P is a product

The V is considered the activator because it exists on both sides of the equation; it catalyzes its own production. 

However, mathematical modeling increases the complexity significantly. The two equations for reaction-diffusion are 

and 

where u and v are concentrations of the two chemical species, corresponding to U and V, respectively, from the reaction equations. The first term for both equations is a diffusion term. The first equation says that the concentration of U will increase proportionally to the Laplacian, which is “a sort of multidimensional second derivative giving the amount of local variation in the gradient,” of U (Reaction-Diffusion by the Gray-Scott Model: Pearson’s Parameterization at MROB, n.d.). Similarly, in the second equation, the concentration of V will increase proportionally to the Laplacian of V. The second term of both equations is the reaction rate—reaction (7) requires 2V’s and 1U. It is negative for the first equation because U is being converted into V, so there is a decrease in V. The opposite is true for the second equation: the change from U to V causes an increase in V. The third term for the first equation is the replenishment term; the U is constantly being used up, so it is necessary to have replenishment by a feed rate of F. Finally, the third term for the second equation is the diminishment term, since the concentration of V cannot be allowed to go to infinity in a real system: the V should diffuse out at high concentrations. The concentration of V is multiplied by F—which represents the V diffusing out the same way U diffuses in—and k, which represents the rate of V being converted into the product. 

These two “simple” equations from the two “simple” reactions, under slightly different conditions, can result in many different patterns. 

Improvements in Modeling: Periodic Patterns from Heterogeneity

Even so, the patterns resulting from this modeling seem random compared to the actual, observed pattern on the mystacial pad. This is likely due to the relative simplicity of the model. Periodic patterns, like the ones seen in whisker mystacial arrangement in mice, can result from spatial  heterogeneity, with the repeats, or oscillations, due to “destabilization of a steady spike pattern due to the creation of a new spike arising from endogenous activator production” (Krause et al., 2018). Many different parameters can result in this periodic patterning, as shown in the figures below (Fig. 13). Due to the numerous parameters and conditions used and provided, only the plots showing solutions of oscillating spatial-temporal patterns are provided, with no further mention of the parameters or conditions. It is still not entirely clear how the whiskers are arranged so squarely, but the solutions shown make clear that it is very much possible for periodic patterns to arise in Reaction-Diffusion systems. 

Fig. 13 Examples of solutions to Gierer-Meinhardt equations—which model a reaction diffusion system—with different parameters and boundary conditions that can have oscillating spatio-temporal patterns. The colors represent different concentrations of U, the activator species.[Adapted from Krause et al., 2018].

Reaction-Diffusion and the Barrel Cortex of Mice

Additionally, Reaction-Diffusion can help explain more complex systems. The barrel cortex in the brain of rodents has a pattern based on the somatotopic map of whiskers. That is, there is a point-to-point correspondence, one whisker to one barrel. The boundaries formed by the barrels result in a Voronoi tessellation. This can possibly be explained by reaction-diffusion and many axons being concentrated in centers then branching out and stopping when there is contact with another center (James et al., 2020). 

Despite the seeming simplicity of the idea, mathematical modeling is much more complicated. In a study, a reaction-diffusion style model developed by Karbowski and Ermentrout was further developed to try and model this phenomenon (James et al., 2020). The model formed a relationship between the fraction of occupied synapses and the density of axon branches. The equations are as follows. 

ci is the fraction of occupied synapses, which depends on space and time. The synaptic connections decay at rate α and are created at a rate proportional to the product of some power of branching density, ai, available neurons, and some constant . There are N nerve projections between the cortical and the thalamus, hence there is a sum to N. The second equation describes the rate of axon branching. 

The ∇ describes the divergence of the flux of axon branching. The flux is the part inside the parentheses, and it contains diffusion at a rate of D as well as molecular signaling x. There are M molecular signals and their effects are dependent on i, j, such that each axon can branch in different directions.

With this simplified modeling done on a hexagonal lattice, a structure similar to the actual arrangement was observed, in which the barrel cortex was formed (Fig. 14).

Fig. 14 A: The real barrels of a rat, stained by cytochrome oxidase. B: The results to the above equations, with parameters N=41 , α=3.6, β=16.67, k=3, D=0.5, γ∈±2, ϵ=1.2 and δt=0.0001. This model is almost perfectly corresponding to the cytochrome oxidase-stained barrels after a 30000 steps run [Adapted from James et al., 2020]. 

