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Energy 102 (2016) 537e549

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Energy

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Thermodynamic analysis of the squid mantle muscles and giant axonduring slow swimming and jet escape propulsion

Bahar Hazal Yalçınkaya a, Sükrü Erikli b, Burak Arda €Ozilgen c, Ali Bahadır Olcay b,Esra Sorgüven b, *, Mustafa €Ozilgen d

a Department of Genetics and Bioengineering, Yeditepe University, Istanbul, Turkeyb Department of Mechanical Engineering, Yeditepe University, Istanbul, Turkeyc The UC Berkeley e UCSF Graduate Program in Bioengineering, USAd Department of Food Engineering, Yeditepe University, Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 10 October 2015Received in revised form1 February 2016Accepted 13 February 2016Available online xxx

Keywords:SquidJet escapeSlow swimmingThermodynamic efficiencyMuscleGiant axon

* Corresponding author.E-mail address: sorguven@yeditepe.edu.tr (E. Sorg

http://dx.doi.org/10.1016/j.energy.2016.02.0770360-5442/© 2016 Elsevier Ltd. All rights reserved.

a b s t r a c t

Squids have two substantially different types of muscle fibers: superficial mitochondria rich fibers, whichperform aerobic respiration during slow swimming, and central mitochondria poor fibers, which performanaerobic respiration during jet escape. A detailed thermodynamic analysis shows that during slowswimming, 3.82 J/(kg s) of chemical exergy is consumed, and a total muscle work of 0.28 J/(kg s) isproduced. 0.27 J/(kg s) of this is produced by the fin to generate lift, and the rest by the mantle volumecontraction. During the jet escape at a speed of 3 mantle length/s, squid consumes an exergy of 9.97 J/(kg s) and produces a muscle work of 0.16 J/(kg s). Exergy destruction rates during slow swimming andjet escape modes are 3.54 and 9.81 J/(kg s), respectively. Exergy destroyed because of the action potentialpropagation in the squid giant axon is calculated as 0.03 and 0.10 J/(kg s) for the slow and fast swimmingmodes, respectively.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

About 600,000 people die of a heart disease only in the UnitedStates every year [1]. One of themajor causes of this is the decline ofthe muscle efficiency [2e4]. People lose approximately 20e30% oftheir muscles between the ages of 20 and 80 [5,6]. The age relatedloss of muscles is accompanied with the loss of efficiency anddepend on mitochondrial functioning and cellular energetics [6].Estimating the theoretical maximummuscle efficiency may help toassess a patient's actual muscle performance, the degree of muscleloss and how close they are to experience a heart problem, if thein vivo muscle efficiency can be determined via appropriate mea-surements. Muscle work efficiency has been a topic of interest tomuscle physiologists and explored by numerous experimentalists[4,7e11]. Studies on muscle mechanics from a thermodynamicalpoint of view made considerable achievement in the analysis ofmuscle heat and work production, muscle efficiency and entropygeneration [12,13]. Efficiency of conversion of chemical energy into

üven).

work in muscles is not constant and may change substantiallydepending on physiological conditions. There are still open ques-tions pertaining to why different energy metabolism pathways areemployed, how the chosen energetic pathway affects aging, howmuscle efficiency can be increased to delay aging, and howknowledge gained from biological systems can be harnessed todevelop mechanical systems such as bio-actuators. Assessing thevalidity of thermodynamical models of human muscles for otherspecies is required to make them rigorous with regards to universalapplications. Squid muscles, which have been widely studied, aresubstantially different when compared to human muscles and thuspresent a good opportunity to reassess the thermodynamicalmodels. Furthermore, selecting the squid as the species of interestallows for the calculation of muscle work with both force emuscleelongation data, pressure e volume work and its comparison withthe metabolic energy utilized to achieve it.

Squids are invertebrates; they belong to the cephalopod classand have bilateral body symmetry. Squid anatomy consists of oneprominent head, eight arms, two tentacles and a mantle. Themantle consists of muscles in a conical shape, encloses vital organsand the mantle cavity. Openings on each side of the prominenthead are used to draw water into the mantle cavity [14], and then

Nomenclature

B empirical constant in the extended Debye-Hückelequation (kg1/2 mol�1/2)

c concentration (mM)e energy (kJ/kg)ex specific exergy (kJ/kg)F force (N)g Gibbs free energy (J/mol)h enthalpy (J/mol)I ionic strength (molL�1)L dorsal mantle length (mm)m mass (kg)n number of atoms (�)P pressure (atm)r reaction rate (mmol g�1 s�1)R ideal gas constant (JK�1 mol�1)T temperature (K)V volume (Liter)

Q heat (kJ)W work (kJ)x displacement (m)z charge number of the ion (�)a Debye-Hückel constant (kg1/2 mol�1/2)h efficiency (�)

Subscripts0 restricted dead stateacc accumulationch chemicalgen generationin inletout outlet

