How mathematics governs the nature of tentacles

Table of Contents

Abstract

The tentacle displays a variety of fascinating properties and functions which scientists have attempted to comprehend via mathematical models over the years. In fact, such numerical modeling emphasizes how the appendage has evolved to become optimized for rapid movement, fine-tuned sensation, and skillful predation via its colour-changing abilities. It also serves as inspiration for bio-inspired robotic structures, such as a soft robot arm inspired by cephalopod tentacles and arms which mimics the mathematical behaviour of their actuators, as well as their response to various forces and pressures. In light of these advancements, the following paper will discuss certain mathematical schemes which model the behaviours of the tentacles of cephalopods and cnidarians, as well as the tentacle-like movements of flagella. The biomechanics of the cephalopod tentacle will be analyzed via modeling of its helical musculature, followed by an investigation of a finite element scheme which aims to simulate the varied movements of the tentacle. Furthermore,  the use of mathematics to predict catch probability via modeling of the tentacle will be discussed, as well as the understanding of the mechanics of flagella. Finally, the deceptive camouflaging mechanism of cephalopods which aids in prey capture will be explored.

Introduction

Many human inventions were inspired by nature like the inspiration of airplanes from birds, submarines from whales, or flippers from ducks. Inspiration from tentacles is no different. For example, cephalopods’ tentacles rapid extension and force generation involved in capturing prey along with the twisting motion can be modelled with several equations and demonstrated in a finite element simulation. This simulation  and its equations allow us to better understand cephalopods’ tentacles, which can help us create robotic arms for prosthetic use. Similarly, zooplankton’s tentacles movement can be modelled to understand the space they occupy and how it impacts the shape of the organism. This, along with modelling tentacle-like flagella found in some microorganisms using wave functions, takes us closer to creating submarine robots that can swim efficiently on their own, which can enable advancements in oceanology and marine biology. Additionally, camouflaging, an essential property of tentacles in cephalopods, can serve as an inspiration to create camouflage material that not only changes color and pattern rapidly and accurately but can also create protrusions that change the texture of the material completely. 

A review of mechanisms in muscular hydrostats

Muscular hydrostats are organs that do not possess a distinct skeletal system and rather achieve movement and support through a complex structure of muscle and soft tissue. These organs, such as the tongues of many terrestrial vertebrates, the appendages of cephalopods, and the trunks of elephants, are capable of remarkably varied movements as a result of their complex musculature and the maintenance of a constant volume (Kier & Smith, 1985). In fact, a change in one dimension will always create a compensatory change in another dimension, and this is a fundamental principle in understanding how cephalopods such as the squid achieve such rapid extension velocities and explosive force generation during prey strike. Indeed, this is the very opposite of auxetic structures, which do not possess a constant volume and thus an increase in one dimension may cause an increase in another dimension (Stavric & Wiltsche, 2019). More details regarding these structures can be found in our previous essay on tentacle biomechanics.

Several mathematical equations have been devised in recent decades that attempt to model the varied movements of muscular hydrostats, with squid tentacles being of particular interest. However, before analyzing these models, it is important to have a basic understanding of the tentacle musculature, which is illustrated in Figure 1.

Fig. 1 Tentacle cross section showing the muscle layout. TM-transverse muscle, LM-longitudinal muscle, HM-helical muscle, CM-circular muscle, AR-artery, DCT-dermal connective tissue, EP-epithelium, IN-intramuscular nerve cord, SLM-superficial longitudinal muscle, TR-trabeculae of transverse muscle, TV-superficial tentacular vein (Kier, 1982).

The musculature of the tentacle surrounds a central nerve cord accompanied by a central artery which both extend along the longitudinal axis of the appendage (Kier, 2016). The nerve cord is surrounded by an extensive mass of transverse muscle, with muscle fiber bundles that are perpendicular to the longitudinal axis extending across the diameter of the tentacle stalk. As they extend towards the periphery, these fibers are intersected by bundles of longitudinal muscle fibers which run parallel to the longitudinal axis. Near the surface of the tentacle, the transverse muscle mass turns and becomes part of a thin layer of circular muscle. This layer is then wrapped by a pair of thin layers of helical muscle fibers responsible for torsion.

