New Perspectives on Collagen Fibers in the SquidMantle

Michael Krieg1,3 and Kamran Mohseni*1,2,3

1Mechanical and Aerospace Engineering Department, Gainesville, Florida2Electrical and Computer Engineering Department, Gainesville, Florida3Institute for Cyber Autonomous Systems, University of Florida, Gainesville, Floriday

ABSTRACT The squid mantle is a complex structurewhich, in conjunction with a highly sensitive sensory sys-tem, provides squid with a wide variety of highly con-trolled movements. This article presents a model describ-ing systems of collagen fibers that give the mantle itsshape and mechanical properties. The validity of themodel is verified by comparing predicted optimal fiberangles to actual fiber angles seen in squid mantle. Themodel predicts optimal configurations for multiple fibersystems. It is found that the tunic fibers (outer collagenlayers) provide optimal jetting characteristics when ori-ented at 318, which matches empirical data from previousstudies. The model also predicted that a set of intramus-cular fibers (IM-1) are oriented relative to the longitudi-nal axis to provide optimal energy storage capacity withinthe limiting physical bounds of the collagen fibers them-selves. In addition, reasons for deviations from the pre-dicted values are analyzed. This study illustrates how thesquid’s reinforcing collagen fibers are aligned to provideseveral locomotory advantages and demonstrates howthis complex biological process can be accurately modeledwith several simplifying assumptions. J. Morphol.273:586–595, 2012. � 2012 Wiley Periodicals, Inc.

KEY WORDS: squid mantle; collagen fibers; structure;energy storage capacity; optimization

INTRODUCTION

Squid jet propulsion produces the fastest swim-ming velocities seen in aquatic invertebrates(O’Dor and Webber, 1991; Anderson and Grosen-baugh, 2005). Although jetting is an inherentlyless efficient form of locomotion than undulatoryswimming (O’Dor and Webber, 1991; Vogel, 2003),squid morphology has evolved to fully exploit it.Soaring and climbing vertically through ocean cur-rents, negotiating prey capture, or hovering nearthe surface are a few of the squid’s many swim-ming capabilities (O’Dor and Webber, 1991). Thefluid dynamics of propulsive jetting in squid, andother jetting invertebrates such as jellyfish hasinspired a great deal of research (Dabiri et al.,2006; Lipinski and Mohseni, 2009; Sahin and Moh-seni, 2009; Sahin et al., 2009). However, the physi-ology of the squid mantle structure, which playsan integral part in creating the propulsive jet, hasreceived less attention.

In general, jetting locomotion begins when thesquid inhales seawater through a pair of vents oraperture behind the head, filling the mantle cavity.The mantle then contracts forcing fluid outthrough the funnel which rolls into a high momen-tum vortex ring and imparts the necessary propul-sive force (Anderson and Grosenbaugh, 2005). Theversatility of the system permits both low-speedsteady swimming or cruising, and fast impulsiveescape jetting. Two distinct gaits are seen insteady swimming as determined by the nature ofthe expelled jet (Bartol et al., 2009) (those beingabove or below the jet formation number). Duringcruising, squid swim at nominal speed with ahigher efficiency; whereas, escape jetting involvesa hyperinflation of the mantle followed by a fastpowerful contraction to impart significant accelera-tion at the cost of fluid dynamic losses; similar tothe loss in efficiency seen in high velocity jet loco-motion of jellyfish (Sahin et al., 2009).

The squid mantle has a complex collagen fibersystem that provides structural support and storeselastic potential energy to reduce reliance on mus-cle force during both the inhalant and exhalantphases (Ward and Wainwright, 1972; Gosline andShadwick, 1983; Thompson and Kier, 2001a). Thisstudy develops a mathematical model to predictthe effect that structural dynamics related to fiberorientation have on both jetting thrust (which werelate to volume constraints) and potential energystorage capacity of the mantle. Although the modelis rather simplistic (treating the mantle as a

Contract grant sponsor: Office of Naval Research; Contract grantnumber: 1545312.

ySome initial testing was performed at the University of Colorado,Boulder.

Kamran Mohseni, University of Florida, PO Box 116250, Gaines-ville, FL 32611. E-mail: mohseni@ufl.edu

Received 30 June 2011; Revised 19 September 2011;Accepted 9 October 2011

Published online 18 January 2012 inWiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/jmor.20003

JOURNAL OF MORPHOLOGY 273:586–595 (2012)

� 2012 WILEY PERIODICALS, INC.

straight uniform cylinder, despite the taperedshape of actual squid mantle), it takes key fea-tures of the structural dynamics into accountwhich have largely gone ignored in previous mod-els; such as the nonuniform strain distribution inmantle thickness and nonzero deformation in thelongitudinal direction. Although these issues mightseem trivial in such a simplified model, they actuallyallow for an optimal arrangement to be determinedwhich helps to explain the regular layout of collagenfibers seen in the squid mantle. This analysis pro-vides new insight to the function of the fibers.