AbbreviationsSUC succinateOAA oxaloacetatePYR pyruvate

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549538

water is ejected from the mantle cavity via contractions of mantlemuscles at different speeds to provide the squid with the necessarymomentum for propulsion (Fig. 1). Muscles of the mantle are ar-ranged in radial and circular orientation [15] (Fig. 2). The radialmuscles are engaged to draw water into the mantle cavity [16].Circular muscles occupy about 90% of the total muscle volume andprovide power for contraction [14]. The circular muscle fibers,which are located along the inner and outer surfaces of the mantle,have a large number of mitochondria. These SMR (superficialmitochondria-rich) fibers are used during the steady state slowswimming mode, when there is sufficient oxygen rich blood supply[17e20]. Since the SMR fibers are located on the inner and outermantle surfaces they uptake oxygen from water through theextremely thin skin. In the slow swimming mode continuousmovement of water and the low rate of the ATP (adenosine triphosphate) consumption reduces the probability of the formationof oxygen poor boundary conditions [20]. In the slow swimmingmode the need for the ATP to support the muscle work is low,therefore the squid is capable of carrying out aerobic catabolism of

Fig. 1. Schematic drawing of the squid to describe locations of th

the nutrients to produce ATP (adenosine tri phosphate) to achievethe muscle contraction (Fig. 3) [20,21]. Calculations in this paperare based on the biological properties of Atlantic brief squid (Lol-liguncula brevis).

It is hypothesized that the CMP fibers provide the muscle powerin cases of emergency such as during hunting or escape, when thesquid has to swim very fast [14,17e19]. The volume occupied by theCMP (centrally-located mitochondria-poor fibers) is about 20 timeslarger than the SMR fibers (Fig. 4) [22]. During the jet escape, theCMP fibers contract and reduce the mantle cavity volume, andwater is expelled to create high-momentum jet flow which even-tually forms a vortex ring. This momentum transfer accelerates thesquid in the opposite direction. Maximum pressure inside themantle cavity changes with the size and swimming speed of thesquid and may reach up to 25 kPa (gauge). During the contraction,the inner pressure of the cavity decreases by approximately 1 kPabelow the pressure of the surroundings, so refilling of water beginsimmediately after the minimum volume is reached. In the center ofthe muscle tissue, the fibers contain only a few mitochondria. The

e muscle fibers, mantle cavity, arms, funnel and the vortex.

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549 539

jet escape is fueled via the anaerobic metabolism [23]. Physiologicalevidence that supports this hypothesis is the accumulation of

DgTf ;i ¼TT0

Dg0f ;_I

þ�1� T

T0

�Dh0f ;i �

RTaðTÞI1=21þ BI1=2

�z2j � nH;j

�� nH;jDg

0f ;Hþ þ 2:303nH;JRTpH

DhTf ;i ¼ Dh0f ;i þRTbI1=2

1þ BI1=2

�z2j � nH;j

� (1)

octopine and succinate, byproducts of the anaerobic catabolism ofcarbohydrate and amino acids [24] in the muscle tissue during fastmotion [25].

The squid giant axon controls the contractions of the mantlemuscles. Since the large size of the giant axon allows easy experi-mentation, it has been a subject of academic research since 1930's[16,26]. Experimental studies measured the role of both the giantaxon and the non-giant axons in both slow swimming and jetescape [27e29]. Hodgkin and Huxley developed a model todescribe the action potential initiation and propagation in the squidgiant axon [30e33]. Action potential propagation is an energyconsuming process. During this, Kþ/Naþ-ATPase pumps 3 Naþ outof the cell and 2 Kþ into the cell (against the ionic concentrationgradient) and 1 mol ATP is hydrolyzed to ADP (adenosine diphos-phate) [34]. The main energy consumption of a mammalian brain isdue to this electrical transmission [35,36]. In the overall usage, theaction potential is the second largest energy consumer [37].

In this paper, both the slow and fast modes of swimming of asquid are analyzed thermodynamically. The thermodynamicalmodel involves the SMR and CMP fibers, as well as the giant axon.The metabolic pathways for the slow and jet escape modes aredescribed as elementary reactions occurring within the systemboundaries. Heat and work production, as well as entropy gener-ation and exergy destruction are calculated for both modes ofmotion.

2. Calculation of the thermodynamical properties

The standard Gibbs energy DgTf ;i and the enthalpy of formationDhTf ;i of the chemical species undergoing biochemical reactions arecalculated based on the method developed by Alberty [38] by

Fig. 2. Muscles of the squid as visualized under the microscope.

following the same procedure as Genc et al. [39] with the followingequations:

At a given pH some biochemical species (for example ATP) canbe at different valence states like ATP4�, ATP3�, and ATP2�, Thestandard Gibbs free energyDGT

f ;i of such species is calculated basedon the sum of the Gibbs free energies of each component:

DgTf ;i ¼ �RT lnX

exp

�DgTf ;iRT

!(2)

The exergy of chemical species is the sum of standard chemicalexergies of its elements, transformed Gibbs energy of formationand the exergy change due to its concentration in the cell atphysiological conditions:

Exj ¼ DgTf ;i þX

nkExch;k þ RT0 ln�cj�

(3)

The biophysical conditions in a squid muscle cell is pH ¼ 7,T¼ 298.15 K and I¼ 0.25 M. Enthalpy, Gibbs free energy and exergyof the metabolites under these conditions was computed withMathematica and listed in Table 1. Thermodynamical data for twometabolites, arginine and octopine could not be found in theliterature, therefore estimated with the group contributionmethod.