This twisting motion of the tentacle is quite important during prey capture. In fact, it has been observed that squids twist their tentacle in order to position their club suckers towards their prey (Liang et al., 2006). This motion is achieved by activating one of the two oppositely handed sets of helical muscle, with the contraction of the right-hand helix causing counter-clockwise torsion (of the tip relative to the base) and the left-hand helix causing clockwise torsion. This musculature can be seen in Figure 2. Simultaneous contraction of both muscles results in an increase in resistance to torsional forces (Kier & Smith, 1985). Indeed, the peripheral location of these muscles results in a greater moment through which torque can be applied. The helically arranged muscle fibers may also contribute to changes in length of the tentacle, with their effect being dependent on fiber angle, which is defined as the angle between the helical muscle fibers and the long axis. The effects of changing this angle—denoted θ below—have been modeled by Kier (1985) via a constant-volume cylinder model. This model can be seen in Figure 3A, where a cylinder is wrapped by a single turn of helical fiber. The length of this fiber, D, can be seen in Figure 3B where the cylinder is slit longitudinally and laid out flat.

Fig. 2 Diagram showing how a tentacle could theoretically rotate about its central axis via helical muscle contraction. The spiral represents a helical muscle, the cylinder represents the tentacle, and the black line is a reference to show rotation

Fig. 3 A. Cylindrical model showing the dimensions of the tentacle, where r is the radius and L is the length of stalk. B. Cylinder slit longitudinally and laid out flat, where D represents the length of the helical muscle and  q is the fibre angle. C. Graph of the helical fibre length as a function of the fibre angle (Adapted from Kier & Smith, 1985).

The length, radius, and constant volume of the cylinder can be expressed as follows:

\begin{equation}
L=D\cos(\theta)
\end{equation}
\begin{equation}
r=\frac{D\sin(\theta)}{2\pi}
\end{equation}
\begin{equation}
V=\pi r^2L
\end{equation}

If we substitute r and L in Equation 3, this gives 

\begin{equation}
V=\frac{D^3\sin^2\theta\cos\theta}{4\pi}
\end{equation}

The helical fiber length of the cylinder of constant volume can therefore be expressed as

\begin{equation}
D=\sqrt[3]{\frac{4\pi V}{\sin^2\theta\cos\theta}}
\end{equation}

The helical fiber length as a function of fiber angle is represented in Figure 3C, where the length is at a minimum when the fiber angle is 54o44′ and reaches maxima as the angle approaches 0o and 90o. The model therefore demonstrates how the fiber angle of the helical muscle layers must increase during shortening and decrease during elongation. In fact, the fiber angle measured in the squid tentacle was found to be approximately 67o in a retracted tentacle and approximately 36o in an extended tentacle (Kier & Smith, 1985). This significant range suggests that helical muscles may produce not only torsional forces but also force for elongation in a retracted state, and retraction in an elongated tentacle.

Finally, the helical muscle layers are surrounded by an outer layer of longitudinal muscle, and a connective tissue sheath surrounds the axial nerve cord. Connective tissue layers can also be found between the above-mentioned muscle groups and within the muscle masses.

Finite element simulation of tentacle movement

After several attempts by scientists to mathematically model the movement of muscular hydrostats over the course of the last half century, Liang et al. (2006) devised a numerical procedure that simulates the varied movements of muscular hydrostats via a finite element code. In other words, the researchers created a program that models the effects of motion on a finite muscle segment (element), and could be implemented into a commercial, general purpose Lagrangian finite element code. In fact, former models were limited by dimension or neglect certain forces—such as those internal to the entity.