The Squid Mantle

The powerful squid mantle primarily consists ofmuscle packed between two helically wound collag-enous tunics which are oriented at an angle of �2786 18 to the longitudinal axis of the squid, for Lolli-goguncula brevis (Ward and Wainwright, 1972).The arrangement of a single layer of collagen fibersin the tunic and a definition of the tunic fiber angle,y, are shown in Figure 1. Circumferential musclesring the mantle and radial muscles run from theinner tunic to the outer tunic (Fig. 2). The robustnature of the collagen fibers in the tunic, theirinelastic properties, and low axial angle suggestthat they act to prevent elongation and deformationof the mantle tissue during jetting.

Wound through the muscle layer, are three sys-tems of intramuscular (IM) collagen fibers conven-tionally dubbed, IM-1, IM-2, and IM-3. IM-1 runs atan oblique angle through the muscle layer that isdifficult to measure unless the angle is known a pri-ori. Measurements of the IM-1 fiber angle relative

to the squid’s long axis, therefore, rely on both sag-ittal and tangential sections (see Fig. 2 for defini-tion of primary sections) to accurately describe thepath. We will refer to the respective fiber angles inthese planes (demarcated by some authors as IM-1sag and IM-1 tan) as b and k. Values differing by asmuch as 208 are reported for both b and k. Wardand Wainright (1972) measured b in L. brevis at288. Bone et al. (1981) measured k at 158 in Alloteu-this subulata. MacGillivray et al. (1999) reportedsimilar values in Loligo pealei. These low anglesare in contrast to those reported by Thompson andKier (2001a), who measured an angle of 438 for band 328 for k in juvenile Sepioteuthis lessoniana(although this value varies significantly throughoutontogeny). Thompson and Kier (2001a) suggest thatthe less streamlined appearance of hatchling andjuvenile squid is related to the larger fiber angles.The differences between findings may also haveresulted from species differences, or largely differ-ent ratio of mantle cavity volume to total volume aswill be discussed in sections Maximizing EnergyStorage and Results.

The exhalant phase of the jetting cycle beginswhen the squid contracts the circumferentialmuscles reducing the circumference of the mantleand thickening the muscle layer, while producing

Fig. 1. The squid tunic fibers are wound in a spiral helixarrangement, and are oriented at a uniform angle (y) to the lon-gitudinal axis. The tunic fibers form a cylindrical tube withlength L and radius a. Although the tunic consists of multiplelayers of spiraling fibers only a single layer is shown for clarity.[Color figure can be viewed in the online issue, which is avail-able at wileyonlinelibrary.com.]

Fig. 2. Conceptual diagram of the squid mantle structure.Depicted are the three primary reference planes defining the(IM) collagen fiber angles, and the muscle structure. The sagit-tal plane cuts through and runs parallel to the longitudinalaxis; the tangential plane runs parallel to the longitudinal axisand is locally tangent to the surface; the transverse plane runsnormal to the longitudinal axis. IM-1 fibers run at obliqueangles through the mantle and form angles b and k with thelongitudinal axis in the sagittal and tangential sections, respec-tively. The IM-2 fibers are found localized in the radial musclesand form an angle u with the circumferential axis in the trans-verse plane. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

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only a small increase in length (Trueman andPackard, 1968; Ward and Wainwright, 1972). Therapid reduction in the mantle cavity volume forcesseawater through the funnel and results in a high-energy jet that rapidly accelerates the squid. Aftercoasting, the inhalant phase begins and the innerand outer tunics are brought closer together, thin-ning the muscle layer (Young, 1938). This isachieved by a combination of radial muscle con-traction and energy transfer from deformed IM-1and IM-2 fibers (Gosline and Shadwick, 1983). Infact, it was shown that the refilling of the mantlecavity can occur in the absence of any radial mus-cle power (Gosline et al., 1983).

Regardless of the measurement discrepancies,the function of the IM-1 fibers is generally agreedon. During the circumferential muscle contraction,the muscle layer thickens, and as the collagenfibers are stretched they store elastic potentialenergy. Once the circumferential muscles relax,the fibers pull the tunics closer together andincrease the mantle circumference. We will showthat the orientation of these fibers allows them tostore an optimal amount of energy during contrac-tion in the Results Section.

The IM-2 measurements have been more consist-ent between studies. When mantle tissue wasviewed in transverse sections, the IM-2 angle rela-tive to the mantle surface has been reported from508 to 558 (Ward and Wainwright, 1972; Gosline andShadwick, 1983; Thompson and Kier, 2001a). How-ever, the exact function of these fibers is less clear.

The IM-3 fibers lie parallel to the circumferentialmuscle fibers, and are observed to be coiled upwhile the mantle is in a resting state (Macgillivrayet al., 1999). Their orientation suggests that theIM-3 fibers are rarely fully extended while thesquid is cruising, but rather aide in the contractionof the mantle after hyperinflation has been used foran especially large jet (Macgillivray et al., 1999).

Problems Addressed

First, the difference between maximizing ejectedjet volume and maximizing total volume must beexamined. Squid draw propulsive power from atransfer of momentum to a fluid jet. The force act-ing on the squid during this process is equal to therate at which the squid transfers momentum tothe jet. This force is equal to the product of the jetmass flux and velocity. Both of these quantities areintrinsically related to the muscle contraction rateand the dynamic response of the mantle geometryassociated with muscle contraction. Jet velocityand mass flux can be determined from the rate ofchange of the mantle cavity volume. In this study,we model the muscular contraction as a geometricconstraint rather than modeling the complicateddynamics of the muscles themselves. Therefore,the thrust experienced by the squid can be explic-

itly determined by the structural kinematics of themantle. The change in mantle cavity volume ismodeled with respect to tunic fiber orientation inthe subsection Maximizing Jet Volume.