3. Muscle models

The SMR and CMP fiber models are described in Figs. 2 and 3. Inboth models nutrient is catabolized and ATP is produced, and thenutilized to contract the muscle fibers. Water is ejected out of themantle and the squid gains momentum in the swimming directionwith the construction of the muscles. For both systems, thermo-dynamic analysis is performed by analytically by solving thefollowing mass, energy and exergy equations:

dmdt

¼X

ð _mÞin �X

ð _mÞout (4)

dEdt

¼X

ð _mhÞin �X

ð _mhÞout þ _Q � _W (5)

dExdt

¼X

ð _mexÞin �X

ð _mexÞout þ _Q�1� T0

T

�� _W þ P0

dVdt

� Ex�

dstrctn

(6)

Actual consequent contractions of a squid may exhibit smallvariations among individual squids in duration and amplitude. Inthis study squid muscle motion is regarded as periodical contrac-tions with the same period and intensity. As a result of thisassumption the governing equations may be simplified by takingthe cyclic integral over one period:

Fig. 3. Schematic representation of SMR fiber model.

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549540

Idmdt

¼I X

ð _mÞin �I X

ð _mÞout (7)

IdEdt

¼I X

ð _mhÞin �I X

ð _mhÞout þI

_Q �I

_W (8)

IdExdt

¼I X

ð _mexÞin �I X

ð _mexÞout þI

_Q�1� T0

T

�

�I

_W þI

P0dVdt

�I

Ex�

dstrctn (9)

Fig. 4. Schematic representation of CMP fiber model. Concentration changes of the metabolmantle length/sec.

During steady-state slow swimming, SMR fibers (Fig. 3) areengaged and ATP is produced via the aerobic pathway. Glucose isoxidized to the final products CO2 and H2O, which are transferredout of the intracellular region; there is no accumulation of massinside the cell during the steady-state slow swimming. Hence theaccumulation terms, (i.e. the left hand side of the governingequations) become zero. The cyclic integral of the surroundingswork

HP0dVdt becomes also zero, since the volume change occurs

periodically. Furthermore,H _Q�1� T0

T

�term becomes zero, since

the boundary temperature remains constant and equal to the deadstate temperature throughout the whole cycle. Table 2 lists the

ites are given as measured by Finke et al. [2] during 15 min of jet escape at a speed of 3

Table 1Concentrations and thermodynamical properties of the metabolites under the physiological conditions (i.e. I ¼ 0.25 M, T ¼ 298.15 K and pH ¼ 7) in the squid mantle muscles.

Metabolite Concentration(mM)

Dh @ I ¼ 0.25 M,pH ¼ 7, T ¼ 298.15 K(kJ/mol)

Dg @ I ¼ 0.25 M,pH ¼ 7, T ¼ 298.15 K(kJ/mol)

Exch(kJ/mol)

RTln(c)(kJ/mol)

Ex(kJ/mol)

Glu 0.012 �1267.12 �426.71 2473.47 �10.96 2035.80G6P 0.100 �2279.30 �1318.92 3343.03 �5.71 2018.40ATP 0.360 �3616.92 �2292.50 6720.93 �2.53 4425.90ADP 0.170 �2627.24 �1424.70 5851.38 �4.39 4422.29NAD 0.0001 �41.83 1120.09 10,372.87 �22.83 11,470.13NADH 0.00001 �10.26 1059.11 10,372.87 �28.54 11,403.44Pi 1.890 �1299.36 �1059.49 871.54 1.58 �186.37H2O 1.000 �286.75 �155.66 1.99 0.00 �153.67PYR 0.00001 �597.04 �350.79 1236.74 �28.54 857.41ARG 29.370 �277.38 25.20 2466.91 8.38 2500.49OCT 0.430 �608.52 �272.85 3701.63 �2.09 3426.69SUC 0.065 �908.70 �530.64 1648.94 �6.78 1111.52CO2 0.026 �692.86 �547.10 414.32 �9.05 �141.83O2 0.960 �11.70 16.40 3.97 �0.10 20.27

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549 541

glucose oxidization and ATP formation reactions, and the changesof their enthalpy, Dhrxn, and Gibbs free energy, Dgrxn. These resultsshow that if the squid muscle would be ideal, i.e. if all the reactionsand fiber motions would occur without any losses (Exdstrctn ¼ 0),then a maximum work of 3930.39 kJ would be produced byconsuming one mole of glucose. A heat removal of 609.95 kJ/mol ofglucose would accompany this work production.

Bartol et al. [18] measured the O2 consumption by squid in slowswimming mode between 14.41 and 40.71 mmol O2/(h g). The mostimportant parameters that effect the O2 consumption rate is thebody size and the swimming velocity. A 65 g squid swimming at 0.5mantle length per second, consumes about 21 mmol O2/(h g). Underthese conditions, glucose oxidization releases an enthalpy of 4.41 J/(kg s) and an exergy of 3.82 J/(kg s). Aerobic metabolism produces30e38 mol of ATP/mol of glucose. The case of 30 mol, e.g.,0.029 mol/(kg s), of ATP production is capable of doing 1.06 J/(kg s)of work, when all of the ATP is hydrolyzed. This means that 28% ofthe available chemical exergy is used to produce ATP. Fig. 5 showsthe variation of the Dgrxn of the reactions SMR1 and SMR2depending on the oxygen consumption rate.

During jet escape, the metabolic pathway switches to anaerobicreactions and CMP fibers are engaged. Finke et al. (1996) measuredthe change of metabolic concentrations during jet escape [40].Their findings show that the squids switch to the anaerobicmetabolism at speeds higher than ca. 2 mantle length s�1 and

Table 2Metabolic reactions occurring during the slow swimming mode in the SMR fibers and d

Slow swimming mode:➢ ATP produced through the aerobic energy metabolism.➢ SMR fibers are engaged.