Van Leeuwen and Kier (1997) derived an expression for the nominal axial stress in a muscle fiber during extension, which is the stress—force per unit area—resulting from the force that creates contraction of a tentacle muscle fiber during elongation. In fact, it is found by dividing the force by the area over which it is applied. The nominal axial stress, σ0m, is represented as 

\begin{equation}
\sigma_m^0=\sigma_{max}f_a(t)f_e\left(\epsilon^0_m\right)f_r\left(\dot{\epsilon}^0_m\right)+\sigma_{pas}^m(\epsilon_m^0)
\end{equation}

where max is the maximum isometric stress–or the stress resulting from muscle tension without contraction (muscle length remains constant, 1)–at optimum fiber length; fa(t) is the activation state, or the pattern of the activation signal as a function of time; fem0) describes the dependence of active stress on the nominal strain; εm0 is the nominal longitudinal strain, also known as the length change divided by the original length; frm0) is the rate dependent function that relates the active muscle stress and the nominal longitudinal strain rate (εm0), or the change in strain over time; and finally, σpasm is the strain-dependent passive nominal stress (Liang et al., 2006). Each of these functions depends on the structure of the muscle fiber which varies amongst different species and muscle types. 

Liang et al. (2006) modeled these functions using a three-dimensional continuum—a continuous sequence in which the adjacent elements are not perceptibly different from each other (Enderton, 2011)—as depicted in Figure 4. In fact, each finite element is sufficiently small so that the muscle fibers of that region have the same properties, orientation, and distribution. Also illustrated in Figure 4 is a muscle fiber and the associated axial Cauchy stress (m), which is defined as the state of stress at a point and thus the resultant of the infinite stress vectors which pass through that point (Mallon & Óbrádaign, 2000). 

Fig. 4 Continuum finite element showing the orientation of muscle fibers within it. Also depicted is a muscle fiber and the Cauchy stress m in the muscle (Liang et al., 2006).

After deriving several formulations of modeling properties such as stress, strain, and velocity for a given muscular hydrostat, Liang et al. (2006) sought to validate their code by comparing their results to the experimental data obtained by Kier & Van Leeuwen (1997) in their study on squid tentacle extension during prey strike. The findings of the latter study were based on the muscle configuration shown in Figure 5, where the tentacular stalk muscles are deformable and thus responsible for elongation, whereas the tentacular club musculature is used for torsion and bending and thus modeled as passive in this study. The stresses due to the transverse and longitudinal muscle groups are incorporated into the code; however, only the mass of the other muscle groups, which were previously discussed, were incorporated because of their inactivity during prey strike and their low volume fraction.

Fig. 5 (a) Finite element mesh of the squid tentacle, with only a quarter shown because of symmetry. (b) Cross-section of the tentacle stalk depicting the arrangement of the orthogonal transverse muscles and the passive longitudinal muscles, which were considered in the finite element simulation. Helical muscles are neglected for simplicity (Liang et al., 2006).

Although the code is based on a complex series of formulations, we will discuss two key functions used to model the movement of the tentacle during elongation in order to better understand the finite element scheme. The first of these is the activation function from Equation 6, fa, which is described by 

\begin{equation}
f_a=\{0\left[0.5\left(1+\sin\sin\left(\frac{\pi(t-t_d)}{t_a}-\frac{1}{2}\pi\right)\right)\right]
\end{equation}
\text{for}\;t< t_d\\ \text{for}\;t_d< t< t_a\\\text{for}\;t^3t_d+t_a

where td is the activation signal delay, ta is the elapsed time between activation initiation and full activation (at fa=1), and q is a parameter for modifying the activation time profile (Liang et al., 2006). Between the signal delay and full activation, the activation strength increases according to a sinusoidal function.

Another important function used in the simulation is the strain-rate dependence function, fr, given by 

\begin{equation}
f_r=\{1.8-0.8\frac{1+\dot\epsilon_m^{^*}}{1-\frac{7.56\dot\epsilon_m^{^*}}{k}}\frac{1-\dot\epsilon_m^{^*}}{1+\frac{\dot\epsilon_m^{^*}}{k}}
\end{equation}
\text{for}\;\dot\epsilon^*_m<0\\\text{for}\;\dot\epsilon \ge0

where k is a constant, εm*= εm0min and εmin is a characteristic strain rate associated with tentacle muscles. The first equation is derived from stretch experiments on the squid tentacle by Van Leeuwen & Kier (1997), where the muscle fibers are undergoing contraction, while the second is derived from a classical muscle model in which the velocity of contraction has an inverse relationship to the strain—which indeed becomes positive as the muscle fibers begin to relax and thus increase in length—up to a maximum velocity when the muscle is unloaded. 