The energy storage capacity of the IM-1 fiberswas modeled next. We considered a squid swim-ming at a steady rate with regular contractionsand without hyperinflation. We modeled the squidmantle as a tube circled by inner and outer walls(the tunics) and determined the energy stored bythe IM fibers according to the mantle stress–straindynamics. In developing the energy storage model,it was determined that the elongation of the squidplayed a crucial role in the energy storagecapacity. The fact that the IM-1 collagen fibers lieat a low angle in the sagittal plane causes thestrain of individual fibers to have a strong depend-ence on longitudinal deformation. Although thisdeformation is small, inclusion in the energy stor-age model resulted in an optimal fiber orientationin the sagittal plane. This methodology is found inthe subsection Maximizing Energy Storage.

METHODSMaximizing Jet Volume

To analyze the effect of collagen geometry, we constructed arigid mathematical definition of the fiber orientation. The squidmantle is essentially a tube of interwoven muscle and collagenfibers. The mantle is encased by the tightly woven spiral stacksof the inner and outer tunics. For the purposes of this analysis,each tunic will be modeled as a perfect cylinder composed ofhelically spiraling fibers (Fig. 1). The parametric equations,

x ¼ a cosðxzÞy ¼ a sinðxzÞ and ð1Þ

describe the layout of a single tunic fiber, where z is the loca-tion of a point along the fiber in the longitudinal direction(starting at the anterior and extending toward the posterior),and x and y are the geometric coordinates of a point on the col-lagen fiber in the plane normal to the longitudinal axis a dis-tance z from the origin (transverse plane at z). The coordinatesin the transverse plane are centered on the longitudinal axis;positive y extends toward the dorsal side, and positive x forms aright handed coordinate system with y and z. The orientation ofthis coordinate system is depicted in Figure 1. In addition, a isthe spiral radius, and x is a parameter which controls the slopeof the spiral (the inverse of x is the spiral wavelength).

This construct allows us to easily determine several geometricparameters of the cylinder that are necessary to model themantle mechanics. The cylinder diameter is simply, D 5 2a, thetotal cylinder length, L, is the maximum value of the paramet-ric length L 5 zmax, and the tunic fiber angle is defined as y 5arctan(ax). The length of the tunic fiber is the total arc lengthof the spiral which is,

s ¼Z L

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_x2 þ _y2 þ 1

pdz ¼ L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2x2 þ 1

p: ð2Þ

With these definitions, the cylinder geometry is defined interms of the tunic fiber angle, y. This allows the cylinder vol-ume to be calculated as,

588 M. KRIEG AND K. MOHSENI

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V ¼ p4D2L ¼ p

4

s

2pmsin u

8: 9;2s cos u ¼ s3

16pm2sin2 u cos u ð3Þ

Here, m is the number of spiral windings in the cylinder and sis the fiber length defined in (Eq. 2). It can be seen from (Eq. 3)that the total cylinder volume is purely a function of fiberlength, s, and fiber angle, y. As the collagen fibers are known tohave a large low extensibility, we hold the fiber length constant.This imposed constraint reduces the cylinder volume to a func-tion of a single variable, y. A similar approach has been used toanalyze the total squid volume (Vogel, 2003). In the next sectionof this manuscript, a model is derived describing the energystorage in the collagen fibers which requires deformation of col-lagen fiber lengths. However, these deformations are very smalland can be neglected when defining mantle geometry with mini-mal error.It is convenient to clarify our naming convention as there are

several characteristic volumes which describe the squid. Equa-tion 3 describes the volume of a cylinder defined by a fiber oflength s and angle y. The total squid volume is the outer tunic’scylindrical volume, V2. The total internal volume is the innertunic’s cylindrical volume, V1. The sum of the mantle cavity vol-ume and the internal organs comprises the total internal vol-ume. The difference between the outer tunic cylinder volumeand the inner tunic cylinder volume is the mantle volume (orvolume of the mantle tissue).Assuming water to have a constant density, q, the mass flux

across the funnel will be proportional to the rate of volumechange of the inner tunic. Jet velocity can be easily determinedfrom the volume flux if the funnel area, A, is known. Thisallows the thrust, T, to be described in terms of the tunic geom-etry to a first-order approximation as:

T ¼ _muj ¼ qA

_m2 ¼ qA

@V1

@t

8>:9>;2

¼ qA

@V1

@C1

8>:9>;2 @C1

@t

8>:9>;2

ð4Þ

Here, _m is the mass flux across the funnel, uj is the jet velocity,A is the funnel cross-sectional area, and C1 5 pD1 is the cir-cumference of the inner tunic. The rate of change of inner tunicvolume, qV1/qt, is decomposed according to the chain rule intothe rate of change of the inner tunic volume with respect tochange in the inner tunic circumference, qV1/qC1, and the timerate of change of the inner tunic circumference itself, qC1/qt. Aswas mentioned previously, the rate at which the circumferencecontracts is purely defined by the dynamics of the ring musclesand will be treated as a constant. Although the funnel area, A,is known to oscillate with the jetting cycle (Anderson andDemont, 2000; Bartol et al., 2001), for simplicity we will assumethat it remains constant. Therefore, the fiber orientation whichmaximizes qV1/qC1 will also maximize the thrust capacity ofthe squid for any given muscle contraction. It should be notedthat maximum qV1/qC1 represents the maximum change in theinner tunic cylinder volume, V1, for a unit differential changein the inner tunic circumference, C1. This partial derivative isdefined here as a function of the inner tunic fiber angle, y1, byuse of the chain rule:

@V1

@C1¼ @V1

@u1

@u1@C1

/ tan u1 3 cos2 u1 � 1� �

where@V1

@u1/ sin u1 3 cos2 u1 � 1

� �

and@u1@C

/ 1

cos u1

; ð5Þ

where constants have been omitted since we only seek to opti-mize with respect to y1, and are somewhat indifferent to theexact value of qV1/qC1 (i.e., the angle which maximizes the vol-

ume derivative will be the optimal tunic fiber angle because itresults in the largest jet volume for some small contraction ofthe circumferential muscles, but the actual jet volume for agiven contraction is less important).

Maximizing Energy Storage

During slow swimming, the power stroke comes from con-tracting the circumferential muscles that ring the mantle andcontribute the bulk of its mass. The inhalant phase is poweredmainly by releasing elastic energy stored during the contractionphase. There is also a set of radial muscles that extend betweenthe inner and outer tunics (Fig. 2); a contraction of thesemuscles will thin out the mantle layer causing its circumferenceto re-expand. The IM collagen fibers IM-1 and IM-2 are pre-dicted in some studies to store the necessary mechanical energywith an efficiency approaching 75% (Gosline and Shadwick,1983). This restoring mechanism allows the mantle compositionto heavily favor the circumferential muscles, with a small num-ber of radial muscles accounting for energy losses and providingpower for the hyperinflation, required for escape jetting andlarge amplitude ventilation. This arrangement gives the squid alarger range of jetting capabilities, as such a large portion ofthe mantle structure is composed of circumferential musclesused actively during jetting.

To model the energy storage process, we investigated thestress–strain dynamics in the mantle structure. We modeledthe mantle as a tube defined by an inner and outer helical shell(the inner and outer tunics), whereby the geometry of each shellis defined by Eqs. 1 and 3, and depicted in Figure 1. The mantlegeometry can be explicitly defined in terms of shell geometries.The mantle volume is,

Vm ¼ p4

D22L

22 �D2

1L21

� �þ p6

D22 �D2D1 þD2

1

� � ¼ f u1; u2ð Þ ð6Þ

Here, D1, L1, D2, L2 are the diameter and length of the innerand outer tunics, respectively, which can be defined in terms ofthe inner and outer tunic fiber angles, y1 and y2, and fiberlengths, s1 and s2, as described in the previous section (weassumed that the inner and outer tunic fibers have the sameangle at rest y1 5 y2). Here, again, the tunic fiber lengths areconsidered to remain constant during the mantle contraction,which means that the tunic fiber angles must change to allowfor any change in tunic length and diameter. Therefore, a defor-mation of the tunic will be modeled by a small shift in the tunicfiber angle, defined as a. It should be noted that a shift in fiberangle will result in coupled changes in volume, length, and di-ameter. As the jet volume will be equal to the change in inter-nal volume, the shift in inner tunic fiber angle, a1, can be deter-mined if the jet volume, initial tunic fiber angle, and initialinner tunic volume are known (i.e., Vj 5 V (y1 1 a1)–V(y1)where V is the volume defined by (Eq. 3) and Vj is the jetvolume). Thus, the shift in the inner tunic fiber angle is calcu-lated as the value which achieves the desired jet volume. Itshould be noted that the actual jet volume, ejected duringswimming, has been minimally studied. Most experiments relyon indirect measurements based on wet and dry weights ofdeceased specimens (Trueman and Packard, 1968; O’Dor andWebber, 1991). The study by Thompson and Kier (2001b) meas-ured the mantle cavity volume more accurately by weighinganesthetized squid with both empty and full mantle cavity. Thismethod should give an appropriate upper bound for the ratiobetween the jet volume and the total volume, but does notaddress the possibility that certain swimming behaviors onlyeject a portion of the fluid in the mantle cavity. This uncer-tainty will be discussed later in the Results section. Andersonand Demont (2000), approximated the jetting volume duringswimming by determining the squid two-dimensional (2D) pro-file in the sagittal plane and interpolating the total squid vol-ume assuming perfect axial symmetry. However, this approachcompletely ignores any oblateness or nonuniformity which