SMR1 Glu þ 6 O2 0 6CO2 þ 6 H2OSMR2 ADP þ Pi 0 ATP þ H2OJet escape mode:➢ ATP produced both through the aerobic and anaerobic energy metabolisms.➢ Mostly CMP fibers are engaged.

CMP1 Glu þ ATP 0 G6P þ ADPCMP2 Glu þ 2 NAD þ 2 ADP þ 2 Pi 0 2 PYR þ 2 NADH þ 2ATP þ 2 H2OCMP3 PYR þ ARG þ NADH 0 OCT þ NAD þ H2OCMP4 2 PYR þ 2 H2O þ 3 NAD 0 SUC þ 2 CO2 þ 3 NADH

a Reaction rate depends on various parameters, such as the swimming velocity, mass offor the slow swimmingmode has been performed assuming a swimming velocity of 0.5 mjet escape, swimming velocity is taken as 3 mantle length/s and the O2 consumption as 1(1996) [2].

b Number of ATP produced through an aerobic pathway varies between 30 and 38 mo

accumulate octopine, glucose-6-phosphate and succinate. In thisstudy, we model the biochemical processes during jet escape withthe reactions listed in Table 2.

It is worth noting that, these reaction scheme represent asimplified model for the anaerobic metabolism. In the actualmetabolism hundreds of reactions occur simultaneously. Forexample, during the succinate production, first OAA (oxaloacetate)is formed from PYR (pyruvate), during which reaction one mole ofATP is consumed.

PYR þ CO2 0 OAA þ ADP þ Pi

ATP þ H2O 0 ADP þ Pi

Later, pyruvate and OAA react to form succinate, during whichone mole of ATP is produced.

PYR þ OAA þ 2 H2O 0 SUC þ 3 CO2

3 NAD 0 3 NADH

ADP þ Pi 0 ATP þ H2O

These 5 reactions are summarized as CMP4 in Table 2. Thepurpose of such a model is to describe the whole pathway with as

uring the jet escape in the CMP fibers.

Reaction rate (mmol/(gs)) Dhrxn (kJ/mol) Dgrxn (kJ/mol)

6.48 � 10�4 e 1.85 � 10�3a �4540.34 �3930.391.94 � 10�2 e 7.04 � 10�2b �22.93 �36.31

Reaction rate (mmol/(gs)) Dhrxn (kJ/mol) Dgrxn (kJ/mol)

1.00 � 10�4 �22.50 �21.006.14 � 10�3 182.04 �381.731.02 � 10�2 �52.42 �18.191.06 � 10�3 �432.13 �779.68

the squid, tail- or arms-first positioning, angle of attack [1]. In this study, calculationsantle length/s and a corresponding O2 consumption of 9.72� 10�4 mmol/(gs). For the6.67 � 10�4 mmol/(gs). These values correspond to the measurements of Finke et al.

l of ATP per mol of glucose.

Fig. 5. Variation of the Dgrxn of the reactions SMR1 and SMR2 depending on the ox-ygen consumption rate.

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549542

few apparent reactions as possible. The model which is presentedin Table 2 is simple enough for through thermodynamic analysis,but also presents sufficient detail to account for the metaboliteconcentration changes measured in vitro [40].

Experimental data shows that squids performing jet escape for15 min accumulate 0.09 mmol G6P, 0.95 mmol succinate and9.16 mmol octopine per g of body mass [40]. Depending on thesemeasurements, the rates of the reactions are calculated and listedin Table 2. Since mass accumulation occurs during jet escape

H dEdt

andH dEx

dt terms are non-zero, the cyclic integral of the surroundings

workHP0dVdt and

H _Q�1� T0

T

�terms are zero, as in the case of the

slow-swimming mode. Multiplying the Dgrxn values listed inTable 2 with the reaction rates, shows us that 3.35 J/(kg s) ofchemical exergy is released during jet escape. In other words, themaximum reversible work that the squid can produce during jetescape is 3.35 J/(kg s). Finke et al. [40] measured a decrease of the

cellular ATP concentration (dcATPdt ¼ �0:002 mmolgs ), which means that

during jet escape the amount of ATP produced via anaerobicrespiration does not meet the demand for the muscular ATP con-sumption. The rate of the ATP consumption is then calculated as:

rATP ¼ 2rCMP2 � rCMP1 �dcATPdt

¼ 14:18mmolgs

(10)

Hydrolyzing this ATP releases an exergy of 0.51 J/(kg s), which isused up to perform work and overcome irreversibilities.

4. Calculation of the work transferred to water

During swimming, a squid has to generate force in both verticaland horizontal directions, since squids are negatively buoyant. Athigh speeds, lift is generated via the kinetic energy difference onthe upper and lower surfaces of the squid. However, at lowswimming speeds, lift generation becomes a major issue and mustbe generated via the movement of fin and arms. Experimentalobservations state that at low speeds about 85% of the total musclework is consumed for lift generation [18,44].

Squid's main mechanism to produce horizontal force is to ejectwater with a high momentum in horizontal direction. The mantlecavity of a squid seems like a pressure vessel since the pressure inthe mantle cavity can reach up to 25 kPa above the atmosphericpressure [14]. Water filled in the mantle cavity is pressurized andejected through squid's funnel. As the squid contracts the mantlecavity, both pressure and volume of the cavity change. A work of!PdV is applied to the water inside the mantle cavity.