These two functions, as well as the nominal fiber strain, fe, and the passive stress, σpasm, constitute the basis of the muscle extension simulations which were superposed on the aforementioned experimental data by Van Leeuwen & Kier (1997) to validate the accuracy of the code, as seen in Figure 6 below. Indeed, as the tentacle cross-section decreases due to contraction of the transverse musculature, the tentacle extends longitudinally (see Figure 6b). As noted, the activation strength fa increases during the early stage of the strike, thus causing the internal stress to rise and thus the velocity to increase (see Figure 6a). The longitudinal strain is still small at this point, and thus the initial passive stress in the longitudinal muscles is small as well. The tentacle therefore has a positive acceleration and thus the velocity increases. However, at a certain time (found experimentally to be t = 43 ms for a squid tentacle), the tentacle length reaches a point where the net tension due to passive stress in the LM (longitudinal muscles) exceeds the longitudinal extension force created by the contraction of the TM (transverse muscles). The tentacle then decelerates, as seen in the corresponding decline in velocity in Figure 6a shortly before maximum tentacle extension is achieved. 

Fig. 6 Comparison between the finite element method simulation and the experimental results of Van Leeuwen & Kier (1997). (a) Velocity of the tentacle tip as a function of time during prey strike. (b) Tentactle length as a function of time during prey strike.

These two functions, as well as the nominal fiber strain, fe, and the passive stress, pasm, constitute the basis of the muscle extension simulations which were superposed on the aforementioned experimental data by Van Leeuwen & Kier (1997) to validate the accuracy of the code, as seen in Figure 6 below. Indeed, as the tentacle cross-section decreases due to contraction of the transverse musculature, the tentacle extends longitudinally (see Figure 6b). As noted, the activation strength fa increases during the early stage of the strike, thus causing the internal stress to rise and thus the velocity to increase (see Figure 6a). The longitudinal strain is still small at this point, and thus the initial passive stress in the longitudinal muscles is small as well. The tentacle therefore has a positive acceleration and thus the velocity increases. However, at a certain time (found experimentally to be t = 43 ms for a squid tentacle), the tentacle length reaches a point where the net tension due to passive stress in the LM (longitudinal muscles) exceeds the longitudinal extension force created by the contraction of the TM (transverse muscles). The tentacle then decelerates, as seen in the corresponding decline in velocity in Figure 6a shortly before maximum tentacle extension is achieved. 

These results suggest that the finite element scheme represents an accurate simulation of tentacular elongation during prey strike and thus can be adapted for other movements such as torsion and bending, which were also investigated by Liang et al. (2006). Altogether, the finite element program is an advantageous approach to modeling tentacle movement because it considers the variety of tentacle topologies and shapes and is adapted to three-dimensional responses. Indeed, this comprehensive approach reflects the fine-tuning of the cephalopod tentacle musculature as it has evolved to its current state of optimal biomechanical function.

Mathematical study of zooplanktons and their tentacles in the aquatic environment

There are several zooplankton species within the animal kingdom. The principal tentacled zooplankton that belongs to the Cnidaria phylum is the jellyfish. Some radiata, especially ctenophores, are also tentacled zooplankton. From a mathematical point of view, it is necessary to model the movement of zooplankton tentacles to better understand how they move in the aquatic environment.

In order to model the movement of these species, we need useful data on their size, their arrangement in space and the deployment pattern of their tentacles in water. This information has been obtained by observing the properties of several jellyfish, siphonophores (a type of jellyfish), and ctenophores. These organisms are displayed in Figure 7, with the specific species studied listed in the left column of Figures 8 and 10. Thus, the objective is to analyze the physical information about tentacled zooplankton and the dimensions of their tentacles in order to determine the volume of their tentacles in space.

Physical analysis

Fig. 7 (A) A siphonophore. (B) A ctenophore. (C) A jellyfish. (NOAA, n.d, 1984).