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might arise during swimming. It has also been qualitativelyobserved that the paralarvae (early development stages) hold aproportionally greater volume of water in their cavities than ju-venile and adult squid (Gilly et al., 1991; Preuss et al., 1997).To calculate the shift in the outer tunic fiber angle, a2, we

assumed that the mantle volume remained constant during con-traction (constant muscle tissue density). The shift in the innertunic fiber angle is directly correlated to the change in volumerequired for jetting. This is coupled to a contraction of the di-ameter and elongation of the length. As a result, the outer tunicmust experience a corrective shift in fiber angle which pre-serves the mantle volume. This shift can be determined by set-ting the initial mantle volume equal to the final mantle volume,

f u1 þ a1; u2 þ a2ð Þ ¼ f u1; u2ð Þ; ð7Þ

where f is the mantle volume function defined in (Eq. 6). a2 cannow be calculated from (Eq. 7) as y1, y2, and a1 are all known.Thus, the geometry of the entire mantle can be determinedbefore and after contraction. The change in geometry will beused to determine the mantle strain characteristics (which willnot be uniform).The energy stored in the mantle structure is directly related

to the strain distribution. Similar to a spring system, theenergy stored in the collagen fibers is equal to the integral ofthe stress (force) applied during stretching over the distance(Pilkey and Pilkey, 1974). Furthermore, the stress applied to amaterial is intrinsically related, by the elastic properties of thematerial, to the strain (stretching) it experiences. The strainexperienced throughout the mantle structure is modeled accord-ing to the change in geometry experienced during contraction,and the strain experienced in the fibers themselves is calculatedaccording to their orientation in the mantle. The axial symme-try of the mantle model allows us to define the 3D strain in cy-lindrical coordinates. In general, the contraction of the mantle’scircumferential muscles not only causes the tunic cylinders todecrease in circumference and volume, but also causes the man-tle to increase in length and thickness. Thus, the strain in theradial and longitudinal directions will be positive, but the strainin the tangential direction (hoop strain) will be negative duringcontraction. We analyzed the orientation of the IM-1 fibers inthe sagittal plane since this involves the radial and longitudinalcomponents of strain, which are both positive.Consider a longitudinal slice through the top of the mantle in

the sagittal plane. Figure 3 shows the strain orientation andprojection of the IM-1 fibers onto this plane. According to theoriginal model construction, the diameter of each tunic isassumed to be constant along its length. This means that thethickness of the mantle will increase uniformly throughout themantle during contraction. Consequently, the radial componentof strain throughout the section will be constant, ey 5 (hf2h0)/h0, where h 5 (D22D1)/2 is the thickness of the mantle, and

the subscripts 0 and f refer to the initial and final states of themantle (before and after contraction), respectively. The lateralstrain is slightly more complicated. Both tunics experience acontraction, which results in elongation. However, the amountsby which they contract are not equal (a1 = a2), so their elonga-tions will not be strictly equal either. The lateral strain of eachtunic can be determined from the length deformation,ei ¼ Lif�Li0

Li0, where ei is the tunic strain, and the subscript i can

take a value of either 1 or 2 and refers to either the inner orouter tunic, respectively. Assuming that the material on thesurface of the tunic experiences the same strain as the tunicitself, and a linear strain distribution, the lateral strain at anylocation in the section is el(y) 5 e2 1 (e12e2)y/hf, where y is thedistance from the inner tunic in the radial direction, and hf isthe final mantle thickness. This gives a complete strain distri-bution in the longitudinal and radial directions, which allowsus to define the total strain imposed on a collagen fiber lying inthis section.

The stress–strain relationship for collagen fibers is onlydefined in the direction of the fibers’ primary axis, as collagenfibers only support tensile loads, and the energy stored in agiven fiber is determined purely by the strain in the direction ofthat fiber. For a fiber of length b, which is oriented at the IM-1sagittal angle b with respect to the longitudinal axis, the nor-mal strain can be calculated as,

efiber ¼ 1

b

Z b

0

e‘ y gð Þð Þ cosbþ ey sinb� �

dg ¼ e1 þ e22

cosbþ ey sinb ;

ð8Þ

where g is a variable which describes position along the lengthof the fiber (see Fig. 3). Given the final strain in a single fiberas defined by (Eq. 8), the energy stored in that fiber is definedby a simple integral equation

Efiber ¼Z df

0

FðeðdÞÞ dd ¼ bAfiber

Z efiber

0

rðeÞ de ð9Þ

In this equation, d is the change in fiber length, df is the totalchange after contraction, Afiber is the cross-sectional area of thefiber, F is the stretching force acting on the fiber (tension), andr is the stress of the fiber which is a function of the strain. Gos-line and Shadwick examined the stress–strain relationship forthe mantle tissue of Loligo opalescens (Gosline and Shadwick,1983; Fig. 7). A section of the mantle tissue was compressed inthe circumferential direction to mimic natural muscle contrac-tion, and the resulting reaction forces were recorded. The man-tle tissue was determined to be relatively stiff with an elasticmodulus of 2 3 106 Nm22. Unfortunately, these findings onlygive the bulk material properties rather than the elastic modu-lus of the collagen fibers themselves, which is the relationshiprequired for our potential energy model (Eq. 9). To the authors’knowledge, there are no studies which present the elastic prop-erties of individual IM fibers; however, Gosline and Shadwick(1985) performed tensile testing on thin isolated sheets of tunicfibers. As the fibers in the tunic are at very acute angles, thisstress–strain relationship should be considered a decent approx-imation for the stress–strain relationship of individual IMfibers, and has been recreated in Figure 4. These experimentsindicated that collagen fibers exhibit a parabolic stress–strainrelationship in the low-strain regime (‘‘toe’’ region), but theslope quickly becomes close to linear and maintains a linearproportionality for the majority of the strain domain. Beforereaching the critical breaking stress, there is a very smallregion where the stress–strain relationship asymptotically pla-teaus, which is a typical behavior for elastic fibers whichdeform plastically at high strains, but the transition in collagenis very sharp. Therefore, we modeled the stress–strain relation-ship as,

Fig. 3. Strain model construction in the sagittal plane. Here,e is the mantle strain (subscripts indicate direction of strain), yis the radial distance from the inner tunic, g is the length inthe direction of the IM-1 fiber.