Reaction-diffusion has great explanatory power because it is, in principle, very simple: just two reactions occurring. However, hidden behind this simplicity is the often-complicated mathematics that must account for boundary conditions and varying initial parameters. With improvements in modeling, different mechanisms can now be better understood through reaction-diffusion. 

The Shape of Antennae 

Antennae are sensory structures found on many different insect species. Their arrangement, shape, and function greatly differ and vary based on environmental and evolutionary conditions. They connect the environment with an animal’s nervous system, enabling them to perceive the world using different senses. Antennae can be used for mechanical, chemical, and thermal sensing, among many other functions (Wang et al., 2020). For many insects with such structures, tactile information gathered with antennae allows for orientation in their environment. Mechanical sensing is often accompanied by active movements to collect information regarding the surroundings. In addition, antennae often fulfill the role of food, reproductive partner, and habitat searching through olfaction (Elgar et al., 2018). 

While many animals and insects carry antennae, the shape of the organ greatly varies between species, often adapted to the specific function they fulfill for the host. For tactile sensation, the morphology of the antenna can have an important impact on its sensory performance. Morphological parameters, such as segment length and joint axis orientation, vary between species to optimize their sensory efficiency in relation to the species’ needs (Krause & Dürr, 2004). A primary example uses the two joints connecting the main segments of an antenna. Some insects, such as crickets and stick insects, have two hinge joints which allow for a singular degree of freedom, while others have a hinge joint and a ball and socket joint, allowing for additional freedom of movement. Crickets and stick insects both have two hinge joints. However, their antennae still have morphological differences that affect their function. Crickets have hinge joint axes parallel to the horizontal and vertical planes of their body, while stick insects have slightly slanted axes (Krause & Dürr, 2004). When it comes to olfaction, geometry can also optimize odor capture and, consequently, sensing efficiency (Jaffar-Bandjee et al., 2020b). Antennae with olfaction as a function can have many different shapes, which are believed to impact sensory experience. They range from simple, filiform antennae with a single cylinder as its main shape to feathery, pectinate antennae with many cylinders used to describe their configuration.  

Tactile sensation and stick antennae 

Basic structure

Every antenna has a basic structure made up of three segments and two joints, one of which connects the structure to the head of the insect. The scape is a short and cylindrical segment connected to both the head of the insect and the second segment, the pedicel, a shorter sub-cylindrical section of the sensory structure (Wang et al., 2020). Together, the scape and the pedicel gather information regarding the movement and position of the antennae. The flagellum is the final section. It is narrow and by far the longest segment. The flagellum carries the majority of sensilla, also referred to as sensory hairs, capable of transmitting different types of sensory information (Krause & Dürr, 2004). Fig. 15 shows the arrangement of the three main segments of insect antennae. The joint connecting the scape and the pedicel is always a hinge joint allowing for a single degree of freedom, while the head-scape joint can be either a ball and socket joint with 2 degrees of freedom or another hinge joint (Krause & Dürr, 2004). In this section, antennae with two hinge joints will be investigated. 

Fig. 15 General view of the head of Stephanitis nashi, showing two antennae (An), the labrum (Lm) and labium (Lb); Sc: scape, Pe: pedicel, Fl: flagellum (Wang et al., 2020).

Effect of Variation of Parameters on Tactile Efficiency

Tactile efficiency, as well as movement in active sensing, can be impacted by both morphological and mechanical parameters. Insects can adopt varying movement strategies to gather tactile information, affecting the quality and nature of the information recovered. However, the interest of this section lies in the effect of the geometrical properties of the antennae on tactile sensing. A basic frame to represent the antennae, along with the parameters of interest, can be constructed to offer a visual representation (Krause & Dürr, 2004). This is shown in Fig. 16.

Fig. 16 Construction of a stick insect antenna and conventions for a generic model of an antenna with two hinge joints (Krause & Dürr, 2004).

The base frame is described by the x0 and y0, and z0 axis system. It represents the connection between the base of the antennae and the head of the insect, also referred to as the head-scape joint. The rα-axis indicates the axis of rotation of this joint. Sα represents the orientation and projection of the scape. The x1, y1, and z1 axis system describes the frame for the second joint, the scape-pedicel joint. The rβ-axis indicates the rotation of this joint. Sβ represents the orientation and projection of the pedicel and flagellum. Varying different parameters in this basic frame can affect both the sampling volume, the volume surrounding the antennae where tactile information can be recovered, and the angular resolution, which describes the precision and quality of the sensory data obtained (Krause & Dürr, 2004).  