First, the measurement of the size of the species listed in Figure 8 was taken by using general blue water diving techniques described by Hamner (1975) and Heine (1986). The data collection also required photographic recordings of the animals in their natural living environment, the ocean. In some cases, the animals were very large, so in order to know their relative size, they were compared to objects of known size in the field of view. The only property that is based on estimation and which was not counted explicitly on the videotapes is the number of tentilla, which are smaller subdivisions of the tentacle. In fact, this number was estimated by counting the tentilla on a section of tentacle and multiplying by the length and number of tentacles.

Fig. 8 Values regarding the shape and size of the marine species studied as well as their physical attributes (Madin, 1988).

Body size is the bell diameter for medusae, stem length for siphonophores, and oral-aboral height for ctenophores (except Cestum = length). Tentacle and tentilla lengths are relative to the body sizes listed. This information about tentacles is important to understand because it is the foundation of the patterns seen in Figure 9 below. In fact, the tentacle dimensions are variables in the formulas used to determine the volume of the encounter zone surrounding an organism.

Fig. 9 Illustrations of several common patterns of tentacle deployment seen in medusae, siphonophores, and ctenophores. The dotted lines describe the encounter zones around the animals. These lines define the shape that the tentacles can take, which will be used to find a formula that finds the volume of the organism when its tentacles are deployed (Madin, 1988).

Modeling encounter zones

Part A of Figure 9 determines the volume of the disk-shaped space wherein the tentacles radiate from the body. The volume of these encounter zones is calculated as a cylinder:

\begin{equation}
V=\pi r^2h \;(\text{cm}^3)
\end{equation}

where r is the sum of the tentacle extension and half the body diameter and h is a nominal thickness of 1 cm. 

Part B of Figure 9 represents when the tentacles extend within a sphere around the body, the volume of which is calculated by

\begin{equation}
V=\frac{4\pi r^3}{3}\;(\text{cm}^3)
\end{equation}

where r is the average tentacle extension. 

Part C of Figure 9 shows the tentacles stream behind the body, filling a truncated cone. The volume of this cone is given by

\begin{equation}
V=\pi\frac{L(r,^2+r_1^2+r_1r_2)}{3}\;(\text{cm}^3)
\end{equation}

where r is half the body diameter, r2 is half the maximum spread of the tentacles, and L is the tentacle extension.

Part D illustrates when the tentacles are held ahead of the body in a cylinder or truncated cone. Volume is calculated as in Equation 11 if it is a truncated cone or as in Equation 12 if it is organized as a cylinder. 

Part E of Figure 9 describes the tentacles that radiate from a long stem, filling a cylindrical space. The volume is calculated as

\begin{equation}
V=\pi r^2\;(\text{cm}^3)
\end{equation}

where r is the average tentacle extension and L is the stem length.

Finally, part F of the picture shows that the tentacles or tentilla form a nearly flat curtain. Some siphonophores may have encounter zones of this shape, with L as the stem length and L2 as the tentacle length. The body surface zone of Cestum veneris (see Figure 9) is of this shape as well. Encounter zones for cydippid ctenophores with this tentacle pattern are considered to be spherical, calculated as in Equation 10.

Specimen and their respective encounter zone shape

After analyzing the different forms that a tentacled zooplankton can take (as seen in Figure 10), it is now possible to establish which species of tentacled zooplankton are associated with which encounter shape.

Fig. 10 is describing the relationship between encounter zone shape of the figure 9 and the different studied species of zooplankton. The numbers of the three columns at the left are obtained with the calculations of the equations of figure 9 with variables of figure 8. (Madin, 1988).

In conclusion, it is now possible to have an idea of the many forms that tentacled zooplankton can take. In addition to having a model of the shape of their movement in water, it is also possible to know the dimensions of the volumes they occupy. Precise techniques are used to measure them in order to know the size of their tentacles. In fact, the tentacles have a significant impact on the shape that the animals can take. Indeed, more than six different encounter zone shapes are used to model the arrangement of the tentacles of various species.