590 M. KRIEG AND K. MOHSENI

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r ¼c1e2 :e � ea

r1 þ Ee :ea � e � ebc2 eþ c3ð Þ1=n : eb � e � em

0 : em < e

8>><>>:

;

where

ea ¼ 0:04eb ¼ 0:13em ¼ 0:14

;

r1 ¼ 5MPar1 ¼ 48MParm ¼ 49MPaE ¼ 540MPa

;

c1 ¼ r1=e2a

c2 ¼ r2m�r2

2

em�eb

8: 9;1=n

c3 ¼ rm=c2ð Þ

ð10Þ

where ea, eb, and em are the critical strains in the stress/strainprofile corresponding to the beginning and end of the linearregion and the critical failure strain, respectively. E is the mod-ulus of elasticity in the linear range (540 MPa), r1, r2, and rm

are the stress values corresponding to strains ea, eb, and em.The values used for all these coefficients were estimated fromGosline and Shadwick (1985, Fig. 5).This form was chosen because it closely matches the shape

of empirical curves obtained for both invertebrate and mam-malian collagen (Rigby et al., 1959; Viidik, 1972; Wainrightet al., 1976; Gosline and Shadwick, 1983). However, it ishypothesized that the ‘‘toe’’ region is due to the fact that thecollagen fibers are still not perfectly aligned with the straindirection, and this is in essence a straightening process.Therefore, the stress–strain relationship was also modeled asa perfect spring with the modulus of elasticity equal to that ofthe linear region; however, this had very little effect on opti-mal fiber angles predicted by the model, which is mostly sensi-tive to the critical stress/strain values, rather than the profilein the low-strain region.Now all the relationships in the mantle model have been

defined so that the total energy stored in a single fiber is foundby numerically approximating the integral of Eq. 9, using thestress relationship defined by Eq. 10.To determine an actual value for total energy storage in the

mantle structure several constraints must be imposed. The ini-tial geometry of the mantle was defined according to the length,diameter, and thickness of S. lessoniana as were reported in(Thompson and Kier, 2001a). We also assumed that the innerand outer tunics start at the same length which gives a rela-tionship between the fiber lengths of each tunic. The predictions

of this model under these constraints will be compared withobserved data in the Results section.

RESULTSTunic Fiber Orientation

To maximize thrust production, the fiber angleshould be aligned so that the ejected volume fluxis maximized rather than the total volume. Therate at which fluid is ejected should be consideredproportional to the rate of change in the total vol-ume with respect to a change in the circumference,as is derived in Eq. 5.

Figure 5 shows the instantaneous change intunic cylinder volume with respect to a differentialchange in circumference, as a function of the ini-tial fiber angle. It can be seen from this figurethat, for a small contraction of the circumferentialring muscles, the squid will expel a maximal jet ifthe initial fiber angle is near 318. This jet willresult in maximum thrust assuming that the ringmuscles have a constant rate of contraction (Eq.4). This angle approaches the actual orientation oftunic fibers measured by Ward and Wainwright(1972).

Intramuscular Fiber Orientation

The squid mantle is oriented so that the circum-ferential ring muscles (which constitute the bulkof the mantle muscle tissue) provide sufficientcompression forces during the jetting phase. How-ever, the refilling phase is driven by sparselypacked radial muscles as well as a release of elas-tic potential energy stored in the deformed mantlefiber structure.

Fig. 5. Differential change in cylinder volume with respectto a contraction of circumference. [Color figure can be viewed inthe online issue, which is available at wileyonlinelibrary.com.]

Fig. 4. Stress versus strain relationship used in the model[estimated from Gosline and Shadwick (1985) for a sheet oftunic collagen fibers]. [Color figure can be viewed in the onlineissue, which is available at wileyonlinelibrary.com.]

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There is an obvious dichotomy between thetightly packed collagen fibers in the tunics and thescarce IM collagen fibers. The tunic fibers arewound in layers of alternating orientation to forma more or less uniform tube of collagen. The IMfibers, by contrast, are arranged more sparselythroughout the muscle tissue, accounting for 0.1–7% of the total mantle volume, depending on theage of the squid (Thompson and Kier, 2001a). Theabundance of collagen fibers in the tunics suggestthat these self-reinforced fibers experience mini-mal stretching compared to the IM fibers. As dis-cussed in the subsection Maximizing Energy Stor-age, energy is directly related to the deformationof the fibers. The small deformation of tunic fibersresults in a low capacity for energy storage, indi-cating that the tunic fibers primarily serve a struc-tural purpose. In contrast, the high deformation ofthe IM-1 fibers suggests that they serve as the pri-mary energy storage devices.