The surfaces generated in Fig. 17 represent possible locations for the tip of the antennae and, consequently, the volume where the antennae can gather spatial information. In part a), the length of the first segment, Sα or the scape, was varied while keeping the total length of the two segments constant. The sampling volume, also referred to as the 3D workspace, forms a ring-like shape that decreases as the length of the first segment increases. In part b), the angle between the head-scape and the scape-pedicel joint axis was varied from 90 to 0 degrees. The sampling volume is compressed and decreases in size as the angle between the two axes approaches 0. In part c), the axis of rotation of the head scape joint, the rα -axis, was rotated with respect to the x0 base axis from 90 to 0 degrees. The variation of this angle leaves the volume of the 3D workspace unaffected. However, the location of the holes in this space changes. Finally, in part d), the angle between the axis of rotation of the scape-pedicel joint, the rβ -axis, and the second segment representing the pedicel and the flagellum varies from 90 to 0 degrees. These changes cause the surface describing the volume to become narrower on one side and wider on the other (Krause & Dürr, 2004). 

Fig. 17 Influence of various morphological parameters on the antennal workspace. For illustration purposes, the shapes indicate the fictive workspace for a complete 360◦ revolution around both joints (Krause & Dürr, 2004).

The graph on the right plots the area of the surface describing the workspace as the different parameters are modified. A general relationship can be drawn between the area of the sampling volume surface and the angular resolution. The antennal joints have a limited angular precision indicating that the antenna movement can be described by a finite number of antennal positions. As the area decreases, the angular resolution increases since the smaller area can be described with many antennal positions, increasing the accuracy of sampling for smaller areas of workspace surfaces (Krause & Dürr, 2004). 

It becomes clear that modifying the morphology of an antenna can change its workspace and positioning accuracy. Both will change the sampling performance and accuracy of the structure. Different species adopt different morphologies to maximize their sensory experience with respect to the environment they live in (Krause & Dürr, 2004). 

Odor Perception and Pectinate Antennae 

Evolutionary explanation 

Olfaction through insect antennae is also believed to be impacted by morphological patterns. The shape of pectinate antennae is widely believed to be the product of natural and sexual selection (Elgar et al., 2018). Investigating the configuration of antennae of males and females of the same species first sparked the hypothesis. As a general observation, the antennae of males are larger and more complex. Fig. 17 compares the shape of males and females in species with a developed sense of olfaction (Elgar et al., 2018).

Fig. 17 Antennae of male and female insects with well-developed olfactory sense: (a) honey bee (Apis mellifera L.); (b) flesh fly (genus Sarcophaga); (c) cariion beetle (genus Necrophorus); (d) scarabid beetle (genus Rhopaea); (e) saturniid moth (genus Antheraea); (f) hawk-moth (sphingidae, genus Pergesa) (g) butterfly (genus Vanessa) (Elgar et al., 2018).

Females of species release small amounts of sex pheromones to attract males. The molecules in this small sample must come in direct contact with chemoreceptors found on the sensory hairs of the antennae, sensilla, in order to be detected (Jaffar-Bandjee et al., 2020a). Natural and sexual selection favored pectinate antennae due to their increased surface area. A larger surface area implies an increased number of sensilla which favors a higher number of interactions between odors and their chemoreceptors. Investigation of morphological traits of sensory structures such as antennae is studied considerably less than the evolution of the signaler, in this case, the female releasing pheromones (Elgar et al., 2018). Physiological characteristics have a major impact on odor perception, which will be discussed in this section. 

Basic structure
The scape, pedicel, and flagellum form the basic structure of most antennae. The flagellum structure differentiates pectinate antennae from other types. The flagellum, the main branch, carries secondary branches perpendicular to its surface called rami. Sensilla are sensory hairs arranged on rami that allow for various types of sensing (Jaffar-Bandjee et al., 2020b). This structure is illustrated in Fig. 18

Fig. 18 Scanning electron micrograph of the macrostructure, consisting of the flagellum and rami and magnification of the tip of a ramus, revealing the sensilla (Jaffar-Bandjee et al., 2020b).

When investigating the morphological parameters of these antennae in relationship with sensing efficiency, many problems arise. First, pectinate antennae contain many irregular segments with varying lengths and curvatures, increasing the difficulty of accurately modeling such structures. Most studies use an array of cylinders to approximate the behavior of these antennae. Second, the important geometric parameters of the antennae range over four different orders of magnitude, from the length of the antennae to the diameter of a sensillum. The structure is often separated into two distinct levels to increase precision and decrease the difficulty of calculations. The macrostructure is made up of the flagellum and the rami, while the microstructure contains the rami as well as the sensilla (Jaffar-Bandjee et al., 2020b). 