Flagella: A comparison to tentacles

The arms of octopus do not show predictable patterns of movement making it very difficult to model them. However, the organelle flagellum, which is similar to a microscopic tentacle, shows highly regular movement in many instances. Flagella are thin structures present on many microorganisms and some of the cells of larger organisms. The primary function of flagella is to provide these cells and organisms a means to move in their environment. Sperm cells are a classic example of cells that utilize flagella for locomotion. The environments that flagella exist in are often fluid, viscous media, which impose some resistance on the flagellum so that it can produce forward motion described by Newton’s second law. 

Flagella are not commonly referred to as tentacles, in fact, neither are the arms of octopus. However, both are structureless, lengthy appendages that have many similarities to tentacles. Also, both are used for swimming in their respective organisms. In the earlier paper “An Advantageous Appendage: the Biomechanical Design of Tentacles and their Adaptation for the Aquatic Environment”, the uniqueness of tentacles to the aquatic environment was elaborated, which arose because of the support that water could provide to their boneless structure. Indeed, flagella are also common to fluid environments, however not due to their morphology, but rather their function. Flagella would not be able to generate the force required for swimming in a less viscous fluid such as air. The modelling of flagella movements in a viscous medium could provide some insights into the movements of octopus arms.Gray (1955) observed the sperm cells of one species of sea urchin, noting that in some instances the wave created by the flagellum could be well approximated by a sine curve; this curve is depicted in Figure 11. However, the flagellum would often make asymmetrical waves. In an accompanying paper, Gray and Hancock (1955) illustrated the forces that would be exerted on a flagellum in a viscous fluid as seen in Figure 12. They also proposed an equation that could be used to calculate the speed of a sperm cell based on the amplitude, wavelength and frequency of the wave created by the flagellum; however, this equation will not be explained here. In an attempt to improve on Gray’s experiments, Brokaw (1965) photographed and modeled the spermatozoa of numerous marine invertebrates. His photographs were often of cells with asymmetrical wave patterns because these cells would swim in circles making it easier to focus on them with a camera, a few photos are shown in Figure 13. Nevertheless, he proposed a new model wave which was composed of circular arcs joined by straight lines. Figure 14 shows a diagram of this wave and compares it to the originally proposed sine wave. Clearly, this wave is more complex than a sine wave, in fact it requires five parameters to be described in comparison to the three parameters for a sine wave.

Fig. 11 This illustration was based off a specific photograph of a sperm cell and the points are a sine curve (Adapted from Gray, 1955).

Fig. 12 The forces acting on a differential portion of a flagellum as it moves at velocity Vx along the x-axis. Ny and Ly are the reactions from the water. B) The forces acting on a differential portion of a flagellum as it is displaced along the x-axis. Nxsin+Lxcos is the resultant drag (Adapted from (Gray & Hancock, 1955)).

Fig. 13 Flash film photomicrographs of sea urchin spermatozoa (Adapted from Brokaw,1965).

Fig. 14 a) Is a wave of straight and circular sections, o=1 radian and so=0.4p. The dashed line in a) is a sine wave with similar parameters. b) Shows the change in curvature of the waves in a), over the length of a flagellum. c) Shows the variation in viscous bending moment, M(s), for the wave of circular arcs and straight segments (Adapted from (Brokaw, 1965)).

Photographs and data from Satir suggested a “sliding filament model” for the movements of cilia which immediately suggested a similar model for the wave-like movements of flagella. The evidence suggested that the filaments in cilia and flagella rather than shortening during bending, slide over one another creating a bend. Interestingly, there was also a sliding filament model for muscle contraction proposed by Hanson and Huxley (1954). These models are widely accepted today, and now the filaments are called actin and myosin. All of this means that in a simplified morphological sense, a structure like an octopus arm could be considered as a scaled-up flagella. Brokaw (1971), taking this theory into account, proposed some ideas on a model for the bending and wave-like movements of flagella. Figure 15 shows how a segment of a flagellum could bend after experiencing an internal shear force created by sliding filaments. Finally, the observation that flagellum waves did not decrease in amplitude from the base to the end of the flagellum motivated Hines and Blume (1978) to “derive the equations of motion” for a flagellum in a viscous fluid. The sliding filament theory also supports this observation of constant amplitude because in this theory, force is generated all along the flagellum with actin myosin interactions, rather than at the base alone. Hines and Blume’s equations will not be presented in this paper, however Figure 16 outlines a starting point to their derivations. These derivations use ideas like a shear and bending moment diagram, unit normal and unit tangent vectors, and polar coordinates,  concepts seen in McGill University courses, such as MECH 210 and MATH 263.