Figure 6 shows the normalized energy storagecapacity of the IM-1 fibers as a function of the sag-ittal plane orientation angle b, as was modeled inthe subsection Maximizing Energy Storage. Thestorage capacity was normalized by the maximumachievable energy storage over the b distribution.Figure 6a shows the fiber storage capacity versusb for a jet volume ratio of 0.25, and Figure 6b fora jet volume ratio of 0.45. It can be seen that asquid expelling a jet with a low volume ratio willstore a maximum potential energy when the IM-1fibers are oriented with an angle b 5 678. Thepeak in the energy storage capacity curve is veryoblate giving a large range of fiber angles withsimilar energy storage capacity. Conversely, theenergy storage capacity for the squid ejecting a jet

with a larger volume ratio has a very distinct peakat b 5 238. This peak does not actually correspondto an equilibrium balance between axial and radialstrain, but rather is associated with the failurestrain of the collagen fibers; as the fiber angleincreases so does the strain in the fiber until thefailure strain is reached and the fiber is ruptured.

Data reported for S. lessoniana in Thompsonand Kier (2001b) was used to define the initial ge-ometry of the mantle (length, diameter, and thick-ness). This data set was chosen because the man-tle geometry and mantle cavity volume ratio,required for the energy storage model, is presentedfor a large range of squid developmental stages. Inaddition, the IM fiber angles are given in Thomp-son and Kier (2001a) corresponding to a similarsquid population, providing a reference to validatethe model. The values for b over this data set areshown as a vertical band in Figure 6 (bounded oneither side by the maximum and minimumobserved fiber angles). The cavity volume ratio canvary quite drastically throughout ontogeny, and ismore precisely, a maximum bound on the jet vol-ume, and ignores the possibility that during cruis-ing the squid might not eject the entire cavity vol-ume. We used our model to predict optimal fiberangle for the entire range of cavity volume ratiosseen in S. lessoniana using the mantle geometryassociated with that cavity volume ratio, andassuming complete evacuation of the cavity. Wealso calculated the optimal fiber angle for thesame range of jet volume ratios using the mantlegeometry of a single adult squid. The optimal fiberangles determined for both ranges of initial condi-tions are shown in Figure 7. It can be seen thatthe optimal fiber angles for both conditions are

Fig. 6. Energy storage capacity of the mantle structure and IM-1 collagen fiber strain versus fiber angle b. Energy storagecapacity as a function of fiber angle is represented by the solid line, fiber strain is shown by the dash-dotted line, and actual distri-butions of fiber angles are bounded by the vertical band. Energy storage capacity for a cavity volume ratio of 0.25 (a) and cavityvolume ratio of 0.45 (b). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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nearly identical, indicating that the squid mantlegrows such that the relationship between jet vol-ume ratio and mantle strain is preserved.

In addition, the longitudinal strain on the innertunic was calculated for the same set of volumefractions, and the result is also shown in Figure 7.For the entire range of volume fractions, the longi-tudinal tunic strain remains very small <8%; afact which has been observed in previous experi-ments (Packard and Trueman, 1974). As a resultmany models have ignored longitudinal tunicstrain entirely, thus losing knowledge of a keyenergy storage mechanism as will be analyzed inthe Discussion Section.

The model predicts an optimal fiber angle withrespect to jet volume ratio (cavity volume ratio),which can be related to dorsal mantle length (viaThompson and Kier (2001b)). Therefore, we candirectly compare the optimal fiber angle predictedby the model with the actual fiber angles observedin the squid (Thompson and Kier, 2001a) over therange of dorsal mantle lengths reported. The largevariability in cavity volume fraction for a givendorsal mantle length, results in the model predict-ing a similarly large range of optimal fiber anglesfor a given dorsal mantle length. To aid in visual-izing this data, the predicted optimal fiber anglewas averaged for three mantle length regions(hatchling, juvenile 1 and juvenile 2) which arecompared to the actual fiber angle distribution inFigure 8.

DISCUSSION

The use of helically wound high-tensile strengthfibers has been examined in the anatomy of sev-eral invertebrates with respect to spiral orienta-tion angles. Harris and Crofton (1957) first lookedinto the effect of the orientation of reinforcingfibers on the length and volume relationship in

nematodes. This analysis was extended andapplied to both nemerteans and turbellarians(both of which are adept at changing shape) inClark and Cowey (1958), and determined a rela-tionship between volume and fiber angle for agiven length of worm. The volume attains a maxi-mum for a fiber angle near 558. Similarly, thisnominal angle was identified by Harris and Crof-ton (1957) as the angle that would maintain a con-stant worm volume for a small deflection in thefiber orientation angle. Vogel (2003) adapted thisanalysis to squid tunic structures and noted thatactual tunic fiber angles will result in a structurethat decreases volume with decreased diameter,despite an increase in length, and that the squidvolume is maximized at the nominal fiber angle of558. However, as is shown in Figure 5, a tunic fiberangle close to 318 will maximize the jet volumeflux for a given circumferential muscle contraction(directly related to the jetting thrust), which isvery close to actual tunic fiber angles. Figure 5also shows that when the fiber angle is 558, therewill be no change in volume for a small contractionin circumference (or equivalently diameter), asobserved by Harris and Crofton (1957).