Fig. 6a and b shows how diameter and thickness of the mantlewall vary for a squid. It was realized that initial diameter of themantle cavity in jet escape mode was larger than the initial diam-eter of the mantle cavity in slow swimming mode. However, thefinal diameter of the mantle cavity in jet escape mode was smallerthan the final diameter of the mantle cavity in slow swimmingmode (Fig. 6a). This implies that a squid inflates her mantle cavitymore in jet escape mode than slow swimming mode to draw largeramount of water to be used for ejection later. Then the mantlecavity diameter in jet escape mode drops about 40% of its initialvalue implying that the amount of water driven in this mode ap-pears to bemaximized. Furthermore, it was noticed that the mantlethickness in jet escape mode is nearly 35% more than the mantlethickness in slow swimming mode because jet escape mode re-quires larger mantle wall thickness to sustain build up pressureinside the mantle cavity (Fig. 6b). The mantle cavity dimensions forfast and slow swimming modes are given in Table 3. Based on thesedata an in-house code is developed to calculate the work trans-ferred to the water inside the mantle cavity. Squid mantle cavityvolume versus time is plotted in Fig. 7a for both fast and slowswimmingmodes. The volume of mantle cavity is V¼ pr2L, where, ris the radius of the mantle cavity at any location and L is being theany length along the squid's dorsal mantle length. Volume changeof the squid's mantle cavity is calculated with the formula belowsince radius (r) varies along the dorsal mantle length,

V ¼Z

phsin�xp2L

�rmax

i2dx (11)

At the end of water ejection the volume decreases by about 35%in slow swimming and nearly 65% in jet escape. Besides, the initialvolume of the jet escape mode is approximately 20% more than theinitial volume of the slow swimming mode, and the duration ofejection is about 40% longer in slow swimming than that of the jetescape. This implies that the momentum of the jet escape, thus theapplied horizontal force, is significantly greater than that of slowswimming. Variation of the pressure inside the mantle cavity withtime during ejection of water is plotted in Fig. 7b, which shows thatpressure inside the mantle cavity can reach up to ten times highervalue during jet escape than that attained during slow swimming.Pressure versus the volume of the mantle cavity is plotted in Fig. 7c.Work transferred to water inside the cavity is calculated for bothfast and slow swimming modes by calculating numerically the areaunder the PeV curve of Fig. 7c with the integral !PdV, as1.6 � 10�4 J/(s g body mass) and 5.32 � 10�6 J/(s g body mass) forfast and slow swimming, respectively.

5. Calculation of the muscle contraction work

Work produced via muscle contractions and the accompanyingheat release has been investigated extensively since the pioneerwork of Hill. During a muscle contractionmyosin filaments pull andgrab actinfilaments and formcross-bridges, so that the entire lengthof the muscle fiber shortens. Hill's muscle model proposes tocalculate the power and heat release rate as functions of the load(tension) in the muscle and the velocity of contraction ( _W ¼ Fv,where _W is power and v is the fiber contraction velocity). In this

Fig. 6. Change of a mantle cavity diameter (a) and wall thickness (b) with time during ejection of water.

Table 3Dimensions of mantle cavity at different swimming modes (adapted from Thompson et al., 2008) [3].

At rest During slow swimming During jet escape

DML (dorsal mantle length in mm) 150 150 150ri (initial mantle cavity radius in mm) 25 25 25rf (final mantle cavity radius in mm) e 23.75 18.75ti (initial mantle wall thickness in mm) 3.5 3.5 3.5tf (final mantle wall thickness in mm) e 3.71 5

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549 543

work, we have calculated themuscle contractionwork based on theexperimentally measured force and displacement values asW ¼ R Fdl. Stress on the mantle wall fibers was extracted for bothfast and slow swimming modes from the muscle contraction elec-tromyography [14,40]. The muscle was divided into 150 parts ofequal length and the thickness muscle thickness of each segment iscalculated. Then, the area, which was calculated by multiplying themuscle thickness with the length of the segment, wasmultiplied bythe average stress at the corresponding radius values to obtain theforce on each fiber group. Since the change in the length (dl) of theeach fiber segment was different; the work (!Fdl) was evaluatedseparately foreachfibergroup. The totalmuscleworkwas calculatedas 4.35� 10�4 J/(s g body mass) and 1.13� 10�4 J/(s g body mass) offast and slow swimmingmodes, respectively. Summary of the mainresults of the muscle work models are given in Table 4 and Fig. 8.

Note that during slow swimming only the SMR fibers areengaged. The CMP fibers, which make up ca. 95% of the mantlevolume, are not activated (Fig. 3). This means that during slowswimming, some of the muscle work generated by the SMR fibershas to be consumed to compress the passive mantle tissue. Duringslow motion 1.13 � 10�4 J/(s g) of work is generated due to musclecontractions. These muscle contractions are used to compress themantle cavity and pressurize the water inside it. The work trans-ferred towater viamantle compression is calculated in the previoussection as 5.32� 10�6 J/(s g), i.e. only 4.7% of the performed musclework. During jet escape, a larger volume of fibers are engaged andthe passive part of the mantle tissue is only 5% (Fig. 4). Therefore, alarger percentage (36.8%) of the generated muscle work is trans-ferred to water. This issue is further discussed in the discussionsection by defining contraction efficiency.