Morphological Changes and Sensing Efficiency 

As mentioned earlier, the capacity with which an antenna reacts to volatile stimuli is dependent on the capture efficiency of the structure. An increased number of odorants captured also increases the strength of the neuronal signal that the antennae will deliver. The capture efficiency can be improved by either increasing the contact area between the air and the antennae or modifying the behavior of the airflow as it passes through the antennae (Jaffar-Bandjee et al., 2020b). The proportion of air that passes through the antennae, as opposed to deflected around it, is also referred to as the leakiness. The leakiness of the structure is extremely important as it provides an estimate for the maximum amount of odorant molecules that can be captured from air flow with a specific concentration and speed  (Jaffar-Bandjee et al., 2020a). The leakiness of an antenna will depend both on the air velocity of the flux directed at the antennae as well as its structure. Fig. 19 presents the effect of air velocity on the leakiness of antennae with varying numbers of rami (Jaffar-Bandjee et al., 2020a). 

Fig. 19 Leakiness of the structures and fitted curves (Jaffar-Bandjee et al., 2020a).

As the air velocity increases, the leakiness of the structure also increases. The figure also demonstrates the increased leakiness for antennae with fewer rami on the flagellum. The number associated with the Str represents the number of rami on the flagellum (Jaffar-Bandjee et al., 2020a).

The efficiency of the sensory structure can be improved via a decrease in the distance between sensilla which will increase the effective surface area of the microstructure and allow for more frequent odorant-receptor interactions. Increasing the leakiness of the antennae can also improve the capture efficiency as it increases the amount of air that passes through the antennae (Jaffar-Bandjee et al., 2020b). One improvement involves an increase in the number of sensilla while the other proposes a decrease. Because of these two contradictory constraints, pectinate antennae are believed to promote a trade-off between these two factors in order to optimize odor perception (Jaffar-Bandjee et al., 2020a). 

Coming back to the evolutionary premise stating that pectinate antennae are the product of natural and sexual selection, the odor capture efficiency of these structures can be compared to the efficiency of cylindrical antennae. The capture efficiency of an antenna is described in the equation:

nant is the capture efficiency of the antennae, ns is the capture efficiency of a single sensillum, and Le is the leakiness of the structure. All these values are computed using v, the far field velocity (Jaffar-Bandjee, 2019). Pectinate antennae are not necessarily more efficient than filiform antennae for the capture of odors since their leakiness is considerably lower. The advantage of pectinate antennae lies in the total amount of odorants captured rather than the capture efficiency. The quantity of odor molecules captured is determined using equation (14):

This equation gives the total number of molecules captured m, as a function of the capture efficiency of the antennae, its surface area s, the far-field concentration of the odorant c,  and the far-field velocity of the flux v. Simple cylindrical antennae have a slightly higher efficiency than pectinate antennae. However, pectinate antennae have a much larger surface area, making this type of structure a good solution to increase the amount of odorant captured, such as pheromones (Jaffar-Bandjee, 2019).

All in all, morphological parameters and special arrangement in antennae has an enormous impact on sensory efficiency and experience. Variations in sizes and shapes of these organs can be attributed to evolutionary development where species evolved to obtain structures aiming to optimize sensory experience for their specific environment. 

Antennal Scales Enhance Signal Detection Efficiency

Insects inhabit a sensory environment rich in odors, which are cues indicating the location of resources or signals that convey specific information to specific receptors (Chapman, 1982). Effective communication requires that receptors accurately interpret signals. In contrast to visual and auditory modalities, olfactory sensory systems require physical contact between the odorant and specific sensory receptors (Kohl et al., 2015). It is well accepted that the shape and size of the antennae have a significant impact in perception efficiency (Chapman, 1982). In fact, Darwin anticipated that sexual selection would favor elaborate, bipectinate antennae in male moths, as they allow for faster detection of female pheromones (Allen et al., 2011). The strength of natural selection will rely on several variables, such as female mating frequency, population density, the type of sex pheromone, and dispersal distances (Symonds et al., 2011). With their multiple branches, bipectinate antennae give a bigger surface area to accommodate more sensilla, hence enhancing the possibility of sensilla-female sex-pheromone molecule interactions. However, moths generally possess antennae composed of simple filaments. By simulating how scales on the filamentous antennae of small and large moths affect the passage of different particles in the airflow through the flagellum, it was possible to determine the effect of scales on the airflow. The scales provide an effective way to enhance the efficacy of filamentous antennae by increasing the concentration of nanoparticles that resemble pheromones around the antennae. According to researchers, the antennal effectiveness of smaller moths improves substantially more than that of larger moths (Wang et al., 2018).