Unlike previous studies (Clark and Cowey, 1958;Ward and Wainwright, 1972) which assert that thelow angles of the tunic and IM fibers prevent themantle from changing length, our model incorpo-rates the variation in mantle length during con-traction. Our analysis predicts that IM-1 fibershave an optimal angle in the sagittal plane thatallows for maximum energy storage. In addition,we find that as length, diameter, and volume areintrinsically coupled, a purely constant mantlelength is an overly restrictive assumption and is

Fig. 8. IM-1 sagittal fiber angle, b, throughout ontogeny.Predicted optimal fiber angles shown by large square, diamond,and triangle markers and actual fiber angles marked by star.[Color figure can be viewed in the online issue, which is avail-able at wileyonlinelibrary.com.]

Fig. 7. Optimal IM-1 sagittal fiber angle b as well as innertunic longitudinal stress e1 shown as a function of the volumeratio Vj/V. [Color figure can be viewed in the online issue, whichis available at wileyonlinelibrary.com.]

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not required to achieve maximal jet volume. Fur-thermore, the large aspect ratio of the mantle(being much longer than it is thick), causes a smalldeformation in length to result in substantialpotential energy storage in the longitudinal direc-tion. In fact, the predicted longitudinal strain inthe tunics is quite small, 4% in the outer tunicand 4.8% in the inner tunic, which is within therange of longitudinal strains measured by Packardand Trueman (1974). These longitudinal tunicstrains were determined assuming a volume ratioof 0.45. In the Results Section, we calculated thelongitudinal strain on the inner tunic for severalother volume ratios. The sensitivity of tunic strainto volume ratio was shown in Figure 7. It can beseen that even as the jet volume approaches amaximum value of 0.8, the longitudinal tunicstrain remains below 8%.

The sensitivity of the optimal IM-1 fiber angle bwith respect to cavity volume fraction and jet vol-ume fraction was also shown in Figure 7. Thelarge variation in the optimal fiber angle can beprimarily attributed to the fact that large volumeratio contractions produce critical strain in the IM-1 fibers with larger orientation angles in the sagit-tal plane. In fact, if the fiber is assumed to have aboundless linear stress–strain relationship, theoptimal fiber angle varies by only 38. This meansthat squid can eject several different size jets withsimilar mantle energy storage properties. As waspreviously mentioned, Figure 6a shows that theenergy storage capacity has a rather broad peak(when critical strain is not a factor), meaning thatthere is a large range of fiber angles, b, with favor-able energy storage characteristics. Moreover, eventhe minimum fiber angle observed throughout on-togeny, still has an energy storage capacity close to70% of predicted maximum for low jet volume ra-tio. Therefore, the IM-1 fibers are most likely ori-ented to provide the maximum energy storage,within the limiting physical bounds of the collagenfibers.

The comparison of IM-1 sagittal fiber angles inFigure 8 shows decent agreement between pre-dicted optimal b and actual measured b; but theoptimal energy storage model predicts b moreacute than that observed, for both hatchling andshorter juvenile squid. First, it should be notedthat both of these age groups have the most uncer-tainty in cavity volume ratio, which will certainlycarry over to uncertainty in predicting optimalfiber angles. In addition, the squid mantle is not aperfect cylinder but tapered (like a conical tube),this shape is more pronounced in younger squid,so the cylinder mantle approximation may not beas valid for these young developmental stages.

In the transverse plane, the components of man-tle strain were quite different. The radial compo-nent of strain (through the thickness) was stilldefined by the mantle thickness expansion. How-

ever, the circumferential component (tangent tothe tunic) was negative due to the contraction ofthe circumferential muscles. As fibers can onlystore energy under tension (not compression), theIM-2 fibers would store a maximum amount ofenergy if they were oriented radially (908). Thefact that these fibers are oriented at an anglebetween 508 and 558 suggests that these fibers arenot purely energy storage components, but alsoserve to transmit forces from the discrete radialmuscles to the rest of the mantle.

The various systems of collagen fibers withinsquid mantle tissue form a complex mechanicalsystem. Several studies have observed a nearlyuniversal orientation of these fiber systems acrossseveral species. We have provided a rigid mathe-matical model to analyze the structural mechanicsof the tunic fiber systems, and have determinedthat the tunic fiber’s angle of incidence maximizesthe expelled jet volume for a given contraction ofcircumferential muscles. We have also modeled theenergy storage dynamics of the IM-1 fiber systemin the sagittal plane. It was shown that the orien-tation of these fibers maximizes their energy stor-age capacity, within the physical limitations of thecollagen fibers themselves. In addition, it wasdetermined that previous assumptions about therole of IM-1 fibers in restricting longitudinal defor-mation are not supported by the energy analysis.

ACKNOWLEDGMENTS

The authors thank Connor Fitzhugh for provid-ing several contributions to the article as well asthe reviewers who provided substantial correctionsand identified studies that measured the requiredstress/strain relationship for invertebrate collagenfibers.

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