Fig. 15 a) Shows a small portion of a flagellum with arrows indicating positive shear. b) Shows the balancing of moments. c) Shows how a flagellum could be bent by this shear as a small section of its length (Adapted from Brokaw, 1971).

Hines and Blume (1978) derived equations of motion for a flagellum in a viscous fluid by first making some assumptions, refer to  Figure 15 for the visual representation:

Fig. 16 Some arbitrary point s on the flagellum at some time t is described by the vector r(s,t). The unit normal and unit tangent vector to the flagellum are illustrated. (s,t) is the angle between the unit tangent vector and the x-axis. The shear and bending moment diagram of the flagellum is also illustrated, with Mint being the moments and FNint and FTint being the shear forces (Adapted from Hines, 1978).

The flagellum is modeled as an inextensible thin filament of length L constrained to move in a plane. The parameter s denotes distance along the flagellum (0 < s < L) and the vector r(s, t) describes the position of point s at time t. A particular resolution of r(s, t) with respect to a set of -x and -y coordinates fixed in the medium is

\begin{equation}
r(s,t)=(x(s,t),y(s,t))
\end{equation}

The inextensibility of the flagellum imposes the constraint that dx2 + dy2 = ds2. If the angle, (s, t) between the flagellum and the x-axis is introduced, then dx = cos αds and dy = sin αds. Then

\begin{equation}\tag{14a}
x(s,t)=x(0,t)+\int_0^s\cos(\alpha(s',t))\;ds'
\end{equation}
\begin{equation}\tag{14b}
y(s,t)=y(0,t)+\int_0^s\sin(\alpha(s',t))\;ds'
\end{equation}

where (x(0, t), y(0, t)) is the motion of the proximal end of the flagellum. A local coordinate system defined by the unit tangent and normal vectors at a point on the flagellum is now defined (Hines & Blum, 1978).

\begin{equation}\tag{15a}
\hat{T}(s,t)=\frac{\partial}{\partial s}=(\cos(\alpha(s,t)),\sin(\alpha(s,t)))
\end{equation}
\begin{equation}\tag{15b}
\hat{N}(s,t)=(-\sin\alpha(s,t),\cos(\alpha(s,t)))
\end{equation}

Camouflaging

Octopuses in many ways are considered intelligent creatures, and one remarkable mechanism that proves this intelligence is the octopus’ ability, alongside other cephalopods, to camouflage within 200 milliseconds, the fastest any creature can do so. Despite being colorblind (Messenger, 1977), octopuses possess the capability of changing colors, patterns, and textures, as shown in Figure 16, which gives them a huge advantage over their prey and predators.

Figure 17. Shows an octopus camouflaged on rocks and corals. In addition to the impressive color changing, the octopus is also able to change its texture, which makes it even more deceiving (Courtesy of Dario Sabljak / Shutterstock, Camouflaged Mediterranean octopus).

How cephalopods see color

We know that cephalopods are able to perceive colors somehow because they control when to camouflage and what colors and patterns to use, but cephalopods’ ability to detect and recognize colors despite being colorblind is not fully understood. Some scientists like Wardill et al. (2015) hypothesize that cephalopods can “see” colors with their skin, which is highly possible because the skin contains light sensitive photoreceptors. Other scientists like Alexander and Christopher Stubbs created the chromatic aberration hypothesis, whereby cephalopods can see color by detecting the bending of the light their eye receives, and different colors can be perceived depending on how much the light bends.