Fig. 7. Variation of volume (a) and pressure in the mantle cavity (b) with time during ejection of water, and pressure versus the volume of the mantle cavity during ejection of water(c). Symbols (þ and B) are the points employed in the calculations (data are adapted from Gosline and DeMont, 1985 [14] and Finke et al., 1996 [40]).

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549544

Table 4Summary of the main results of the muscle models.

Slow swimming Jet escape

Respiration mode Steady state aerobic respiration Aerobic þ anaerobic respiration accompaniedwith accumulation of chemicals

Engaged muscle group SMR fibers Mainly CMP fibersRate of the exergy release due to aerobic respiration 3.82 J/(kg s) 6.55 J/(kg s)Rate of the exergy release due to anaerobic respiration 0 3.35 J/(kg s)Rate of the chemical exergy consumed from the cellular reserves 0 0.072 J/(kg s)Total exergy consumed 3.82 J/(kg s) 9.97 J/(kg s)Rate of the exergy release due to ATP hydrolysis 1.06 J/(kg s) 2.33 J/(kg s)Respiration efficiency 27.7% Only aerobic: 27.7%

Only anaerobic: 13.2%Overall: 23.37%

Fin power 0.27 J/(kg s) 0Muscle contraction power 0.11 J/(kg s) 0.44 J/(kg s)Power transferred to the ejected water 0.0053 J/(kg s) 0.16 J/(kg s)Total work output 0.28 J/(kg s) 0.16 J/(kg s)Ratio of the work output to the available ATP exergy 26.41% 6.87%Contraction efficiency 4.7% 36.8%Exergy destruction rate 3.54 J/(kg s) 9.81 J/(kg s)

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549 545

6. Calculation of the ATP consumption during neural actionpotential propagation

Squid has both giant and non-giant axons. Here, the neuronalmodel of the squid is simplified as one giant axon. The action po-tential propagation is modeled via the Hodgkin and Huxley model[30]. Our neuronal model consists of branching degree of three andsix trees. Deviations from the resting potential during action po-tential propagation were calculated as a function of space for anyinstance along the dendritic tree. ATP consumption duringinstantaneous deviations from the resting membrane potentialwere calculated from the Naþ influx [35]. The total Naþ load iscalculated using initial segment Naþ current parameters as gmax(mS/cm2) gNa ¼ 120 and ENa (mV) ¼ 115 as reported by Hodgkinand Huxley [30]. Sodium channel activation and deactivation ratesare simulated via an in-house developed MATLAB code. The totalNaþ load for an individual action potential is integrated and thenmultiplied with NA/3F, where NA is Avogadro's number and F isFaraday's constant. Our calculations show that during a single ac-tion potential propagation 1.0163 � 1015 ATP molecules areconsumed in the giant squid neuron. This result is in agreementwith the findings of Sengupta et al. [35], who showed that the ac-tion potential Naþ charge may consume 2.3 � 1012 ATP molecules/cm2.

ATP consumed during the action potential propagation is pro-duced via the cellular energy metabolism. We assume that duringslow swimming, oxygen concentration in the squid is high, so thatATP production in the squid cells occurs under normoxic condi-tions. However, during jet escape, oxygen level decreases, so thatATP production has to occur under hypoxic conditions. Genc et al.[40] based on the procedure suggested by Alberty [38,41e46] andthe database developed by Li et al. [47] investigated the neuronalenergy metabolism to estimate the energy and exergy utilized toproduce 1 mol of ATP under various metabolic conditions. Theirresults show that the exergy destruction to produce 1 mol of ATP is131.3 kJ/mol and 129.8 kJ/mol under normoxic and hypoxic con-ditions, respectively. Multiplication of the number of ATP molesconsumed during the propagation of one action potential with theexergy destruction to produce 1 mol of ATP, results in the exergydestruction during the propagation of one action potential(Table 5). Neurons can emit 10 to 100 signals per second [48]. Weassumed that the neuronal signal emission speed is 10 signals persecond for the sustained slow swimming mode and 30 signals per

second for the jet escape mode. As a result, the exergy destructionand the entropy generation rates due to the action potentialpropagation in our model 65 g squid is calculated as 0.034 J/(kg s)and 0.101 J/(kg s) for the slow swimming and jet escape modes,respectively (Tables 5 and 6).

7. Discussion

Table 4 summarizes the main results of this work. Thermody-namic analysis of the steady-state slow swimming shows thatoxidation of glucose within the squid mantle releases an exergy of3.82 J/(kg s). Depending whether 30 or 38 mol of ATP is producedper 1 mol of glucose, 0.029e0.037 mmol/(kg s) of ATP is produced.We assume that during slow swimming, ATP concentration re-mains steady. Accordingly, an exergy of 1.05e1.34 J/(kg s) is storedin the produced ATP, which can be used to perform musclecontraction or maintain cellular activities. If we define the exergeticefficiency of the respiration as:

hrespiration ¼ exergy released via ATP hydrolysisexergy released during respiration

(12)

Then, the exergetic efficiency of the squid aerobic respiration is:

28% ¼ 1:05 J=ðkg sÞ3:82 J=ðkg sÞ � hrespiration;aero � 1:34 J=ðkg sÞ

3:82 J=ðkg sÞ ¼ 35%

(13)

Mantle muscle fiber contraction work (Fdl) is calculated as0.11 J/(kg s) based on empirical data. This means that 8e11% of theexergy stored in ATP is used to perform mechanical work via themantle SMR fibers. This result is in agreement with the previousobservations [18,49], which state that during slow swimmingapproximately 85% of the muscle work is performed by the fin andthe arms to generate lift for the negatively buoyant squid. If weassume that 85% of the ATP is consumed for the finmotion, and thatATP chemical exergy is converted into useful work with a 30% ef-ficiency, than fin power is estimated as 0.27 J/(kg s) [12] (Table 4).