Air Flow Pattern of Moth Antennae

Numerous scales, typically organized in one of four ways (Fig. 20 a–d), cover the filamentous antennae of many moth families. (i) Parallel to or covering the flagellum (Par); (ii) forming an angle with two complete rings per segment (Com); (iii) occasionally present in every two rings (Int); or (iv) lacking a row on both rings (Int) (Mis). The role of scales on the body and wings of moths has been established (Müller et al., 2016), but the function of scales on the antennae has received little study. They may offer limited protection to the sensilla because many of them are exposed in locations where scales are lacking (Fig. 20 a–d; electronic supplementary material, Fig. 20 c,f,g). Although the scales increase the diameter of the antennas and must thus increase aerodynamic drag, there is likely a gain that compensates for this cost. One theory is that the scales interrupt the airflow around the antennae, increasing the likelihood of odor molecule exposure to the receptors (Wang et al., 2018). 

The airflow streamlines reveal the airflow conditions surrounding the antennas. The scales significantly affect the airflow velocity and direction surrounding the antennas (Fig. 20 g,h). Flow velocity decreases sharply upstream of antennas with scales and then progressively increases downstream (Fig. 20 h). Flow velocity for antennae without scales (according to the Par configuration) decreases upstream and climbs fast downstream (Fig. 20 g) (Wang et al., 2018). 

The airflow streamlines are deformed in the direction perpendicular to the surface of the antenna by the scales before resuming their direction downstream. Flow streamlines between the scales indicate that air can enter the space between each ring of scales (Fig. 20 h), indicating that odor molecules can enter the space between the scales and the antenna. The disruption induced by the scales generates airflow bundles downstream of the antenna, which are absent in Par-arrangement antennas (Fig. 20 g) (Wang et al., 2018).

Fig. 20 The development of antenna models and boundary conditions. (a–d) The various scale arrangements of Heliozelidae moth antennae: (a) parallel to flagellum (Par), (b) entire rings (Com), (c) missing scales intermediately in every two rings (Int) or (d) missing a row of scales (Mis), scale bar 50 µm. (e) Dimensions of the antennal model of the 60 µm-diameter antennae; green and red represent different scale rings. (f) Computer-simulated boundary conditions. (g) Airflow velocity and flow-field pattern of an antenna with a Par scale configuration. (h) Airflow velocity and flow-field pattern of an antenna with Com scale arrangement: the scales disrupt and slow the flow field [Presented by: rspb.royalsocietypublishing.org Proc. R. Soc. B285: 201728322].

Modeling the relative antennal detection efficiency

The detection performance of various antennae sizes and scale arrangements was tested by translating particle trajectories to pheromone concentrations using the Particle-Source-In cell (PSI-C) method (Li et al., 2015). The model hypothesized that particles interact with receptors situated on the sensilla (and are therefore sensed by the moth) if they are located in the “detection zone”, which is defined as the space between the flagellum and the scale tips. The spatial distribution of the relative concentration of different particles has been plotted for both small and large antennae (Fig. 21 a–h), and the relative number of particles that the moth can perceive has been calculated by integrating the relative concentration of particles against angles in the detection zone (Wang et al., 2018).

Fig. 21 The distribution of nanoparticle and microparticle concentrations around antennas with a 60 µm diameter, based on various scale arrangements. The antennal scales raise the nanoparticle concentration (a–d) and decrease the microparticle concentration (black semicircle) in the perceiving zone (e–h) (Wang et al., 2018).

Estimating the pheromone concentration in each cell by approximating the total residence time (Wang et al., 2018):

where Cj and Vj represent the local pheromone concentration and volume in the jth cell, respectively, M represents the particle mass, and Δt(i,j) represents the residence duration of the ith particle trajectory in the jth cell.

Based on the ratio between local and far-field concentrations, a relative concentration RCj is established to stress the spatial heterogeneity of pheromone (Wang et al., 2018):

In this model, the detection zone is defined as the space between the surface of the flagellum and the scale tips to pick particles that the sensilla can sense. The size of the detection zone of antennas with parallel scales (Pa) is defined as that of other scale arrangements with the same antennal size, allowing for uniform comparisons across all scale arrangements.  