How cephalopods change color

Cephalopod’s skin contains millions of organs called chromatophores, which each contains a sac of pigment that can be of different colors including yellow, orange, red, black, and brown. These chromatophores also consist of radial muscles, which when contracted, expand the sac of pigment making it visible on the skin and when they relax, the pigment’s sac shrinks, and the color disappears (Messenger, 2001). The chromatophores, shown in Figure 17 are a neuromuscular organ controlled by the brain. Their radial muscles are connected to nerves with no synapses that go directly to lobes in the brain specifically for controlling chromatophores, which is what allows cephalopods to change colors so rapidly. In addition to the chromatophores, Iridophores, very thin electron dense platelets, are responsible for reflecting green and blue light by acting as what is suggested to be diffraction gratings where white light can be diffracted into different colors and the desirable color is reflected back. Alongside the chromatophores and iridophores, the leucophores create the white spots in cephalopods by producing white light using scattered reflection. All of these organs are controlled by the brain, which allows the cephalopod to create different patterns on different parts of the skin to better mimic its surroundings (Hanlon & Messenger, 2018).

Figure 17. Shows different species of cephalopods with different patterns, arrangement, and colors of chromatophores (Hanlon & Messenger, 2018).

The accuracy of cephalopod’s color change was studied by Josef et al. (2012) by using an objective and automated image analysis algorithm of images of free ranging octopuses camouflaged in their natural surroundings. The octopus’ mantle was taken as a sample and compared to the overall image using Equation 16 where RA-fft is the rotational-averaged Fast Fourier Transformation, which basically gives the wavelength and frequency of a signal. In this case, the signal is light.

\begin{equation}\tag{16}
\text{RAfft Similarity Percentage}=\frac{[1-(\text{Octopus RA Slope}-\text{Subsample of Overall Image RA slope)}]}{\text{Maximum Difference RA Slope}}
\end{equation}

A low difference in (Octopus RA Slope)- (Subsample of overall image RA Slope) will result in a higher similarity. Using this equation and superimposing the results on the original image as shown in Figure 18, Josef et al. were able to identify where the most similarities in color between the octopus and object surrounding it occurred, and it turns out that the most similarities appeared in a mix of significant objects like algae, corals, protruding rocks, and unevenly colored sand. This means that octopuses sample specific features of their surroundings as opposed to using their whole surroundings, which is a type of mimicry called “deceptive resemblance.”

Figure 18. A and C show the original image of 2 octopuses camouflaging. B and D show the areas where similarities are greater than 90% superimposed on the image. These areas are specific features of the entire surrounding like corals, rocks, sand, and algae (Josef et al., 2012).

Changing texture

In addition to changing colors and patterns, cephalopods, mainly octopuses and cuttlefish, can also change the texture of their skin. This is possible through different variations of expressing the papillae, protrusions of the skin. Papillae, shown in Figure 19, are controlled by a hydrostatic muscular system underneath the skin where some of the muscles control the intensity of the protrusion; i.e., how far vertically it goes and other muscles control the shape of the protrusion, which can be round or spikey. This mechanism allows the skin to go from smooth to rough in under 2 seconds and further aids cephalopods in creating a deceiving camouflage (Allen et al., 2014).

Fig. 19 Shows how an octopus changes its appearance by expressing various intensity and patterns of the papillae (Allen et al., 2014).

Conclusion

Mathematically modeling complex biological systems or structures is a difficult task because of their unpredictability, however, the capabilities of the field are in progress. In this paper, the earliest literature reviewed is the modeling of the relatively simple movements of flagella. In contrast, some of the more recent literature is modeling the movements of muscular hydrostats such as squid tentacles. Computer simulations have allowed for much more complex and accurate schemes for modelling these behaviours displayed by organisms. Unlike the movements of flagella, an equation cannot likely be derived for any given biological process, in this paper we have discussed a few ways that some biological processes can be modelled. First looking at the tentacles of cephalopods, it’s been shown that equations can loosely describe the geometry of some muscle groups, such as the helical muscle of the tentacle, during extension and contraction. Later, a finite element simulation was used to model the movements of these complicated organs. The movements of jellyfish, as well as how their size and shape can be approximated were also discussed, taking their tentacles into consideration. A comparison between the simple organelle flagellum and the tentacles of octopus was made and the regularity of the movements of flagellum so that they can be written in a closed form wave equation was discussed. Finally, the interesting topic of camouflage in octopuses was explored with a specific look at color and texture changes.

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