The work transferred to the water inside the cavity (PdV) iscalculated as 0.0053 J/(kg s). This means that about 0.5% of theexergy stored in ATP is transferred to the water inside the mantlecavity. This is a negligible amount, proving that the squid does notuse water jet momentum transfer as its primary mode of

Fig. 8. Exergy (Grassman) charts describing how and where the exergy input, e.g., exergy of the nutrients is utilized or destroyed during slow swimming (top) and the jet escape(bottom).

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549546

propulsion during slow swimming. The total work output duringslow swimming is the sum of the Pdv and finwork, which is 0.28 J/(kg s). This means that 26.41% of the exergy stored in ATP is con-verted into useful work.

Another interesting point becomes apparent, if we define thecontraction efficiency as follows:

hcontraction ¼ work transferred to the ejected water ðPdvÞmuscle contraction work ðFdlÞ

(14)

hcontraction;slow ¼ 0:00530:11

¼ 4:7% (15)

The contraction efficiency during slow swimming is only ca. 5%.During the slow swimming, the SMR fibers are engaged, and therest of the mantle tissue is passive. This result shows that, about95% of the work done by the SMR fibers is used up to compress thepassive mantle tissue, and only about 5% can be employed to in-crease the cavity pressure.

During jet escape, energy comes from both from the aerobic andanaerobic respiration. Oxygen consumption rate increases nearlylinearly between 0.5 and 4 mantle length/s. Therefore, at a speed of

Table 5Summary of the main results of the squid giant axon model.

Slow swimming Jet escape

Cellular condition Normoxic HypoxicSignal emission speed 10 Hz 30 HzExergy destruction rate 2.22 � 10�3 J/s 6.57 � 10�3 J/s

0.03 J/(kg s) 0.10 J/(kg s)

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549 547

3 mantle length/s, squid consumes a large amount of oxygen; andabout 6.55 J/(kg s) of exergy is released via aerobic respiration,which is nearly twice the amount of exergy as the previous case.The chemical exergy released via the anaerobic pathway is 3.35 J/(kg s). About 0.072 J/(kg s) of the chemical exergy comes viadepleting the cellular ATP reserves.

The exergetic efficiency of the squid anaerobic respiration is:

hrespiration;anaero ¼ 0:44 J=ðkg sÞ3:35 J=ðkg sÞ ¼ 13:2% (16)

A squid swimming with a speed of 3 mantle length/s, producesan exergy of 2.33 J/(kg s) via ATP hydrolysis. 0.44 J/(kg s) of thiscomes from ATP produced via anaerobic respiration, 0.07 J/(kg s)comes from the ATP used up from cellular reserves, and 1.82 J/(kg s)comes from ATP produced via aerobic respiration. Since the jetescape swimming speed (3 mantle length/s) is very close to thecritical speed (2 mantle length/s) at which anaerobic respirationbegins; the squid still uses the aerobic pathway extensively toproduce the required ATP. One can speculate that if the swimmingspeed would be higher, than the ATP production rate from theanaerobic pathway would increase dramatically, so that a muchhigher percent of the consumed ATP would come from the anaer-obic pathway. Here, at this low jet escape speed, still 78% of theconsumed ATP comes from the aerobic pathway. Therefore, therespiration efficiency of the overall system, which is 23.37%, is closeto the aerobic respiration efficiency.

The animals in the nature generally prefer to stay in steady stateconditions. The squids also not depart from this rule. Squidgenerally prefers the steady state swimming mechanism in theirlifetime unless they are in danger or when they are hunting. Thesquid switch its swimming behavior for survival or for capture theprey. When the squid switches to the jet escape swimmingmode, itmeans a drop in the respiration efficiency. At very high speeds,squids may be forced to use the anaerobic pathway exclusively,which would drop the respiration efficiency to 13.2%.

Mantle muscle fiber contraction work during jet escape (Fdl) iscalculated as 0.44 J/(kg s) based on empirical data. This means that84% of the chemical exergy stored in ATP is converted into me-chanical work by the mantle CMP muscles. Rest of this exergy maybe used up as muscle work in other parts of the squid body, or toovercome the irreversibility's. The high ratio of contraction work toATP hydrolysis exergy shows that, during jet escape the squidconsumes nearly all of its resources to contract the mantle.

The contraction efficiency during jet escape is much higher thanthe slow speed contraction efficiency:

Table 6Total exergy destruction and entropy generation rates.

Exergy destruction rate in the muscleExergy destruction rate in the giant axonTotal exergy destruction rateTotal entropy generation rate

hcontraction;jet ¼0:160:44

¼ 36:8% (17)

The volume of the engagedmuscles during jet escape is 20 timeslarger than the slow swimming. Thus, the passive volume in themantle tissue during jet escape is only about 11.8% (as opposed to95.6% during slow swimming). Percentage of the passive tissue, andcontraction efficiency seems to be proportional. As the passivetissue decreases to 1/8th, contraction efficiency increases 8 times.Biologically advanced species, e.g., human beings, do not have sucha drawback to reduce their muscle work efficiency.