Assume that the pores on the surface of the sensilla that lead to the receptors are scattered linearly from the root to the tip of the sensilla. Then, the density of pores can be approximated by (Wang et al., 2018):

where A is a constant, r is the distance between a point in the detection zone and the flagellum’s central axis, and r0 is the flagellum’s radius. The overall detection efficiency can thus be obtained by integrating across the entire detection zone (Wang et al., 2018):

  where r1 is the upper detection zone limit. The relative signal concentration is defined as the ratio between the particle concentration at any site and the environmental particle concentration at airflow velocity of 0 ms-1. Normalized by the detection efficiency in a static environment, the relative total detection efficiency can be defined as follows (Wang et al., 2018):

In addition, the angular relative detection effectiveness at angle θ (θ = 0 at the front and θ = π at the back of the sensilla) was defined as follows (Wang et al., 2018):

Therefore, the relative signal intensity in the detection zone of large and small antennae with all four-scale arrangements are integrated to measure the relative overall detection efficiency.

Discussions on the Relative Antenna Detection Efficiency and Function of Scales

       The influence of antennal scale arrangements on the possibility that receptors on the sensilla detect pheromone molecules has been evaluated by plotting the relative concentration of nano- and microparticles within the detection zone (antennal detection efficiency) against different angles of the antennae on both small (Fig. 22 a,c) and large (Fig. 22 b,d) antennae from 180° (downstream side) to 0° (upstream side). Regardless of antennal size, the Par configuration corresponding to antennae without scales, has the lowest antennal detection effectiveness of sex pheromones. Depending on the scale arrangement and antennal size, scales improve the antennal detection efficacy of sexual pheromones. The antennal detection effectiveness of sex pheromones is consistently higher for smaller antennae with the same size configuration than for larger antennae (Wang et al., 2018). This indicates that scales are more effective on smaller antennae than on larger ones.

Fig. 22 Relative antennal efficiency of particles against different orientations of antennas with various size configurations. (a) Nanoparticles, 60 µm diameter antenna, (b) nanoparticles, 120 µm diameter antenna, (c) microparticles, 60 µm diameter antenna and (d) microparticles, 120 µm diameter antenna (Wang et al., 2018).

The antennal scales of moths with filamentous antennae give a relatively simple solution for increasing signal reception, with the improvement being more apparent in smaller moths than in bigger ones. Scales concentrate signal molecules on the downwind side of the antennae and at the distal end of the sensilla, where there are often more pores allowing chemicals to enter the sensilla and interact with the receptors (Fig. 20 f–h) (Greenfield, 1981). By producing a region of slow airflow around the antennae, the scales ensure the pheromones remain within the detection zone, thus enhancing their interaction with receptors. In addition, the scales are efficient in deflecting larger particles, such as dust. Small particles have a lower Brownian movement strength and are therefore kept away from the antennas by the airflow. This means that these particles are less likely to encounter the sensilla, reducing interference with signal reception caused by sensilla contamination (Wang et al., 2018). This is crucial for species, such as moths, that cannot trim their antennae to their full length.

Conclusion

It is well-known in the field of biology that structure typically dictates function. It is unsurprising, therefore, that the geometry, arrangement, and morphology of sensors such as whiskers and antennae play an essential role in the sensory experience of the species to which these organs belong. The specific curvature of whiskers, modeled using the Euler spiral, the pattern of whisker arrangement, the parameters in two-joint antennae, and the shape of pectinate antennae and their scales are all examples of anatomical features that affect the physiology, function, and efficiency of sensory organs.  

Understanding the mechanics and physiology of sensing organs such as whiskers or antennae could lead to improvements in biomimetic sensors. Sensors for robots created using the active sensing techniques of species with whiskers and antennae, coupled with the new understanding of these organs’ morphology and geometric properties, would have enhanced performance, as their efficiency could be maximized for specific tasks through mathematical modeling. Additionally, the complex design of pectinate antennae and their scales could be used to create varying types of mechanical sensors to improve their overall efficiency. In fields other than biosensors, concepts linked to the geometry of these sensory organs have been utilized. The Euler spiral has been used to design smoother train track curves and improve microwaves’ operation (Starostin et al., 2020a). All in all, the elegant and efficient geometry of whiskers and antennae enhances animals’ and insects’ survival, and research into these sensory structures has promising applications in biomimetic design applications.

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