Calculations show that a squid swimming slowly (0.5 mantlelength/s) consumes an exergy of 3.82 J/(kg s) and produces 0.28 J/(kg s). The difference between these values gives us an estimate ofthe exergy destruction rate, which is 3.54 J/(kg s). In addition tothat, the squid destroys an exergy of 0.034 J/(kg s) during the actionpotential propagation in the giant axon. The total exergy destruc-tion rate during slow swimming is 3.57 J/(kg s). This corresponds toan entropy generation rate of 0.01198 J/(kg s K).

In the jet escape mode, anaerobic respiration begins. Extent ofthe anaerobic respiration depends on the swimming speed. As theswimming speed increases, ATP consumption rate increases. Theoverall system tries to generate ATP through the aerobic pathway aslong as oxygen concentration allows; depletes the cellular ATP re-serves; and generates the additional ATP through the inefficientanaerobic pathway. Even though the efficiency of the biochemicalreactions drop, the mechanical efficiency during jet escape is high.During jet escape a much larger volume of the muscle fibers areengaged, and the passive tissue, that the contracted fibers have tocompress occupy a much smaller volume. The contraction effi-ciency during jet escape is about 8 times larger than that of the slowswimming.

During jet escape at 3 mantle length/s, the squid consumes achemical exergy of 9.97 J/(kg s), produces a work of 0.16 J/(kg s),which lead to an exergy destruction rate of 9.81 J/(kg s). The exergydestruction rate in the giant axon increases with respect to the slowswimming mode, because of two reasons: ATP production underhypoxic conditions is less exergy efficient, and a higher rate ofsignal emission is necessary. 0.101 J/(kg s) of exergy is destroyed forthe action potential propagation. The total exergy destruction rateis 9.91 J/(kg s) and the entropy generation rate is 0.03291 J/(kg s K).Squids are able to swim at very high speeds, such as 25 mantlelength/s during jet escape. In such a case the respiration efficiencywould approach to 13.2%, and the entropy generation would occurat a much higher rate.

Muscle work efficiency is widely studied with both analyticalmethods [12,50,51] and experiments [52e58]. In these studies,muscle efficiency is usually defined either via the first law effi-ciency, i.e. ratio of the produced work to the chemical energy(hI¼W/DH), or the second law efficiency, i.e. ratio of the producedwork to themaximumwork (hII¼W/Wmax¼W/DG). Sorgüven et al.[12] recently calculated the theoretically maximum first and secondlaw efficiencies as 0.84 and 1.0, respectively, and compared thesevalues to the experimentally measured muscle efficiencies, whichranged between 0.35 and 0.14 for the first law efficiencies, and

Slow swimming Jet escape

3.54 J/(kg s) 9.81 J/(kg s)0.03 J/(kg s) 0.10 J/(kg s)3.57 J/(kg s) 9.91 J/(kg s)0.01198 J/(kg s K) 0.03326 J/(kg s K)

B.H. Yalçınkaya et al. / Energy 102 (2016) 537e549548

between 0.42 and 0.17 for the second law efficiencies. These areefficiencies measured from isolated muscle groups. In this study,the ratio of the produced work to the ATP exergy is calculated forthe fast and slow swimming modes as 0.26 and 0.069, respectively.These values are in agreement with the experimentally measuredsecond law muscle efficiencies. Nevertheless, a direct comparisonwith the published muscle efficiencies should be made withcaution. In this study, calculations are performed based on themeasurements made from the whole body of the squid. Therefore,the consumed chemical exergy is not only used to generate musclework, but also to perform basal metabolic activities. Hence, thecalculated efficiencies are expected to be lower than the efficienciesof isolated muscle fibers.

8. Conclusion

This paper presents a thermodynamic evaluation of the fast andslow swimming modes of a squid. Squid engages different types ofmuscle fibers and switches between aerobic and anaerobic respi-ration modes as its swimming speed changes. Main findings of thisstudy are:

1. During fast swimming 9.97 J/(kg s) of chemical exergy isconsumed, which is ca. 2.6 times larger than the exergy con-sumption during slow swimming (3.82 J/(kg s)).

2. Respiration efficiency decreases as more ATP is produced viaanaerobic respiration.

3. During slow swimming most of the power is generated via finand the power transferred to water inside the mantle cavity isonly 0.0053 J/(kg s).

4. During jet escape power transferred to water rises to 0.16 J/(kg s).

5. Contraction efficiency decreases from 36.8% to 4.7% as the vol-ume of the passive tissue increases from 5% to 95%.

6. The ratio of the produced work to the available ATP exergy is26.41% for the slow and 6.87% for the fast swimming modes.

All of these conclusions point that slow swimming is exergeti-cally more efficient, and leads to about 3 times less exergydestruction. ATP consumption calculations during neural actionpotential propagation showed that during jet escape the giant axondestroyed nearly 3 times neuron exergy as that of slow swimming,suggesting another clue why squid prefers slow swimming undernon-emergency conditions.

Acknowledgments

This work has been partially supported by the TUBITAK (TheScientific and Technological Research Council of Turkey) 3501Program with the Grant number 111M598.

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